Instructions to Authors

www.elsevier.com/locate/physletb

Instructions to authors

Aims and Scope Physics Letters B ensures the rapid publication of letter-type communications in the fields of Nuclear Physics, Particle Physics and Astrophysics. Articles should influence the physics community significantly. Submission Electronic submission is strongly encouraged. The electronic file, accompanied by a covering message, should be e-mailed to one of the Editors indicated below. Easy submission via the LANL-preprint server is certainly possible; please visit http://www.elsevier.com/locate/plbsubmission. If electronic submission is not feasible, submission in print is possible, but it will delay publication. In the latter case manuscripts (one original + two copies), accompanied by a covering letter, should be sent to one of the following Editors: L. Alvarez-Gaumé, Theory Division, CERN, CH-1211 Geneva 23, Switzerland, E-mail address: Luis.Alvarez-Gaume@CERN. CH Theoretical High Energy Physics (General Theory) J.-P. Blaizot, ECT*, Strada delle Tabarelle, 266, I-38050 Villazzano (Trento), Italy, E-mail address: [email protected]. FR Theoretical Nuclear Physics M. Cvetic,ˇ David Rittenhouse Laboratory, Department of Physics, University of Pennsylvania, 209 S, 33rd Street, Philadelphia, PA 19104-6396, USA, E-mail address: [email protected] Theoretical High Energy Physics M. Doser, EP Division, CERN, CH-1211 Geneva 23, Switzerland, E-mail address: [email protected] Experimental High Energy Physics D.F. Geesaman, Argonne National Laboratory, 9700 South Cass Avenue, Argonne, IL 60439, USA, E-mail address: [email protected] Experimental Nuclear Physics H. Georgi, Department of Physics, Harvard University, Cambridge, MA 02138, USA, E-mail address: Georgi@PHYSICS. HARVARD.EDU Theoretical High Energy Physics G.F. Giudice, CERN, CH-1211 Geneva 23, Switzerland, E-mail address: [email protected] Theoretical High Energy Physics N. Glover, Institute for Particle Physics Phenomenology, Department of Physics, Science Laboratories, University of Durham, South Road, Durham DH1 3LE, UK, E-mail address: [email protected] Theoretical High Energy Physics W. Haxton, Institute for Nuclear Theory, Box 351550, University of Washington, Seattle, WA 98195-1550, USA, E-mail address: [email protected] Theoretical Nuclear Physics and Nuclear Astrophysics V. Metag, II. Physikalisches Institut, Universität Giessen, Heinrich-Buff-Ring 16, D-35392 Giessen, Germany, E-mail address: [email protected]. UNI-GIESSEN.DE Experimental Nuclear Physics L. Rolandi, EP Division, CERN, CH-1211 Geneva 23, Switzerland, E-mail address: [email protected] Experimental High Energy Physics W.-D. Schlatter, CERN, CH-1211 Geneva 23, Switzerland, E-mail address: [email protected] Experimental High Energy Physics

0370-2693/2005 Published by Elsevier B.V. doi:10.1016/S0370-2693(05)00827-0 vi Instructions to authors

H. Weerts, 3247 Biomedical and Physical Sciences Building, Department of Physics and Astronomy, Michigan State University, East Lansing, MI 48824-1111, USA, E-mail address: [email protected] Experimental High Energy Physics T. Yanagida, Department of Physics, Faculty of Science, University of Tokyo, Tokyo 113-0033, Japan, E-mail address: [email protected] Theoretical High Energy Physics The authors should indicate in which of the following four sections they would like to see their article published: Astrophysics & Cosmology, covered by L. Alvarez-Gaumé, M. Cveticˇ and W. Haxton. Experiments, covered by M. Doser, D.F. Geesaman, V. Metag, L. Rolandi, W.-D. Schlatter and H. Weerts. Phenomenology, covered by J.-P. Blaizot, M. Cvetic,ˇ H. Georgi, G.F. Giudice, N. Glover, W. Haxton and T. Yanagida. Theory, covered by L. Alvarez-Gaumé, J.-P. Blaizot, M. Cvetic,ˇ H. Georgi, G.F. Giudice, N. Glover, W. Haxton and T. Yanagida. For submissions in High Energy Physics authors are encouraged to provide the number of their Los Alamos preprint to the Physics Letters B Editor at the moment of submission. Original material. By submitting a paper for publication in Physics Letters B the authors imply that the material has not been published previously nor has been submitted for publication elsewhere and that the authors have obtained the necessary authority for publication. Refereeing. Submitted papers will be refereed and, if necessary, authors may be invited to revise their manuscript. If a submitted paper relies heavily on unpublished material, it would be helpful to have a copy of that material for the use of the referee. Publication speed The Editors and Publisher cooperate closely to ensure minimal publication delays. All proofreading will be done by the Publisher and proofs are not sent to the author(s). In order to keep delays to a minimum it is of utter importance for the author(s) closely to observe the guidelines given in the “Preparation of Manuscripts” below before submission of the manuscript and to supply an E-mail address and/or telefax number of the corresponding author. Preparation of manuscripts The following requirements as regards presentation of the manuscript should be met: (1) Manuscript: All manuscripts should be written in good English. The original typescript should be typed on one side of the paper, with double spacing and a wide margin. Instead of the original typescript, a copy of high quality (not a carbon copy) is also acceptable. Please adhere to the following order of presentation: Article title, Author(s), Affiliation(s), Abstract, Classification codes and keywords, Main text, Acknowledgements, Appendices, References, Figure captions, Tables. The name, complete postal address, telephone and fax numbers and the E-mail address of at least the corresponding author should be given on the first page of the manuscript. The title page should contain title, author(s), address(es) and abstract. The main text should start on a new page. All pages should be numbered. (2) Length: The total length of the paper should preferably not exceed six journal pages, equivalent to ten typewritten pages with double spacing, including the list of authors, abstract, references, figure captions and three figures. In the case that more figures are required, the text should be shortened accordingly. (3) Title: The title should be brief and such that it conveys to the informed reader the particular nature of the contents of the paper. (4) Address: The name(s) of the author(s) and the name and address of the institute where the research work was done should be indicated on the manuscript. The name of the author to whom correspondence is to be addressed should be underlined and an E-mail address and/or a telefax number supplied. (5) Abstract: An abstract of less than 60 words is required. It should contain the keywords of the paper as well as the essence of the results achieved. (6) Classification codes and keywords: Supply one to four classification codes (PACS and/or MSC) and up to six keywords of your own choice that describe the content of your article in more detail. (7) Formulae: Displayed formulae should be numbered and typed or clearly and unambiguously written by hand. Symbols requiring bold-face type, like vectors, etc., should be identified properly in the margin of the manuscript. Long equations should be avoided as much as possible by introduction of suitable abbreviations of component expressions. Special attention should be paid to symbols that can easily be misread, such as i (lower case), I (cap.), 1 (el), 1 (one), (prime), ◦ o (lower case), O (cap.), 0 (zero), (degree), u,v (vee), ν (Greek nu), V (cap.), x, ×,X,z,Z,p,P,ρ (Greek rho), etc. (8) Footnotes: The footnotes may be typed at the end of the page on which they are alluded to, or at the end of the paper, or on a separate sheet. Please do not mix footnotes and references. Instructions to authors vii

(9) References: In the text, reference to other parts of the paper should be made by section (or equation) number, not by page number. References to other papers should be consecutively numbered in the text using square brackets and should be listed by number on a separate sheet at the end of the paper. Please do not combine multiple references to different papers into one numbered reference. The references should be as complete as possible and be presented as follows: For a book: B. de Wit and J. Smith, Field theory in particle physics, Vol. 1 (North-Holland, Amsterdam, 1986). For a paper in a journal: UAl Collab., G. Arnison et al., Phys. Lett. B 177 (1986) 244. For a paper in a contributed volume: R. Jackiw, in: Progress in quantum field theory, eds. H. Ezawa and S. Kamefuchi (North-Holland, Amsterdam, 1986) p. 83. For an unpublished paper: J.F. Gunion and H.E. Haber, UCD Report 86-12 (1986), unpublished. For a preprint: A. Lahanas and D.V. Nanopoulos, CERN preprint CERN-TH 4400/86 (1986). For a conference report: M.B. Green, Superstrings and the unification of forces and particles, in: Proc. fourth Marcel Gross- mann Meeting on General relativity (Rome, June 1985), Vol. 1, ed. R. Ruffini (North-Holland, Amsterdam, 1986) p. 203. (10) Figures: Each figure should also be submitted in triplicate: one master figure and two copies, the figure must be referred to in the text, be numbered and have a caption. The captions should be collected on a separate sheet. The appropriate place of each figure should be indicated in the margin. Axes of figures must be labelled properly. The (line) drawings for the figures must be submitted on separate sheets, drawn in black India ink and carefully lettered (with the use of stencils). The lettering as well as the essential details should have proportionate dimensions so as not to become illegible or unclear after the usual reduction by the printers (ideal lettering size after reduction of the drawing to one-column width is 1.8 mm). The drawings should preferably be of the same size as the typescript and designed for a reduction factor in print of two to three. The photographs should be originals, with somewhat more contrast than is required in the printed version. They should be unmounted unless part of a composite figure. Any scale markers should be inserted on the photograph, not drawn below it. The figures should be identified by the name of the first author, the journal name and the figure number. Instead of original drawings, sharp and contrasty glossy prints of about typescript size or high quality laserprints are also acceptable. If requested, original drawings will be returned to the author(s) upon publication of the paper. For detailed instructions on the preparation of electronic artwork, consult the Author Gateway from Elsevier at http:// authors.elsevier.com. (11) Colour illustrations: Illustrations in colour will be accepted in cases when the use of colours is judged by the Editor to be essential for the presentation. The Publisher and the author will each bear part of the extra costs involved. The costs charged to the authors of articles containing colour figures will be € 635 (approximately US$ 760) for the first page containing colour and € 318 for each additional page with colour, independent of the number of colour figures on each page. These prices are exclusive of Value Added Tax (VAT). Authors will be billed in Euros; the dollar price is for guidance only. The author receives 200 reprints of an article with colour illustrations free of charge. More reprints can be ordered at the usual rates for (black and white) reprints, there will be no additional charge for reprints containing colour illustrations. Colour illustrations should be submitted in the form of good quality colour photographs, transparencies, colour printer output, or 35 mm slides. Polaroid colour prints should be avoided. When supplying colour photographs and transparencies they should be close to the final size expected for publication. It should be noted that, in some cases, printing requirements will prevent figures from being located in the most preferred position in the text. (12) Tables: Tables should be typed on separate sheets and each table should have a number and a title. The appropriate places for the insertion of the tables should be indicated in the margin.

After acceptance Notiﬁcation. You will be notiﬁed by the Editor of the journal of the acceptance of your article and invited to supply an electronic version of the accepted text, if this is not already available. Copyright transfer. You will be asked to transfer the copyright of the article to the Publisher. This transfer will ensure the widest possible dissemination of information. No proofs. In order to speed up publication, all proofreading will be done by the Publisher and proofs are not sent to the author(s). Enquiries. Visit the Author Gateway from Elsevier (http://authors.elsevier.com) for the facility to track accepted articles and set up e-mail alerts to inform you of when an article’s status has changed. The Author Gateway also provides detailed artwork guidelines, copyright information, frequently asked questions and more. Contact details for questions arising after acceptance of an article, especially those relating to proofs, are provided when an article is accepted for publication. viii Instructions to authors

Instructions for LATEX manuscripts The Publisher welcomes the receipt of an electronic version of your accepted manuscript (preferably encoded in LATEX). If you have not already supplied the final, accepted version of your article to the journal Editor, you are requested herewith to send a file with the text of the accepted manuscript directly to the Publisher by E-mail to the address given below. If the electronic file is suitable for processing by the Publisher, the article will be published without rekeying the full text. The article should be encoded in LATEX, preferably using the Elsevier document class ‘elsart’, or alternatively the standard document class ‘article’ or the document style ‘revtex’. The Elsevier LATEX package (including detailed instructions for LATEX preparation) can be obtained from Elsevier’s web site: www.elsevier.com/locate/latex, or from the Comprehensive TEX Archive Network (CTAN). The Elsevier package consists of the files: elsart.cls (use this file if you are using LaTeX2e, the current version of LATEX), elsart.sty and elsart12.sty (use these two files if you are using LaTeX2.09, the previous version of LATEX), instraut.dvi and/or instraut.ps and/or instraut.pdf (instruction booklet), model-harv.tex or model-num.tex (model files with instructions), template-harv.tex or template-num.tex (template files). To obtain the package from CTAN, use direct access via FTP at ftp.dante.de (Germany), ftp.tex.ac.uk (UK), or ctan.tug.org (Massachussets, USA) and go to the directory /tex-archive/macros/latex/contrib/supported/elsevier, or search for Elsevier with one of the CTAN search engines (http://ctan.tug.org/CTANfind.html, http://www.tex.ac.uk/CTANfind.html or http://www.dante.de/cgi-bin/ctan-index). CTAN is a mirrored network of ftp.tex.ac.uk, ftp.dante.de and ctan.tug.org, which are widely mirrored (see ftp://ctan.tug.org/ tex-archive/README.mirrors) and hold up-to-date copies of all the public-domain versions of TEX, LATEX, Metafont and ancillary programs. Questions concerning the LATEX author-prepared article and requests for the booklet with instructions to authors should be directed to the address given below. No changes from the version accepted by the Editor of the journal are permissible, without the prior and explicit approval by the Editor. Such changes should be clearly indicated on an accompanying printout of the file. The Publisher reserves the right to decide whether to process a manuscript from authors’ files or not. Articles coded in a simple manner with no user-defined macros are most likely to be handled this way. If sent via electronic mail, files should be accompanied by a clear identification of the article (name of journal, Editor’s reference number) in the “subject field” of your electronic-mail message. Authors should include an ASCII table (available from the Publisher) in their files, to enable any transmission errors to be detected. Addresses Ð Mail: Drs. E.J. van der Wolk, Physics Letters B, Editorial-Production Journals Department, Elsevier B.V., P.O. Box 2759, 1000 CT Amsterdam, The Netherlands. E-mail: [email protected]. Author benefits No page charges. Publishing in Physics Letters B is free. Free offprints. The corresponding author will receive 25 offprints free of charge. An offprint order form will be supplied by the Publisher for ordering any additional paid offprints. Discount. Contributors to Elsevier journals are entitled to a 30% discount on all Elsevier books. Contents Alert. Physics Letters B is included in Elsevier’s pre-publication service Contents Alert (for information, please contact: [email protected]). ContentsDirect. Physics Letters B is included in Elseviers free E-mail service ContentsDirect. Please register at http://www. elsevier.com/locate/ContentsDirect. Publication scheme Physics Letters B will be published weekly. All correspondence with the Editors or Publisher should contain full reference to the paper concerned, i.e., the names of all the contributors, the full title, as well as the reference number. Contributors are reminded that, once their contribution has been accepted for publication, all further correspondence should be sent directly to the Publisher and not to the Editors (Drs. E.J. van der Wolk, Physics Letters B, Editorial-Production Journals Department, Elsevier B.V., P.O. Box 2759, 1000 CT Amsterdam, The Netherlands; telephone number +31 20 485 2634; telefax number +31 20 485 2431; electronic mail: [email protected] (the receipt of an E-mail message will be acknowledged; in the absence of this acknowledgement, the mailer should assume that his message was never received by the Publisher)). A prompt reply may be expected whenever appropriate. If no reaction is received within three weeks, authors are kindly requested to seek contact again. PHYSICS LETTERS B

EDITORS

L. ALVAREZ-GAUMÉ J.-P. BLAIZOT M. CVETICˇ GENEVA VILLAZZANO (TRENTO) PHILADELPHIA, PA

M. DOSER D.F. GEESAMAN H. GEORGI GENEVA ARGONNE, IL CAMBRIDGE, MA

G.F. GIUDICE N. GLOVER W. HAXTON GENEVA DURHAM SEATTLE, WA

V. METAG L. ROLANDI W.-D. SCHLATTER GIESSEN GENEVA GENEVA

H. WEERTS T. YANAGIDA EAST LANSING, MI TOKYO

VOLUME 619, 2005

Amsterdam – Boston – Jena – London – New York – Oxford Paris – Philadelphia – San Diego – St. Louis Physics Letters B 619 (2005) 1–4 www.elsevier.com/locate/physletb

Future universe with w < −1 without big smash

S.K. Srivastava

Department of Mathematics, North Eastern Hill University, Shillong 793022, India Received 17 February 2005; received in revised form 12 May 2005; accepted 20 May 2005 Available online 1 June 2005 Editor: T. Yanagida

Abstract It is demonstrated that if cosmic dark energy behaves like a ﬂuid with equation of state p = wρ (p and ρ being pressure and energy density respectively) as well as generalized Chaplygin gas simultaneously, big rip or big smash problem does not arise even for equation of state parameter w < −1. Unlike other phantom models, here, the scale factor for the future universe is found regular for all time.  2005 Elsevier B.V. All rights reserved.

PACS: 98.80.Cq

Keywords: Dark energy; Phantom model; Big rip and accelerated universe

Experimental probes, during last few years sug- state (EOS) p = wρ with p as isotropic pressure, ρ as gest that the present universe is spatially ﬂat as well energy density and −1 w < −1/3. as it is dominated by yet unknown form of dark en- In the recent past, it was pointed out that the current ergy [1,2]. Moreover, studies of Ia Supernova [3,4] data also allowed w < −1 [7]. Rather, in Refs. [8–10], and WMAP [5,6] show accelerated expansion of the it is discussed that these data favor w < −1 being EOS present universe such that a>¨ 0 with a(t) being the parameter for phantom dark energy. Analysis of recent scale factor of the Friedmann–Robertson–Walker line- Ia Supernova data also support w < −1 strongly [11– element 13]. Soon after, Caldwell [8] proposed the phantom dS2 = dt2 − a2(t) dx2 + dy2 + dz2 . (1) dark energy model exhibiting cosmic doomsday of the future universe, cosmologists started making efforts Theoretically accelerated expansion of the universe to avoid this problem using w < −1 [14,15].Inthe is obtained when the cosmological model is supposed braneworld scenario, Sahni and Shtanov has obtained to be dominated by a ﬂuid obeying the equation of well-behaved expansion of the future universe without big rip problem with w < −1. They have shown that E-mail address: [email protected] (S.K. Srivastava). acceleration is a transient phenomenon in the current

0370-2693/$ – see front matter  2005 Elsevier B.V. All rights reserved. doi:10.1016/j.physletb.2005.05.056 2 S.K. Srivastava / Physics Letters B 619 (2005) 1–4 universe and the future universe will re-enter matter- and integrating, it is obtained that dominated decelerated phase [16]. + ρ(1 α)/α(t) It is found that GR (general relativity)-based phantom model encounters “sudden future singularity” = + (1+α)/α − 3(1+α)/α A ρ0 A a0/a(t) (6) leading to divergent scale factor a(t), energy density with ρ0 = ρ(t0) and a0 = a(t0), where t0 is the present and pressure at finite time t = ts . Thus the classical approach to phantom model yields big-smash problem. time. For models with “sudden future singularity” Elizalde Eqs. (3) and (4) yield w as et al. [17] argued that, near t = ts , curvature invariants A w(t) =− . (7a) become very strong and energy density is very high. ρ(1+α)/α(t) So, quantum effects should be dominant for |t − t| < s = one unit of time, like early universe. This idea is pur- So, evaluation of Eq. (7a) at t t0 leads to sued in Refs. [18–20] and it is shown that an escape (1+α)/α A =−w0ρ , (7b) from the big-smash is possible on making quantum corrections to energy density ρ and pressure p in with w0 = w(t0).FromEqs.(6) and (7a), (7b),itis Friedmann equations. obtained that In the framework of Robertson–Walker cosmol- + + = − + + 3(1 α)/α α/(1 α) ogy, Chaplygin gas (CG) is also considered as a good ρ ρ0 w0 (1 w0) a0/a(t) source of dark energy for having negative pressure, (8) given as with w0 < −1. In the homogeneous model of the universe, a scalar A p =− (2) field φ(t) with potential V(φ)has energy density ρ 1 ˙2 with A>0. Moreover, it is the only gas having super- ρφ = φ + V(φ) (9a) symmetry generalization [21,22]. Bertolami et al. [12] 2 have found that generalized Chaplygin gas (GCG) is and pressure better fit for latest Supernova data. In the case of GCG, 1 ˙2 Eq. (2) looks like pφ = φ − V(φ). (9b) 2 A Using Eqs. (3), (4), (7a), (7b) and (8), it is obtained p =− , (3) ρ1/α that ∞ = (1+α)/α (1+α)/α where 1 α< .α 1 corresponds to Eq. (2). ρ + ρ w0 φ˙2 = 0 . (10) In this Letter, a different prescription for GR-based ρ future universe, dominated by the dark energy with w < −1, is proposed which is not leading to the Connecting Eqs. (8) and (10), it is obtained that catastrophic situations mentioned above. The scale (1+α)/α 3(1+α)/α (1 + w0)ρ (a0/a) factor, obtained here,does not possess future singular- φ˙2 = 0 . [− + + 3(1+α)/α]α/(1+α) ity. In the present model, it is assumed that the dark w0 (1 w0)(a0/a) (11) energy behaves like GCG, obeying Eq. (3) as well as ˙ fluid with equation of state This equation shows that φ2 > 0 (giving positive kinetic energy) for w0 > −1, which is the case p = wρ with w < −1 (4) of quintessence and φ˙2 < 0 (giving negative kinetic energy) for w < −1, being the case of super- simultaneously. 0 quintessence (phantom). As a reference, it is rele- Connecting Eq. (3) with the hydrodynamic equa- vant to mention that, long back, Hoyle and Narlikar tion used C-field (a scalar called creation field) with neg- a˙ ρ˙ =−3 (ρ + p) (5) ative kinetic energy for steady-state theory of the uni- a verse [23]. S.K. Srivastava / Physics Letters B 619 (2005) 1–4 3

Thus, it is shown that dual behaviour of dark energy The horizon distance for this case (a(t) given by fluid, obeying Eqs. (3) and (4) is possible for scalars, Eq. (16)) is obtained as frequently used for cosmological dynamics. So, this + 3(1 + α)a(t) 2(1 + α)|w | α/3(1 α) assumption is not unrealistic. d (t) 0 H αa α + (α + 2)|w | Now the Friedmann equation, with dominance of 0 0 α/2(1+α) dark energy having double fluid behaviour, is × exp 6H0|w0| Ω0 αt/3(1 + α) a˙ 2 (15a) = H 2Ω |w | a 0 0 0 showing that 3(1+α)/α α/(1+α) + 1 −|w0| a0/a(t) , dH (t) > a(t). (15b) (12a) So, horizon grows more rapidly than the scale factor impling colder and darker universe. It is like flat or where |w0| > 1. H0 is the present value of Hubble’s = = 2 open universe without dominance of dark energy. constant and Ω0 ρ0/ρcr,0 with ρcr,0 3H0 /8πG (G being the Newtonian gravitational constant). In this case, Hubble’s distance is −| | 3(1+α) 1 w0 3(1 + α) Neglecting higher powers of (a0/a(t)) α , −1 |w0| H = √ | |α/2(1+α) Eq. (12a) is written as αH 0 Ω0 w0 α(1 −|w0|) a˙ + × 1 − H Ω |w |α/2(1 α) + + | | 0 0 0 α (α 2) w0 a −| | α/2(1+α) α(1 w0 ) 3(1+α)/α × exp −H0|w0| Ω0(t − t0) (16) × 1 + a0/a(t) . (12b) 2(1 + α)|w0| −1 → Eq. (12b) is integrated to showing its growth with time such that H 3(1+√α) −α/2(1+α) −1 |w0| = 0ast →∞. Here, H∞ is a αH0 Ω0 a(t) = 0 found large and finite. It means that, in the present α/3(1+α) [2(1 + α)|w0|] case, galaxies will not disappear when t →∞.Itis √ | |α/2(1+α) − × + + | | 6H0 w0 Ω0(t t0) unlike phantom models with future singularity ex- α 2(1 α) w0 e n panding as |t − ts| for n<0, where galaxies are α/3(1+α) − α 1 −|w0| , (13) expected to vanish near future singularity time ts [8] as H −1 → 0fort → t . In Barrow’s model [24] yielding accelerated expansion of the universe with s q n →∞ →∞ − B + Ct + D(t − t) a(t) as t , supporting observational evi- H 1 = s , q−1 n−1 (17) dences of Ia Supernova [3,4] and WMAP [5,6].Itis qCt − Dn(ts − t) interesting to see that expansion, obtained here, is free where B,C,D are positive constants and q>0. from “finite time future singularity” unlike other GR- −1 Eq. (17) shows that, for n<1,H → 0ast → ts based phantom models. It is due to GCG behaviour of −1 and at t = ts,H is finite for n>1. In the model, phantom dark energy. taken by Nojiri and Odintsov [18] Moreover, Eqs. (8) and (13) that energy density −1 ˜ n −1 grows with time for w0 < −1 and decreases for H = H(t)+ A |ts − t| , (18) − → | |α/3(1+α) → w0 > 1. Also ρ ρ0 w0 (finite) and p ˜ −p /|w |α/3(1+α) as t →∞.Eqs.(7a), (7b) and (8) where H(t)is a regular function of t and A > 0. This 0 0 −1 → → imply time-dependence of EOS parameter equation shows that, for n<0, H 0ast ts and it is finite at t = t for n>0. s 3(1+α)/α −1 Thus, it is found that if phantom fluid behaves like w =−|w0| |w0|− |w0|−1 a0/a(t) , GCG and fluid with p = wρ, it is possible to get (14) accelerated growth of scale factor of the future uni- with a(t), given by Eq. (13). This equation shows that verse for time t0 ρ0 and −p0/|w0| > −p0, P.H. Frampton, hep-th/0302007; respectively. It is unlike GR-based models, driven by S.M. Carroll, et al., Phys. Rev. D 68 (2003) 023509; EOS p = wρ, with w < −1 having future singular- P. Singh, gr-qc/0502086. ity at t = ts , where ρ and p are divergent [8,14] or ρ [10] J.M. Cline, et al., hep-ph/0311312. is finite and p is divergent [18,24]. Based on Ia Su- [11] U. Alam, et al., astro-ph/0311364; U. Alam, et al., astro-ph/0403687. pernova data, Singh et al. [13] have estimated w0 for − − [12] O. Bertolami, et al., Mon. Not. R. Astron. Soc. 353 (2004) 329, models in the range 2.4 < w0 < 1.74 upto 95% astro-ph/0402387. confidence level. Taking this estimate as an example [13] P. Singh, M. Sami, N. Dadhich, Phys. Rev. D 68 (2003) with α = 3, ρ∞ = ρ(t→∞) is found in the range 023522; 1.15ρ0 <ρ∞ < 1.24ρ0. This does not yield much in- M. Sami, A. Toporesky, gr-qc/0312009. crease in ρ as t →∞. But if this model is realistic and [14] B. McInnes, JHEP 0208 (2002) 029, hep-th/0112066. [15] P.F. González-Díaz, Phys. Rev. D 68 (2003) 021303(R); | | ∞ future experiments support large w0 , ρ will be very V.K. Onemli, et al., Class. Quantum Grav. 19 (2002) 4607, gr- high. In both cases, small or large values of |w0|,in- qc/0204065; crease in ρ indicates creation of phantom dark energy V.K. Onemli, et al., Phys. Rev. D 70 (2004) 107301, gr-qc/ in future. It may be due to decay of some other compo- 0406098; nents of energy in universe, which is not dominating, V.K. Onemli, et al., Class. Quantum Grav. 22 (2005) 59, gr-qc/ 0408080. for example, cold dark matter. [16] V. Sahni, Yu.V. Shtanov, JCAP 0311 (2003) 014, astro-ph/ It is interesting to see that big-smash problem does 0202346; not arise in the present model. In Refs. [17–20],for G. Calcagni, Phys. Rev. D 69 (2004) 103508; models with future singularity, escape from cosmic V. Sahni, astro-ph/0502032. doomsday is demonstrated using quantum corrections [17] E. Elizalde, S. Noriji, S.D. Odintsov, Phys. Rev. D 70 (2004) = 043539, hep-th/0405034. in field equations near t ts . Here, using classical [18] S. Nojiri, S.D. Odintsov, Phys. Lett. B 595 (2004) 1, hep- approach, a model for phantom cosmology, with ac- th/0405078; celerated expansion, is explored which is free from S. Nojiri, S.D. Odintsov, Phys. Rev. D 70 (2004) 103522, hep- catastrophic situations. This model is derived from th/0408170. Friedmann equations using the effective role of GCG [19] S.K. Srivastava, hep-th/0411221. [20] S. Nojiri, S.D. Odintsov, S. Tsujikawa, hep-th/0501025. behaviour in a natural way. [21] R. Jackiw, physics/0010042. [22] M.C. Bento, O. Bertolami, A.A. Sen, Phys. Rev. D 66 (2002) References 043507, gr-qc/0202064; N. Bilic, G.B. Tupper, R. Viollier, Phys. Lett. B 535 (2002) 17; [1] A.D. Miller, et al., Astrophys. J. Lett. 524 (1999) L1; J.S. Fabris, S.V. Goncalves, P.E. de Souza, astro-ph/0207430; P. de Bernadis, et al., Nature (London) 400 (2000) 955; V. Gorini, A. Kamenshchik, U. Moschella, Phys. Rev. D 67 A.E. Lange, et al., Phys. Rev. D 63 (2001) 042001; (2003) 063509, astro-ph/0210476; A. Melchiorri, et al., Astrophys. J. Lett. 536 (2000) L63. C. Avelino, L.M.G. Beca, J.P.M. de Carvalho, C.J.A.P. Martins, [2] S. Hanay, et al., Astrophys. J. Lett. 545 (2000) L5. P. Pinto, Phys. Rev. D 67 (2003) 023511, astro-ph/0208528. [3] S. Perlmutter, et al., Astrophys. J. 517 (1999) 565. [23] F. Hoyle, J.V. Narlikar, Mon. Not. R. Astron. Soc. 108 (1948) [4] A.G. Riess, et al., Astron. J. 116 (1998) 1009. 372; [5] D.N. Spergel, et al., astro-ph/0302209. F. Hoyle, J.V. Narlikar, Mon. Not. R. Astron. Soc. 109 (1949) [6] L. Page, et al., astro-ph/0302220. 365; [7] V. Faraoni, Phys. Rev. D 68 (2003) 063508; F. Hoyle, J.V. Narlikar, Proc. R. Soc. London, Ser. A 282 R.A. Daly, et al., astro-ph/0203113; (1964) 191; R.A. Daly, E.J. Guerra, Astron. J. 124 (2002) 1831; F. Hoyle, J.V. Narlikar, Mon. Not. R. Astron. Soc. 155 (1972) R.A. Daly, astro-ph/0212107; 305; S. Hannestad, E. Mortsell, Phys. Rev. D 66 (2002) 063508; J.V. Narlikar, T. Padmanabhan, Phys. Rev. D 32 (1985) 1928; A. Melchiorri, et al., Phys. Rev. D 68 (2003) 043509; F. Hoyle, G. Burbidge, J.V. Narlikar, A Different Approach to P. Schuecker, et al., astro-ph/0211480. Cosmology, Cambridge Univ. Press, Cambridge, 2000. [8] R.R. Caldwell, Phys. Lett. B 545 (2002) 23; [24] J. Barrow, Class. Quantum Grav. 21 (2004) L79, gr-qc/ R.R. Caldwell, M. Kamionkowski, N.N. Weinberg, Phys. Rev. 0403084. Lett. 91 (2003) 071301. Physics Letters B 619 (2005) 5–10 www.elsevier.com/locate/physletb

Viscous dark energy and phantom evolution

Mauricio Cataldo a, Norman Cruz b, Samuel Lepe c

a Departamento de Física, Facultad de Ciencias, Universidad del Bío–Bío, Avenida Collao 1202, Casilla 5-C, Concepción, Chile b Departamento de Física, Facultad de Ciencia, Universidad de Santiago, Casilla 307, Santiago, Chile c Instituto de Física, Facultad de Ciencias Básicas y Matemáticas, Pontiﬁcia Universidad Católica de Valparaíso, Avenida Brasil 2950, Valparaíso, Chile Received 29 April 2005; received in revised form 10 May 2005; accepted 11 May 2005 Available online 23 May 2005 Editor: M. Cveticˇ

Abstract In order to study if the bulk viscosity may induce a big rip singularity on the ﬂat FRW cosmologies, we investigate dissipative processes in the universe within the framework of the standard Eckart theory of relativistic irreversible thermodynamics, and in the full causal Israel–Stewart–Hiscock theory. We have found cosmological solutions which exhibit, under certain constraints, a big rip singularity. We show that the negative pressure generated by the bulk viscosity cannot avoid that the dark energy of the universe to be phantom energy.  2005 Elsevier B.V. All rights reserved.

PACS: 98.80.Jk; 04.20.Jb

1. Introduction The dark energy is considered a fluid characterized by a negative pressure and usually represented by the The existence of an exotic cosmic fluid with nega- equation of state w = p/ρ, where w lies very close tive pressure, which constitutes about the 70 percent of to −1, most probably being below −1. Dark energy the total energy of the universe, has been perhaps the with w<−1, the phantom component of the uni- most surprising discovery made in cosmology. This verse, leads to uncommon cosmological scenarios as dark energy is supported by the astrophysical data ob- it was pointed out in [1]. First of all, there is a vio- tained from Wilkinson Microwave Anisotropy Probe lation of the dominant energy condition (DEC), since (WMAP) (Map) and high redshift surveys of super- ρ + p<0. The energy density grows up to infinity novae. in a finite time, which leads to a big rip, characterized by a scale factor blowing up in this finite time. E-mail addresses: [email protected] (M. Cataldo), These sudden future singularities are, nevertheless, not [email protected] (N. Cruz), [email protected] (S. Lepe). necessarily produced by a fluids violating DEC. Bar-

0370-2693/$ – see front matter  2005 Elsevier B.V. All rights reserved. doi:10.1016/j.physletb.2005.05.029 6 M. Cataldo et al. / Physics Letters B 619 (2005) 5–10 row [2] has shown, with explicit examples, that exist The organization of the Letter is as follows. In Sec- solutions which develop a big rip singularity at a finite tion 2 we present the field equations for a flat FRW time even if the matter fields satisfy the strong-energy universe filled with a bulk viscous fluid within the conditions ρ>0 and ρ + 3p>0. A generalization framework of the Eckart theory. We indicate that un- of Barrow’s model has been realized in [3], giving its der a constraint for the parameters of the fluid, one of Lagrangian description in terms of scalar tensor the- the Barrow’s solutions presents a future singularity in ory. It was also proved by Chimento et al. [4] that a finite time. In Section 3 we obtain big rip solutions exists a duality between phantom and flat Friedmann– using the approach of the full Israel–Stewart–Hiscock Robertson–Walker (FRW) cosmologies with nonex- causal thermodynamics. In Section 4 we discuss our otic fluids. This duality is a form-invariance transfor- results in relation to the nature of the dark energy of mation which can be used for constructing phantom the universe. cosmologies from standard scalar field universes. Cos- mological solutions for phantom matter which violates the weak energy condition were found in [5]. 2. Eckart theory The role of the dissipative processes in the evolution of the early universe also has been extensively The FRW metric for an homogeneous and isotropic studied. In the case of isotropic and homogeneous cos- flat universe is given by mologies, any dissipation process in a FRW cosmol- 2 =− 2 + 2 2 + 2 2 + 2 2 ogy is scalar, and therefore may be modelled as a bulk ds dt a(t) dr r dθ sin θdφ , (3) viscosity within a thermodynamical approach. where a(t) is the scale factor and t represents the cos- A well-known result of the FRW cosmological so- mic time. In the following we use the units 8πG= 1. lutions, corresponding to universes filled with perfect In the first-order thermodynamic theory of Eckart [8] fluid and bulk viscous stresses, is the possibility of the field equations in the presence of bulk viscous violating DEC [6]. The bulk viscosity introduces dis- stresses are sipation by only redefining the effective pressure, Peff, a˙ 2 ρ according to = H 2 = , (4) a 3 Peff = p + Π = p − 3ξH, (1) ¨ a ˙ 2 1 where Π is the bulk viscous pressure, ξ is the bulk = H + H =− (ρ + 3Peff), (5) viscosity coefficient and H is the Hubble parameter. a 6 Since the equation of energy balance is with = + ρ˙ + 3H(ρ+ p + π)= 0, (2) Peff p Π, (6) and the violation of DEC, i.e., ρ + p + Π<0 implies an increasing energy density of the fluid that fills the Π =−3Hξ. (7) universe, for a positive bulk viscosity coefficient. The condition ξ>0 guaranties a positive entropy produc- The conservation equation is tion and, in consequence, no violation of the second ρ˙ + 3H(ρ+ p + Π)= 0. (8) law of the thermodynamics [7]. In the present Letter we show that the above results Assuming that the dark component obey the state are straightforward to obtain from the exact cosmolog- equation ical solutions already found by Barrow in [6]. These = − solutions were obtained using non-causal thermody- p (γ 1)ρ, (9) namics. Nevertheless, we consider a more physical where 0 γ 2, we can obtain from Eqs. (4)–(9) a approach like the full Israel–Stewart–Hiscock causal single evolution equation for H : thermodynamics, showing that it is also possible to ob- ˙ 2 tain big rip type solutions. 2H + 3γH = 3ξH. (10) M. Cataldo et al. / Physics Letters B 619 (2005) 5–10 7

This equation may be integrated directly as a func- ξ>0, and if γ<0Eq.(12) implies that we can have tion of the bulk viscosity. For γ = 0 the solution has a big rip singularity at a finite value of cosmic time. the form Let us consider some examples to see this more clearly. From Eq. (12) the well-known standard case exp 3 ξ(t)dt H(t)= 2 , (11) for a perfect fluid, i.e., ξ = 0, takes the form a(t) = + 3 3 2/(3γ) C 2 γ exp 2 ξ(t)dt dt D(C + (3/2)γ t) . This scale factor may be rewritten as where C is an integration constant. From this equation we find the following expression for the scale factor: 3 2/(3γ) a(t) = a 1 + H γt , (15) 0 2 0 3 3 2/(3γ) a(t) = D C + γ exp ξ(t)dt dt , 2 2 and the energy density is given by (12) ρ0 ρ = , (16) + 3 2 where D is a new integration constant. Thus for a (1 2 H0γt) given ξ(t) we have the expressions for a(t), ρ(t) and where ρ = 3H 2, in order to have H(t = 0) = p(t). 0 0 0 H > 0. If γ<0 we have a big rip singularity at a For the case γ = 0wehavefromEq.(10) 0 finite value of cosmic time tbr =−2/(3H0γ)>t0 = 0. ˙ 2 H In the special case of ξ(t) = ξ0 = const we have ξ = , (13) 3 H from Eq. (12) for the scale factor and substituting this expression into Eq. (8) we have γ 2/(3γ) a(t) = D C + e(3/2)ξ0t . ξ ρ˙ = 6HH,˙ (14) 0 We can rewrite it into the form from which we conclude that ρ = 3H 2 + const. Com- H 2/(3γ) paring this expression with (4) we have that the inte- 0 3ξ0t/2 a(t) = a0 1 + γ e − 1 , (17) gration constant is zero. ξ0 =− Thus we have that for the state equation p ρ, from which we obtain for the energy density i.e., for γ = 0, the scale factor is not defined by the field equations. So for a given a(t) we can write H e3ξ0t ρ(t)= ρ , (18) 0 H 2 and then obtain the expressions for the energy den- 1 + 0 γ(e3ξ0t/2 − 1) sity from Eq. (4) and the bulk viscosity from Eq. (13). ξ0 = = 2 Clearly, if ξ 0, the well-known de Sitter scale factor where ρ0 3H0 . As before, for γ<0wehaveabig a(t) = eH0t is obtained, where p =−ρ and both are rip singularity at a finite value of cosmic time constants. 2 ξ0 Notice that the solution of the field equations may tbr = ln 1 − >t0 = 0. be written through ξ(t) or a(t) because there are three 3ξ0 H0γ independent equations for the four unknown func- Note that any additional condition on the system tions a(t), ρ(t), ξ(t) and p(t). of the field equations will fix the unknown functions. Now we are interested in the possibility that there So for instance, for a variable ξ(t) we can take the are cosmological models with viscous matter which condition ξ(t)= ξ(ρ(t)). present in its development a big rip singularity. Another example in this line is given by the solution obtained by Barrow [6] for the case ξ ∼ ρ1/2. 2.1. The case for γ = 0 Effectively, Barrow [6] assumed that the viscosity has a power-law dependence upon the density Firstly, let us consider the case γ = 0. If the viscous ξ = αρ s,α 0, (19) fluid satisfies DEC, then the condition 0 γ 2must be satisfied. Thus for γ<0 we have a phantom cos- where α and s are constant parameters, and exact cosmology. Now from the thermodynamics we know that mological solutions for a variety of ξ(ρ) in the form 8 M. Cataldo et al. / Physics Letters B 619 (2005) 5–10 givenbyEq.(19). In particular, for the case s = 1/2, 2.2. The case for γ = 0 i.e., ξ = αρ 1/2, yields a power-law expansion for the scale factor. Nevertheless, none condition was im- Notice that the structure of Eq. (10) changes if posed upon the parameters α and γ in order to obtain γ = 0 and ξ is an arbitrary function of the density, solutions with big rip. since the quadratic term in H disappears. Neverthe- For the case s = 1/2, the integration of Eq. (10) less, if ξ ∼ ρ1/2, the structure of Eq. (10) is the same yields the following expression for H(t) for any value√ of γ in the range 0 γ 2, except in the case 3 α = γ , where Eq. (10) becomes H˙ = 0. √ 1 1 3 Then, the solution with γ = 0 can be obtained directly = − 3 α − γ (t − t0), (20) H H0 2 from the general solution given by Eq. (21).Inthis case there is a big rip singularity at a finite value of where H = H(t = t ) and t correspond to the time 0 0 0 cosmic time where dark component begins to become dominant. The scale factor becomes 2 tbr = √ >t0 = 0. 3 3 H α 2√ 0 t − t0 3(γ − 3 α) a(t) = a0 1 − , (21) tbr 3. Israel–Stewart–Hiscock theory where a0 = a(t = t0). If we demand to have the occurrence of a big rip in the future cosmic time then We now consider the dissipative process in the uni- we have the following constraint on the parameters α verse within the framework of the full causal theory and γ of Israel–Stewart–Hiscock. In this case we have the √ same Friedmann equations but instead of Eq. (7),we 3 α>γ, (22) have an equation for the causal evolution of the bulk viscous pressure, which is given by leading the scale factor blow up to infinity at a finite ˙ ˙ time tbr >t0, which expression is 1 τ˙ ξ T τΠ˙ + Π =−3ξH − τΠ 3H + − − , 2 τ ξ T 2 −1 tbr = √ H . (23) (27) 3( 3 α − γ) 0 where T is the temperature and τ the relaxation time. In terms of time tbr, the Hubble parameter is given by In order to close the system we have to give the equation specifying T − −1 t t0 r H(t)= H0 1 − . (24) T = βρ . (28) tbr The relaxation time is defined by the expression From Eq. (4) and the parameterized equations (21) and (24) for the scale factor and Hubble parameter, re- ξ − τ = = αρ s 1, (29) spectively, we obtain the expression for the increasing ρ density of the dark component in terms of scale factor where β 0. This model imposes the constraint √ 3( 3 α−γ) γ − 1 = 2 a r = , (30) ρ(a) 3H0 . (25) γ a0 in order to have the entropy as a state function. Notice We reproduce completely this solution if we put into that the above constraint exclude the range 0 <γ <1, the Eq. (12) the bulk viscosity given by which implies that quintessence fluids are not allowed √ −1 in this approach. With the above assumptions the field t − t0 ξ(t)= 3 αH0 1 − . (26) equations and the causal evolution equation for the tbr bulk viscosity lead to the following evolution equation M. Cataldo et al. / Physics Letters B 619 (2005) 5–10 9 for H [9]: where the discriminant ∆ has the expression: 3 − − − 2 H¨ + 1 + (1 − r)γ HH˙ + 31 sα 1H 2 2sH˙ b 1 2 ∆ ≡ − 4 . (40) a aγ − 9 − (1 + r)H 1H˙ 2 + (γ − 2)H 3 4 Since we are interested only in positive solutions 1 − − − for A, the coefficient a must be negative. We have two + 32 sα 1γH4 2s = 0. (31) 2 cases of interest. As in the non-causal case we will choose s = 1/2 and ; the above equation becomes Case 1. a<0 γ>0. In this case only A+ correspond to a solution with big rip. The parameters α and γ sat- 1 − isfy the following constraint: H¨ + bHH˙ − 2 − H 1H˙ 2 + aH3 = 0, (32) γ − √ γ 1 where a is defined by 3 α>γ 1 − . (41) 2 9 2 a ≡ 1 + √ γ − 2 , (33) Notice that there is no big rip solution if the cosmic 4 3 α fluid representing the dark component is stiff matter and b by (γ = 2). The factor (1−γ/2)−1 is the correction introduced by the causal thermodynamics to the constraint 1 b ≡ 3 1 + √ . (34) given by Eq. (22). The solution for A+ is given by 3 α + √1 + 1 + 2 1/2 Solutions of Eq. (32) were obtained in [10].Inthis 1 1 2 γ = 3α 3α work only was considered γ in the range 1 γ 2. A+ γ , (42) 3 1 − 1 + √2 Some of these solutions presents an increasing energy 3α 2 density and accelerated expansion. which implies that a big rip will occurs at a time Inspired in the solution for the Hubble parameter given by Eq. (24) in the non-causal scheme, we use the = −1 τbr A+H0 . (43) following ansatz, where for simplicity we take t0 = 0 The expressions for a = a(t) and ρ = ρ(a) can be eas- −1 H(t)= A(τbr − t) , (35) ily evaluate from Eqs. (36) and (37), respectively. where A ≡ H τ . With this ansatz the scale factor a(t) 0 br ; evolutes as Case 2. a<0 γ<0. Since we need ∆ 0 in order to have real solutions, the parameters α and γ must −A a(t) ∼ (τbr − t) , (36) satisfy the following constraint: and the energy density, ρ, of the dark component as a √ |γ | function of the scale factor becomes 3 α . (44) 2 √ ∼ 2/A √ ρ(a) a . (37) If ∆ = 0, i.e., 3α = |γ |/2, the solution for A, Using the ansatz (35) in Eq. (32) we obtain a second which we shall call A0, has the following expression: grade equation for A 1 + √1 2 3α 1 A0 = . (45) aA2 + bA + = 0. (38) 3 2 + 1 + √2 |γ | γ 3α The solutions for A are given by If ∆>0, the solutions for A can be written as b √ 1√ 2A± =− ± ∆, (39) A± = A ± ∆. (46) a 0 2 10 M. Cataldo et al. / Physics Letters B 619 (2005) 5–10

4. Discussion also reobtained the Barrow’s solution, for γ = 0, considered here. In the framework of the standard Eckart Within the framework of the non-causal thermo- theory [8], the authors show that fluids which lie in the dynamics we have showed that the power law solu- quintessence region (w>−1) can reduce its thermo- tion, found by Barrow in [6] for dissipative universes dynamical pressure and cross the barrier w =−1, and with ξ = αρ 1/2, yields cosmologies which present big behave like a phantom fluid (w<−1) with the inclu- rip singularity when the constraint given in Eq. (22) sion of a sufficiently large bulk viscosity. The case for holds. If we consider that the dark component is γ = 0 was not considered by these authors. quintessence, i.e., 0 γ 2/3, with a sufficiently large bulk viscosity will make this quintessence behaves like a phantom energy. In the range 2/3 >γ 2 Acknowledgements it is possible, at least from the mathematical point of view, to obtain solutions with big rip even with a mat- N.C. and S.L. acknowledge the hospitality of the ter fluid. It is not clear for us how can be interpreted a Physics Department of Universidad de Concepción radiation fluid, for example, with a large bulk viscos- where an important part of this work was done last Jan- ity leading to high negative pressures and increasing uary. S.L. acknowledges the hospitality of the Physics densities. Department of Universidad de Santiago de Chile. We At the boundary between the quintessence sector thank the suggestion of a new reference given by the = =− and the phantom sector, i.e., γ 0orp ρ,also referee, in order to improve the presentation of this there exist cosmologies with a big rip singularity. Letter. We acknowledge the partial support to this re- Using a more accurate approach like the full causal search by CONICYT through grants Nos. 1051086, theory of Israel–Stewart–Hiscock, we have also found 1030469 and 1040624 (M.C.); No. 1040229 (N.C. and cosmological solutions with big rip. S.L.); grant MECESUP USA0108 (N.C.). It also was If 1 γ 2 the parameters α and γ satisfy the supported by the Direccion de Investigación de la Uni- constraint given in Eq. (41). Due to the constraint stiff versidad del Bío–Bío (M.C.), PUCV Grant 123.771/04 matter is not allowed. As we mentioned above, this (S.L.). correspond to matter fluids that can lead to a phantom behavior. Quintessence region are not allowed. If γ<0 the cosmological solutions can be computed di- References rectly from Eqs. (45) and (46). The main conclusion, in the context of the full causal thermodynamics, is [1] R.R. Cadwell, M. Kamionkowski, N.N. Weinberg, Phys. Rev. that in order to obtain physically reasonable big rip so- Lett. 91 (2003) 071301. lutions, the dark component must be phantom energy. [2] J.D. Barrow, Class. Quantum Grav. 21 (2004) L79; J.D. Barrow, Class. Quantum Grav. 21 (2004) 5619. [3] S. Nojiri, S. Odintsov, Phys. Rev. D 70 (2004) 103522. Note added [4] L. Chimento, R. Lazkoz, Phys. Rev. Lett. 91 (2003) 211301. [5] M.P. Dabrowski, T. Stachowiak, M. Szydlowski, hep-th/ 0307128. While this manuscript was being written we no- [6] J.D. Barrow, Phys. Lett. B 180 (1987) 335; ticed about the work of Brevik and Gorbunova [11]. J.D. Barrow, Nucl. Phys. B 310 (1988) 743. The authors also consider the possibility of big rip [7] W. Zimdahl, D. Pavón, Phys. Rev. D 61 (2000) 108301. in viscous fluids with p = wρ, by a different formal- [8] C. Eckart, Phys. Rev. 58 (1940) 919. [9] R. Maartens, Class. Quantum Grav. 12 (1995) 1455. ism. They consider the case where the bulk viscosity [10] M.K. Mak, T. Harko, Gen. Relativ. Gravit. 30 (1998) 1171. is proportional to the scalar expansion.√ This is equiv- [11] I. Brevik, O. Gorbunova, gr-qc/0504001. alent to the Barrow’s choice ξ(t)∝ ρ, and then they Physics Letters B 619 (2005) 11–16 www.elsevier.com/locate/physletb

Energy conditions and Segre classiﬁcation of phantom ﬁelds

Janilo Santos a,J.S.Alcanizb

a Universidade Federal do Rio Grande do Norte, Departamento de Física, C.P. 1641, 59072-970 Natal, RN, Brazil b Observatório Nacional, Rua Gal. José Cristino 77, 20921-400 Rio de Janeiro, RJ, Brazil Received 20 April 2005; accepted 24 May 2005 Available online 1 June 2005 Editor: N. Glover

Abstract Recent discoveries in the field of observational cosmology have provided increasing evidence that the Universe is undergoing a late time acceleration, which has also stimulated speculations on the nature of the dark component responsible for such a phenomenon. Among several candidates discussed in the current literature, phantom fields, an exotic scalar field with a negative kinetic term and that violates most of the classical energy conditions, appear as a real possibility according to recent observational analysis. In this Letter we examine the invariant characterization for the energy–momentum tensor of phantom fields through the Segre algebraic classification in the framework of general relativity. We also discuss some constraints which are imposed on the values of V(φ)from the classical energy conditions.  2005 Elsevier B.V. All rights reserved.

PACS: 98.80.Jk; 98.80.-k; 04.20.Cv

1. Introduction criteria one chooses to group them into equivalence classes. There are, however, criteria that prove to be The algebraic classification of symmetric second- more important than others. The great appeal of Segre order tensors locally defined on a 4-dimensional classification in general relativity is that it incorpo- Lorentzian manifold, such as the Ricci tensor Rab, rates, ab initio, the Lorentzian character of space–time. the Einstein tensor Gab and the energy–momentum It is of interest in several contexts such as, for example, tensor Tab, is known as Segre classification [1].The in understanding purely geometrical features of space– major idea underlying most of classifications in sci- times [2], in classifying and interpreting matter fields ence is the concept of equivalence. Clearly the objects in general relativity [3] and in higher-dimensional the- may be classified in different ways according to the ories [4] (e.g., 5D brane-worlds [5]) or still as part of the procedure for checking whether apparently different space–times are in fact locally the same up to E-mail addresses: [email protected] (J. Santos), [email protected] (J.S. Alcaniz). coordinate transformations (equivalence problem [6]).

Because of Einstein’s equations Gab = κTab the Ein- by strange properties such as, for instance, the fact stein tensor (and so Rab) has the same algebraic clas- that its energy density increases with the expansion sification as the energy–momentum tensor. This can of the Universe (in contrast with quintessence fields); be used to decide, on base of generic features such as the possibility of a rip-off of large and small scale the Segre class, which energy–momentum tensors do structures of matter; a possible occurrence of future couple to a given geometry. curvature singularity, etc. [12]. Although having these A great deal of difficulty in Segre classifying sec- unusual characteristics, a phantom type behavior is ond order tensors in the context of general relativity is predicted by several scenarios, e.g., kinetically driven that while Gab and Rab are universal functions of the models [14] and some versions of brane world cos- space–time geometry, Tab depends on the symmetries mologies [15]. From the observational point of view, of the model as well as on the particular type or com- phantom dark energy is found to be compatible with bination of matter fields present in the cosmological most of the classical cosmological tests and provide a scenario. In this regard, a powerful way of imposing better fit to type Ia supernovae observations than do physical constraints and searching realistic forms for ΛCDM or quintessence scenarios (w > −1). There- the energy–momentum tensor is through cosmologi- fore this means that, although exotic, phantom fields cal observations. The high degree of isotropy observed may be the dominant form of energy in our Universe. in the cosmic microwave background radiation [7],for In this Letter we examine the invariant character- instance, restricts the general form of Tab while results ization for the energy–momentum tensor of phantom from distance measurements using type Ia supernovae fields through the Segre algebraic classification in the (which constitute the most direct evidence for the cur- framework of general relativity. In Section 2,byusing rent cosmic acceleration) [8] impose important con- real null tetrad technique, we show that phantom fields straints on the physical quantities of the cosmic fluid. can be classified in two different subclasses of equiva- In light of an impressive convergence of observa- lence which are represented by Segre types [1,(111)] tional results, several groups have recently tested the and [(1, 111)]. In Section 3 we present the so-called viability of different matter fields (or, equivalently, energy conditions of general relativity for the whole different forms of Tab) as a realistic description for Segre class [1, 111] and also examine the further re- the matter content of our Universe. The two favorite strictions they impose on the values of the phantom candidates are the energy density stored on the true field potential V(φ). We end this Letter by summariz- vacuum state of all existing fields in the Universe, ing the main results in the conclusion section. Λ = i.e., the cosmological constant (Tab Λgab) and the potential energy density V(φ) associated with a dynamical scalar field φ, usually called dynamical dark 2. Segre classification energy (see, e.g., [9] for a recent review on this topic). Physically, the role of such a dynamical dark energy A classification of a generic symmetric second or- can be played by any scalar field violating the so- der tensor Tab can be cast in terms of the eigenvalue called strong energy condition (see Section 3 for a problem discussion). Two possibilities, however, have been ex- T a − λδa vb = 0, (1) haustively explored in the current literature, namely, b b quintessence scalar fields with positive kinetic term where λ are eigenvalues, va are eigenvectors, and the a and an equation-of-state parameter w −1 [10], and mixed tensor T b may be thought of as a linear map phantom fields with negative kinetic term and w<−1 Tp(M) → Tp(M). M is a real 4-dimensional space– [11–13]. In terms of the parameter w, the former case time manifold locally endowed with a Lorentzian met- can be seen as a very general scenario, which includes ric of signature (−+++), Tp(M) denotes the tangent cold dark matter models with a cosmological constant space to M at a point p ∈ M and Latin indices range (ΛCDM), w =−1, and cosmologies dominated by from 0 to 3. Because of the Lorentzian character of the topological defects (e.g., domain walls, strings and metric the mixed form of the energy–momentum ten- =−n a textures) for which w 3 , n being the dimension sor is no more symmetric and T b may not have a di- of the defect. The latter case in turn is characterized agonal matrix representation, i.e., it need not have four J. Santos, J.S. Alcaniz / Physics Letters B 619 (2005) 11–16 13 linearly independent eigenvectors. However, using the taken into account by Segre class [zz¯11], where the zz¯ Jordan canonical forms of the matrix of an operator, refers to a pair of complex eigenvalues. Degeneracy and imposing the Lorentzian character of the metric on amongst eigenvalues in different Jordan blocks will be M, it has been shown [1,3] (for detailed calculations indicated by enclosing the corresponding digits inside in n 5 dimensional space–times and a review on this round brackets, as in [1,(111)], which indicate that topic see [20]) that any energy–momentum tensor de- three out of the four eigenvalues are degenerate. So, fined on Tp(M) reduces to one of the four canonical to each Segre class there may be several subclasses forms: depending on the degeneracies of the eigenvalues. [ ] = + + 1, 111 Tab 2σ1l(amb) σ2(lalb mamb) 2.1. Segre classification of phantom fields + σ3xaxb + σ4yayb, The energy–momentum tensor for the phantom [211] Tab = 2σ1l(amb) ± lalb + σ2xaxb field has the form [13] + σ3yayb, 1 cd [31] Tab = 2σ1l(amb) + 2l(axb) + σ1xaxb T =−φ φ + g g φ φ − V(φ) , (4) ab a b ab 2 c d + σ2yayb, where V(φ) is the phantom potential, φ ≡ φ; and [ ¯ ] = + − a a zz11 Tab 2σ1l(amb) σ2(lalb mamb) the semicolon denotes covariant derivative. Here, we + σ3xaxb + σ4yayb, (2) assume that φ = φ(t)is a function of time alone evolving in an isotropic and homogeneous space–time, so where σ ,...,σ ∈ R, having different values for dif- 1 4 g φaφb =−φ˙2, that is, φa is a time-like vector (in- ferent formulae, and in the Segre class [zz¯11] σ = 0. ab 2 deed an eigenvector of T ). In this case it is always In the above “catalog” the first column is the notation ab possible to find out two null vectors la and ma such commonly used for indicating the Segre class, which a that l ma = 1, and φa can be written as is a list [r1r2 ...rn] of the dimensions of the Jordan blocks of the corresponding Jordan canonical matrix. ˙ = √φ − {l,m,x,y} φa (la ma). (5) The basis vectors form a real null tetrad 2 basis such that the only non-vanishing inner products are Besides, we choose two spacelike vectors xa and ya, belonging to the 2-space orthogonal to the 2-space a a a l ma = x xa = y ya = 1. (3) generated by la and ma, so as to form the real null tetrad basis {l,m,x,y} defined by (3). In terms of this This basis is constructed from the preferred directions basis the metric tensor is written as intrinsically defined by the tensor, i.e., from the Jor- [ ] dan basis. The Segre class 1, 111 distinguishes ten- gab = 2l(amb) + xaxb + yayb, (6) sors that have diagonal matrix representation and is the unique that admits a timelike eigenvector. The comma where the round brackets indicate symmetrization. in this case is used to separate timelike from spacelike Taking into account Eqs. (5) and (6), the canonical eigenvectors. Regardless of the dimension of a Jordan form for the energy–momentum tensor (4) is then writ- block, there is only one eigenvector associated to each ten as block, and the eigenvector associated to a block of di- T = 2σ l m + σ (l l + m m ) mension r>1 is a null vector [4]. Energy–momentum ab 1 (a b) 2 a b a b tensors which belong to Segre class [211], for exam- + σ3(xaxb + yayb), (7) ple, have only three linearly independent eigenvectors where (one of which is a null vector), and its characteristic polynomial obtained from (1) has three roots: one 1 ˙2 σ1 =−V(φ), σ2 =− φ , of multiplicity 2 and the others of multiplicity 1. So, 2 1 the digits inside the brackets give also the multiplic- σ =− φ˙2 − V(φ), (8) ity of the real eigenvalues, the complex one being 3 2 14 J. Santos, J.S. Alcaniz / Physics Letters B 619 (2005) 11–16 indicating that phantom fields belong to Segre class here only on the Segre class [1, 111] of Eq. (2).The [1, 111] (see Eq. (2))forσ3 = σ4. In order to find the most common energy conditions are [17,18] (see also subclasses we now determine its eigenvalues and cor- [19] for a recent discussion on this topic): responding eigenvectors. For the sake of brevity we present our results without going into details of calcu- (i) The null energy condition (NEC). NEC states that a b a lations, which can be easily verified from Eq. (7) and Tabn n 0 for null vectors n ∈ Tp(M) which, the expressions below. We find that the set of linearly for Segre class [1, 111], is equivalent to require independent eigenvectors and associated eigenvalues that σ2 − σ1 + σα 0(α = 3, 4). a of T b is given by (ii) The weak energy condition (WEC). WEC states that T tatb 0 for timelike vectors ta ∈ T (M). 1 ab p la − ma → φ˙2 − V(φ) , This will also imply, by continuity, the NEC. The 2 WEC for Segre class [1, 111] means that σ2 − 1 la + ma →− φ˙2 + V(φ) , σ1 0. 2 (iii) The strong energy condition (SEC). SEC is the assertion that for any timelike vector (T − a 1 ab x →− φ˙2 + V(φ) , a b T/2gab)t t 0, where T is the trace of Tab.If 2 Tab belongs to the Segre class [1, 111] then we a 1 ˙2 y →− φ + V(φ) . (9) must have 2σ2 + σ3 + σ4 0. 2 (iv) The dominant energy condition (DEC). DEC a b Note that three out of the four eigenvalues are degen- requires that Tabt t 0 for timelike vectors a erate, making it apparent that the corresponding Segre t ∈ Tp(M) and the additional requirement that b subclass is [1,(111)]. This is the same Segre type as Tabt be a non-spacelike vector. By continuity a that for a perfect fluid [16] (we shall comment this this will also hold for null vectors n ∈ Tp(M). latter). We emphasize that this classification is inde- For energy–momentum tensors of Segre class pendent of the functional form of the potential V(φ) [1, 111] this requires that σ2 − σ1 0, and ˙ as well as the time derivative φ except for φ = const. σ1 − σ2 σα σ2 − σ1 (α = 3, 4). In this latter case the Segre subclass for the phantom field is [(1, 111)] with −V(φ) a fourfold-degenerate When imposed on the energy–momentum tensor of the eigenvalue. This is the same Segre subclass of energy– phantom field these energy conditions require: momentum tensors of Λ-term type (cosmological con- 1 stant) [16]. Since a Λ-term can always be incorporated NEC ⇒− φ˙2 0, into an energy–momentum tensor of the perfect fluid, 2 1 we shall banish this subclass from further considera- WEC ⇒− φ˙2 + V(φ) 0, tion. 2 SEC ⇒ φ˙2 + V(φ) 0, DEC ⇒−φ˙2 0 and V(φ) 0. (10) 3. Classical energy conditions for phantom fields From the above conditions, it should be noticed that In this section we investigate possible constraints although the Segre subclass [1,(111)] for the phan- that the classical energy conditions may impose on tom field is consistent with the Lorentzian signature ˙ the values of the potential V(φ) as well as on φ for of the metric tensor gab, imposition of NEC, as well phantom fields. Restrictions imposed by energy con- as DEC, generates at least one contradiction in the ˙2 a b ditions on energy–momentum tensors of general rel- sense that the timelike character (−φ = gabφ φ ) of ativity theory for matter fields like those represented the vector φa is violated. Maintaining the Lorentzian by Segre classes [211], [31] and [zz¯11] in Eq. (2), timelike character of φa WEC can be preserved only have been presented in the literature [1,17].Aswe for positive potentials since that V(φ) φ˙2/2. SEC, have found the Segre specializations [1,(111)] and on the other hand, cannot be satisfied unless V(φ) is [(1, 111)] for phantom fields, we focus our attention negative and V(φ) −φ˙2. There is however an inter- J. Santos, J.S. Alcaniz / Physics Letters B 619 (2005) 11–16 15

phantom Tab belongs to Segre subclasses [1,(111)] or [(1, 111)]. As is well known, all energy–momentum tensors belonging to these Segre subclasses (e.g., perfect fluid and Λ-term type energy–momentum tensors) couple to Friedmann–Robertson–Walker geometries. We also have found some constraints which are imposed on the values of V(φ)from the classical energy conditions. Although the SEC is being violated right Fig. 1. Energy conditions constraints on phantom fields potential. In now, according recent observational data regarding the interval (I) phantom fields do not violate SEC but do violate WEC. accelerating universe, we see from our analysis that In interval (II) both (SEC and WEC) are violated while in region an evolving potential function V(φ)can, in principle, (III) phantom fields do not violate WEC but do violate SEC. be constructed such that this condition is not violated in the past (interval I of Fig. 1), while in that epoch ˙2 ˙2 val (−φ

second-order tensor on 4-dimensional Lorentzian space–times J.A.S. Lima, J.S. Alcaniz, Phys. Lett. B 600 (2004) 191; manifolds. J.M. Aguirregabiria, L.P. Chimento, R. Lazkoz, Phys. Rev. [2] R.V. Churchill, Trans. Amer. Math. Soc. 34 (1932) 784; D 70 (2004) 023509. W.J. Cormack, G.S. Hall, J. Phys. A 12 (1979) 55. [12] R.R. Caldwell, M. Kamionkowski, N.N. Weinberg, Phys. Rev. [3] G.S. Hall, D.A. Negm, Int. J. Theor. Phys. 25 (1986) 405; Lett. 91 (2003) 071301; J. Santos, M.J. Rebouças, A.F.F. Teixeira, J. Math. Phys. 34 P.F. Gonzalez-Diaz, Phys. Rev. D 68 (2003) 021303; (1993) 186. S. Nesseris, L. Perivolaropoulos, Phys. Rev. D 70 (2004) [4] J. Santos, M.J. Rebouças, A.F.F. Teixeira, J. Math. Phys. 36 123529; (1995) 3074; P.F. Gonzalez-Diaz, J.A. Jimenez-Madrid, Phys. Lett. B 596 J. Santos, M.J. Rebouças, A.F.F. Teixeira, Gen. Relativ. (2004) 16. Gravit. 27 (1995) 989. [13] P. Singh, M. Sami, N. Dadhich, Phys. Rev. D 68 (2003) [5] M.J. Rebouças, J. Santos, Mod. Phys. Lett. A 18 (2003) 2807. 023522. [6] A. Karlhede, Gen. Relativ. Gravit. 12 (1980) 693; [14] T. Chiba, T. Okabe, M. Yamaguchi, Phys. Rev. D 62 (2000) M.A.H. MacCallum, Computer-aided classification of exact 023511. solutions in general relativity, in: R. Cianci, R. de Ritis, M. [15] V. Sahni, Y. Shtanov, Int. J. Mod. Phys. D 11 (2002) 1515; Francaviglia, G. Marmo, C. Rubano, P. Scudellaro (Eds.), Gen- V. Sahni, Y. Shtanov, JCAP 0311 (2003) 014. eral Relativity and Gravitational Physics, 9th Italian Confer- [16] D. Kramer, H. Stephani, E. Herlt, M. MacCallum, Exact So- ence, World Scientific, Singapore, 1991. lutions of Einstein’s Field Equations, Cambridge Univ. Press, [7] D.N. Spergel, et al., Astrophys. J. Suppl. 148 (2003) 175. Cambridge, 1980, p. 67. [8] A.G. Riess, et al., Astrophys. J. 607 (2004) 665. [17] S.W. Hawking, G.F.R. Ellis, The Large Scale Structure of [9] V. Sahni, A. Starobinsky, Int. J. Mod. Phys. D 9 (2000) 373; Spacetime, Cambridge Univ. Press, Cambridge, 1973. J.E. Peebles, B. Ratra, Rev. Mod. Phys. 75 (2003) 559; [18] M. Visser, Lorentzian Wormholes, AIP, New York, 1996. T. Padmanabhan, Phys. Rep. 380 (2003) 235; [19] M. Visser, Science 276 (1997) 88; J.A.S. Lima, Braz. J. Phys. 34 (2004) 194. M. Visser, C. Barceló, Energy conditions and their cosmologi- [10] B. Ratra, P.J.E. Peebles, Phys. Rev. D 37 (1988) 3406; cal implications, in: Cosmo99, Proceedings of the Third Inter- R.R. Caldwell, R. Dave, P.J. Steinhardt, Phys. Rev. Lett. 80 national Workshop on Particle Physics and the Early Universe, (1998) 1582; ICTP, Trieste, Italy, September–October 1999, World Scien- P.G. Ferreira, M. Joyce, Phys. Rev. D 58 (1998) 023503. tific, Singapore, 2000, gr-qc/0001099; [11] R.R. Caldwell, Phys. Lett. B 545 (2002) 23; C. Barceló, M. Visser, Int. J. Mod. Phys. D 11 (2002) 1553. B. McInnes, astro-ph/0210321; [20] M.J. Rebouças, J. Santos, A.F.F. Teixeira, Braz. J. Phys. 34 V. Faraoni, Int. J. Mod. Phys. D 11 (2002) 471; (2004) 535. S.M. Carroll, M. Hoffman, M. Trodden, Phys. Rev. D 68 [21] L. Perivolaropoulos, astro-ph/0412308. (2003) 023509; [22] J.D. Barrow, Class. Quantum Grav. 21 (2004) L79; S. Nojiri, S.D. Odintsov, Phys. Lett. B 562 (2003) 147; J.D. Barrow, Class. Quantum Grav. 21 (2004) 5619; J.S. Alcaniz, Phys. Rev. D 69 (2004) 083521; J.D. Barrow, C.G. Tsagas, gr-qc/0411045. Physics Letters B 619 (2005) 17–25 www.elsevier.com/locate/physletb

Cosmic acceleration and the string coupling

John Ellis a, N.E. Mavromatos b, D.V. Nanopoulos c,d,e

a TH Division, Physics Department, CERN, CH-1211 Geneva 23, Switzerland b Theoretical Physics, Physics Department, King’s College London, Strand WC2R 2LS, UK c George P. and Cynthia W. Mitchell Institute for Fundamental Physics, Texas A&M University, College Station, TX 77843, USA d Astroparticle Physics Group, Houston Advanced Research Center (HARC), Mitchell Campus, Woodlands, TX 77381, USA e Academy of Athens, Division of Natural Sciences, 28 Panepistimiou Avenue, Athens 10679, Greece Received 24 January 2005; accepted 20 May 2005 Available online 31 May 2005 Editor: L. Alvarez-Gaumé

Abstract In the context of a cosmological string model describing the propagation of strings in a time-dependent Robertson–Walker background space–time, we show that the asymptotic acceleration of the Universe can be identiﬁed with the square of the string coupling. This allows for a direct measurement of the ten-dimensional string coupling using cosmological data. We conjecture that this is a generic feature of a class of non-critical string models that approach asymptotically a conformal (critical) σ model whose target space is a four-dimensional space–time with a dilaton background that is linear in σ -model time. The relation between the cosmic acceleration and the string coupling does not apply in critical strings with constant dilaton ﬁelds in four dimensions.  2005 Elsevier B.V. All rights reserved.

String theory [1,2] was ﬁrst developed as a the- were found to construct low-energy models that could ory of the strong interactions, but it soon turned out be consistent with the current particle physics phe- that mathematical consistency (world-sheet confor- nomenology, but string models of this type had zero mal invariance) required the theory to live in higher- predictability, in the sense that they were unable to dimensional space times. Even target-space supersym- make predictions for the parameters of the Standard metry was not successful in reducing the number of Model, and there were many string models with indis- space–time dimensions below ten. Thus, enormous ef- tinguishable low-energy limits. fort has been expended on the compactiﬁcation of the Although in principle string theory has no free pa- extra dimensions, with the eventual aim of accommo- rameters, and the ground state corresponding to the dating the Standard Model at low energies. Many ways observable low-energy world is supposed to be chosen dynamically, a detailed understanding of mechanism for choosing the ground state has not been achieved so E-mail address: [email protected] (N.E. Mavromatos). far. Lacking a microscopic, dynamical mechanism for

0370-2693/$ – see front matter  2005 Elsevier B.V. All rights reserved. doi:10.1016/j.physletb.2005.05.047 18 J. Ellis et al. / Physics Letters B 619 (2005) 17–25 specifying the various string model parameters, such the context of the minimal supersymmetric extension as the compactification radii and the four-dimensional of the Standard Model. gauge couplings, one has had simply to fix them by Modern developments in string theory [2] make hand, so as to match the results with experimental ob- possible consistent quantum treatments of domain- servations in particle physics. In this framework, the wall structures in string theory (D-branes). These have mechanism whereby one particular model is chosen opened up novel ways of looking at both the micro- from among the complicated string ‘landscape’ [3] is cosmos and the macrocosmos, offering new insights still unclear. into both particle phenomenology and the cosmic evo- The most important and fundamental string para- lution of our Universe. In the microcosmos, there are meter is the string coupling, gs , which determines the novel ways of compactification, either via the observa- regime of validity of string perturbation theory, and tion [4] that large (compared to the string scale) extra hence the world-sheet σ -model scheme for low-energy dimensions are consistent both with the foundations computations of the low-energy string effective action. of string theory and phenomenology, or by viewing Since gs is connected to the unified ten-dimensional our four-dimensional world as a brane embedded in gauge coupling of the effective supersymmetric low- a bulk space–time. This would allow for large ex- energy theory, its value is usually inferred from par- tra bulk dimensions, which could even be infinite in ticle phenomenology. The string coupling is not a size [5], offering new ways to analyze the large hier- constant but, like any other dynamical coupling in a archy between the Planck scale and the electroweak supersymmetric field theory, is related to the vacuum symmetry-breaking scale. In this modern approach, expectation value of a field, in this particular case fields in the gravitational (super)multiplet of the (su- the dilaton field Φ, which belongs to the gravitational per)string are allowed to propagate in the bulk, but not multiplet obtained from the string [1] the gauge fields, which are attached to the brane world. In this way, the weakness of gravity as compared to the 2 = 2Φ gs e . (1) rest of the interactions is a result of the large extra dimensions. Their compactification is not necessar- Usually, upon compactification the dilaton field is split ily achieved through conventional means, i.e., closing into a product of two factors, one depending on the up the extra dimensions in compact spatial manifolds, compact six-dimensional space coordinates and the but might also involve shadow brane worlds with spe- other on the four-dimensional space–time coordinates, cial reflecting properties (such as orientifolds), which which are supposed to correspond to the large, unbound the bulk dimension [6]. In such approaches, compactified coordinates of our observable world. In the string scale Ms is not necessarily identical to the most of the phenomenological approaches to model four-dimensional Planck mass scale MP , but instead building, the four-dimensional dilaton field has been they are related through the large compactification vol- assumed to be constant and therefore trivial, since this ume V6: constant value could be absorbed in an innocuous shift 8M8V in the field. M2 = s 6 . P 2 (2) In this approach, neither the string coupling nor gs the unified gauge coupling are accessible directly to As for the macrocosmos, there are novel ways of experimental measurement. It is consistency of the discussing cosmology in brane worlds, which may available phenomenological model with low-energy revolutionize our way of approaching issues such as observational data that leads to an indirect fixing of inflation [7,8]. 2 the string coupling. A popular value is gs 0.52, Mounting experimental evidence from diverse as- which, upon compactification to small dimensions (of trophysical sources presents important cosmological the order of a tenth of the four-dimensional Planck puzzles that string theory must address if it is to pro- 19 mass, MP ∼ 10 GeV), yields a four-dimensional vide a realistic description of Nature. Observations of 2 ∼ unified gauge coupling strength gU /4π 1/24 at large-scale structures, distant type-1a supernovae [9], 16 scales MU ∼ 10 GeV, as suggested by extrapolat- and the cosmic microwave background fluctuations ing the measured gauge couplings to high energies in (by WMAP [10] in particular) have established that J. Ellis et al. / Physics Letters B 619 (2005) 17–25 19 the Hubble expansion of our Universe is currently ac- positive cosmological constant Λ>0 has a non-zero celerating, and that 70% of its energy density consists Ricci tensor Rµν = Λgµν , where gµν is the metric ten- of unknown dark energy that appears in ‘empty’ space sor. and does not clump with ordinary matter. An interesting proposal for obtaining a non-zero These observations have great potential signifi- cosmological constant in string theory was made in cance for string theory, and may even revolutionize [11]. It was suggested that dilaton tadpoles in higher- the approach to it that has normally been followed so genus world-sheet surfaces, which produce additional far. If the dark energy leads to an asymptotic de Sit- modular infinities whose regularization leads to extra ter horizon, as would occur if it turns out to be a true world-sheet structures in the σ -model not appearing cosmological constant, then the entire concept of the at the world-sheet level, modify the string β-function scattering S-matrix breaks down, and hence the con- in such a way that the Ricci tensor of the space–time ventional approach to string theory. On the other hand, background is now that of a de Sitter Universe, with if there is some quintessential mechanism for relaxing a cosmological constant specified by the dilaton tad- the vacuum energy, so that the vacuum energy density pole graph. The problem with this approach is the vanishes at large cosmic times in a manner consis- above-mentioned existence of an asymptotic horizon tent with the existence of an S-matrix, there is still the in the de Sitter case, which prevents the proper defini- open issue of embedding such models in (perturbation of asymptotic states, and hence an S-matrix. Since tive) string theory. One would need, in particular, to the perturbative world-sheet formalism is based on the develop a consistent σ -model formulation of strings existence of such an S-matrix, there is a question of propagating in such time-dependent, relaxing space– consistency in this approach. time backgrounds. It was proposed in [12] that a way out of this dif- We here propose a resolution of this dilemma, ficulty would be to assume specific time-dependent based on string theory in a time-dependent dilaton dilaton backgrounds, with a linear dependence on time background, in which the asymptotic acceleration of in the so-called σ -model frame the Universe is directly related to the string coupling. Φ = const − Qt, (4) The world-sheet conformal-invariance conditions of critical string theory are equivalent to the target- where Q is a constant, and Q2 > 0 is a deficit in space equations of motion for the background fields the σ -model central charge. Such backgrounds, even through which the string propagates. These conditions when the σ -model metric is flat, lead to exact solu- are very restrictive, allowing only for vacuum solutions (in all orders in α) of the conformal-invariance tions of (critical) strings to be constructed in this way. conditions of the pertinent stringy σ -model, thereby The main problem may be expressed as follows. Con- constituting acceptable solutions from a perturbative sider the graviton world-sheet β function, which is string viewpoint. The appearance of Q allowed this nothing but the Ricci tensor of the target space–time supercritical string theory [12] to be formulated in background to lowest order in α: spaces with numbers of dimensions different from the critical case. This was actually the first example of a β = α R , (3) µν µν non-critical string, with the target-space coordinates where we ignore the possible presence of other fields, Xi , i = 1,...,D − 1, playing the rôles of the perti- for simplicity. Conformal invariance requires the van- nent σ -model fields. This non-critical string was not ishing condition βµν = 0, which implies that the back- conformally invariant, and hence required Liouville ground must be Ricci flat, which is a solution of the dressing [13]. The Liouville field had time-like signa- vacuum Einstein equations. The issue then arises how ture in target space, since the central charge deficit Q2 to describe in string theory cosmological backgrounds, was positive in the model of [12], and its zero mode which are not vacuum solutions, but require the pres- played the rôle of target time. ence of a matter fluid and hence a non-vanishing Ricci As a result of the existence of a non-trivial dilaton tensor. In this respect, we see that a cosmological con- field, the Einstein term in the effective D-dimensional stant is inconsistent with the conformal invariance of low-energy field theory action is conformally rescaled string since, for instance, a de Sitter Universe with a by e−2Φ . This requires a specific redefinition of tar- 20 J. Ellis et al. / Physics Letters B 619 (2005) 17–25 get time in order that the metric acquires the standard conformal invariance. Such backgrounds were also al- Robertson–Walker (RW) form in the normalized Ein- lowed to be time-dependent, and the target time was stein frame for the effective action identified with the Liouville world-sheet zero mode, thereby not increasing the target space–time dimen- 2 =− 2 + 2 2 + 2 2 dsE dtE aE(tE) dr r dΩ , (5) sionality. We have provided several justifications and where we have only exhibited a spatially-flat RW checks of this identification [14], which is possible metric for definiteness, and aE(tE) is an appropri- only when the initial σ -model is supercritical, so that ate scale factor, which is simply a function of the the Liouville mode has time-like signature [12,13].For Einstein-frame time tE in the homogeneous cosmolog- example, in certain models [15,16], such an identifica- ical backgrounds that we assume throughout. tion was energetically preferable from a target-space The Einstein-frame time is related to the σ -model- viewpoint, since it minimized certain effective poten- frame time [12] by tials in the low-energy field theory corresponding to the string theory at hand. t Such non-critical σ models relax asymptotically in = −Φ → = −Φ(t) dtE e dt tE e dt. (6) cosmic Liouville time to conformal σ models, the latter viewed as equilibrium points in string theory space. The linear dilaton background (4) yields then the In some interesting cases of relevance to cosmology, following relation between the Einstein- and σ -model- which were particularly generic, the asymptotic con- frame times formal field theory was that of [12], with a linear c0 Qt tE = c1 + e , (7) dilaton and a flat Minkowski target-space metric in the Q σ -model frame. where c1,0 are appropriate (positive) constants. Thus, One such model was considered in detail in [17]. a dilaton background that is linear in σ -model-frame The model was originally formulated within a specific time (4) will scale logarithmically with the Einstein- string theory, namely ten-dimensional type-0 [18], frame time tE, which is just the Robertson–Walker which leads to a non-supersymmetric target-space cosmic time spectrum, as a result of a special projection of the supersymmetric partners out of the spectrum. However, Q Φ(t ) = const − ln t . (8) the basic properties of its cosmology, which are those E c E 0 interest to us in this work, are sufficiently generic that In this regime, the string coupling (1) varies with the they can be extended to the bosonic sector of any other cosmic time tE as effective low-energy supersymmetric field theory of supersymmetric strings, including those relevant to 2 ∝ 1 gs (tE) , (9) unified particle physics phenomenology. t2 E The ten-dimensional metric configuration consid- implying that the effective string coupling vanishes ered in [17] was asymptotically in cosmic time. In the linear-dilaton background of [12], the asymptotic space–time met- (4) gµν 00 ric in the Einstein frame reads 2σ GMN = 0 e 1 0 , (11) 2σ 2 =− 2 + 2 2 2 + 2 2 00e 2 I5×5 ds dtE a0tE dr r dΩ , (10) where a0 a constant, which is a linearly-expanding where lower-case Greek indices are four-dimensional Universe. Clearly, there is no acceleration in the Uni- space–time indices, and I5×5 denotes the 5 × 5 unit verse (10). matrix. We have chosen two different scales for the In [14] we went one step further than the analy- internal space. The field σ1 sets the scale of the fifth di- sis in [12], and considered more complicated σ -model mension, while σ2 parametrizes a flat five-dimensional metric backgrounds, which did not satisfy the σ - space. In the context of the cosmological models we (4) model conformal-invariance conditions, and therefore treat here, the fields gµν , σi , i = 1, 2, are assumed to were in need of Liouville dressing in order to restore depend on the time t only. J. Ellis et al. / Physics Letters B 619 (2005) 17–25 21

Type-0 string theory, as well as its supersymmetric runs with the local world-sheet renormalization group versions appearing in other scenarios including brane scale, namely the zero mode of the Liouville field, models, contains appropriate form fields with non- which is identified [14] with the target time in the σ - trivial gauge fluxes (flux-form fields), which live in the model frame. The supercriticality [12] Q2 > 0ofthe higher-dimensional bulk space. In the specific model underlying σ model is crucial, as already mentioned. of [18], one such field was considered to be non-trivial. Physically, the non-critical string provides a frame- As was demonstrated in [17], a consistent background work for non-equilibrium dynamics, which may be the choice for the flux-form field has the flux parallel to result of some catastrophic cosmic event, such as a the fifth dimension σ1. This implies that the internal collision of two brane worlds [7,15,16], or an initial space is crystallized (stabilized) in such a way that this quantum fluctuation. dimension is much larger than the remaining five di- In the generic class of non-critical string models mensions σ2, demonstrating the physical importance considered in this work, the σ model always asymp- of the flux fields for large radii of compactification. totes, for long enough cosmic times, to the linear- Considering the fields to be time-dependent only, dilaton conformal σ -model field theory of [12].But i.e., considering spherically-symmetric homogeneous it is important to stress that this is only an asymptotic backgrounds, restricting ourselves to the compactifica- limit. In this respect, the current era of our Universe is tion (11), and assuming a Robertson–Walker form of viewed as being close, but still not quite at the relax- the four-dimensional metric with scale factor a(t),the ation (equilibrium) point, in the sense that the dilaton generalized conformal-invariance conditions and the is almost linear in the σ -model time, and hence varies Curci–Paffuti σ -model renormalizability constraint logarithmically with the Einstein-frame time (8).Itis [19] yield a set of differential equations which were expected that this slight non-equilibrium will lead to solved numerically in [17]. The generic form of these a time-dependence of the unified gauge coupling and equations reads [13,14,17] other constants such as the four-dimensional Planck length (2) that characterize the low-energy effective ï + ˙i =−˜i g Q(t)g β , (12) field theory, mainly through the time-dependence of where the β˜i are the Weyl-anomaly coefficients of the the string coupling (1) that results from the time- stringy σ -model on the background {gi}. In the model dependent linear dilaton (4). of [17],the{gi} include graviton, dilaton, tachyon, The asymptotic regime of the type-0 cosmological flux and moduli fields σ1,2, whose vacuum expecta- string model of [17] has been obtained analytically, tion values control the sizes of the extra dimensions. by solving the pertinent equations (12) for the various The detailed analysis of [17] indicated that the fields. As already mentioned, at late times the theory moduli fields σi froze quickly to their equilibrium val- becomes four-dimensional, and the only non-trivial ues. Thus, together with the tachyon field which also information is contained in the scale factor and the decays to a constant value rapidly, they decouple from dilaton, given that the topological flux field remains the four-dimensional fields at very early stages in the conformal in this approach, and the moduli and ini- evolution of this string Universe.1 There is an infla- tial tachyon fields decouple very fast during the initial tionary phase in this scenario and a dynamical exit stages after inflation in this model. For times that are from it. The important point to guarantee the exit is long after the initial fluctuations, such as the present the fact that the central-charge deficit Q2 is a time- epoch when the linear approximation is valid, the so- dependent entity in this approach, obeying specific lution for the dilaton in the σ -model frame, as derived relaxation laws determined by the underlying confor- from the equations (12), takes the form mal field theory [15–17]. In fact, the central charge αA Φ(t) =−ln cosh(F1t) , (13) F1 1 The presence of the tachyonic instability in the spectrum is due where F1 is a positive constant, α is a numerical con- to the fact that in type-0 strings there is no target-space supersym- stant of order one, and metry, by construction. From a cosmological viewpoint the tachyon s01 fields are not necessarily bad features, since they may provide the = C√5e initial instability leading to cosmic expansion [17]. A , (14) 2V6 22 J. Ellis et al. / Physics Letters B 619 (2005) 17–25 where s01 is the equilibrium value of the modulus field For large tE, e.g., present or later cosmological time σ1 associated with the large bulk dimension, and C5 is values, one has the corresponding flux of the five-form field. Notice F1 that A is independent of this large bulk dimension. a (t ) 1 + γ 2t2 . (19) E E γ E For very large times F1t 1 (in string units), one therefore approaches a linear solution for the dilaton: At very large (future) times, a(tE) scales linearly Φ ∼ const − F1t.From(13), (1) and (2), we then see with the Einstein-frame cosmological time tE [17], that the asymptotic weakness of gravity in this Uni- and hence the cosmic horizon expands logarithmi- verse [17] is due to the smallness of the internal space cally. From a field-theory viewpoint, this would allow V6 as compared with the flux C5 of the five-form field. for a proper definition of asymptotic states and thus The constant F1 is related to the central-charge deficit a scattering matrix. As we mentioned briefly above, of the underlying the non-conformal σ -model by [17] however, from a stringy point of view, there are restrictions on the asymptotic values of the central-charge q0 dΦ deficit q , and it is only a discrete spectrum of values Q = q0 + F1 + , (15) 0 F1 dt of q0 which allow for a full stringy S-matrix to be de- where q0 is a constant, the parenthesis vanishes as- fined, respecting modular invariance [12]. ymptotically, and the numerical solution√ of (12) stud- Asymptotically in time, therefore, the Universe re- ied in [17]) requires that q0/F1 = (1+ 17)/2 2.53. laxes to its ground-state equilibrium situation and For this behaviour of Φ, the central-charge deficit (15) the non-criticality of the string, caused by the initial tends to a constant value q0. In this way, F1 is related quantum fluctuation or other initial condition, disap- to the asymptotic constant value of the central-charge pears, giving way to a critical (equilibrium) string deficit, up to an irrelevant proportionality factor of Universe with a Minkowski metric and a linear-dilaton order one, in agreement with the conformal model background. These are the generic features of the mod- of [12], to which this model asymptotes. This value els we consider here, which can include strings with should be, for consistency of the underlying string target-space supersymmetry as well as the explicit theory [12], some discrete value for which the fac- bosonic type-0 string considered here for simplicity. torization property (unitarity) of the string scattering The Hubble parameter of such a Universe becomes amplitudes is valid. Notice that this asymptotic string for large tE theory, with a constant (time-independent) central- 2 γ tE charge deficit, q2 ∝ c∗ − 25 (or c∗ − 9 for superstring) H(t ) . (20) 0 E + 2 2 is considered as an equilibrium situation, and an S ma- 1 γ tE trix can be defined for specific (discrete) values of the On the other hand, the Einstein-frame effective four- central charge c∗. The standard critical (super)string dimensional ‘vacuum energy density’, which is deter- corresponds to a central charge c∗ = 25 (c∗ = 9), but mined by the running central-charge deficit Q2 after ∗ in our case c differs from that critical value. compactifying to four√ dimensions the ten-dimensional 10 −2Φ 2 Defining the Einstein-frame time tE through (6),we expression d x −ge Q (tE),is[17] obtain in the case (13) q2γ 2 2Φ−σ1−5σ2 2 0 αA ΛE(tE) = e Q (tE) , (21) t = (F t). F 2(1 + γ 2t2 ) E 2 sinh 1 (16) 1 E F1 where, for large tE, Q is given in (15), and approaches In terms of the Einstein-frame time, one obtains a log- its equilibrium value q0. Thus, we see explicitly how arithmic time-dependence [12] for the dilaton the dark energy density relaxes to zero for tE →∞. Finally, and most importantly for our purposes ΦE = const − ln(γ tE), (17) here, the deceleration parameter in the same regime where of tE becomes 2 2 2 F1 (d aE/dt )aE 1 γ ≡ . (18) q(t ) =− E − . (22) αA E 2 2 2 (daE/dtE) γ tE J. Ellis et al. / Physics Letters B 619 (2005) 17–25 23

The key point about this expression is that, as is clear for the present era: from (17) and (1), up to irrelevant proportionality con- 1 1 6.56 1 stant factors which by conventional normalization are Φ˙ 2 ∼ ,V(Φ)∼ , (27) 2 2 2 2 set to unity, it can be identified with the square of the 2tE tE string coupling where the numerical factor is a consequence of the numerical result of [17]. This implies an equation of state =− − =− 2 q(tE) exp 2(Φ const) gs . (23) (26) This is our central result. w (t 1) −0.74 (28) To guarantee consistency of perturbation theory, Φ E one must have gs < 1, which can be achieved in our for (large) times tE in string units corresponding to the approach if one defines the present era by the time present era (24). Correspondingly, we have a cosmic regime deceleration parameter ∼ −1 1 γ tE (24) q = (1 + 3w ) =−0.61. (29) 2 Φ in the Einstein frame. This is compatible with large This fixes the string coupling to perturbative values, enough times tE (in string units) for consistent with naive scenarios for grand unification. − So far the model did not include ordinary matter, |C |e 5s02 1, (25) 5 as only fields from the gravitational string multiplet as becomes clear from (14) and (18). This condition have been included. The inclusion of ordinary matter can be guaranteed either for small radii of the five ex- is not expected to change qualitatively the result. We tra dimensions or by a large value of the flux |C5| of conjecture that the fundamental relation (23) will con- the five-form of the type-0 string. We recall that the tinue to hold, the only difference being that probably relatively large extra dimension, s01, which extends the inclusion of ordinary matter will tend to reduce the in the direction of the flux, decouples from this con- string acceleration, due to the fact that matter is sub- dition. Therefore, effective five-dimensional models ject to attractive gravity, and resists the acceleration of with a large uncompactified fifth dimension may be the Universe. In such a case, one has constructed consistently with the condition (24). 1 We next turn to the equation of state in such a q = ΩM − ΩΛ, (30) 2 Universe. As discussed in [17], this model resembles quintessence, with the dilaton playing the rôle of the where ΩM (ΩΛ) denote the matter (vacuum) energy quintessence field. Hence the equation of state for our densities, normalized to the critical energy density of type-0 string Universe reads [20] a spatially flat Universe. There is a remarkable coincidence in numbers for p 1 (Φ)˙ 2 − V(Φ) this non-supersymmetric type-0 string Universe with w = Φ = 2 , (26) Φ 1 ˙ the astrophysical observations, which yield also a q ρΦ (Φ)2 + V(Φ) 2 close to this value, since the ordinary matter content where pΦ is the pressure and ρΦ is the energy den- of the Universe (normalized with respect to the energy sity, and V(Φ)is the effective potential for the dilaton, density of a flat Universe) is Ωordinary matter 0.04 and which in our case is provided by the central-charge the dark matter content is estimated to be ΩDM = 0.23, deficit term. Here the dots denote Einstein-frame dif- while the dark energy content is ΩΛ 0.73. This ferentiation. In the Einstein frame, the potential V(Φ) yields q =−0.595, which is only a few per cent away is given by ΛE in (21). In the limit Q → q0, which we from (29). Conversely, if one used naively in the ex- assume characterizes the present era to a good approx- pression (30) the value (29) for q, obtained in our 2 2 −2 imation, V(Φ) is of order (q0 /2F1 )tE , where we case where ordinary matter was ignored, one would recall that q0/F1 is of order one, as discussed above. find ΩΛ 0.74, indicating that the contribution of the The exact normalization of the dilaton field in the Ein- dilaton field to the cosmic acceleration is the dominant stein frame is ΦE = const − ln(γ tE). We then obtain one. 24 J. Ellis et al. / Physics Letters B 619 (2005) 17–25

If the relation (23) were to hold also upon the inclu- The initial state of our cosmos may correspond to a sion of matter, even in a realistic case with (broken) certain ‘random’ Gaussian fixed point in the space supersymmetry, one would arrive at a value of the of string theories, which is then perturbed in the Big 2 string coupling, gs 0.55, which would be quite con- Bang by some ‘random’ relevant (in a world-sheet sistent with the unification prediction of the minimal sense) deformation, making the theory non-critical supersymmetric extension of the Standard Model at and taking it out of equilibrium from a target space– scales ∼ 1016 GeV. The only requirement for the as- time viewpoint. The theory then flows, following a ymptotic condition (23) to hold is that the underlying well-defined renormalization-group trajectory, and as- stringy σ model theory is non-critical and asymp- ymptotes to the specific ground state corresponding to totes for large times to the linear-dilaton conformal the infrared fixed point of this perturbed world-sheet field theory of [12]. It should be understood, though, σ -model theory. This approach allows for many par- that the precise relation of the four-dimensional gauge allel universes to be implemented of course, and our coupling with the ten-dimensional string coupling de- world would be just one of these. Each Universe may pends on the details of compactification, which we did flow between a different pair of fixed points, as it may not discuss in this work. be perturbed by different operators. It seems to us that We close this discussion by stressing once more the this scenario is much more attractive (no pun intended) importance of non-criticality in order to arrive at (23). and specific than the static ‘landscape’ scenario [3], In critical strings, which usually assume the absence which is currently advocated as an attempt to parame- of a four-dimensional dilaton, such a relation cannot trize our ignorance of the true structure of the string/M be obtained, and the string coupling is not directly theory vacuum and its specification. measurable. The logarithmic variation of the dilaton field with the cosmic time at late times implies a slow variation of the string coupling (23), |˙gs/gs|=1/tE ∼ Acknowledgements 10−60 for the present era, and hence a correspondingly slow variation of the gauge couplings. N.E.M. wishes to thank Juan Fuster and IFIC- From a physical point of view, the use of critical University of Valencia (Spain) for their interest and strings to describe the evolution of our Universe seems support, and P. Sodano and INFN-Sezione di Perugia desirable, whilst non-critical strings may be associ- (Italy) for their hospitality and support during the final ated with non-equilibrium situations, as undoubtedly stages of this work. The work of D.V.N. is supported occur in the early Universe. The space of non-critical by DOE grant DE-FG03-95-ER-40917. string theories is much larger that of critical strings. Therefore, it is remarkable that the departure from criticality has the potential to enhance the predictability References of string theory to such a point that the string coupling may become accessible to experiment. A similar situa- [1] M.B. Green, J.H. Schwarz, E. Witten, Superstring Theory, vols. I, II, Cambridge Univ. Press, Cambridge, 1987. tion arises in a non-critical string approach to inflation, [2] J. Polchinski, String Theory, vol. 2, Cambridge Univ. Press, in the scenario where the Big Bang is identified with Cambridge, 1998; the collision [7] of two D-branes [16]. In such a sce- J.H. Schwarz, hep-th/9907061. nario, astrophysical observations may place important [3] L. Susskind, hep-th/0302219. bounds on the recoil velocity of the brane worlds after [4] N. Arkani-Hamed, S. Dimopoulos, G.R. Dvali, Phys. Lett. B 429 (1998) 263, hep-ph/9803315; the collision, and lead to an estimate of the separation I. Antoniadis, N. Arkani-Hamed, S. Dimopoulos, G.R. Dvali, of the branes at the end of the inflationary period. Phys. Lett. B 436 (1998) 257, hep-ph/9804398. The approach of identifying target time in such a [5] L. Randall, R. Sundrum, Phys. Rev. Lett. 83 (1999) 4690, hep- framework with a world-sheet renormalization-group th/9906064. scale, the Liouville mode [14], provides a novel way [6] See, for instance: L.E. Ibanez, R. Rabadan, A.M. Uranga, Nucl. Phys. B 576 (2000) 285, hep-th/9905098; of selecting the ground state of the string theory, which L.E. Ibanez, hep-ph/9905349; may not necessarily be associated with minimization C.A. Scrucca, M. Serone, JHEP 9912 (1999) 024, hep- of energy, but could be a matter of cosmic ‘chance’. th/9912108; J. Ellis et al. / Physics Letters B 619 (2005) 17–25 25

D.M. Ghilencea, G.G. Ross, Nucl. Phys. B 595 (2001) 277, [14] J.R. Ellis, N.E. Mavromatos, D.V. Nanopoulos, Phys. Lett. hep-ph/0006318, and references therein. B 293 (1992) 37, hep-th/9207103; [7] J. Khoury, B.A. Ovrut, P.J. Steinhardt, N. Turok, Phys. Rev. C. Castro, M.S. El Naschie (Eds.), Invited Review for the Spe- D 64 (2001) 123522, hep-th/0103239. cial Issue of J. Chaos, Solitons and Fractals, vol. 10, Elsevier, [8] D. Langlois, Prog. Theor. Phys. Suppl. 148 (2003) 181, hep- Amsterdam, 1999, p. 345, hep-th/9805120; th/0209261. J.R. Ellis, N.E. Mavromatos, D.V. Nanopoulos, Phys. Rev. [9] Supernova Search Team Collaboration, B.P. Schmidt, et al., D 63 (2001) 024024, gr-qc/0007044. Astrophys. J. 507 (1998) 46, astro-ph/9805200; [15] E. Gravanis, N.E. Mavromatos, Phys. Lett. B 547 (2002) 117, Supernova Cosmology Project Collaboration, S. Perlmutter, et hep-th/0205298; al., Astrophys. J. 517 (1999) 565, astro-ph/9812133; N.E. Mavromatos, Beyond the Desert, in: H.V. Klapdor- Supernova Search Team Collaboration, J.P. Blakeslee, et al., Kleingrothaus (Ed.), Oulu, Finland, 2002, Institute of Physics, Astrophys. J. 589 (2003) 693, astro-ph/0302402; 2003, p. 3, hep-th/0210079. Supernova Search Team Collaboration, A.G. Riess, et al., As- [16] J. Ellis, N.E. Mavromatos, D.V. Nanopoulos, A. Sakharov, trophys. J. 560 (2001) 49, astro-ph/0104455. New J. Phys. 6 (2004) 171, gr-qc/0407089. [10] WMAP Collaboration, D.N. Spergel, et al., Astrophys. J. [17] G.A. Diamandis, B.C. Georgalas, N.E. Mavromatos, E. Pa- Suppl. 148 (2003) 175, astro-ph/0302209. pantonopoulos, Int. J. Mod. Phys. A 17 (2002) 4567, hep- [11] W. Fischler, L. Susskind, Phys. Lett. B 171 (1986) 383; th/0203241; W. Fischler, L. Susskind, Phys. Lett. B 173 (1986) 262. G.A. Diamandis, B.C. Georgalas, N.E. Mavromatos, E. Pa- [12] I. Antoniadis, C. Bachas, J.R. Ellis, D.V. Nanopoulos, Nucl. pantonopoulos, I. Pappa, Int. J. Mod. Phys. A 17 (2002) 2241, Phys. B 328 (1989) 117; hep-th/0107124. I. Antoniadis, C. Bachas, J.R. Ellis, D.V. Nanopoulos, Phys. [18] I. Klebanov, A.A. Tseytlin, Nucl. Phys. B 546 (1999) 155; Lett. B 257 (1991) 278. I. Klebanov, A.A. Tseytlin, Nucl. Phys. B 547 (1999) 143. [13] F. David, Mod. Phys. Lett. A 3 (1988) 1651; [19] G. Curci, G. Paffuti, Nucl. Phys. B 286 (1987) 399. J. Distler, H. Kawai, Nucl. Phys. B 321 (1989) 509; [20] S.M. Carroll, Living Rev. Relativ. 4 (2001) 1, astro- J. Distler, Z. Hlousek, H. Kawai, Int. J. Mod. Phys. A 5 (1990) ph/0004075. 391; See also: N.E. Mavromatos, J.L. Miramontes, Mod. Phys. Lett. A 4 (1989) 1847; E. D’Hoker, P.S. Kurzepa, Mod. Phys. Lett. A 5 (1990) 1411. Physics Letters B 619 (2005) 26–29 www.elsevier.com/locate/physletb

Effective cosmological constant from supergravity arguments and non-minimal coupling

Remi Ahmad El-Nabulsi

Plasma Application Laboratory, Department of Nuclear and Energy Engineering, and Faculty of Mechanical, Energy and Production Engineering, Cheju National University, Ara-dong 1, Jeju 690-756, South Korea Received 15 April 2005; accepted 1 June 2005 Available online 13 June 2005 Editor: W. Haxton

Abstract It was shown that in the case of non-minimal coupling between the scalar curvature and the density of the scalar field, and for a particular scalar potential field inspired from supergravities inflation theories, ultra-light masses are implemented naturally in Einstein field equations, leading to a cosmological constant in accord with recent astrophysical observations.  2005 Published by Elsevier B.V.

It is well believed today that the cosmological approaching de Sitter (dS) regime. In most of the mod- constant describes the energy density of the vacuum els of dark energy it is assumed that the cosmological (empty space), and it is a potentially important contrib- constant is equal to zero and the potential energy V(φ) utor to the dynamical history of the Universe. Recent of the scalar field driving the present stage of acceler- observations of type Ia supernovae and the CMB indi- ation, slowly decreases and eventually vanishes as the cates that the Universe is in accelerated regime [1–3]. field rolls to φ =∞[4–8]. In this case, after a tran- The total energy of the universe consists in fact of or- sient dS-like stage, the speed of expansion of the Uni- dinary matter and dark matter. One can interpret the verse decreases, and the Universe reaches Minkowski dark energy as the vacuum energy corresponding to regime. Of course, depending on the choice of the the cosmological Einstein constant or as the slowly model, the flat Universe will become dS space, or changing energy of a certain scalar field with a vac- Minkowski space, or collapse [9–12]. uum φ corresponding to the equation of state p =−ρ. However, it was found that one can describe dark In both cases the Universe is accelerated with time and energy in some extended supergravities that have a dS solutions [13,14]. These dS solutions correspond to the extrema of the effective potentials V(φ)for some scalar fields φ. An interesting result of these solutions E-mail addresses: [email protected], [email protected] (R.A. El-Nabulsi). is that the squared mass of these scalars in all theo-

0370-2693/$ – see front matter  2005 Published by Elsevier B.V. doi:10.1016/j.physletb.2005.06.002 R.A. El-Nabulsi / Physics Letters B 619 (2005) 26–29 27

√ ries with N = 2(extended supergravities with unstable We set R˜ = gR in what follows where g is the dS vacua) is quantized in units of the Hubble constant metric scalar. The stress constrained tensor in the non- 2 = 2 H0. That is m nH0 where n are integers of order minimal coupling is then of unity (in units of unity Planck Mass). ˜ ∗ − 2ξ δ(Rφφ ) In extended supergravities with a positive cosmo- T n c =−√ 2 = µν g δgµν logical constant, one always has 3m nΛ where Λ being the cosmological constant. For the N = 8 = µ ∗ − ∗ supergravity, dS vacuum corresponds to an unsta- 2ξ gµν∂µ∂ φφ Dν∂µφφ ble maximum m2 =−6H 2 at |φ|1 and V(φ)= 0 1 2 − 2 = − ∗ − 3H0 (1 φ ). Meanwhile for N 2 gauged super- φφ Rµν gµνR , (1) 2 = 2 2 gravity with stable dS vacuum, one has m 6H0 for = 2 − one of the scalars and in this case V(φ) 3H0 (1 where Dν is the covariant derivative. In this non- φ2) [15–18]. minimal coupling, the field equations read In application to the cosmological constant prob- 1 − lem, this leads to the conclusion that there are ultra R − g R + Λg + T n c + T M + tm = 0, (2) µν 2 µν µν µν µν µν light scalars with the mass of the order m ≈ H ≈ −33 M m 10 eV. The significance of this fact and the pos- where Tµν is the matter stress-energy tensor and tµν sibility to use these supergravity models in modern is the microscopic gauge field stress energy tensor de- cosmology still have to be well studied and under- fined in our model as stood. The existence of such ultra light fields may be ˜ 2 δ(Lm) a desirable feature for the description of the acceler- tm = √ , (3) µν g δgµν ated universe. Their presence signals that the corresponding potentials are very shallow. In extended su- where pergravity theories ultra light fields necessarily come √ ˜ 1 µν ∗ ∗ ∗ in a package with too small Λ. Due to the presence Lm = g g ∂µφ ∂νφ + ∂µφ∂νφ − V φφ . 2 of Λ, supersymmetry in dS vacua is broken spontaneously, the scale of SUSY breaking here is 10−3 eV. (4) Before it is coupled to a ‘visible sector’, both the tiny The trace of (3) is then Λ as well as the ultra light masses of scalars, that ∗ ∗ tm =−2∂ φ ∂ φ + 4V φ φ . (5) is m, are protected from large quantum corrections. µν µ µ Coupling of these theories to real universe is a big We now introduce our quartic potential as [21–28] problem, of course. If they play a role of a ‘hidden ∗ = 2 − 2 ∗2 sector’, one may ask whether the tiny m ≈ H will be V φφ pm 1 ωφ φ , (6) preserved after coupling to the ‘visible sector’. The p = 3/4 for cosmological considerations [29], m ∝ H preservation of the small Λ may imply preservation (the Hubble parameter), the constant or false vacuum of small scalar masses m. Thus, extended supergrav- energy which is the leading term in the potential, and ities suggest a new perspective for investigation of ω is a positive parameter less than unity. the cosmological constant problem, intertwined with The trace of (2) gives ultra-light scalars [19]. From here came our motiva- ∗ ∗ tions. −R + 4Λ + 4V − 2∂µφ ∂µφ + 2ξRφ φ In this Letter, for some scalar field φ, we introduce ∗ µ 2 ∗ + 12ξ∂µφ ∂ φ − 12ξ Rφ φ a non-minimal coupling between the scalar curvature − ∗ ∂V + ∂V = and the density√ of the scalar field in the following form 6ξ φ ∗ φ 0. (7) L =−ξ gRφ∗φ, ξ = 1/6. R is the scalar curvature ∂φ ∂φ and φ∗ is the complex conjugate of φ [20].Froma For conformal coupling, that is ξ = 1/6, implanting view point of quantum field theory in curved space- Eq. (6) into (7), reduces to R = 4Λ − 3m2. time, it is natural to consider such a non-minimal cou- In this particular case and when ω 1, a possible pling. candidate field equations for the scalar curvature R, 28 R.A. El-Nabulsi / Physics Letters B 619 (2005) 26–29

= = 2 will be that in the flat case, for ρ ρc 3H0 /8πG, one has Λ = 3(H 2 − m2)<3H 2 as required by inflation. 1 ¯ 0 0 Rµν − gµνR + Λgµν = 0, (8) ≈ 2 ≈ 2 2 From recent observations, Λ 2H0 so that Λ 6m . We mention that in N = 2 gauged supergravities with ¯ = − 3m2 where Λ Λ 4 . The trace of Eq. (8) in four- stable de Sitter vacua, the effective potential near its = − dimensional spacetime leads certainly to R 4Λ extremum can be represented by the quadratic poten- 2 3m . tial V(φ)= Λ + m2φ2/2. However, in extended su- This case corresponds to a static spacetime. An- pergravities with positive cosmological constant, one other interesting possible candidate, is the dynamical 2 = = 2 | | = has 3m nΛ. When Λ 3H0 and φ 1, V(φ) one, which is Λ(1 + n2φ2/6). In both theories, the universalities of the relation − 1 + =− 2 Rµν gµνR Λgµν 3m uµuν (9) 2 = 2 2 2 2 m nH0 may be attributed to the fact that m /H0 is an eigenvalue of the Casimir operator of the dS group with R = 4Λ − 3m2. In general, this equation is iden- [8–10]. In this case, the ultra-light masses are imple- tical to the Einstein field equations mented naturally in Einstein field equations. This later, 1 as we have seen, could play an important role in cos- R − g R =−8πG (p + ρ)u u + pg (10) µν 2 µν µ ν µν mology. Finally, if we introduce a second mass scale M such with p ≡ p = Λ/8πG and ρ ≡ ρ = (3m2 − Λ)/ √ Λ m that pm2 = M2, p2ω = 1/2, that is m2 = 2ωM2 8πG. Notice that if the cosmological is set equal (m M) and we let x = φφ∗, our potential described to zero, than the pressure vanishes and ρ ≡ ρ = m in Eq. (6) takes the simple form V = M2(1 − x2).The 3m2/8πG.IfΛ<3m2, both the density and the pres- potential has a minimum at x = 0 where it takes a pos- sure remains positive. If 2Λ = 3m2, than the pressure itive value M2. This later plays an important role in and the density are equal. While if 2Λ =−3m2 < 0, new inflation theory, in particular brane-world infla- than p =−ρ/3. tion [28]. Cosmology with this potential shares some We believe that ρ ≡ ρ = 3m2/8πG could play an m common features with the cosmology of the “inverse” important role in standard cosmology. If this latter is harmonic oscillator potential (cosmology with nega- implemented naturally in Einstein field equations in tive potential) [31]. the presence of the cosmological constant, that is It is interesting to investigate in the future how 1 much the presence of the ultra light masses in Einstein Rµν − gµνR + Λgµν 2 field equations will contribute in most of the well- 3m2 known cosmological and astrophysical models. =−8πG p + ρ + u u + pg . (11) 8πG µ ν µν These equations are the same of that of Einstein ones References but there is an additional energy density. In fact, in standard cosmology, the cosmological fluid is in fact [1] A.G. Riess, et al., Astrophys. J. 116 (1998) 1009. not unicomponent; instead mater and radiation in dis- [2] I. Zavadi, et al., Astrophys. J. 403 (1998) 483. equilibrium coexist in many elementary subvolumes of [3] P. Garnavich, Astrophys. J. 493 (1998) 53. the Universe. The new additional term could play the [4] A.D. Dogulov, in: G.W. Gibbons, S.W. Hawking, S. Siklos (Eds.), The Very Early Universe, Cambridge Univ. Press, Cam- role of the density of the dark matter. bridge, 1983. When applying Eq. (9) to the FRW cosmology, the [5] C. Wetterich, Nucl. Phys. B 302 (1998) 668. Friedman equation will be [30] [6] E.J. Copeland, A.R. Liddle, D. Wands, Phys. Rev. D 57 (1998) ˙ 4686. R2 k 8πGρ Λ [7] B. Ratra, P.J.E. Peebles, Phys. Rev. D 37 (1988) 3406. + = + + m2, (12) R2 R2 3 3 [8] C. Armendariz-Picon, V.Mukhanov, J.P. Steinhardt, Phys. Rev. Lett. 85 (2000) 4438. where R is the scale factor and k = (−1, 0, +1) corre- [9] R. Kallosh, A. Linde, S. Prokushkin, M. Shmakova, Phys. Rev. sponding to open, flat or closed spacetime. It is clear D 65 (2002) 105016. R.A. El-Nabulsi / Physics Letters B 619 (2005) 26–29 29

[10] A. Linde, JHEP 0111 (2001) 052. [22] A. Albrecht, P.J. Steinhardt, Phys. Rev. Lett. 48 (1982) 1220. [11] P.J. Steinhardt, N. Turok, Phys. Rev. D 65 (2002) 126003. [23] A.R. Liddle, D.H. Lyth, Phys. Rep. 231 (1993) 1. [12] G.N. Felder, A. Frolov, L. Kolman, A. Linde, Phys. Rev. D 66 [24] A. Linde, Phys. Rev. D 49 (1994) 748. (2002) 023507. [25] S. Mollerach, S. Matarrese, F. Lucchin, Phys. Rev. D 50 (1994) [13] R. Kallosh, hep-th/0205315. 4835. [14] C.M. Hull, N.P. Warner, Class. Quantum Grav. 5 (1988) 1517. [26] E.J. Copeland, A.R. Liddle, D.H. Lyth, E.D. Stewart, [15] M. Dine, W. Fischler, D. Nemeschansky, Phys. Lett. B 136 D. Wands, Phys. Rev. D 49 (1994) 6410. (1984) 169. [27] P.H. Frampton, T. Takahashi, Phys. Rev. D 70 (2004) 083530. [16] G.D. Coughlan, R. Holman, P. Ramond, G.G. Ross, Phys. Lett. [28] A. Lukas, D. Skinner, JHEP 0109 (2001) 020. B 140 (1984) 44. [29] A. Linde, Particle Physics and Inﬂationary Cosmology, Har- [17] M. Dine, L. Randall, S. Thomas, Nucl. Phys. B 458 (1996) 291. wood, Chur, 1990. [18] A. Linde, Phys. Rev. D 53 (1996) 4129. [30] S. Weinberg, Gravitation and Cosmology, Wiley, New York, [19] S.M. Carroll, Phys. Rev. Lett. 81 (1998) 3067, astro- 1972. ph/9806099. [31] G. Felder, A. Frolov, L. Kofman, A. Linde, Phys. Rev. D 66 [20] E. Elbaz, Cosmologie, Ellipses, 1992. (2002) 023507. [21] A.D. Linde, Phys. Lett. B 108 (1982) 389. Physics Letters B 619 (2005) 30–42 www.elsevier.com/locate/physletb

The effects of unstable particles on light-element abundances: Lithium versus deuterium and 3He

John Ellis a, Keith A. Olive b, Elisabeth Vangioni c

a TH Division, Physics Department, CERN, CH-1211 Geneva 23, Switzerland b Theoretical Physics Institute, University of Minnesota, Minneapolis, MN 55455, USA c Institut d’Astrophysique de Paris, F-75014 Paris, France Received 7 March 2005; received in revised form 21 May 2005; accepted 26 May 2005 Available online 6 June 2005 Editor: G.F. Giudice

Abstract We reconsider the effects of the radiation from the decays of unstable particles on the production and destruction of the primordial light elements, with a view to reconciling the high primordial 7Li abundance deduced from big bang nucleosynthesis (BBN), as implied by the baryon-to-photon ratio now inferred from the anisotropies of the cosmic microwave background (CMB), with the lower abundance of 7Li observed in halo stars. The potential destruction of 7Li is strongly constrained by observations of deuterium (D), 3He and 6Li. We identify ranges for the unstable particle abundance and lifetime which would deplete 7Li while remaining consistent with the abundance of 6Li. However, in these regions either the D abundance is unacceptably low or the ratio 3He/D is unacceptably large. We conclude that late particle decay is unable to explain both the discrepancy of the calculated 7Li abundance and the observed 7Li plateau. In the context of supersymmetric theories with neutralino or gravitino dark matter, we display the corresponding light-element constraints on the model parameters.  2005 Elsevier B.V. All rights reserved.

1. Introduction to-photon ratio η [2]. Recent high-precision measurements of the cosmic microwave background (CMB) The observed abundances of light elements are radiation by WMAP and other experiments now com- generally in good agreement with the predictions of plement the BBN in important ways [3]. For example, big bang nucleosynthesis (BBN) calculated assum- they impose strong constraints on η and weaker con- ing a homogeneous Robertson–Walker–Friedman cos- straints on the number of light particle species. Of mology [1]. Within this framework, the success of particular interest are the very precise predictions of BBN calculations imposes important constraints on the light element abundances from BBN that are made the number of light particle species and on the baryon- with the CMB value of η [4]. Based on these predictions, there is now tension E-mail address: [email protected] (K.A. Olive). between some observed light-element abundances and

0370-2693/$ – see front matter  2005 Elsevier B.V. All rights reserved. doi:10.1016/j.physletb.2005.05.066 J. Ellis et al. / Physics Letters B 619 (2005) 30–42 31 those that would be calculated using the CMB value in which heavy particles have late decays into radi- of η and assuming no additional light particles beyond ation, which excludes the region of parameter space three light neutrino species. In particular, the predic- where they could have the desired impact on the pri- tion for the primordial abundance of 7Li made using mordial abundance of 7Li.1 We display this and other the CMB value of η [5–9] is somewhat higher (by a constraints on the lifetime and abundance of any mas- factor of 2–3) than the primordial abundance inferred sive unstable relic particle such as a gravitino, pointing from astrophysical observations. The significance of out also the potential impact of weakening the lower this discrepancy should not be over-emphasized, in limit on the primordial deuterium abundance. We also view of the potential systematic errors in the inter- display the effects of the 3He constraint on the parame- pretation of the astrophysical data. However, it has ter spaces of models in which heavier supersymmetric stimulated theoretical explorations of mechanisms for sparticles decay into gravitinos. The overall reductions modifying the CMB/BBN prediction for 7Li, for ex- in the allowed parameter spaces are often small, but ample, via the late decays of massive particles [10,11]. they do exclude the regions where the abundance of The effects of such decays have been studied exten- 7Li could be brought into line with the astrophysi- sively, and the constraints imposed on them by the ob- cal observations. This may motivate a reassessment of served abundances of 4He, deuterium and 6Li are well their interpretation. understood [12–17]. Previous studies had shown no in- compatibilities between these constraints and the suggestion that late-decaying particles might have modi- 2. Is there a problem with lithium-7? fied the BBN prediction for the abundance of 7Li so as to agree better with the astrophysical observations. The most direct and accurate estimate of the baryon- Supersymmetric models with conserved R parity nat- to-photon ratio η is currently provided by the acoustic urally predict such a particle, either the gravitino if structures in the CMB perturbations [3], namely, it is not the lightest supersymmetric particle (LSP), −10 or a neutralino or stau slepton if the gravitino is the η = 6.14 ± 0.25 × 10 . (1) LSP [18,19], which may well have the appropriate 6 7 This range may be used an input into homogeneous abundance and lifetime to affect both the Li and Li BBN calculations [5–9], yielding the following abun- abundances. Later in this Letter, we identify these re- dances for the elements of principal interest, which are gions in the parameter space of a constrained version taken from [9]: of the minimal supersymmetric extension of the Stan- dard Model (CMSSM), as well as in very constrained Yp = 0.2485 ± 0.0005, versions of the model [20] motivated by supergravity D +0.21 −5 considerations, which predict a gravitino LSP in parts = 2.55− × 10 , H 0.20 of the parameter space. 3 He − However, we also point out in this Letter that such = 1.01 ± 0.07 × 10 5, scenarios typically yield an abundance of 3He that may H 7 be more than an order of magnitude larger than the Li + − = 4.26 0.73 × 10 10, deuterium abundance. The importance of 3He produc- H −0.60 3 6 tion, and in particular the ratio of He to D, was first Li − = 1.3 ± 0.1 × 10 14, (2) pointed out in this context in [16], and this constraint H has subsequently been used in [14,17]. Since it appears 4 that the abundance of 3He has remained relatively con- where Yp is the He mass fraction, and the other abun- stant in time, whereas deuterium (D) would have been dances are expressed in terms of their numbers relative destroyed in stars, it seems unlikely that the ratio of to H, as shown. the 3He to D abundances could have been significantly These abundances may then be compared with the larger than it present value in the early history of the abundances of the same elements that are inferred Universe [14,16,17]. Imposing this constraint yields a further restriction on the parameter space of models 1 We comment later on the potential impact of hadronic decays. 32 J. Ellis et al. / Physics Letters B 619 (2005) 30–42 from those observed in the most primitive astrophysi- The comparison of (2) and (3) also shows a dis- cal sites [21–25]: crepancy for 6Li. It has generally been assumed that 6Li is produced in post-BBN processes such as galac- Yp = 0.232 to 0.258, tic cosmic-ray nucleosynthesis [33]. Until recently, the abundance of 6Li had been observed only in a few D −5 = 2.78 ± 0.29 × 10 , metal-poor halo stars with metallicity [Fe/H] larger H − 3 than 2.3. However, new observations of this isotope He − = 1.5 ± 0.5 × 10 5, have now been obtained in halo stars. New values of H the ratio 6Li/7Li have been measured with UVES at 7 Li + − the VLT-UT2 Kueyen ESO telescope, in halo stars = 1.23 0.68 × 10 10, H −0.32 with metallicity ranging from −2.7to−0.5 [24,25]. 6 These observations indicate the presence of a plateau Li + − = 6 7 × 10 12. (3) in 6Li/H 10−11, suggesting a pregalactic origin for H −3 the formation of 6Li [34]. These data provide interest- Note that, for 3He, we have at our disposal only lo- ing new constraints in the present context. cal data coming from star-forming HII regions in the The relation of the observed 7Li abundance to its galactic disk [26] or from the proto-solar value [27]. primordial value may be debated, but for the moment Comparing the two sets of abundances, (2) and (3), we take the discrepancy between the observed and we see no significant discrepancies, except in the case calculated 7Li abundances at face value, and explore of 7Li and 6Li. However, whereas 7Li has to be de- its possible theoretical interpretation in terms of late- stroyed, one must produce a factor 1000 more 6Li. decaying massive particles, depleting 7Li and possi- The value quoted above for the 7Li abundance bly producing 6Li, without negative effects on D and assumes that Li depletion is negligible in the stars 3He. observed. Indeed, standard stellar evolution models predict Li depletion factors which are very small (less than 0.05 dex) in very metal-poor turnoff stars [28]. 3. Possible impact of late-decaying particles However, there is no reason to believe that such simple models incorporate all effects which could The decays of massive particles X with lifetimes lead to depletion, such as rotationally-induced mixing > 102 s could, in principle, have modified the BBN and/or diffusion. Including these effects, current esti- predictions in either of two ways. First, their decay mates for possible depletion factors are in the range products would have increased the entropy in the pri- ∼ 0.2–0.4dex[29]. However, the data sample [30] mordial plasma, implying that its value during BBN used in deriving the abundance in (3) shows a negli- was lower than that inferred from the CMB. How- gible intrinsic spread in Li, leading to the conclusion ever, this effect is negligible compared with the second that depletion in these stars is in fact quite low (less effect, which is the modification of the BBN light- than 0.1 dex). element abundances by the interactions of decay prod- Another important source of potential systematic ucts [12,13]. uncertainty is related to the assumed surface tempera- The latter possibility has been explored recently as ture of the star. A recent study [31] found significantly a mechanism for reducing the primordial 7Li abun- and systematically higher temperatures for stars used dance [10,11]. The existence of such late-decaying in 7Li observations, specifically at low metallicity. massive particles X is a generic possibility in super- This result leads to 7Li/H = (2.34 ± 0.32) × 10−10, symmetric models, in particular [18,19]. Examples in- which is still, however, nearly a factor of 2 smaller clude a massive gravitino weighing ∼ 100 GeV, if than the BBN/WMAP prediction. We note, finally, that it is not the lightest supersymmetric particle (LSP), another potential source for theoretical uncertainty lies or some other next-to-lightest supersymmetric parti- in the BBN calculation of the 7Li abundance. How- cle (NSP) if the gravitino is the LSP. Cosmologi- ever, this too has been shown to be incapable of re- cal constraints on such scenarios have been explored solving the 7Li discrepancy [6,32]. previously [12,13,15,17]. However, the potential 7Li J. Ellis et al. / Physics Letters B 619 (2005) 30–42 33 problem motivates a re-examination of the astrophys- 3.1. The D and 4He abundances ical and cosmological constraints on such scenarios. In particular, we wish to determine whether or not a Fig. 1 shows the current constraints in the (τX,ζX) possible solution to this problem can be found in the plane, updating those shown in [15]. The green lines context of motivated and well-studied supersymmetric are the contours models. − D − Specifically, we apply the results of [15], which are (1.3or2.2) × 10 5 < < 5.3 × 10 5. (4) H based on the decays into radiation of a massive particle with a lifetime longer than 104 s. For a given The first of the lower bounds is the higher line to lifetime, τX, the observed abundances provide us with the left of the cleft, and represents the very conserv- constraints on the abundance of relic particles, or more ative lower limit on D/H assumed in [15]. The range × −5 precisely on the quantity ζX ≡ mXnX/nγ , where nγ 1.3–5.3 10 effectively brackets all recent observa- + − is the density of photons after e e annihilation, mX tions of D/H in quasar absorption systems. The second is the mass of the decaying particle, and ζX is re- of the lower bounds is the lower line on the left side, lated to the density of relic particles prior to decay by and represents what now seems a reasonable lower 0 2 = × 7 −1 bound, which is obtained from the 2σ lower limit in ΩXh 3.9 10 GeV ζX. Our treatment of the electromagnetic shower produced by the decay was (3). The upper bound in (4) is the line to the right of the described in detail in [19]. Specifically, we include the cleft, and is the same upper limit as was used in [15]. decays of the NSP into a gravitino and either a pho- A priori, there is also a narrow strip at larger ζX and ton, Z, or a Higgs boson. In the case that the stau τX where the D/H ratio also falls within the range (4), 4 is the NSP, the dominant decay mode is to a grav- but this is excluded by the observed He abundance. itino and tau lepton. Explicit expressions for the de- The solid red lines in the upper right part of Fig. 1 cays widths for each channel considered were given correspond to the limits in [19]. The other NSP decay modes listed above in- Y > 0.227 or 0.232, (5) ject electrons, muons and hadrons into the primordial p medium, as well as photons. Electromagnetic showers develop similarly, whether they are initiated by electrons or photons, so we can apply the analysis of [15] directly also to electrons. As in [19], we treat the decays of µ, π and K as if their energies were equipartitioned among their decay products. In this approximation, we estimate that the fractions of particle energies appearing in electromagnetic showers are π 0: 100%, µ:1/3, π ±:1/4, K±:0.3,K0:0.5. Using the measured decay branching ratios of the τ , we then estimate that ∼ 0.3 of its energy also appears in electromagnetic showers. In the case of generic hadronic showers from Z or Higgs decay, we estimate that ∼ 0.6 of the energy is electromagnetic, due mainly to π 0 and π ± production. We note that we have not included the effects of hadronic decays which are definitely important for lifetimes shorter than 104 s. The effects of hadronic decays on element abundances have been discussed recently in [10,17], where it was found that, for lifetimes around 103 s and densities of Fig. 1. The constraints imposed by the astrophysical observations of 4He (red lines), D/H (green lines), 6Li (yellow line), 6Li/7Li (blue ∼ −12 7 ζX 10 , some destruction of Li and production lines), 7Li (blue band) and 3He (black lines). (For interpretation of 6 of Li is possible. We comment below on this possi- the references to colour in all figures legends, the reader is referred bility. to the web version of this Letter.) 34 J. Ellis et al. / Physics Letters B 619 (2005) 30–42 where the lower number (corresponding to the higher high as 0.15, and we display that upper limit here. The line) was used in [15], and the higher number (corre- main new effect of this constraint is to disallow a responding to the lower line) is a lower limit that has gion in the near-vertical cleft between the upper and been advocated recently [21]. It is apparent that, for lower limits on D/H, as seen in Fig. 1. our purposes, the third significant figure in the 4He abundance is unimportant: the narrow D/H strip is 3.3. The 7Li abundance in any case excluded, and there are always stronger bounds on ζX at large τX. The main region of interest in Fig. 1 is the blue shaded band that represents the inferred 7Li abun- 3.2. The 6Li abundance dance: 7 − Li − As said above, recent observations of 6Li in halo 0.9 × 10 10 < <(2or3) × 10 10, (7) stars have provided new insight into the origin and the H 6 7 evolution of this isotope. We recall that Li is a pure with the Li abundance decreasing as ζX increases and product of spallation, and many studies have followed the intensity of the shading changing at the interme- the evolution of 6Li in our Galaxy [33]. Of particular diate value. In [15], only the lower bound was used importance in this context is the α + α reaction that due the existing discrepancy between the primordial leads to the synthesis of this isotope as well as 7Li, and observationally determined values. It is apparent and is efficient very early in the evolutionary history that 7Li abundances in the lower part of the range (7) of the Galaxy. The new values of 6Li/7Li that have are possible only high in the deuterium cleft, and even been measured in halo stars with UVES at the VLT- then only if one uses the recent and more relaxed limit UT2 Kueyen ESO telescope indicate the presence of on the 6Li/7Li ratio (6). Values of the 7Li abundance a plateau in 6Li, which suggests a pregalactic origin in the upper part of the range (7) are possible, how- for the formation of this isotope. The evolution of 6Li ever, even if one uses the more stringent constraint on with redshift was calculated [34] following an initial 6Li/7Li. In this case, the allowed region of parame- burst of cosmological cosmic rays up to the formation ter space would also extend to lower τX, if one could − of the Galaxy. This process is able to produce the re- tolerate values of D/H between 1.3 and 2.2 × 10 5. quired abundance of 6Li observed in metal-poor halo For the convenience of the subsequent discussion, 7 stars without the additional over-production of Li. In the region of the (τX,ζX) plane that is of interest for this Letter, we have to consider the new constraint lowering the 7Li abundance is shown alone in panel (a) brought by the existence of this plateau. The poten- of Fig. 2. The blue region in the arc at low τX is the tial destruction of 7Li by unstable particles must not region excluded by the stronger lower limit on the deu- − lead to over-production of 6Li. terium abundance: D/H > 2.2 × 10 5, and the red The constraint imposed by the 6Li abundance is region to its right is the extra domain that is excluded shown as a solid yellow line in Fig. 1, which is the by the 3He/D ratio, as we discuss below. same as that discussed in [15]. Also shown, as solid blue lines, are two contours representing possible upper limits on the 6Li/7Li ratio: 4. The importance of the 3He abundance

6 Li 3 < 0.07 or 0.15, (6) We now come to the constraint from He, which 7Li was previously discussed in [14,16,17]. We ﬁnd that with the upper (lower) contour corresponding to the the 3He/D ratio is absurdly high in the deuterium cleft. upper (lower) number in (6). The lower number was Panel (b) of Fig. 2 shows a histogram of the values of used in [15] and represented the upper limit available the 3He/D ratio found in a dense sample of scenar- at the time, which was essentially based on multiple ios in the interesting regions shown in panel (a). Since observations of a single star. The most recent data [24, deuterium is more fragile than 3He, whose abundance 25] includes observations of several stars. The Li iso- is thought to have remained roughly constant since tope ratio for most metal-poor stars in the sample is as primordial nucleosynthesis when comparing the BBN J. Ellis et al. / Physics Letters B 619 (2005) 30–42 35

7 Fig. 2. (a) The region of the (τX,ζX) plane in which a decaying relic particle could have the desired impact on the Li abundance. To derive the blue (darker grey) region, the only abundance cuts applied are: 0.9 < 7Li/H × 1010 < 3.0, 1.3 < D/H × 105 < 5.3, and 6Li/7Li < 0.15. − To obtain the red (lighter grey) region, the lower bound on D/H was increased to 2.2 × 10 5. (b) A histogram of the 3He/D ratios found in scenarios sampling the region displayed in panel (a), with similar colour coding. value to it proto-solar abundance, one would expect, It is interesting to note that in the red region in principle, the 3He/D ratio to have been increased which has acceptable D/H and a 7Li/H abundance by stellar processing. Indeed, there is considerable un- low enough to match the observed values, the 6Li certainty in the evolution of 3He [35]. This uncertainty abundance is relatively high: 7.3 × 10−12 < 6Li/H < is largely associated with the degree to which 3He is 1.6 × 10−11. This matches the new 6Li observations produced or destroyed in stars. Since D is totally de- quite well, and would circumvent the need for an early stroyed in stars, the ratio of 3He/D can only increase period of 6Li production by cosmological cosmic rays. in time or remain constant if 3He is also completely Unfortunately, 3He/D ranges from 17–37 for these pa- destroyed in stars. The present or proto-solar value of rameter values. 3 He/D can therefore be used to set an upper limit on The previous upper limit on ηX [15] corresponded −12 the primordial value. Fig. 1 displays the upper limits to the constraint mXnX/nγ < 5.0 × 10 GeV for 8 3 τX = 10 s. The weaker (stronger) version of the He constraint adopted here corresponds to 3He < 1or2 (8) D n − m X < 2.0(0.8) × 10 12 GeV, (9) X n as solid black lines. Above these contours, the value γ 3 of He/D increases very rapidly, and points high in 8 3 for τX = 10 s. However, the impact of the He con- the deuterium cleft of Fig. 1 have absurdly high val- = 7 3 straint is even stronger for τX 10 s, the location ues of He/D, exceeding the limit (8) by an order of the previous Deuterium cleft. The analysis of [15] of magnitude or more. These are the red points pro- −12 would have given mXnX/nγ < 360 × 10 GeV, ducing the high-end peak of the histogram shown in whereas the weaker (stronger) 3He constraint adopted panel (b) of Fig. 2, whereas the blue points are those here corresponds to excluded by the lower limit D/H > 2.2 × 10−5 that is now preferred. We see that these points mostly have nX −12 3 mX < 9.3(3.8) × 10 GeV, (10) acceptably low values of He/D, though some large nγ values are found near the boundary with the red region 7 in panel (a) of Fig. 2. for τX = 10 s. 36 J. Ellis et al. / Physics Letters B 619 (2005) 30–42

5. Applications to supersymmetric scenarios supersymmetry-breaking scalar masses m0 and gaug- ino masses m1/2 at the GUT scale before renormaliza- We now discuss some examples of the conse- tion. The magnitude of the higgsino mixing parameter 3 quences of the He constraint for various supersym- |µ| and the pseudoscalar Higgs mass mA are fixed by metric scenarios in which R parity is conserved. In the electroweak vacuum conditions. We consider sce- such models, if the gravitino is not the LSP it will narios with µ>0, which are favoured by gµ − 2 and, generically decay gravitationally with a long lifetime. to a lesser extent, b → sγ. The scenarios that we study If the gravitino is the LSP, the next-to-lightest super- differ in their assumptions about the relationship of the symmetric particle (NSP) will decay gravitationally gravitino mass m3/2 to m0, but they all share the com- into the gravitino LSP, again with a long lifetime. mon feature that the LSP is the gravitino in generic domains of parameter space. 5.1. Models with an unstable gravitino In the first set of scenarios, shown in Fig. 3,we fix the ratio of supersymmetric Higgs vacuum expec- = We first consider the possibility that the gravitino is tation values tan β 10, which is among the lower not the lightest supersymmetric particle (LSP), which values consistent with our hypotheses, and assume = = = is instead the lightest neutralino χ. In this case, the (a) m3/2 10 GeV, (b) m3/2 100 GeV, (c) m3/2 = gravitino is unstable, with a lifetime that could well 0.2m0 and (d) m3/2 m0. In each panel of Fig. 3, fall within the range considered here. In such a sce- we display accelerator, astrophysical and cosmologi- nario, the light-element abundances impose an impor- cal constraints in the corresponding (m1/2,m0) planes, tant upper limit on the possible temperature of the concentrating on the regions to the right of the near- Universe, e.g., during reheating after inflation, which vertical black lines, where the gravitino is the LSP. The vertical black dashed and (red) dot-dashed lines we denote by TR [12,13,15,17]. We recall that thermal reactions are estimated to produce an abundance represent the lower limits on m1/2 implied for each of gravitinos given by [15]: value of m0 by the non-observation of a chargino and a Higgs boson at LEP, the latter having a theoretical nm − T uncertainty δm ∼ 50 GeV. The (pale green) nar- 3/2 = ( . . ) × 11 × R . 1/2 0 7–2 7 10 10 (11) nγ 10 GeV row diagonal strips represent the regions where the relic density of the NSP would have lain in the range 8 Assuming that m3/2 = 100 GeV and τX = 10 s, and 0.094 Ωh2 0.129 favoured by WMAP and other imposing the constraints (9), we now find measurements of the cold dark matter density, if the gravitino had not been the LSP. In fact, the gravitino is T <(0.8–2.8) × 107 GeV, R always the LSP in the scenarios considered. The NSP (0.3–1.1) × 107 GeV , (12) may be either the lightest neutralino χ or the lighter supersymmetric partner of the τ lepton: this is lighter 3 for the weaker (stronger) version of the He con- than the neutralino χ below the (red) dotted line. straint. This becomes an even more significant con- Below and to the right of the upper (purple) dashed straint on inflationary models, which were already lines, the density of relic gravitinos produced in the somewhat embarrassed by the previous upper limit decays of other supersymmetric particles is always ∼ × 7 2 TR 2 10 GeV. below the WMAP upper limit: Ω3/2h 0.129. To the right of the lower black solid lines, the lifetime 5.2. Models with gravitino dark matter of the next-to-lightest supersymmetric particle (NSP) falls below 104 s, and the analysis of [15] cannot We now consider the possibility that the gravitino is evaluate the astrophysical constraints from the light- itself the LSP, in which case the next-to-lightest super- element abundances, in the absence of a suitably modi- symmetric particle (NSP) would instead be unstable, fied BBN code. The code used in [15], when combined also with a long lifetime. We will work in the con- with the observational constraints used in [15], yielded text of the constrained MSSM (CMSSM) [36–38],so the astrophysical constraint represented by the dashed that all the scenarios we consider have universal soft grey-green lines in the different panels of Fig. 3. J. Ellis et al. / Physics Letters B 619 (2005) 30–42 37

Fig. 3. The (m1/2,m0) planes for µ>0, tanβ = 10 and (a) m3/2 = 10 GeV, (b) m3/2 = 100 GeV, (c) m3/2 = 0.2m0 and (d) m3/2 = m0.We restrict our attention to the regions between the solid black lines, where the gravitino is the LSP and the NSP lifetime exceeds 104 s. In each panel, the near-vertical dashed black (dash-dotted red) line is the constraint mχ± > 104 GeV (mh > 114 GeV), the upper (purple) dashed line 2 2 is the constraint Ω3/2h < 0.129, and the light green shaded region is that where the NSP would have had 0.094 Ωh 0.129 if it had not decayed. The solid red (dashed grey-green) line is the region now (previously) allowed by the light-element abundances: r<1 as described in the text. The red (blue) shaded region is that where the 7Li abundance could have been improved by NSP decays, but which is now excluded by the 3He (D) constraint.

These constraints on the CMSSM parameter plane computed. Then for each τX, the limit on ζX is found were computed in [19]. For each point in the (m1/2, from the results shown in Fig. 1. The region to the right ˜ = limit m0), the relic density of either χ or τ is computed and of this curve where r ζX/ζX < 1 is allowed. 2 7 ζX is determined using ΩXh = 3.9 × 10 GeV ζX. The astrophysical constraints obtained with the When X =˜τ , ζX is reduced by a factor of 0.3, as only newer abundance limits used here yields the solid red 30% of stau decays result in electromagnetic showers lines in Fig. 3. The examples where τX and ζX for the which affect the element abundances at these lifetimes. NSP decays fall within the ranges shown in Fig. 2(a), In addition, at each point, the lifetime of the NSP is and hence are suitable for modifying the 7Li abun- 38 J. Ellis et al. / Physics Letters B 619 (2005) 30–42 dance, are shown as red and blue shaded regions in Fig. 4 shows the corresponding (m1/2,m0) planes each panel of Fig. 3. We see that they straddle the erst- for the choice tan β = 57, which is among the larger while WMAP strips.2 If we had been able to allow a values allowed in the context of the constrained deuterium abundance as low as D/H ∼ (1–2) × 10−5, MSSM. Although the shapes of the allowed regions the blue shaded region would have been able to resolve are rather different from the previous tan β = 10 case, the Li discrepancy in the context of the CMSSM with the qualitative conclusions are similar. The 3He con- gravitino dark matter. The blue region that we now re- straint again has relatively modest impact. However, gard as excluded by the lower limit on D/H, which is in all cases the red shaded 7Li regions are excluded 3 stronger than that used in [15], extends to large m1/2. by the He constraint, and the blue regions by the D The red shaded region, which is consistent even with abundance. We note that, in models with m3/2 = m0 3 this limit on D/H, but yields very large He/D, is close as in Fig. 4(d), very little of the (m1/2,m0) plane ad- to the Higgs lower limit on m1/2 for small m3/2,mov- mits gravitino dark matter and, in the viable corner, 7 ing to larger m1/2 for larger m3/2,soastokeepτNSP there were no possibilities for depleting Li. within the desired range. Finally, we consider very constrained models moti- We displayed in Fig. 1 the impact of the improved vated by minimal supergravity (mSUGRA), in which 3 lower limit on D/H and the new He constraint on not only is m3/2 = m0, but also the trilinear soft the abundance of an unstable particle, as a function supersymmetry-breaking parameter A determines the of its lifetime. Interpreting this as a constraint on NSP bilinear soft supersymmetry-breaking Higgs-mixing decay into a gravitino, the panels in Fig. 3 show as parameter: B = A − m3/2. This is compatible with solid red lines the additional restrictions these con- the values of µ and mA specified by the electroweak straints impose on the (m1/2,m0) planes for different vacuum conditions for only one value of tan β for values of m3/2. The effects for small m3/2 = 10 GeV any given pair of values of (m1/2,m0) [20]. Such [in panel (a)] m3/2 = 100 GeV when m0 is large [in mSUGRA models are then specified by a choice of ≡ ˆ ˆ = panel (b)] and 0.2m0 [in panel (c)] are relatively mod- A Am√ 3/2: the panels of Fig. 5 assume (a) A est. This is because the limit occurs in a region where (3 − 3) as found in the simple Polonyi model of the NSP is a neutralino, and the relic density varies supersymmetry breaking√ in mSUGRA [39], and (b) ˆ ˆ relatively rapidly. Hence a small change in the m1/2 or A = 2. For A = (3 − 3), the contours of constant m0 results in a large change in ζX, and the old and new tan β are approximately vertical, and range from about bounds are relatively close. However, they do bite in 10 at low m1/2 to about 30 at high values of m1/2.In the neighbourhood of the shaded 7Li blobs, and have the interesting region of panel (a) where 7Li can be the effect of excluding them entirely. The effects for depleted, tan β ∼ 20–30. As before, we consider here large m3/2 = 100 GeV and small m0 [in panel (b)] and only regions of the (m1/2,m0) planes between the two m3/2 = m0 [in panel (d)] are relatively large, mainly solid black lines: above the higher one, the gravitino is due to the slow variation of ζX near the limit which is no longer the LSP, and below the lower one the life- characteristic of the τ˜ NSP region, as can be seen from time falls below 104 s. In addition to the constraints the relatively wide WMAP strips in this region. Re- discussed earlier, panel (b) also displays a small green flecting this wide separation between the (old) dashed shaded region at low m1/2 that is excluded by b → sγ grey-green lines and the (new) solid red lines, we see decay. large red and blue swaths in panel (d), where the 7Li We see that there is a large difference between the abundance could have been reduced, but the 3He/D effects of implementing the old and new light-element√ and/or D/H ratios are unacceptable.3 constraints in panel (a) of Fig. 5 for Aˆ = (3 − 3), the Polonyi value, whereas the effect in panel (b) for Aˆ = 2 is smaller. In the Polonyi case, there are large 2 This raises the possibility that a discovery of supersymmetry 7Li-friendly regions that are excluded by the 3He and might have an ambiguous interpretation—neutralino LSP or grav- D constraints. This reflects the fact that tan β is rela- itino LSP—in the absence of supplementary information. tively small in this case, so the model is qualitatively 3 We note in passing that in several panels there are parts of the similar to the m = m case for tan β = 10 shown in blue and/or red regions at large m0 that are also excluded by the 3/2 0 WMAP relic density limit. panel (d) of Fig. 3. On the other hand, tan β is typically J. Ellis et al. / Physics Letters B 619 (2005) 30–42 39

Fig. 4. As in Fig. 3, but now for tan β = 57.

ˆ larger for A = 2, and when combined with a smaller Fig. 4(a), (b) or (d). When m3/2 = 0.2m0,wedofinda 7 −12 stau mass, we find no visible Li-friendly region, as handful of cases, all with ζX between 4–7 × 10 and a result of small yet significant shifts in the values of all with m0 20 GeV and m1/2 1.7TeV.Amore both ζX and τX. complete treatment of the effects of hadronic decays As noted earlier, the 7Li abundance can be reduced will be given elsewhere. slightly by hadronic decays when the lifetime is ap- 3 ∼ −12 proximately 10 s with a density ζX 10 [10,17]. 6. Conclusions We have searched the parameter spaces of both the CMSSM and mSUGRA models discussed above. For In the absence of a convincing astrophysical ex- lifetimes between 300 and 3000 s, we find no models planation for the apparent discrepancy between the −13 −11 7 with ζX between 10 and 10 when tan β = 10, observed abundance of Li and that calculated on the or in any of the mSUGRA models. When tan β = 57, basis of the baryon-to-photon ratio inferred from CMB we also find no parameters for the models described in observations, it has been natural to explore the possi- 40 J. Ellis et al. / Physics Letters B 619 (2005) 30–42

Fig. 5. As in Fig. 3, but now for very constrained models motivated by mSUGRA. The value of tan β is fixed by the vacuum conditions, and ≡ ˆ varies across the√ (m1/2,m0) planes with values indicated by the steep black contours. These models are specified by the choices of A Am3/2: (a) Aˆ = (3 − 3), the Polonyi model, and (b) Aˆ = 2. ble effects of late-decaying massive particles. Indeed, growth of its abundance between BBN and the for- they could suppress the primordial abundance of 7Li, mation of halo stars could also have been explained but at some price. Either the abundance of deuterium by the decays of the NSP in supersymmetric mod- should be very low compared with the latest avail- els, were it not for either the low resulting abundance able measurements, and/or the primordial 3He/D ratio of D/H or the high ratio of 3He/D. We have also must have been very high. considered the possibility that radiation from decays The latest observations of remote cosmological may be responsible for the observed 6Li abundance clouds along the lines of sight of high-redshift quasars at low metallicity. We find that, indeed, for lifetimes −5 9 −12 suggest that D/H > 2.2 × 10 , ruling out much of longer than 3 × 10 s with ζX 1–3 × 10 , suffi- the parameter space for unstable particles that would cient 6Li production is possible. However, this range 7 otherwise have been suitable for diminishing the Li of τX and ζZ does not correspond to any range of abundance to agree with observations. A significant (m1/2,m0) in the supersymmetric models we have part of this parameter space would have been allowed considered. by the more relaxed limit D/H > 1.3 × 10−5 consid- So what is the interpretation of the apparent dis- ered previously [15]. crepancy between the calculations of the primordial The remaining part of the parameter space for un- 7Li abundance and the Spite plateau? We have ar- stable particles that is consistent with the current lower gued that the origin of the discrepancy cannot be the limit would yield a 3He/D ratio at least an order of possible existence of unstable particles able to de- magnitude higher than the proto-solar value. Since D stroy the primordial nucleus. This leaves the problem has been destroyed by stars, reducing its abundance open. Systematic uncertainties in nuclear effects such from approximately 2.5 to 1.5 ×10−5, while the 3He as higher 7Be + D reaction rates have been consid- abundance is thought to have remained roughly con- ered [6,32], but seem unable to modify substantially stant, a primordial ratio of 3He/D > 1 or 2 is unaccept- the abundance of 7Li. Stellar mechanisms of depletion able. This closes the remaining loophole for suppress- maybethelastresort[29]. Perhaps other new and ex- ing 7Li without running into conflict with the other citing astrophysical or physical effects will have to be light-element abundances. considered. We have also analyzed the potential bounds im- A significant output of this analysis has been the posed by the 6Li abundance. In principle, the high demonstration of the importance of the 3He/D con- J. Ellis et al. / Physics Letters B 619 (2005) 30–42 41 straint on late-decaying massive particles as argued in K.A. Olive, G. Steigman, T.P. Walker, Phys. Rep. 333 (2000) [14,16,17]. Can one make the 3He constraint more pre- 389; cise? This would require considering in more detail the B.D. Fields, S. Sarkar, Phys. Rev. D 66 (2002) 010001. 3 [2] G. Steigman, D.N. Schramm, J. Gunn, Phys. Lett. B 66 (1977) cosmic evolution of D and He. One should allow for 202; the possibility of exotic effects such as large-scale de- For a recent analysis see: R.H. Cyburt, B.D. Fields, K.A. Olive, struction of this isotope in primitive structures such E. Skillman, astro-ph/0408033. as massive Population-III stars, followed by moder- [3] C.L. Bennett, et al., Astrophys. J. Suppl. 148 (2003) 1, astro- ate production by normal galactic evolution. However, ph/0302207; D.N. Spergel, et al., Astrophys. J. Suppl. 148 (2003) 175, astro- any such scenario should consider simultaneously the ph/0302209. 3 cosmic evolution of D. As it is more fragile than He, [4] R.H. Cyburt, B.D. Fields, K.A. Olive, Phys. Lett. B 567 (2003) D would also be destroyed in any Population-III stars. 227, astro-ph/0302431. As we have seen, the 3He constraint sharpens the [5] R.H. Cyburt, B.D. Fields, K.A. Olive, New Astron. 6 (1996) embarrassment of supersymmetric models with heavy 215. ∼ 7 8 [6] A. Coc, E. Vangioni-Flam, P. Descouvemont, A. Adahchour, particles whose lifetimes are 10 to 10 s. This in- C. Angulo, Astrophys. J. 600 (2004) 544, astro-ph/0309480. terest motivates more detailed studies of the cosmic [7] P. Descouvemont, A. Adahchour, C. Angulo, A. Coc, E. evolution of 3He and D, as well as 6Li and 7Li. Vangioni-Flam, At. Data Nucl. Data Tables 88 (2004) 203, Finally, it would be very useful if the observed astro-ph/0407101. abundances of D/H in quasar absorption systems were [8] A. Cuoco, F. Iocco, G. Mangano, G. Miele, O. Pisanti, P.D. Ser- pico, Int. J. Mod. Phys. A 19 (2004) 4431, astro-ph/0307213. improved. There is currently considerable dispersion [9] R.H. Cyburt, Phys. Rev. D 70 (2004) 023505, astro- in the observed abundances. We note that two such ph/0401091. systems in the directions of Q2206-199 at z = 2.0762 [10] K. Jedamzik, Phys. Rev. D 70 (2004) 063524, astro- with D/H = 1.65 ± 0.35 × 10−5 [40] and PKS1937- ph/0402344; +0.25 −5 K. Jedamzik, Phys. Rev. D 70 (2004) 083510, astro- 1009 at z = 3.256, with D/H = 1.6− × 10 [41] 0.30 ph/0405583. have quite low abundances of D, similar to that ob- [11] J.L. Feng, A. Rajaraman, F. Takayama, Phys. Rev. D 68 (2003) served at present in the ISM of our Galaxy [42].If 063504, hep-ph/0306024. these measurements were to represent the correct D/H [12] D. Lindley, Astrophys. J. 294 (1985) 1; abundance in those clouds, 7Li depletion and 6Li pro- J.R. Ellis, D.V. Nanopoulos, S. Sarkar, Nucl. Phys. B 259 duction by sparticle decay would be a viable option, (1985) 175; J. Ellis, et al., Nucl. Phys. B 337 (1992) 399; though by solving one problem we would open two M. Kawasaki, T. Moroi, Prog. Theor. Phys. 93 (1995) 879, hep- new problems. Why is the D/H abundance in most of ph/9403364; the other absorption systems significantly higher, and M. Kawasaki, T. Moroi, Astrophys. J. 452 (1995) 506, astro- how can we account for the D/H abundances in the so- ph/9412055. lar system, which are also in the range 1.5–2.5×10−5? [13] D. Lindley, Phys. Lett. B 171 (1986) 235; M.H. Reno, D. Seckel, Phys. Rev. D 37 (1988) 3441; S. Dimopoulos, R. Esmailzadeh, L.J. Hall, G.D. Starkman, Nucl. Phys. B 311 (1989) 699; Acknowledgements K. Kohri, Phys. Rev. D 64 (2001) 043515, astro-ph/0103411. [14] E. Holtmann, M. Kawasaki, K. Kohri, T. Moroi, Phys. Rev. D 60 (1999) 023506, hep-ph/9805405; We thank R. Cyburt, B. Fields, Y. Santoso and M. Kawasaki, K. Kohri, T. Moroi, Phys. Rev. D 63 (2001) V. Spanos for collaborations on related topics. The 103502, hep-ph/0012279. work of K.A.O. and E.V. was supported by the Project [15] R.H. Cyburt, J.R. Ellis, B.D. Fields, K.A. Olive, Phys. Rev. “CNRS/USA”, and the work of K.A.O. was also sup- D 67 (2003) 103521, astro-ph/0211258. ported partly by DOE grant DE-FG02-94ER-40823. [16] G. Sigl, K. Jedamzik, D.N. Schramm, V.S. Berezinsky, Phys. Rev. D 52 (1995) 6682, astro-ph/9503094. [17] M. Kawasaki, K. Kohri, T. Moroi, astro-ph/0402490; M. Kawasaki, K. Kohri, T. Moroi, astro-ph/0408426. References [18] J.L. Feng, S. Su, F. Takayama, Phys. Rev. D 70 (2004) 063514, hep-ph/0404198; J.L. Feng, S. Su, F. Takayama, Phys. Rev. D 70 (2004) 075019, [1] T.P. Walker, G. Steigman, D.N. Schramm, K.A. Olive, K. Kang, Astrophys. J. 376 (1991) 51; hep-ph/0404231. 42 J. Ellis et al. / Physics Letters B 619 (2005) 30–42

[19] J.R. Ellis, K.A. Olive, Y. Santoso, V.C. Spanos, Phys. Lett. [36] J.R. Ellis, T. Falk, G. Ganis, K.A. Olive, M. Srednicki, Phys. B 588 (2004) 7, hep-ph/0312262. Lett. B 510 (2001) 236, hep-ph/0102098; [20] J.R. Ellis, K.A. Olive, Y. Santoso, V.C. Spanos, Phys. Lett. V.D. Barger, C. Kao, Phys. Lett. B 518 (2001) 117, hep- B 573 (2003) 162, hep-ph/0305212; ph/0106189; J.R. Ellis, K.A. Olive, Y.Santoso, V.C. Spanos, Phys. Rev. D 70 L. Roszkowski, R. Ruiz de Austri, T. Nihei, JHEP 0108 (2001) (2004) 055005, hep-ph/0405110. 024, hep-ph/0106334; [21] K.A. Olive, E.D. Skillman, Astrophys. J. 617 (2004) 29, astro- A.B. Lahanas, V.C. Spanos, Eur. Phys. J. C 23 (2002) 185, hep- ph/0405588. ph/0106345; [22] D. Kirkman, D. Tytler, N. Suzuki, J.M. O’Meara, D. Lubin, A. Djouadi, M. Drees, J.L. Kneur, JHEP 0108 (2001) 055, hep- Astrophys. J. Suppl. 149 (2003) 1, astro-ph/0302006. ph/0107316; [23] S.G. Ryan, T.C. Beers, K.A. Olive, B.D. Fields, J.E. Norris, U. Chattopadhyay, A. Corsetti, P. Nath, Phys. Rev. D 66 (2002) Astrophys. J. 530 (2000) L57, astro-ph/9905211. 035003, hep-ph/0201001; [24] M. Asplund, et al., 2004, in preparation. H. Baer, C. Balazs, A. Belyaev, J.K. Mizukoshi, X. Tata, Y. [25] D.L. Lambert, AIP Conf. Proc. 743 (2005) 206, astro- Wang, JHEP 0207 (2002) 050, hep-ph/0205325; ph/0410418. R. Arnowitt, B. Dutta, hep-ph/0211417; [26] T.M. Bania, R.T. Rood, D.S. Balser, Nature 415 (2002) 54. J.R. Ellis, K.A. Olive, Y. Santoso, V.C. Spanos, Phys. Lett. [27] J. Geiss, G. Gloeckler, Space Sci. Rev. 84 (1998) 239; B 573 (2003) 163, hep-ph/0308075. G. Gloeckler, J. Geiss, Space Sci. Rev. 84 (1998) 275. [37] J.R. Ellis, K.A. Olive, Y. Santoso, V.C. Spanos, Phys. Lett. [28] C.P. Deliyannis, P. Demarque, S.D. Kawaler, Astrophys. J. B 565 (2003) 176, hep-ph/0303043. Suppl. 73 (1990) 21. [38] H. Baer, C. Balazs, JCAP 0305 (2003) 006, hep-ph/0303114; [29] S. Vauclair, C. Charbonnel, Astrophys. J. 502 (1998) 372; A.B. Lahanas, D.V. Nanopoulos, Phys. Lett. B 568 (2003) 55, M.H. Pinsonneault, T.P. Walker, G. Steigman, V.K. Narayanan, hep-ph/0303130; Astrophys. J. 527 (1998) 180, astro-ph/9803073; U. Chattopadhyay, A. Corsetti, P. Nath, Phys. Rev. D 68 (2003) M.H. Pinsonneault, G. Steigman, T.P. Walker, V.K. Narayanan, 035005, hep-ph/0303201; Astrophys. J. 574 (2002) 398, astro-ph/0105439; C. Munoz, hep-ph/0309346; O. Richard, G. Michaud, J. Richer, Astron. Astrophys. 431 R. Arnowitt, B. Dutta, B. Hu, hep-ph/0310103. (2005) 1, astro-ph/0409672. [39] J. Polonyi, Budapest preprint KFKI-1977-93, 1977; [30] S.G. Ryan, J.E. Norris, T.C. Beers, Astrophys. J. 523 (1999) R. Barbieri, S. Ferrara, C.A. Savoy, Phys. Lett. B 119 (1982) 654, astro-ph/9903059. 343. [31] J. Melendez, I. Ramirez, Astrophys. J. 615 (2004) L33, astro- [40] M. Pettini, D.V. Bowen, Astrophys. J. 560 (2001) 41, astro- ph/0409383. ph/0104474. [32] R.H. Cyburt, B.D. Fields, K.A. Olive, Phys. Rev. D 69 (2004) [41] N.H.M. Crighton, J.K. Webb, A. Ortiz-Gill, A. Fernandez- 123519, astro-ph/0312629. Soto, Mon. Not. R. Astron. Soc. 355 (2004) 1042, astro- [33] B.D. Fields, K.A. Olive, New Astron. 4 (1999) 255; ph/0403512. E. Vangioni-Flam, M. Cassé, R. Cayrel, J. Audouze, M. Spite, [42] H.W. Moos, et al., Astrophys. J. Suppl. 140 (2002) 3, astro- F. Spite, New Astron. 4 (1999) 245. ph/0112519. [34] E. Rollinde, E. Vangioni-Flam, K.A. Olive, astro-ph/0412426. [35] E. Vangioni-Flam, K.A. Olive, B.D. Fields, M. Cassé, Astro- phys. J. 585 (2003) 611, astro-ph/0207583. Physics Letters B 619 (2005) 43–49 www.elsevier.com/locate/physletb

CP violation in a light Higgs boson decay from τ-spin correlations at a linear collider

André Rougé

Laboratoire Leprince-Ringuet, Ecole Polytechnique-IN2P3/CNRS, F-91128 Palaiseau cedex, France Received 9 May 2005; received in revised form 30 May 2005; accepted 31 May 2005 Available online 6 June 2005 Editor: L. Rolandi

Abstract + − We present a new method to measure the transverse spin correlation in the H → τ τ decay. The method has been devised to be insensitive to the beamstrahlung which affects the deﬁnition of the beam energy at a linear collider. In the case of two ± ± τ → π ν¯τ (ντ ) decays, using the anticipated detector performance of the TESLA project, we get a promising estimation of the error expected on the measurement of a CP violating phase.  2005 Elsevier B.V. All rights reserved.

1. Introduction is a CP =+1 state for ξ = 0, a CP =−1 state for ξ = π, and a mixed CP state otherwise. Such a state The possibility to determine the CP properties of a is produced by the decay of a CP =+1 Higgs with a light Higgs boson through the spin correlations in its coupling [5,9] H → τ +τ − decay has been often considered [1–9]. gτ(¯ cos ψ + i sin ψγ )τH. (2) The principle is simple. Let ± denote the projection of 5 O 2 2 the spins of the τ ’s in their respective rest frames on a In this case, neglecting (mτ /mH ), the phase is z-axis oriented in the direction of the τ − line of ﬂight ξ = 2ψ. for the τ + and opposite to the τ + line of ﬂight for the The spin correlations for the state (1) are: τ −.Theτ +τ − spin state Czz =−1, 1 C = C = cos ξ = cos 2ψ, √ |+− + eiξ|−+ (1) xx yy 2 Cxy =−Cyx = sin ξ = sin 2ψ. (3) The way to measure ψ (ξ) is transparent in the case ± ± E-mail address: [email protected] (A. Rougé). of two τ → π ν¯τ (ντ ) decays. Using (1) and the τ

± ± decay amplitudes, one gets the correlated decay distri- τ → π ν¯τ (ντ ). This vector can be computed [12,13] bution from the measured momenta in the case of the above ± → ± ¯ + − decay modes. For example, if τ ρ ντ (ντ ) and W3 cos θ , cos θ ,ϕ ρ± → π ±π 0, 1 = + + − i = N · i − · i 1 cos θ cos θ a 2(q pν)q (q q)pν , (6) 8π + − − sin θ sin θ cos(ϕ − 2ψ) , (4) where N is a normalization factor, pν = pτ ± − pρ± is the four-momentum of the neutrino, and q = pπ± − where θ ± is the polar angle in the τ ± rest frame be- p 0 is the difference of the four-momenta of the two ± ˆ π tween the π direction (π) and the above defined z- pions. The distribution (4), where πˆ is replaced by aˆ in = + − − axis. The relative azimuthal angle ϕ ϕ ϕ is the definition of the angles, contains all the available the angle between the two planes defined by the τ − di- + − information on the spin correlation. The three decay rection and the π (π ) direction respectively in the modes (i.e., (55%)2 of the τ +τ − pairs) can therefore Higgs rest frame. The distribution of the azimuthal an- be used in the same way for the measurement of the gle is obtained by integrating out the polar angles: CP violating parameter ψ [5]. 1 π 2 Unfortunately, it has been shown by Monte Carlo W1(ϕ) = 1 − cos(ϕ − 2ψ) . (5) studies [6] that the standard reconstruction (Section 2) 2π 16 of the τ four-momenta is critically impaired by the The merits of the two distributions for the measure- effects of beamstrahlung. A new estimator has been ment of ψ√can be quantified by their sensitivities proposed [7–9], which is less sensitive to the quality of = + − Sψ 1/σψ N, where σψ is the error on ψ expected the τ reconstruction but requires that both τ and τ ± ± from a maximum likelihood fit of the distribution for decay by the process τ → ρ ν¯τ (ντ ). However, the a sample of N events. The sensitivities measure the ideal sensitivity of the new method, when all the four- information per event on ψ, contained in the distribu- momenta are exactly known, is 0.48 to be compared tions [10]; their computation is straightforward when with 0.92 for the standard method. Besides it takes ad- an analytical or numerical expression of the distribu- vantage of (25%)2 only of the τ +τ − pairs. There is 3 = tions is known. They are Sψ 1.15 for the distribu- therefore a possible improvement of the error on ψ by 1 = tion (4) and Sψ 0.92 for (5). The superiority of (4) a factor of at least four, if a method of reconstruction is not very large and it decreases when experimental of the τ ’s insensitive to beamstrahlung is found. effects are introduced. For that reason and for the sake The aim of the present Letter is to devise such a of simplicity, we will only consider the distribution (5) method. To check the sensitivity to beamstrahlung, we in the following. use a simple Monte Carlo procedure. First the energies The best place to study a light Higgs at a linear of the e± after beamstrahlung are generated using the + − + − e e collider is the Higgsstrahlung process e e → program circe [14] with the parameters of the TESLA ZH. We assume that the mass of the Higgs is well project and the cross section for Higgsstrahlung in the measured by the analysis of its dominant decay modes. case of a CP =+1 Higgs [4]; next the production and The four-momentum of the Higgs is determined by the decay angles of the Z are generated according to their measurement of the Z and therefore the four-momenta correlated distribution under the same hypothesis [4]. of the τ + and the τ − can, in principle, be recon- The Higgs decay angles are generated according to an structed in the case of two hadronic decays (Section 2). isotropic distribution. For the correlated decay of the It is known [10] that under such circumstances the τ ’s we consider three cases: two decays into πν (ππ), ± ± three main hadronic decay modes, τ → π ν¯τ (ντ ), a decay into πν and a decay into ρν (ρπ), and two ± → ± ¯ ± → ± ¯ τ ρ ντ (ντ ), and τ a1 ντ (ντ ) have the same decays into ρν (ρρ). For each case, the azimuthal an- analysis power for the measurement of spin effects. gles and the cosines of the polar angles in the decays This is due to the fact [11–13] that, in the τ rest frame, of τ ’s and ρ’s are generated uniformly. From Eq. (1) ± ± all the information on the spin is embodied in the dis- and the decay amplitudes for τ → π ν¯τ (ντ ) and ± ± ± 0 tribution of a unit vector aˆ, the polarization analyser, τ → ρ ν¯τ (ντ ) → π π ν¯τ (ντ ), the correlated decay which is equal to πˆ (pion direction) in the case of probabilities are computed for diverse values of ψ and A. Rougé / Physics Letters B 619 (2005) 43–49 45 the events are accepted or rejected accordingly. The The direction of the τ −, τˆ ≡ˆτ − is on the intersection four-momenta of all the particles are then computed between the cone around hˆ− with angle δ− and the and, finally, the decay lengths of the τ ’s are generated. cone around −hˆ+ with angle δ+. 1/2 The Higgs mass is assumed to be 120 GeV√ and In general there are two solutions τˆ =n ±n⊥, three values√ of the total energy are√ considered: s = where the vectors n and n⊥,showninFig. 1,are 230 GeV, s = 350 GeV and s = 500 GeV. The given by natural energy for a detailed study of the Higgs is the + − − cos δ cos δ cos δ ˆ+ energy of the largest cross section (i.e., near 230 GeV). n =− h 2 The consideration of higher energies allows estimating sin δ − + cos δ − cos δ cos δ − the robustness of the method and possibly the effect of + hˆ , 2 an underestimation of radiative effects. sin δ −2 1 n ˆ+ ˆ− ˆ+ ˆ− n⊥ = h ∧ h , δ =−h · h . 2 with cos 2. Reconstruction of the τ’s in the Higgs rest sin δ (8) frame The ambiguity can be resolved using the information from a vertex detector but, because the main vertex The reconstruction of the τ ’s in the τ +τ − rest is known from the Z decay, the use of the detector is frame for hadronic decays τ ± → h±ν¯ (ν ) has been τ τ rather different and simpler than at LEP [16,17].One known for a long time [15] and was used at LEP to im- needs only to make a chi-square test on the distances prove the measurement of the τ polarization [16].Its in the laboratory between the reconstructed τ lines of principle is sketched in Fig. 1.Bothτ ’s have the same flight and the trajectories of the charged pion’s. energy E± = m /2 and the energies and momenta of τ H The situation is degraded in the presence of beam- the hadrons are measured. The angle δ± between the strahlung and/or other experimental effects because direction of a τ ± displayed by the unit vector τˆ± and the intersection of the two cones is no more granted. the direction of the hadron hˆ± is therefore fixed: As a result the acceptance is reduced and the dis-

± ± − 2 − 2 tribution of ϕ strongly deformed. The reconstruc- ± 2Eτ Eh mτ m ± cos δ = h . (7) tion is especially awkward in the case of two τ ± → 2pτ ± ph± ± π ν¯τ (ντ ) decays, because the polarization analyser is then aˆ± =ˆπ ± = hˆ± and thus ϕ is the angle between the vectors τˆ− ∧ hˆ− and τˆ− ∧ hˆ+.FromEq.(8), one gets 2 2 2 sin δ 1 − cos ϕ = 1 −n , (9) sin2 δ+ sin2 δ− which shows that the two cones are tangent when | cos ϕ|=1, feature that can also√ be observed in Fig. 1. As a consequence, even for s = 230 GeV, where the closeness to the threshold reduces the effect of beamstrahlung, the acceptance becomes very small when cos ϕ is near ±1 and the distribution (5) can hardly be used to test the CP properties of the Higgs.

3. Reconstruction of the τ’s in the laboratory

Owing to the observation of the Z decay products, Fig. 1. Reconstruction of the τ direction in the Higgs rest frame. which allows the reconstruction of the main vertex, 46 A. Rougé / Physics Letters B 619 (2005) 43–49

± ± ± ± 0 Fig. 2. Reconstruction of the τ direction in the laboratory frame: (a) for a τ → π ν¯τ (ντ ) decay mode, (b) for a τ → π π ν¯τ (ντ ) decay mode. one may envisage to perform the reconstruction of the 4. A simplified algorithm τ ’s in the laboratory frame. ± Let us assume that the τ energies in the labora- Implementing the last method by a fit would require L ± tory Eτ ± are known. The angles α between the di- a good knowledge of the errors and their correlations. rections of the τ ’s and the hadrons are given by the This is not possible with our simple simulation. For relation (7), which reads here that reason, we will use a new procedure, which com- bines elements of the approaches followed in the two L L − 2 − 2 ± 2E ± E ± mτ m ± previous sections and gives good results, without the cos α = τ h h . (10) L L intricacies of a fit. 2pτ ± ph± Taking pH =−pZ, we start the reconstruction in The reconstruction of the τ ± direction is then very the Higgs rest frame but use the approximation τˆ = ± ± ± simple in the case of the τ → π ν¯τ (ντ ) decay mode n/|n|. This is always possible if cos δ is replaced (Fig. 2(a)). by 1 when it is found greater than 1. Denoting by ιˆ the unit vector of the perpendicular As the τ energies in the Higgs rest frame are ˆ ± = L from the vertex to the pion trajectory and by π, the unit known: Eτ mH /2, we can compute Eτ ± and per- vector of the pion momentum, the vector τˆ is given by form the τ reconstruction in the laboratory. For that, we replace cos α± by 1 when it is found greater than 1 ± ± τˆ = cos απˆ + sin αι.ˆ (11) and, in the case of a τ → ρ ν¯τ (ντ ) decay, use the projection of hˆ on the (V, π,ˆ ι)ˆ plane to define τˆ when ± ± 0 In the case of a τ → π π ν¯τ (ντ ) decay mode, τˆ is the cone and the plane do not intersect. In the ππ on the intersection of the plane defined by the vertex channel, an event is rejected if both cos α+ and cos α− ˆ and the charged π trajectory with the cone around h are greater than 1. The ambiguities are resolved by with angle α (Fig. 2(b)). There are in general two so- choosing the solution with the smallest missing |p⊥|. lutions. The problem of the reconstruction of the two We can now redefine the Higgs frame as the rest frame L + − τ ’s is therefore a problem with two unknowns: Eτ + of the τ τ pair and compute ϕ. L and Eτ − , but even in the presence of beamstrahlung The reconstructed√ distributions for the three chan- we still have three constraints: the conservation of nels at an energy of s = 350 GeV are shown in Fig. 3 the components of the momentum orthogonal to the and their sensitivities to ψ at the three considered en- beams and the equality of the τ +τ − effective mass ergies are given in Table 1. with the Higgs mass. They are sufficient to deter- Both the curves in Fig. 3 and the numbers in Table 1 mine the two energies and resolve the ambiguities if include the effect of the small loss of acceptance due needed. to the rejection of events in the ππ channel. A. Rougé / Physics Letters B 619 (2005) 43–49 47

For the ππ channel, the sensitivity is nearly the straints can be used, like the τ +τ − effective mass ideal one (0.92) up to 500 GeV. For the ρπ and and the positivity of the decay length. It should also ρρ channels, the sensitivities are slightly reduced at be noted that the conservation of p⊥ is an impor- 230 GeV and decrease more rapidly with the energy tant but not vital point in the method. For example, than for the ππ channel. Two effects contribute to smearing the p⊥ of the Z with σ(px) = σ(py) = = = that. The ﬁrst is the closing of the τ decay angle 1 GeV√ yields sensitivities Sρπ 0.66 and Sρρ 0.52 when the hadron mass increases. The second is the at s = 350 GeV. imperfect resolution of the ambiguities. The second Finally, a few remarks are in order about the τ ± → ± ¯ point can be improved because the criterion used for a1 ντ (ντ ) decay channel, which is not included in the choice of the solution is not optimal. Other con- our unsophisticated simulation, for the reason that its description is more complex than a simple angular ± → ± 0 0 Table 1 distribution. The a1 π π π decay mode is re- The sensitivities to ψ of the reconstructed distributions when only constructed by the same method that the ρ±,butthe beamstrahlung is taken into account. The effect of the small loss of sensitivity will probably be worsened by the closing acceptance in the ππ channel is included ± → ± + − √ of the τ decay angle. For the a1 π π π de- s(GeV) Sensitivity (Sψ ) cay mode, the determination of τˆ in the laboratory ππ πρ ρρ is in principle possible from vertexing information 230 0.92 0.88 0.83 only, consequently a good sensitivity can certainly be 350 0.91 0.73 0.66 achieved by using this determination of τˆ or by an 500 0.88 0.64 0.55 adaptation of the method used for the ρ±.

Fig. 3. The distributions of ϕ: (a) at the generation level, (b), (c), and (d), reconstructed by the method of Section 4 for the three channels ππ, ρπ,andρρ at an energy of 350 GeV. Beamstrahlung effects only are taken into account. The histograms are normalized to the number of generated events and multiplied by the number of bins. With this normalization, the distribution (a) is 2πW1(ϕ). The full lines correspond to ψ = 0, the dotted lines to ψ = π/8. 48 A. Rougé / Physics Letters B 619 (2005) 43–49

Fig. 4. The reconstructed distributions of ϕ for the ππ channel at an energy of 350 GeV, when all the experimental effects are taken into + − account: (a) for the decay of the Z into µ µ , (b) for its decays into two jets. The convention of normalization and the values of ψ are the same as in Fig. 3. .

5. A semi-realistic simulation of the ππ channel Table 2 The sensitivities to ψ of the reconstructed distributions for the ππ channel, when all the experimental effects are taken into account To get a realistic estimation of the sensitivities, it is √ necessary to take into account the performance of the s(GeV) Sensitivity (Sψ ) detector. We use for that the parameters of the TESLA Z → µ+µ− Z → qq¯ 0 project [18,19]. Since the precision of the π measure- 230 0.69 0.71 ment depends not only on the accuracy of the detector 350 0.60 0.61 but also on the quality of the reconstruction algorithm, 500 0.58 0.58 we consider here the ππ channel only. For the charged tracks an independent Gaussian smearing is performed on the ﬁve parameters: θ, φ, Because of the key role of the vertex detector in the reconstruction, we have done again the simulations 1/p⊥ and the two components of the impact parameter. We use for the widths of the Gaussians the follow- with the vertex detector errors multiplied by two. The ing values [19]: reduction of the sensitivities that results from the increased uncertainties is always smaller than 0.1. σ(θ)= σ(φ)= 0.1mrad, −5 −1 σ(1/p⊥) = 5 × 10 GeV , 6. Conclusion σ(rφ)= σ(rz)= 4.2 ⊕ 4.0/(p sin3/2 θ) µm.

The energy√ of the jets is smeared according to σ(E)/ We have studied the production of a light Higgs E = 0.3/ E(GeV). The position of the vertex is de- boson by the process of Higgsstrahlung and its sub- termined by the shape of the beam [18] for the x and y sequent decay into τ +τ −, under the conditions of a coordinates and by the charged decay products of the linear collider. Z for the z coordinate. The smearing of these coordi- We have described a method, which by the joint use nates is done accordingly. of kinematics and vertexing allows the measurement The reconstructed distributions at an energy of of the transverse spin correlations of the two τ ’s. This 350 GeV, with all the experimental effects included in method is not impaired by beamstrahlung and can be the simulation, are shown in Fig. 4 for the decays of applied for the main hadronic decay modes of the τ the Z both into µ+µ− and into qq¯. Their sensitivities and most of the visible decay modes of the Z. ± ± to ψ at the three considered energies are given in Ta- In the case of two τ → π ν¯τ (ντ ) decays, a com- ble 2. It is clear from these values that a large part of plete simulation of the detector effects with the para- the sensitivity is retained. meters of the TESLA project [19] has been performed. A. Rougé / Physics Letters B 619 (2005) 43–49 49

A realistic study of the reconstruction of the π 0 is still [8]K.Desch,Z.W¸as, M. Worek, Eur. Phys. J. C 29 (2003) 491. to be done, nevertheless it appears that a reasonable [9]K.Desch,A.Imhof,Z.W¸as, M. Worek, Phys. Lett. B 579 goal for the measurement of the phase ψ that parame- (2004) 157. [10] M. Davier, L. Duﬂot, F. Le Diberder, A. Rougé, Phys. Lett. trizes a possible CP violation should be to use all the B 306 (1993) 411. above mentioned ﬁnal states√ and get a sensitivity bet- [11] Y.S. Tsai, Phys. Rev. D 4 (1971) 2821. ter than 0.5, i.e., σψ < 0.6π/ Nevt.. [12] H. Kühn, F. Wagner, Nucl. Phys. B 236 (1984) 16. [13] S. Jadach, J.H. Kühn, Z. W¸as, Comput. Phys. Commun. 64 (1991) 275. References [14] T. Ohl, Comput. Phys. Commun. 101 (1997) 269. [15] Y.-S. Tsai, A.C. Hearn, Phys. Rev. 140 (1965) B721. [16] ALEPH Collaboration, A. Heister, et al., Eur. Phys. J. C 20 [1] J.R. Dell’Aquila, C.A. Nelson, Nucl. Phys. B 320 (1989) 61. (2001) 401. [2] C.A. Nelson, Phys. Rev. D 41 (1990) 2805. [17] J.H. Kühn, Phys. Lett. B 313 (1993) 458. [3] M. Krämer, J. Kühn, M.L. Stong, P.M. Zerwas, Z. Phys. C 64 [18] R. Brinkman, K. Flöttmann, J. Roßbach, P. Schmüser, (1994) 21. N. Walker, H. Weise (Eds.), TESLA Technical Design Report, [4] V. Barger, K. Cheung, A. Djouadi, B.A. Kniehl, P.M. Zerwas, Part II: The Accelerator, DESY 2001-011. Phys. Rev. D 49 (1994) 79. [19] T. Behnke, S. Bertolucci, R.D. Heuer, R. Settles (Eds.), TESLA [5] B. Grz¸adkowski, J.F. Gunion, Phys. Lett. B 350 (1995) 218. Technical Design Report, Part IV: A Detector for TESLA, [6] Z. W¸as, M. Worek, Acta Phys. Pol. B 33 (2002) 1875, hep- DESY 2001-011. ph/0202007. [7] G.R. Bower, T. Pierzchała, Z. W¸as, M. Worek, Phys. Lett. B 543 (2002) 227. Physics Letters B 619 (2005) 50–60 www.elsevier.com/locate/physletb

First measurement of the π +π − atom lifetime

B. Adeva p,L.Afanasyevl,1, M. Benayoun e, A. Benelli q,Z.Berkab, V. Brekhovskikh o, G. Caragheorgheopol m, T. Cechak b, M. Chiba k, S. Constantinescu m, C. Detraz a, D. Dreossi g, D. Drijard a, A. Dudarev l,I.Evangeloud, M. Ferro-Luzzi a, M.V. Gallas p,a, J. Gerndt b,R.Giacomichg, P. Gianotti f,D.Goldinq,F.Gómezp,A.Gorino, O. Gorchakov l,C.Guaraldof, M. Hansroul a,R.Hosekb, M. Iliescu f,m, V. Karpukhin l, J. Kluson b, M. Kobayashi h, P. Kokkas d, V. Komarov l,V.Kruglovl, L. Kruglova l, A. Kulikov l, A. Kuptsov l, I. Kurochkin o, K.-I. Kuroda l,A.Lambertog,A.Lanaroa,f, V. Lapshin o,R.Lednickyc,P.Lerustee, P. Levi Sandri f, A. Lopez Aguera p, V. Lucherini f,T.Makij,N.Manthosd, I. Manuilov o, L. Montanet a, J.-L. Narjoux e, L. Nemenov a,l, M. Nikitin l, T. Núñez Pardo p,K.Okadai, V. Olchevskii l,A.Pazosp, M. Pentia m,A.Penzog, J.-M. Perreau a,C.Petrascuf,m,M.Plóp, T. Ponta m,D.Popm, G.F. Rappazzo g, A. Rodriguez Fernandez p,A.Romerop, A. Ryazantsev o,V.Rykalino, C. Santamarina p,q,a, J. Saborido p, J. Schacher r, Ch.P. Schuetz q,A.Sidorovo, J. Smolik c,F.Takeutchii, A. Tarasov l,L.Tauscherq, M.J. Tobar p,S.Trusovn, V. Ut k i n l, O. Vázquez Doce p, P. Vázquez p,S.Vlachosq,V.Yazkovn, Y. Yoshimura h, M. Zhabitsky l,P.Zrelovl

a CERN, Geneva, Switzerland b Czech Technical University, Prague, Czech Republic c Institute of Physics ACSR, Prague, Czech Republic d Ioannina University, Ioannina, Greece e LPNHE des Universites Paris VI/VII, IN2P3-CNRS, France f INFN, Laboratori Nazionali di Frascati, Frascati, Italy g INFN, Trieste and Trieste University, Trieste, Italy h KEK, Tsukuba, Japan i Kyoto Sangyo University, Kyoto, Japan j UOEH-Kyushu, Japan k Tokyo Metropolitan University, Japan l JINR, Dubna, Russia m IFIN-HH, National Institute for Physics and Nuclear Engineering, Bucharest, Romania n Skobeltsin Institute for Nuclear Physics of Moscow State University, Moscow, Russia o IHEP, Protvino, Russia p Santiago de Compostela University, Spain q Basel University, Switzerland r Bern University, Switzerland

Received 22 April 2005; accepted 18 May 2005 Available online 31 May 2005 Editor: M. Doser

Abstract + − The goal of the DIRAC experiment at CERN (PS212) is to measure the π π atom lifetime with 10% precision. Such a measurement would yield a precision of 5% on the value of the S-wave ππ scattering lengths combination |a0 − a2|. Based on =[ +0.49]× −15 part of the collected data we present a ﬁrst result on the lifetime, τ 2.91−0.62 10 s, and discuss the major systematic | − |= +0.033 −1 errors. This lifetime corresponds to a0 a2 0.264−0.020mπ .  2005 Elsevier B.V. All rights reserved.

PACS: 36.10.-k; 32.70.Cs; 25.80.E; 25.80.Gn; 29.30.Aj

Keywords: DIRAC experiment; Elementary atom; Pionium atom; Pion scattering

1. Introduction 0.220 ± 0.005, a2 =−0.0444 ± 0.0010, a0 − a2 = 0.265 ± 0.004 in units of inverse pion mass) and lead −15 The aim of the DIRAC experiment at CERN [1] is to the prediction τ1S = (2.9 ± 0.1) × 10 s. The to measure the lifetime of pionium, an atom consist- generalized chiral perturbation theory though allows + − ing of a π and a π meson (A2π ). The lifetime is for larger a-values [12]. Model independent measure- dominated by the charge-exchange scattering process ments of a0 have been done using Ke4 decays [13,14]. (π +π − → π 0π 0)2 and is thus related to the relevant Oppositely charged pions emerging from a high scattering lengths [4]. The partial decay width of the energy proton–nucleus collision may be either pro- atomic ground state (principal quantum number n = 1, duced directly or stem from strong decays (“short- orbital quantum number l = 0) is [2,5–9] lived” sources) and electromagnetic or weak decays 1 2 (“long-lived” sources) of intermediate hadrons. Pion Γ = = α3p|a − a |2(1 + δ) (1) pairs from “short-lived” sources undergo Coulomb fi- 1S τ 9 0 2 1S nal state interaction and may form atoms. The region with τ1S the lifetime of the atomic ground state, α of production being small as compared to the Bohr 0 the fine-structure constant, p the π momentum in radius of the atom and neglecting strong final state the atomic rest frame, and a0 and a2 the S-wave ππ n interaction, the cross section σA for production of scattering lengths for isospin 0 and 2, respectively. atoms with principal quantum number n is related to The term δ accounts for QED and QCD corrections the inclusive production cross section for pion pairs = ± × −2 [6–9]. It is a known quantity (δ (5.8 1.2) 10 ) from “short lived” sources without Coulomb correla- ensuring a 1% accuracy for Eq. (1) [8]. A measure- 0 tion (σs ) [15] ment of the lifetime therefore allows to obtain in a model-independent way the value of |a −a |.Theππ 0 2 n 2 0 dσ EA ∗ 2 d σ scattering lengths a0, a2 have been calculated within A = ( π)3 Ψ C r = s 2 n 0 (2) the framework of standard chiral perturbation theory dpA MA dp+ dp− p+=p− [10] with a precision better than 2.5% [11] (a0 = with pA, EA and MA the momentum, energy and mass of the atom in the lab frame, respectively, and p+, E-mail addresses: [email protected], p− the momenta of the charged pions. The square of [email protected] (L. Afanasyev). 1 PH Division, CERN, CH 1211 Geneva 23, Switzerland. the Coulomb atomic wave function for zero distance ∗ | C |2 = 2 Annihilation into two photons amounts to ≈ 0.3% [2,3] and is r between them in the c.m. system is Ψn (0) 3 3 = neglected here. pB/πn , where pB mπ α/2 is the Bohr momentum 52 B. Adeva et al. / Physics Letters B 619 (2005) 50–60 of the pions and mπ the pion mass. The production of atoms occurs only in S-states [15]. Final state interaction also transforms the “unphys- 0 ical” cross section σs into a real one for Coulomb correlated pairs, σC [16,17]: 2 2 0 d σC = C ∗ 2 d σs Ψ ∗ r , (3) dp+ dp− −k dp+ dp− where Ψ C (r∗) is the continuum wave function and −k∗ 2k∗ ≡q with q being the relative momentum of the π + and π − in the c.m. system.3 |Ψ C (r∗)|2 de- −k∗ scribes the Coulomb correlation and at r∗ = 0 coincides with the Gamov–Sommerfeld factor AC(q) with q =|q| [17]:

2πmπ α/q AC(q) = . (4) 1 − exp(−2πmπ α/q)

For low q,0 q q0,Eqs.(2)–(4) relate the num- + − Fig. 1. Relative momentum distributions (q, qL) for atomic π π ber of produced A2π atoms, NA, to the number of pairs at the point of break-up and at the exit of the target. Note that Coulomb correlated pion pairs, NCC [18] qL is almost not affected by multiple scattering in the target. ∞ 1 σ tot 3 = NA = A = (2παmπ ) n 1 n3 q tot| 0 3 NCC σC q q0 π 0 AC(q) d q + − = kth(q0). (5) break up. The π π pairs from break-up (atomic pairs) exhibit specific kinematical features which al- Eq. (5) defines the theoretical k-factor. Throughout the low to identify them experimentally [15], namely very Letter we will use low relative momentum q and qL (the component of + q0 = 2MeV/c and kth(q0) = 0.615. (6) q parallel to the total momentum p+ p−)asshown in Fig. 1. After break-up, the atomic pair traverses the In order to account for the finite size of the pion pro- target and to some extent loses these features by mul- duction region and of the two-pion final state strong tiple scattering, essentially in the transverse direction, interaction, the squares of the Coulomb wave func- while qL is almost not affected. This is one reason for tions in Eqs. (2) and (3) must be substituted by the considering distributions in QL as well as in Q when square of the complete wave functions, averaged over analyzing the data. ∗ the distance r and the additional contributions from Excitation/deexcitation and break-up of the atom 0 0 → 0 0 → + − π π A2π as well as π π π π [17].It are competing with its decay. Solving the transport should be noticed that these corrections essentially equations with the cross sections for excitation and cancel in the k-factor (Eq. (5)) and lead to a correc- break-up, [20–31] leads to a target-specific relation tion of only a fraction of a percent. Thus finite size between break-up probability and lifetime which is corrections can safely be neglected for kth. estimated to be accurate at the 1% level [22,32,33]. Once produced, the A2π atoms propagate with rel- Measuring the break-up probability thus allows to de- ¯ ≈ ativistic velocity (average Lorentz factor γ 17 in termine the lifetime of pionium [15]. our case) and, before they decay, interact with tar- The first observation of the A2π atom [34] has al- get atoms, whereby they become excited/deexcited or lowed to set a lower limit on its lifetime [18,19] of τ>1.8 × 10−15 s (90% CL). In this Letter we present 3 For the sake of clarity we use the symbol Q for the experimen- a determination of the lifetime of the A2π atom, based tally reconstructed and q for the physical relative momentum. on a large sample of data taken in 2001 with Ni targets. B. Adeva et al. / Physics Letters B 619 (2005) 50–60 53

Fig. 2. Schematic top view of the DIRAC spectrometer. Upstream of the magnet: target, microstrip gas chambers (MSGC), scintillating ﬁber detectors (SFD), ionization hodoscopes (IH) and iron shielding. Downstream of the magnet: drift chambers (DC), vertical and horizontal scintillation hodoscopes (VH, HH), gas Cherenkov counters (Ch), preshower detectors (PSh) and, behind the iron absorber, muon detectors (Mu).

2. The DIRAC experiment binatorials and inefficiencies of the SFD, the distributions for the transverse components have substantial The DIRAC experiment uses a magnetic double- tails, which the longitudinal component does not ex- arm spectrometer at the CERN 24 GeV/c extracted hibit [37]. This is yet another reason for analyzing both proton beam T8. Details on the set-up may be found Q and QL distributions. in [35]. Since its start-up, DIRAC has accumulated Data were analyzed with the help of the DIRAC about 15 000 atomic pairs. The data used for this work analysis software package ARIANE [39]. were taken with two Ni foils, one of 94 µm thickness The tracking procedures require the two tracks ei- (76% of the π +π − data), and one of 98 µm thick- ther to have a common vertex in the target plane ness (24% of the data). An extensive description of the (“V-tracking”) or to originate from the intersect of DIRAC set-up, data selection, tracking, Monte Carlo the beam with the target (“T-tracking”). In the fol- procedures, signal extraction and a first high statistics lowing we limit ourselves to quoting results obtained demonstration of the feasibility of the lifetime mea- with T-tracking. Results obtained with V-tracking do surement, based on the Ni data of 2001, have been not show significant differences, as will be shown published in [36]. later. The set-up and the definitions of detector acronyms The following cuts and conditions are applied (see are shown in Fig. 2. The main selection criteria and [36]): performance parameters [36] are recalled in the following. • at least one track candidate per arm with a confi- Pairs of oppositely charged pions are selected by dence level better than 1% and a distance to the beam means of Cherenkov, preshower and muon counters. spot in the target smaller than 1.5 cm in x and y; Through the measurement of the time difference be- • “prompt” events are defined by the time differ- tween the vertical hodoscope signals of the two arms, ence of the vertical hodoscopes in the two arms of the | | time correlated (prompt) events (σ t = 185 ps) can be spectrometer of t 0.5ns; distinguished from accidental events (see [36]). The • “accidental” events are defined by time intervals resolution of the three components of the relative mo- −15 t −5 ns and 7 t 17 ns, determined mentum Q of two tracks, transverse and parallel to by the read-out features of the SFD detector (time de- the c.m. flight direction, Qx , Qy and QL, is about pendent merging of adjacent hits) and exclusion of − 0.5 MeV/c for Q 4MeV/c. Due to charge com- correlated π p pairs. [36]; 54 B. Adeva et al. / Physics Letters B 619 (2005) 50–60

• gen ∝ 2 × protons in “prompt” events are rejected by time- dNCC /dq q AC(q). Processing them with of-flight in the vertical hodoscopes for momenta of GEANT-DIRAC and then analyzing them using the the positive particle below 4 GeV/c. Positive particles full detector and trigger simulation leads to the Cou- MC with higher momenta are rejected; lomb correlated distribution dNCC /dQ. • e± and µ± are rejected by appropriate cuts on Non-correlated π +π − pairs (NC-background). the Cherenkov, the preshower and the muon counter π +π − pairs, where at least one pion originates from information; the decay of a “long-lived” source (e.g., electromag- • cuts in the transverse and longitudinal compo- netically or weakly decaying mesons or baryons) do nents of Q are QT 4MeV/c and |QL| < 15 MeV/c. not undergo any final state interactions. Thus they are gen ∝ 2 The QT cut preserves 98% of the atomic signal. The generated according to dNNC /dq q , using slightly QL cut preserves data outside the signal region for softer momentum distributions than for short-lived defining the background; sources (difference obtained from FRITIOF-6). The • MC only events with at most two preselected hits per Monte Carlo distribution dNNC /dQ is obtained as SFD plane are accepted. This provides the cleanest above. possible event pattern. Accidental π +π − pairs (acc-background). π +π − pairs, where the two pions originate from two different proton–nucleus interactions, are generated according gen 2 3. Analysis to dNacc /dq ∝ q , using measured momentum distri- MC butions. The Monte Carlo distribution dNacc /dQ is The spectrometer including the target is fully simu- obtained as above. lated by GEANT-DIRAC [38], a GEANT3-based sim- All the Monte Carlo distributions are normalized, Qmax MC = MC = ulation code. The detectors, including read-out, ineffi- 0 (dNi /dQ)dQ Ni , i CC, NC, acc, with ciency, noise and digitalization are simulated and im- statistics about 5 to 10 times higher than the experi- MC plemented in the DIRAC analysis code ARIANE [39]. mental data; similarly for atomic pairs (nA ). The triggers are fully simulated as well. The measured prompt distributions are approxi- The simulated data sets for different event types can mated by appropriate shape functions. The functions therefore be reconstructed with exactly the same pro- for atomic pairs, FA(Q), and for the backgrounds, cedures and cuts as used for experimental data. FB(Q), (analogously for QL) are defined as The different event types are generated according to the underlying physics. Atomic pairs. Atoms are generated according to rec MC nA dnA Eq. (2) using measured total momentum distributions FA(Q) = , + − MC dQ for short-lived pairs. The atomic π π pairs are gen- nA erated according to the probabilities and kinematics rec MC rec MC described by the evolution of the atom while propa- NCC dNCC NNC dNNC FB(Q) = + gating through the target and by the break-up process N MC dQ N MC dQ + − CC NC (see [40]). These π π pairs, starting from their spa- MC ωaccNpr dN tial production point, are then propagated through the + acc (7) N MC dQ remaining part of the target and the full spectrome- acc ter using GEANT-DIRAC. Reconstruction of the track rec rec rec pairs using the fully simulated detectors and triggers with nA , NCC, NNC the reconstructed number of leads to the atomic pair distribution dnMC/dQ. atomic pairs, Coulomb- and non-correlated back- + − A Coulomb correlated π π pairs (CC-background, respectively, and ωacc the fraction of acciden- ground). The events are generated according to tal background out of all prompt events Npr. Analyz- Eqs. (3), (4) using measured total momentum dis- ing the time distribution measured with the vertical tributions for short-lived pairs. The generated q-dis- hodoscopes (see [36]) we find ωacc = 7.1% (7.7%) for tributions are assumed to follow phase space modi- the 94 µm (98 µm) data sets [36,37] and keep it fixed fied by the Coulomb correlation function (Eq. (4)), when fitting. The χ2 function for Q (analogously for B. Adeva et al. / Physics Letters B 619 (2005) 50–60 55

Fig. 3. Top: experimental Q and QL distributions after subtraction of the prompt accidental background, and ﬁtted Monte Carlo backgrounds (dotted lines). The peak at Q = 4 MeV/c is due to the cut QT 4 MeV/c. Bottom: residuals after background subtraction. The dotted lines represent the expected atomic signal shape. The bin-width is 0.25 MeV/c.

QL) to minimize is meters found, the background is subtracted from the dN 2 measured prompt distribution, resulting in the resid- νmax pr − [ + ] dQ Q ν ( FA(Q) FB(Q) Q)ν ual spectra. For the signal region, defined by the cuts χ2 = dN = = pr Q + (σ )2 + (σ )2 Q 4MeV/c and QL 2MeV/c, we obtain the to- νmin dQ ν A ν B ν residual (8) tal number of atomic pairs, nA and of Coulomb sig with Q the bin width and σA, σB the statistical er- correlated background events, NCC. Results of fits for rors of the Monte Carlo shape functions, which are Q and QL together are shown in Table 1. much smaller than that of the measurement. The fit CC-background and NC- or acc-backgrounds are rec rec rec distinguishable due to their different shapes, most pro- parameters are nA , NCC, NNC (see Eq. (7)). As a constraint the total number of measured prompt events nounced in the QL distributions (see Fig. 3, top). − = rec + Accidental and NC-background shapes are almost is restricted by the condition Npr(1 ωacc) NCC rec + rec identical for Q and fully identical for Q (uniform NNC nA . The measured distributions as well as the L background are shown in Fig. 3 (top). distributions). Thus, the errors in determining the ac- The data taken with 94 and 98 µm thick targets were cidental background ωacc are absorbed in fitting the analyzed separately. The total number of events in the NC background. The correlation coefficient between − prompt window is Npr = 471 290. CC and NC background is 99%. This strong correla- rec rec First, we determine the background composition by tion leads to equal errors for NCC and NNC. The CC- minimizing Eq. (8) outside of the atomic pair signal background is determined with a precision better than region, i.e., for Q>4MeV/c and QL > 2MeV/c. 1%. Note that the difference between all prompt events rec = and the background is N − N rec − N rec − ω N = For this purpose we require nA 0. As a constraint, pr CC NC acc pr the background parameters N rec and N rec represent- 6590, hence very close to the number of residual CC NC residual ing the total number of CC- and NC-events, have to atomic pairs (nA ) as expected. This relation is be the same for Q and QL. Then, with the para- also used as a strict constraint for fits outside of the sig- 56 B. Adeva et al. / Physics Letters B 619 (2005) 50–60

Table 1 rec Fit results (94 and 98 µm targets together, background shapes from Monte Carlo (MC)) for the parameters NCC (total number of CC-events), rec rec residual NNC (total number of NC-events) and nA (atomic pairs) and deduced results for the number of atomic pairs from the residuals (nA )and sig the number of CC-background events in the signal region (NCC). MC-a: background fit excluding the signal region. MC-b: fit of the entire momentum range including Monte Carlo shape for atomic pairs (“shape fit”). The cuts were at Qcut = 4 MeV/c and QL,cut = 2 MeV/c. Q 2 and QL-distributions were fitted together. The normalized χ were 0.9 for MC-a and MC-b rec rec residual rec sig NCC NNC nA nA NCC MC-a Q 374 022 ± 3969 56 538 6518 ± 373 106 500 ± 1130 QL same same 6509 ± 330 82 289 ± 873 MC-b Q 374 282 ± 3561 56213 6530 ± 294 106 549 ± 1014 QL same same same 82 345 ± 783

> − rec> − rec> − > = nal region (>), Npr NCC NNC (ωaccNpr) 0 ber of atomic pairs is obtained from the measured pairs = rec cut and, hence, the fit requires only one free parameter, by nA nA (Q Qcut)/A . rec> NCC . Number of produced A2π atoms.Hereweusethe Second, the atomic pair signal may be directly known relation between produced atoms and Coulomb obtained by minimizing Eq. (8) over the full range correlated π +π − pairs (CC-background) of Eq. (5). gen and including the Monte Carlo shape distribution FA Using the generator for CC pairs, NCC events, of gen (“shape fit”). The signal strength has to be the same which N (q q0) (see Eq. (6))haveq below rec CC for Q and QL. The result for the signal strength nA q0, are generated into the same acceptance window sig Ω as for atomic pairs and processed analogously as well as the CC-background below the cuts, NCC, gen are shown in Table 1. The errors are determined by to the paragraph above to provide the number of MINOS [41]. reconstructed CC-events below the same arbitrary MC-rec The consistency between the analysis in Q with the cut in Q as for atomic pairs, NCC (Q Qcut). one in QL establishes the correctness of the QT recon- These CC-events are related to the originally gener- cut = MC-rec struction. A 2D fit in the variables (QL,QT ) confirms ated CC-events below q0 through CC NCC (Q gen the results of Table 1. Qcut)/NCC (q q0). The number of produced atoms = rec cut thus is NA kth(q0)NCC(Q Qcut)/CC (see Eq. (6)). The break-up probability P thus becomes 4. Break-up probability br

rec n n (Q Qcut) P = A = A with In order to deduce the break-up probability, Pbr = br rec NA k(Qcut)NCC(Q Qcut) nA/NA, the total number of atomic pairs nA and the total number of produced A atoms, N ,havetobe cut 2π A k(Q ) = k (q ) A . (9) known. None of the two numbers is directly measured. cut th 0 cut CC The procedure of obtaining the two quantities requires reconstruction efficiencies and is as follows. In Table 2 the k-factors are listed for different cuts Number of atomic pairs. Using the generator for in Q and QL for the two target thicknesses (94 and gen atomic pairs a large number of events, nA , is gener- 98 µm) and the weighted average of the two, corre- ated in a predefined large spatial acceptance window sponding to their relative abundances in the Ni data Ωgen, propagated through GEANT-DIRAC including of 2001. The accuracy is of the order of one part per the target and reconstructed along the standard proce- thousand and is due to Monte Carlo statistics. dures. The total number of reconstructed Monte Carlo With the k-factors of Table 2 and the measurements MC-rec atomic pairs below an arbitrary cut in Q, nA (Q listed in Table 1, the break-up probabilities of Table 3 Qcut) defines the reconstruction efficiency for atomic are obtained. The simultaneous fit of Q and QL with cut = MC-rec gen pairs A nA (Q Qcut)/nA . The total num- the atomic shape results in a single value. B. Adeva et al. / Physics Letters B 619 (2005) 50–60 57

Table 2 k(Qcut) factors as a function of cuts in Q and QL for the 94 and 98 µm thick Ni targets, and the weighted average of the two for a relative abundance of 76% (94 µm) and 24% (98 µm)

k94 µm k98 µm kaverage Qcut = 2 MeV/c 0.5535 ± 0.0007 0.5478 ± 0.0007 0.5521 ± 0.0007 Qcut = 3 MeV/c 0.2565 ± 0.0003 0.2556 ± 0.0003 0.2563 ± 0.0003 Qcut = 4 MeV/c 0.1384 ± 0.0002 0.1383 ± 0.0002 0.1384 ± 0.0002 QL,cut = 1 MeV/c 0.3054 ± 0.0004 0.3044 ± 0.0003 0.3050 ± 0.0004 QL,cut = 2 MeV/c 0.1774 ± 0.0002 0.1776 ± 0.0002 0.1774 ± 0.0002

Table 3 The break-up probability has to be corrected for Break-up probabilities for the combined Ni 2001 data, based on the the impurities of the targets. Thus, the 94 µm thick = results of Table 1 and the k-factors of Table 2 for the cuts Qcut target has a purity of only 98.4%, while the 98 µm 4 MeV/c and Q = 2 MeV/c. Errors are statistical L,cut thick target is 99.98% pure. The impurities (C, Mg, residual rec sig nA nA NCC Pbr Si, S, Fe, Cu) being mostly of smaller atomic num- Q 6518 ± 373 106 500 ± 1130 0.442 ± 0.026 ber than Ni lead (for the weighted average of both ± ± ± QL 6509 330 82 289 873 0.445 0.023 targets) to a reduction of the break-up probability of Q & QL 6530 ± 294 106 549 ± 1004 0.447 ± 0.023 1.1% as compared to pure Ni, assuming a lifetime of 3 fs. Therefore, the measured break-up probability has to be increased by 0.005 in order to correspond to pure The break-up probabilities from Q and QL agree Ni. The final result is within a fraction of a percent. The values from shape fit and from background fit are in perfect agreement Pbr = 0.452 ± 0.023stat. (10) (see Table 1). We adopt the atomic shape fit value of Pbr = 0.447 ± 0.023stat, because the fit covers the full rec Q, QL range and includes correlations between nA 5. Systematic errors sig and NCC. Analyzing the data with three allowed hit candi- Systematic errors may occur through the analy- dates in the SFD search window instead of two, re- sis procedures and through physical processes which sults in more atomic pairs (see Ref. [36], T-tracking). are not perfectly under control. We investigate first The break-up probabilities obtained are 0.440 ± 0.024 procedure-induced errors. and 0.430 ± 0.021 for Q and QL, respectively. They The break-up probability will change, if the ratio are not in disagreement with the adopted value of rec rec NCC/NNC depends on the fit range. If so, the Monte 0.447. Despite the larger statistics, the accuracy is not Carlo distributions do not properly reproduce the mea- improved, due to additional background. This back- sured distributions and the amount of CC-background ground originates from additional real hits in the up- may not be constant. In Fig. 4 the dependence is shown stream detectors or from electronic noise and cross- for the fits in Q, QL and both together. The ratio is rea- talk. This has been simulated and leads essentially to sonably constant within errors, with the smallest errors a reduced reconstruction efficiency but not to a dete- for a fit range of Q = QL = 15 MeV/c. At this point rioration of the reconstruction quality. The additional the difference between Q and QL fits leads to a differ- sources of systematic uncertainties lead us not to con- CC = ence in break-up probability of Pbr 0.023. sider this strategy of analysis further on. Consistency of the procedure requires that the V-tracking provides a slightly different data sam- break-up probability does not depend on Qcut.In ple, different k-factors and different signal strengths Fig. 5 the dependence on the cut is shown for break-up and CC-background. The break-up probability, how- residual probabilities deduced from nA . There is a sys- V-tracking = ever, does not change significantly and is Pbr tematic effect which, however, levels off for large cut 0.453±0.025stat, only 0.3σ off from the adopted value momenta. This dependence indicates that the shape of 0.447. the atomic pair signal as obtained from Monte Carlo 58 B. Adeva et al. / Physics Letters B 619 (2005) 50–60

Fig. 4. Ratio of CC-background over NC-background as a function Fig. 5. Pbr as a function of cut momentum for Q and QL. of ﬁt range.

In fact we have measured the multiple scattering for (and used for the k-factor determination) is not in per- all scatterers (upstream detectors, vacuum windows, fect agreement with the residual shape. This may be target) and found narrower angular distributions than due to systematics in the atomic pair shape directly expected from the standard GEANT model [42].This, and/or in reconstructed CC-background for small rel- however, may be due also to errors in determining the ative momenta. The more the signal is contained in thickness and material composition of the upstream the cut, the more the Pbr values stabilize. As a con- detectors. Based on these studies we conservatively sequence, we chose a cut that contains the full signal attribute a maximum error of +5% and −10% to mul- (see Eq. (10)). This argument is also true for sharper tiple scattering. cuts in QT than the one from the event selection. Cut Another source of uncertainty may be due to the momenta beyond the maximum cut of Fig. 5 would presence of unrecognized K+K− and pp¯ pairs that only test background, as the signal would not change would fulﬁll all selection criteria [43]. Such pairs may anymore. be as abundant as 0.5% and 0.15%, respectively, of To investigate whether the atomic pair signal shape π +π − pairs as estimated for K+K− with FRITIOF- is the cause of the above cut dependence, we studied 64 and for pp¯ from time-of-ﬂight measurements in a two extreme models for atom break-up: break-up only narrow momentum interval with DIRAC data. Their from the 1S-state and break-up only from highly ex- mass renders the Coulomb correlation much more cited states. The two extremes result in a difference in peaked at low Q than for pions, which leads to a shape = + − break-up probability of Pbr 0.008. change in effective π π Coulomb background at Sources of systematic errors may also arise from small Q, thus to a smaller atomic pair signal and there- uncertainties in the genuine physical process. We have fore to a decrease of break-up probability. The effect ¯ ¯ investigated possible uncertainties in multiple scatter- leads to a change of P KK,pp =−0.04. We do not ing as simulated by GEANT by changing the scat- br tering angle in the GEANT simulation by ±5%. As a result, the break-up probability changes by 0.002 4 FRITIOF-6 reproduces well production cross sections and mo- per one percent change of multiple scattering angle. mentum distributions for 24 GeV/c proton interactions. B. Adeva et al. / Physics Letters B 619 (2005) 50–60 59

Table 4 Summary of systematic effects on the measurement of the break- up probability Pbr. Extreme values have been transformed into σ assuming uniform distributions Source Extreme values σ CC-background +0.012/−0.012 ±0.007 Signal shape +0.004/−0.004 ±0.002 + − +0.006 Multiple scattering 0.01/ 0.02 −0.013 + − ¯ + − +0 K K and pp 0/ 0.04 −0.023 + − +0 Finite size 0/ 0.03 −0.017 +0.009 Total −0.032

apply this shift but consider it as a maximum systematic error of Pbr. Admixtures from unrecognized e+e− pairs from photon conversion do not contribute because of their different shapes. Finally, the correlation function Eq. (3) used in the Fig. 6. Break-up probability Pbr as a function of the lifetime of the analysis is valid for pointlike production of pions, cor- atomic ground state τ1S for the combined 94 and 98 µm thick Ni related only by the Coulomb final state interaction targets. The experimentally determined Pbr with statistical and to- (Eq. (4)). However, there are corrections due to finite tal errors translates into a value of the lifetime with corresponding errors. size and strong interaction [17]. These have been studied based on the UrQMD transport code simulations − − 6. Lifetime of pionium [44] and DIRAC data on π π correlations. The parameters of the underlying model are statistically fixed The lifetime may be deduced on the basis of the with data up to 200 MeV/c relative momentum. For relation between break-up probability and lifetime for Q 30 MeV/c, the DIRAC data are too scarce to a pure Ni target (Fig. 6). This relation, estimated to serve as a test of the model. The corrections lead to a be accurate at the 1% level, may itself have uncer- finite-size =− change of Pbr 0.02. Due to the uncertain- tainties due to the experimental conditions. Thus the ties we conservatively consider 1.5 times this change target thickness is estimated to be correct to better than as a maximum error, but do not modify Pbr. ±1 µm, which leads to an error in the lifetime (for The systematics are summarized in Table 4.Theex- Pbr = 0.45) smaller than ±0.01 fs, less than 1% of the treme values represent the ranges of the assumed uni- expected lifetime and thus negligible. The result for form probability density function (u.p.d.f.), which, in the lifetime is case of asymmetric errors, were complemented sym- = +0.45 +0.19 × −15 metrically for deducing the corresponding standard τ1S 2.91−0.38 −0.49 10 s stat syst deviations σ . Convoluting the five u.p.d.f. results in + − = 2.91 0.49 × 10 15 s. (12) bell-shaped curves very close to a Gaussian, and the −0.62 ±σ (Table 4, total error) correspond roughly to a The errors are not symmetric because the Pbr–τ rela- 68.5% confidence level and can be added in quadra- tion is not linear, and because finite size corrections ture to the statistical error. and heavy particle admixtures lead to possible smaller The final value of the break-up probability is values of Pbr. The accuracy achieved for the lifetime is about +17%, almost entirely due to statistics and −21%, due to statistics and systematics in roughly = ± +0.009 = +0.025 Pbr 0.452 0.023stat−0.032 syst 0.452−0.039. equal parts. With full statistics (2.3 times more than (11) analysed here) the statistical errors may be reduced 60 B. Adeva et al. / Physics Letters B 619 (2005) 50–60 accordingly. The two main systematic errors (particle J. Gasser, H. Leutwyler, Nucl. Phys. B 250 (1985) 517; admixtures and finite size correction) will be studied J. Gasser, H. Leutwyler, Nucl. Phys. B 250 (1985) 539. in more detail in the future program of DIRAC. [11] G. Colangelo, J. Gasser, H. Leutwyler, Nucl. Phys. B 603 (2001) 125. Using Eq. (1), the above lifetime corresponds to [12] M. Knecht, et al., Nucl. Phys. B 457 (1995) 513. | − |= +0.033 −1 a0 a2 0.264−0.020mπ . [13] L. Rosselet, et al., Phys. Rev. D 15 (1977) 547. [14] S. Pislak, et al., Phys. Rev. Lett. 87 (2001) 221801. [15] L.L. Nemenov, Yad. Fiz. 41 (1985) 980, Sov. J. Nucl. Phys. 41 Acknowledgements (1985) 629. [16] A.D. Sakharov, Zh. Eksp. Teor. Fiz. 18 (1948) 631. [17] R. Lednicky, DIRAC note 2004-06, nucl-th/0501065. We are indebted to the CERN PS crew for pro- [18] L.G. Afanasyev, O.O. Voskresenskaya, V.V. Yazkov, Commu- viding a beam of excellent quality. This work was nication JINR P1-97-306, Dubna, 1997. supported by CERN, the Grant Agency of the Czech [19] L.G. Afanasyev, et al., Phys. Lett. B 338 (1994) 478. Republic, grant No. 202/01/0779 and 202/04/0793, [20] L.S. Dulian, A.M. Kotsinian, Yad. Fiz. 37 (1983) 137, Sov. J. the Greek General Secretariat of Research and Tech- Nucl. Phys. 37 (1983) 78. [21] S. Mrówczynski,´ Phys. Rev. A 33 (1986) 1549; nology (Greece), the University of Ioannina Research S. Mrówczynski,´ Phys. Rev. D 36 (1987) 1520; Committee (Greece), the IN2P3 (France), the Istituto K.G. Denisenko, S. Mrówczynski,´ Phys. Rev. D 36 (1987) Nazionale di Fisica Nucleare (Italy), the Grant-in-Aid 1529. for Scientific Research from Japan Society for the Pro- [22] L.G. Afanasyev, A.V. Tarasov, Yad. Fiz. 59 (1996) 2212, Phys. motion of Science 07454056, 08044098, 09640376, At. Nucl. 59 (1996) 2130. [23] Z. Halabuka, et al., Nucl. Phys. B 554 (1999) 86. 09440012, 11440082, 11640293, 11694099, [24] A.V. Tarasov, I.U. Khristova, JINR-P2-91-10, Dubna, 1991. 12440069, 14340079, and 15340205, the Ministry [25] O.O. Voskresenskaya, S.R. Gevorkyan, A.V. Tarasov, Phys. At. of Education and Research, under project CORINT Nucl. 61 (1998) 1517. No.1/2004 (Romania), the Ministery of Industry, Sci- [26] L. Afanasyev, A. Tarasov, O. Voskresenskaya, J. Phys. G 25 ence and Technologies of the Russian Federation and (1999) B7. [27] D.Yu. Ivanov, L. Szymanowski, Eur. Phys. J. A 5 (1999) 117. the Russian Foundation for Basic Research (Russia), [28] T.A. Heim, et al., J. Phys. B 33 (2000) 3583. under project 01-02-17756, the Swiss National Sci- [29] T.A. Heim, et al., J. Phys. B 34 (2001) 3763. ence Foundation, the Ministerio de Ciencia y Tecnolo- [30] M. Schumann, et al., J. Phys. B 35 (2002) 2683. gia (Spain), under projects AEN96-1671 and AEN99- [31] L. Afanasyev, A. Tarasov, O. Voskresenskaya, Phys. Rev. D 65 0488, the PGIDT of Xunta de Galicia (Spain). (2002) 096001, hep-ph/0109208. [32] C. Santamarina, M. Schumann, L.G. Afanasyev, T. Heim, J. Phys. B 36 (2003) 4273. [33] L. Afanasyev, et al., J. Phys. B 37 (2004) 4749. References [34] L.G. Afanasyev, et al., Phys. Lett. B 308 (1993) 200. [35] B. Adeva, et al., Nucl. Instrum. Methods A 515 (2003) 467. [1] B. Adeva, et al., DIRAC proposal, CERN/SPSLC 95-1, SP- [36] B. Adeva, J. Phys. G 30 (2004) 1929. SLC/P 284 (1995). [37] Ch.P. Schuetz, Measurement of the breakup probability of + − [2] J. Uretsky, J. Palfrey, Phys. Rev. 121 (1961) 1798. π π atoms in a nickel target with the DIRAC spectrom- [3] H.-W. Hammer, J.N. Ng, Eur. Phys. J. A 6 (1999) 115. eter, PhD Thesis, Basel, March 2004, http://cdsweb.cern.ch/ [4] S. Deser, et al., Phys. Rev. 96 (1954) 774. searc.py?recid=732756. [5] S.M. Bilenky, et al., Yad. Phys. 10 (1969) 812, Sov. J. Nucl. [38] P. Zrelov, V. Yazkov, The GEANT-DIRAC Simulation Pro- Phys. 10 (1969) 469. gram, DIRAC note 1998-08, http://zrelov.home.cern.ch/zrelov/ [6] H. Jallouli, H. Sazdjian, Phys. Rev. D 58 (1998) 014011; dirac/montecarlo/instruction/instruct26.html. H. Jallouli, H. Sazdjian, Phys. Rev. D 58 (1998) 099901, Erra- [39] D. Drijard, M. Hansroul, V. Yazkov, The DIRAC offline user’s tum. guide, http://dirac.web.cern.ch/DIRAC/Userguide.html. [7] M.A. Ivanov, Phys. Rev. D 58 (1998) 094024. [40] C. Santamarina, Ch.P. Schuetz, DIRAC note 2003-9. [8] J. Gasser, et al., Phys. Rev. D 64 (2001) 016008, hep- [41] F. James, M. Roos, Function Minimization and Error Analysis, ph/0103157. Minuit, CERN Program Library, D506 MINUIT. [9] A. Gashi, et al., Nucl. Phys. A 699 (2002) 732. [42] A. Dudarev, et al., DIRAC note 2005-02. [10] S. Weinberg, Physica A 96 (1979) 327; [43] O.E. Gortchakov, V.V. Yazkov, DIRAC note 2005-01. J. Gasser, H. Leutwyler, Phys. Lett. B 125 (1983) 325; [44] S.A. Bass, et al., Prog. Part. Nucl. Phys. 41 (1998) 225. J. Gasser, H. Leutwyler, Nucl. Phys. B 250 (1985) 465; Physics Letters B 619 (2005) 61–70 www.elsevier.com/locate/physletb

0 A direct search for the CP-violating decay KS → 3π with the KLOE detector at DANE

KLOE Collaboration F. Ambrosino d, A. Antonelli a, M. Antonelli a, C. Bacci i,P.Beltramea,G.Bencivennia, S. Bertolucci a,C.Binig, C. Bloise a, V. Bocci g, F. Bossi a,D.Bowringa,k, P. Branchini i, R. Caloi g, P. Campana a,G.Capona, T. Capussella d,F.Ceradinii,S.Chia, G. Chiefari d, P. Ciambrone a, S. Conetti k,E.DeLuciaa,A.DeSantisg, P. De Simone a, G. De Zorzi g, S. Dell’Agnello a,A.Denigb, A. Di Domenico g,C.DiDonatod, S. Di Falco e, B. Di Micco i,A.Doriad, M. Dreucci a, G. Felici a, A. Ferrari b, M.L. Ferrer a,G.Finocchiaroa,C.Fortia,P.Franzinig, C. Gatti g, P. Gauzzi g, S. Giovannella a, E. Gorini c, E. Graziani i, M. Incagli e,W.Klugeb, V. Kulikov 1, F. Lacava g,G.Lanfranchia, J. Lee-Franzini a,1, D. Leone b, M. Martini a,∗, P. Massarotti d,W.Meia,S.Meolad, S. Miscetti a,∗, M. Moulson a,S.Müllerb, F. Murtas a, M. Napolitano d, F. Nguyen i, M. Palutan a, E. Pasqualucci g,A.Passerii, V. Pat er a a,f,F.Perfettod, L. Pontecorvo g, M. Primavera c, P. Santangelo a, E. Santovetti h, G. Saracino d, B. Sciascia a, A. Sciubba a,f,F.Scurie,I.Sﬁligoia, T. Spadaro a,M.Testag,L.Tortorai,P.Valenteg, B. Valeriani b, G. Venanzoni a, S. Veneziano g,A.Venturac, R. Versaci i,G.Xua,2

a Laboratori Nazionali di Frascati dell’INFN, Frascati, Italy b Institut für Experimentelle Kernphysik, Universität Karlsruhe, Germany c Dipartimento di Fisica dell’Università e Sezione INFN, Lecce, Italy d Dipartimento di Scienze Fisiche dell’Università “Federico II” e Sezione INFN, Napoli, Italy e Dipartimento di Fisica dell’Università e Sezione INFN, Pisa, Italy f Dipartimento di Energetica dell’Università “La Sapienza”, Roma, Italy g Dipartimento di Fisica dell’Università “La Sapienza” e Sezione INFN, Roma, Italy h Dipartimento di Fisica dell’Università “Tor Vergata” e Sezione INFN, Roma, Italy i Dipartimento di Fisica dell’Università “Roma Tre” e Sezione INFN, Roma, Italy j Physics Department, State University of New York at Stony Brook, USA k Physics Department, University of Virginia, USA Received 7 May 2005; received in revised form 23 May 2005; accepted 26 May 2005 Available online 2 June 2005 Editor: L. Rolandi

Abstract 0 + − We have searched for the decay KS → 3π with the KLOE experiment at DANE using data from e e collisions at a 2 −1 center of mass energy W ∼ mφc for an integrated luminosity L = 450 pb . The search has been performed with a pure KS beam obtained by tagging with KL interactions in the calorimeter and detecting six photons. We ﬁnd an upper limit for the − branching ratio of 1.2 × 10 7 at 90% C.L.  2005 Elsevier B.V. All rights reserved.

+ − Keywords: e e collisions; DANE; KLOE; Rare KS decays; CP; CPT

0 −7 1. Introduction ing to BR(KS → 3π ) 7.4 × 10 at 90% C.L. The sensitivity to CPT violation via unitarity [5] → 0 The decay KS 3π violates CP invariance. The is now limited by the error in η+− = A(KL → + − + − parameter η000, defined as the ratio of KS to KL de- π π )/A(KS → π π ). cay amplitudes, can be written as: η000 = A(KS → We report in the following an improved limit from 0 → 0 = + 0 3π )/A(KL 3π ) , where quantifies a direct search for the 3π decays of the KS . Apart 000 the KS CP impurity and 000 is due to a direct CP- from the interest in confirming the Standard Model, violating term. Since we expect 000 [1],itfol- knowledge of η000 allows tests of the validity of CPT lows that η000 ∼ . In the Standard Model, therefore, invariance using unitarity. 0 −9 BR(KS → 3π ) ∼ 1.9 × 10 to an accuracy of a few %, making the direct observation of this decay quite a challenge. 2. DANE and KLOE 0 The best upper limit on BR(KS → 3π ) from a search for the decay was obtained by the SND exper- 0 The data were collected with the KLOE detec- iment at Novosibirsk. They find BR(KS → 3π ) − tor [6–9] at DANE [10], the Frascati φ factory. 1.4 × 10 5 at 90% C.L. [2]. CPLEAR has pioneered DANE is an e+e− collider operated at a center-of- the method of searching for interference between K S mass energy W ∼ 1020 MeV, the mass of the φ me- and K decays. Interference results in the appear- L son. Positron and electron beams of equal energy col- ance of a term (η ) cos(mt) −(η ) sin(mt) 000 000 lide at an angle of π − 0.025 rad, producing φ mesons in the decay intensity. (η ) and (η ) are ob- 000 000 nearly at rest (p ∼ 12.5MeV). φ mesons decay 34% tained from a fit, without discriminating between φ of the time into nearly collinear K0K¯ 0 pairs. Because K or K → 3π 0 decays. In this way CPLEAR L S J PC(φ) = 1−−, the kaon pair is in a C-odd antisym- finds η = (0.18 ± 0.15) + i(0.15 ± 0.20) [3]. 000 metric state, so that the final state is always K –K . The NA48 Collaboration [4] has recently reached S L Detection of a K signals the presence of a K of much higher sensitivity. By fitting the K /K → L S S L known momentum and direction. We say that detec- 3π 0 interference pattern at small decay times, they tion of a K “tags” the K . find (η ) =−0.002 ± 0.011 ± 0.015 and L S 000 stat sys The KLOE detector consists of a large cylindrical (η ) =−0.003 ± 0.013 ± 0.017 , correspond- 000 stat sys drift chamber (DC), surrounded by a lead/scintillating- fiber electromagnetic calorimeter (EMC). A supercon- * Corresponding authors. Mailing address: INFN, LNF, Casella ducting coil around the calorimeter provides a 0.52 T postale 13, 00044 Frascati (Roma), Italy. field. The drift chamber, 4 m in diameter and 3.3 m E-mail addresses: [email protected] (M. Martini), long, is described in Ref. [6]. The momentum reso- [email protected] (S. Miscetti). ≈ 1 Permanent address: Institute for Theoretical and Experimental lution is σ(p⊥)/p⊥ 0.4%. Two track vertices are Physics, Moscow, Russia. reconstructed with a spatial resolution of ∼3 mm. The 2 Permanent address: Institute of High Energy Physics of Acad- calorimeter, described in Ref. [7], is divided into a emica Sinica, Beijing, China. barrel and two endcaps, for a total of 88 modules, KLOE Collaboration / Physics Letters B 619 (2005) 61–70 63 and covers 98% of the solid angle. The modules are (IP) to the EMC. Each cluster is required to satisfy read out at both ends by photomultipliers providing the condition |T − R/c| < min(3σT , 2ns), where T is energy deposit and arrival time information. The read- the photon flight time and R the path length; σT also out segmentation provides the coordinates transverse includes a contribution from the finite bunch length to the fiber plane. The coordinate along the fibers is (2–3 cm), which introduces a dispersion in the colli- obtained by the difference between the arrival times of sion time. the signals at either end. Cells close in time and space In order to retain a large control sample for the are grouped into calorimeter clusters.√ The energy and background while preserving high efficiency for the time resolutions√ are σE/E = 5.7%/ E (GeV) and signal, we keep all photons satisfying Eγ > 7MeV σT = 54 ps/ E (GeV) ⊕ 50 ps, respectively. and | cos(θ)| < 0.915. The photon detection efficiency The KLOE trigger, described in Ref. [9],uses is ∼90% for Eγ = 20 MeV, and reaches 100% above calorimeter and chamber information. For this analy- 70 MeV. The signal is searched for by requiring six sis, only the calorimeter signals are used. Two en- prompt photons after tagging. ergy deposits above threshold (E>50 MeV for The normalization is provided by counting the 0 the barrel and E>150 MeV for the endcaps) are KS → 2π events in the same tagged sample. required. Recognition and rejection of cosmic-ray events is also performed at the trigger level. Events with two energy deposits above a 30 MeV threshold 3. Monte Carlo simulation in two of the outermost calorimeter planes are rejected. The response of the detector to the decay of inter- During 2002 data taking, the maximum luminos- est and the various backgrounds is studied using the ity reached by DANE was 7.5 × 1031 cm−2 s−1, and Monte Carlo (MC) program GEANFI [11]. GEANFI in September 2002, DANE delivered 91.5pb−1.We accounts for changes in machine operation and back- collected data in 2001–2002 for an integrated luminos- ground conditions, following the machine conditions ity L = 450 pb−1. A total of 1.4 billion φ mesons were run by run, and has been calibrated with Bhabha scat- produced, yielding 450 million KS –KL pairs. Assum- tering events and other processes. The response of the 0 −9 ing BR(KS → 3π ) = 1.9 × 10 , ∼1 signal event is EMC to KL interactions is not simulated but has been expected to have been produced. obtained from a large sample of KL-mesons tagged by + − The mean decay lengths of the KS and KL are identifying KS → π π decays. This not only gives λS ∼ 0.6 cm and λL ∼ 340 cm at DANE. About accurate representation of the EMC response to the 50% of KL’s reach the calorimeter before decaying. KL crash, but also results in an effective 40% increase The KL interaction in the calorimeter (“KL crash”) is in MC statistics. The KL-crash efficiency cancels in identified by requiring a cluster with energy greater the final 3π 0/2π 0 ratio to better than 1% and we as- than 100 MeV that is not associated to any track and sign a 0.9% systematic error to the final result due to whose time corresponds to a velocity in the φ rest this source. ∗ frame, β ,of∼0.2. The KL-crash provides a very Backgrounds are obtained from MC φ → KSKL clean KS tag. The average value of the center-of-mass events corresponding to an integrated luminosity L = energy, W , is obtained with a precision of 30 keV for 900 pb−1.WealsouseaMCsampleofφ → K+K− each 100 nb−1 running period (of duration ∼1h)us- events for L = 450 pb−1 and a MC sample of radiative ing large-angle Bhabha events. The value of W and φ decays for L = 2250 pb−1.Asampleof∼340 000 0 the KL-crash cluster position allows us to establish, KS → 3π MC events is used to obtain the signal ef- for each event, the trajectory of the KS with an angu- ficiency. lar resolution of 1◦ and a momentum resolution better than 2 MeV. Because of its very short lifetime, the displacement 4. Photon counting for data and Monte Carlo of the KS from the φ decay position is negligible. We therefore identify as KS decay photons neutral parti- To test how well the MC reproduces the observed cles that travel with β = 1 from the interaction point photon multiplicity after tagging, we determine the 64 KLOE Collaboration / Physics Letters B 619 (2005) 61–70

Table 1 background composition when comparing data and Measured values of F for data and Monte Carlo samples, in percent MC samples. Data 2001 MC 2001 Data 2002 MC 2002 F(3) 30.95 ± 0.16 30.31 ± 0.11 30.79 ± 0.12 30.06 ± 0.08 F(4) 67.35 ± 0.23 67.93 ± 0.17 67.93 ± 0.18 68.15 ± 0.12 F(5) 1.55 ± 0.01 1.80 ± 0.01 1.19 ± 0.01 1.66 ± 0.01 5. Data analysis F(6) 0.15 ± 0.01 0.14 ± 0.01 0.08 ± 0.01 0.13 ± 0.01 0 KS → 3π candidates consist of a KL crash plus six photons. In our data sample of L = 450 pb−1,we Table 2 find 39 538 events, essentially all background. After Measured values of the probabilities PA and PS removing background, we obtain the branching ratio F(K) Data 2001 MC 2001 Data 2002 MC 2002 0 by normalizing to the number of KS → 2π events. 2 10 × PA(1) 0.75 ± 0.30 1.03 ± 0.16 0.38 ± 0.17 0.89 ± 0.08 The latter are found by asking for three to five prompt 2 × ± ± ± ± 10 PA(2) 0.14 0.05 0.16 0.03 0.07 0.02 0.10 0.03 photons plus the K -crash. 3 L 10 × PS (1) 3.6 ± 0.23.8 ± 0.33.7 ± 0.23.3 ± 0.1 4 According to the MC, the six-photon sample is 10 × PS (2) 1.5 ± 0.41.5 ± 0.30.9 ± 0.21.7 ± 0.2 0 dominated (95%) by KS → 2π decays plus two additional photon clusters. These clusters are due to fragmented or split showers (2S, 1S + 1A, 34%) and = fraction of events of given multiplicity, Nγ k, de- to accidental photons from machine background (2A, = = 6 = fined as F(k) Nev(Nγ k)/ i=3 Nev(Nγ i).As 64%). About 2% of the background events are due to + − 0 shown in Table 1, there is a significant discrepancy false KL-crash tags from φ → KSKL → π π ,3π between data and Monte Carlo for events with mul- events. In such events, charged pions from KS de- tiplicity five and six. These samples are dominated by cays interact in the low-beta insertion quadrupoles,3 → 0 KS 2π decays plus additional clusters due either ultimately simulating the KL-crash signal, while KL to shower fragmentation (split clusters) or the acci- decays close to the IP produce six photons. Similarly, dental coincidence of machine background photons φ → K+K− events give a false signal (∼1%), as well (accidental clusters). To understand this discrepancy, as φ → ηγ → 3π 0γ events (∼0.3%). The cuts de- we have measured the probability, PA(1, 2),ofhav- scribed in the following make the latest two sources ing one, or more than one, accidental cluster passing of contamination negligible. our selection by extrapolating the rates measured in an To reduce the background, we first perform a kine- out-of-time window, (−68 T −14) ns, that is ear- matic fit with 11 constraints: energy and momentum lier than the bunch crossing. In Table 2, we list the conservation, the kaon mass and the velocity of the six average values of these probabilities. The observed photons. The χ2 distribution of the fit to data and MC discrepancy has been traced to an understood prob- background is shown in Fig. 1. In the same plot, we lem with the procedure for the selection of machine- also show the expected shape for signal events. Cutting background clusters. at a reasonable χ2 value (χ2/11 < 3) retains 71% of The MC-true fraction of events with a given mul- the signal while considerably reducing the background tiplicity, fMC, is obtained by ignoring clusters due to from false KL-crash events (33%), in which the direc- machine background and counting at most one cluster tion of the KS and KL are not correlated. However, per simulated particle incident on the calorimeter. Us- this cut is not as effective on the 2S, 2A background, ing the fractions fMC, together with the values of PA due to the soft energy spectrum of the fake clusters. obtained as discussed above, we fit the observed F(k) In order to gain rejection power over the background distribution to get the probability for a cluster to gen- for events with split and accidental clusters, we look erate fragments, PS (see Table 2). This fit accurately at the correlation between the following two χ2-like reproduces the observed fractions in the multiplicity bins five and six. More details on these measurements can be found in Ref. [12]. The results of this study 3 The first quadrupoles are located approximately 45 cm on ei- demonstrate the need for careful calibration of the ther side of the IP. KLOE Collaboration / Physics Letters B 619 (2005) 61–70 65

0 to zero for a KS → 3π event and large for six- photon background events.

For each estimator, the photon pairing with smallest ζ value is kept. Fig. 2a shows the distribution of events in the ζ2–ζ3 plane for the MC background. Most of the events are concentrated at low values of ζ2,as 0 expected for KS → 2π events plus some additional isolated energy deposits in the EMC. A clear signal/background separation is achieved as can be seen by comparing the background and signal distributions in Figs. 2a and b. We subdivide the ζ2-ζ3 plane into the six regions B1, B2, B3, B4, B5, and S as indicated in Fig. 2a. Region S, with the largest signal-to- background value, is the “signal” box. The scatter plot in the ζ2–ζ3 plane for the data is χ2 Fig. 1. Distribution of for the tagged six-photon sample for data shown in Fig. 3a. Our MC simulation does not accu- (points), MC background (solid line), and 10 000 events of MC signal (dashed line). rately reproduce the absolute number of 2S and 2A background events. This is also true of the predicted number of false KL-crash events. However, the simu- estimators: lation does describe the kinematical properties of these events quite well. The two-dimensional ζ2–ζ3 distri- • ζ2, defined as bution allows us to calibrate the contributions from m2 m2 (E − E )2 the different backgrounds. The MC shapes for each of ζ = 1 + 2 + KS i γi 2 2 2 2 the three categories are shown in Figs. 3b–d. We per- σm σm σ E form a binned likelihood fit of a linear combination (P KS − P γi )2 (P KS − P γi )2 of these shapes to the data, excluding the signal-box + x i x + y i y 2 2 region. From the fit we find the composition of the six- σP σP x y photon sample to be (37.9 ± 1.0)%, (57.4 ± 1.3)%, γ (P KS − P i )2 (π − ϑ∗ )2 and (4.7 ± 0.3)% for the 2S, 2A, and false K -crash + z i z + ππ , L 2 2 categories, respectively. σP σϑ∗ z ππ As a check, we compare data and MC for the pro- selecting the four out of six photons that provide jected distribution in ζ3 for the three bands in ζ2,as → 0 the best kinematic agreement with the KS 2π shown in Figs. 4a, b. Excellent agreement is observed. decay hypothesis. This variable is quite insensitive The large peak at low values of ζ3 in the central band to fake clusters. It is constructed using the two val- is due to the false K -crash events. As a final test, we = − L ues of m mi mπ0 (where mi is the invariant compare data and MC in the signal box and the five mass of a photon pair), the opening angle between surrounding control regions. The agreement is better 0 π ’s in the KS rest frame, and 4-momentum con- than 10% in all regions, as seen from Table 3. servation. The resolutions on these quantities have Although cutting on χ2 substantially suppresses been evaluated using a control sample of events the false KL-crash background, we reduce this back- with a KL-crash and four prompt photons. ground to a negligible level by vetoing events with • ζ3, defined as tracks coming from the IP. This effectively elimi- 2 2 2 nates events in which the false KL-crash is due to m1 m2 m3 + − ζ3 = + + , a K → π π decay with the pion secondaries in- σ 2 σ 2 σ 2 S m m m teracting in the quadrupoles. The effect on the signal where the pairing of the six photons into π 0’s is region can be appreciated by comparison of Figs. 4a performed by minimizing this variable. ζ3 is close and c. Moreover, in order to improve the quality of 66 KLOE Collaboration / Physics Letters B 619 (2005) 61–70

Fig. 2. Scatter plot of ζ2 vs. ζ3 plane for the tagged six-photon sample: (a) MC background, (b) MC signal.

Fig. 3. Scatter plots of ζ2 vs. ζ3 for the tagged six-photon sample: data (a), MC sample with two split clusters (b), two accidental clusters (c), and false KL-crash events (d).

Table 3 Comparison between data and MC expectations in the different regions of the ζ2–ζ3 plane for the entire sample with a KL-crash and six prompt photons. The boxes are deﬁned as in Fig. 2a B1 B2 S B3 B5 B4 Data 452 ± 21 10132 ± 101 282 ± 17 5037 ± 71 326 ± 18 22309 ± 149 MC 419 ± 19 9978 ± 104 282 ± 13 4816 ± 43 380 ± 10 22682 ± 190 KLOE Collaboration / Physics Letters B 619 (2005) 61–70 67

Fig. 4. Distributions in ζ3 for the tagged six-photon sample. Plots on the left are for events in the central band in ζ2; plots on the right are for events in all other regions of the plane. The plots in the top row are for the entire sample, before any cuts are made. The plots in the bottom row are after the application of the track veto. In all cases, black points represent data; solid line represents MC.

Fig. 5. Distributions of ζ3 for the central band 12.1 <ζ2 < 60 (a), the side-bands ζ2 < 12.1, ζ2 > 60 (b), after all cuts. Points represent data, solid line MC. the photon selection using ζ2, we cut on the variable and ∆>1.7. The signal box is defined by 12.1 <ζ2 < 2 ∆ = (mφc /2 − Ei)/σE, where i = 1–4 stands for 60 and ζ3 < 4.6. the four chosen photons in the ζ2 estimator and σE is Figs. 5a and b show the ζ3 distributions for the cen- 0 the appropriate resolution. For KS → 2π decays plus tral band and the sidebands in ζ2. two background clusters, we expect ∆ ∼ 0, while for In Table 4, we also list the number of events ob- → 0 0 0 2 KS π π π , ∆ mπ0 c /σE. tained in each of the six regions of the ζ2–ζ3 plane Before opening the signal box, we refine our cuts at this final stage of the analysis. In Figs. 6a, b we 2 on χ , ζ2, ζ3, and ∆ using the optimization procedure show the ζ2–ζ3 scatter plots for data and Monte Carlo. described in Ref. [13]. We end up choosing χ2 < 40.4 The rectangular region illustrates the boundaries of 68 KLOE Collaboration / Physics Letters B 619 (2005) 61–70

−1 −1 Fig. 6. Distribution of ζ2 vs. ζ3 after cuts: MC background 900 pb (a), data 450 pb (b).

Table 4 Same as Table 3, after all cuts. The background in the signal box is expected by MC to be composed of (2.30 ± 0.64), (0.28 ± 0.20) and (0.47 ± 0.47) events for the 2S, 2A and false KL-crash category respectively B1 B2 S B3 B4 B5 Data 0 4 ± 2 2.0 ± 1.4 520 ± 23 3 ± 2 326 ± 18 MC 0 3.2 ± 0.8 3.1 ± 0.8 447 ± 10 2.5 ± 0.8 389 ± 10 the optimized signal box. Seventeen MC events are 6. Systematic uncertainties counted in this region before applying the data-MC scale factors resulting from the calibration procedure Systematics arise from uncertainties in estimation described above. Contributions to the scale factors of the acceptance, backgrounds, and the analysis effi- include the fact that the simulated integrated lumi- ciency. The evaluation of the systematic uncertainties nosity is greater than that for the data set (×2), is described in detail in Ref. [14]. the increased KL-crash efficiency in the simulation Concerning the acceptance of the event selection (×1.4), and the increased probability of having acci- for both the 2π 0 and 3π 0 samples, we estimate the dental or split clusters in the simulation (on average, systematic errors in photon counting by comparing ∼× 1.9). data and MC values for the PA and PS probabilities The selection efficiency at each step of the analysis described above. The photon reconstruction efficiency has been studied using the MC. After tagging, the effi- for both data and MC is evaluated using a large sample + − 0 0 ciency for the six-photon selection is (47.8±0.1stat)%. of φ → π π π , π → γγ events. The momentum Including all cuts, we estimate a total efficiency of of one of the photons is estimated from tracking infor- 3π = (24.4 ± 0.1stat)%. At the end of the analysis mation and position of the other cluster. The candidate chain, we have two candidates with an expected back- photon is searched for within a search cone. The ef- exp ground of B = 3.13 ± 0.82stat. ficiency is parameterized as a function of the photon In the same tagged sample, we also count events energy. Systematics related to this correction are ob- with photon multiplicities of three, four, or five. tained from the variation of the efficiency as a function The corresponding efficiency is (91.8 ± 0.2stat)%for of the width of the search cone. The results are listed 0 0 KS → π π events. The residual background con- in Table 5 under the heading cluster. The total uncer- tamination is estimated to be (0.77 ± 0.24stat+sys)% tainty is smaller for the normalization sample since an and (0.65 ± 0.10stat+sys)% in the 2001 and 2002 run- inclusive selection criterion is used in this case. ning periods, respectively. Subtracting the background The normalization sample also suffers a small and correcting for the efficiency, we count 3.78 × 107 (0.4%) loss due to the use of a filter during data re- 0 0 KS → π π events. We use this number to normal- construction to reject cosmic rays, Bhabha fragments ize the number of signal events when obtaining the from the low-beta quadrupole, and machine back- branching ratio. ground events. This loss is estimated using the MC. KLOE Collaboration / Physics Letters B 619 (2005) 61–70 69

Table 5 Table 6 Systematic acceptance uncertainties, α,forthe2π0 and 3π0 event Systematic uncertainties on the expected background and analysis selection criteria efficiency, ana 0 0 exp exp α/α (KS → 2π ) α/α (KS → 3π ) B /B ana/ana Cluster 0.16% 0.70% Background composition 2.4% − Trigger 0.08% 0.08% Track veto 4.8% 0.2% Background filter 0.20% 0.08% Energy resolution 6.6% 0.5% Energy scale 6.7% 1.0% Total 0.27% 0.71% χ2 5.0% 1.8% Total 11.5% 2.1% We correct for it and add a 0.2% systematic error to the selection efficiency. The trigger and cosmic-ray utions for data and MC. An error of 5% is obtained. veto efficiencies have been estimated with data for A summary of all the systematic errors on the back- the normalization sample and extrapolated by MC to ground estimate is given in Table 6. Adding in quadra- the signal sample. These efficiencies are very close to ture all sources we obtain a total systematic error of unity and the related systematics are negligible. 12% on the background estimate. For the tagged six-photon sample, we have inves- To determine the systematics related to the analysis tigated the uncertainties related to the estimate of the cuts for the signal, we have first evaluated the effect exp background in the signal box after all cuts, B .We of the track veto. Using the MC signal sample, we es- have first considered the error related to the calibration timate a vetoed event fraction of (3.7 ± 0.1)%. The of the MC background composition by propagating data-MC ratio of the cumulative distributions for the the errors on the scale factors obtained from the fit. track-vetoed events in the tagged six-photon sample is exp This corresponds to a relative error of 2.4% on B . RTV = 1.06 ± 0.03, which translates into a 0.2% sys- Moreover, we have investigated the extent to which tematic error on the track-veto efficiency. the track-veto efficiency influences the residual false Because of the similarity of the χ2 distributions KL-crash contamination. To do so, we examine the ∗ for the tagged four- and six-photon samples, as con- data-MC ratio, Rβ , of the sidebands in β for events ∗ firmed by MC studies, an estimate of the systematic rejected by this veto, since for true KL’s β peaks at error associated with the application of the χ2 cut can ∼ 0.2 while false KL-crashes are broadly distributed be obtained from the data-MC comparison of the cu- ∗ = ± in β . We obtain Rβ 1.10 0.01. Knowing that in mulative χ2 distributions for the four-photon sample. the MC only 24% of the fakes survive the veto, we The systematic error arising from data-MC discrepan- find a fractional error of 32% on the fake background. cies in the χ2 distribution is estimated to be 1.8% by Since false KL-crash events account for 15% of the this comparison. exp total background, the error on B from data-MC dif- Moreover, the efficiency changes related to differ- ferences in the track veto efficiency is 4.6%. ences between the calorimeter resolution and energy → 0 A control sample of KS 2π with four prompt scale for data and MC events have been studied in photons has been used to compare the energy scale a manner similar to that previously described for the and resolution of the calorimeter in data and in the evaluation of the systematics on the background. The MC. The distributions of the m and ∆ variables have systematic uncertainties on the analysis efficiency are also been compared by fitting them with Gaussians. summarized in Table 6. Adding all sources in quadra- By varying the mass and energy resolution by ±1σ ture we quote a total systematic error of 2.1% on the in the definitions of ζ2 and ζ3, we observe a relative estimate of the analysis efficiency. change of 6.6% in the background estimate. Similarly, correcting for small differences in the energy scale for data and MC, we derive a systematic uncertainty of 7. Results 6.7% on Bexp. Finally, we have tested the effect of the cut on χ2 by At the end of the analysis, we find 2 events in the constructing the ratio between the cumulative distrib- signal box with an estimated background of Bexp = 70 KLOE Collaboration / Physics Letters B 619 (2005) 61–70

3.13 ± 0.82stat ± 0.37syst. To derive an upper limit detector; A. Balla, M. Gatta, G. Corradi and G. Pa- on the number of signal counts, we build the back- palino for the maintenance of the electronics; M. San- ground probability distribution function, taking into toni, G. Paoluzzi and R. Rosellini for the general sup- account our finite MC statistics and the uncertain- port the detector; C. Piscitelli for his help during major ties on the MC calibration factors. This function is maintenance periods. folded with a Gaussian of width equivalent to the en- This work was supported in part by DOE grant tire systematic uncertainty on the background. Using DE-FG-02-97ER41027; by EURODANE contract the Neyman construction described in Ref. [15],we FMRX-CT98-0169; by the German Federal Ministry 0 limit the number of KS → 3π decays observed to of Education and Research (BMBF) contract 06-KA- 3.45 at 90% C.L., with a total reconstruction efficiency 957; by Graduiertenkolleg ‘H.E. Phys. and Part. Astro- of (24.36 ± 0.11stat ± 0.57sys)%. In the same tagged phys.’ of Deutsche Forschungsgemeinschaft, Contract 7 0 0 sample, we count 3.78×10 KS → π π events. This No. GK 742; by INTAS, contracts 96-624, 99-37; and number is used for normalization. Finally, using the by the EU Integrated Infrastructure Initiative Hadron 0 0 value BR(KS → π π ) = 0.3105 ± 0.0014 [16] we Physics Project under contract number RII3-CT-2004- obtain: 506078. 0 −7 BR KS → 3π 1.2 × 10 at 90% C.L., (1) which represents an improvement by a factor of ∼6 References with respect to the best previous limit [4], and by a factor of 100 with respect to the best limit obtained [1] G. D’Ambrosio, et al., in: L. Maiani, et al. (Eds.), The Second with a direct search [2]. DANE Handbook, vol. 63, Frascati, 1995. [2] SND Collaboration, M.N. Achasov, et al., Phys. Lett. B 459 The limit on the BR can be directly translated into (1999) 674. a limit on |η000|: [3] CPLEAR Collaboration, A. Angelopoulos, et al., Phys. Lett. B 425 (1998) 391. A(K → 3π 0) τ BR(K → 3π 0) [4] NA48 Collaboration, A. Lai, et al., Phys. Lett. B 610 (2005) |η |= S = L S 000 0 0 165. A(KL → 3π ) τS BR(KL → 3π ) [5] G.B. Thomson, Y. Zou, Phys. Rev. D 51 (1995) 1412. < 0.018 at 90% C.L. (2) [6] KLOE Collaboration, M. Adinolfi, et al., Nucl. Instrum. Meth- ods A 488 (2002) 51. This result describes a circle of radius 0.018 cen- [7] KLOE Collaboration, M. Adinolfi, et al., Nucl. Instrum. Meth- teredatzerointhe (η000), (η000) plane and rep- ods A 482 (2002) 364. resents a limit 2.5 times smaller than the result of [8] KLOE Collaboration, M. Adinolfi, et al., Nucl. Instrum. Meth- Ref. [4]. As follows from the discussion in that ref- ods A 483 (2002) 649. erence, our result confirms that the sensitivity of the [9] KLOE Collaboration, M. Adinolfi, et al., Nucl. Instrum. Meth- ods A 492 (2002) 134. CPT test from unitarity is now limited by the uncer- [10] S. Guiducci, Status report on DANE, in: P. Lucas, S. Weber tainty on η+−. (Eds.), Proceedings of the 2001 Particle Accelerator Confer- ence, Chicago, IL, 2001. [11] KLOE Collaboration, F. Ambrosino, et al., Nucl. Instrum. Acknowledgements Methods A 453 (2004) 403. [12] M. Martini, S. Miscetti, Determination of the probability of accidental coincidence between machine background and col- We thank the DANE team for their efforts in lision events and fragmentation of electromagnetic showers, maintaining low-background running conditions and KLOE note 201, 2005, http://www.lnf.infn.it/kloe. their collaboration during all data taking. We would [13] J.F. Grivaz, F. Le Diberder, LAL 92-37, 1992. → 0 like to thank our technical staff: G.F. Fortugno for his [14] M. Martini, S. Miscetti, A direct search for KS 3π decay, dedicated work to ensure efficient operations of the KLOE note 200, 2005, http://www.lnf.infn.it/kloe. [15] G.J. Feldman, R. Cousins, Phys. Rev. D 57 (1998) 57. KLOE computing facilities; M. Anelli for his contin- [16] S. Eidelman, et al., Phys. Lett. B 592 (2004). uous support of the gas system and the safety of the Physics Letters B 619 (2005) 71–81 www.elsevier.com/locate/physletb

Measurement of the cross section for open-beauty production in photon–photon collisions at LEP

L3 Collaboration P. Achard t, O. Adriani q, M. Aguilar-Benitez y, J. Alcaraz y, G. Alemanni w, J. Allaby r, A. Aloisio ac, M.G. Alviggi ac, H. Anderhub aw, V.P. Andreev f,ah,F.Anselmoh, A. Areﬁev ab, T. Azemoon c, T. Aziz i,P.Bagnaiaam,A.Bajoy,G.Baksayz,L.Baksayz, S.V. Baldew b,S.Banerjeei,Sw.Banerjeed, A. Barczyk aw,au, R. Barillère r, P. Bartalini w, M. Basile h,N.Batalovaat, R. Battiston ag,A.Bayw, F. Becattini q, U. Becker m,F.Behneraw, L. Bellucci q, R. Berbeco c, J. Berdugo y,P.Bergesm, B. Bertucci ag,B.L.Betevaw,M.Biasiniag, M. Biglietti ac,A.Bilandaw, J.J. Blaising d, S.C. Blyth ai, G.J. Bobbink b,A.Böhma, L. Boldizsar l,B.Borgiaam,S.Bottaiq, D. Bourilkov aw, M. Bourquin t, S. Braccini t,J.G.Bransonao,F.Brochud,J.D.Burgerm, W.J. Burger ag,X.D.Caim, M. Capell m, G. Cara Romeo h, G. Carlino ac, A. Cartacci q, J. Casaus y, F. Cavallari am, N. Cavallo aj, C. Cecchi ag, M. Cerrada y,M.Chamizot, Y.H. Chang ar,M.Chemarinx,A.Chenar,G.Cheng,G.M.Cheng,H.F.Chenv, H.S. Chen g, G. Chiefari ac, L. Cifarelli an, F. Cindolo h,I.Clarem,R.Clareal, G. Coignet d,N.Colinoy, S. Costantini am,B.delaCruzy, S. Cucciarelli ag, R. de Asmundis ac, P. Déglon t, J. Debreczeni l,A.Degréd,K.Dehmeltz, K. Deiters au, D. della Volpe ac, E. Delmeire t, P. Denes ak, F. DeNotaristefani am,A.DeSalvoaw, M. Diemoz am, M. Dierckxsens b, C. Dionisi am, M. Dittmar aw,A.Doriaac,M.T.Dovaj,5, D. Duchesneau d, M. Duda a, B. Echenard t,A.Eliner,A.ElHagea, H. El Mamouni x, A. Engler ai,F.J.Epplingm, P. Extermann t, M.A. Falagan y, S. Falciano am,A.Favaraaf, J. Fay x,O.Fedinah, M. Felcini aw, T. Ferguson ai, H. Fesefeldt a, E. Fiandrini ag, J.H. Field t, F. Filthaut ae,P.H.Fisherm, W. Fisher ak, I. Fisk ao, G. Forconi m, K. Freudenreich aw,C.Furettaaa, Yu. Galaktionov ab,m, S.N. Ganguli i, P. Garcia-Abia y, M. Gataullin af, S. Gentile am,S.Giaguam, Z.F. Gong v,G.Grenierx,O.Grimmaw, M.W. Gruenewald p, M. Guida an, V.K. Gupta ak,A.Gurtui,L.J.Gutayat, D. Haas e, D. Hatzifotiadou h, T. Hebbeker a,A.Hervér,J.Hirschfelderai,H.Hoferaw, M. Hohlmann z,G.Holzneraw,S.R.Houar,B.N.Jing, P. Jindal n, L.W. Jones c,

0370-2693/$ – see front matter  2005 Published by Elsevier B.V. doi:10.1016/j.physletb.2005.05.072 72 L3 Collaboration / Physics Letters B 619 (2005) 71–81

P. de Jong b, I. Josa-Mutuberría y, M. Kaur n, M.N. Kienzle-Focacci t,J.K.Kimaq, J. Kirkby r, W. Kittel ae, A. Klimentov m,ab, A.C. König ae, M. Kopal at, V. Koutsenko m,ab, M. Kräber aw, R.W. Kraemer ai,A.Krügerav, A. Kunin m, P. Ladron de Guevara y, I. Laktineh x, G. Landi q, M. Lebeau r,A.Lebedevm,P.Lebrunx,P.Lecomteaw, P. Lecoq r,P.LeCoultreaw,J.M.LeGoffr,R.Leisteav,M.Levtchenkoaa, P. Levtchenko ah,C.Liv, S. Likhoded av,C.H.Linar,W.T.Linar,F.L.Lindeb, L. Lista ac, Z.A. Liu g, W. Lohmann av, E. Longo am,Y.S.Lug,C.Luciam, L. Luminari am, W. Lustermann aw,W.G.Mav,L.Malgerir, A. Malinin ab,C.Mañay,J.Mansak, J.P. Martin x, F. Marzano am, K. Mazumdar i, R.R. McNeil f,S.Meler,ac,L.Merolaac, M. Meschini q, W.J. Metzger ae,A.Mihulk,H.Milcentr, G. Mirabelli am,J.Mnicha, G.B. Mohanty i,G.S.Muanzax, A.J.M. Muijs b,B.Musicarao,M.Musyam, S. Nagy o, S. Natale t, M. Napolitano ac, F. Nessi-Tedaldi aw,H.Newmanaf,A.Nisatiam, T. Novak ae,H.Nowakav, R. Oﬁerzynski aw,G.Organtiniam,I.Palat,C.Palomaresy, P. Paolucci ac, R. Paramatti am,G.Passalevaq,S.Patricelliac,T.Paulj, M. Pauluzzi ag, C. Paus m,F.Paussaw, M. Pedace am,S.Pensottiaa, D. Perret-Gallix d,D.Piccoloac, F. Pierella h, M. Pioppi ag,P.A.Pirouéak, E. Pistolesi aa, V. Plyaskin ab, M. Pohl t, V. Pojidaev q, J. Pothier r,D.Prokoﬁevah, G. Rahal-Callot aw, M.A. Rahaman i, P. Raics o,N.Rajai, R. Ramelli aw,P.G.Rancoitaaa,R.Ranieriq, A. Raspereza av, P. Razis ad,D.Renaw, M. Rescigno am, S. Reucroft j, S. Riemann av,K.Rilesc,B.P.Roec, L. Romero y,A.Roscaav, C. Rosemann a, C. Rosenbleck a,S.Rosier-Leesd,S.Rotha, J.A. Rubio r, G. Ruggiero q, H. Rykaczewski aw,A.Sakharovaw, S. Saremi f,S.Sarkaram, J. Salicio r, E. Sanchez y, C. Schäfer r, V. Schegelsky ah, H. Schopper u, D.J. Schotanus ae, C. Sciacca ac,L.Servoliag, S. Shevchenko af,N.Shivarovap, V. Shoutko m, E. Shumilov ab, A. Shvorob af,D.Sonaq,C.Sougax, P. Spillantini q, M. Steuer m, D.P. Stickland ak,B.Stoyanovap,A.Straessnert, K. Sudhakar i, G. Sultanov ap, L.Z. Sun v,S.Sushkova,H.Suteraw,J.D.Swainj, Z. Szillasi z,3,X.W.Tangg,P.Tarjano, L. Tauscher e, L. Taylor j, B. Tellili x, D. Teyssier x,C.Timmermansae, Samuel C.C. Ting m, S.M. Ting m, S.C. Tonwar i,J.Tóthl,C.Tullyak,K.L.Tungg, J. Ulbricht aw, E. Valente am,R.T.VandeWalleae, R. Vasquez at, V. Veszpremi z, G. Vesztergombi l, I. Vetlitsky ab,G.Viertelaw, S. Villa al, M. Vivargent d,S.Vlachose, I. Vodopianov z,H.Vogelai,H.Vogtav, I. Vorobiev ai,ab, A.A. Vorobyov ah, M. Wadhwa e, Q. Wang ae,X.L.Wangv,Z.M.Wangv, M. Weber r, S. Wynhoff ak,L.Xiaaf,Z.Z.Xuv, J. Yamamoto c,B.Z.Yangv,C.G.Yangg,H.J.Yangc,M.Yangg,S.C.Yehas, An. Zalite ah, Yu. Zalite ah, Z.P. Zhang v,J.Zhaov,G.Y.Zhug,R.Y.Zhuaf, H.L. Zhuang g,A.Zichichih,r,s,B.Zimmermannaw, M. Zöller a

a III. Physikalisches Institut, RWTH, D-52056 Aachen, Germany 1 b National Institute for High Energy Physics, NIKHEF, and University of Amsterdam, NL-1009 DB Amsterdam, The Netherlands c University of Michigan, Ann Arbor, MI 48109, USA d Laboratoire d’Annecy-le-Vieux de Physique des Particules, LAPP, IN2P3-CNRS, BP 110, F-74941 Annecy-le-Vieux cedex, France L3 Collaboration / Physics Letters B 619 (2005) 71–81 73

e Institute of Physics, University of Basel, CH-4056 Basel, Switzerland f Louisiana State University, Baton Rouge, LA 70803, USA g Institute of High Energy Physics, IHEP, 100039 Beijing, China 6 h University of Bologna and INFN, Sezione di Bologna, I-40126 Bologna, Italy i Tata Institute of Fundamental Research, Mumbai (Bombay) 400 005, India j Northeastern University, Boston, MA 02115, USA k Institute of Atomic Physics and University of Bucharest, R-76900 Bucharest, Romania l Central Research Institute for Physics of the Hungarian Academy of Sciences, H-1525 Budapest 114, Hungary 2 m Massachusetts Institute of Technology, Cambridge, MA 02139, USA n Panjab University, Chandigarh 160 014, India o KLTE-ATOMKI, H-4010 Debrecen, Hungary 3 p Department of Experimental Physics, University College Dublin, Belﬁeld, Dublin 4, Ireland q INFN, Sezione di Firenze and University of Florence, I-50125 Florence, Italy r European Laboratory for Particle Physics, CERN, CH-1211 Geneva 23, Switzerland s World Laboratory, FBLJA Project, CH-1211 Geneva 23, Switzerland t University of Geneva, CH-1211 Geneva 4, Switzerland u University of Hamburg, D-22761 Hamburg, Germany v Chinese University of Science and Technology, USTC, Hefei, Anhui 230 029, China 6 w University of Lausanne, CH-1015 Lausanne, Switzerland x Institut de Physique Nucléaire de Lyon, IN2P3-CNRS, Université Claude Bernard, F-69622 Villeurbanne, France y Centro de Investigaciones Energéticas, Medioambientales y Tecnológicas, CIEMAT, E-28040 Madrid, Spain 4 z Florida Institute of Technology, Melbourne, FL 32901, USA aa INFN, Sezione di Milano, I-20133 Milan, Italy ab Institute of Theoretical and Experimental Physics, ITEP, Moscow, Russia ac INFN, Sezione di Napoli and University of Naples, I-80125 Naples, Italy ad Department of Physics, University of Cyprus, Nicosia, Cyprus ae Radboud University and NIKHEF, NL-6525 ED Nijmegen, The Netherlands af California Institute of Technology, Pasadena, CA 91125, USA ag INFN, Sezione di Perugia and Università Degli Studi di Perugia, I-06100 Perugia, Italy ah Nuclear Physics Institute, St. Petersburg, Russia ai Carnegie Mellon University, Pittsburgh, PA 15213, USA aj INFN, Sezione di Napoli and University of Potenza, I-85100 Potenza, Italy ak Princeton University, Princeton, NJ 08544, USA al University of Californa, Riverside, CA 92521, USA am INFN, Sezione di Roma and University of Rome, “La Sapienza”, I-00185 Rome, Italy an University and INFN, Salerno, I-84100 Salerno, Italy ao University of California, San Diego, CA 92093, USA ap Bulgarian Academy of Sciences, Central Lab. of Mechatronics and Instrumentation, BU-1113 Soﬁa, Bulgaria aq The Center for High Energy Physics, Kyungpook National University, 702-701 Taegu, Republic of Korea ar National Central University, Chung-Li, Taiwan, Republic of China as Department of Physics, National Tsing Hua University, Taiwan, Republic of China at Purdue University, West Lafayette, IN 47907, USA au Paul Scherrer Institut, PSI, CH-5232 Villigen, Switzerland av DESY, D-15738 Zeuthen, Germany aw Eidgenössische Technische Hochschule, ETH Zürich, CH-8093 Zürich, Switzerland

Received 1 March 2005; received in revised form 13 May 2005; accepted 16 May 2005

Available online 8 June 2005

Editor: L. Rolandi

Abstract The cross section for open-beauty production in photon–photon collisions is measured using the whole high-energy and − high-luminosity data sample collected by the L3 detector at LEP. This corresponds to 627 pb 1 of integrated luminosity for 74 L3 Collaboration / Physics Letters B 619 (2005) 71–81 electron–positron centre-of-mass energies from 189 to 209 GeV. Events containing b quarks are identified through their semi- + − + − leptonic decay into electrons or muons. The e e → e e bbX¯ cross section is measured within our fiducial volume and then extrapolated to the full phase space. These results are found to be in significant excess with respect to Monte Carlo predictions and next-to-leading order QCD calculations.  2005 Published by Elsevier B.V.

1. Introduction

The production of b quarks through hard processes constitutes a unique environment for the study of perturbative QCD, as the mass of the b quark, mb, largely exceeds the typical non-perturbative scale of hadronic interactions. High-energy hadron colliders are copi- ous sources of b quarks and therefore extensive experimental studies and QCD calculations have been performed. Much debate has taken place on the apparent disagreement between the measured cross section for b-quark production in pp¯ collisions at the Teva- tron [1] and the next-to-leading order (NLO) QCD calculations [2]. The first measurements of open beauty production in e±p collisions at HERA were found to be markedly higher than NLO QCD predictions [3]. Some more recent measurements were in better agreement [4], while others still showed an excess [5,6]. A comparison of these different measurements with NLO QCD predictions is shown in Ref. [6]. + − Fig. 1. Dominant diagrams contributing to open-beauty production Photon–photon collisions at e e colliders also in photon–photon collisions at LEP. give access to the hard production of√ b quarks. The LEP e+e− centre-of-mass energy, s, was around 200 GeV. In this environment b quarks are expected to be produced with comparable rates by the direct and the direct and the single-resolved process depend on single-resolved processes [7], illustrated in Fig. 1.The mb, while the latter also depends on the gluon density main contribution to the resolved-photon cross section in the photon. is the photon–gluon fusion process. The rates of both The first measurement of the cross section for the e+e− → e+e−bbX¯ process was published by the L3 −1 Collaboration√ using 410 pb of data collected at s = 189–202 GeV [8]. The results were found to 1 Supported by the German Bundesministerium für Bildung, be in excess of the QCD prediction by a factor of Wissenschaft, Forschung und Technologie. 2 Supported by the Hungarian OTKA fund under contract three. Since these first findings, compatible prelimi- No. T019181, F023259 and T037350. nary results were obtained by other LEP collabora- 3 Also supported by the Hungarian OTKA fund under contract tions [9]. In this Letter, we extend our measurement to No. T026178. the whole high-energy and high-luminosity data sam- 4 Supported also by the Comisión Interministerial de Ciencia y ple collected at LEP with√ the L3 detector [10], corre- Tecnología. sponding to 627 pb−1 at s = 189–209 GeV. 5 Also supported by CONICET and Universidad Nacional de La Plata, CC 67, 1900 La Plata, Argentina. Hadronic events from photon–photon interactions 6 Supported by the National Natural Science Foundation of are selected through their specific multiplicity and China. topology. The production of b quarks is then tagged by L3 Collaboration / Physics Letters B 619 (2005) 71–81 75 the detection of electrons7 or muons from their semi- lisions are selected by means of three criteria. First, at leptonic decays. The cross section of the e+e− → least five charged tracks are required, thus suppress- e+e−bbX¯ process is measured in a phase space which ing background from the e+e− → e+e−τ +τ − and + − + − reflects the energy thresholds used in the analyses and e e → τ τ processes. Second, the visible energy√ the fiducial volume for lepton identification: the lep- of the event, Evis, is required to satisfy Evis < s/3, ton momentum must exceed 2 GeV and the angle, in order to reject events from the e+e− → qq¯ annihi- θ, between the leptons and the beam line must sat- lation process and further eliminate events from the isfy |cos θ| < 0.725 for electrons and |cos θ| < 0.8for e+e− → τ +τ − process. Finally, possible instrumental muons, respectively. For the first time the experimen- background and uncertainties in the trigger procedure tal results are compared to Monte Carlo predictions in are reduced by requiring the event visible mass, Wvis, this fiducial volume. An extrapolation factor is then to satisfy Wvis > 3GeV.Wvis is calculated from the applied to compare the measured cross section with four momenta of reconstructed tracks and of isolated the QCD predictions in the full phase space. calorimetric clusters. In this calculation, the pion mass is associated to the tracks while the clusters are treated as massless. Clusters in the low-angle luminosity mon- 2. Monte Carlo simulations itor are included in this calculation. In addition to these cuts, the analysis is restricted The PYTHIA [11] Monte Carlo generator is used to events with small photon virtuality by removing√ to model hadron production in photon–photon colli- events with clusters with energy greater than 0.2 s in the low-angle calorimeter, covering a polar angle sions. Final states without b quarks are generated with ◦ ◦ massless matrix elements [12] while massive matrix from 1.4 to 3.7 . This criteria corresponds to retain- 2 2 elements are used for b-quark production. Resolved ing quasi-real photons with Q 0.015 GeV . processes are described by means of the SaS1d parton About two million photon–photon events are se- density function [13]. The photon–photon luminosity lected by these cuts, with a background contamination function is implemented in the equivalent photon ap- of 0.1%. Events are further analysed if they have an proximation [14] with a cutoff for the virtuality of the identified electron or muon. 2 2 Electrons are identified as clusters in the elec- interacting photons Q

Muon candidates are selected from tracks in the muon spectrometer in the range |cos θ| < 0.8. A minimal muon momentum of 2 GeV is required to ensure the muons reach the spectrometer after having crossed the calorimeters. The background from annihilation processes is suppressed√ by requiring the muon momentum to be less than 0.1 s. Background from cosmic muons is rejected by requiring the muons to be associated with a signal in the scintillator time-of- ﬂight system in time with the beam crossing. After these cuts, 166 events√ with muon candidates are selected in data with s = 202–209 GeV.√ In- cluding the 269 events previously selected at s = 189–202 GeV [8], a total of 435 events with muons are retained. The estimated background from the e+e− → qq,¯ e+e− → τ +τ − and e+e− → e+e−τ +τ − is 5.7% and the signal efﬁciency is 2.2%. Fig. 2 presents the Wvis spectra of the selected events for the electron and muon samples.

4. Results

The cross section for the e+e− → e+e−bbX¯ process is determined from the distribution of the transverse momentum of the lepton with respect to the nearest jet, Pt. As a consequence of the large value of mb, the distribution of this variable is enhanced for high values as compared to the background. The jets are reconstructed using the JADE algorithm [20] with ycut = 0.1. The identified lepton is not included in the jet. Fig. 3 presents the observed distributions of Pt for electrons and muons. The data distributions are fitted using the least- squares method to the sum of four contributions, whose shapes are fixed by Monte Carlo simulations. The first describes the background from annihilation processes and the e+e− → e+e−τ +τ − reaction. Its normalisation, Nbkg, is fixed to the Monte Carlo predictions listed in Table 1. The three other contributions are those from b quarks, c quarks and lighter flavours. + − + − Their normalisations, Nbb¯ , Ncc¯ and Nuds, respectively, Fig. 2. Visible-mass spectra for the selected e e → e e hadrons are the free parameters of the fit. The results of the fits √events containing (a) an electron or (b) a muon candidate at are given in Table 1: a b-quark fraction of 46.2 ± 5.1% s = 189–209 GeV. The points are the data while the dotted line + − + − + − is observed for electrons and 41.2 ± 3.8% for muons, represents the background from the e e → qq,¯ e e → τ τ , + − → + − + − → + − + − where the uncertainties are statistical. The χ2 per de- e e W W and e e e e τ τ processes. The dashed lines are the sum of this background and the light-quark contri- gree of freedom of the fits is acceptable, with values bution, while the solid lines also include b-quark production. The of 13.7/6 for electrons and 6.4/6 for muons. A corre- normalisation follows from the fit discussed in the text. L3 Collaboration / Physics Letters B 619 (2005) 71–81 77

Table 1 Results of the ﬁt to the distribution of the transverse momentum of the lepton with respect to the nearest jet. The ﬁt parameters are constrained to be positive. The correlation between Nbb¯ and Ncc¯ is 75% Electrons Muons

Nbkg 4.4 (ﬁxed) 24.8 (ﬁxed) ± ± Nbb¯ 94.3 18.3 172.0 31.0 Ncc¯ 105.4 ± 17.9 220.5 ± 35.4 +12.0 +52.3 Nuds 0.0−0.0 0.0−0.0 χ2/d.o.f. 13.7/66.4/6

lation coefficient of about 75% between Nbb¯ and Ncc¯ is observed. The results of the fits are also graphically shown in Fig. 3. Fig. 4 presents the distributions of the lepton momentum, transverse momentum and cosine of polar angle. The measured fractions of b quarks correspond to observed cross sections√ for the luminosity-averaged centre-of-mass energy s=198 GeV of + − → + − ¯ observed σ e e e e bbX electrons = 0.41 ± 0.08 ± 0.08 pb, + − → + − ¯ observed σ e e e e bbX muons = 0.56 ± 0.10 ± 0.10 pb. The first uncertainties are statistical and the second systematic, and arise from the sources discussed below. These cross sections correspond to the phase space of the selected leptons, without any extrapolation: lepton momenta above 2 GeV and polar angles in the ranges |cos θ| < 0.725 for electrons and |cos θ| < 0.8 for muons, respectively.

5. Systematic uncertainties

Several potential sources of systematic uncertainty are considered and their impact on the observed cross section is detailed in Table 2. The largest sources of Fig. 3. Distributions of the transverse momentum of (a) the electron uncertainty arises from the event-selection procedure candidate and (b) the muon candidate with respect to the closest jet and the Monte Carlo modelling of the detector re- for the data and the results of the fit. The points are the data while sponse. Several components contribute to these uncer- + − the dotted line represents the background from the e e → qq,¯ tainties: the event-selection criteria, the lepton identi- + − → + − + − → + − + − → + − + − e e τ τ ,ee W W and e e e e τ τ fication and the detector response and resolution on processes. The dashed lines are the sum of this background and the light-quark contribution, while the solid lines also include b-quark the energy and angular variables which identify the production. The normalisation follows from the fit discussed in the fiducial volume. The effect of these systematic un- text. certainties is estimated by varying the corresponding 78 L3 Collaboration / Physics Letters B 619 (2005) 71–81

Fig. 4. Distribution of (a) the lepton momentum, (c) its transverse momentum and (e) the cosine of its polar angle for events containing electrons + − and (b), (d) and (f) for events containing muons. The points are the data while the dotted line represents the background from the e e → qq,¯ + − + − + − + − + − + − + − e e → τ τ ,e e → W W and e e → e e τ τ processes. The dashed lines are the sum of this background and the light-quark contribution, while the solid lines also include b-quark production. The normalisation follows from the ﬁt discussed in the text. L3 Collaboration / Physics Letters B 619 (2005) 71–81 79

Table 2 CCFM [22] equation. The most important difference Systematic uncertainties on the observed values of the cross section as compared to NLO QCD calculations is the use of + − → + − ¯ of the process e e e e bbX for events tagged by electrons or an unintegrated gluon density function taking explic- muons. An additional uncertainty of 3% affects the extrapolation to the total cross section itly into account the transverse momentum distribution of initial state gluons in hard scattering processes. In Source of uncertainty Uncertainty on cross section (%) NLO QCD, all initial state partons have vanishing Electrons Muons transverse momentum. CASCADE was shown [23] Event selection 6.010.4 to give a consistent description of b-quark production Lepton identification 7.92.2 at the Tevatron, whereas H1 electro-production data Fiducial volume 12.310.0 Jet reconstruction 8.28.2 was found to be in excess by a factor of 2.6. Better Massive/massless charm 3.03.0 agreement was found with ZEUS electro-production Trigger efficiency 2.02.0 data. Monte Carlo statistics 1.61.4 The comparison of measurements and expectations Direct/resolved ratio 0.11.0 in the actual phase space of the selected leptons has Total 18.317.2 the advantage of providing an assessment of the agreement before any extrapolation is performed. Summing statistical and systematic uncertainties in quadrature, cuts and repeating the fits for the newly selected event one finds samples. The second most important source of systematic uncertainty is the jet-reconstruction method. It is assessed by varying the value of y used in the + − → + − ¯ observed = ± cut σ e e e e bbX electrons 0.41 0.11 pb, reconstruction of the jets, and performing the fits for + − + − CASCADE the different Pt distributions which are obtained after → ¯ = ± σ e e e e bbX electrons 0.11 0.02 pb, the corresponding variation of the jet direction. This variation also addresses uncertainties in the hadroni- + − → + − ¯ observed = ± σ e e e e bbX muons 0.56 0.14 pb, sation process by excluding or adding soft clusters to the jets. The impact of the modelling of c quarks in the + − → + − ¯ CASCADE = ± σ e e e e bbX muons 0.14 0.02 pb, event generation is estimated by repeating fits by using Monte Carlo events generated with massive matrix el- where the uncertainty on the CASCADE predictions ements. The trigger efficiency is determined from the corresponds to a variation of m in the range 4.75 ± ± b data themselves and found to be (95.6 2.0)%, this 0.25 GeV [24]. A disagreement of about three stan- uncertainty is also propagated to the final results. The dard deviations is observed for both flavours of the limited Monte Carlo statistics has a small impact on final-state leptons. This disagreement is mostly due to the total systematic uncertainty. In the fits, the signal the overall normalisation of the sample rather than to events are produced in two separate samples for the a difference in shape of the most relevant kinematic direct and resolved processes and then combined in a variables, as also shown in Figs. 3 and 4. 1 : 1 ratio [7]. Systematic uncertainties on this predic- The total cross section for open-beauty production tion are estimated by repeating the fits with ratios of in photon–photon collisions is determined by an ex- 1:2and2:1. trapolation of the observed cross section to the full phase space of the process and by correcting for the semi-leptonic branching ratio of b quarks. The ex- 6. Discussion and conclusions trapolation factors are determined with the PYTHIA Monte Carlo program, and similar results are obtained The b-production cross sections measured in the if the CASCADE Monte Carlo is used. Their differ- phase space of the selected leptons are compared ence, which amounts to 3%, is considered as an ad- with the predictions obtained with the CASCADE ditional systematic uncertainty. The experimental un- Monte Carlo program [21]. This generator employs certainties on the semi-leptonic branching ratio of b a backward-evolving parton cascade based on the quarks [25] is also propagated to the measurement. 80 L3 Collaboration / Physics Letters B 619 (2005) 71–81

The results for the electron and muon final states read + − → + − ¯ total σ e e e e bbX electrons = 12.6 ± 2.4 ± 2.3pb, + − → + − ¯ total = ± ± σ e e e e bbX muons 13.0 2.4 2.3pb, where the first uncertainty is statistical and the second systematic. These results are in perfect agreement with each other and their combination gives + − + − σ e e → e e bbX¯ total = 12.8 ± 1.7 ± 2.3pb, where, again, the first uncertainty is statistical and the second systematic. This result is in agreement with our previous measurement performed with just a subset of the data investigated here [8] and has an improved precision. Fig. 5. The open-charm, upper, and open-beauty, lower, produc- As a cross check, the values of Ncc¯ found by the tion cross sections in photon–photon collisions measured with the fit are used to extract the total cross section for the L3 detector. Statistical and systematic uncertainties are added in production of open charm√ at the luminosity-averaged quadrature. The dashed lines correspond to the direct-process con- centre-of-mass energy s=198 GeV as tribution and the solid lines represent the NLO QCD prediction for the sum of the direct and single-resolved processes. The effects of a + − + − σ e e → e e ccX¯ different choice of the values of the quark masses, mc and mb,are electrons illustrated. = (10.4 ± 1.8) × 102 pb, + − → + − ¯ = ± × 2 and 5.0 GeV. The threshold for open-beauty produc- σ e e e e ccX muons (9.8 1.6) 10 pb, tion is set at 10.6 GeV. The theory prediction for the where uncertainties are statistical. These values agree resolved process is calculated with the GRV parton well, and their average density function [26]. The same results are obtained if + − + − the Drees–Grassie parton density function [27] is used. σ e e → e e ccX¯ = (10.0 ± 1.2) × 102 pb For completeness, Fig. 5 also compares the cross sec- agrees with the dedicated measurement of Ref. [8], tions for open-charm production measured in Refs. [8, + − + − 2 σ(√e e → e e ccX¯ ) = (10.2 ± 0.3) × 10 pb for 28] with√ the corresponding predictions. s=194 GeV, where the uncertainties are statis- For s=198 GeV and mb = 4.75 GeV, the cross tical only. section expected from NLO QCD is 4.1 ± 0.6pb, An additional cross check showed that values of the where the uncertainty is dominated by uncertainties open-beauty cross section determined with the fit pro- on the renormalisation scale and on mb. Our measure- cedure discussed above or with a counting method [8] ment is a factor of three, and three standard deviations, are compatible. In the latter case experimental crite- higher than expected. In this respect it is interesting ria were chosen to optimise the charm cross section to remark that the prediction of CASCADE, when ex- measurement yielding a result essentially uncorrelated trapolated to the full phase space, 3.5 pb, agrees with with the b-quark production rate. those from NLO QCD [24], and the excess of our data The total cross section for open-beauty produc- with respect to the expectations is consistent before tion is compared in Fig. 5 to NLO QCD calcula- and after the extrapolation from the fiducial volume to tions [7]. The dashed line corresponds to the direct the full phase-space. process while the solid line shows the prediction for In conclusion, all high-energy data collected by the sum of direct and resolved processes. The cross L3 at LEP is investigated and the e+e− → e+e−bbX¯ section depends on mb, which is varied between 4.5 cross sections are measured within the detector fidu- L3 Collaboration / Physics Letters B 619 (2005) 71–81 81 cial volume and found to be in excess with respect to [10] L3 Collaboration, B. Adeva, et al., Nucl. Instrum. Methods Monte Carlo predictions. The cross sections are ex- A 289 (1990) 35; trapolated to the full phase space and found to be in L3 Collaboration, O. Adriani, et al., Phys. Rep. 236 (1993) 31; M. Acciarri, et al., Nucl. Instrum. Methods A 351 (1994) 300; excess with respect to next-to-leading order QCD cal- M. Chemarin, et al., Nucl. Instrum. Methods A 349 (1994) 345; culations. This confirms our previous findings based G. Basti, et al., Nucl. Instrum. Methods A 374 (1996) 293; on a subset of the full data-sample. I.C. Brock, et al., Nucl. Instrum. Methods A 381 (1996) 236; A. Adam, et al., Nucl. Instrum. Methods A 383 (1996) 342. [11] PYTHIA version 5.722 is used; T. Sjöstrand, preprint CERN-TH/7112/93 (1993), revised in References August 1995; T. Sjöstrand, Comput. Phys. Commun. 82 (1994) 74. [1] CDF Collaboration, F. Abe, et al., Phys. Rev. Lett. 71 (1993) [12] M. Cacciari, et al., Nucl. Phys. B 466 (1996) 173. 500; [13] G.A. Schuler, T. Sjöstrand, Z. Phys. C 68 (1995) 607; CDF Collaboration, F. Abe, et al., Phys. Rev. D 50 (1994) G.A. Schuler, T. Sjöstrand, Phys. Lett. B 376 (1996) 193. 4252; [14] V.M. Budnev, et al., Phys. Rep. 15 (1975) 181. CDF Collaboration, F. Abe, et al., Phys. Rev. Lett. 75 (1995) [15] J.A.M. Vermaseren, Nucl. Phys. B 229 (1983) 347. 1451; [16] KORALZ version 4.02 is used; CDF Collaboration, D. Acosta, et al., Phys. Rev. D 65 (2002) S. Jadach, B.F.L. Ward, Z. W¸as, Comput. Phys. Commun. 79 052005; (1994) 503. CDF Collaboration, D. Acosta, et al., Phys. Rev. D 66 (2002) [17] KORALW version 1.33 is used; 032002; S. Jadach, et al., Comput. Phys. Commun. 94 (1996) 216; D0 Collaboration, S. Abachi, et al., Phys. Rev. Lett. 74 (1995) S. Jadach, et al., Phys. Lett. B 372 (1996) 289. 3548; [18] GEANT version 3.15 is used; D0 Collaboration, B. Abbott, et al., Phys. Lett. B 487 (2000) R. Brun, et al., preprint CERN DD/EE/84-1 (1984), revised 264; 1987. D0 Collaboration, B. Abbott, et al., Phys. Rev. Lett. 85 (2000) [19] H. Fesefeldt, RWTH Aachen Report PITHA 85/2 (1985). 5068. [20] JADE Collaboration, W. Bartel, et al., Z. Phys. C 33 (1986) 23; [2] P. Nason, S. Dawson, K.R. Ellis, Nucl. Phys. B 303 (1988) 607; JADE Collaboration, S. Bethke, et al., Phys. Lett. B 213 (1988) P. Nason, S. Dawson, K.R. Ellis, Nucl. Phys. B 327 (1989) 49; 235. W. Beenakker, et al., Nucl. Phys. B 351 (1991) 507; [21] CASCADE version 1.1 is used; M. Cacciari, et al., JHEP 0407 (2004) 033, and references H. Jung, Comput. Phys. Commun. 143 (2002) 100; therein; H. Jung, G.P. Salam, Eur. Phys. J. C 19 (2002) 351. M.L. Mangano, hep-ph/0411020, and references therein; [22] M. Ciafaloni, Nucl. Phys. B 296 (1988) 49; S. Frixione, in: Proceedings of DIS’2004, hep-ph/0408317. S. Catani, F. Fiorani, G. Marchesini, Nucl. Phys. B 336 (1990) [3] H1 Collaboration, C. Adloff, et al., Phys. Lett. B 467 (1999) 18. 156; [23] H. Jung, Phys. Rev. D 65 (2002) 034015. H1 Collaboration, C. Adloff, et al., Phys. Lett. B 518 (2001) [24] H. Jung, M. Hanson, private communication. 331, Erratum; [25] S. Eidelman, et al., Phys. Lett. B 592 (2004) 1. ZEUS Collaboration, J. Breitweg, et al., Eur. Phys. J. C 18 [26] M. Glück, E. Reya, A. Vogt, Phys. Rev. D 46 (1992) 1973. (2001) 625. [27] M. Drees, K. Grassie, Z. Phys. C 28 (1985) 451. [4] H1 Collaboration, A. Aktas, et al., hep-ex/0411046; [28] L3 Collaboration, M. Acciarri, et al., Phys. Lett. B 453 (1999) ZEUS Collaboration, S. Chekanov, et al., Phys. Rev. D 70 83; (2004) 012008. L3 Collaboration, M. Acciarri, et al., Phys. Lett. B 514 (2001) [5] ZEUS Collaboration, S. Chekanov, et al., Phys. Lett. B 599 19; (2004) 173. L3 Collaboration, M. Acciarri, et al., Phys. Lett. B 536 (2002) [6] H1 Collaboration, A. Aktas, et al., hep-ex/0502010. 217. [7] M. Drees, et al., Phys. Lett. B 306 (1993) 371. [8] L3 Collaboration, M. Acciarri, et al., Phys. Lett. B 503 (2001) 10. [9] DELPHI Collaboration, W. Da Silva, in: Proceedings of PHO- TON2003, Nucl. Phys. B (Proc. Suppl.) 126 (2004) 185; OPAL Collaboration, A. Csilling, in: Proceedings of PHO- TON2000, hep-ex/0010060. Physics Letters B 619 (2005) 82–87 www.elsevier.com/locate/physletb

Important pickup coupling effect on 8He(p, p) elastic scattering

F. Skaza a,N.Keeleya, V. Lapoux a, N. Alamanos a,F.Augera,D.Beaumelb, E. Becheva b, Y. Blumenfeld b, F. Delaunay b,A.Drouarta, A. Gillibert a, L. Giot c, K.W. Kemper d, R.S. Mackintosh e, L. Nalpas a,A.Pakouf, E.C. Pollacco a,R.Raabea,1, P. Roussel-Chomaz c, J.-A. Scarpaci b, J.-L. Sida a,2,S.Stepantsovg,R.Wolskig,h

a CEA-Saclay, DSM/DAPNIA/SPhN, F-91191 Gif-sur-Yvette, France b Institut de Physique Nucléaire, IN2P3-CNRS, F-91406 Orsay, France c GANIL, Bld. Henri Becquerel, BP 5027, F-14021 Caen cedex, France d Department of Physics, Florida State University, Tallahassee, FL 32306-4350, USA e Department of Physics and Astronomy, The Open University, Milton Keynes, MK7 6AA, UK f Department of Physics, University of Ioannina, 45110 Ioannina, Greece g Flerov Laboratory of Nuclear Reactions, JINR, RU-141980 Dubna, Russia h The Henryk Niewodnicza´nski Institute of Nuclear Physics, PL-31342 Kraków, Poland Received 20 April 2005; received in revised form 11 May 2005; accepted 25 May 2005 Available online 2 June 2005 Editor: V. Metag

Abstract The 8He(p, p) and (p, d) reactions were measured in inverse kinematics at 15.7 A MeV and analyzed within the coupled reaction channels framework, the (p, d) cross section being particularly large. We ﬁnd that coupling to 8He(p, d) pickup has a profound effect on the 8He(p, p) elastic scattering, and that these strong coupling effects should be included in analyses of proton elastic and inelastic scattering. Through its modiﬁcation of the elastic scattering wave functions this coupling will strongly affect the extraction of spectroscopic information such as the relationship between neutron and proton nuclear deformations, with important consequences for our understanding of the structure of exotic nuclei.  2005 Elsevier B.V. All rights reserved.

PACS: 25.60.Bx; 25.60.Je; 24.10.Eq

Keywords: 8He(p, p); (p, d); Coupled reaction channels calculations; Dynamic polarization potential

E-mail address: [email protected] (V. Lapoux). 1 Present address: IKS, University of Leuven, B-3001 Leuven, Belgium. 2 Present address: CEA DIF/DPTA/SPN, B.P. 12, F-91680 Bruyères-le-Châtel, France.

Strong coupling effects in low-energy nuclear re- detection system consisted of the MUST array [10] actions are well established for heavy-ion collisions, to detect the light charged particles, a plastic wall for and lead to important modifications of the effective the detection of the projectile-like fragment, and two nucleus–nucleus interaction. The 16O + 208Pb system beam tracking detectors (CATS) upstream of the tar- is a well documented example, with coupled reaction get. The position sensitive CATS detectors [11] were channels (CRC) calculations showing how inelastic used to improve the definition of the beam position scattering and transfer channels generate a dynamic and incident angle on target. They provided particle polarization potential (DPP) with a substantial real by particle position and time tracking of the beam. part [1,2], having important consequences for elastic The MUST array consists of eight three-stage tele- scattering and fusion. scopes, each 6 × 6cm2. The first stage is a 300 µm- Important effects on (p, p) elastic scattering due to thick double-sided Si-strip detector which provides coupling to (p, d) pickup have been demonstrated for horizontal and vertical position, time-of-flight (TOF) stable nuclei [3–5]. The effect is particularly large for with respect to the beam detectors, and energy loss of light nuclei [4], reducing with increasing target mass the recoil proton. The second stage is a 3 mm-thick and incident proton energy, although remaining sig- Si(Li) giving the energy for protons up to 25.4 MeV, nificant for 50 MeV protons incident on 64Zn. Pickup and the third stage a 1.5 cm-thick CsI allowing the decoupling was also found to significantly affect inelas- tection of protons up to 75 MeV in energy. The array tic scattering, mainly through the modification of the was assembled in a wall configuration located 15 cm elastic scattering wave functions [3], leading to signifi- from the target. The wall was placed in two positions, cant changes in the extracted deformation parameters. covering the angular range between 30◦–90◦ (lab.). At However, the possibility of such strong coupling ef- this distance, the 1 mm wide strips result in an angular fects has come to be ignored in analyses of proton resolution of 0.4◦ (lab.) for the detection of the scat- elastic and inelastic scattering, although a recent study tered particle. of 6He(p, p) postulated the existence of a repulsive For the less energetic recoil particles stopped in the real DPP due to breakup that gave improved agree- first stage, e.g., protons with energies below 6 MeV, ment with the data [6], subsequently further investi- mass identification was obtained using the energy ver- gated through coupled discretized continuum channels sus TOF technique. Particles were identified in the (CDCC) calculations [7]. correlation plot of their energy loss, E, in the Si- We report here a measurement of 8He(p, p) scat- strip detector versus their TOF. The TOF was mea- tering at 15.7 A MeV incident energy. Data for sured between the Si-stage and the start signal given 8He(p, d) populating the 3/2− ground state resonance by the passage of the incident particle through the sec- of the unbound 7He measured in the same experi- ond CATS detector. Protons from 6 to 25 MeV were ment have been previously reported [8], and the cross unambiguously identified by the E–E method us- section is found to be very large. This should there- ing energy loss measurements in the Si strip and the fore be a case where (p, d) coupling will have an Si(Li) detectors. The energy resolution obtained var- important influence on the 8He(p, p) scattering. We ied between 600 keV and 1 MeV, depending on target present CRC calculations including 8He(p, d) pickup thickness and the reaction kinematics. Events with a p to the 3/2− ground state of 7He which demonstrate the or d in coincidence with the heavy ejectile, plus a par- profound influence of this coupling on the elastic scat- ticle detected in the two CATS detectors to provide tering and, hence, on the nucleon–nucleus interaction the incident beam trajectory, were retained to build in a way that falls outside the scope of local-density the kinematical spectra and subsequently extract the folding models. (p, p) and (p, d) angular distributions. The 8He beam was produced by the ISOL tech- The elastic data extend from 20◦–110◦, and the nique and accelerated to 15.7 A MeV by the CIME transfer data from 27◦–85◦, in their respective center cyclotron at the SPIRAL facility [9], with no conta- of mass (c.m.) systems. To measure angular distrib- minants. The maximum (average) intensity in the ex- utions from 40◦ down to 20◦ (c.m.) where the en- periment was 14 000 (5000) p/s. The proton target was ergy of the recoiling protons decreases to 1.5 MeV, 2 2 a8.25mg/cm thick polypropylene (CH2)n foil. The a1.48mg/cm polypropylene target was used. To ob- 84 F. Skaza et al. / Physics Letters B 619 (2005) 82–87 tain good statistics at large angles, from 40◦ to 110◦ (c.m.), a 8.25 mg/cm2 target was used. The overall values for the statistical plus systematic errors in the angular distributions arise from the detection efficiency and reconstruction process, which gives ±5% uncertainty, including the effect of background subtraction (±2%); the target thickness (±5%); and the efficiency in the detection of the incident particles (±2%). This results in a total uncertainty of ±7.5% in the normalization of the data for elastic scattering and transfer to the 7He ground state. In Fig. 1 the measured elastic scattering angular distribution is compared to optical model calculations performed within the framework of the microscopic nucleon–nucleus JLM potential [12], using a no-core shell model 8He density [13]. The JLM potential is complex and the data for well-bound nuclei were found to be well reproduced with slight variations of Fig. 1. Optical model calculations using the JLM potential compared to the 8He + p elastic scattering data. See text for details. the real and imaginary parts, V and W . The required normalization factors, λV and λW , respectively, are found to be close to unity. For well-bound light nuclei in determining which states should be included in the 8 (A 20), the only modification required is λW = 0.8 coupling scheme, or not. In the He(p, p ) experiment [14], adopted as a “standard” normalization. at 72 A MeV reported in Ref. [16], the first excited The standard JLM (dotted curve) does not repro- state of 8He was found to be a 2+ located at 3.6 MeV. ◦ ◦ duce the data. Best agreement was obtained with λV = The cross sections measured between 20 to 50 (c.m.) 1.11, λW = 1.06 (solid curve), but the data at an- were found to lie below 1 mb/sr; a weak excitation gles smaller than 40◦ (c.m.) are significantly underpre- of the 2+ 8He was found [17]. In our experiment, as dicted. It should be emphasized that simply modeling mentioned in Ref. [8], inelastic (p, p) to the 2+ ex- the DPP by a renormalization of the JLM potential is cited state was also selected. These cross sections at unable to reproduce the whole angular range of the 15.7 A MeV will be presented and analyzed in a forth- data. coming article. Compared to the angular distributions Clearly, we need to include explicitly in our calcu- of the (p, d) transfer reaction, they were found to be lations the effect of coupling to other reaction chan- twice up to 5 times lower in the angular range from nels. To investigate the effect of coupling to (p, d) 20◦ to 80◦ (c.m.). We also face the problem of the exit pickup on 8He(p, p) scattering a series of CRC calcu- channel of the (p, d) reaction. It is beyond the scope of lations was carried out using the code FRESCO [15]. present CDCC calculations to include within the cou- The JLM prescription was retained for the p + 8He pling scheme the continuum of the unbound 7He states optical potential. We should include in the coupling and calculate the transfer reaction. The best calcula- scheme, a priori, the following reactions: elastic, in- tion which can be performed, at the present stage, is to elastic scattering and transfer reactions to the ground consider the deuteron states within the continuum. In or excited states of the nuclei produced in the exit Ref. [18], Halderson showed that the recoil corrected channel, either in bound or resonant states. But this re- continuum shell model predictions support a low-lying quires the corresponding inputs, transition strengths to 1/2− excited state for 7He at 1 MeV,as found by Meis- the excited states and spectroscopic factors. To sim- ter et al. [19]. Our recent results [8] also indicated this plify, we limitate the coupling scheme to the main low-lying excited state of 7He; it is weakly excited, channels which may contribute significantly in terms and roughly the cross sections are 10 times lower than 7 of angular distributions in the domain treated in our the (p, d) Hegs ones. In Ref. [20],at50A MeV, a res- analysis. The experimental observations can help us onance at 2.9 MeV was observed in 7He, the cross F. Skaza et al. / Physics Letters B 619 (2005) 82–87 85

Fig. 2. Coupling scheme used in the CRC calculations. sections (from 5◦ to 15◦ (c.m.)) were found to be 5 7 times less than the (p, d) Hegs. Consequently, in our analysis, we did not explicitly include the coupling to the 7He excited states and we considered (p, p) and 7 (p, d) Hegs as the main coupled reactions. The CDCC formalism was employed in the exit channel, as described in Ref. [21]. The bare d + 7He potential was of Watanabe type [22],then and p plus 7He optical potentials being calculated using the global parametrization of Koning and Delaroche [23]. Couplings to deuteron breakup with the neutron and Fig. 3. 8He(p, p) (upper panel) and 8He(p, d) (lower panel) calcu- proton in relative S and D states were explicitly in- lations compared to the data. The solid curves denote the full CRC cluded using the CDCC formalism and the coupling calculation with λV = 1.05, λW = 0.2 and the dotted curve indi- scheme presented in Fig. 2. cates the no-coupling calculation with the same bare potential. The For the transfer step, the neutron–proton over- dashed curves denote the result of a CRC calculation omitting the non-orthogonality correction. lap was calculated using the Reid soft-core potential [24], including the D-state component of the deuteron ground state. The same interaction was used to calcu- In Fig. 3 we present the calculated angular distri- late the exit channel deuteron potentials. The n + 7He butions for 8He(p, p) and 8He(p, d) compared to the binding potential was a Woods–Saxon well with the data. The results shown are for the ﬁnal calculation 1/3 “standard” geometry of R0 = 1.25 × A fm, a = with JLM normalization factors λV = 1.05, λW = 0.2. 0.65 fm, the well depth being adjusted to give the A 8He(0+)/7He(3/2−) spectroscopic factor of C2S = correct binding energy. The spin–orbit term was omit- 3.3 gave the best agreement with the data, slightly ted as it has no effect on the calculated cross section. smaller than the value (4.1 ± 1.3) obtained in the Transfers to unbound states of the “deuteron” were in- CCBA analysis of Ref. [8], but within the quoted un- cluded in addition to that to the deuteron ground state. certainty. The full complex remnant term and non-orthogonality Excellent agreement between the calculated and correction were also included. measured elastic scattering is obtained over the whole There were three adjustable parameters, the real angular range, which was not possible in the optical and imaginary normalizations of the JLM entrance model calculations shown in Fig. 1.Theverylarge channel potential and the spectroscopic factor for the effect of the (p, d) coupling on the elastic scatter- 8He(0+)/7He(3/2−) overlap. All three were adjusted ing is evident. Note that in the full CRC calculation to obtain the optimum simultaneous agreement with the pickup coupling generates a considerable fraction the elastic scattering and transfer data. The normaliza- of the total absorption; only a small component of tion of the (real) JLM spin–orbit potential was con- the JLM imaginary potential is retained (λW = 0.2), strained to be the same as that of the real central po- which may be mostly attributed to compound nucleus tential. effects. For comparison, the no-coupling calculation 86 F. Skaza et al. / Physics Letters B 619 (2005) 82–87

Table 1 Volume integrals per nucleon pair/(MeV fm3), and rms radii/fm of the bare potential (OM) and the potentials found by inversion for the full CRC calculation and for the CRC calculation in which the non-orthogonality term was omitted (NONO) 21/2 21/2 JR r R JI r I JSOR JSOI OM 704.14 3.092 55.37 3.336 26.60 0.005 CRC 653.94 2.938 307.47 4.138 40.27 1.25 NONO 571.28 2.840 252.62 4.360 33.15 6.55

sion procedure Slj → V(r). The inversion is carried out using the iterative-perturbative inversion method of Kukulin and Mackintosh [25] which can give very reliable potentials, including spin–orbit potentials for the spin-half case, for all relevant radii. The bare diagonal proton potential (i.e., without coupling) of the CRC calculation is then subtracted from V(r) and the remainder is identified as the DPP. The result is shown in Fig. 4 for two cases, the solid line being the DPP in the case of the full CRC calculation and the dashed line the DPP from the CRC calculation with the non-orthogonality correction omitted. Previous calculations [3–5,26] omitted the latter, but the qualitative finding that pickup leads to substantial repulsion as well as absorption is confirmed. We Fig. 4. DPP generated by the 8He(p, d) coupling obtained as ex- find that the non-orthogonality correction changes the plained in the text. shape of the real DPP, in particular, so that for a 8He target it is largely in the nuclear center. For this rea- using the bare JLM potential with λV = 1.05, λW = son, the effect on the real central volume integral, as 0.2 is also shown in Fig. 3. presented in Table 1,isjust7%. The agreement between the calculated and mea- Other features of the DPP are a significant imag- sured (p, d) angular distributions is less good, the inary spin–orbit term and an emissive imaginary cen- calculations overpredicting the data for angles greater tral term at the nuclear center. Emissivity at the nuclear than 50◦ in the c.m. system. This is probably due to center often occurs in local representations of a fun- the use of global potentials as a basis for the exit chan- damentally non-local and, in principle, L-dependent nel bare potential and could be improved by tuning the potential [25]. This emissivity and the other character- potential parameters, although we have chosen not to istics of the radial form of the DPP (accounting for the do so to show the quality of agreement that may be better fit to elastic scattering than renormalized JLM obtained with such potentials. potentials), can be traced to the fact that the contribu- The large change in the elastic scattering induced tion of the pickup coupling to the effective nucleon– by the pickup coupling may be represented as a sub- nucleus potential lies outside the scope of what could stantial DPP. To obtain the local and L-independent be described within the framework of folding models representation of this DPP, we followed the procedure based on an underlying local-density approximation. which was used to obtain the DPP for the 6He + p We therefore conclude that the inclusion of pickup system in Ref. [7]. The elastic scattering S-matrix is coupling is essential for a complete understanding of generated by the full CRC calculations (including cou- proton scattering. pling processes), and the total local optical potential The modification of the elastic scattering wave is obtained by subjecting this S-matrix to an inver- functions by the pickup coupling also has important F. Skaza et al. / Physics Letters B 619 (2005) 82–87 87 implications for proton inelastic scattering and the in- References formation that may be drawn therefrom. If one follows the usual conventions and renormalizes the transition potentials by the same factors as the entrance channel [1] I.J. Thompson, et al., Nucl. Phys. A 505 (1989) 84. optical potential, be they of phenomenological form [2] S.G. Cooper, R.S. Mackintosh, Nucl. Phys. A 513 (1990) 373. or calculated microscopically from theoretical tran- [3] R.S. Mackintosh, Nucl. Phys. A 209 (1973) 91. [4] R.S. Mackintosh, Nucl. Phys. A 230 (1974) 195. sition densities, the effect on the level of agreement [5] R.S. Mackintosh, A.A. Ioannides, I.J. Thompson, Phys. Lett. with data will be important. A full investigation of the B 178 (1986) 1. magnitude of this effect for the L = 2 8He(p, p ) tran- [6] V. Lapoux, et al., Phys. Lett. B 517 (2001) 18. sition to the 2+ first excited state is left for a later [7] R.S. Mackintosh, K. Rusek, Phys. Rev. C 67 (2003) 034607. comprehensive article, but test calculations using col- [8] F. Skaza et al., Phys. Rev. C, submitted for publication. [9] A.C. Villari, et al., Nucl. Phys. A 693 (2001) 465. lective model form-factors show a decrease of 14% in [10] Y. Blumenfeld, et al., Nucl. Instrum. Methods A 421 (1999) the nuclear deformation length extracted from a CRC 471. calculation compared to that obtained from a DWBA [11] S. Ottini, et al., Nucl. Instrum. Methods A 431 (1999) 476. calculation. [12] J.-P. Jeukenne, A. Lejeune, C. Mahaux, Phys. Rev. C 16 (1977) To summarize, we have shown that for a particu- 80. 8 (p, d) [13] P. Navrátil, B.R. Barrett, Phys. Rev. C 57 (1998) 3119. lar case, He, the explicit inclusion of coupling [14] J.S. Petler, et al., Phys. Rev. C 32 (1985) 673. has a profound influence on the calculated elastic scat- [15] I.J. Thompson, Comput. Phys. Rep. 7 (1988) 167. tering. Combined with the results of previous studies [16] A.A. Korsheninnikov, et al., Phys. Lett. B 316 (1993) 38. for stable nuclei [3–5,26], we may infer that this ef- [17] L.V. Chulkov, C.A. Bertulani, A.A. Korsheninnikov, Nucl. fect is probably general throughout the chart of the Phys. A 587 (1995) 291. [18] D. Halderson, Phys. Rev. C 70 (2004) 041603. nuclides. Evidently, an investigation of the systematics [19] M. Meister, et al., Phys. Rev. Lett. 88 (2002) 102501. of the effect to determine under what circumstances it [20] A.A. Korsheninnikov, et al., Phys. Rev. Lett. 82 (1999) 3581. is most pronounced would be of great interest. How- [21] N. Keeley, N. Alamanos, V. Lapoux, Phys. Rev. C 69 (2004) ever, it appears necessary to measure the (p, d) re- 064604. action and include this effect in analyses of proton [22] S. Watanabe, Nucl. Phys. 8 (1958) 484. [23] A.J. Koning, J.P. Delaroche, Nucl. Phys. A 713 (2003) 231. scattering for radioactive beams if correct inferences [24] R.V. Reid Jr., Ann. Phys. (N.Y.) 50 (1968) 441. are to be drawn. Through its modification of the elas- [25] V.I. Kukulin, R.S. Mackintosh, J. Phys. G: Nucl. Part. Phys. 30 tic scattering wave function the pickup coupling will (2004) R1. also have an important influence on the calculated in- [26] S.G. Cooper, R.S. Mackintosh, A.A. Ioannides, Nucl. Phys. elastic proton scattering, with all that this implies for A 472 (1987) 101. the extraction of information such as Mn/Mp ratios from data for this process. Physics Letters B 619 (2005) 88–94 www.elsevier.com/locate/physletb

Hindered E4 decay of the 12+ yrast trap in 52Fe

A. Gadea a, S.M. Lenzi b,D.R.Napolia, M. Axiotis a,C.A.Urb,c, G. Martínez-Pinedo d, M. Górska e, E. Roeckl e,E.Caurierf,F.Nowackif, G. de Angelis a,L.Batistg, R. Borcea e,F.Brandolinib,D.Cano-Otth, J. Döring e, C. Fahlander i,E.Farneab, H. Grawe e, M. Hellström i,Z.Janase,j, R. Kirchner e, M. La Commara e, C. Mazzocchi e,k, E. Nácher h, C. Plettner l, A. Płochocki j, B. Rubio h, K. Schmidt e, R. Schwengner l,J.L.Tainh,J.Zylicz˙ j

a Laboratori Nazionali di Legnaro, I-35020 Legnaro, Italy b Dipartimento di Fisica and INFN, I-35100 Padova, Italy c H. Hulubei NIPNE, Bucharest, PO Box MG-6, Romania d University of Aarhus, DK-8000 Aarhus, Denmark e Gesellschaft für Schwerionenforschung, D-64229 Darmstadt, Germany f Institut Recherches Subatomiques, F-67037 Strasbourg cedex 2, France g PNPI, 188-350 Gatchina, Russia h Instituto de Física Corpuscular, E-46071 Valencia, Spain i Lund University, S-22100 Lund, Sweden j University of Warsaw, PL-00681 Warsaw, Poland k Università degli Studi di Milano, I-20133 Milano, Italy l Institut für Kern- und Hadronenphysik, FZ Rossendorf, 01314 Dresden, Germany

Received 31 March 2005; received in revised form 21 May 2005; accepted 26 May 2005

Available online 9 June 2005

Editor: V. Metag

Abstract + + The γ decay of the 12 yrast trap in 52Fe has been measured for the ﬁrst time. The two E4 γ -branches to the 8 states are hindered with respect to other B(E4) reduced transition probabilities measured in the f7/2 shell. The interpretation of the data is given in the full pf shell model framework, comparing the results obtained with different residual interactions. It is shown that measurements of hexadecapole transition probabilities constitute a powerful tool in discriminating the correct conﬁguration of the involved wavefunctions.  2005 Elsevier B.V. All rights reserved.

E-mail address: [email protected] (A. Gadea).

PACS: 21.10.-k; 21.10.Re; 21.60.Cs; 23.20.Lv

Keywords: Nuclear structure; Yrast trap; Shell model calculations

High multipole moments in nuclei are considered one often encounters isomeric states and even inver- to be a vital source of information in nuclear struc- sion of states in the yrast line creating spin traps that ture studies connected with shape phenomena [1].In decay by high multipolarity transitions. particular electric hexadecapole moments and transi- Recently, Ur et al. [9] have studied the high tion strengths are experimentally accessible and pro- spin structure of 52Fe with the γ -ray detector array vide information on features that are independent of GASP [10]. The level scheme of 52Fe has been ex- the quadrupole structure of the nucleus and therefore tended up to the 10+ state at 7.4 MeV excitation contribute to test the theoretical models and, in partic- energy, lying above the yrast 12+ isomer, thereby con- ular, to reduce the degrees of freedom in the effective firming the predicted inversion [11,12] of the yrast nuclear interaction. 10+ and 12+ states. From a β+-decay end-point As pointed out in Refs. [1,2], the calculation of measurement the excitation of the 12+ state was de- E4 transition strength seems to be more sensitive to termined with an accuracy of the order of hundred model details than the E2 transitions and therefore it keV, and the half-life of the isomer was measured has a higher discrimination power when identifying to be 45.9(6) s [12]. The 12+ isomer mainly decays individual components in the nuclear wave function. (99.98%), by Gamow–Teller transitions, into excited In particular, in the sd shell, the B(E2) values for the states of the daughter nucleus 52Mn. transition to the ground state in even–even nuclei do In this Letter we report on the measurement of the not change very much from nucleus to nucleus while E4 γ -decay of the 12+ yrast trap in 52Fe to the two the B(E4) values show drastic changes. Interestingly, known 8+ states. The experiment was performed at the investigation of E4 transitions has allowed to de- the GSI on-line mass separator, where a 2.5 mg/cm2 termine a significant hexadecapole collectivity of 4+ thick natSi target was bombarded by a 170 MeV 36Ar states in closed shell nuclei (A ∼ 132,A∼ 208) (see beam delivered by the UNILAC accelerator. The esti- Ref. [3] for a systematic survey). mate of the cross section, performed with HIVAP [13], Electric hexadecapole moments in stable N = Z for this reaction, gives a population of ≈ 13 mb for nuclei were studied in the past in sd-shell nuclei by 52Fe, above the 12+ isomer. inelastic scattering with several probes (electrons, pro- The recoiling reaction products were stopped in the tons, α-particles) [2,4]. Such experiments cannot be graphite catcher of a FEBIAD-E type ion source [14]. done for heavier N = Z nuclei without resorting to After ionisation and extraction from the ion source, the radioactive beam facilities. With the recent develop- mass separated A = 52 beam was implanted in a tape ments in detection techniques, information of tran- which moved every 80 s, taking away the undesirable sition strengths can be directly obtained by γ -ray long-lived activity. spectroscopy. Whenever γ -transition probabilities are The implantation position was surrounded by a compatible with the detection sensitivity, these mea- plastic scintillator, with a β-detection efficiency of surements give more complete information than that ∼ 85% (measured with a 24Na source), two composite deduced by using scattering techniques. Moreover, germanium (Ge) detectors of the Cluster [15] and large γ -ray spectroscopy allows to measure hexadecapole Clover [16] type, and a 60% single Ge crystal. The transitions from high spin states. setup included a second single crystal low-energy Ge In the past few years, considerable effort has been detector, as shown in Fig. 1, but was not relevant for put into the study of high spin states in f7/2-shell nu- the present analysis. The photopeak efficiency of the clei. It has been shown that near the middle of the shell Ge setup was 3.9% for a γ -ray energy of 1.33 MeV, (48Cr) nuclei present strong deformation [5–8]. Heav- which improved the detection sensitivity limit by a ier nuclei, like 52Fe, are less deformed due to their factor of 100 compared to that achieved in the previ- proximity to the N,Z = 28 shell closure. In such cases ous study [12]. The large segmentation of the detection 90 A. Gadea et al. / Physics Letters B 619 (2005) 88–94 system (12 independent large volume Ge crystals) was of the setup performed with the GEANT3 library [17], essential to keep the summing losses within a reason- the summing perturbation to all measured quantities able limit, i.e., below 10% for large multiplicity cas- was estimated to be far below the respective experi- cades. By using a complete Monte Carlo simulation mental uncertainties. The total measurement time amounted to 32 hours with a production rate of ≈ 4.5×104 atoms/s. β–γ –γ and γ –γ coincidence events were recorded and af- terwards sorted into 3D-cubes and 2D-matrices. The analysis of the γ –γ coincidences, including the “add- back” of the composite detectors and a veto condition derived from the β counter allowed us for the first time to observe the γ de-excitation of the 12+ isomer + + to the 81 and 82 at 6360 and 6493 keV states via E4 transitions of 597 and 465 keV, respectively. The anti- coincidence with the β counter served to reduce the background contribution from β-delayed γ -rays. The resulting spectrum is shown in Fig. 2. Fig. 1. Sketch of the β–γ detection setup at the GSI on-line mass The new transitions fix the excitation energy of the separator. The A = 52 beam is implanted into a tape which is not 12+ isomer at 6957.5(4) keV (see Fig. 3). This value shown. The implantation position is in the center of the β-detector. is significantly more accurate than the previous result Collimation system, tape and β-detector are mounted in a vacuum chamber, while the Ge detectors are positioned around the chamber. deduced from β-decay measurements [12].

Fig. 2. Spectrum obtained in coincidence with strong transitions in 52Fe and in anti-coincidence with the β-counter. In the spectrum there + is a small “leak” of the large β annihilation peak, due to the background subtraction procedure, and a peak coming from the Compton back-scattering of the 1461 keV background transition (1461–850 keV). The gamma transitions belonging to 52Fe are marked by their energies in keV. A. Gadea et al. / Physics Letters B 619 (2005) 88–94 91

detector. The intensity of the β-decay branch has been obtained from the total spectrum without any condition. All the intensity populating the 2+ in 52Fe is expected to go through the two E4 transitions, and since the relative intensities of these two transitions are easily obtained from the γ –γ coincidence matrix, it is possible to evaluate the intensity of each transition compared to the total isomer decay rate. The second method to determine the intensities is based exclusively on the γ –γ coincidences. Considering a 100% intensity for the E2 850 keV transition to the ground state and the measured absolute efﬁciencies of the setup for this and the observed E4 transition, the determination of the intensity of the latter is straightforward. Also in this case the β-decay branch intensity is determined from the total spectrum. Both methods gave the same values. The evaluated intensities reﬂect very low E4 transition probabilities: 1.1(4) e2 fm8 (4.6(17) × 10−4 W.u.) and 8(3) e2 fm8 (3.5(13)×10−3 W.u.) for the 597 keV and 465 keV transitions, respectively. If one compares the 52Fe data with the B(E4) observed in other f7/2-shell nuclei (see Table 1), to obtain the lowest value, corresponding to 52Mn (0.138 W.u.), partial deexcitation branches that are ∼ 300 and ∼ 40 times higher than those observed for the 597.1 keV and 465.0 keV transitions, respectively, would be needed. This explains why these transitions where not ob-

+ served in previous studies [12]. Fig. 3. Level scheme of the 52Fe 12 isomer decay. Transitions + To interpret these results we have performed cal- from the higher lying 10 state at 7381 keV observed in a in-beam study [9] are shown. culations in the shell model framework with the code ANTOINE [18] in the full pf model space. Three different residual interactions have been used, namely the From the spectrum shown in Fig. 2 it is evident that FPD6 [19], the KB3G [20], and the recently intro- the 465 and 597 keV transitions have similar inten- duced GXPF1 [21] interactions. The effective charges sities. Their E4 transition probabilities, however, are used to calculate the B(E4) reduced transition proba- strikingly different due to the strong dependence on bilities are the same as those used to obtain the B(E2) the latter quantities upon the transition energy. The values, i.e., ep = 1.5 and en = 0.5 [9]. A recent mea- + transition intensities per isomer decay have been esti- surement of the 2 → 0+ B(E2) value in 52Fe, using − + 1 mated to be 1.2(4) × 10 4 for the 597 keV (12 → Coulomb excitation techniques [22], is in excellent + × −4 81 ) transition and 0.9(3) 10 for the 465 keV agreement with the calculation performed in Ref. [9]. + → + (12 82 ) transition. These results are based on the The calculated energies and reduced transition combined information of γ –γ coincidence matrices probabilities of the two E4 transitions in 52Fe are with and without β-detector veto. Two methods have confronted with the experimental data in Table 2.All been used to determine the intensities. The ﬁrst one calculations overestimate the experimental values. The consisted on determining the absolute intensity of the best description is achieved by the FPD6 interaction 850 keV 2+ → 0+ transition in 52Fe starting from while both the KB3G and GXPF1 calculations fail in the γ -ray spectrum in anti-coincidence with the β- reproducing even the order of magnitude of the B(E4) 92 A. Gadea et al. / Physics Letters B 619 (2005) 88–94

Table 1 Experimental E4 systematics for f7/2-shell nuclei

Eγ (keV) Ji → Jf T1/2 γ branch B(E4) (W.u.) + + 44Sc 271 6 → 2 58.61 h 0.988 1.42 + + 46Ti 2010 4 → 0 1.62 ps 1.6a + + 52Mn 378 2 → 6 21.1 min 0.0175 0.138 52 + → + × −4 × −4 Fe 597 12 81 45.9 s 1.2(4) 10 4.6(17) 10 + + − − 52Fe 465 12 → 8 45.9 s 9(3) × 10 5 3.5(13) × 10 3 − 2 − 53Fe 701 19/2 → 11/2 2.52 min 0.9866 0.256 + + 54Fe 3578 10 → 6 364 ns 0.019 0.79 + + a The experimental B(E4) value for the 4 → 0 in 46Ti obtained from the γ -intensity measurements reported in Ref. [23] (B(E4) = 400(300) W.u.) is inconsistent with the values expected in the region. This discrepancy is not understood. However, agreement with the expectations is obtained by using the B(E4) value extracted from the hexadecapole deformation measured in Ref. [24] (B(E4) ≈ 1.6W.u.).

Table 2 Experimental and calculated energies and reduced transition probabilities of the two E4 transitions in 52Fe and previously known E4 transitions in f7/2-shell nuclei

Ji → Jf Eγ (keV) B(E4) (W.u.) Exp FPD6 KB3G GXPF1 Exp FPD6 KB3G GXPF1 52 + → + × −4 × −3 × −1 × −2 Fe 12 81 597 1227 907 888 4.6(17) 10 2.4 10 3.3 10 6.5 10 + + − − − − 52Fe 12 → 8 465 519 700 756 3.5(13) × 10 3 4.7 × 10 3 2.6 × 10 2 2.3 × 10 2 + +2 44Sc 6 → 2 271 674 373 281 1.42 1.96 1.79 1.65 + + 46Ti 4 → 0 2010 1966 1819 2000 1.610.77.97.39 + + 52Mn 2 → 6 378 205 91 213 0.138 0.272 0.422 0.728 − − 53Fe 19 → 11 701 990 883 776 0.256 0.151 1.23 0.84 2 + +2 54Fe 10 → 6 3578 3660 3838 3306 0.79 1.80 0.98 1.25 values. Both interactions yield a higher value for the + → + B(E4) 12 81 transition, in contrast with the experimental ﬁndings. It is interesting to see how these interactions reproduce the other B(E4) values known in the f7/2 shell. The results obtained with the three interactions are listed in Table 2 together with the corresponding experimental data. A full pf calculation has been performed for 44Sc, 46Ti and 52Mn, whereas for 53Fe nine of the thirteen valence particles have been allowed to 54 be excited to orbitals above the f7/2 one, and for Fe a truncation to eight of the fourteen valence particles has been made. As shown in Table 2, all interactions reproduce with the same good accuracy the experimental B(E4) data of these nuclei. In Fig. 4, the ratios between the experimental and theoretical reduced transition probabilities are shown Fig. 4. Ratio between experimental and theoretical B(E4) values for nuclei in the f7/2 shell. Results obtained by using the FPD6, GXPF1 for all the measured E4 transitions in the f7/2-shell and KB3G interactions are shown by squares (full line), full circles nuclei. It is evident that both the KB3G and GXPF1 (dashed-line) and triangles (dotted-line), respectively. See Table 2 interactions fail in the case of 52Fe. A possible ori- for details. A. Gadea et al. / Physics Letters B 619 (2005) 88–94 93

+ Table 3 laps are obtained between the yrast 8 state and be- + 1 Proton (or neutron) occupation numbers for the states of interest in tween the yrare 8 state. On the contrary, an overlap of 52Fe 2 + ∼ 0.93 is obtained between the 8 FPD6 state and the + 1 + f7/2 p3/2 f5/2 p1/2 8 KB3G state, and vice versa between the 8 FPD6 2 + 2 FPD6 state and the 8 KB3G state. This inversion of the + + 1 81 4.64 0.50 0.71 0.15 + 8 states could explain the fact that the B(E4) values 82 5.27 0.35 0.32 0.06 + obtained with the KB3G interaction, when compared 121 5.43 0.26 0.26 0.05 with experiment, are inverted in strength. In the case + KB3G of the GXPF1 interaction, the two 8 states are simi- + 8 5.70 0.15 0.21 0.04 lar, which translates in similar overlaps (∼ 0.60–0.70) 1 + + + with the 8 and 8 states of FPD6 and KB3G wave 82 5.02 0.30 0.57 0.10 1 2 + functions. 121 5.63 0.14 0.19 0.03 As mentioned above, the B(E4) values have been GXPF1 + obtained with the effective charges used to reproduce 81 5.20 0.30 0.41 0.08 52 + the quadrupole transition probabilities in Fe and 82 5.37 0.30 0.26 0.06 + N = Z B( ) 12 5.67 0.19 0.12 0.02 neighboring nuclei. As for nuclei, the E4 1 2 transition probability is proportional to (ep + en) , it is the square of the sum of the effective charges gin of this failure could arise from the fact that both which enters as a multiplicative factor. In a very re- 8+ states in 52Fe are very close in excitation energy cent work [25], the polarization charges have been de- (see Table 2), and therefore these calculations could duced from B(E2) values measured for the mirror pair = = = mix the configurations of the two levels or invert their A 51, obtaining ep 1.15 and en 0.8. The use of order. A possible way to check the quality of the inter- these effective charges would not change the present 52 actions in describing the two 8+ states is to calculate results for the B(E4) values in Fe. The need of us- the quadrupole transition probabilities B(E2) for their ing very large or even negative polarization charges to + reproduce the systematics of the B(E4) values in this decay to the 61 state and compare the predictions with the measured values [9]. Unfortunately, all the reduced mass region has been discussed by Yokoyama [26]. transition probabilities are of the same order and the In the latter work, however, shell model calculations n experimental uncertainties do not allow a discrimina- were performed in strongly truncated spaces (f7/2 or n + n−1 1 tion. f7/2 f7/2 (p3/2,p1/2,f5/2) ). Even if it is out of To get a better understanding of the wave functions, the scope of the present study to fit the polarization we report in Table 3 the occupation numbers (protons charges, it is interesting to note that enlarging the and neutrons are equivalent in an N = Z nucleus) of model space has allowed us to reproduce on the same the different orbits for the states involved in the E4 footing all the hexadecapole transition probabilities + decay. While the 12 states have similar occupations known for f7/2-shell nuclei by using the same polar- with any of the interactions, the 8+ states are different. ization charges. + The 81 level obtained with FPD6 is the most collective Finally, the origin of the hindrance of the B(E4) + 52 one, followed by the 82 state obtained with KB3G. values in Fe, can be understood from the hexadeca- To elucidate these discrepancies, we have com- pole strength distribution. Using the different residual puted the wave function overlaps between the 8+ interactions, we have calculated the E4 strength from states and between the 12+ states calculated by us- the 12+ isomer to all the I π = 8+ states in the pf ing different interactions. As expected, the overlap be- shell model space. As expected, the results indicate tween the yrast 12+ wave functions obtained with any that most of the E4 strength is located at excitation en- of the interactions are 0.96, which implies that the ergies higher than the 12+ state. In fact, only up to few 12+ states described by all the Hamiltonians almost per cent (10% for KB3G, 2% for GXPF1 and 0.2% for coincide. When comparing the 8+ states calculated FPD6) of the E4 strength is predicted to feed the first with the FPD6 and KB3G interactions, vanishing over- and second experimentally observed 8+ states. 94 A. Gadea et al. / Physics Letters B 619 (2005) 88–94

In conclusion, the combination of in-beam and off- [7] E. Caurier, et al., Phys. Rev. Lett. 75 (1995) 2466. beam (ISOL) experiments has allowed us for the ﬁrst [8] G. Martínez-Pinedo, et al., Phys. Rev. C 54 (1996) R2150. time to observe the γ -decay of the 52Fe 12+ yrast trap [9] C.A. Ur, et al., Phys. Rev. C 58 (1998) 3163. [10] D. Bazzacco, in: Proceedings of the International Conference and to establish its excitation energy. Two γ -rays of on Nuclear Structure at High Angular Momentum, Ottawa, 597 keV and 465 keV have been assigned to be the E4 AECL Report 10613, vol. II, 1992, p. 376. + + transitions feeding the 81 and 82 states, respectively. [11] D.F. Geesaman, et al., Phys. Rev. Lett. 34 (1975) 326. These two transitions are strongly hindered compared [12] D.F. Geesaman, et al., Phys. Rev. C 19 (1979) 1938. to any other E4 transition in the region. From large [13] W. Reisdorf, et al., Z. Phys. A 343 (1992) 47. [14] R. Kirchner, et al., Nucl. Instrum. Methods 186 (1981) 295. scale shell model calculations performed in the full pf [15] J. Eberth, Prog. Part. Nucl. Phys. 28 (1992) 495; space it has been shown that the hexadecapole transi- J. Eberth, Nucl. Instrum. Methods A 369 (1996) 135. tions can give vital information to distinguish the more [16] J. Gerl, et al., in: Proceedings of the Conference on Physics realistic wave function from those predicted by differ- from Large γ -ray Detector Arrays, Berkeley, LBL 35687, ent interactions, which might be indistinguishable on CONF 940888, UC 413, 1994, p. 159. [17] GEANT—Detector Description and Simulation Tool, CERN the basis of B(E2) measurements. Program Library Writeup W5013. [18] E. Caurier, Code ANTOINE, Strasbourg, 1989; E. Caurier, F. Nowacki, Acta Phys. Pol. 30 (1999) 705. References [19] W.A. Richter, et al., Nucl. Phys. A 523 (1991) 325. [20] A. Poves, E. Caurier, F. Nowacki, Nucl. Phys. A 694 (2001) [1] B.H. Wildenthal, B.A. Brown, I. Sick, Phys. Rev. C 32 (1985) 157. 2185. [21] M. Honma, T. Otsuka, B.A. Brown, T. Mizusaki, Phys. Rev. [2] B.A. Brown, R. Radhi, B.H. Wildenthal, Phys. Rep. 101 (1983) C 65 (2002) 061301(R). 313. [22] K.L. Yurkewicz, et al., Phys. Rev. C 70 (2004) 034301. [3] P.C. Sood, R.K. Sheline, B. Singh, Phys. Rev. C 51 (1995) [23] M. Fujishiro, Y. Satoh, K. Okamoto, T. Tsujimoto, Can. J. 2798. Phys. 58 (1980) 1712. [4] J. Fritze, et al., Phys. Rev. C 43 (1991) 2307. [24] M. Fujiwara, et al., Phys. Rev. C 35 (1987) 1257. [5] S.M. Lenzi, et al., Z. Phys. A 354 (1996) 117. [25] R. du Rietz, et al., Phys. Rev. Lett. 93 (2004) 222501. [6] S.M. Lenzi, et al., Phys. Rev. C 56 (1997) 1313. [26] A. Yokoyama, Phys. Rev. C 55 (1997) 1282. Physics Letters B 619 (2005) 95–104 www.elsevier.com/locate/physletb

Charm production in antiproton–nucleus collisions at the J/ψ and the ψ thresholds

L. Gerland a, L. Frankfurt b, M. Strikman c

a SUBATECH, Laboratoire de Physique Subatomique et des Technologies Associées, University of Nantes, IN2P3/CNRS, Ecole des Mines de Nantes, 4 rue Alfred Kastler, F-44072 Nantes cedex 03, France b School of Physics and Astronomy, Raymond and Beverly Sackler Faculty of Exact Science, Tel Aviv University, Ramat Aviv 69978, Tel Aviv, Israel c Pennsylvania State University, University Park, PA 16802, USA Received 31 January 2005; received in revised form 3 May 2005; accepted 9 May 2005 Available online 16 May 2005 Editor: J.-P. Blaizot

Abstract We discuss the production of charmonium states in antiproton–nucleus collisions at the ψ threshold. It is explained that measurements in pA¯ collisions will allow to get new information about the strengths of the inelastic J/ψN and ψ N interaction, ¯ on the production of Λc and D in charmonium–nucleon interactions and for the ﬁrst time about the nondiagonal transitions ψ N → J/ψN. The inelastic J/ψ-nucleon cross section is extracted from the comparison of hadron–nucleus collisions with hadron–nucleon collisions. We extract the total J/ψ-nucleon cross section from photon–nucleon collisions by accounting for the color transparency phenomenon within the frame of the GVDM (generalized vector meson dominance model). We evaluate within the GVDM the inelastic ψ -nucleon cross section as well as the cross section for the nondiagonal transitions. Predictions for the ratio of J/ψ to ψ yields in antiproton–nucleus scatterings close to the threshold of ψ production for different nuclear targets are presented.  2005 Elsevier B.V. All rights reserved.

1. Introduction tion is played by the value of the total and the elastic cross sections for charmonium–nucleon interactions as well as the amplitude for the inelastic transition be- During the last two decades signiﬁcant attention was given to the absorption of charmonium states pro- tween J/ψ and ψ states characterizing the role of duced in heavy ion collisions, see, e.g., Ref. [1] and color transparency phenomena. The aim of this Letter references therein. An important role in such evalua- is to extract these cross sections from photoproduction data following Ref. [2] and to make predictions for antiproton–nucleus collisions at the ψ threshold. We E-mail address: [email protected] (L. Gerland). demonstrate that in these collisions the cross section

0370-2693/$ – see front matter  2005 Elsevier B.V. All rights reserved. doi:10.1016/j.physletb.2005.05.013 96 L. Gerland et al. / Physics Letters B 619 (2005) 95–104 for the nondiagonal transition ψ + N → J/ψ + N sible enhancement of the charmonium–nucleus cross can be measured. We account for the dependence section near threshold of charmonium–nucleus inter- of the cross sections on energy, and the dependence actions like that described in Ref. [8]. of the elastic cross section on the momentum trans- The amplitude of the J/ψ photoproduction close fer. to the threshold Eγ ∼ 9 GeV is dominated by the gen- The charmonium production at the ψ threshold eralized gluon density at large x1 − x2 ∼ 0.5. In such is well suited to measure the genuine charmonium– a kinematic, Fermi motion effects may lead to a sig- nucleon cross sections. At higher energies formation nificant enhancement. However, similar to the quark time effects makes the measurement of these cross sec- distribution functions one may expect a suppression tions more difficult [3]. These cross sections and the reflecting medium modifications of the nucleon struc- cross section for the nondiagonal transition ψ + N → ture functions (the analogue of the EMC effect). In a J/ψ + N are important for the analysis of charmo- genuine photoproduction experiment it would be very nium production data at SPS-energies [4,5]. At col- difficult to distinguish the EMC type effect from the lider energies, i.e., at RHIC and LHC, the forma- absorption due to the final state interaction. However, tion time effects will become dominant and charmo- the combination of a measurement at the GSI of the nium states will be produced only far outside of the p¯ + A → J/ψ + X reaction and of photoproduction nuclei [6]. However, measurements of the genuine at 12 GeV at the Jefferson-lab will make it possible to charmonium–nucleon cross sections as well as the measure the A dependence of the nuclear generalized cross section for the nondiagonal transition ψ + N → gluon distributions at large x. J/ψ + N are also important at collider energies for The ratio of the elastic to the total J/ψN cross sec- the evaluation of the interaction of charmonium states tion has been evaluated long ago in Ref. [10] in the with the produced secondary particles. vector meson dominance model (VDM), where the en- By using the appropriate incident energy in antipro- ergy dependence and the real part of the forward scat- ton–nucleus scattering, the case when the scattering tering amplitude were neglected and the t dependence occurs off the nucleons with small internal momenta of the elastic cross section was effectively adjusted to can be selected. Correspondingly, in this situation off the data on the soft QCD process of the ρ photopro- shell effects in the amplitude should be very small and, duction off the nucleon target. We will show in this hence, there will be no significant nuclear corrections Letter that all these effects and color transparency phe- due to possible modification of the nucleons in nuclei. nomenon should be taken into account. In addition, due to the large antiproton–nucleon cross In the beginning of this Letter we examine the en- section, those antiprotons which do not undergo ab- ergy dependence of the ratio of the elastic to the total sorption at the surface of the nuclear target will lose a J/ψN cross section as well as the influence of the real significant fraction of their energy. Therefore, at ener- part of the amplitude and treat the t dependence of gies close to the threshold the charmonium production the elastic cross section within a charmonium model. is almost impossible inside of the nuclear target. The final result is that the ratio of the elastic to the to- We discuss in this Letter the production of charm tal J/ψN cross section is still small (approximately at the pp¯ → J/ψ,ψ thresholds. To avoid difficulties 5–6.5%) but by a factor ≈ 2.5 larger than that given in with the specifics of low energy initial state interac- the previous evaluation. tion effects, which are actually included into the partial In Section 2 the amplitudes of the GVDM, the elas- width of the J/ψ → pp¯ decay, like those discussed tic form factor and the two-gluon-form factor are de- in Ref. [7], we will discuss ratios of cross sections, scribed. The amplitudes of the GVDM are described in in which the factor σ(pp¯ → J/ψ,ψ) is canceled. more detail in Appendix A. In Section 3 the semiclas- We demonstrate that these ratios are well suited to sical Glauber model is described and the predictions measure the nondiagonal (ψ → ψ) cross section as for the future GSI experiment are shown. In Section 4 well as the inelastic J/ψ and ψ cross section. At the the results of this Letter are summarized. The phenom- same time the momentum of charmonium in the fi- ena considered in this Letter are complementary to the nal state is 5 GeV/c in the rest frame of the nuclear program of antiproton–nucleus scattering experiments target. Hence, one cannot probe in this reaction a pos- at the GSI range outlined in the recent review [11]. L. Gerland et al. / Physics Letters B 619 (2005) 95–104 97

Fig. 1. The four leading graphs that contribute to the elastic form factor of the J/ψ.

2. Model description and results the two-gluon-form factor [12], which is 1 In Ref. [10] the elastic and the total J/ψ-nucleon F 2 (t) = (6) 2g − t 4 (1 2 ) cross sections were evaluated within the vector me- m2g son dominance model (VDM). In this model, the J/ψ with m2 ≈ 1.1GeV2. photoproduction amplitude fγψ and the J/ψ-nucleon 2g And the two-gluon form factor of the J/ψ, calcu- elastic scattering amplitude fψψ are related as lated as the nonrelativistic limit of the diagrams shown e f = f . (1) in Fig. 1 is γψ f ψψ ψ − 2 dz Ψ(z,kt )∆(kt )Ψ (z, kt zqt )d kt z(1−z) Here, e is the charge of an electron and f is the Fψ (t) = . ψ Ψ(z,k )∆(k )Ψ (z, k )d2k dz J/ψ–γ coupling given by t t t t z(1−z) (7) e2 2 3 Γ(V → ee)¯ Here, Ψ is the wave function of the J/ψ, z is the = . (2) 4πfψ 4π mψ fraction of the longitudinal momentum of the charmonium state carried by the c-quark, while kt is the A similar relation like Eq. (1) can be written also for relative transverse momentum of the c-quark and the the ψ. From the optical theorem c¯-quark. ∆(kt ) is two-dimensional Laplace operator. Iψψ(t = 0) qt is the sum of the momenta of the two gluons. This σ (J/ψN) = , (3) tot 2p E form factor unambiguously follows from the analysis cm cm of Feynman diagrams for hard exclusive processes. By where Iψψ is the imaginary part of fψψ and the dif- deﬁnition it is equal to one at zero momentum trans- ferential elastic cross section fer Fψ (t = 0) = 1. To evaluate this form factor we use here the nonrelativistic wave functions of Ref. [13]. dσel 1 2 = |fψψ| (4) In the gluon exchange between the charmonium dt 64πp2 E2 cm cm and the target only one gluon polarization dominates. follows In QCD evolution only this contribution contains the 2 2 | | + 2 large logarithm ln(mc). Using the QCD Ward iden- σtot(J/ψN) teff (1 η ) σel = . (5) tity one can express the obtained formulae in terms of 16π the exchange by transversely polarized gluons like in |teff| comes from the integration of Eq. (4) over t. η is the derivation of the Weizsäcker–Williams approxima- the ratio of the real part to the imaginary part of the tion. In the nonrelativistic approximation the binding amplitude of J/ψN scattering. The t dependence of is dominated by a Coulomb potential. The Yang– the differential cross section is given by the square of Mills vertex between the Coulomb potential and trans- 98 L. Gerland et al. / Physics Letters B 619 (2005) 95–104

Fig. 2. The form factors squared of the diagonal (J/ψ → J/ψ Fig. 3. The form factors squared of the diagonal (J/ψ → J/ψ and and ψ → ψ ) and the nondiagonal transitions (J/ψ → ψ and ψ → ψ ) for two different nonrelativistic charmonium models. ψ → J/ψ). versely polarized gluons is zero. Therefore, only the glect the last term. Taking into account the elastic interaction between the two gluons and the two heavy form factor of the J/ψ reduces this value to |teff|= quarks of Fig. 1 have to be taken into account in this 0.3GeV2. calculation. Eq. (9) differs from the power law that arises in the The nonrelativistic approximation is justified at limit of large t, i.e., in large angle scattering where small momentum transfer because of the large mass of −t/s ∼ 1/2. In this regime, the selection of domi- the c-quark. Small momentum transfers are the most nant diagrams follows from the requirement to obtain important domain because the two-gluon form factor the lowest power of t. In the literature this is known decreases rather quickly with the momentum transfer. as power counting rules. However, in the processes The result for the elastic J/ψ form factor is shown in considered in this Letter, this integral is dominated − 2 ∼ Fig. 2. Additionally, Fig. 2 depicts the elastic ψ form by ( t)rN 1, where rN is the radius of a nucleon. factor as well as the nondiagonal transition from the This kinematical region does not overlap with high- J/ψ into the ψ.InFig. 3 the dependence of the form momentum transfers. factor on the charmonium model is shown. The elas- Two important phenomena are neglected in the tic form factors of the J/ψ and the ψ calculated in VDM model. One is the color transparency phenom- two different charmonium models are depicted. The enon due to production of cc¯ in configurations sub- charmonium models are from Refs. [13,14]. One can stantially smaller than the mean J/ψ size. As a result see that the dependence on the charmonium model is the effective cross section σtot(J/ψN) as extracted small in comparison to other uncertainties. The elastic from the J/ψ photoproduction off a nucleon is much cross section is then proportional to smaller than the genuine cross section of the J/ψN interaction. Another neglected effect is the hard con- | |= 2 2 teff dt F2g(t)Fψ (t). (8) tribution to σtot which rapidly increases with energy [9]. Therefore, we use the correspondence between the Integrating the two-gluon form factor of (J/ψ → GVDM and the QCD dipole model which leads to the J/ψ) over t yields parametrization of cross section see Ref. [2] s + m2 −3 2 1 2 2g dt F (t) = m 1 − 0.08 0.2 2g 3 2g m2 s s 2g σ (J/ψN) = 3.2mb + 0.3mb tot s s ≈ 0.4GeV2. (9) 0 0 (10) The approximation at the end of this equation is for 2 sufficiently high energies, where it is possible to ne- with s0 = 39.9GeV . L. Gerland et al. / Physics Letters B 619 (2005) 95–104 99

It is worth noting here that such a parametrization is cross section and the nondiagonal cross section (ψ + reasonable only for the energies where inelastic non- N → J/ψ + N). One can see that the nondiagonal diffractive channels (the lowest√ nondiffractive channel cross section (ψ + N → J/ψ + N) is comparable ¯ is Λc + D) are open, that is for s>4.15 GeV, in the with the elastic J/ψ-nucleon cross section. rest system of the nucleon this is ω>3.61 GeV. The amplitudes within the GVDM are related by e e fγψ = fψψ + fψψ , fψ fψ e e fγψ = fψψ + fψψ . (11) fψ fψ

The amplitudes, fψψ and fψψ , that appear here additionally in comparison to the VDM in Eq. (1) are the amplitudes for the nondiagonal transitions J/ψ → ψ and ψ → J/ψ, respectively. The amplitudes following from Eqs. (10) and (11) are given in Appendix A. The results of Eqs. (5) and (10) (the total and the elastic cross section for J/ψN collisions) are shown in Fig. 4. Fig. 5 shows the same for ψN collisions. The ratio of the elastic to the total cross section is depicted in Fig. 6. The elastic cross section calculated with and without the real part of the amplitude is shown in Fig. 7. The real part contributes approximately 2% to the elas- Fig. 5. The elastic and the total ψ -nucleon cross section in depen- tic cross section in the discussed energy range. dence of the energy of the ψ in the rest frame of the nucleon. Fig. 8 shows the energy dependence of the elastic J/ψ-nucleon cross section, the elastic ψ-nucleon

Fig. 6. The ratio of the elastic charmonium–nucleon cross section Fig. 4. The elastic and the total J/ψ-nucleon cross section in de- to the total charmonium–nucleon cross section in dependence of the pendence of the energy of the J/ψ in the rest frame of the nucleon. energy of the charmonium in the rest frame of the nucleon. 100 L. Gerland et al. / Physics Letters B 619 (2005) 95–104

tion of charmonium states in the antiproton–nucleus collisions at the ψ threshold. The direct production of J/ψ’s is suppressed here. However, a ψ is produced and becomes an J/ψ in a further collision with a nucleon in the nuclear target. Since the produced hidden charm state has a large momentum relative to the nucleus target the semiclassical Glauber- approximation can be used. In our calculation we will neglect color transparency effects in the initial state for the production of J/ψ and ψ mesons [18], since the coherence length for the ﬂuctuation of the incoming antiproton into a small conﬁguration is very small at the relevant energies practically completely washing out the CT effect [19]. The production of a J/ψ at the threshold in a pA¯ collision and the subsequent production of a ψ in a rescattering of the J/ψ, is not well suited for the mea- Fig. 7. The elastic J/ψ-nucleon cross section in dependence of the surements of the nondiagonal√ cross sections. This is energy of the J/ψ in the rest frame of the nucleon is shown with because in a pA¯ collision at s = mψ = 3.1GeV and without the real part of the amplitude. the J/ψ is produced at rest in the center of mass system. This means the energy in the center of mass of the J/ψ and the nucleon is 4.5 GeV. The threshold for the production of a ψ in such a collision is mψ + mN = 4.626 GeV. This boundary is extended when nucleon Fermi motion within the nuclear target is taken into account (see Ref. [19] for the discussion of the role of Fermi motion effects in the production of charmonium states). However, this process is strongly suppressed by the phase space. At the same time, the process J/ψ+p → ¯ Λc + D is likely to dominate the inelastic cross section. Hence the measurement of the process p¯ + A → + ¯ + = 2 Λc D X in the vicinity of spp¯ mJ/ψ will allow a direct measurement of σinel(J/ψN). In the semiclassical Glauber-approximation the cross section to produce a ψ in an antiproton–nucleus collision is σ(p¯ + A → ψ ) Fig. 8. The elastic J/ψ-nucleon cross section, the elastic n ψ -nucleon cross section and the nondiagonal cross section = · p ¯ + → 2π db bdz1 ρ(b,z1)σ (p p ψ ) (ψ + N → J/ψ + N) in dependence of the energy of the char- A monium in the rest frame of the nucleon. z1

× exp − dzσpN¯ inelρ(b,z) ¯ 3. pA collisions at the ψ and the J/ψ threshold −∞ ∞ A program of studies of charmonium production × − dzσ ρ(b,z) . in a pA¯ collisions at a p¯ accumulator is planned exp ψ Ninel (12) [15]. Hence we discuss in this section the produc- z1 L. Gerland et al. / Physics Letters B 619 (2005) 95–104 101

In this formula, b is the impact parameter of the antiproton–nucleus collision, np is the number of protons in the nuclear target, z1 is the coordinate of the production point of the ψ in beam direction, and ρ is the nuclear density. σ(p¯ + p → ψ) is the cross section to produce a ψ in an antiproton–proton collision. σpN¯ inel is the inelastic antiproton–nucleus cross section. σψNinel is the inelastic ψ -nucleon cross section. All the factors in Eq. (12) have a rather direct interpretation z1 exp − dzσpN¯ inelρ(b,z) −∞ gives the probability to ﬁnd an antiproton at the coor- Fig. 9. The ratio σ(p¯ + A → ψ + nuclear fragments)/σ (p¯ + A → ψ + nuclear fragments) is shown for 5 different sets of parameters dinates (b, z1), which accounts for its absorption, and np ρ(b,z )σ (p¯ + p → ψ) is the probability to create (see text for further details). Shown are the nuclear targets O, S, Cu, A 1 W, and Pb. The lines are just to guide the eye. np a ψ at these coordinates. The factor A accounts for the fact that close to the threshold the antiproton can The factor produce a ψ only in an annihilation with a proton but z not with a neutron. The term 2 exp − dzσ ρ(b,z) ∞ ψ Ninel z1 exp − dzσψinelρ(b,z) is the probability to ﬁnd, at the coordinate (b, z2), z1 a ψ that was produced at (b, z1). σ(ψ + N → ψ + gives the probability that the produced ψ has no in- N)ρ(b,z2) is the probability that a ψ collides at the elastic collision in the nucleus, i.e., that it survives on coordinate (b, z2) with a nucleon and that a J/ψ is the way out of the nucleus. produced. Finally, Then, in the semiclassical Glauber-approximation ∞ J/ψ the cross section to subsequently produce a in an − antiproton–nucleus collision is exp dzσψinelρ(b,z) z2 ¯ + → + σ(p A J/ψ X) is the probability that the outgoing J/ψ has no in- np elastic interactions. θ(z2 − z1) = 0forz2 z1, this takes into account that z1 a ψ has to be produced before it can collide again. In Fig. 9 the ratio of the production cross section × σ(p¯ + p → ψ ) exp − dzσpN¯ inelρ(b,z) of the J/ψ of Eq. (13) to that of the ψ Eq. (12) ver- −∞ sus the size of the nuclear target A is plotted. Shown z 2 are the nuclear targets O, S, Cu, W, and Pb. The den- × − exp dzσψ Ninelρ(b,z) sity distributions are from Ref. [16]. In contrast to

z1 Eqs. (12) and (13), the ratio does not depend on the cross section σ(p¯ + p → J/ψ+ X), which is not well × σ(ψ + N → ψ + N)ρ(b,z2) ∞ known at the threshold. In Fig. 9 we used ﬁve sets of parameters. “Normal” × − dzσ ρ(b,z) . exp ψNinel (13) means that the inelastic antiproton–nucleon cross sec- z 2 tion is σpN¯ inel = 50 mb, the inelastic cross section of 102 L. Gerland et al. / Physics Letters B 619 (2005) 95–104

the ψ is σψNinel = 7.5 mb, the inelastic cross section rapidity, with transverse momentum transfer the shift of the J/ψ is σψNinel = 0 mb, and the cross section would be larger. In the center of mass system this is a for the nondiagonal transition ψ + N → J/ψ + N is shift of pcms = 0.6 GeV. In the laboratory system a σψ+N→ψ+N = 0.2 mb. The other sets differ by only J/ψ produced at ycms = 0 has momentum of 6.1 GeV one of these parameters each: the J/ψ’s produced in nondiagonal transitions have an average momentum of at most 4.2 GeV. • in “ψ absorption” σψNinel = 3.1mb; At the J/ψ threshold the only possibility for an in- • in “large ψ absorption” σψNinel = 15 mb; elastic interaction of the produced J/ψ is the channel • in “small nondiagonal” σψN→ψN = 0.1mb; J/ψ + N → Λc + D.Attheψ threshold next to the ¯ • in “large nondiagonal” σψN→ψN = 0.4mb. channel ψ + N → Λc + D, ψ + N → N + D + D is also kinematically allowed. However, in the strange One can see that the result depends much more sector the channel φ + N → N + K + K¯ is strongly strongly on the nondiagonal cross section than on suppressed versus the channel φ +N → Λ+K, there- the absorption cross sections of the J/ψ and the ψ . fore it is very likely that the Λc +D channel dominates Therefore, this process is well suited to measure the at both energies, the J/ψ threshold and the ψ thresh- nondiagonal cross section. However, it will be nec- old. essary to differ between J/ψ’s produced in rescat- The cross section for the production of ψ that does terings of ψ and those which come from the decays not undergo an inelastic rescattering is σ(p¯ + A → of the ψ.1 However, J/ψ’s produced in ψ decays ψ + nuclear fragments) is given by Eq. (12).The have longitudinal momenta which differ strongly from cross section for the production of ψ, whether they those produced in the two step processes and hence have subsequent inelastic scatterings or not is given could be easily separated. by In antiproton–nucleus collisions at the ψ thresh- σ(p¯ + A → ψ ) old, predominantly ψ mesons are produced, i.e., the w/oinel ¯ → direct (pp J/ψ) production of J/ψ mesons is np = 2π db · bdz ρ(b,z )σ (p¯ + p → ψ ) strongly suppressed since this would require huge 1 A 1 Fermi momenta of the nucleons. Therefore, there are z1 two sources of J/ψ production in this case. One × exp − dzσ ¯ ρ(b,z) . (14) source the nondiagonal transitions where a produced pinel ψ interacts in the final state via ψN → J/ψN.The −∞ → + other source is the decay of ψ J/ψ X, where Assuming that the Λc channel is the only possible fi- X could be two pions or other hadrons. Because the nal state in inelastic collisions (i.e., the DD¯ channel as ψ’s decay outside of the nucleus, the second reaction well as the nondiagonal transition is neglected as a cor- can be eliminated experimentally. The momenta of the rection here), the fraction of the initially produced ψ J/ψ’s in these two mechanisms are quite different. that ends up in the Λc channel is In particular, in the second mechanism ycms ∼ 0 and the distribution is symmetric around y = 0, while in NΛc the nondiagonal mechanism there is a shift to rapidi- Nψinitial ties closer to yA. For a rescattering without transfer ¯ + → + = − σ(p A ψ nuclear fragments) of transverse momentum, this is a shift of 0.2 units of 1 . σ(p¯ + A → ψ + nuclear fragments)w/oinel (15) 1 The experiments also have to be able to detect radiative decays, Here we neglected the final state interactions of Λc because the ψ decays with a probability of 8.4% and 6.4% respec- as they may only effect the momentum distribution tively into a gamma and a χ and a χ respectively, which decay c1 c2 of Λc since the Λc energy is below the threshold for with a probability of 31.6% and 20.2% respectively into a gamma + → + + and a J/ψ [17]. In total this gives 8.4% · 31.6% + 6.4% · 20.2% = the process p Λc N N D. For this reac- 3.9%, which is of the same order of magnitude as the nondiagonal tion the Λc would need an energy of 4.2 GeV in the transitions. rest frame of the proton, while it has in average less L. Gerland et al. / Physics Letters B 619 (2005) 95–104 103

This result is in agreement with naive expectations. The ratio of the elastic cross section to the total cross section known for the pion or the proton projectiles is approximately 25%. The total J/ψ-nucleon cross section is signiﬁcantly smaller than the total π-nucleon cross section, because the J/ψ has a much smaller size than the π (the radius of the J/ψ is a factor 4 smaller the one of the π). Therefore the interaction of the J/ψ with a nucleon is much weaker than the π-nucleon interaction due to a stronger screening of color charge of the constituents. That is also why the J/ψN interaction is further from the black disk limit than the π. Measurements in pA¯ collisions will allow one to get new information about the strength of the inelastic J/ψN and ψN interactions leading to the Fig. 10. The ratio of the number of Λc divided by the number of ¯ produced J/ψ and ψ respectively states at the threshold of J/ψ production of Λc and D and for the ﬁrst time about and ψ production, respectively. Shown are the nuclear targets O, S, nondiagonal transitions, ψ N → J/ψN. Cu, W, and Pb. The lines are just to guide the eye. than 3 GeV. The change of the momentum distribu- Acknowledgements tion of Λc would provide unique information about the ΛcN interaction and could be a promising method We thank Ted Rogers for discussions. This work for forming charmed hypernuclei. Obviously Eq. (15) was supported in part by BSF and DOE grants. L.G. is valid also for D¯ production. The fraction for the ψ thanks the School of Physics and Astronomy of the and the J/ψ threshold is depicted in Fig. 10. Note that Raymond and Beverly Sackler Faculty of Exact Sci- in Eq. (15) and Fig. 10 the denominator is the number ence of the Tel Aviv University for support and hospi- of produced particles, while in Fig. 9 the denominator tality. is the number of ψ that do not undergo subsequent inelastic scatterings only. Appendix A. Amplitudes from the GVDM

4. Conclusion The imaginary part of the amplitude of the forward scattering of a J/ψ on a nucleon is given by the cross We found that accounting for color transparency section of Eq. (10) and the optical theorem phenomena within the GVDM leads to a ratio of the √ 0.08 0.2 s s −2 elastic J/ψ nucleon cross section to the total J/ψ Iψψ = 2 spcm 8 + 0.75 GeV . nucleon cross section of approximately 5–6.5%. This s0 s0 value is signiﬁcantly larger than that which follows (A.1) from the analysis based on VDM cf. [10], where 2% The real part of the amplitude can be evaluated with was found. Note that in Ref. [10] |t |=1/6GeV2 has the help of the Gribov–Migdal relation eff √ been taken from soft QCD process of the photopro- spcmπ ∂ Iψψ duction of ρ mesons, while the value following from Rψψ = √ . (A.2) 2 ∂ ln s sp the two-gluon form factor extracted from the photo- cm production of J/ψ mesons within the framework of This yields √ the QCD factorization theorem and the form factor of Rψψ = π spcm | |= 2 the J/ψ is closer to teff 0.4GeV. Also, the to- 0.08 0.2 s s − tal J/ψ-nucleon cross section ﬁtted with the VDM to × 0.64 + 0.15 GeV 2. the data is approximately a factor two smaller than the s0 s0 value obtained within the GVDM. (A.3) 104 L. Gerland et al. / Physics Letters B 619 (2005) 95–104

The nondiagonal amplitudes and the amplitudes for References the ψ is given in the GVDM by

e e [1] R. Vogt, Phys. Rep. 310 (1999) 197. fγψ = fψψ + fψψ , fψ fψ [2] L. Frankfurt, L. Gerland, M. Strikman, M. Zhalov, Phys. Rev. e e C 68 (2003) 044602. fγψ = fψψ + fψψ , (A.4) [3] L. Gerland, L. Frankfurt, M. Strikman, H. Stöcker, W. Greiner, fψ fψ Phys. Rev. Lett. 81 (1998) 762, nucl-th/9803034; e is the charge of an electron and f is the J/ψ–γ L. Gerland, L. Frankfurt, M. Strikman, H. Stöcker, W. Greiner, ψ Nucl. Phys. A 663 (2000) 1019, nucl-th/9908052; coupling. fγψ is the J/ψ photoproduction amplitude L. Gerland, L. Frankfurt, M. Strikman, H. Stöcker, Phys. Rev. and fψψ the J/ψ-nucleon elastic scattering ampli- C 69 (2004) 014904, nucl-th/0307064. tude. The amplitudes fψψ and fψψ are the ampli- [4] C. Spieles, R. Vogt, L. Gerland, S.A. Bass, M. Bleicher, tudes for the nondiagonal transitions J/ψ → ψ and H. Stöcker, W. Greiner, Phys. Rev. C 60 (1999) 054901, hep- ψ → J/ψ, respectively. ph/9902337. [5] E.L. Bratkovskaya, A.P. Kostyuk, W. Cassing, H. Stöcker, We assume that the real part of the amplitudes fγψ Phys. Rev. C 69 (2004) 054903, nucl-th/0402042. and fγψ can be neglected. This is realistic, because [6] L. Gerland, L. Frankfurt, M. Strikman, H. Stöcker, W. Greiner, fγψ fψψ. Note that fψψ = fψψ , because of the J. Phys. G 27 (2001) 695, nucl-th/0009008. CPT invariance of the amplitude. Then real parts of the [7] S.J. Brodsky, A.H. Hoang, J.H. Kuhn, T. Teubner, Phys. Lett. two other amplitudes are B 359 (1995) 355, hep-ph/9508274. [8] S.J. Brodsky, I.A. Schmidt, G.F. de Teramond, Phys. Rev. fψ Lett. 64 (1990) 1011. Rψψ =− Rψψ =−1.7Rψψ, [9] L.L. Frankfurt, M.I. Strikman, Phys. Rep. 160 (1988) 235. fψ [10] B.L. Ioffe, L.B. Okun, V.I. Zakharov, Sov. Phys. JETP 41 fψ (1975) 819, Zh. Eksp. Teor. Fiz. 68 (1975) 1635. =− =− = Rψ ψ Rψ ψ 1.7Rψ ψ 2.9Rψψ. (A.5) [11] S.J. Brodsky, hep-ph/0411046. f ψ [12] L. Frankfurt, M. Strikman, Phys. Rev. D 66 (2002) 031502, For their imaginary parts follows hep-ph/0205223. [13] E. Eichten, K. Gottfried, T. Kinoshita, K.D. Lane, T.M. Yan, 0.2 fψ fψ √ s Phys. Rev. D 21 (1980) 203. Iψψ = Iγψ − Iψψ = 2 spcm · 136.5 [14] W. Buchmüller, S.H.H. Tye, Phys. Rev. D 24 (1981) 132. e f 1.2 ψ s0 [15] PANDA Collaboration, H. Koch, Nucl. Instrum. Methods − 1.7Iψψ, B 214 (2004) 50. [16] C.W. De Jager, H. De Vries, C. De Vries, At. Data Nucl. Data fψ fψ fψ Iψψ = Iγψ − Iψψ = Iγψ − 1.7Iψψ Tables 14 (1974) 479. e fψ e [17] Particle Data Group Collaboration, S. Eidelman, et al., Phys. Lett. B 592 (2004) 1. fψ fψ = Iγψ − 1.7 Iγψ + 2.9Iψψ [18] S.J. Brodsky, A.H. Mueller, Phys. Lett. B 206 (1988) 685. e e [19] G.R. Farrar, L.L. Frankfurt, M.I. Strikman, H. Liu, Nucl. Phys. √ s0.2 B 345 (1990) 125. = sp · . + . I . 2 cm 175 5 1.2 2 9 ψψ (A.6) s0 Physics Letters B 619 (2005) 105–114 www.elsevier.com/locate/physletb

Mass and decay constant of I = 1/2 scalar meson in QCD sum rule

Dong-Sheng Du a,b, Jing-Wu Li b, Mao-Zhi Yang a,b

a CCAST (World Laboratory), PO Box 8730, Beijing 100080, China b Institute of High Energy Physics, Chinese Academy of Sciences, PO Box 918(4), Beijing 100049, China 1 Received 22 September 2004; received in revised form 5 May 2005; accepted 13 May 2005 Available online 31 May 2005 Editor: T. Yanagida

Abstract We calculate the mass and decay constant of I = 1/2 scalar mesons composed of quark–antiquark pairs based on QCD sum rule. The quark–antiquark pairs can be sq¯ or qs¯ (q = u, d) in quark model, the quantum numbers of spin and orbital angular momentum are S = 1, L = 1. We obtain the mass of the ground sate in this channel is 1.410 ± 0.049 GeV. This result favors that ∗ ¯ ¯ ¯ K0 (1430) is the lowest scalar state of sq or qs. We also predict the ﬁrst excited scalar resonance of sq is larger than 2.0 GeV.  2005 Elsevier B.V. All rights reserved.

1. Introduction

Glueball and scalar mesons should exist according to QCD and quark model. Some scalar mesons below 2 GeV have been observed, such as, (i) for isospin I = 0, 1 states: f0(600) or σ , a0(980), f0(980), f0(1370), f0(1500), = ∗ f0(1710); (ii) for I 1/2 states: κ(900) and K0 (1430) [1–4]. The number of these scalar mesons exceeds the particle states which can be accommodated in one nonet in the quark model. It is believed that there are two nonets below and above 1 GeV [5,6]. The components of the meson states in each nonet have not been completely determined yet. For the scalar mesons below 1 GeV there are several interpretations. They are interpreted as meson– meson molecular states [7] or multi-quark states qqq¯q¯ [8], etc. However, from the theoretical point of view there must be quark–antiquark SU(3) scalar nonet. Therefore it is important to determine the masses of the ground states of qq¯ with quantum number J P = 0+ based on QCD. For isospin I = 0, 1 states different quark ﬂavor may mix, and scalar qq¯ states may also mix with scalar glueball if they have the same quantum number of J PC and similar masses [9–14]. Some authors have tried to determine the mixing angles of the glueball with qq¯ scalar mesons by using decay patterns of some scalar mesons [14–17]. These works imply that glueball possibly mix with qq¯

E-mail addresses: [email protected] (D.-S. Du), [email protected] (J.-W. Li), [email protected] (M.-Z. Yang). 1 Mailing address.

0370-2693/$ – see front matter  2005 Elsevier B.V. All rights reserved. doi:10.1016/j.physletb.2005.05.043 106 D.-S. Du et al. / Physics Letters B 619 (2005) 105–114 scalar mesons. For I = 1/2 states, they cannot mix with glueball because they have strange quantum number. The physical state is directly the sq¯ and qs¯ bound state. Therefore the mass of the ground state of sq¯ or qs¯ can be determined without necessity for considering mixing effect. QCD sum rule is a powerful tool to calculate hadronic nonpertubative parameters based on QCD [18]. It has been used to calculate the masses and decay constants of 0−+, 1−+, 2++ mesons before and give satisfactory results [18–21]. In this Letter, we calculate the mass and decay constant of I = 1/2 scalar meson with QCD sum rule. We find that it is impossible to obtain sq¯ scalar meson mass below 1 GeV from QCD sum rule. The most favorable result for the mass of sq¯ scalar meson is 1.410 ± 0.049 GeV. Therefore, if κ(900) is sq¯ scalar bound state, this would be a big problem for QCD. This problem can be solved by assuming that κ(900) is irrelevant to sq¯ scalar channel, 0|¯sq|κ(900)∼0, and K∗(1430) is the scalar ground state of sq¯ or qs¯. With this assumption, calculation based 0 ∗ on QCD will be consistent with experiment. Therefore, our result favors that K0 (1430) is the lowest scalar bound state of sq¯. If this is correct, then from the approximate SU(3) flavor symmetry, the masses of the other J P = 0+ mesons in the scalar nonet should be slightly above or below 1.4 GeV. This result would imply that scalars with masses below 1 GeV are not dominated by quark–antiquark pairs. This is consistent with the calculation of lattice QCD which implies that a nonet of quark–antiquark scalars is in the range 1.2–1.6 GeV [22]. The remaining part of this Letter is organized as follows. In Section 2, we briefly introduce the process to calculate the scalar meson with QCD sum rule and get the Wilson coefficients for the corresponding two-point scalar current correlation function. Section 3 is devoted to numerical analysis and conclusion.

2. The method

To calculate the mass of scalar sq¯ or qs¯ meson, the two-point correlation function should be taken as · + Π q2 = i d4xeiq x0|T j(x)j (0) |0, (1) where j(x)=¯s(x)q(x), j +(0) =¯q(0)s(0). On one hand, the correlation function can be expressed based on the dispersion relation in terms of hadron states 1 dsIˆ Π(s) Π h q2 = m , (2) π s − q2 ˆ where ImΠ(s) is the imaginary part of the two-point correlation function, which can be obtained by inserting a complete set of quantum states |nn| into Eq. (1). The result is ˆ = − 2 | | | + | 2ImΠ(s) 2πδ s mn 0 j(0) n n j (0) 0 . (3) n

For the scalar states S, its decay constant fS can be deﬁned through

0|j(0)|S=mSfS, (4) where mS is the mass of the scalar state. Based on Eqs. (2)–(4), and explicitly separating out the lowest scalar state, the correlation function can be expressed as ∞ m2 f 2 1 dsρh(s) Π h q2 = S S + , (5) 2 − 2 − 2 mS q π s q s0 D.-S. Du et al. / Physics Letters B 619 (2005) 105–114 107 where ρh(s) expresses the contribution of higher resonances and continuum state, s0 is the threshold of higher resonances and continuum state. On the other hand, the correlation function can be expanded in terms of operator-product expansion at large negative value of q2. · + Π QCD q2 = i d4xeiq x0|T j(x)j (0) |0

= + | ¯ | + | a aαβ| + | ¯ a aαβ | C0I C3 0 ΨΨ 0 C4 0 Gαβ G 0 C5 0 Ψσαβ T G Ψ 0 ¯ ¯ + C60|ΨΓΨΨΓ Ψ |0+···, (6) ¯ where Ci , i = 0, 3, 4, 5, 6,...are Wilson coefficients, I is the unit operator, ΨΨ is the local fermion field operator a of light quarks, Gαβ is gluon strength tensor, Γ and Γ are the matrices appearing in the procedure of calculating the Wilson coefficients. For convenience later, we reexpress the above equation as 1 dsρpert Π QCD q2 = + ρnonp + ρnonp + ρnonp + ρnonp +···, (7) π s − q2 3 4 5 6 nonp nonp where ρ3 ,...,ρ6 ,...are contributions of condensates of dimension 3, 4, 5, 6, ...inEq.(6). Matching Π h(q2) with Π QCD(q2) we can get the equation which relates mass of scalar meson with QCD parameters and a few condensate parameters. In order to suppress the contribution of higher resonances and that of condensate terms, we make Borel transformation over q2 in both sides of the equation, the Borel transformation is defined as − 2 n n ˆ 2 ( q ) ∂ 2 B| 2 2 f q = lim f q . p ,M n→∞ (n − 1)! ∂(q2)n q2→−∞ −q2/n=M2 After assuming quark–hadron duality, i.e., by assuming that the contribution of higher resonance and continuum states can be approximately canceled by the perturbative integration over the threshold s0 [23], the resulted sum rules for the mass and decay constant of the scalar meson are R1 mS = , (8) R2

1 m2 /M2 fS = e S R2, (9) mS where

s0 2 ˆ nonp 2 ˆ nonp 1 − 2 ∂(M Bρ ) ∂(M Bρ ) R = dssρpert(s)e s/M + M4 3 + M4 4 1 π ∂M2 ∂M2 2 (m1+m2)

∂(M2Bρˆ nonp) ∂(M2Bρˆ nonp) + M4 5 + M4 6 , (10) ∂M2 ∂M2 s0 1 − 2 R = dsρpert(s)e s/M + M2Bρˆ nonp + M2Bρˆ nonp + M2Bρˆ nonp + M2Bρˆ nonp, (11) 2 π 3 4 5 6 2 (m1+m2) 108 D.-S. Du et al. / Physics Letters B 619 (2005) 105–114

Fig. 1. Diagrams for the contribution to Wilson coefﬁcients. (a) diagrams contribute to unit operator; (b) diagrams contribute to bi-quark ¯ a aµν ¯ a operators Ψ(x)Ψ(0); (c) diagrams contribute to Gµν G ; (d) diagrams contribute to quark–gluon mixing Ψ(x)Ψ(0)Gµν ; (e) diagrams contribute to four-quark operators ΨΨ¯ 2. ˆ nonp nonp where Bρi express Borel transformation of ρi , M is Borel parameter, and m1 and m2 are the masses of the two light quarks. We need to calculate the Wilson coefﬁcients in Eq. (6) to get the mass and decay constant of scalar meson. ˆ nonp Collecting the contribution of diagrams in Fig. 1, we get the result of Bρi which is listed in Appendix A.

3. Numerical analysis and conclusion

The numerical parameters used in this Letter are taken as [18,21] ¯ =− ± 3 ¯ = 2¯ qq (0.24 0.01 GeV) , ss m0 qq , D.-S. Du et al. / Physics Letters B 619 (2005) 105–114 109

= 4 ¯ = 2 ¯ αs GG 0.038 GeV ,gΨσTΨ m0 ΨΨ , ¯ 2 = × −5 6 2 = ± 2 αs ΨΨ 6.0 10 GeV ,m0 0.8 0.2GeV , ms = 0.14 GeV,mu ≈ md = 0.005 GeV. (12) 2 2 2 For the choice of Borel parameter M ,asin[18,24], we define fthcorr(M ) as m(M ) in Eq. (8) without the 0 2 2 continuum contribution (s =∞) and mnopower(M ) as m(M ) in Eq. (8) without power corrections, then define 2 2 2 2 2 fnopower(M ) as m(M )/mnopower(M ) and fcont as m(M )/fthcorr(M ). To get reliable prediction of the mass in QCD sum rule, fcont should be limited to above 90% to suppress the contribution of higher resonance and contin- 2 uum, and fnopower(M ) be limited to less than 10% deviation from 1, which can ensure condensate contribution much less than perturbative contribution. There are two low mass scalar meson states with isospin I = 1/2 and strange number |S|=1 found in exper- ∼ ∗ ∗ = iment. They are κ(900) with mass mκ about 800 900 MeV [2–4], and K0 (1430) with mass m(K0 (1430)) 1.412 ± 0.006 GeV [1]. In theory, taking appropriate value for the threshold parameter s0, one can separate out the contribution of the lowest resonance in QCD sum rule. We vary the value of the threshold parameter s0, and find that it is impossible to obtain the mass of κ(900) with the sum rule in Eq. (8). There is no stable ‘window’ for the Borel parameter in this mass region. Therefore, if κ(900) is the lowest scalar state in the sq¯ channel, it would be a big problem for QCD sum rule. However, if we increase the value of s0, i.e., for s0 = 4.0–4.8GeV2, we does find the stable ‘window’ for Borel parameter, which is shown in Fig. 2. The resulted stable window is in the range 1.0

Fig. 2. (a) The region between the arrow A and B is reliable for determining the mass (s0 = 4.4GeV2). (b) The curves correspond to the mass of scalar sd¯ meson for the continuum threshold s0 = 4.0GeV2, s0 = 4.4GeV2,4.8GeV2, respectively. The central one is for s0 = 4.4GeV2. 110 D.-S. Du et al. / Physics Letters B 619 (2005) 105–114

Fig. 3. The possible mass result by varying the value of the threshold s0. The solid curve is for the Borel parameter M2 = 1.0GeV2,andthe dashed one for M2 = 1.2GeV2. is canceled between the numerator and denominator of Eq. (8). We use 2.2% to estimate the uncertainty caused by the higher order αs corrections. On one hand, it is impossible to obtain the mass of lower scalar state κ(900) fromQCDsumruleforsq¯ channel. If regard κ(900) as sq¯ scalar bound state, it would be a big problem for QCD. On the other hand, QCD sum rule ∗ can give most favorable mass which is consistent with the mass of K0 (1430). Therefore it is acceptable to assume that κ(900) is irrelevant to sq¯ scalar bound state, and

0|¯sq κ(900) ∼ 0. (14) ∗ ¯ With this assumption, K0 (1430) can be accepted as the lowest scalar bound state of sq. Then there will be no problem between QCD and experiment. One may still be afraid that there are contributions of the lower mass state κ(900) mixedintheresultofEq.(13) in fact. If this is indeed the case, the result of the sum rule may be some weighted average of the two resonances of κ(900) and K∗(1430). Therefore this situation should be carefully checked. Because the sum rule for the mass 0 s0 of the scalar bound state in Eqs. (8), (10) and (11) includes the spectrum integration 2 ds, in principle one (m1+m2) can lower the value of s0 to separate the lowest bound state. Therefore, we checked what result for the mass can be got by lower the value of s0 within the stable window 1.0 960 MeV. (15) Therefore the possible effect of κ(900) can be safely ruled out in the sum rule result in Eq. (13). Note that the most recent experimental result for the mass of κ(900) from E791 Collaboration is mκ = 797 ± 19 ± 42 MeV [3]. ∗ ¯ ¯ If K0 (1430) is the ground state of sq or qs, from the approximate SU(3) ﬂavorsymmetry,themassesofthe other J P = 0+ mesons in the scalar nonet should be also around 1.4 GeV. This implies that the scalars with masses less than 1 GeV, i.e., f0(600), a0(980), f0(980), etc., cannot be dominated by quark–antiquark bound states. This is consistent with the calculation of lattice QCD which implies that a nonet of quark–antiquark scalars is in the region 1.2–1.6 GeV [22]. Our result can be further checked by experiment. From the threshold√ parameter s0, we can predict that the mass of the ﬁrst excited resonance in sq¯ scalar channel should be larger than s0, that is ∗ ∗ m K0 > 2.0GeV. (16) This prediction can be tested by experiment. Next we discuss the decay constant of the two-quark scalar bound state sq¯. From the above analysis, we take the 0 = 2 ∗ ¯ threshold parameter s 4.0–4.8GeV . Consider K0 (1430) as the only resonance below 2 GeV in the sq scalar D.-S. Du et al. / Physics Letters B 619 (2005) 105–114 111

∗ 2 0 = 2 Fig. 4. The decay constant of K0 (1430) as a function of the Borel parameter M . The solid curve is for s 4.0GeV , and the dashed one for s0 = 4.8GeV2.

∗ 2 channel, we can obtain the decay constant of K0 (1430) as a function of Borel parameter M (see Eq. (9)). The numerical result is shown in Fig. 4. Fig. 4 shows that the decay constant is very stable. The determined stable ‘window’ is still in 1.0

Fig. 5. The decay constants in two-resonance ansatz below 2 GeV. The solid curve is for s0 = 4.0GeV2, and the dashed one for s0 = 4.8GeV2. ∗ (a) The decay constant of the low resonance κ(900). (b) The decay constant of the higher resonance K0 (1430). the stability existing in the one-resonance ansatz, which is shown in Fig. 4. From the requirement of numerical stability of QCD sum rule, the numerical analysis of the decay constant does not favor to include κ(900) in sq¯ scalar channel. In addition, we can see from Fig. 5(a) that the decay constant of the lower scalar resonance κ(900) tend to be zero at M2 ∼ 1.01 and 1.05 GeV. This is consistent with the requirement that 0|¯sq|κ(900)∼0inthe one-resonance ansatz, where the stability window is located in the range 1.0

Acknowledgements

This work is supported in part by the National Science Foundation of China, and by the Grant of BEPC National Laboratory.

Appendix A

ˆ nonp Borel transformed coefﬁcients of perturbative and nonperturbative contributions Bρi in Eqs. (10) and (11) are listed below 2 2 2 −3[(m + m ) − s] (−(m − m ) + s)(−(m + m ) + s) 3s 13 α (µ) − 2 ρpert(s) = 1 2 1 2 1 2 + s e s/M , (A.1) 8πs 8π 3 π D.-S. Du et al. / Physics Letters B 619 (2005) 105–114 113 where the term with αs(µ) is the radiative correction to the perturbative contribution [24], and the scale is taken to be µ = M. −m2/M2 e 2 Bρˆ nonp = 3M4m m2 + 3M2m2m3 + m3m4 + 3M6(m + 2m ) ¯ss 3 1 2 1 2 1 2 1 2 6M8 −m2/M2 e 1 + 3M4m2m + 3M2m3m2 + m4m3 + 3M6(2m + m ) dd¯ , (A.2) 1 2 1 2 1 2 1 2 6M8 − + 2 ˆ nonp 3(m1 m2) Bρ = 4παsGG 4 ((m +m )2/M2) 2 2 256e 1 2 M m1m2π + 4 2 + 2 3 2 + 4 3 + 6 + 1 3M m1m2 3M m1m2 m1m2 3M (2m1 m2) 2 (m /M2) 8 2 288e 1 M m2π + 4 2 + 2 2 3 + 3 4 + 6 + 1 3M m1m2 3M m1m2 m1m2 3M (m1 2m2) 2 (m /M2) 8 2 288e 2 M m1π −12m (m − m )2m + M2(−7m2 + 26m m − 7m2) + 1 1 2 2 1 1 2 2 ((m −m )2/M2) 4 2 768e 1 2 M m1m2π ∞ 3(m + m )4 × dt 1 2 [(m +m )2/M2] 4 2 2 2 128e 1 2 M π (m + 2m1m2 + m − t) 2 1 2 (m1+m2) m m (m2 − m m + m2 − t)t2 + 1 2 1 1 2 2 8e(t/M2)M4π 2(m2 − 2m m + m2 − t)2(m2 + 2m m + m2 − t) 1 1 2 2 1 1 2 2 − (m − m )2 4m (m − m )2m m2 − 2m m + m2 − t 1 2 1 1 2 2 1 1 2 2 + M2 3m4 − 16m3m + 26m2m2 − 16m m3 + 3m4 − 3m2t + 14m m t − 3m2t 1 1 2 1 2 1 2 2 1 1 2 2 × 1 − 2 2 128e((m1 m2) /M )M6π 2(m2 − 2m m + m2 − t)2 1 1 2 2 1 × , (A.3) 2 2 [(−(m1 − m2) + t)(−(m1 + m2) + t)] 4 3 2 m1[−6M + m m2 + 3M m1(m1 + m2)] Bρˆ nonp = gΨσTΨ¯ − 1 5 (m2/M2) 8 12e 1 M 4 3 2 m2[−6M + m1m + 3M m2(m1 + m2)] − 2 , (A.4) (m2/M2) 12e 2 M8 4(m + m )2 Bρˆ nonp = 4πα ΨΨ¯ 2 1 2 + − m2m6 + m8 + 36M6m (m + 2m ) + 84M4m2 m2 − m2 6 s 2 2 2 1 2 2 1 1 2 2 1 2 9M m1m2 1 + 15M2m4 m2 − m2 2 1 2 (m2/M2) 81e 2 M8m2(−m2 + m2) 2 1 2 + 36M6m (2m + m ) + m6 m2 − m2 − 84M4 m4 − m2m2 − 15M2 m6 − m4m2 2 1 2 1 1 2 1 1 2 1 1 2 1 × , 2 2 (A.5) (m1/M ) 8 2 2 − 2 81e M m1(m1 m2) where m1 = ms , m2 = mq . 114 D.-S. Du et al. / Physics Letters B 619 (2005) 105–114

References

[1] Particle Data Group, S. Eidelman, et al., Phys. Lett. B 592 (2004) 1. [2] LASS Collaboration, D. Aston, et al., Nucl. Phys. B 296 (1988) 493. [3] E791 Collaboration, E. Aitala, et al., Phys. Rev. Lett. 16 (2003) 121801. [4] M. Ishida, Prog. Theor. Phys. Suppl. 149 (2003) 190. [5] F.E. Close, N.A. Törnqvist, J. Phys. G 28 (2002) R249. [6] E. Klempt, in: Phenomenology of Gauge Interactions, Zuoz, 2000, p. 61, hep-ex/0101031. [7] N. Isgur, J. Weinstein, Phys. Rev. Lett. 48 (1982) 659. [8] R.L. Jaffe, Phys. Rev. D 15 (1977) 267. [9] C. Amsler, F.E. Close, Phys. Lett. B 353 (1995) 385; C. Amsler, F.E. Close, Phys. Rev. D 53 (1996) 295. [10] W. Lee, D. Weingarten, Phys. Rev. D 61 (2000) 014015. [11] D. Li, H. Yu, Q. Shen, Commun. Theor. Phys. 34 (2000) 507. [12] L. Celenza, et al., Phys. Rev. C 61 (2000) 035201. [13] M. Strohmeier-Presicek, et al., Phys. Rev. D 60 (1999) 054010. [14] F.E. Close, A. Kirk, Phys. Lett. B 483 (2000) 345. [15] C. Amsler, F.E. Close, Phys. Lett. B 353 (1995) 385. [16] F.E. Close, G. Farrar, Z.P. Li, Phys. Rev. D 55 (1997) 5749. [17] D. Weingarten, Nucl. Phys. B (Proc. Suppl.) 53 (1997) 232; D. Weingarten, Nucl. Phys. B (Proc. Suppl.) 63 (1998) 194; D. Weingarten, Nucl. Phys. B (Proc. Suppl.) 73 (1999) 249. [18] M.A. Shifman, A.I. Vainshtein, V.I. Zakharov, Nucl. Phys. B 147 (1979) 385; M.A. Shifman, A.I. Vainshtein, V.I. Zakharov, Nucl. Phys. B 147 (1979) 448. [19] A.I. Vainshtein, M.B. Voloshin, V.I. Zakharov, M.A. Shifman, Sov. J. Nucl. Phys. 28 (1978) 237, Yad. Fiz. 28 (1978) 465. [20] L.J. Reinders, H. Rubinstein, S. Yazaki, Phys. Rep. 127 (1985) 1. [21] S. Narison, QCD Spectral Sum Rules, World Scientiﬁc, Singapore, 1989. [22] SESAM Collaboration, G. Bali, et al., Nucl. Phys. B (Proc. Suppl.) 63 (1997) 209. [23] The general explanation of quark–hadron duality can be found in the review article by P. Colangelo, A. Khodjamirian, in: M. Shifman (Ed.), At the Frontier of Particle Physics/Handbook of QCD, World Scientiﬁc, Singapore, 2001, hep-ph/0010175. [24] L.J. Reinders, S. Yazaki, Nucl. Phys. B 196 (1982) 125. Physics Letters B 619 (2005) 115–123 www.elsevier.com/locate/physletb

SU(3) baryon resonance multiplets in large Nc QCD

Thomas D. Cohen a, Richard F. Lebed b

a Department of Physics, University of Maryland, College Park, MD 20742-4111, USA b Department of Physics and Astronomy, Arizona State University, Tempe, AZ 85287-1504, USA Received 27 April 2005; accepted 30 May 2005 Available online 6 June 2005 Editor: H. Georgi

Abstract

We extend the recently developed treatment of baryon resonances in large Nc QCD to describe resonance multiplets collected according to the SU(3) ﬂavor symmetry that includes strange quarks. As an illustration we enumerate the SU(3) partners of a ± P = 1 = hypothetical J 2 resonance in the SU(3) representation that reduces to 10 when Nc 3, and reproduce results hitherto obtained only in the context of a large Nc quark picture. While these speciﬁc quantum numbers represent one favored set for + the possible pentaquark state Θ (1540), the method is applicable to baryon resonances with any quantum numbers.  2005 Elsevier B.V. All rights reserved.

PACS: 11.15.Pg; 14.20.Gk; 14.20.Jn

1. Introduction

A recent series of papers [1–3] by a collaboration led by the current authors shows how baryon resonances may be properly treated in the context of large Nc QCD. For reviews of this literature, see Ref. [4]. In the standard treatment of baryons at arbitrary Nc, the ground-state spin–flavor multiplet is taken to be the completely symmetric Nc-tableau representation, which is the analog to the SU(6) 56. Notationally, we denote such arbitrary-Nc generalizations of Nc = 3 representations with quotes, as in “56”. It should be pointed out that the spin–flavor symmetry of the ground state is not a rigorous result following directly from manipulations of the QCD action, but rather an assumption whose phenomenological predictions concur with all available experimental measurements. Turning this result around, one can show that the successes of the old baryon SU(6) spin–flavor =−3 symmetry, such as µp 2 µn or the relative closeness of N and ∆ masses, are actually consequences of the 1/Nc 1 expansion [5–7]. Equivalently, assuming that the baryon masses and the πN axial-current coupling gA scale as Nc

E-mail addresses: [email protected] (T.D. Cohen), [email protected] (R.F. Lebed).

(as naturally arises in quark and Skyrme models) leads to a degenerate “56” multiplet through an analysis using “consistency relations” in πN scattering, which are obtained by the imposition of unitarity order-by-order in Nc [5]. = 1 3 Nc The ground-state band “56” decomposes into multiplets with spins SB 2 , 2 ,..., 2 , with corresponding Nc − SU(3) representations [in the Dynkin weight notation (p, q)] (2SB , 2 SB ). The first two members of this + + PB = 5 7 series are denoted, as expected, “8” and “10”, while we may label the SB 2 , 2 , etc. members (those that → = 0 disappear as Nc 3) as “large-Nc exotic”. The mass splittings between the multiplets with J O(Nc ) are only O(ΛQCD/Nc), which for sufficiently large Nc are smaller than mπ ; hence, such members of the ground-state multiplet are stable against strong interactions and have widths that vanish in the large Nc limit. One may argue that it is a fluke of our universe that the chiral limit is more closely realized than the large Nc limit, allowing the decay ∆ → πN. Once baryon states are determined to be stable in this way, they may be analyzed using a Hamiltonian formalism, in which the spin–flavor symmetry is broken perturbatively in powers of 1/Nc by operators with specific quantum numbers under spin and flavor. This method has a long history and been dubbed the “operator approach” [1]. Baryon resonances, on the other hand, are completely different entities. Appearing in meson–baryon scattering 0 amplitudes, whose generic size is O(Nc ), resonances arise as poles at complex values of energy in the analytic continuation of these amplitudes. Of course, the real and imaginary part of each such value represents the excitation 0 mass and width, respectively, of the resonance, both of which are typically O(Nc ). Resonances are unstable against strong decay even in the large Nc limit, and require a treatment distinct from that of the stable baryons. Nevertheless, if a mechanism other than the 1/Nc expansion can be invoked to suppress the creation of quark– antiquark pairs—the mechanism through which a baryon resonance decays—then the treatment of resonances as almost stable baryons becomes more reasonable. For example, if the quarks comprising the baryon are heavy compared to ΛQCD then pair creation is suppressed, and an approach analogous to that used for the ground-state baryons applies. The analysis of ordinary baryon resonances treated as almost stable in the 1/Nc expansion, using the operator approach, has been carried out in great detail in the literature [8]. However, there exists a model-independent treatment [1,2] using the 1/Nc expansion in which the resonances 0 have natural [O(Nc )] widths and yet retains a good deal of predictive power. This “scattering approach” was originally inspired [9] by the observation that numerous results obtained in chiral soliton models such as the Skyrme model appeared to be purely group-theoretical in origin. A series of papers in the 1980s by Mattis and collaborators [10,11] showed that such results are in fact independent of any dynamical details of the models. Indeed, in the case of two light quark flavors the dominant S matrix amplitudes were found to be precisely those with t-channel exchange quantum numbers It = Jt [11]. This result can in turn be shown to follow directly from the analysis of consistency relations derived from scattering processes in large Nc [1]. Exploiting crossing relations, the It = Jt rule can be used to express observable meson–baryon scattering amplitudes in terms of a smaller set of reduced amplitudes labeled in the s-channel by eigenvalues K of the “grand spin” K = I + J. In particular, a resonant pole appearing in one scattering amplitude must appear in at least one of the reduced amplitudes, which in turn appears in other scattering amplitudes. Baryon resonances therefore appear in multiplets degenerate in both mass and width for large Nc. The group theory for the three-flavor case relevant to hyperon physics is of course more involved, but the necessary exercise was carried out [12] by Mattis and Mukerjee in 1989 in the context of soliton models. When three flavors are included, the It = Jt rule no longer holds in its original form, but as we discuss below a more complicated set of constraints applies. In fact, in repeating the derivation of Ref. [12], we find small discrepancies, and discuss them below. Nevertheless, the (suitably modified) Mattis–Mukerjee relation is the proper SU(3) generalization of the SU(2) scattering relation at large Nc. It predicts degenerate SU(3) multiplets of baryon resonances at large Nc and in the SU(3) limit. This observation and its practical implementation (which required the computation of relevant SU(3) Clebsch–Gordan coefficients (CGC) [3]) are the purposes of this Letter. As a first illustration of the power of the scattering method, we consider the SU(3) partners to a hypothetical + − P = 1 1 J 2 or 2 isosinglet baryon resonance in an SU(3) “10”. These are none other than two theoretically favored T.D. Cohen, R.F. Lebed / Physics Letters B 619 (2005) 115–123 117 sets of quantum numbers for the purported pentaquark state Θ+(1540). We discerned the partners of this I = 0 state using the two-flavor formalism in Ref. [2]. We of course make no claims whether this state does indeed exist, but rather conclude that if any baryon resonance with these quantum numbers exists, then it must have partners P degenerate in mass and width at leading order in the 1/Nc expansion that carry specific J and SU(3) quantum numbers. Moreover, we show below that the particular pattern of partners to this state (for P =+) is precisely the one recently derived by Jenkins and Manohar [13]. They employed a 1/Nc operator approach originally inspired by the rigid rotor Skyrme model, but consistent with the group theory of quark models as well. One might have thought that such a coincidence is trivial. However, since the physical picture in Ref. [13] is based upon stable states at large Nc, one can easily imagine the possibility that when the widths of the states become order unity, mixing between the multiplets given in Ref. [13] could occur. This is not unheard of in the scattering approach; indeed, the old nonrelativistic SU(2Nf ) × O(3) quark model multiplets were shown [1] to form reducible collections of complete distinct multiplets in the scattering approach. It is, therefore, quite heartening that the two methods agree so well in this case. This Letter is organized as follows. In Section 2, we present the master expression for three-flavor meson– baryon scattering and explain its origin and relation to previous work. Section 3 presents a specific example: the enumeration of quantum numbers of resonances degenerate in the large Nc limit with a hypothetical isoscalar, − + + P 1 1 strangeness 1 resonance in the SU(3) “10” representation, for the J values 2 and 2 . These are of course the favored theoretical preferences for the purported pentaquark Θ+(1540) state; however, even if this state should turn out not to survive the current experimental scrutiny, the example presented here should be viewed as an indication of the power of the method. In Section 4 we summarize, indicate future directions of research, and conclude.

2. The SU(3) amplitude relation

We now present the expression for the S-matrix amplitude in the meson–baryon scattering process φ(Sφ,Rφ, Iφ,Yφ) + B(SB ,RB ,IB ,YB ) → φ (Sφ ,Rφ ,Iφ ,Yφ ) + B (SB ,RB ,IB ,YB ), where S, R, I , and Y stand, respectively, for the spin, SU(3) representation, isospin, and hypercharge of the mesons φ and φ and the baryons B and B. Primes here indicate ﬁnal-state quantum numbers. The total spin angular momentum (added vectorially) among the meson–baryon pairs are denoted by S and S, and the relative angular momenta between the meson– baryon pairs are denoted by L and L . The amplitude is described in terms of s-channel angular momentum Js , SU(3) representation Rs , isospin Is , and hypercharge Ys . In addition, multiple copies of Rs can arise in the products RB ⊗ Rφ and RB ⊗ Rφ , and the quantum numbers deﬁned to lift this degeneracy are labeled by γs and γs (which need not be equal). That the other s-channel quantities are conserved can be demonstrated explicitly = (e.g., Rs Rs ), and thus the primes on such quantities are suppressed. The amplitude is reduced, in the sense of the Wigner–Eckart theorem, in that the SU(2) quantum numbers Jsz and Iz do not appear explicitly. The notation [X] refers to the dimension of a given representation, whether X is labeled by I or J in SU(2), or by the actual dimension in SU(3) (i.e., [J = 1]=3, but [R = 8]=8). The master expression for such scattering amplitudes in the large-Nc limit then reads

− 1/2 + + = − SB SB [ ][ ][ ][ ] [ ] − I I Y [ ] SLL SS Js Rs γs γs Is Ys ( 1) RB RB S S / Rs ( 1) I ∈ ∈ I Rφ ,I Rφ , I ∈Rs ,Y∈Rφ ∩R φ RB Rφ Rs γs RB Rφ Rsγs × N S Nc IY I Y + c I Y I Y I Y B 3 3 B B φ φ s s R R × B φ Rsγs RB Rφ Rsγs N N c + c I Y I Y I Y SB 3 I Y I Y 3 B B φ φ s s 118 T.D. Cohen, R.F. Lebed / Physics Letters B 619 (2005) 115–123

˜ ˜ LIK L I K 1/2 {IIY } × [ ] [ ˜ ][ ˜ ] K K K SSB Sφ S SB Sφ τ ˜ ˜ . (1) KKK LL K,K,˜ K˜ Js I K Js I K

The quantities containing double vertical bars are SU(3) isoscalar CGC [3], while those in braces are ordinary SU(2) 9j symbols. This expression should be compared with the original Mattis–Mukerjee result [Ref. [12], Eq. (12)]. Since its derivation was a primary result of that Letter, we do not present a detailed rederivation here, but merely discuss its structure, and then detail differences between the present expression and that of Ref. [12] (in particular, why the expression in Ref. [12] is not suitable for physical processes). The two-flavor scattering formula is derived in the earlier chiral soliton-type treatment [10] by starting with a fundamental soliton in the conventional hedgehog configuration, which is an eigenstate of the grand spin K ≡ I+J. Scattering is accomplished by the standard linear expansion of the soliton in terms of pion field fluctuations. However, physical hadrons are of course specified not by K but by I and J , and hence one must allow for multiple values of K in a full physical scattering process in order to form a linear superposition that is an eigenstate of I and J ; nevertheless, one may treat K as a hidden degree of freedom conserved in the underlying scattering processes, which therefore attaches as a label to the reduced scattering amplitudes τ . In generalizing the process to allow for mesons φ, φ of arbitrary isospin I and spin S, one requires also the intermediate quantum numbers ˜ ˜ ˜ ˜ K ≡ Iφ + L and K ≡ Iφ + L (so that K = K + Sφ = K + Sφ )usedinEq.(1).The9j symbols simply arise through the combination of the numerous SU(2) CGC that arise in this procedure from the vector addition of multiple SU(2)-valued quantities. The three-flavor generalization is conceptually quite straightforward, if mathematically more cumbersome: One simply rotates the full initial and final states into their nonstrange partners in the same irreducible SU(3) representation, and uses the two-flavor expression for the nonstrange scattering process. The inclusion of SU(3) rotation matrices of course introduces the SU(3) CGC in Eq. (1). In repeating the derivation of Ref. [12] Eq. (12) to obtain our Eq. (1), we find a few small but significant − differences. First, the overall phase of the original result lacks our phase (−1)SB SB . Second, Ref. [12] appears to average over baryons and mesons in the external states with all possible quantum numbers within the given SU(3) multiplets (rendering their expression phenomenologically less useful); if we do the same with Eq. (1), two of our SU(3) CGC are absorbed through an orthogonality relation, matching the older result. Finally, their explicit unity Nc values for the nonstrange baryon hypercharges must be modified to 3 , in light of the proper quantization [14] of the Wess–Zumino term for arbitrary Nc. While these agree for Nc = 3, it is important to keep the general form so that consistency in Nc scaling can be verified. A difference in our interpretation relative to Ref. [12] also helps resolve a paradox of that work. In Ref. [12] it was noted that the It = Jt rule does not hold for meson–baryon scattering in soliton models with SU(3) flavor at leading order in Nc, even for processes with no exchange of strangeness. This is worrying, since as noted in Refs. [1,2] the It = Jt rule can be derived for such processes directly from large Nc QCD with no additional model assumptions. The origin of this perplexing discrepancy is that the SU(3) representations used for the baryons of interest in Ref. [12] are those which occur for Nc = 3 (e.g., the literal 8 and 10). However, as noted in the introduction, the appropriate representations in a large Nc world are not these but rather the “8”, “10”, and so on. Strictly speaking the relations derived here hold for the large Nc world, and thus one expects the It = Jt rule to hold for meson–baryon scattering with strangeness and the baryons in their large Nc representations. Thus, by using the Nc = 3 representations, Ref. [12] implicitly includes specific 1/Nc corrections. However, since this was done with Nc set to 3, one could not cleanly isolate the numerically small violation of the It = Jt rule as a 1/Nc correction. Using the proper large Nc representations and formulae derived here, the It = Jt rule indeed holds. For phenomenological purposes, the most interesting special case of Eq. (1) is that in which the baryons belong to the parity-positive ground-state “56” multiplet, and the mesons are both pseudoscalar SU(3)-octet pseudo-Nambu–Goldstone bosons. In this case, parity (P ) conservation demands that L − L is an even integer, and the 9j symbols collapse to 6j symbols. The master scattering amplitude for the process φ(Sφ = 0, T.D. Cohen, R.F. Lebed / Physics Letters B 619 (2005) 115–123 119

Rφ = 8,Iφ,Yφ) + B(SB ,RB ,IB ,YB ) → φ (Sφ = 0,Rφ = 8,Iφ ,Yφ ) + B (SB ,RB ,IB ,YB ) then reads − 1/2 + + S = (− )SB SB [R ][R ] /[R ] (− )I I Y [I ] LL SB SB Js Rs γs γs Is Ys 1 B B s 1 I,I,Y∈8, ∈ I Rs × RB 8 Rsγs RB 8 Rsγs S Nc IY I Y + Nc I Y I Y I Y B 3 3 B B φ φ s s RB 8 Rsγs RB 8 Rsγs × N S c I Y I Y + Nc I Y I Y I Y B 3 3 B B φ φ s s × [ ] KI Js KI Js {II Y } K τKKKLL . (2) SB LI SB L I K While these expressions were originally derived within the context of a chiral soliton picture, they are model- independent consequences of QCD in the large Nc limit. By employing crossing relations, one first shows [11] that the conservation of K in s-channel processes for the two-flavor case is equivalent to the rule It = Jt in the t channel; and as discussed in Section 1, this rule is a direct large Nc consequence [1]. The three-flavor generalization, in turn, is simply an SU(3) rotation of the SU(2) result with no additional dynamics, and therefore is also a model- independent large Nc result.

3. Explicit example

To illustrate the utility of the approach, we apply it to ﬁnd SU(3) partners of the reported narrow Θ+ exotic pentaquark resonance. A few caveats are useful before proceeding. First, there is considerable controversy as to whether these states are real; in this work we take an agnostic position. The issue addressed here is that if the resonance is real, then at large Nc and in the SU(3) limit it must have degenerate partners, which at ﬁnite Nc would correspond to some nearly degenerate partners; our goal is to enumerate them. Second, the analysis is based on exact SU(3) symmetry. While small SU(3) violations can be accounted for perturbatively, if for some reason large SU(3) violations occur then the present formalism breaks down. For example, it has been suggested by Jaffe and Wilczek [15] in the context of a diquark model that nearly ideal mixing may occur between different SU(3) multiplets (such as for φ and ω mesons), which could lead to large SU(3) violations. The present work assumes that such a scenario does not occur; indeed, sound phenomenological arguments [16] oppose it. Third, if the Θ+ does in fact exist, then apart from its strangeness and isospin we do not know its quantum numbers directly from experiment. Since predictions of SU(3) partners depend upon the quantum numbers, we assume here that the Θ+ 1 is a spin- 2 isoscalar, which seems to be the most natural possibility from a theoretical perspective. The parity of the Θ+ is unknown, and various plausible theoretical arguments can be made to suggest either parity; accordingly, we consider both cases. Finally, it remains possible that the Θ+ exists and has different quantum numbers. In such a case one could perform an analysis entirely analogous to the one considered here.

3.1. “Seed” amplitudes

= = 1 We suppose that a pole corresponding to a baryon resonance appears in an NK partial wave (N: SB SB 2 , = =+ = = 1 = = Nc = = = = 1 = = = = PB PB , IB IB 2 , YB YB 3 , RB RB “8”; K: Iφ Iφ 2 , Yφ Yφ 1, Rφ Rφ 8) = = Nc + = 1 = with quantum numbers Is 0, Ys 3 1, Js 2 , Rs “10”, and examine the consequences of Eq. (2).The first task is to determine which reduced amplitudes τ contribute to partial waves carrying these quantum numbers, { 1 1 } 2 2 1 and therefore act as “seed” amplitudes to produce poles in other partial waves. As we now show, only τ 1 1 1 2 2 2 LL appears (with L = 0, 1forPs =∓), implying that the assumed resonant pole must lie in that reduced amplitude; 120 T.D. Cohen, R.F. Lebed / Physics Letters B 619 (2005) 115–123 had several amplitudes arisen, it would have necessary to perform a more delicate analysis to look for degenerate poles in multiple partial waves in order to determine which reduced amplitudes they have in common. The triangle rules imposed by Eq. (2) are δ(SB II ), δ(SB I I ), δ(KI Js), δ(KLI), δ(KL I ), δ(SB LJs), and δ(SB L Js). Imposing the substitutions listed above, the last two imply L = L = 0or1forPs =−or +, = = ± 1 = = respectively. If L L 0, then I equals either 2 added to the common value I I K. On the other hand, = = + 1 − 1 if L L 1, then satisfying the triangle rules requires that each of I , I , and K differ from I by 2 or 2 .The sums in Eq. (2) are truncated by the requirement that (I, Y ) and (I ,Y) must be the quantum numbers of states 1 1 − within a literal SU(3) octet, which are ( 2 , 1), (1, 0), (0, 0), and ( 2 , 1). The current case is simplified considerably by noting that “10” =[0,(Nc + 3)/2] contains only singly- degenerate states; in particular, one finds using the variables of Eq. (2) that 2I + Y = 1. To each isospin multiplet (I, Y ) or (I ,Y)within the 8 one therefore identifies a unique value I = (1 − Y)/2. The required SU(3) CGC then assume the form “8” 8 “10” 1 Nc { } 1−Y + Nc . (3) 2 , 3 I,I ,Y 2 ,Y 3 All of these CGC are compiled in Table I of Ref. [3]. From this source one readily determines that the only such { } = 1 − coefficient nonvanishing in the large Nc limit has ( I,I ,Y) ( 2 , 1), for which the CGC equals 1. But then also = = = 1 = = 1 I 0, which forces not only I I 2 and Y 1, but also K 2 . It follows that the unique reduced amplitude { 1 1 1} = = 2 2 contributing in the each of the L 0 and L 1 cases is, as promised, τ 1 1 1 . 2 2 2 LL To complete the simplification of Eq. (2) for this case, we note that [K]=2, the dimension of any SU(3) 1 + + + + [ ]= 1 + + [ ]= representation (p, q) assumes the usual value 2 (p 1)(q 1)(p q 2): “8” 4 (Nc 5)(Nc 1) and “10” 1 (Nc + 7)(Nc + 5), and the 6j symbols give 8 1 0 1 1 2 2 =∓ 1 { } 1 . (4) 2 0, 1 2 2 { 1 1 } 2 2 1 In the large Nc limit, one then finds the numerical coefficient of τ 1 1 1 to be unity. 2 2 2 LL 3.2. Degrees of “exoticness”

Before continuing, it is important to point out that the concept of “exoticness” can be used in three different 3 but related senses here: First, we label the members of the “56” with SB > 2 as “large-Nc exotic” because their Nc − = corresponding SU(3) representations (2SB , 2 SB ) are not allowed for Nc 3. We denote processes in which the initial ground-state baryon is large-Nc exotic by E, nonexotic by N . Second, the product representation in which the baryon resonances appear may be nonexotic (N ∗) or exotic in one of two distinct ways: Either it is a perfectly E∗ ordinary SU(3) representation that cannot be produced through a qqq state, such as the “10” (denoted by 0 ), or E∗ NN∗ NE∗ NE∗ EN∗ EE∗ it may also be large-Nc exotic (denoted by 1 ). All 6 types of scattering process, , 0, 1, , 0, EE∗ and 1, may occur in a large Nc world. The possibility of some of the mixed N –E combinations may come as a bit of a surprise. As an example of an ∗ NE process, note that the product “8” ⊗ 8 contains the large-Nc exotic SU(3) representation [2,(Nc − 5)/2].On 1 + EN∗ 5 the other hand, also can occur: The 2 ground-state baryon can scatter a pseudoscalar 8 mesontogivea“10” E∗ NE∗ resonance. For our present purposes, we are interested in 0 processes, both “singly exotic” ( 0) and “doubly EE∗ = exotic” ( 0). That is, we are interested in exotic resonances that lie in SU(3) representations existing at Nc 3, but allow for the possibility that the ground-state baryon representation needed to produce them in scattering with a pseudoscalar 8 meson might itself not occur for Nc = 3. We make this choice to mirror the terminology of Ref. [13]. EE∗ One can readily show that there exists an upper limit to ground-state baryon spins SB in “56” allowing 0 EE∗ processes via scattering with 8 mesons (beyond which only 1 occurs). As SB increases, the second row of its T.D. Cohen, R.F. Lebed / Physics Letters B 619 (2005) 115–123 121

Nc − SU(3) tableau (length 2 SB ) becomes so short that the 3 boxes in the 8 are insufﬁcient to produce a large-Nc NE∗ ⊗ → nonexotic resonance. By direct computation, one ﬁnds that the 0 possibilities are “8” 8 “10” and “27”, ⊗ → =[ + ] =[ − ] EE∗ “10” 8 “27” and ”35” (where “27” 2,(Nc 1)/2 and “35” 4,(Nc 1)/2 ), and the 0 possibilities + 5 ⊗ → =[ − ] 7 + ⊗ → are 2 8 “28” and “35” (where “28” 6,(Nc 3)/2 ), and 2 8 “28”.

3.3. Finding SU(3) partners

Having isolated the reduced amplitudes containing the desired resonant pole, we now reverse the process in order to determine the full set of partial waves to which these reduced amplitudes contribute. The triangle rules imposed by Eq. (2) with I = I = K = 1 force each of L, L to equal either 0 or 1; and the fact that all baryons 2 in the ground-state “56”havePB =+again forces L = L . Note in particular that this procedure obtains only degenerate partners all carrying the same parity.

3.3.1. Negative parity We first analyze the case Ps =−case, for which L = L = 0. Then SB = SB = Js , so that RB = RB , and the 1 only remaining nontrivial triangle rule is δ(SB 2 I );Eq.(2) collapses to [ ] { 1 1 } RB 2 2 1 S = δR R δS S δS J τ [I ] 00SB SB Js Rs γs γs Is Ys B B B B B s [ ][ ] 1 1 1 00 Rs SB 2 2 2 ∈ I Rs × RB 8 Rsγs RB 8 Rsγs S Nc 1 1 I Nc + 1 I Y I Y I Y B 3 2 3 B B φ φ s s RB 8 Rsγs RB 8 Rsγs × . (5) Nc 1 Nc + SB 3 2 1 I 3 1 IB YB Iφ Yφ IsYs E∗ 7 = 1 = In order to study only 0 processes, as discussed above we limit SB 2 . The CGC for SB 2 (RB “8”) and 3 = →∞ = 1 2 (RB “10”) again all appear in Ref. [3]. One finds that the only CGC surviving as Nc for SB 2 have = = − = = + = 3 = = either Rs “10”, I 0 (giving 1), or Rs “27”, I 1( 1). For SB 2 , we have either Rs “27”, I 1 − = = + = 5 7 ( 1), or Rs “35”, I 2( 1). Ref. [3] does not compile CGC for SB 2 or 2 baryons, but for our purposes it 0 = 5 = = 7 is only necessary to know that there exist O(Nc ) couplings for SB 2 to Rs “35” and “28”, and for SB 2 to = E∗ [ − ] Nc Rs “28” and the 1 representation 8,(Nc 5)/2 . Indeed, for the states of maximal hypercharge in RB ( 3 ), 8 + Nc + ( 1), and Rs ( 3 1) [as required by the first CGC in Eq. (2)], it is straightforward to show that only one SU(3) = + 1 = + Nc − + = − 1 representation occurs for I SB 2 [(p, q) (2SB 1, 2 SB 1)], and only one occurs for I SB 2 = − Nc − + [(p, q) (2SB 1, 2 SB 2)], which are the representations listed above. The CGC in each of these cases + − must therefore be either 1or 1. − − − 1 1 Collecting these results, one then finds the set of degenerate multiplets (Rs,Js ) to be (“10”, ), (“27”, ), − − − − 2 2 3 3 NE∗ 5 5 7 − (“27”, 2 ), and (“35”, 2 ) (singly exotic, via 0 processes), and (“35”, 2 ), (“28”, 2 ), and (“28”, 2 ) (doubly EE∗ exotic, via 0 processes).

3.3.2. Positive parity The case Ps =+,forwhichL = L = 1, is only a bit more complicated. Now one may have SB = SB and RB = RB , and Js must be separately specified. In this case, Eq. (2) becomes 1/2 2([R ][R ]) { 1 1 1} SB −S B B 2 2 S = (−1) B τ [I ] 11SB SB Js Rs γs γs Is Ys [ ] 1 1 1 11 Rs 2 2 2 ∈ I Rs × RB 8 Rsγs RB 8 Rsγs Nc 1 Nc + SB 3 2 1 I 3 1 IB YB IφYφ IsYs 122 T.D. Cohen, R.F. Lebed / Physics Letters B 619 (2005) 115–123 RB 8 Rsγs RB 8 Rsγs × N S c 1 1 I Nc + 1 I Y I Y I Y B 3 2 3 B B φ φ s s 1 I J 1 I J × 2 s 2 s . (6) 1 1 SB 1 2 SB 1 2 Note that precisely the same set of SU(3) CGC are relevant to the case Ps =+, meaning that the enumeration of SU(3) representations carries over verbatim from the case Ps =−; only the angular momenta need be con- 1 sidered more carefully. The remaining independent triangle rules imposed by the 6j symbols are δ(SB 2 I ), 1 1 = 1 = = → δ(SB 2 I ), and δ(2 I Js). One then finds the following combinations. From SB 2 , RB “8”: I 0 = = 1 = → = = 1 3 = 3 = = → = Rs “10”, Js 2 and I 1 Rs “27”, Js 2 , 2 .FromSB 2 , RB “10”: I 1 Rs “27”, = 1 3 = → = = 3 5 = 5 =[ − ] = → = Js 2 , 2 and I 2 Rs “35”, Js 2 , 2 .FromSB 2 , RB 5,(Nc 5)/2 : I 2 Rs “35”, = 3 5 = → = = 5 7 = 7 =[ − ] = → = Js 2 , 2 and I 3 Rs “28”, Js 2 , 2 . And from SB 2 , RB 7,(Nc 7)/2 : I 3 Rs “28”, 5 7 7 9 Js = , and I = 4 → Rs =[8,(Nc − 5)/2], Js = , . 2 2 2 2 + + + 1 1 Collecting these results, one then finds the set of degenerate multiplets (Rs,Js ) to be (“10”, ), (“27”, ), + + + + 2 2 3 3 5 NE∗ 5 7 + (“27”, 2 ), (“35”, 2 ), and (“35”, 2 ) (singly exotic, via 0 processes), and (“28”, 2 ), and (“28”, 2 ) (doubly EE∗ exotic, via 0 processes). As promised, these multiplets precisely match those obtained in Ref. [13] via counting using Young tableaux, once a consistent definition of degree of exoticness is included.

4. Conclusions

We have generalized to three flavors the two-flavor large Nc meson–baryon scattering method that relates different scattering partial waves. In particular, resonant poles occurring in one such amplitude appear in others, creating multiplets of resonances degenerate in both mass and width at leading order in the 1/Nc expansion limit. We illustrated the method by finding the partners of a resonance carrying the quantum numbers suggested by the purported + + P = 1 Θ (1540) particle, and found that the results (for J 2 ) agree with those obtained using a large Nc method that does not recognize the instability of the resonances. One may immediately apply this formalism to numerous other problems involving three-flavor baryon resonances. For example, in it has been shown [1] that the suppression of the N(1535)πNpartial width is due to the fact that the reduced amplitude appearing in the S11 partial wave in the large Nc limit does not couple to spinless isovector mesons. What interesting analogous consequences arise for the Λ and Ξ resonances? Finally, the It = Jt rule was used [1] to parametrize 1/Nc corrections to the leading-order large Nc results, an absolute must for useful phenomenological studies. As discussed above, the crossing constraint for three flavors cannot be described so simply. We defer to future work the description and application of this important concept.

Acknowledgements

T.D.C. was supported by the DOE through grant DE-FGO2-93ER-40762; R.F.L. was supported by the NSF through grant PHY-0140362.

References

[1] T.D. Cohen, R.F. Lebed, Phys. Rev. Lett. 91 (2003) 012001; T.D. Cohen, R.F. Lebed, Phys. Rev. D 67 (2003) 096008; T.D. Cohen, R.F. Lebed, Phys. Rev. D 68 (2003) 056003; T.D. Cohen, D.C. Dakin, A. Nellore, R.F. Lebed, Phys. Rev. D 69 (2004) 056001; T.D. Cohen, R.F. Lebed / Physics Letters B 619 (2005) 115–123 123

T.D. Cohen, D.C. Dakin, A. Nellore, R.F. Lebed, Phys. Rev. D 70 (2004) 056004; T.D. Cohen, D.C. Dakin, R.F. Lebed, D.R. Martin, Phys. Rev. D 71 (2005) 076010. [2] T.D. Cohen, R.F. Lebed, Phys. Lett. B 578 (2004) 150. [3] T.D. Cohen, R.F. Lebed, Phys. Rev. D 70 (2004) 096015. [4] T.D. Cohen, hep-ph/0501090; R.F. Lebed, hep-ph/0501021. [5] R. Dashen, A.V. Manohar, Phys. Lett. B 315 (1993) 425; R.F. Dashen, E. Jenkins, A.V. Manohar, Phys. Rev. D 49 (1994) 4713. [6] E. Jenkins, Phys. Lett. B 315 (1993) 441. [7] R.F. Dashen, E. Jenkins, A.V. Manohar, Phys. Rev. D 51 (1995) 3697; E. Jenkins, R.F. Lebed, Phys. Rev. D 52 (1995) 282. [8] C.D. Carone, H. Georgi, L. Kaplan, D. Morin, Phys. Rev. D 50 (1994) 5793; J.L. Goity, Phys. Lett. B 414 (1997) 140; D. Pirjol, T.-M. Yan, Phys. Rev. D 57 (1997) 1449; D. Pirjol, T.-M. Yan, Phys. Rev. D 57 (1998) 5434; C.E. Carlson, C.D. Carone, J.L. Goity, R.F. Lebed, Phys. Lett. B 438 (1998) 327; C.E. Carlson, C.D. Carone, J.L. Goity, R.F. Lebed, Phys. Rev. D 59 (1999) 114008; C.E. Carlson, C.D. Carone, Phys. Lett. B 441 (1998) 363; C.E. Carlson, C.D. Carone, Phys. Rev. D 58 (1998) 053005; C.E. Carlson, C.D. Carone, Phys. Lett. B 484 (2000) 260; C.L. Schat, J.L. Goity, N.N. Scoccola, Phys. Rev. Lett. 88 (2002) 102002; C.L. Schat, J.L. Goity, N.N. Scoccola, Phys. Rev. D 66 (2002) 114014; C.L. Schat, J.L. Goity, N.N. Scoccola, Phys. Lett. B 564 (2003) 83; C.L. Schat, J.L. Goity, N.N. Scoccola, Phys. Rev. D 71 (2005) 034016; D. Pirjol, C.L. Schat, Phys. Rev. D 67 (2003) 096009; D. Pirjol, C.L. Schat, Phys. Rev. D 71 (2005) 036004; M. Wessling, Phys. Lett. B 603 (2004) 152; N. Matagne, Fl. Stancu, Phys. Rev. D 71 (2005) 014010; J.L. Goity, N.N. Scoccola, hep-ph/0504101; J.L. Goity, hep-ph/0504121; N.N. Scoccola, hep-ph/0504150. [9] A. Hayashi, G. Eckart, G. Holzwarth, H. Walliser, Phys. Lett. B 147 (1984) 5; M.P. Mattis, M. Karliner, Phys. Rev. D 31 (1985) 2833. [10] M.P. Mattis, M.E. Peskin, Phys. Rev. D 32 (1985) 58; M.P. Mattis, Phys. Rev. Lett. 56 (1986) 1103; M.P. Mattis, Phys. Rev. Lett. 63 (1989) 1455; M.P. Mattis, Phys. Rev. D 39 (1989) 994. [11] M.P. Mattis, M. Mukerjee, Phys. Rev. Lett. 61 (1988) 1344. [12] M.P. Mattis, M. Mukerjee, Phys. Rev. D 39 (1989) 2058. [13] E. Jenkins, A.V. Manohar, Phys. Rev. Lett. 93 (2004) 022001; E. Jenkins, A.V. Manohar, JHEP 0406 (2004) 039. [14] E. Guadagnini, Nucl. Phys. B 236 (1984) 35; P.O. Mazur, M.A. Nowak, M. Praszałowicz, Phys. Lett. B 147 (1984) 137; A.V. Manohar, Nucl. Phys. B 248 (1984) 19; M. Chemtob, Nucl. Phys. B 256 (1985) 600; S. Jain, S.R. Wadia, Nucl. Phys. B 258 (1985) 713. [15] R.L. Jaffe, F. Wilczek, Phys. Rev. Lett. 91 (2003) 232003. [16] T.D. Cohen, Phys. Rev. D 70 (2004) 074023. Physics Letters B 619 (2005) 124–128 www.elsevier.com/locate/physletb

An estimate of the chiral condensate from unquenched lattice QCD

C. McNeile

Theoretical Physics Division, Department of Mathematical Sciences, University of Liverpool, Liverpool L69 3BX, UK Received 21 April 2005; received in revised form 31 May 2005; accepted 31 May 2005 Available online 9 June 2005 Editor: N. Glover

Abstract Using the parameters in the chiral Lagrangian obtained by MILC from their unquenched lattice QCD calculations with 2 + 1 ¯ 3 ﬂavours of sea quarks, I estimate the chiral condensate. I obtain the result ψψ(2GeV)/nf = −(259 ± 27 MeV) in the MS scheme. I compare this value to other determinations.  2005 Elsevier B.V. All rights reserved.

1. Introduction used to date. The results from MILC’s lattice calculation have been successfully compared against ex- The spontaneous breaking of chiral symmetry plays periment for many quantities that are stable to strong an important role in the dynamics of low energy QCD. decay [10]. MILC’s lattice calculations use the im- The non-zero value for the chiral condensate is caused proved staggered fermion action. The method of per- by spontaneous chiral symmetry breaking. The chiral forming unquenched calculations with improved stag- condensate is a basic parameter in the QCD sum rule gered quarks has potential problems with non-locality approach to computing hadronic quantities [1,2] so a (see [11] for a review), however this problem has numerical value from lattice QCD is a valuable check not shown up in the comparison of results currently on that formalism. computed against experiment. The unquenched cal- There have been many quenched lattice QCD calculations use 2 light quarks in the sea and one sea culations that have reported a value for the chiral quark ﬁxed at approximately the strange quark mass. condensate [3–6]. The MILC Collaboration [7–9] The data set from MILC has two lattice spacings have been performing unquenched lattice QCD cal- (0.09 fm and 0.125 fm). All the volumes were larger culations with the most realistic set of parameters than 2.5 fm. The lightest pion mass used in MILC’s calculation is 250 MeV. Although the data from MILC’s unquenched cal- E-mail address: [email protected] (C. McNeile). culations has been used do much important phenom-

0370-2693/$ – see front matter  2005 Elsevier B.V. All rights reserved. doi:10.1016/j.physletb.2005.05.078 C. McNeile / Physics Letters B 619 (2005) 124–128 125 enology, it has not been used to estimate a physical the GMOR and scalar correlators to extract a consis- value for the chiral condensate. In this Letter, I esti- tent result for the chiral condensate [3]. mate the chiral condensate from the latest published The use of the parameters from the chiral La- MILC data. grangian that have been fitted from lattice data to estimate the chiral condensate may miss some interesting physics. Stern and collaborators [18–20] have 2. Extracting the chiral condensate using chiral proposed a scenario where the chiral condensate for perturbation theory three flavours is very small, so the higher order terms in Eq. (1) contribute the majority of the meson mass. In an extensive calculation [9] the MILC Collabora- The possibility that the two flavour chiral condensate tion fitted the squared masses of pseudo-scalar mesons is small has been ruled out in the theory with nf = 2 and pseudo-scalar decay constants to the expressions by comparison of chiral perturbation theory with ππ from chiral perturbation theory [12–15] scattering [21]. 2 + = +··· MPS/(mx my) µ(1 ), (1) where MPS is the mass of the pseudo-scalar meson 3. Extracting the result made of quarks x and y with masses mx and my respectively. In Eq. (1) the dots represent higher order MILC have recently reported a fit of the numerical terms that MILC included in the fits. data for light pseudo-scalar meson masses and decay Chiral perturbation theory relates the chiral con- constants to expressions derived from staggered chiral ¯ densate ψψ/nf to µ perturbation theory [9]. A simultaneous fit was done to the data at the two different lattice spacings. The ¯ 1 2 fit functions for the masses and decay constants used ψψ/nf =− µf , (2) 2 NNLO analytic terms, as well as lattice artifact terms where f is the pion decay constant in the chiral limit. arising from working at fixed lattice spacing and fixed I use the normalisation of the axial current such that lattice volume. the physical pion decay constant is 131 MeV. When IuseEq.(2) with the results from MILC’s exten- there are no correction terms in Eq. (1), these two sive fits to chiral perturbation theory. One important equations are known as the Gell-Mann–Oakes–Renner issue is the renormalisation of Eq. (2). The MILC Col- formulae (GMOR). Strictly speaking the GMOR rela- laboration used a conserved axial current, so no renor- tion can only be used to extract the chiral condensate malisation factor is needed for the pion decay constant if there are no higher order corrections to Eq. (1).Chi- in the chiral limit (f ). The µ term does need to be ral perturbation theory is the generalisation of GMOR renormalised. The µ term is renormalised with ZS to higher orders in the pion mass. The extraction of that is related to the renormalisation of the mass via the chiral condensate from the parameters of the chi- Z = 1 .TheZ renormalisation factor computed S Zm m ral Lagrangian obtained by fits to lattice data in the by the HPQCD, MILC, and UKQCD Collaborations continuum large volume limits may be an empirical is reported in [22]. approach, but I believe it is valuable. ¯ 1 4 2 In Eq. (2), ψψ is the sum of the chiral conden- Zm(Λ) = 1 + α b − − ln(aΛ) , (3) sates for each sea quark. In the appropriate limit where u0 3π π the masses of all the sea quarks go to zero in the infi- where α is the QCD coupling, b = 0.5432, and u0 is nite volume limit, then the chiral condensate of each the tadpole factor. I always quote numbers at the scale sea quark is the same [16,17]. 2 GeV in the MS scheme. Another technique to extract the chiral condensate The MILC analysis [9] is a combined fit to data from lattice QCD calculations is to compute the scalar at two lattice spacings. The convention for the quark correlator directly [3,5,6]. The results are then extrap- mass renormalisation is to convert the quark masses to olated to the zero quark mass limit. In a quenched lat- the lattice scheme on the fine lattice. Hence, the ZS tice QCD calculation, Becirevi´ c´ and Lubicz used both factor for the quark masses on the fine lattice must 126 C. McNeile / Physics Letters B 619 (2005) 124–128

Table 1 Results for chiral condensate in the MS scheme at a scale of 2 GeV for various fits that MILC did to their data. The coarse and fine columns correspond to the mass ranges used in the fits with the data on the coarse and fine lattice spacing. The ms is the mass of the strange sea quark in MILC’s unquenched calculation. The points column is the number of data used in the fit ¯ Fit Points Coarse Fine ψψ(2GeV)/nf + + ± 3 A94mx my < 0.40ms mx my < 0.54ms (269 17 MeV) + + ± 3 B 240 mx my < 0.70ms mx my < 0.80ms (250 21 MeV) + + ± 3 C 316 mx my < 1.10ms mx my < 1.14ms (249 15 MeV) be used to convert µ into the MS scheme at a scale strange quark mass. In the latter case the three flavour of 2 GeV. Using the numbers quoted by MILC [9] I chiral condensate could be extracted with the help of get Zm(2GeV) = 1.195. The two loop computation of chiral perturbation theory [25]. In the MILC calcula- Zm in lattice perturbation theory is underway [23].A tion the mass of the strange quark is fixed. However, non-perturbative estimate of Zm, using similar tech- the data is analysed with three flavour chiral perturba- niques to those used by JLQCD [24] to renormalise tion theory. The mass of the strange quark was slightly the quark mass with Kogut–Susskind fermions, would incorrect on the coarse lattice, so some small extrapo- be useful. I use the estimate of 9% from MILC [9] lation in the data is done for the strange quark mass. for the error due to the truncation of the perturbative It is hard to give a definitive answer to the flavour series. dependence of the condensate extracted in this Letter One advantage of the MILC Collaboration’s calcu- without further analysis. lation is that a consistent lattice spacing is obtained I could not find any recent calculations of the chi- from many different quantities that are stable against ral condensate from unquenched lattice QCD calcu- strong decay [10]. The chiral condensate involves the lations with two flavours of sea quarks. Early work third power of the lattice spacing, so any errors in the is reviewed in [26]. I have used the chiral perturba- choice of scale are amplified. I used MILC’s value [8] tion theory approach to extract the chiral condensate r1 = 0.317(7) fm to convert the µ and f parameters from a recent unquenched lattice QCD calculation by into physical units. the JLQCD Collaboration [27]. This lattice calcula- MILC’s analysis [22] of their data used a number tion was done at a fixed lattice spacing of 0.089 fm. of fits that included various subsets of their data. In The lightest sea quark mass was half the strange quark ¯ Table 1, I compute ψψ(2GeV)/nf using Eq. (2) mass. The axial vector current used in JLQCD’s cal- and the perturbative matching factor with the values culation needs to be renormalised. I used tadpole im- for µ and f in Table IV of [9]. The results for the proved perturbation theory with a simple boosted cou- three main fits (called A, B and C) are in Table 1.The pling to estimate the required renormalisation (using errors for the chiral condensate are dominated by the the summary of results in the appendix of [28]). The error on the pion decay constant in the chiral limit (f ). numerical value of renormalisation factor is 0.45, so MILC use combinations of the results from the fits some kind of non-perturbative renormalisation is re- A, B, and C to estimate the central values and the sys- quired for a definitive answer, hence the error for tematic errors. I take the average of the result for fit A JLQCD estimate is unreliable. Dürr [29] has previ- and B as the central value. ously noted the problems with extracting the chiral condensate from unquenched calculations done by the ¯ 3 ψψ(2GeV)/nf =−0.018(5) GeV (4) CP-PACS and UKQCD Collaborations. =−(259 ± 27 MeV)3. (5)

I now discuss the important issue as to whether 4. Conclusion and comparison to other work 2 + 1 = 3. There are two main possibilities, that the chiral perturbation theory analysis is sensitive to the In Table 2, I compare my analysis of the MILC and three flavour chiral condensate or to the two flavour JLQCD data to a selection of recent lattice results for chiral condensate with one sea quark fixed at the the chiral condensate. C. McNeile / Physics Letters B 619 (2005) 124–128 127

Table 2 Results for chiral condensate in the MS scheme at a scale of 2 GeV ¯ ¯ Group nf ψψ(2GeV)/nf ψψ(2GeV)/nf This work, MILC 2 + 1 −0.017(5) GeV3 −(259 ± 27 MeV)3 This work, JLQCD 2 −0.009(1) GeV3 −(209 ± 8 MeV)3 Becirevi´ c´ and Lubicz [3] 0 −(273 ± 19 MeV)3 Becirevi´ c´ and Lubicz [3] 0 −(312 ± 24 MeV)3 Giusti et al. [4] 0 −0.0147(8)(16)(12) GeV3 −(245(4)(9)(7) MeV)3 Gimenez et al. [5] 0 −(265 ± 5 ± 22 MeV)3 Hernandez et al. [30] 0 −(268(12) MeV)3 DeGrand [6] 0 −(282(6) MeV)3 Giusti et al. [31] 0 −(267(5)(15) MeV)3 Blum et al. [32] 0 −(256(8) MeV)3

Giusti et al. [4] note that their numbers are compa- to other determinations [38–40], however, this calcu- rable to estimates of the chiral condensate from sum lation is the first large scale lattice calculation with rules [1,2]. The first entry in Table 2 from Becire-´ 2 + 1 flavours of dynamical light quarks. The only vic´ and Lubicz comes from a GMOR analysis and other group to have published results for the strange the second is from the pseudo-scalar vertex. Penning- quark mass from unquenched lattice QCD calculations ton [33] reviews various calculations of the chiral con- with 2 + 1 flavours of sea quarks is the JLQCD/CP- densate from lattice, and sum rules and estimates the PACS Collaboration. At a fixed lattice spacing, they 3 size of the chiral condensate to be ∼−(270 MeV) . obtain ms(2GeV) between 80 and 90 MeV [41].The Jamin [34] obtained a value for the chiral condensate JLQCD/CP-PACS Collaboration are planing to com- of ∼−(267 ± 16 MeV)3 fromQCDsumrules. pute the strange quark mass at other lattice spacings to From Table 2, I note that the result from MILC is do a continuum extrapolation. essentially consistent with the other results. Descotes- Unfortunately, the data in Table 2 are not precise Genon et al. have argued that the chiral condensate enough to understand the systematics of quark mass with three sea quarks should be less than that from determinations between lattice QCD calculations with QCD with two light sea quarks [19].Giventheas- 2 and 3 flavours of sea quarks. A reduction in the sumptions in this analysis it does not look as though size of errors in the estimates of the chiral conden- the chiral condensate has a strong dependence on the sate from lattice calculations with a different number number of quarks in the sea. The Columbia [35,36] of sea quarks would help compare the results for quark group claimed to see a reduction in chiral symme- masses from different calculations. try breaking from unquenched calculations with 0, 2, The main theoretical concern with unquenched cal- and 4 flavours of sea quarks, but the analysis of their culations with improved staggered fermions is that data was complicated by finite size effects. From lat- the formalism requires taking the fourth of the deter- tice QCD calculations Iwasaki et al. [37] find that the minant that controls the sea quark dynamics. There theory becomes deconfined for nf > 6. have been a number of theoretical papers on this From Eq. (1), the higher value for the chiral con- topic [42–48] (the issues are adroitly explained by De- densate is correlated with smaller quark masses. This Grand [11]). None of the theory papers on the locality trend was noted by Gupta and Bhattacharya in a re- of improved staggered fermions have satisfactorily review of lattice data before 1997 [26]. Although when solved the issue for QCD. One of the main tests of higher order mass corrections are included in Eq. (1) the fourth root trick is comparison of the lattice data this is not so obvious. with the results from chiral perturbation theory [9], From the same data set, the MILC, HPQCD and hence it is important to fully understand all aspects of UKQCD [22] have obtained the mass of the strange chiral perturbation theory applied to the MILC data. MS = Crosschecks on the chiral perturbation theory analysis quark to be ms (2GeV) 76(0)(3)(7)(0) MeV. This value for the strange quark mass is low relative of MILC’s data are also very valuable, because of the 128 C. McNeile / Physics Letters B 619 (2005) 124–128 important phenomenology extracted from their work. [19] S. Descotes-Genon, L. Girlanda, J. Stern, JHEP 0001 (2000) An independent computation of the magnitude of the 041, hep-ph/9910537. chiral condensate using different correlators would be [20] S. Descotes-Genon, Eur. Phys. J. C 40 (2005) 81, hep- a useful crosscheck [49,50] on the estimate from the ph/0410233. [21] H. Leutwyler, Czech. J. Phys. 52 (2002) B9, hep-ph/0212325. chiral perturbation theory fits. MILC do study the chi- [22] HPQCD, MILC, UKQCD Collaborations, C. Aubin, et al., ral condensate in their work on finite temperature [51]. Phys. Rev. D 70 (2004) 031504, hep-lat/0405022. [23] H.D. Trottier, Nucl. Phys. B (Proc. Suppl.) 129 (2004) 142, hep-lat/0310044. [24] JLQCD Collaboration, S. Aoki, et al., Phys. Rev. Lett. 82 Acknowledgements (1999) 4392, hep-lat/9901019. [25] J. Gasser, H. Leutwyler, Nucl. Phys. B 250 (1985) 465. [26] R. Gupta, T. Bhattacharya, Phys. Rev. D 55 (1997) 7203, hep- I thank Chris Michael for discussions and for read- lat/9605039. ing the paper. Thanks to Claude Bernard for an ex- [27] JLQCD Collaboration, S. Aoki, et al., Phys. Rev. D 68 (2003) planation of the quark mass conventions in the MILC 054502, hep-lat/0212039. data. [28] T. Bhattacharya, R. Gupta, W.-J. Lee, S.R. Sharpe, Phys. Rev. D 63 (2001) 074505, hep-lat/0009038. [29] S. Dürr, Eur. Phys. J. C 29 (2003) 383, hep-lat/0208051. [30] P. Hernandez, K. Jansen, L. Lellouch, H. Wittig, JHEP 0107 References (2001) 018, hep-lat/0106011. [31] L. Giusti, C. Hoelbling, C. Rebbi, Phys. Rev. D 64 (2001) 114508, hep-lat/0108007. [1] H.G. Dosch, S. Narison, Phys. Lett. B 417 (1998) 173, hep- [32] T. Blum, et al., Phys. Rev. D 69 (2004) 074502, hep- ph/9709215. lat/0007038. [2] S. Narison, hep-ph/0202200. [33] M.R. Pennington, hep-ph/0207220. [3] D. Becirevi´ c,´ V. Lubicz, Phys. Lett. B 600 (2004) 83, hep- [34] M. Jamin, Phys. Lett. B 538 (2002) 71, hep-ph/0201174. ph/0403044. [35] F.R. Brown, et al., Phys. Rev. D 46 (1992) 5655, hep- [4] L. Giusti, F. Rapuano, M. Talevi, A. Vladikas, Nucl. Phys. lat/9206001. B 538 (1999) 249, hep-lat/9807014. [36] R.D. Mawhinney, hep-lat/9705030. [5] V. Gimenez, V. Lubicz, F. Mescia, V. Porretti, J. Reyes, hep- [37] Y. Iwasaki, K. Kanaya, S. Kaya, S. Sakai, T. Yoshie, Phys. Rev. lat/0503001. D 69 (2004) 014507, hep-lat/0309159. [6] MILC Collaboration, T. DeGrand, Phys. Rev. D 64 (2001) [38] V. Lubicz, Nucl. Phys. B (Proc. Suppl.) 94 (2001) 116, hep- 117501, hep-lat/0107014. lat/0012003. [7] C.W. Bernard, et al., Phys. Rev. D 64 (2001) 054506, hep- [39] H. Wittig, Nucl. Phys. B (Proc. Suppl.) 119 (2003) 59, hep- lat/0104002. lat/0210025. [8] C. Aubin, et al., Phys. Rev. D 70 (2004) 094505, hep- [40] P.E.L. Rakow, Nucl. Phys. B (Proc. Suppl.) 140 (2005) 34, hep- lat/0402030. lat/0411036. [9] MILC Collaboration, C. Aubin, et al., Phys. Rev. D 70 (2004) [41] CP-PACS Collaboration, T. Ishikawa, et al., Nucl. Phys. B 114501, hep-lat/0407028. (Proc. Suppl.) 140 (2005) 225, hep-lat/0409124. [10] HPQCD, UKQCD, MILC, Fermilab Lattice Collaborations, [42] S. Dürr, C. Hoelbling, Phys. Rev. D 69 (2004) 034503, hep- C.T.H. Davies, et al., Phys. Rev. Lett. 92 (2004) 022001, hep- lat/0311002. lat/0304004. [43] S. Dürr, C. Hoelbling, U. Wenger, Phys. Rev. D 70 (2004) [11] T. DeGrand, hep-ph/0501183. 094502, hep-lat/0406027. [12] W.-J. Lee, S.R. Sharpe, Phys. Rev. D 60 (1999) 114503, hep- [44] S. Dürr, C. Hoelbling, Phys. Rev. D 71 (2005) 054501, hep- lat/9905023. lat/0411022. [13] MILC Collaboration, C. Bernard, Phys. Rev. D 65 (2002) [45] HPQCD, UKQCD Collaborations, E. Follana, A. Hart, C.T.H. 054031, hep-lat/0111051. Davies, Phys. Rev. Lett. 93 (2004) 241601, hep-lat/0406010. [14] C. Aubin, C. Bernard, Phys. Rev. D 68 (2003) 074011, hep- [46] Y. Shamir, Phys. Rev. D 71 (2005) 034509, hep-lat/0412014. lat/0306026. [47] D.H. Adams, hep-lat/0411030. [15] C. Aubin, C. Bernard, Phys. Rev. D 68 (2003) 034014, hep- [48] F. Maresca, M. Peardon, hep-lat/0411029. lat/0304014. [49] S.J. Hands, M. Teper, Nucl. Phys. B 347 (1990) 819. [16] S. Scherer, hep-ph/0210398. [50] G. Salina, A. Vladikas, Nucl. Phys. B 348 (1991) 210. [17] C. Vafa, E. Witten, Nucl. Phys. B 234 (1984) 173. [51] MILC Collaboration, C. Bernard, et al., Phys. Rev. D 71 (2005) [18] S. Descotes-Genon, N.H. Fuchs, L. Girlanda, J. Stern, Eur. 034504, hep-lat/0405029. Phys. J. C 34 (2004) 201, hep-ph/0311120. Physics Letters B 619 (2005) 129–135 www.elsevier.com/locate/physletb

Importance of threshold corrections in quark–lepton complementarity

Sin Kyu Kang a,C.S.Kimb, Jake Lee b

a School of Physics, Seoul National University, Seoul 151-741, Republic of Korea b Department of Physics, Yonsei University, Seoul 120-749, Republic of Korea Received 7 February 2005; received in revised form 18 May 2005; accepted 21 May 2005 Available online 4 June 2005 Editor: M. Cveticˇ

Abstract

The recent experimental measurements of the solar neutrino mixing angle θsol and the Cabibbo mixing angle θC reveal + π a surprising relation, θsol θC 4 . We note that the lepton mixing matrix derived from quark–lepton uniﬁcation can lead to a shift of the complementarity relation at low energy. While the renormalization group effects generally lead to additive contribution on top of the shift, in this Letter, we show that the threshold corrections which may exist in some intermediate scale new physics such as supersymmetric standard model can diminish it, so we can achieve the complementarity relation at a low energy. Finally, we discuss a possibility to achieve the complementarity relation at a high energy by taking particular form of non-symmetric form of down Yukawa matrix.  2005 Elsevier B.V. All rights reserved.

PACS: 14.60.Pq; 12.15.Ff; 12.10.Dm; 11.10.Gh

Recently, it has been noted that the solar neutrino mixing angle θsol required for a solution of the solar neutrino problem and the Cabibbo angle θC reveal a surprising relation π θ + θ , (1) sol C 4 ◦ ◦ which is satisfied by the experimental results θsol + θC = 45.4 ± 1.7 to within a few percent accuracy [1–3]. This quark–lepton complementarity (QLC) relation (1) has been interpreted as an evidence for certain quark– lepton symmetry or quark–lepton unification as shown in Refs. [4–6]. Yet, it can be a coincidence in the sense that reproducing the exact QLC relation (1) at low energy scale in the framework of grand unification depends on the renormalization effects whose size can vary with the choice of parameter space. But, we believe that such a

E-mail address: [email protected] (C.S. Kim).

0370-2693/$ – see front matter  2005 Elsevier B.V. All rights reserved. doi:10.1016/j.physletb.2005.05.065 130 S.K. Kang et al. / Physics Letters B 619 (2005) 129–135 coincidence does not necessarily mean that attempts to catch a deep meaning behind the QLC relation were in vain. Anyway, establishing the origin of the QLC relation may be a challenge that underlying theory of ﬂavor should address. To effectively describe the deviation from maximal mixing of solar neutrino, small mixing element Ue3 and possible deviation from maximal mixing of atmospheric neutrino, a parametrization of the PMNS mixing matrix in terms of a small parameter whose magnitude can be interestingly around sin θC has been proposed as follows [7–10]:

† UPMNS = U (λ)Ubimax. (2)

Here U(λ) is a mixing matrix parameterized in terms of a small parameter λ and Ubimax corresponds to the bi- maximal mixing matrix [11]. Among possible generic structures of the matrix U(λ) which are compatible with experimental results on the neutrino oscillations, the “CKM-like” form of U(λ) has rather profound implication in view of the connection between quarks and leptons. In this Letter, we first of all show that the lepton mixing matrix given in the form of Eq. (2) with the “CKM-like” U(λ)∼ UCKM can be indeed realized in the framework of grand unification with symmetric Yukawa matrices when we incorporate seesaw mechanism, and examine whether or not UPMNS reflecting quark–lepton unification given by (2) can predict the QLC relation (1) exactly. We see that the solar mixing angle derived from UPMNS leads to correction to the QLC relation which can be more than 1σ deviation from the QLC relation as similarly shown in [5]. Notice that while the QLC relation holds at a low energy, the corresponding relation derived from the mixing matrix given in the form of Eq. (2) is in fact realized at a high scale such as seesaw scale or unification scale. Thus, it is necessary to take into account the renormalization effects on the lepton mixing matrix when one compares the prediction at a high energy scale with the QLC relation observed at low energy scale. One can then expect that the deviation from the QLC relation is explained by renormalization effects. In the SM, the renormalization effect through the renormalization group (RG) running down to the weak scale is negligible because of small charged lepton Yukawa couplings. But, in some models such as the minimal supersymmetric standard model (MSSM), the renormalization effect on the leptonic mixing angle θ12 depends on the type of light neutrino mass spectrum as well as model parameters [12–14]. It is also known that in MSSM with large tan β and the quasi-degenerate neutrino mass spectrum the RG effects are generally large and can enhance the mixing angle θ12 at low energy [12–14]. Such an enhancement of θ12 is not suitable for achieving the QLC relation (1) at low energy because we need to diminish the deviation so as to get the exact QLC relation (1) at a low energy. In this Letter, we show that the sizeable threshold corrections which may exist in the MSSM [15–17] can diminish the deviation from the QLC relation while keeping the mixing angle θ23 almost maximal and θ13 small, so that the QLC relation at low energy can be achieved in the case that the RG effects are suppressed. Some conditions on the parameters to realize the QLC relation will be discussed. Finally, we propose a possible way to achieve the QLC relation by taking a non-symmetric form of the down Yukawa matrix. The flavor mixings stem from the mismatch between the left-handed rotations of the up-type and down-type quarks, and the charged leptons and neutrinos. For our purpose, it is useful to work in a basis where the quark and diag † lepton Yukawa matrices are related. In general, the quark Yukawa matrices Yu,Yd are given by Yu = UuYu Vu , = diag † = † Yd Ud Yd Vd , from which the observable quark mixing matrix is deduced as UCKM Uu Ud . For the charged = diag † lepton sector, the Yukawa matrix is given by Yl UlYl Vl . For the neutrino sector, we introduce one right- handed singlet neutrino per family which leads to the seesaw mechanism, according to which the light neutrino mass matrix after the breakdown of the electroweak symmetry is given by

= 1 T = diag † 1 ∗ diag T Mν MDirac MDirac U0MDiracV0 V0 MDiracU0 , (3) MR MR S.K. Kang et al. / Physics Letters B 619 (2005) 129–135 131 where U0 and V0 are the left-handed and right-handed mixing matrices of the Dirac neutrino mass matrix, respectively. We can then rewrite Mν as follows = diag T T Mν U0VM Mν VM U0 , (4) where V represents the rotation of Mdiag V † 1 V ∗Mdiag . The observable PMNS mixing matrix can then be M Dirac 0 MR 0 Dirac written as [10] = † = † UPMNS Ul Uν Ul U0VM . (5)

Note that equating the above expression for UPMNS with Eq. (2), we get the “CKM-like” form of U(λ): † = † † U (λ) Ul U0VM Ubimax. (6) In order for U †(λ) to have “CKM-like” small mixing angles we have three possible choices of U †(λ):   †  Ul , U †(λ) = U †U , (7)  l 0 † Ul U0V , = † † where V VM Ubimax. The first choice of U (λ) indicates that the maximal mixing angles in Ubimax are cancelled 0 out by the mixing angles in the combination of U VM , while the second choice implies VM = Ubimax.Thelast form is the most general one, which shows that the maximal mixing angles in Ubimax are not completely cancelled out, and its actual form is not unique. In view of the quark–lepton unification, the second case is more natural than others because down-type (up-type) quarks are related with charged lepton (Dirac neutrino) sector in grand unification. In such a case, the form of VM is taken to be almost bi-maximal mixing because it is natural to suppose that the structure of the lepton mixing matrix to a leading order is the bi-maximal mixing, whereas the CKM matrix is an identity matrix, which corresponds to U †(λ) = IinEq.(2), and then the QLC relation can emerge from quark–lepton unification. However, the bi-maximal mixing pattern of VM is not necessarily required. It is in fact possible to take VM to be small mixing or even identity matrix at GUT scale and then to generate two large mixing angles in the lepton mixing matrix by radiative magnification through evolving RG equations down to the weak scale [13,14]. Now, let us consider how the PMNS mixing matrix given by Eq. (5) can be related with CKM mixing matrix in the context of quark–lepton unification. (A) Minimal quark–lepton unification Since the down-type quarks and the charged leptons are in general assigned into a multiplet in grand unification, = T = T we assume that the following simple relations hold in the minimal models of grand unification, Ye Yd , Yu Yu . = ∗ Then, we deduce that Ul Vd from which = T UPMNS Vd U0VM . (8)

From this expression for UPMNS, we see that the contribution of UCKM may appear in UPMNS if we further assume Yν = Yu which can be realized in some larger unified gauge group such as SO(10). Then, one can obtain [10] = T † UPMNS Vd Ud UCKMVM . (9) In addition, requiring symmetric form of the down-type quark Yukawa matrix, we finally obtain = † UPMNS UCKMVM , (10) where the mixing matrix VM has bi-maximal mixing pattern as explained above. In this way, UPMNS can be connected with UCKM. We note that taking the bi-maximal mixing form of VM is equivalent to taking MR in Eq. (3) 132 S.K. Kang et al. / Physics Letters B 619 (2005) 129–135 as follows, in the basis where V0 is absorbed into the heavy Majorana neutrino field:  2  αm βmD mD −βmD mD D1 1 2 1 3 1  1 2 1  MR =  βmD mD α + γ m − α + γ mD mD  , (11) 2 1 2 2 D2 2 2 3 1 1 2 −βmD mD − α + γ mD mD α + γ m 1 3 2 √2 3 2 D3 = −1 + −1 = − −1 + −1 = −1 where α m1 m2 , β ( m1 m2 )/ 2, γ m3 and mDi , mi stand for the mass eigenvalues of Dirac mass matrix and light neutrino mass matrix, respectively. To see whether the parametrization of UPMNS given by (10) can lead to the QLC relation (1), it is convenient to present UPMNS for the CP-conserving case as follows:

U = U † U m U m PMNS CKM 23 12 π ≡ U (θ )U (θ )U − θ , (12) 23 23 13 13 12 4 12 m m where U12 and U23 correspond to the maximal mixing between (1, 2) and (2, 3) generations, respectively. Then, the mixing angles θij in the second line of Eq. (12) can be presented in terms of Wolfenstein parameter λ as follows: 1 3 1 1 2 1 sin θ12 √ λ + O λ , sin θ23 −√ 1 − λ , sin θ13 −√ λ. (13) 2 2 2 2

The solar neutrino mixing parameter sin θsol in this parametrization becomes π λ√ sin θ sin − θ + 2 − 1 . (14) sol 4 C 2 Thus, we see that the neutrino mixing matrix (12) originating from the quark–lepton unification obviously leads ◦ 2 to a shift of the relation (1). Numerically, the shift amounts to δθsol 3 and thus δ sin θsol 0.05 which is more than 1σ deviation from the recent measurement of the solar neutrino experiment. While a dedicated experiment to 2 measure θ12 with a sensitivity of a few % to sin θ12 would be expected to confirm or rule out the deviation, we can expect that renormalization effects on the neutrino mixing matrix (12) may fill the gap between the QLC relation and the prediction (14) from high energy mixing matrix. Later, we will discuss the renormalization effect in detail. (B) Realistic quark–lepton unification Although the minimal quark–lepton unification can lead to an elegant relation between PMNS mixing and CKM mixing as shown in the above, it indicates undesirable mass relations between quarks and leptons at the GUT scale diag = diag † = † such as md ml . Recently, the following form for Ul U0 U (λ) has been suggested based on a well-known empirical relation |V | md 3 me [18], us ms mµ   1 − λ 1 θλ2  3 3  † = †  λ  U (λ) Ul U0 3 12θλ , (15) −θλ2 −2θλ 1 where the deviation from unitarity is just of order θλ3. This form of mixing matrix can be obtained by introducing the Higgs sector transforming under the representation 45 of SU(5) or 126 of SO(10) [19]. In this case, the solar neutrino mixing sin θsol is then given by π λ √ 1 sin θ sin − θ + 2 − . (16) sol 4 C 2 3 S.K. Kang et al. / Physics Letters B 619 (2005) 129–135 133

◦ Numerically, the deviation from the QLC relation amounts to δθsol 7 , much more than in the observed QLC relation. One possible way to generate the correct prediction for sin θsol based on the lepton mixing with Ul given by Eq. (15) is to abandon the exact bi-maximal form of the neutrino mixing matrix VM and to consider the generic corrections to the bi-maximal neutrino mixing matrix that can account for the QLC relation [18]. But, we consider an alternative possibility that the threshold corrections can diminish the deviation from the QLC relation. Now, let us examine how the renormalization effects can diminish the deviation from the QLC relation. In general, the radiative corrections to the effective neutrino mass matrix can be presented as follows: = · 0 · Mν I Mν I = · T ∗ † · I UCKMUbimaxMDUbimaxUCKM I = · T m∗ m† · I UCKMU23 MD12U23 UCKM I, (17) = [ ] = m∗ m† ≡ = where MD Diag m1,m2,m3 , MD12 U12 MDU12 , and the matrix I IAδAB , (A, B e,µ,τ) stands for the radiative corrections. The correction I generally consists of two parts I = I RG + I TH where I RG are the renormalization group corrections which are explicitly presented in Ref. [20] and I TH are electroweak scale threshold corrections [15]. We note that the flavor blind interactions such as gauge interactions contribute to overall scale of neutrino masses whereas the charged lepton Yukawa interactions generate flavor dependent radiative corrections I in the standard model (SM) and in MSSM. The typical size of RG corrections I RG is known to be about 10−6 in the SM and MSSM with small tan β, and thus negligible. In addition, supersymmetry can induce flavor dependent threshold corrections related with slepton mass splitting which can dominate over the charged lepton Yukawa corrections [16].EvenifIτ is the dominant contribution in SM, it is not guaranteed at all in MSSM due to the threshold corrections. We have numerically checked that RG evolution from the seesaw scale to the weak scale enhances the size of θ12 in the case that θ13 and θ23 are kept to be small and almost maximal mixing, respectively. Thus, the case of sizable RG effects is not suitable for our purpose. Instead, we examine whether the threshold corrections can be suitable for diminishing the deviation from the QLC relation while keeping θ23 nearly maximal and θ13 small in the case that RG effect is negligible. To achieve our goal, we note that the contribution Ie should be dominant over Iµ,τ because only Ie can lead to the right amount of the shift of θ12 while keeping the changes of θ23 and θ13 small. Taking |Ie||Iµ,τ |, the neutrino mass matrix corrected by the leading contributions is rewritten as follows: T m∗ [ + ∗] + † m† Mν UCKMU23 ID IeΛλ MD12 ID IeΛλ U23 UCKM, (18) 2 where ID is 3 × 3 identity matrix, and the matrix Λλ is given up to λ order by   1 − √λ − √λ 2 2  2 2  Λ =  − √λ λ λ  for minimal unification, (19) λ  2 2 2  − √λ λ2 λ2  2 2 2  1 − √λ − √λ 3 2 3 2  2 2  =  − √λ λ λ  for realistic unification. (20)  3 2 18 18  − √λ λ2 λ2 3 2 18 18 Let us discuss how much the lepton mixing angles can be shifted by the renormalization effects. We first of all do numerical analysis in a model independent way based on the form given by Eqs. (19)–(20). For our numerical 2 ≡ 2 − 2 × −5 2 2 ≡ 2 − 2 × −3 2 analysis, we take msol m2 m1 7.1 10 eV and matm m3 m2 2.5 10 eV . Regarding Mν as the light neutrino mass matrix at low energy scale and varying the parameter Ie and the smallest light neutrino mass m1, we find which parameter set (Ie,m1) can lead to the QLC relation exactly and the results are presented in Table 1. We note that the required values for Ie are negative. The first and the second column in Table 1 correspond 134 S.K. Kang et al. / Physics Letters B 619 (2005) 129–135

Table 1 Parameter set (Ie,m1) leading to the QLC relation. The columns (A) and (B) correspond to the minimal uniﬁcation and realistic case, respectively (A) (B)

m1 (eV) Ie Ie − − 0.15 −4.0 × 10 5 −1.0 × 10 4 − − 0.1 −8.5 × 10 5 −2.2 × 10 4 − − 0.05 −3.4 × 10 4 −8.6 × 10 4

to the minimal unification and realistic case, respectively. In our analysis, we have also checked that θ23 is almost ◦ unchanged, whereas the shift of θ13 is about 1 for both cases (A) and (B). From Table 1, we see that a larger value of Ie is required to achieve the relation (1) as m1 goes down. How can we obtain such a value of Ie while keeping |Ie||Iµ,τ |?AsshowninRef.[17], it requires the existence of new states which gives a dominant contributions to Ie. In MSSM, it can easily be realized by taking into account the contribution of chargino (pure W-ino). In the case of a diagonal slepton mass matrix in the same basis where the charged lepton mass matrix is diagonal, the contribution is presented by [17] g2 1 1 x2 − 1 x2 − 1 I − + + f ( − x ) − µ ( − x ) , f 2 2 ln 1 f 2 ln 1 µ (21) 32π xf xµ xf xµ 2 where xf = 1 − (Mf /m)˜ and Mf , m˜ are the f th charged slepton mass and W-ino mass, respectively. We see that the size of |Ie| is about 10 times larger than that of |Iµ,τ | for Me ∼ 2Mµ,τ , and the value of Ie becomes negative −4 −3 and of the order of 10 ∼ 10 for xe 0.65, which are required to achieve the exact QLC relation at low energy. We also note that the contribution of the tau Yukawa coupling Yτ in MSSM to Iτ can be as much as Ie for large tan β. Thus, our estimation is suitable for small tan β so that the contribution of Yτ should be negligible compared to Ie. In passing, we discuss a possible way to diminish the correction to QLC appeared in Eq. (14). While we have considered the case with symmetric form of fermion Yukawa matrices so far, non-symmetric types of Yd and Yl are T generally allowed. Then, the term Vd Ud in Eq. (9) is not trivial unity matrix in the non-symmetric Yukawa structure and it can generate additional correction to the solar neutrino mixing parameter sin θsol, which can diminish the correction to QLC. We have checked that sin θsol sin(π/4 − θC) for the case of the minimal unification (A) when T Vd Ud has the form in the leading order of λ, √ √ 1√ (1 − 2)λ −(1 − 2)λ T = − − Vd Ud (1 √ 2)λ 10. (22) (1 − 2)λ 01 For the realistic case (B), one can obtain the QLC relation by taking the matrix form (22) but replacing λ with λ/3. We note that in order to achieve the QLC relation at low energy scale in those cases, renormalization effects should T be negligibly small. But, it is quite arbitrary to generate the form of Vd Ud given above from some underlying symmetries or flavor structure. + = π In summary, while the QLC relation, θsol θC 4 , itself can be an evidence for the quark–lepton unification, it can be a coincidence in the sense that the relation achieved at low energy in the framework of grand unification strongly depends on the renormalization effects whose size can vary with the choice of parameter space. In this Letter, we have found that the lepton mixing matrix derived from quark–lepton unification can lead to a shift of the complementarity relation at a high energy. While the renormalization group effects generally lead to additive contribution on top of the shift, we show that the threshold corrections which may exist in the supersymmetric standard model diminish it, so we can achieve the complementarity relation at a low energy. In addition, we commented on how to achieve the QLC relation by taking a non-symmetric form of the down Yukawa matrix. S.K. Kang et al. / Physics Letters B 619 (2005) 129–135 135

Acknowledgements

We would like to thank G. Cvetic for careful reading of the manuscript and his valuable comments. The work of S.K.K. was supported in part by BK21 program of the Ministry of Education in Korea and in part by KOSEF grant R02-2003-000-10085-0. The work of C.S.K. was supported in part by CHEP-SRC Program and in part by Grant No. R02-2003-000-10050-0 from BRP of the KOSEF. The work of J.L. was supported in part by BK21 program of the Ministry of Education in Korea and in part by Grant No. F01-2004-000-10292-0 of KOSEF-NSFC International Collaborative Research Grant.

References

[1] S. Hukuda, et al., Super-Kamiokande Collaboration, Phys. Rev. Lett. 86 (2001) 5656; S. Hukuda, et al., Super-Kamiokande Collaboration, Phys. Lett. B 539 (2002) 179. [2] Q. Ahmad, et al., SNO Collaboration, Phys. Rev. Lett. 87 (2001) 071301; Q. Ahmad, et al., SNO Collaboration, Phys. Rev. Lett. 89 (2002) 011301; S. Ahmed, et al., SNO Collaboration, nucl-ex/0309004. [3] G.L. Fogli, et al., hep-ph/0310012; A. Bandyopadhyay, et al., hep-ph/0309174. [4] M. Raidal, hep-ph/0404046. [5] H. Minakata, A.Yu. Smirnov, hep-ph/0405088. [6] P.H. Frampton, R.N. Mohapatra, hep-ph/0407139. [7] C. Giunti, M. Tanimoto, Phys. Rev. D 66 (2002) 053013. [8] W. Rodejohann, Phys. Rev. D 69 (2004) 033005. [9] P.H. Frampton, S.T. Petcov, W. Rodejohann, Nucl. Phys. B 687 (2004) 31. [10] A. Datta, F.-S. Ling, P. Ramond, Nucl. Phys. B 671 (2003) 383; P. Ramond, hep-ph/0405176. [11] F. Vissani, hep-ph/0978483; V.D. Barger, S. Pakvasa, T.J. Weiler, K. Whisnant, Phys. Lett. B 437 (1998) 107; A.J. Baltz, A.S. Goldhaber, M. Goldhaber, Phys. Rev. Lett. 81 (1998) 5730; R.N. Mohapatra, S. Nussinov, Phys. Rev. D 60 (1999) 013002; R.N. Mohapatra, S. Nussinov, Phys. Lett. B 441 (1998) 299; S.K. Kang, C.S. Kim, Phys. Rev. D 59 (1999) 091302; H. Fritzsch, Z.-z. Xing, Prog. Part. Nucl. Phys. 45 (2000) 1, and references therein. [12] J.A. Casas, J.R. Espinosa, A. Ibarra, I. Navarro, Nucl. Phys. B 573 (2000) 652; N. Haba, Y. Matsui, N. Okamura, Eur. Phys. J. C 17 (2000) 513; S. Antusch, J. Kersten, M. Lindner, M. Ratz, Nucl. Phys. B 674 (2003) 401; S. Antusch, J. Kersten, M. Lindner, M. Ratz, M.A. Schmidt, hep-ph/0501272. [13] K.R.S. Balaji, A.S. Dighe, R.N. Mohapatra, M.K. Parida, Phys. Lett. B 481 (2000) 33; K.R.S. Balaji, A.S. Dighe, R.N. Mohapatra, M.K. Parida, Phys. Rev. Lett. 84 (2000) 5034; K.R.S. Balaji, R.N. Mohapatra, M.K. Parida, E.A. Pascos, Phys. Rev. D 63 (2001) 113002; R.N. Mohapatra, M.K. Parida, G. Rajasekaran, Phys. Rev. D 69 (2004) 053007. [14] S. Antusch, M. Ratz, JHEP 0211 (2002) 010; C. Hagedorn, J. Kersten, M. Lindner, Phys. Lett. B 597 (2004) 63. [15] P.H. Chankowski, S. Pokorski, Int. J. Mod. Phys. A 17 (2002) 575; P.H. Chankowski, P. Wasowicz, Eur. Phys. J. C 23 (2002) 249. [16] E.J. Chun, S. Pokorsky, Phys. Rev. D 62 (2000) 053001; B. Brahmachari, E.J. Chun, Phys. Lett. B 596 (2004) 184. [17] P.H. Chankowski, A.N. Ioannisian, S. Pokorski, J.W.F. Valle, Phys. Rev. Lett. 86 (2001) 3488. [18] J. Ferrandis, S. Pakvasa, hep-ph/0409204; J. Ferrandis, S. Pakvasa, hep-ph/0412038. [19] H. Georgi, C. Jarlskog, Phys. Lett. B 86 (1979) 297; K.S. Babu, S.M. Barr, Phys. Rev. Lett. 85 (2000) 1170. [20] J. Ellis, S. Lola, Phys. Lett. B 458 (1999) 310. Physics Letters B 619 (2005) 136–144 www.elsevier.com/locate/physletb

High energy neutrino spin light

A.E. Lobanov

Moscow State University, Department of Theoretical physics, 119992 Moscow, Russia Received 18 February 2005; received in revised form 18 May 2005; accepted 20 May 2005 Available online 31 May 2005 Editor: P.V. Landshoff

Abstract The quantum theory of spin light (electromagnetic radiation emitted by a Dirac massive neutrino propagating in dense matter due to the weak interaction of a neutrino with background fermions) is developed. In contrast to the Cherenkov radiation, this effect does not disappear even if the medium refractive index is assumed to be equal to unity. The formulas for the transition rate and the total radiation power are obtained. It is found out that radiation of photons is possible only when the sign of the particle helicity is opposite to that of the effective potential describing the interaction of a neutrino (antineutrino) with the background medium. Due to the radiative self-polarization the radiating particle can change its helicity. As a result, the active left-handed polarized neutrino (right-handed polarized antineutrino) converting to the state with inverse helicity can become practically “sterile”. Since the sign of the effective potential depends on the neutrino flavor and the matter structure, the spin light can change a ratio of active neutrinos of different flavors. In the ultra relativistic approach, the radiated photons averaged energy is equal to one third of the initial neutrino energy, and two thirds of the energy are carried out by the final “sterile” neutrinos.  2005 Elsevier B.V. All rights reserved.

A Dirac massive neutrino has nontrivial electromagnetic properties. In particular, it possesses nonzero magnetic moment [1]. Therefore a Dirac massive neutrino propagating in dense matter can emit electromagnetic radiation due to the weak interaction of a neutrino with background fermions [2,3]. As a result of the radiation, neutrino can change its helicity due to the radiative self-polarization. In contrast to the Cherenkov radiation, this effect does not disappear even if the refractive index of the medium is assumed to be equal to unity. This conclusion is valid for any model of neutrino interactions breaking spatial parity. The phenomenon was called the neutrino spin light in analogy with the effect, related with the synchrotron radiation power depending on the electron spin orientation (see [4]). The properties of spin light were investigated basing upon the quasi-classical theory of radiation and self- polarization of neutral particles [5,6] with the use of the Bargmann–Michel–Telegdi (BMT) equation [7] and its

E-mail address: [email protected] (A.E. Lobanov).

0370-2693/$ – see front matter  2005 Elsevier B.V. All rights reserved. doi:10.1016/j.physletb.2005.05.046 A.E. Lobanov / Physics Letters B 619 (2005) 136–144 137 generalizations [8,9]. This theory is valid when the radiated photon energy is small as compared with the neutrino energy, and this narrows the range of astrophysical applications of the obtained formulas. In the present Letter, the properties of spin light are investigated basing upon the consistent quantum theory, and this allows the neutrino recoil in the act of radiation to be considered for. The above mentioned restriction is eliminated in this way. On the other hand, the detailed analysis of the results of our investigations shows that the features of the effect depend on the neutrino flavor, helicity and the matter structure [10]. This fact leads to the conclusion that the spin light can initiate transformation of a neutrino from the active state to a practically “sterile” state, and the inverse process is also possible. When the interaction of a neutrino with the background fermions is considered to be coherent, the propagation of a massive neutrino in the matter is described by the Dirac equation with the effective potential [11,12]. In what follows, we restrict our consideration to the case of a homogeneous and isotropic medium. Then in the frameworks of the minimally extended standard model, the form of this equation is uniquely determined by the assumptions similar to those adopted in [13] 1 i∂ˆ − fˆ 1 + γ 5 − m Ψ = 0. (1) 2 ν ν The function f µ is a linear combination of fermion currents and polarizations. The quantities with hats denote µ scalar products of Dirac matrices with 4-vectors, i.e., aˆ ≡ γ aµ. If the medium is at rest and unpolarized then f = 0. The component f 0 calculated in the first order of the perturbation theory is as follows [14–16] √ 0 = + (f ) − (f ) 2 − f 2GF Ieν T3 2Q sin θW (nf nf¯) . (2) f (f ) Here, nf ,nf¯ are the number densities of background fermions and antifermions, Q is the electric charge of (f ) the fermion and T3 is the third component of the weak isospin for the left-chiral projection of it. The parameter Ieν is equal to unity for the interaction of electron neutrino with electrons. In other cases Ieν = 0. Summation is performed over all fermions f of the background. Let us obtain a solution of Eq. (1). Since function f µ = const, Eq. (1) commutes with operators of canonical momentum i∂µ. However, the commonly adopted choice of eigenvalues of this operator as quantum numbers in this problem is not satisfactory. Kinetic momentum components of a particle, related to its group 4-velocity uµ by µ = µ 2 = 2 the relation q mνu , q mν , are more suitable to play the role of its quantum numbers. This choice can be justified, since it is the particle kinetic momentum that can be really observed. The explicit form of the kinetic momentum operator for the particle with spin is not known beforehand, and hence, in order to find the appropriate solutions, we have to use the correspondence principle. It was shown in [8] that when the effects of the neutrino weak interaction are taken into account, the Lorentz invariant generalization of the BMT equation for spin vector Sµ is as follows: ˙µ µν µν µ νλ νλ S = 2µ0 F + G Sν − u uν F + G Sλ , (3) where

µν 1 µνρλ G = e fρuλ, (4) 2µ0 and a dot denotes the differentiation with respect to the proper time τ . Let us introduce the quasi-classical spin wave functions. Such wave functions can be constructed as follows [6,9]. Suppose the Lorentz equation is solved, i.e., the dependence of particle coordinates on proper time is found. Then the BMT equation transforms to ordinary differential equation, whose resolvent determines a one-parametric 138 A.E. Lobanov / Physics Letters B 619 (2005) 136–144 subgroup of the Lorentz group. The quasi-classical spin wave function is represented by a spin-tensor, whose evolution is determined by the same one-parametric subgroup. In the case when the effect of an external electromagnetic field can be neglected as compared with the effect of the neutrino interaction with the background matter, the equation for the neutrino quasi-classical wave function Ψ(τ)is ˙ 5 µν Ψ(τ)= iµ0γ G uνγµuΨˆ (τ), (5) µν =−1 µνρλ µν where G 2 e Gρλ is a tensor dual to G . Obviously, the quasi-classical density matrix of a polarized neutrino takes the form 1 − ρ(τ,τ ) = U(τ,τ ) q(τˆ ) + m 1 − γ 5S(τˆ ) U 1(τ ,τ ), (6) 2 0 0 0 0 where U(τ,τ0) is the resolvent of Eq. (5), and the equation for U(τ,τ0) is i U(τ,τ˙ ) = γ 5(fûˆ −ûf)U(τ,τˆ ). (7) 0 4 0 −iF(x) We note that the operator U(τ,τ0) is defined up to a phase factor e , with the derivative of the exponent with respect to the proper time is equal to zero: F(x)˙ = 0. (8) Let us choose the solution of Eq. (1) in the form [9] Ψ(x)= U τ(x) Ψ0(x), (9) where Ψ0 is a solution of the Dirac equation for a free particle −i(qx) 5 ˆ 0 Ψ0(x) = e (qˆ + mν) 1 − γ S0 ψ . (10) 0 Here ψ is constant bispinor and Ψ0(x) normalized by the condition ¯ Ψ0(x)Ψ0(x) = 2mν. Substitution of the expression (9) in Eq. (1) results in the relation

1 1 1 − qˆ + (∂F)ˆ − fˆ + γ 5fˆ + γ 5N(ˆ fûˆ −ûf)ˆ − m e iF(x)U τ(x) Ψ = 0, (11) 2 2 4 ν 0 where N µ = ∂µτ . Since the commutator [ˆq,U]=0, and the matrix U is nondegenerate, then for this relation to hold the following condition is required 1 1 1 (∂F)ˆ − fˆ + γ 5fˆ + γ 5N(ˆ fûˆ −ûf)ˆ = 0. (12) 2 2 4 It is easy to find out that Eq. (12) is valid only if 1 ∂λF = f λ,eµνρλN f u = 0, 1 − (Nu) f λ + (Nf )uλ = 0. (13) 2 µ ν ρ From two latter equations it follows that f µ(f u) − uµf 2 N µ = . (14) (f u)2 − f 2u2 So f µ = const, then 1 τ = (Nx), F = (f x), (15) 2 A.E. Lobanov / Physics Letters B 619 (2005) 136–144 139 and we can write −i(fx)/2 iζϕ U(x)= e e Λζ . (16) ζ =±1 Here 1 Λ = 1 − ζγ5Sˆ q/mˆ ,ζ± 1, (17) ζ 2 tp ν are spin projection operators with eigenvalues ζ ± 1 respectively, and τ (f q)(f x) − f 2(qx) qµ(f q)/m − f µm ϕ = (f q)2 − f 2m2 = ,Sµ = ν ν . (18) ν 2 − 2 2 tp 2 − 2 2 2 2 (f q) f mν (f q) f mν From the obtained formulas it follows that the eigenvalues of the operator of canonical momentum i∂µ are ζf 2 f µ ζ(fq) P µ = qµ 1 + + 1 − . (19) 2 − 2 2 2 − 2 2 2 (f q) f mν 2 (f q) f mν The dispersion law follows from Eq. (19) in the form 2 = 2 + − 2 − − 2 2 − 2 2 P mν (Pf ) f /2 ζ (Pf ) f /2 f mν. (20) If the medium is at rest and unpolarized then the neutrino total energy and canonical momentum are determined by the formulas

0 0 ε = q + f /2, P = q∆qζ, (21) 0 where ∆qζ = 1 + ζf /2|q|, and µ = 1 | | 0 | | Stp q ,q q/ q , (22) mν i.e., the eigenvalues ζ =±1 determine the helicity of the particle. Consequently, the dispersion law is = | |− 0 2 + 2 + 0 ε ∆ P ζf /2 mν f /2, (23) where ∆ = sign(∆qζ ). Obviously

∂ε = q ∂P q0 is the particle group velocity. The relation (23) differs those used in previous papers (see, for example, [18]) by the multiplier ∆. This is due to the fact that, in these papers the projection of the particle spin on the canonical momentum P and not the helicity of the particle was used as the spin quantum number ζ . The helicity is the projection of the spin on the direction of its kinetic momentum [19–21], because the rest frame of the particle is determined by the condition that its group velocity is equal to zero. In our problem the directions of canonical and kinetic momenta, generally speaking, are different, and hence, the projection of particle spin on the canonical momentum does not coincide with its helicity. From formulas (21), it is seen that if the energy is expressed in terms of the kinetic momentum, then it does not depend on the particle helicity, while the particle canonical momentum is a function of the helicity. Therefore, the statement of the authors of [17], i.e., that the radiation of photons in the process of the spin light emission takes place due to neutrino transitions from the “exited” helicity state to the low-lying helicity state in matter, is not correct. 140 A.E. Lobanov / Physics Letters B 619 (2005) 136–144

Let us consider the process of emitting photons by a massive neutrino in unpolarized matter at rest. In this case, the orthonormalized system of solutions for Eq. (1) is

|∆ | 0 0 0 qζ −i(q +f /2)x iqx∆qζ 5 ˆ 0 Ψ(x)= e e (qˆ + mν) 1 − ζγ Stp ψ . (24) 2q0 The formula for the spontaneous radiation transition probability of a neutral fermion with anomalous magnetic 1 moment µ0 is 4 4 1 4 4 d qd k 2 2 2 P =− d xd y δ k δ q − m Sp Γµ(x)i(x, y; p,ζi)Γν(y)f (y, x; q,ζf ) 2p0 (2π)6 ν × µν ; ph(x, y k). (25) ; ; µν ; Here, i(x, y p), f (y, x q) are density matrices√ of the initial (i) and ﬁnal (f ) states of the fermion, ph(x, y k) µ µν is the radiated photon density matrix, Γ =− 4πµ0σ kν is the vertex function. The density matrix of longitu- dinally polarized neutrino in the unpolarized matter at rest constructed with the use of the solutions of Eq. (1) has the form 1 − 0− 0 0+ 0 + − (x,y; p,ζ) = ∆2 (pˆ + m ) 1 − ζγ5Sˆ e i(x y )(p f /2) i(x y)p∆pζ . (26) 2 pζ ν p After summing over photon polarizations2 and integrating with respect to coordinates we obtain the expression for the transition rate under investigation: µ2 d4qd4k W = 0 δ k2 δ q2 − m2 δ p0 − q0 − k0 δ3(p∆ − q∆ − k)T (p, q), (27) p0 (2π) ν pζi qζf where T(p,q)= 4∆2 ∆2 (pk)(qk) − ζ ζ k0|p|−p0(pk)/|p| k0|q|−q0(qk)/|q| . (28) pζi qζf i f After integrating over k, k0, |q| we obtain the spectral-angular distribution of the ﬁnal neutrino

p0 2 µ0 0 0 0 2 2 2 2 2 W =−ζiζf dq ∆pζ ∆qζ dOδ p − q + 2|p||q|∆pζ ∆qζ cos ϑν −|p| ∆ −|q| ∆ πp0|p| i f i f pζi qζf mν × 0 2 | || |+ 2 − 0 0 2 + 0 0| |− 0| | + 2 0 − 0 2 f /2 ζf p q ζi mν p q f /2 ζiq p ζf p q mν p q , (29) where | |= 0 2 − 2 q q mν. It is convenient to express the results of integrating over angular variables using dimensionless quantities. Intro- ducing the notations 0 0 0 ¯ 0 x = q /mν,γ= p /mν,d= f /2mν, ζi,f = ζi,f sign f (30)

1 In the expression for the radiation energy E, the additional factor k, i.e., the energy of radiated photon, appears in the integrand. 2 We do not consider the polarization of spin light photons here. In the quasi-classical approximation, this problem was investigated in [17]. A.E. Lobanov / Physics Letters B 619 (2005) 136–144 141 we have µ2m3 dx = 0 ν √ 2 ¯ 2 − 2 − − ¯ − 2 Wζ¯ d ζf γ 1 x 1 ζi(γ x 1) f γ(γ2 − 1) x2 − 1 ¯ 2 ¯ 2 2 + γ − x + d ζix γ − 1 − ζf γ x − 1 . (31) The integration bounds in the formula (31) are

x ∈∅,γ∈[1, ∞), (32) ¯ if ζi = 1,

x ∈∅,γ∈[1,γ0),

x ∈[ω1,ω2],γ∈[γ0,γ1),

x ∈[1,ω2],γ∈[γ1,γ2),

x ∈∅,γ∈[γ2, ∞), (33) ¯ ¯ if ζi =−1, ζf =−1, and

x ∈∅,γ∈[1,γ1),

x ∈[1,ω1],γ∈[γ1,γ2),

x ∈[ω2,ω1],γ∈[γ2, ∞), (34) ¯ ¯ if ζi =−1, ζf = 1. Here

1 − 1 − ω = z + z 1 ,ω= z + z 1 , (35) 1 2 1 1 2 2 2 2 where 2 2 z1 = γ + γ − 1 − 2d, z2 = γ − γ − 1 + 2d, (36) and 2 γ0 = 1 + d , 1 − γ = (1 + 2d)+ (1 + 2d) 1 , 1 2 1 − γ = (1 − 2d)+ (1 − 2d) 1 ,d<1/2, 2 2 γ2 =∞,d 1/2. (37) The integration is carried out elementary. The transition rate under investigation is deﬁned as

2 3 µ0mν W¯ = (1 + ζ¯ ) Z(z , 1)Θ(γ − γ ) + Z(z , −1)Θ(γ − γ ) ζf f 1 1 2 2 4 ¯ ¯ + (1 − ζf ) Z(z1, 1)Θ(γ1 − γ)+ Z(z2, −1)Θ(γ2 − γ) Θ(γ − γ0) (1 − ζi). (38) 142 A.E. Lobanov / Physics Letters B 619 (2005) 136–144

Here

¯ 1 2 2 2 1 2 −2 2 2 2 Z(z,ζf ) = ln z γ + d γ − 1 + d + 1/2 + z − z d 2γ − 1 + d γ − 1 + 1/2 γ(γ2 − 1) 4 ¯ ζf − − + z − z 1 2 2d γ 2 − 1 + 1 dγ − z − z 1 d2 + d γ 2 − 1 + 1 γ 4 ¯ −1 2 2 − ζf z + z − 2 d γ − 1 + γ d . (39)

Therefore, the transition rate after summation over polarizations of the ﬁnal neutrino becomes 2 3 µ0mν ¯ W¯ = + W¯ =− = (1 − ζi) Z(z1, 1) + Z(z2, −1) Θ(γ − γ0). (40) ζf 1 ζf 1 2 If dγ 1, then expression (38) leads to the formula 2 3 3 16µ0mνd 2 3/2 ¯ ¯ W¯ = γ − 1 (1 − ζi)(1 + ζf ), (41) ζf 3γ obtained in the quasi-classical approximation in [3]. In the ultra-relativistic limit (γ 1,dγ 1), the transition rate is given by the expression 2 3 2 W¯ = µ m d γ(1 − ζ¯ )(1 + ζ¯ ). (42) ζf 0 ν i f Let us consider now the radiation power. If we introduce the function ˜ ¯ ¯ ¯ Z(z,ζf ) = γZ(z,ζf ) − Y(z,ζf ), (43) where

¯ 1 2 2 1 2 −2 2 2 Y(z,ζf ) = − ln z d + d γ − 1 + 1 γ − z − z d + d γ − 1 + 1 γ γ(γ2 − 1) 4 1 − + z − z 1 3 d2 2γ 2 − 1 + d γ 2 − 1 + 1/2 12 1 − + z − z 1 2d2γ 2 + 2d γ 2 − 1 + γ 2 + 1 2 ¯ ¯ ζf − ζf − + z + z 1 3 − 8 2d γ 2 − 1 + 1 dγ − z − z 1 2 d γ 2 − 1 + γ 2 d , (44) 12 12 ¯ then the formula for the total radiation power can be obtained from (38), (40) by the substitution Z(z,ζf ) → ˜ ¯ Z(z,ζf ). It can be veriﬁed that if dγ 1 then the radiation power is 2 4 4 32µ0mνd 2 2 ¯ ¯ I¯ = γ − 1 (1 − ζi)(1 + ζf ). (45) ζf 3 This result was obtained in the quasi-classical approximation in [2]. In the ultra-relativistic limit, the radiation power is equal to

1 2 4 2 2 ¯ ¯ I¯ = µ m d γ (1 − ζi)(1 + ζf ). (46) ζf 3 0 ν It can be seen from Eqs. (42) and (46) that in the ultra-relativistic limit the averaged energy of emitted photons is εγ =εν/3. It should be pointed out that the obtained formulas are valid both for a neutrino and for an antineutrino. The charge conjugation operation leads to the change of the sign of the effective potential and the replacement of the left-hand projector by the right-hand one in Eq. (1). Thus the sign in front of the γ 5 matrix remains invariant. A.E. Lobanov / Physics Letters B 619 (2005) 136–144 143

Using Eq. (27), it is possible to find the dependence of the radiated photon energy on the angle ϑγ between the direction of the neutrino propagation and the photon wave vector k0 βX − d/γ = 2d . (47) m (X + d/γ )(X − d/γ) ν 2 Here β = γ − 1/γ is the neutrino velocity and X = 1 − (β − d/γ)cos ϑγ . In the quasi-classical approximation, this formula reduces to the relation k0 2dβ = , (48) mν 1 − β cos ϑγ which follows from the results of [3] after Lorentz transformation to the laboratory frame. The following conclusions can be made from the obtained results. A neutrino (antineutrino) can emit photons due to coherent interaction with matter only when its helicity has the sign opposite to the sign of the effective potential f 0. Otherwise, radiation transitions are impossible. In the case of low energies of the initial neutrino, only radiation without spin-flip is possible and the probability of the process is very low. At high energies, the main contribution to radiation is given by the transitions with the spin-flip, the transitions without spin-flip are either absent or their probability is negligible. This leads to total self-polarization, i.e., the initially left-handed polarized neutrino (right-handed polarized antineutrino) is transformed to practically “sterile” right-handed polarized neutrino (left-handed polarized antineutrino). For “sterile” particles, the situation is opposite. They can be converted to the active form in the medium “transparent” for the active neutrino. With the use of the effective potential calculated in the first order of the perturbation theory (2), the following conclusions can be made. If the matter consists only of electrons then, in the framework of the minimally extended standard model in the ultra-relativistic limit (here we use Gaussian units), we have for the transition rate 2 ˜ 2 αεν µ0 GFne W¯ = (1 − ζ¯ )(1 + ζ¯ ), (49) ζf 2 i f 32h¯ µB mec and for the total radiation power 2 2 ˜ 2 αεν µ0 GFne I¯ = (1 − ζ¯ )(1 + ζ¯ ). (50) ζf 2 i f 96h¯ µB mec Here εν is the neutrino energy, µB = e/2me is the Bohr magneton, α is the fine structure constant, me is the electron ˜ 2 mass and GF = GF(1+4sin θW), where GF, θW are the Fermi constant and the Weinberg angle respectively. Thus, after the radiative transition, two thirds of the initial active neutrino energy are carried away by the final “sterile” one. At the same time, as it can be seen from Eq. (2), a muon neutrino in the electron medium does not emit any radiation. Moreover, a muon neutrino does not emit radiation in an electrically neutral medium, when the number density of protons is equal to the electron number density. And an electron neutrino can emit radiation if the electron number density is greater than the neutron number density. An example of such medium is provided by the Sun. Therefore the spin light can change the ratio of active neutrino of different flavors. It is obviously that the above conclusions change to opposite if the matter consists of antiparticles. Therefore the neutrino spin light can serve as a tool for determination of the type of astrophysical objects, since neutrino radiative transitions in dense matter can result in radiation of photons of super-high energies, even exceeding the GZK cutoff. Indeed, the neutron medium is “transparent” for all active neutrinos, but an active antineutrino emits radiation in such a medium, the transition rate and the total radiation power can be obtained from Eqs. (49) and ˜ 38 (50) after substitution GF → GF, ne → nn. If the density of the neutron star is assume to be n ≈ 10 , the transition rate is estimated as ε µ 2 W = 1022 ν 0 , (51) εGZK µB 144 A.E. Lobanov / Physics Letters B 619 (2005) 136–144

19 where εGZK = 5 × 10 eV is GZK cutoff energy. Although the transition rate determined by Eq. (51) is extremely low, this effect can still serve as one of a possible explanations of the cosmic ray paradox. The spin light can also be important for the understanding of the “dark matter” formation mechanism in the early stages of evolution of the Universe. When the present Letter was already submitted for publication, we came across an article [22], where the spin light theory was also considered. The formulas of [22] in the ultra-relativistic limit of physical interest reproduce the results for the transition rate and the total power of spin light already obtained in our earlier publication [10].

Acknowledgements

The author is grateful to V.G. Bagrov, A.V. Borisov, and V.Ch. Zhukovsky for fruitful discussions. This work was supported in part by the grant of President of Russian Federation for leading scientiﬁc schools (Grant SS-2027.2003.2).

References

[1] K. Fujikawa, R. Shrock, Phys. Rev. Lett. 45 (1980) 963. [2] A. Lobanov, A. Studenikin, Phys. Lett. B 564 (2003) 27, hep-ph/0212393. [3] A. Lobanov, A. Studenikin, Phys. Lett. B 601 (2004) 171, astro-ph/0408026. [4] V.A. Bordovitsyn, I.M. Ternov, V.G. Bagrov, Sov. Phys. Usp. 38 (1995) 1037, Usp. Fiz. Nauk 165 (1995) 1084. [5] A.E. Lobanov, O.S. Pavlova, Phys. Lett. A 275 (2000) 1, hep-ph/0006291. [6] A.E. Lobanov, hep-ph/0311021. [7] V. Bargmann, L. Michel, V. Telegdi, Phys. Rev. Lett. 2 (1959) 435. [8] A. Lobanov, A. Studenikin, Phys. Lett. B 515 (2001) 94, hep-ph/0106101. [9] A.E. Lobanov, O.S. Pavlova, Moscow Univ. Phys. Bull. 54 (4) (1999) 1, Vestnik MGU Fiz. Astron. 40 (4) (1999) 3. [10] A.E. Lobanov, hep-ph/0411342. [11] L. Wolfenstein, Phys. Rev. D 17 (1978) 2369. [12] S.P. Mikheyev, A.Yu. Smirnov, Sov. J. Nucl. Phys. 42 (1985) 913, Yad. Fiz. 42 (1985) 1441. [13] L.L. Foldy, Phys. Rev. 87 (1952) 688. [14] P.B. Pal, T.N. Pham, Phys. Rev. D 40 (1989) 259. [15] J.F. Nieves, Phys. Rev. D 40 (1989) 866. [16] D. Nötzold, G. Raffelt, Nucl. Phys. B 307 (1988) 924. [17] A. Studenikin, A. Ternov, hep-ph/0410297; A. Studenikin, A. Ternov, hep-ph/0412408. [18] J. Pantaleone, Phys. Lett. B 268 (1991) 227. [19] A.A. Sokolov, I.M. Ternov, Synchrotron Radiation, Academic Verlag, Berlin, 1968. [20] V.G. Bagrov, D.M. Gitman, Exact Solutions of Relativistic Wave Equations, Kluwer, Dordrecht, 1990. [21] I.M. Ternov, Introduction to Physics of Spin of Relativistic Particles, Moscow State University, Moscow, 1997 (in Russian). [22] A. Grigoriev, A. Studenikin, A. Ternov, hep-ph/0502231. Physics Letters B 619 (2005) 145–148 www.elsevier.com/locate/physletb

Mixing the strong and EW Higgs sectors

Nils A. Törnqvist

Department of Physical Sciences, University of Helsinki, PO Box 64, FIN-00014, Finland Received 3 May 2005; accepted 24 May 2005 Available online 1 June 2005 Editor: N. Glover

Abstract After noting the well-known similarity of the minimal electro-weak Higgs sector with the linear sigma model for the pion and the sigma, it is found that a small mixing term between the two models generates a pion mass. Although the custodial SU(2)L ×SU(2)R, and the gauged SU(2)L ×U(1) symmetry for the whole model remains intact, the mixing breaks the relative chiral symmetry between the two sectors. The mixing should be calculable from light quark masses as a quantum correction. This simple mechanism of “relative symmetry breaking” is believed to have applications for other forms of symmetry breaking.  2005 Elsevier B.V. All rights reserved.

PACS: 11.15.Ex; 12.39.Fe; 14.80.Bn

Spontaneous symmetry breaking in the vacuum is vacuum values are orders of magnitudes different, v/vˆ certainly a very beautiful and important concept in is about 2656. many areas of physics. The prototype for this mech- The analogy is more evident if we represent the anism in particle physics is given by the linear sigma conventional complex Higgs-doublet composed of φ+ model (Lσ M) [1], (which can be looked upon as an ef- and φ0 by a seemingly redundant matrix form (we fol- fective theory at sufficiently low energy of a more fun- low the notations of Willenbrock [2]) damental theory as QCD). The minimal electro-weak + ∗ + φ 1 φ0 φ scalar sector was essentially copied from the Lσ M, → √ − = Φ. (1) φ0 −φ φ0 except for a much higher value for the vacuum expec- 2 tation value, v = 246 GeV, instead of the 92.6 MeV The Higgs Lagrangian then takes the form in the Lσ M, which we here denote by vˆ. The latter † 2 † LHiggs(Φ) = Tr (DµΦ) DµΦ + µ Tr Φ Φ value, vˆ, is also the pion decay constant (fπ ), and is 2 proportional to the qq¯ condensate of QCD. Thus the − λ Tr Φ†Φ , (2) from which (at the tree level) the vacuum value is fixed by µ2 (which is assumed to have the right E-mail address: nils.tornqvist@helsinki.fi (N.A. Törnqvist). sign for spontaneous symmetry breaking) and λ, v =

[µ2/(2λ)]1/2 = 246 GeV, and where the covariant Now add the two Lagrangians with a small mixing derivative is term between the Φ and the Φˆ proportional to 2: g g L = L + L ˆ + 2 † ˆ + D Φ = ∂ Φ + i τ¯ · W Φ − i B Φτ . (3) Higgs(Φ) Lσ M(Φ) Tr Φ Φ h.c. /2. µ µ 2 µ 2 µ 3 (7) (The τ3 matrix shows explicitly [2] in this matrix rep- This is, in fact, similar to the two-Higgs doublet resentation that the right handed gauge group is only model [3], although the application here is different U(1).) The vacuum value of the Φ field is and much more down to earth, i.e., not the usual supersymmetric or technicolor extensions of the standard 1 v 0 Φ=√ . (4) model.1 0 v 2 2 The term breaks a relative (global) SU(2)L × The Lagrangian is, as well known, invariant under SU(2)R in the sense that if only one of the two Φ’s the gauge symmetry SU(2) ×U(1) (Φ → LΦ and is transformed by, say, a left-handed rotation (LΦ or L Y ˆ − 1 τ ϕ L Φ), the relative symmetry is broken. One could, Φ → Φe 2 3 ), which is broken spontaneously down of course, write down many other terms [3], which to U(1)EM by the v of Eq. (4). But, in the limit of similarly break another relative symmetry.2 But, for g → 0, i.e., θ → 0, (or if one disregards the gaug- W our discussion here the simplest possible choice as in ing) there is, in fact, also a global SU(2) × SU(2) L R Eq. (7) is sufficient. (Φ → LΦR). Thus in this limit the electro-weak Neglecting for a moment the gauging, and with our Higgs sector has a global SU(2) ×SU(2) symmetry, L R choice of the term the Lagrangian (7) has an overall which is spontaneously broken by v to SU(2) + = L R global SU(2) × SU(2) symmetry (i.e., when Φ → SU(2) with W and Z degenerate. This symmetry is L R V LΦR is transformed simultaneously as Φˆ → LΦRˆ usually called the custodial symmetry. In our discus- with the same L and R). Also the overall SU(2) × sion we treat this as an exact global symmetry, as the L U(1) gauge symmetry is left intact. Let us first discuss g term (or the isospin breaking) is not essential here. how the mixing term breaks the relative global sym- This is just like the simplest Lσ M (without isospin metry, neglecting the gauging. breaking) for the pion and σ . The analogue of Φ is For = 0 and v = 0, vˆ = 0, there is a triplet of + 0 + Goldstone (or would-be Goldstone) bosons in each ˆ = √1 σ iπ i(π1 iπ2) Φ − − − 0 sector. With = 0 one of these triplets, (the pion) gets 2 i(π1 iπ2)σiπ mass proportional to , since the relative symmetry is 1 2 − = √ (σ · 1 + iπ¯ ·¯τ). (5) broken. The pseudoscalar mass matrix m (0 ) gets 2 contributions in two ways (1) from the fact that the

0 0 + vacuum values are disturbed, and (2) directly from the Thus φ√ corresponds to (σ −iπ ) and φ to i(π1 + ˆ + mixing term. The corrections to v and v obey the rela- iπ2) = i 2π in conventional notation. (The gen- tions: eralization of Eq. (5) to full scalar and pseudoscalar ˆ nonets, sk,pk, k = 0to8,isΦ ∝ (sk + ipk)λk, v( )ˆ k v2( ) = v2(0) + 2 , where λk are Gell-Mann matrices.) The gauged Lσ M 2v( )λ is apart from different constants identical to Eq. (2) 2 2 2 v( ) vˆ ( ) =ˆv (0) + . (8) ˆ ˆ † ˆ 2 ˆ † ˆ 2v( )ˆ λˆ LLσ M(Φ) = Tr (DµΦ) DµΦ +ˆµ Tr Φ Φ − λˆ Tr Φˆ †Φˆ 2, (6) 1 A similar effective Lagrangian as in Eq. (7) was also suggested from which vˆ is given by vˆ =[ˆµ2/(2λ)ˆ ]1/2 = previously [4], but with a quite different phenomenological applica- ≈ ˆ = tion in mind, which involved two light scalar meson nonets and only 92.6 MeV. (With a σ mass of 600 MeV λ strong interactions. [ ˆ ]2 ≈ 2 2 † ˆ mσ /(2v) 10, but we actually do not need that E.g., a small term δ Tr(Φ Φτ3), which breaks SU(2)R (cf. number here.) the g term in (3)) would be allowed by the overall SU(2)L × U(1). N.A. Törnqvist / Physics Letters B 619 (2005) 145–148 147

The pseudoscalar squared mass matrix becomes mass. In fact, it is natural to assume that the 2 term arises because of quantum corrections involv- − 2λv2( ) − µ2 − 2 m2 0 = ing virtual quark loops (Φˆ → qq¯ → Φ or even Φˆ → − 2 2λˆvˆ2( ) −ˆµ2 ˆ qq¯ → VV → Φ, which would involve ABJ anomaly v( ) −1 =+ 2 v( ) , (9) graphs). In principle, such terms should be calculable − v( ) + 1 v( )ˆ from QCD EW gauge theory. which is diagonalized by a small rotation θ = A few concluding ﬁnal remarks are in order. I dis- 1 [ ˆ 2 +ˆ2 ]≈ˆ = × −4 cussed a way of how to look at a certain kind of sym- 2 arctan 2vv/(v v ) v/v 3.764 10 (we leave out from now on the small dependence on in metry breaking, which I named “relative symmetry v and vˆ). Thus the eigenvalues are 0 and 2(v/vˆ + breaking”. By mixing two models with the same sym- × v/v)ˆ ≈ 2v/vˆ: metry group SU(2)L SU(2)R, one for the scalar sector of strong interactions, and the other for the Higgs cs − c −s m2 0 sector of electro-weak interactions, a relative symme- −sc sc try is broken giving in our example rise to a pion mass, v2 +ˆv2 00 00 although for the whole model the chiral symmetry is = 2 ≈ . ˆ 2 v (10) still exact and unbroken in the Lagrangian, except for vv 01 0 vˆ spontaneous symmetry breaking in the vacuum. I be- (Here s = sin θ and c = cos θ.) Note that the mass ma- lieve that this as a general idea has not received proper trix only depends on v,vˆ and , not on λ nor λˆ , which attention in the literature. It need not be limited to the are not well known from experiment as they are re- application discussed here. lated to the Higgs or σ(600) masses. Note also that For example, it can be applied to the perhaps sim- the pseudoscalar which gets mass is the one which is plest spontaneous symmetry breaking one can think related to the pion in the Lσ M sector, (i.e., it is not the of, that of O(2) or U(1) symmetry models. One needs “would-be Goldstone” of the EW sector. To get the two U(1) models, each described by a “Mexican hat” right pion mass should be about 2.70 MeV.) On the potential, which are coupled to each other by a mixing other hand, the Higgs and σ bare masses and mixings term. In this system of “coupled spontaneous sym- are only very little affected since the corrections are metry breaking” the coupled potentials are “tilted” proportional to the very small number 3 2. compared to each other. Therefore, because of the As the whole Lagrangian (7) still has the full mixing, one combination of the two would-be Gold- SU(2) × U(1) gauge symmetry the remaining Gold- L stones gets mass. The whole system would still have stone is turned, in the usual way, into the longitudinal one massless Goldstone, which is a linear combina- components of the W and Z bosons. But, these masses tion of the original massless states. Then, if one gauges get a small contribution also from the Lσ M sector the whole system also this massless scalar is “eat- M2 = g2 v2 +ˆv2 /4, en” by the vector boson, and no massless states re- W main. M2 = g2 + g 2 v2 +ˆv2 /4. (11) Z In this simple U(1) example, our relative symme- One can also easily see that the 2 term mixes the lon- try breaking may seem trivial, almost self-evident, gitudinal W and the pion in the expected way by the but its possible generalization within more compli- same angle θ. cated structures is often missed in current litera- Thus the pion gets mass from a small breaking of ture. Our main conjecture is thus that a symmetry a relative symmetry between the EW and strong in- of the standard model for strong and electro-weak teractions, and through a small mixing with the lon- interactions combined can remain exact in the to- gitudinal W , described by θ = 3.764 × 10−4. Still, tal Lagrangian, although the symmetry looks “as the symmetry remains intact, although spontaneously if it were broken explicitly by one of the inter- broken by the vacuum, for the combined strong plus actions”. If this conjecture turns out to be true it electroweak Higgs sectors. should have consequences for a better understanding Our result is in no way in conﬂict with the usual of, say, PCAC, chiral symmetry and isospin break- reference to light quark masses as the source of pion ing. 148 N.A. Törnqvist / Physics Letters B 619 (2005) 145–148

Acknowledgements [2] S. Willenbrock, Symmetries of the standard model, Lectures given at Theoretical Advance Study Institute in Elementary Par- ticle Physics (TASI 2004), Boulder, CO, 6 June–2 July 2004, I thank Prof. Shou-hua Zhu for a comment during hep-ph/0410370; my recent talk at Peking University that this conjec- See also M. Veltman, et al., Reﬂections on the Higgs system, ture may lead to calculable radiative corrections to, Lectures at the Academic Training Program at CERN, CERN e.g., standard model Higgs mass estimates. Support Yellow report 97-05. from EU RTN Contract CT2002-0311 is gratefully ac- [3] See, e.g., J.F. Gunion, The Higgs Hunter’s Guide, in: Frontiers in Physics, Addison–Wesley, Reading, MA, 1990, p. 191. knowledged. [4] F.E. Close, N.A. Törnqvist, J. Phys. G: Nucl. Part. Phys. 28 (2002) R249, hep-ph/0204205; N.A. Törnqvist, hep-ph/0204215; References D. Black, A. Fariborz, S. Moussa, S. Nasri, J. Schechter, Phys. Rev. D 64 (2001) 014031; M. Napsuciale, S. Rodriguez, Phys. Rev. D 70 (2004) 094043, [1] J. Schwinger, Ann. Phys. 2 (1957) 407; hep-ph/0407037. M. Gell-Mann, M. Levy, Nuovo Cimento XVI (1960) 705; B.W. Lee, Nucl. Phys. B 9 (1969) 64. Physics Letters B 619 (2005) 149–154 www.elsevier.com/locate/physletb

Compactiﬁcations on twisted tori with ﬂuxes and free differential algebras

Gianguido Dall’Agata a, Riccardo D’Auria b,c, Sergio Ferrara a,d,e

a Physics Department, Theory Unit, CERN, CH-1211 Geneva 23, Switzerland b Dipartimento di Fisica, Politecnico di Torino, C.so Duca degli Abruzzi 24, I-10129 Torino, Italy c Istituto Nazionale di Fisica Nucleare, Sezione di Torino, Italy d Istituto Nazionale di Fisica Nucleare, Sezione di Frascati, Italy e Department of Physics and Astronomy University of California, Los Angeles, USA Received 23 March 2005; accepted 1 April 2005 Available online 13 April 2005 Editor: L. Alvarez-Gaumé

Abstract We describe free differential algebras for non-Abelian one and two form gauge potentials in four dimensions deriving the integrability conditions for the corresponding curvatures. We show that a realization of these algebras occurs in M-theory compactifications on twisted tori with constant four-form flux, due to the presence of antisymmetric tensor fields in the reduced theory.  2005 Published by Elsevier B.V.

1. Introduction ing gauge algebraic structures emerge which in most cases have the interpretation of a gauged Lie alge- Flux compactifications on twisted tori provide in- bra [5,9–12]. teresting examples of string and M-theory compact- In this case the Maurer–Cartan equations (zero cur- ifications where most of the moduli fields are stabi- vature conditions) read lized [2–8]. Particular cases of such compactifications 1 dAΛ + f Λ AΣ ∧ AΓ = 0, (1.1) include heterotic string, type II orientifold models and 2 ΣΓ M-theory in the presence of constant p-form fluxes where integrability implies the Jacobi identities (where p depends on the particular string model and Λ Π p = 4 in M-theory). When fluxes and (or) Scherk– f [ΣΓ f ∆]Λ = 0. (1.2) Schwarz geometrical fluxes are turned on, interest- This comes from the vanishing of the cubic term

2 Π 1 Λ Π Σ Γ ∆ E-mail address: [email protected] (G. Dall’Agata). d A =− f f A ∧ A ∧ A = 0. (1.3) 2 ΣΓ ∆Λ 0370-2693/$ – see front matter  2005 Published by Elsevier B.V. doi:10.1016/j.physletb.2005.04.005 150 G. Dall’Agata et al. / Physics Letters B 619 (2005) 149–154

When fundamental tensor fields are present in the A ∧ A and from A4 terms: theory, in absence of gauge couplings in the supergrav- (j Λk) = ity theory, one can transform them into scalars and this (TΛ)i m 0, (2.7) j Λ k j is the way the full duality symmetry (sometimes called (TΛ)i f ΣΓ − 2(T[Σ )i (TΓ ])k U-duality) is recovered. However, in presence of non- Λj + 6m kiΛΣΓ = 0, (2.8) abelian gauge couplings, an obstruction can arise in Λ − j = the dualization of such antisymmetric tensors, so that 3f [ΣΓ kiΠ∆]Λ 2(TΠ )i kjΣΓ∆] 0. (2.9) the theory only preserves some subalgebra of the full When mΛi = 0, the condition (2.5) implies for the duality group. Moreover, the gauged algebra structure AΛ the ordinary Lie algebra Jacobi identities. Eq. (2.8) j may be more complicated than an ordinary Lie alge- tells us that (TΛ)i is a representation of the Lie algebra and in fact, as noted in [10] for generic Scherk– bra and (2.9) states that kiΛΣΓ is a cocycle of the Lie Λi Schwarz and form flux couplings it turns out to be a algebra. When m kiΓ Π∆ = 0 (2.5) gives the depar- free differential algebra (FDA) [13–17]. ture from an ordinary Lie algebra for the f structure In the case of M-theory, we will show that its constants. Maurer–Cartan equations are equivalent to the integrability conditions for the 4-form GIJKL and for the vielbein 1-form in D = 11. This also will explain how 3. FDA from M-theory on twisted tori with fluxes the Lie algebra part of the free differential algebra is deformed in the presence of generic Scherk–Schwarz As an example of a concrete realization of the free and form flux couplings. differential algebra (2.1) and (2.2), we will now describe the one obtained by compactification of M- theory on twisted tori in the presence of fluxes con- 2. The free differential algebra and its sidered in [10]. The compactification of M-theory to Maurer–Cartan equations I 4 dimensions provides 28 vector fields Gµ, AµIJ and 7 2-form tensor fields AµνI . This means that we can The generalization of (1.1) to a free differential al- identify the generic indices Λ,i of our FDA as follows gebra including 2-form gauge fields Bi consists of the Λ ={I,IJ}, i = I . Furthermore, one has to write the following (zero-curvature) system single indices I,J in the same position as Λ,i,butthe 1 antisymmetric couples IJ, KL, ... are written as up- F Λ = dAΛ + f Λ AΣ ∧ AΓ + mΛiB = 0, (2.1) 2 ΣΓ i per indices if Λ,Σ,... are lower ones and as lower j Λ indices if Λ,Σ,... are upper ones. Hi = dBi + (TΛ)i A ∧ Bj If one considers first the case when only form fluxes + Λ ∧ Σ ∧ Γ = kiΛΣΓ A A A 0, (2.2) are turned on, the Lie algebra is where f Λ , (T ) j , mΛi and k are the struc- ΣΓ Λ i iΛΣΓ [Z ,Z ]=g W KL, ture constants of the FDA. I J IJKL JK IJ KL The integrability condition of this system comes ZI ,W = W ,W = 0, (3.1) from the Bianchi identities which is the central extension of an Abelian gauge al- dF Λ = 0, (2.3) gebra. In this case the only non-vanishing structure constants are [10] dHi = 0. (2.4) Λ = = From (2.3), by setting to zero the terms proportional to f ΣΓ f[IJ]KL gIJKL, (3.2) A3 and A ∧ B polynomials we get 1 kiΛΣΓ = kIJKL = gIJKL, (3.3) Λ Σ Λi 6 f Σ[Γ f Π∆] + 2m kiΓ Π∆ = 0, (2.5) Λi = j = Λ Σj Λi j while m (TΛ)i 0. It then follows that (2.5) and f ΣΓ m + m (TΓ )i = 0, (2.6) (2.6) are trivially satisfied and gIJKL is arbitrary. This respectively. From (2.4) we get three conditions from result is a consequence of the very degenerate structure the vanishing of the terms proportional to B ∧ B, B ∧ of the Lie algebra (3.1). G. Dall’Agata et al. / Physics Letters B 619 (2005) 149–154 151

Λ An intermediate richer example comes in the case f ΣΓ define an ordinary Lie algebra. This happens K K of Scherk–Schwarz fluxes τIJ and vanishing 4-form if the Scherk–Schwarz fluxes τIJ have the K index flux. This is the case considered in the pioneering pa- complementary to the flux coupling gIJKL. This can pers of Scherk–Schwarz [1,2]. In this case kiΛΣΓ = 0, actually be realized in certain type II orientifold mod- Λi j but m and (TΛ)i do not vanish. In fact, the non- els. vanishing parts of these structure constants are To summarize, we have shown that for generic Scherk–Schwarz couplings τ K and 4-form flux mΛi = 0forΛ =[IJ],i= K, IJ gIJKL, the M-theory gauge algebra is a free differ- K = K mIJ τIJ, (3.4) ential algebra rather than an ordinary Lie algebra. The equations j = = = = (TΛ)i 0forΛ I, i J, j K, M L = τ[IJτK]M 0, (3.10) K =− K (TI )J τIJ. (3.5) N = τ[IJgKLM]N 0, (3.11) The other non-vanishing structure constants occur for Λ are the integrability conditions for the FDA. When f ΣΓ when N = the stronger condition τIJgKLMN 0 holds then the = = = I = I Λ Λ I, Σ J, Γ K, f JK τJK, f ΣΓ define an ordinary Lie algebra whose commutators read [10] Λ =[IJ],Σ= K, Γ =[LM], [ ] [L M] KL K LM =− [ZI ,ZJ ]=gIJKLW + τ ZK , f[IJ]K 2τK[I δJ ] . (3.6) IJ [ ] Z ,WJK = 2τ J W K L, In this case (2.7) is identically satisfied and (2.6), (2.8) I IL L M = IJ KL are identical to (2.5), which reads as τ[IJτK]L 0. W ,W = 0. (3.12) Λi Note that m corresponds to a “magnetic” mass term It is interesting to note that in M-theory compactified for the Bi field. Λi Λ on a twisted torus with 4-form flux turned on m and The f ΣΓ structure constants in (3.6) define the gPQRS have the physical interpretation of magnetic Scherk–Schwarz algebra for M-theory: and electric masses for the antisymmetric tensors B . I IJ KL K This is clear looking at the covariant field strength W ,W = 0, [ZI ,ZJ ]=τ ZK , IJ [ ] JK = J K L Λ Λ 1 Λ Σ Γ ΛI ZI ,W 2τILW . (3.7) F = dA + f A ∧ A + m B . (3.13) 2 ΣΓ I K Let us now consider the general case when both τIJ This expression appears quadratically in the (kinetic and gIJKL are non-vanishing. In this case the last term part of the) Lagrangian together with the coupling in (2.5) is non-vanishing for Λ =[IJ], Σ = K, Γ = IJKLMNP L and Π = M. It reads gIJKLBM ∧ dANP , (3.14) N which comes from the 11-dimensional Chern–Simons τ gKLMN. (3.8) IJ term F ∧ F ∧ A. It is amusing to note that the con- If this term does not vanish the f structure constants sistency condition [18,19] for electric and magnetic do not define a Lie algebra. In this case (2.5) (as also contributions to the mass is in this case a consequence (2.9)) becomes of (3.9). N The M-theory FDA also includes a 3-form gauge τ g ] = 0. (3.9) [IJ KLM N field C which is a singlet. The zero-curvature condi- This condition has the 11-dimensional interpretation for this 3-form is tion of the integrability condition of the 4-form field dC + mij B ∧ B + mi AΛ ∧ AΣ ∧ B strength [10]. i j ΛΣ i Λ Λ Σ Γ ∆ All other equations are satisfied as a consequence + tΛA ∧ C + kΛΣΓ ∆A ∧ A ∧ A ∧ A = 0. L M = of the τ Jacobi identities τ[IJτK]L 0. These follow (3.15) from (2.5) by taking Λ,Σ,Γ,Π = IJKL.Itisobvi- In the M-theory FDA, the only non-vanishing terms ∼ I ∼ I ous that if the stronger condition (3.8) holds then the are kIJKL gIJKL and mJK τJK, with all the 152 G. Dall’Agata et al. / Physics Letters B 619 (2005) 149–154

ij other components and tΛ and m vanishing. In this Then the HI curvature reads case the Bianchi identity is trivially satisfied because H = dB + F J ∧ A a 5-form in D = 4 identically vanishes. However, the I I IJ 1 J K L curvature of C can be determined by demanding its + gIJKLA ∧ A ∧ A , (4.5) full invariance under all gauge transformations. 6 and the coefficient of the F ∧ A term is fixed, relative 3 J to the A term in such a way that dHI = F ∧ FIJ. H 4. Non-zero curvature case Now I and its Bianchi identity are invariant under the gauge transformations The previous Maurer–Cartan equations (2.1) and 1 δB = dΛ − F J + ωM g AJ ∧ AK , (4.6) (2.2), which entail the “structure constants” relations I I IJ 2 MIJK (2.5)–(2.9) can be lifted to non-zero curvature, so ob- δAI = dωI , (4.7) taining covariant Bianchi identities for the curvatures. δA = d − g ωK AL. (4.8) In the case of M-theory with Scherk–Schwarz fluxes IJ IJ IJKL turned on this procedure essentially reproduces the co- Analogously, the threefold antisymmetric tensor C variant curvatures G of Section 3.4 of [2]. When also curvature is the constant 4-form fluxes F = g are turned 1 IJKL IJKL dC − F I ∧ B + g AI ∧ AJ ∧ AK ∧ AL, on, then one gets generalized curvatures which are co- I ! IJKL 4 (4.9) variant under the combined 1-form and 2-form gauge which is invariant under the gauge transformations transformations considered in Section 2 of [10]. I An interesting new feature of the curvatures is the δC = dΣ + F ∧ ΛI presence in HI of a “contractible generator” [13], i.e., 1 − g ωI ∧ AJ ∧ AK ∧ AL, (4.10) in physical language, of a curvature itself (which also 6 IJKL exists in the ungauged theory) J 1 M J K δBI = dΛI − IJF + ω gMIJKA ∧ A , H = + F J ∧ 2 I dBI AIJ, (4.1) (4.11) I = I where F J = dAJ . This is a kind of Green–Schwarz δA dω . (4.12) (mixed) Chern–Simons term which modifies the gauge Note that the dC field strength is a Lagrange multi- transformations of BI so that HI is invariant under the plier and can be algebraically eliminated from the La- gauge transformations grangian giving a contribution to the scalar potential.

J δBI = dΛI − IJF , 5. Concluding remarks δAI = dωI ,

δAIJ = d IJ. (4.2) In the present Letter we have considered the free differential algebra which comes from M-theory com- The (ungauged) Bianchi identity is now pactified on a twisted torus with constant 4-form fluxes. This is just a special case of the Maurer–Cartan H = F J ∧ F d I IJ, (4.3) equations described by (2.1) and (2.2). A similar sit- 2 uation arises in type IIA theories since in this case which satisfies d HI = 0 and is also invariant under the gauge transformations (4.2). charged antisymmetric tensor fields are also present. Let us now consider the case when gIJKL = 0(but However, in this case one can find a particular set of K = geometrical fluxes which can be consistently set to τIJ 0), so that vanish and then the Lie algebra structure is recovered 1 K L because the condition FIJ = dAIJ + gIJKLA ∧ A , 2 Λi m kiΣΓ Π = 0, (5.1) F I = dAI . (4.4) is satisfied. Such examples were described in [10]. G. Dall’Agata et al. / Physics Letters B 619 (2005) 149–154 153

Λ The FDA given by the system of curvatures F , Hi and the zero curvatures conditions read can be recast in the form of an ordinary Lie algebra if dA0 = 0, (5.8) (some of the) Bi are redefined so that the quadratic Σ Γ ˜ α + α 0 ∧ β = term in A ∧ A is absorbed in the new Bi [13].This dA t β A A 0, (5.9) β ˜ can be done at most for rank(m) tensors fields, which dA0α + t αBβ = 0, (5.10) can be the same as the range of the i indices provided γ 0 dA + 2t [ A ∧ A ] = 0, (5.11) that this is smaller than that of the vector fields Λ,asin αβ α β γ ˜ = the M-theory case. Explicitly, for those Bα for which dBα 0, (5.12) the subblock mαβ is invertible, one can introduce the ˜ β α γ dB0 − t αA ∧ A ∧ Aβγ = 0. (5.13) definition (Λ ={α, A}) Note that the Jacobi identities of the τ do not set any ˜ 1 −1 β Λ Σ constraint on the t matrices. If we split the generators Bα ≡ Bα + m f ΛΣ A ∧ A , (5.2) 0α αβ 2 αβ into Z0, Zα, W , W , it is immediate to see that the index α goes over six values and the gauge fields so that the new zero curvature conditions read Aµ0α disappear from the gauge algebra. The generator ˜ = algebra becomes then dBα 0, (5.3) αβ [α β]γ γ F α = α + αβ ˜ = Z0,W = 2τ W , [Z0,Zα]=τ Zγ , dA m Bβ 0, (5.4) 0γ 0α βγ =[ ]= αβ γδ = A A 1 A B C Zα,W Zα,Zβ W ,W 0, (5.14) F = dA + f BCA ∧ A = 0. (5.5) 2 which is the usual (22-dimensional) flat Scherk– The new Lie algebra is defined by the structure con- Schwarz algebra. This algebra becomes 24- or 26- A α stants f BC and this is obtained by deleting the A dimensional if one or two eigenvalues of the t matrix generators from the original algebra. This is the quo- vanish. The same reasoning applies when form fluxes tient of the original algebra with the subalgebra related are present. In this case the commutators of the Zα are to the Aα vectors. It is an obvious consequence of the not vanishing and the gauge algebra get modified. A A Jacobi identities for F that f ΛΣ = 0 whenever Λ Note that the physical interpretation of this reduc- or Σ take values in the α range. tion of the FDA to a minimal part and a contractible In the M-theory case, the rank of mΛi is encoded in one [13] corresponds to the anti-Higgs mechanism K × where antisymmetric tensors absorb vector fields to the Scherk–Schwarz fluxes τIJ regarded as a 7 21 triangular matrix. A quadratic submatrix can have at become (dual to) massive vectors. The quotient Lie almost rank 7 so the Lie algebra spanned by the AA is at gebra is the unbroken gauge algebra. It is interesting to least 21-dimensional. When describing the algebra in see that, due to the cubic terms of the 1-forms AΛ in terms of its generators, one must delete the generators the Hi curvature (this only happens when the 4-form ˜ ˜ F Λ W LK whose gauge fields are absorbed by the antisym- flux is present), the quadratic part of the curvature metric tensors. The resulting Lie algebra is obtained does not correspond to an ordinary Lie algebra before ˜ ˜ the quotient has been taken. by all Z , W LK generators but the W LK , which is K Another interesting generalization is to extend such an Abelian subalgebra. A simple example is the case FDA to the fermionic sector of the theory, since when τ K correspond to a “flat group”. In this case IJ the D = 4 theory has N = 8 local supersymmetry. B ={B ,B } and AΛ ={Aα,AA}, with Aα = A I 0 α 0α Such program was originally carried out in D = 11 and AA ={A0,Aα,A }. The original structure con- αβ in [14] and its extension to the present compactifica- stants follow from τ K = τ α = tα , where α, β = IJ 0β β tion should be possible. 1,...,6, and t is an invertible antisymmetric matrix, We finally remark that the different structures of (this means that the 3 skew eigenvalues are non-zero). the 4-dimensional effective theories obtained when the The redefined tensor fields are gauge algebra is a FDA or an ordinary Lie algebra are ˜ δ −1β γ 0 reflected in different scalar potentials. This fact may Bα = t γ t Aβδ ∧ A + A α ∧ A + Bα, (5.6) α 0 have important consequences when looking for com- ˜ α B0 = B0 − A ∧ A0α, (5.7) plete moduli stabilization in such compactifications. 154 G. Dall’Agata et al. / Physics Letters B 619 (2005) 149–154

Acknowledgements [5] N. Kaloper, R.C. Myers, JHEP 9905 (1999) 010, hep- th/9901045. [6] J.-P. Derendinger, C. Kounnas, P.M. Petropoulos, F. Zwirner, We would like to thank R. Stora for an enlight- hep-th/0411276. ening discussion. The work of R.D. and S.F. has [7] I. Antoniadis, T. Maillard, hep-th/0412008. been supported in part by the European Community’s [8] M. Bianchi, E. Trevigne, hep-th/0502147. Human Potential Programme under contract HPRN- [9] L. Andrianopoli, R. D’Auria, S. Ferrara, M.A. Lledó, Nucl. Phys. B 640 (2002) 63, hep-th/0204145. CT-2000-00131 Quantum Spacetime, in which R.D. [10] G. Dall’Agata, S. Ferrara, hep-th/0502066. is associated to Torino University and S.F. to INFN [11] L. Andrianopoli, M.A. Lledó, M. Trigiante, hep-th/0502083. Frascati National Laboratories. The work of S.F. has [12] C.M. Hull, R.A. Reid-Edwards, hep-th/0503114. also been supported in part by DOE grant DE-FG03- [13] D. Sullivan, Inst. Hautes Études Sci. Publ. Math. 47 (1977) 91ER40662, Task C. S.F. would like to thank the De- 269. [14] R. D’Auria, P. Fre, Nucl. Phys. B 201 (1982) 101; partment of Physics of the Politecnico di Torino for the R. D’Auria, P. Fre, Nucl. Phys. B 206 (1982) 496, Erratum. kind hospitality. [15] L. Castellani, R. D’Auria, P. Fre, Supergravity and Super- strings: A Geometric Perspective, vol. 2, World Scientiﬁc, Sin- gapore, 1991. [16] P. van Nieuwenhuizen, CERN-TH-3499. References [17] J.A. de Azcarraga, J.M. Izquierdo, M. Picon, O. Varela, Nucl. Phys. B 662 (2003) 185, hep-th/0212347. [18] R. D’Auria, L. Sommovigo, S. Vaulà, JHEP 11 (2004) 028, [1] J. Scherk, J.H. Schwarz, Phys. Lett. B 82 (1979) 60. hep-th/0409097. [2] J. Scherk, J.H. Schwarz, Nucl. Phys. B 153 (1979) 61. [19] R. D’Auria, S. Ferrara, Phys. Lett. B 606 (2005) 211, hep- [3] S. Kachru, M.B. Schulz, P.K. Tripathy, S.P. Trivedi, JHEP 0303 th/0410051. (2003) 061, hep-th/0211182. [4] M.B. Schulz, Fortschr. Phys. 52 (2004) 963, hep-th/0406001. Physics Letters B 619 (2005) 155–162 www.elsevier.com/locate/physletb

Singlet fermions on curved extra dimension tori

I. Vilja

Department of Physics, FIN-20014 University of Turku, Finland Received 4 March 2005; received in revised form 31 May 2005; accepted 31 May 2005 Available online 9 June 2005 Editor: N. Glover

Abstract A model with two curved, compact extra dimensions is introduced. The model is based on a four-brane immersed in a six- dimensional space, where the extra dimensions are compact but not ﬂat. They have topology of two-torus. The form of metric in the empty bulk is studied and the gauge singlet fermion structure showed to be very simple. It contain only one massless low energy mode which couples to brane matter. The massive modes are not related to the volume of torus while the Planck mass is related to the volume of extra dimensions. In the model “our” brane is situated on a nearly singular line on the torus.  2005 Elsevier B.V. All rights reserved.

1. Introduction

The intriguing idea, that the physical space may have more that three dimensions has got a lot attention recently. The idea itself dates back to the beginning of the 20th century, to the works of Nordström, Kaluza and Klein [1]. They attempted to unify gravity and electromagnetism by introducing a fifth dimension. The idea reappeared in string theories, where unification of all four known fundamental interactions was made in ten or eleven-dimensional space–time. Originally the string theorists supposed, that the left-over dimensions are compactified to extremely short distances. Therefore the extra dimensions would have only a little (direct) impact to low energy physics and particle phenomenology. Later it was found that these extra dimensions, or some of them, can be relatively large, at an intermediate scale [2]. This is related to the large uncertainty of the measurements of gravitational forces. They indicate, that the extra dimensions may even be as large as ∼ 1mm[3–5]. These considerations have boosted a new wave of research on the large extra dimensions: they may have testable consequences in present day or near future experiments. In the core of this research is the notion, that the Planck scale MPl is not necessarily the fundamental gravity scale, but only an effective one related to our four-dimensional space–time M4 of our natural perception [6–8].

E-mail address: vilja@utu.ﬁ (I. Vilja).

The gravity scale M∗ of the space–time which includes M4 and n large extra dimensions, is related to Planck n+2 (n) (n) scale by a well-known relation MPl = M∗ V , where V is an extra dimension size related factor. Depending (n) on V , M∗ may be considerably lower scale than MPl. Moreover, the particles of the Standard Model (SM) or other gauge symmetries are supposed to live only on a four-dimensional slices of all large dimensions (e.g. on M4) [7]. These slices, called the branes, may have at least two possible constructions. They may be defined by solitonic configurations of underlying (string) theory. In this case the brane has tiny but finite width being effectively four-dimensional. The brane may also be constructed as a singular point of an orbifold, when it is truly a four-dimensional manifold [9]. In these large extra dimension models gauge singlet particles (fields) may propagate outside the brane(s), in so-called bulk [7], forming the bulk matter. Right-handed neutrinos may be this kind singlet fermion fields [10,11]. Interactions between the brane and bulk matter may rise interesting consequences. They would explain quite naturally e.g. the lightness of neutrinos: their masses would be suppressed by factor ∼ M∗/MPl.Alotofworkon particle phenomenology of the extra dimension models has done during last years [12,13]. Also the cosmological implications of the extra dimensions have been subject of numerous studies [9,14]. In practice, there are two fundamental types of extra dimension constructions. One may have compact extra dimensions in so-called ADD-models [6], or the extra dimensions may be noncompact but “wrapped”, when one speaks of RS-models [15]. In the ADD-models the factor V (n) is simply the volume of the extra dimensions, while it in the RS-models is merely related to the normalization of graviton states. In either case the particles living in the bulk show up to form to an infinite Kaluza–Klein tower of excitations observed on the brane. It should be emphasized however, that no satisfactory extra dimension model has been constructed but only numerous more or less incomplete scenarios have been suggested. There has also been doubts, if string inspired models allow (semi)classical description of gravity and branes at all. For discussion about these issues, see [16] and references therein. Here we suppose, that (semi)classical description is, at least, approximately correct. However, we make no direct reference to any string inspired scenario, so it in the present Letter is only a possible motivation. The underlying physics may have a different origin, too. In the present Letter we study a model which lies somewhere in-between ADD- and RS-model. The manifold of the extra dimensions is a compact one, and the bulk is empty. We study a four-brane embedded on a six-dimensional space of large dimensions. However we do not suppose it to be a flat manifold but allow it to be curved, wrapped like in RS-models. This makes possible to obtain a considerably simple spectrum of excitations of the bulk matter, in particular bulk fermions as right handed neutrinos. We also study the general features of this model and propose a more detailed, simple construction.

2. Geometry on a torus

We coordinatise the six-dimensional space with zM , where M = 0,...,5, where the ﬁrst four coordinates are associated to the Minkowski space. The two extra dimensions have topology of the two-torus T 2 with coordinates a = 2 = a b y , a 1, 2. The metric of these extra dimensions, dsE gab dy dy , can quite generally be written in the form (y1 = φ, y2 = θ)

2 = 2 2 + 2 2 dsE a dφ f(φ,θ) dθ , (1) using suitable coordinate transformations. Here we deﬁne a as a dimensional constant determining the circumference of the torus at the φ-direction and f is a dimensional function determining the rest of the geometry. The range of the coordinates, the angle variables φ and θ is [0, 2π). The metric of the extra dimensions do not depend on Minkowski space coordinates (in particular on time) and therefore Riemann and Ricci curvature tensors do not mix Minkowski and extra dimension components. Thus the only nontrivial components are Rθφθφ =−ffφφ, 2 2 Rθθ = ffφφ/a and Rφφ = fφφ/f and hence the curvature scalar of the extra dimensions reads R =−2fφφ/(a f). I. Vilja / Physics Letters B 619 (2005) 155–162 157

The volume of the torus can also be calculated, giving

2π 2π

V2 = a dθ dφf(φ,θ). (2) 0 0 µ = µ − 1 µR µ =− µR The Minkowski space components of the Einstein tensor Eν Rν 2 gν reads now Eν δν while extra a = a = dimension components are equal to zero: Eb Eµ 0. The six-dimensional Einstein equation A = A EB 8πG6TB (3) implies thus, that the bulk is empty and possible branes are located at some points φi in φ-direction. Also one reads out from Eq. (3) that there are four-branes instead of a three-branes because there are not discontinuities at θ-direction. Thus the geometry describes empty bulk with branes of tension σi located at the points φi .The piecewise deﬁned bulk solution, i.e. the scale function f consists of simple positive linear pieces φ − φi f(φ,θ)= fi(θ) + fi+1(θ) − fi(θ) ,φi φ<φi+1, (4) φi+1 − φi where, without loss of generality can be chosen φ1 = 0,φK+1 = 2π representing the same point. Note, that due to periodicity the number of branes is 0 or larger than 1. Naturally, if it equals zero, f is a function of θ only and this function can be absorbed to the coordinate itself. The positions of branes, functions fi and the brane tensions σi are related by the crossing rule, the jump condition, which reads now as

f i f i−1 −4 2 − =−4πM∗ σia fi, (5) φi φi−1 where f i = fi+1 − fi and φi = φi+1 − φi and M∗ is the six-dimensional fundamental gravity scale related to −4 six-dimensional gravity constant by G6 = M∗ . With this notation the volume of the extra dimensions reads

2π f + + f V = a dθ φ i 1 i , (6) 2 i 2 0 i and relation between the four-dimensional √ gravitational scale (the Planck scale MPl) and six-dimensional scale M∗ 2 = 4 2 − = 4 reads now as MPl M∗ d y g6 V2M∗ . Of course, if functions fi are independent on coordinate θ,the integration contributes only with factor 2π. Moreover, if there are only two branes, the jump conditions are simply, 1 1 f + =−16πG a2σ f = 16πG a2σ f , (7) 2π − φ φ 6 1 1 6 2 2 and the volume is

2 V2 = 2π a(f1 + f2), (8) where f = f2 − f1, and φ = 0 is the position of the other brane. Should be again emphasized, that one single brane is not possible because it leads to constant f at whole bulk: it implies no discontinuities, no tension and no matter. Because our aim is to calculate the equations of motion for fermion ﬁelds, we need the spin connection deﬁned by 1 Ω = Γ AB V N ∂ V , (9) M 2 A M BN 158 I. Vilja / Physics Letters B 619 (2005) 155–162

N N = AB = i [ A B ] A where VA is the moving frame (sixbein) VA diag(1, 1, 1, 1,a,f), Γ 2 Γ ,Γ and matrices Γ are 8 × 8 gamma matrices for six-dimensional space. We use the gamma matrices µ 5 µ = 0 γ 4 = 014 5 =− 0 γ Γ µ ,Γ ,Γ i 5 , (10) γ 0 −14 0 γ 0 where γ µ’s are any four-dimensional gamma matrices. For the metric of 2-torus (1), however, all components of 1 D = M the spin connection vanish. Thus the covariant derivative for spin 2 -ﬁeld is now A VA ∂M . Note also, that the = 1 ± 7 7 =− 0 1 ··· 5 = − six-dimensional chirality operators are given by P± 2 (I8 Γ ), where Γ Γ Γ Γ diag(14, 14) is the generalization of γ 5 in six dimensions. If the dimension coordinated by θ is a maximally symmetric subspace, one can even choose a coordinate system where f in independent of θ-coordinate itself. It is naturally equivalent with the assumption, that the function f can be split to product of separate θ and φ dependent parts.

3. Singlet fermions on a general torus

In particle theory models the gauge-singlet fermions living on the whole six-dimensional space. The most natural candidates for such particles are the singlet neutrinos, which may appear even in the simplest extensions of the 2 Standard Model. To study the properties of the singlet ﬁelds on the manifold M4 × T we have to write down the action for them. The free ﬁeld Lagrangian reads as 4 2 ¯ A ¯ SN = d xd zV iNΓ DAN + h.c. − MNN , (11)

= N = ¯ = † 0 where V det(VA ) af , N is the fermion ﬁeld at six dimensions, N N Γ and M is the mass of the fermion ﬁeld. We also denote the chiral six-dimensional, eight component fermions by N± = P±N and identify them with four component chiral fermions, i.e. N+ N = . (12) N−

Note, that N± are identifiable to four-dimensional Dirac fermions. Using these components, the action reads as 4 2 ¯ ¯ 01 µ N+ i ¯ ¯ 01 1 N+ SN = d xd zV i(N−N+) ⊗ γ ∂µ + (N−N+) ⊗ 1 ∂φ + h.c. 10 N− −10 N− 2 a i ¯ ¯ 0 γ 5 −i N+ ¯ ¯ N+ + (N−N+) ⊗ 1 ∂θ + h.c. + M(N−N+) , (13) 2 γ 5 0 f N− N− ¯ † 0 where N± = N±γ . The corresponding equation of motion is given by − µ N+ i −N+ 5 i N+ i ∂φf −N+ N− −iγ ∂µ = ∂φ + iγ ∂θ + + M . (14) N− a N− f N− 2a f N− N+ This equation can be solved by separating Minkowski space and torus variables coordinates. It leads to tedious calculations, because the components mix strongly. Therefore, we assume that the six-dimensional field N is chiral field in the sense of six-dimensional theory, e.g. N− = 0. This corresponds the assumption M = 0 that decouples the six-dimensional chiral fields. It should be also noted, that for chiral six-dimensional spinors there are no separate Majorana-couplings, because Majorana fermions at six dimensions are readily chiral. Hence we may rewrite the equation of motion for N+ field alone as

µ i i 1 5 −iγ ∂ N+ =− ∂ N+ − (∂ ln f)N+ + γ ∂ N+. (15) µ a φ 2a φ f θ I. Vilja / Physics Letters B 619 (2005) 155–162 159

The right side of Eq. (15) is set equal to −M+N+, where M+ = m+ + iµ+γ 5 and m+ and µ+ are real numbers. Then we write the field as a sum of its chiral components N+ = N++ + N+− with γ 5N+± =±N+±, so that the equation of motion converts to equations i 1 i − ∂ N++ + ∂ N++ − (∂ ln f)N++ =−M+N++, a φ f θ 2a φ i 1 i − ∂ N+− − ∂ N+− − (∂ ln f)N+− =−M+N+−. (16) a φ f θ 2a φ After the separation of variables N+± = H±(φ, θ)ψ±(x), where ψ± are usual four-dimensional fermion fields, one finds that the equations of motion for H± are i 1 i − ∂ H± ± ∂ H± − (∂ ln f)H± =−(m+ ± iµ+)H±, (17) a φ f θ 2a φ where upper/lower signs belong to same equation. Immediately one sees, that

iν±θ H±(θ, φ) = e K±(φ), (18) where ν± are (due to θ-periodicity) integers and K± are functions of φ only. Inserting (18) to (17), one ﬁnds solution φ i ν± K±(φ) = exp ia du −m+ + ∂ ln f(u)± i −µ+ − K±(0). (19) 2a u f(u) 0 Again, consistency with periodicity requires, that K±(2π) = K±(0) and we obtain relation for mass parameters m+ and µ+:

−am+ = n, n ∈ Z (20) and 2π ν± dφ µ+ + = 0. (21) f(φ) 0 2π = − Note, that 0 dφ∂φ ln f(φ) ln f(2π) ln f(0) vanish automatically due to periodicity of f . Moreover, Eq. (21) should hold for both ν+ and ν−. Therefore they have to be equal, ν+ = ν− ≡ ν, and

2π ν dφ µ+ =− ,ν∈ Z. (22) 2π f(φ) 0 Thus two integers n and ν determine the solution besides the multiplicative constant K±(0), which can be absorbed to four-dimensional spinors ψ±. If we define a matrix 1/2 φ nν 1 f(0) iνθ+inφ a dφ(µ++ ν )γ 5 H+ = √ e e 0 f(φ) , (23) V2 f(φ) the solution of general linear equation (15), can be written as a generalized Fourier-expansion nν nν N+ = H+ ψ , (24) n,ν nν nν nν where ψ = ψ+ +ψ− is the Fourier coefficient of the mode (n, ν). They are the four-dimensional fermion fields. 160 I. Vilja / Physics Letters B 619 (2005) 155–162

4. Effective four-dimensional singlet fermions

The next task is to calculate the fermion spectrum of the four-dimensional effective model. Indeed there will be a double tower of exited states ψnν when the extra dimensions are integrated out. These modes mix with each ± other, because the functions H+ are not orthogonal. There appear complicated mixings in kinetic and mass terms, which should, in principle, be diagonalized. We consider again chiral (M = 0, N− = 0) version of Eq. (11): 4 2 i ¯ A i ¯ A S = d xd zV N+Γ D N+ − D N+Γ N+ , (25) N 2 A 2 A which now can be cast in the form 4 ¯ µ 1 ¯ 1 S = d xdφdθaf(φ) iN+γ ∂ N+ − N+M+N+ − (M+N+)N+ . (26) N µ 2 2 By inserting the expansion (24) into preceding equation one is able to integrate over the angular coordinates φ and ψ, i.e. write the action in the form of effective four-dimensional theory. However, this four-dimensional action is rather complicated: neither potential nor kinetic term are diagonal with respect the index n. One obtain the action for the four-dimensional spinors, = 4 + ¯ nν nν + − ¯ nν nν − ¯ ¯ nν nν + S d x iAnnνψ+ ∂ψ/ + iAnnνψ− ∂ψ/ − Bmψ− ψ+ h.c. , (27) nnµ nν where m¯ stands for m¯ = m+ + iµ+ and, ﬁnally, the coefﬁcients of the various terms are

2π + 2πaf(0) − + φ + ν = iamφ 2a 0 (µ+ f(u))du Annν dφe e , V2 0 2π − 2πaf(0) − − φ + ν = iamφ 2a 0 (µ+ f(u))du Annν dφe e , V2 0 4π 2af (0) B = , (28) V2 where m = (n − n)/a. Note, that B is independent on all indices. Thus we have obtained a infinite double tower of fermionic excitations with mixings of the kinetic terms. However, in this base the mass terms are diagonal. Of course, if f is a constant function, as on flat torus, all + ν ≡ nondiagonal terms vanish, because µ+ f 0, and modes decouple from each other. To proceed, one needs a viable approximation which simplifies the mixed up situation. First, we assume, that there is only two branes. Then the θ-independent scale function defined on the interval φ ∈[0, 2π) is now φ 2π − φ f(φ)= H(φ − φ) f + f + H(φ− φ ) f + f , (29) 1 φ 1 2π − φ where f and φ are as previously and H(φ − φ) is the step function. The jump condition and the volume are given by Eqs. (7) and (8), respectively. From Eq. (22) one reads now

=− ν f2 =−ν µ+ ln ¯, (30) f f1 f where f¯−1 = 1 ln f2 is an effective size of the φ-direction of the torus. f f1 I. Vilja / Physics Letters B 619 (2005) 155–162 161

2 Suppose now, that f2 f1, we find that the volume of extra dimensions is essentially V2 = 2π af2 and the six- 4 = 2 2 dimensional gravity scale reads M∗ MPl/(2π af2). There are now some physical requirements which should be fulfilled. The scale of the new physics may not be to low, i.e. we require that M∗ 10 TeV. On the other hand, the four-brane where usual nongauge-singlet particles are bounded has one compact extra dimension of radius a (φ-direction), the massive Kaluza–Klein modes along it are to be pushed beyond observational limit. The lightest massive Kaluza–Klein mode has mass 2π/a which has to be high enough, say 2π/a 10 TeV, i.e. a 0.6TeV−1. Taking the upper limit for a and lower limit for M∗, and combining these two physical conditions one finds, that 24 −1 8 f2 is large, f2 ∼ 1.2 × 10 GeV = 2.3 × 10 m. By inserting these values to Eqs. (28), we find that the ratios ± ± ∼ −16 ∼ = ± ± = Annν/Annν 10 (φ π, n n) and Annν are in practice independent on ν with Annν B. Thus, for most practical purposes the four-dimensional effective spinor action (27) is diagonal. We may once more write the effective spinor action using assumptions given above. The physically meaningful action reads 4 ¯ nν nν ¯ nν nν ¯ nν nν S = d x iψ+ ∂ψ/ + + iψ− ∂ψ/ − −¯mnνψ− ψ+ + h.c. , (31) nµ √ nν nν where the fields are rescaled as ψ± → ψ± / B and to each mode a suitable chiral rotation is made in purpose to remove complex masses. The mass m¯ nν of an individual mode is simply n 2 ν 2 m¯ nν = + . (32) a f¯ Thus n = 0 excitations are readily heavy and above direct observational limits. When one requires that also n = 0, ν = 0 excitations are not directly observable, one has to require f¯ to be small enough. Because f¯ f / ln f2 , 2 f1 24 −1 where f2 ∼ 10 GeV , one has to push f1 very low, nearly zero. Indeed one finds that φ = 0 brane lies almost at a gravitational singularity. The geometry of this torus can be compared to real doughnut submerged to R3:the perimeter of the inner ring is smaller than the perimeter of the outermost ring while at the perpendicular direction the perimeter is independent of the position.

5. Conclusions and discussion

In the present Letter a six-dimensional low energy model has been presented. The model contains four-branes and empty bulk. It appears, that in a two four-brane case the requirement of the usual four-dimensional Einstein gravity and nonexistence of unwanted light Kaluza–Klein modes constraint the model so, that the various modes of gauge-singlet fermion ﬁelds decouple. The mass spectrum of these modes, given by (32), contains only heavy effective fermions (besides the zero mode), if the effective size of both directions of the torus are small enough. Thus for each gauge-singlet fermion ﬁeld there is only one light fermionic mode which couples to ordinary matter, e.g. to left-handed neutrinos. In this model it is thus possible to have large Kaluza–Klein masses and keep the volume of the torus large enough, i.e. pull the gravitational scale M∗ down near to its experimental limit. Mixing with ordinary left-handed neutrinos on the brane(s) results hierarchy of the neutrino masses and mixings as studied by several authors. The model has, of course, many open questions, including the stability of the extra dimensions, the mass spectrum of graviton excitations, the corrections to Newton’s law, the neutrino masses and their mixings on the brane. In the present Letter, we have introduced only ideas, how viable models can be possibly constructed on a curved, compact manifold. More work is certainly needed to verify the properties of the model and also to construct a realistic model. 162 I. Vilja / Physics Letters B 619 (2005) 155–162

References

[1] G. Nordström, Phys. Z. 15 (1914) 504; T. Kaluza, Sitzungsber. Preuss. Akad. Wiss. Math. Phys. Kl. (1921) 966; O. Klein, Phys. Z. 37 (1926) 895. [2] For early considerations towards large extra dimensions, see I. Antoniadis, Phys. Lett. B 246 (1990) 377. [3] J.C. Long, H.W. Chan, J.C. Price, Nucl. Phys. B 539 (1999) 23. [4] J.C. Long, A.B. Churnside, J.C. Price, hep-ph/0009062; C.D. Hoyle, et al., Phys. Rev. Lett. 86 (2001) 1418. [5] For a review, see E.G. Adelberg, et al., Annu. Rev. Nucl. Part. Sci. 53 (2003) 77. [6] N. Arkani-Hamed, S. Dimopoulos, G. Dvali, Phys. Lett. B 429 (1998) 263. [7] I. Antoniadis, N. Arkani-Hamed, S. Dimopoulos, G. Dvali, Phys. Lett. B 436 (1998) 257; See also D. Cremades, L.E. Ibanez, F. Marchesano, Nucl. Phys. B 643 (2002) 93; C. Kokorelis, Nucl. Phys. B 677 (2004) 115. [8] N. Arkani-Hamed, S. Dimopoulos, G. Dvali, Phys. Rev. D 59 (1999) 086004. [9] For a review, see e.g. V.A. Rubakov, Phys. Usp. 44 (2001) 871. [10] N. Arkani-Hamed, S. Dimopoulos, G. Dvali, J. March-Russell, Phys. Rev. D 65 (2002) 024032. [11] K.R. Dienes, E. Dudas, T. Gherghetta, Nucl. Phys. B 557 (1999) 25. [12] A.E. Faraggi, M. Pospelov, Phys. Lett. B 458 (1999) 237; G. Dvali, A.Yu. Smirnov, Nucl. Phys. B 563 (1999) 63; A. Das, O.C.W. Kong, Phys. Lett. B 470 (1999) 149; R.N. Mohapatra, S. Nandi, A. Perez-Lorenzana, Phys. Lett. B 466 (1999) 115; R.N. Mohapatra, A. Perez-Lorenzana, Nucl. Phys. B 576 (2000) 466; G.C. McLaughlin, J.N. Ng, Phys. Lett. B 470 (1999) 157; A. Ioannisian, A. Pilaftsis, Phys. Rev. D 62 (2000) 066001; R. Barbieri, P. Creminelli, A. Strumia, Nucl. Phys. B 585 (2000) 28; A. Lukas, P. Ramond, A. Romanino, G.G. Ross, Phys. Lett. B 495 (2000) 136; A. Ioannisian, J.W.F. Valle, Phys. Rev. D 63 (2001) 073002; R.N. Mohapatra, A. Perez-Lorenzana, Nucl. Phys. B 593 (2001) 451; K.R. Dienes, I. Sarcevic, Phys. Lett. B 500 (2001) 133; K. Agashe, G.-H. Wu, Phys. Lett. B 498 (2001) 230; N. Cosme, et al., Phys. Rev. D 63 (2001) 113018; D.O. Caldwell, R.N. Mohapatra, S.J. Yellin, Phys. Rev. Lett. 87 (2001) 041601; D.O. Caldwell, R.N. Mohapatra, S.J. Yellin, Phys. Rev. D 64 (2001) 073001; A.S. Dighe, A.S. Joshipura, Phys. Rev. D 64 (2001) 073012; C.S. Lam, J.N. Ng, Phys. Rev. D 64 (2001) 113006; A. Lukas, P. Ramond, A. Romanino, G.G. Ross, JHEP 0104 (2001) 010; J.-M. Frere, M.V. Libanov, S.V. Troitsky, JHEP 0111 (2001) 025; C.S. Lam, Phys. Rev. D 65 (2002) 053009; A. de Gouvea, G.F. Giudice, A. Strumia, K. Tobe, Nucl. Phys. B 623 (2002) 395; R. Mohapatra, A. Perez-Lorenzana, Phys. Rev. D 67 (2003) 075105. [13] J. Maalampi, V. Sipiläinen, I. Vilja, Phys. Lett. B 512 (2001) 91; J. Maalampi, V. Sipiläinen, I. Vilja, Phys. Rev. D 67 (2003) 113005. [14] T. Gherghetta, M. Shaposhnikov, Phys. Rev. Lett. 85 (2000) 240; P. Kanti, R. Madden, K. Olive, Phys. Rev. D 64 (2001) 044021; I. Kogan, et al., Phys. Rev. D 64 (2001) 124014; Z. Chagko, A. Nelson, Phys. Rev. D 62 (2000); T. Multamäki, I. Vilja, Phys. Lett. B 535 (2002) 170; T. Multamäki, I. Vilja, Phys. Lett. B 559 (2003) 1. [15] L. Randall, R. Sundrum, Phys. Rev. Lett. 83 (1999) 3370; L. Randall, R. Sundrum, Phys. Rev. Lett. 83 (1999) 4690; For recent discussion about neutrinos in RS-models see also G. Moreau, hep-ph/0407177. [16] J.P. Lykken, hep-ph/0503148. Physics Letters B 619 (2005) 163–170 www.elsevier.com/locate/physletb

Strings in twistor superspace and mirror symmetry

S. Prem Kumar a,b, Giuseppe Policastro a

a DAMTP, University of Cambridge, Wilberforce Road, Cambridge CB3 0WA, UK b Department of Physics, University of Wales Swansea, Singleton Park, Swansea SA2 8PP, UK Received 8 April 2005; accepted 27 May 2005 Available online 6 June 2005 Editor: L. Alvarez-Gaumé

Abstract We obtain the super-Landau–Ginzburg mirror of the A-twisted topological sigma model on a twistor superspace—the quadric | | in CP3 3 × CP3 3 which is a Calabi–Yau supermanifold. We show that the B-model mirror has a geometric interpretation. In a particular limit for one of the Kähler parameters of the quadric, we show that the mirror can be interpreted as the twistor | superspace CP3 4. This agrees with the recent conjecture of Neitzke and Vafa proposing a mirror equivalence between the two twistor superspaces.  2005 Elsevier B.V. All rights reserved.

1. Introduction

Recently Witten [1] has argued that perturbative N = 4 supersymmetric U(N) Yang–Mills theory can be formulated as topological string theory with the supertwistor space CP3|4 as target. Among other things this has led to new and interesting observations about certain Calabi–Yau supermanifolds [2,3] which happen to be supertwistor spaces and these form the focus of our note. In particular, in [1] it was demonstrated that the N = 4 Yang–Mills amplitudes, when transformed to the supertwistor space CP3|4, are supported on holomorphic curves which were then interpreted as D1-instantons of the topological B-model on CP3|4. On the other hand, it was shown long ago [4] that the classical equations of motion of the N = 4 gauge theory follow from integrability of gauge fields on supersymmetric lightlike lines. The space of all such lightlike lines in (complexified) compactified Minkowski space is the quadric in CP3|3 × CP3|3.Itis natural to ask if there is some relation between these two pictures. In [2] it was conjectured that these two twistorial formulations could possibly be related by mirror symmetry between CP3|4 and the quadric in CP3|3 ×CP3|3.(They

E-mail addresses: [email protected], [email protected] (S.P. Kumar), [email protected] (G. Policastro).

0370-2693/$ – see front matter  2005 Elsevier B.V. All rights reserved. doi:10.1016/j.physletb.2005.05.068 164 S.P. Kumar, G. Policastro / Physics Letters B 619 (2005) 163–170 are both Calabi–Yau supermanifolds.) This conjecture was prompted by a combination of two observations. First, the authors of [2] argued that the N = 4 Yang–Mills amplitudes could also be obtained from the A-model topological string on CP3|4 to be understood as S-dual (see also [5] and [6]) to the B-model picture of [1]. In this picture the D1-instantons of [1] are replaced by worldsheet instantons of the A-model (and the D5-branes by NS5-branes). Secondly, given such an A-model description, it is natural to expect that a potential mirror B-model description will have no instantons and the perturbative Yang–Mills amplitudes will be realized classically. The observations of [4] outlined above suggest a candidate for such a mirror. Specifically, one expects that the B(A)-model on CP3|4 is mapped by a mirror transformation to the A(B)-model on the quadric in CP3|3 × CP3|3. In a recent work [3] it was demonstrated that the A-model on the former supermanifold is mirror to the B-model on the latter in the limit where the Kähler class t of CP3|4 is sent to minus infinity. In this Letter we show that the A-model on the quadric in CP3|3 × CP3|3 is mirror to the B-model on CP3|4 in a certain limit for one of the two Kähler parameters of the quadric. The motivation for this study is two-fold. One would like to understand if indeed the two twistorial formulations above are related by mirror symmetry, independently of the conjectured S-duality for topological strings. Further, one would like to shed light on the relationship between Kähler class and complex structure deformations of the two supermanifolds in question. It should be pointed out that we are studying the closed topological model (without any extra branes) which corresponds to the gravitational theory. Our results show that the B-model mirror of the quadric should be understood as a complex deformation of the twistor superspace CP3|4 (with a line at infinity removed). It has been argued that complex deformations of twistor space get mapped to C4 (complexified Minkowski space) with points blown up (see [5] and references therein). It would be extremely interesting to develop this idea further. In the following section we review some essential results in the context of mirror symmetry for supermanifolds. In Section 3 we apply these to the A-model on the quadric in CP3|3 × CP3|3 and obtain the mirror B-model which has a geometric interpretation.

2. Supermanifolds and hypersurfaces in toric manifolds

We begin by reviewing the results of [7] and [3] which are relevant for our computation. We are interested in computing the mirror transform of the topological sigma model of the A-type on the quadric1 which is realized as a hypersurface in a toric supermanifold.

2.1. Degree d hypersurface in CPd−1

To understand how this proceeds we first recall the well-known fact [8] that the observables of the A-model on a bosonic Calabi–Yau manifold M realized as a hypersurface in a compact toric manifold are simply related to the observables of the A-model on a corresponding non-compact toric manifold V . As a simple example, consider a hypersurface obtained from a degree d polynomial equation in CPd−1. This is realized as a U(1)-gauged linear sigma model with N = (2, 2) supersymmetry and d chiral superfields {Φi} of charge +1 each. In addition there is a field P of charge −d and a superpotential,

W = PG(Φi) (2.1) where G(Φi) is a degree d polynomial (weight d) [9]. These lead in the infrared to a non-linear sigma model description with the Calabi–Yau M as target via the vacuum equations

G(Φi) = 0; P = 0, (2.2)

| | 1 For the sake of brevity we will refer to the quadric in CP3 3 × CP3 3 simply as “the quadric”. S.P. Kumar, G. Policastro / Physics Letters B 619 (2005) 163–170 165 modulo complex gauge transformations. However, the observables of the A-model (the (a,c)-ring) cannot depend on the superpotential which is a (c,c)-ring deformation. Therefore we expect that as → 0 and the superpotential disappears, the observables of the A-model on M are simply related to the observables of the non-compact Calabi– Yau manifold V which is the O(−d) line bundle over CPd−1 (the field P is then identified with the coordinate on the fibre). Of course, this reasoning is not strictly correct since the two theories differ drastically, as the one with vanishing superpotential has 2 extra complex dimensions. Nevertheless, one indeed obtains the following correspondence between states of the A-twisted theory on M and those on V ,

|ΣV →|1M (2.3) where Σ is the twisted chiral Abelian ﬁeld strength and the state |ΣV is a normalizable ground state of the non- compact theory obtained by an insertion of Σ which corresponds to the Kähler form controlling the size of the compact part of the geometry. Alternatively, such Σ-insertions can be achieved by taking derivatives of correlators of the A-twisted model on V with respect to the complexiﬁed Kähler parameter t, yielding A-model observables on the compact manifold M.

2.2. Realization as the supermanifold WCPd−1|1(1,...,1|d)

It was shown in [7] that the above procedure of computing A-model observables of the compact manifold M is equivalent to A-model computations on a (d − 1|1)-dimensional toric supermanifold. Specifically, in the example above, it simply amounts to replacing P by a fermionic chiral superfield Ψ of charge d. Thus we have an O(d) (fermionic) bundle over CPd−1 which can also be viewed as the toric supermanifold Mˆ namely, WCPd−1|1(1,...,1|d). The D-term constraint is now, d 2 ¯ |Φi| + dΨΨ = Re[t] (2.4) i=1 where the Kähler class parameter of Mˆ is the same as that of the compact Calabi–Yau M. Since Mˆ has U(1) isometries one can perform T-duality to obtain the super-Landau–Ginzburg mirror. Techniques for doing this have been discussed in [10] and [3]. Interestingly the upshot of this procedure is that the super-Landau–Ginzburg directly yields the observables, such as periods of the compact bosonic Calabi–Yau M. Put another way, the fermionic fields of the sigma model on Mˆ automatically incorporate the projection (the t-derivative) that was required above to translate the observables of the non-compact manifold V into those of the compact Calabi–Yau M.

2.3. Review of T-duality for fermionic coordinates

The implementation of T-duality for fermionic coordinates has only recently been discussed in [3]. Since it is not part of standard literature we review the main results here which will be used subsequently. Just as in the case of bosonic coordinates with a U(1) isometry [11] we wish to dualize the phase for a fermionic superfield Ψ with a U(1)-charge q. The phase is bosonic and hence it will dualize into a bosonic twisted chiral multiplet Y .The real part of Y is determined as Y + Y¯ = ΨΨ¯ while its imaginary part is periodic. In addition, the usual twisted chiral superpotential is also generated for Y which gives the winding modes a mass qΣ. However, this is not all. The original theory had one fermionic coordinate Ψ whose momentum modes have mass qΣ. Hence the dualized theory cannot simply have one bosonic degree of freedom. In fact, it should have two fermionic superfields η,χ with the same mass qΣ as the winding modes of the dual bosonic coordinate Y . This ensures that one boson and one fermion cancel in the partition function. In sum then, the T-dual of the fermionic superfield Ψ yields the bosonic twisted chiral multiplet Y and two fermion superfields η,χ with a superpotential − W =−qΣ(Y − ηχ) + e Y . (2.5) 166 S.P. Kumar, G. Policastro / Physics Letters B 619 (2005) 163–170

This superpotential gives the same mass −dΣ to the winding modes of Y and the excitations of η,χ. We can rewrite this superpotential in a different form after a shift Y → Y + ηχ so that − W =−qΣY + e Y (1 − ηχ). (2.6) Now we see what the effect of the fermions is on the partition function of the dual theory. Integrating them out brings down a factor of e−Y in the measure turning e−Y into a good coordinate. In the context of the example discussed above this is precisely the effect of taking a t-derivative of the partition function of the theory corresponding to the bosonic non-compact manifold V .

3. Mirror of the quadric

We are now ready to apply the results above to the case of interest. We take two copies of CP3|3 each with ˜ ˜ ˜ the homogeneous coordinates {XI ,ΨA} and {XI , ΨA} respectively where {XI , XI },(I = 1,...,4) are bosonic ˜ coordinates and {ΨA, ΨA},(A = 1, 2, 3) are the fermionic coordinates. Let t1 and t2 be the complexified Kähler class parameters for the two CP3|3s. The quadric Q in CP3|3 × CP3|3 is then defined by the bilinear equation 4 3 ˜ ˜ G := XI XI + ΨAΨA = 0. (3.1) I=1 A=1 3|3 There is a Z2 symmetry which exchanges the two CP s and their Kähler classes t1 ↔ t2. This quadric can be realized as a U(1) × U(1) gauged linear sigma model with the charge assignments (1, 0) for the fields {XI ,ΨA} ˜ ˜ and (0, 1) for the second copy of coordinates {XI , ΨA}. In addition we introduce a bosonic chiral superfield P of charge (−1, −1) and a superpotential ˜ ˜ W = PG[XI , XI ΨA, ΨA]. (3.2)

The field content and charge assignments ensure that there is no U(1)A anomaly and the Calabi–Yau condition is satisfied. (Note that due to the reversed statistics the fermionic coordinates contribute to the anomaly with a sign opposite to that of the bosonic coordinates.) The theory flows to a (super-)Calabi–Yau phase where it is a non-linear sigma model with the quadric Q as target, corresponding to the vacuum manifold ˜ ˜ G[XI , XI ,ΨA, ΨA]=0 (3.3) with P = 0 in the D-term constraints, 4 3 2 2 2 |XI | + |ΨA| −|P | = Re[t1], I=1 A=1 4 3 ˜ 2 ˜ 2 2 |XI | + |ΨA| −|P | = Re[t2], (3.4) I=1 A=1 modulo U(1) × U(1) gauge transformations (Re[t1], Re[t2] > 0). The quadric is a hypersurface in a (6|6)- dimensional toric supermanifold. The bosonic hypersurface equation (3.3) means that the quadric Q has complex (super)dimension (5|6). In order to implement the mirror transform we need to realize the topological A-model observables on Q in terms of a sigma model on a toric (super)manifold i.e. one without the superpotential which imposes the hypersurface constraint. One is naturally tempted to use the ideas of [8] outlined in the previous section namely, to study the A-model on the “non-compact” Calabi–Yau where we send W → 0. However, this immediately leads to a puzzle—the sigma model with W = 0 has a (7|6)-dimensional target space. On the other hand, the quadric has S.P. Kumar, G. Policastro / Physics Letters B 619 (2005) 163–170 167 bosonic minus fermionic dimension −1 and its mirror must naturally have dimension (n − 1|n). The resolution is straightforward and we simply need to employ the ideas of [7] and [3] as explained earlier. We must not only send W to zero but we must also replace the bosonic field P with a fermionic chiral superfield ΨP with charge (1, 1) under the U(1) × U(1) gauge symmetry. In summary, the topological A-model on the quadric Q is equivalent to the A-model on the O1(1) ⊗ O2(1) 3|3 3|3 3|3 fermionic bundle over CP × CP (by O1(1) we mean the pullback of the O(1) line bundle on the first CP factor, and similarly for the second).

3.1. B-model mirror

The Landau–Ginzburg B-model dual of the above can be obtained as follows. T-duality replaces each bosonic ˜ ˜ superfield XI and XI with the cylinder-valued coordinates YI and YI respectively, (I = 1,...,4). Further, using the rules for dualizing the fermionic coordinates where each such field yields a bosonic coordinate and a pair of ˜ ˜ fermionic fields, the {ΨA} and {ΨA} dualize to the set {MA,ηA,χA} and {MA, ηÃ, χÃ} respectively, (A = 1, 2, 3). In our notation the η and χ are fermion superfields. Finally, the A-model fermion ΨP with charge (1, 1) dualizes to (YP ,η,χ). The Landau–Ginzburg mirror of the quadric is given by the path integral (for the holomorphic sector)

4 3 ˜ ˜ Z = [dYI dYI ] [dMA dMA dηA dχA dηÃ dχÃ] dYP dηdχ I=1 A=1 ˜ ˜ × δ YI − MA − YP − t1 δ YI − MA − YP − t2 ˜ ˜ −YI −YI −MA −MA −YP × exp e + e + e (1 + ηAχA) + e (1 +˜ηAχÃ) + e (1 + ηχ) . (3.5)

The Landau–Ginzburg model has 13 bosonic (taking into account the two delta-function constraints) and 14 fermionic degrees of freedom. Note that the Z2 exchange symmetry of the quadric is explicit in the B-model partition function above. To arrive at a mirror super-Calabi–Yau interpretation for this Landau–Ginzburg we perform a sequence of manipulations that involve integrating out some of the fields and successive field redefinitions. We first integrate out the fermions ηÃ, χÃ, η3, χ3, η, χ, and solve the delta-function constraints for YP and M3. This breaks the symmetry that exchanges the two CP3|3s of the A-model. We will come back to this point later. At this stage we have the following B-model integral with 4 fermions and 13 bosons

4 3 ˜ ˜ ˜ M1+M2− YI − MA+t1 Z = [dYI dYI ] dM1 dM2 [dMA] dη1 dχ1 dη2 dχ2 e I=1 A=1 ˜ ˜ −YI −YI −M1 −M2 −MA × exp e + e + e (1 + η1χ1) + e (1 + η2χ2) + e − + − + ˜ − ˜ ˜ − ˜ + et1 t2 eM1 M2 YI YI MA + et2 e MA YI . (3.6)

We see that integrating out the fermions leads to non-trivial factors in the measure. These measure factors turn the − − ˜ ∗ fields e YI and e MA which were C -valued, into good coordinates, so that we can define the new C-valued fields −Y −M˜ yI = e I and m˜ A = e A . In fact it is convenient to make a similar change of variables for all the bosonic fields:

˜ ˜ −M1,2 −MA −YI −YI m1,2 := e ;˜mA := e ; yI := e ;˜yI := e . (3.7) 168 S.P. Kumar, G. Policastro / Physics Letters B 619 (2005) 163–170

In terms of these new ﬁelds the Landau–Ginzburg model is

4 4 3 dy˜I dm1 dm2 Z = et1 [dy ] [dm˜ ] dη dχ dη dχ I 2 2 A 1 1 2 2 y˜I m m I=1 I=1 1 2 A=1

4 4 2 3 y˜ m˜ y t2 I I t1−t2 A A I I × exp yI + y˜I + ma(1 + ηaχa) + m˜ A + e + e . (3.8) m˜ A m1m2 y˜I I=1 I=1 a=1 A=1 A I A further change of variable

yÃ yam˜ a yb xÃ := (A = 1, 2, 3);˜x4 :=y ˜4; xa := (a = 1, 2); xb := (b = 3, 4) (3.9) m˜ A yãma x˜b allows us to bring the Landau–Ginzburg superpotential in the exponent in (3.8) to a polynomial form which will lead us to the interpretation as a super-Calabi–Yau manifold: 4 4 dm dm 3 t1 1 2 Z = e [dxI ] [dx˜I ] [dm˜ A] dη1 dχ1 dη2 dχ2 m1 m2 I=1 I=1 A=1

3 × exp m˜ A(1 +˜xA) + ma(1 + xax˜a + ηaχa) + x3x˜3 + x4x˜4 +˜x4 A=1 a=1,2

4 4 t2 t1−t2 + e x˜I + e xI . (3.10) I=1 I=1

One can now see that the fields m˜ A and x˜4 are Lagrange multipliers and their equations of motion set xÃ =−1 and t x4 = e 2 −1. It is clear from the measure that all the variables except m1,2 are “good” variables. The situation can be rectified following a procedure that is often useful for getting a geometric description from the Landau–Ginzburg B- model mirrors of Calabi–Yau manifolds (for instance see [12]). We introduce additional fields (ua,va), (a = 1, 2) 2 to absorb the non-trivial measure for ma. In the resulting expression m1 and m2 become Lagrange multipliers enforcing algebraic constraints. Integrating out the Lagrange multipliers we finally arrive at the interesting part of the story t1 t1 −t2 Z = e [dxa dua dva dηa dχa] δ(uava + ηaχa − xa + 1)δ e e − 1 x1x2 − 1 . (3.11) a=1,2 a The δ-functions inside the integral contain the information on the geometry of the mirror manifold. The first thing to note is that the putative mirror geometry has dimension (3|4) (the six bosonic coordinates have three delta- functions constraints) consistent with the conjecture of [2]. How do we understand and interpret this mirror geometry? Often a geometrical interpretation of the Landau– Ginzburg model emerges upon taking a limit of the A-model Kähler parameters. In other situations homogenizing the algebraic constraints allows one to interpret the mirror as a complete intersection in projective superspace.3 In the present situation we find that a natural interpretation emerges in a particular limit for the Kähler parameters.

2 uvm = 1 The idea is to make use of the relation dudve m for a suitable choice of contour. 3 Even though strictly speaking the constraints describe a non-compact manifold, the original geometry we started from, i.e. the quadric, is compact and therefore one expects that what we see in (3.11) is only an afﬁne coordinate patch inside a compact geometry; but one needs to check that the measure is consistent with such an interpretation. S.P. Kumar, G. Policastro / Physics Letters B 619 (2005) 163–170 169

We first perform a simple rescaling of the fields to rewrite the Landau–Ginzburg “period” as Z = [dua dva dxa dηa dχa] δ(u1v1 + η1χ1 − x1 + ν)δ(u2v2 + η2χ2 − x2 + ν)δ(x1x2 − µ), (3.12) a=1,2 − − where µ = e t1 and ν = (e t2 − 1)1/2. As pointed out earlier, the Z2 symmetry under the exchange of t1 and t2, corresponding to the exchange of the two CP3|3 factors in the A-model, has been broken. The Landau–Ginzburg integral (3.12) is a period integral over a supermanifold of dimension (3|4) which we want to identify. (We point out that our identification of the Landau–Ginzburg integral as a “period” is purely a formal analogy with the case of bosonic Calabi–Yaus. For supermanifolds the integrals above vanish unless there are suitable insertions of fermionic coordinates. For a discussion of related issues see [10].) We start by considering the limit ν → 0 ∼ t2 → 0. Then, solving the constraints for x ,x 1 2 Z = dua dva dηa dχa δ [u1v1 + η1χ1][u2v2 + η2χ2]−µ , (3.13) a=1,2 we can actually perform the delta-function integral by introducing additional variable changes as follows. In a patch where u1 = 0 we can introduce the variables z2 = u2/u1, z3 = v1/u1, z4 = v2/u1, ψ1 = η1/u1, ψ2 = η2/u1, ψ3 = χ1/u1 and ψ4 = χ2/u1. With these new variables the Landau–Ginzburg path integral is = du1 4 − − − Z dz2 dz3 dz4 dψ1 dψ2 dψ3 dψ4 δ u1(z2 ψ1ψ2)(z3z4 ψ3ψ4) µ u1 ≡ du1 4 − Ω1δ Au1 µ u1 1 = Ω , (3.14) µ 1 where we introduce the form Ω1 = dz2 dz3 dz4 dψ1 dψ2 dψ3 dψ4. Note that various factors of u1 in the measure, induced by the above variable change, cancel out precisely because of the presence of fermionic coordinates. The 3|4 form Ω1 is the natural holomorphic form on CP , given in affine coordinates z2,z3,z4,ψI , in a patch where one of the homogeneous coordinates (identified with u1) is set to 1. We can see that this interpretation is valid by trying to write the integral in a different patch. Then we can solve the δ-function constraints for u2, and we will define ˜ 3|4 some new coordinates z˜i, ψI related to zi,ψI in precisely the way affine coordinates in different patches of CP are related. This will lead to a holomorphic form Ω2 in the second affine patch, where Ω1 and Ω2 are the same on the intersection of the two patches. This is a confirmation of the conjecture of [2] that CP3|4 is the mirror supermanifold of the quadric Q and that this interpretation only emerges in a limit of the Kähler moduli of the quadric. The Landau–Ginzburg partition t function computed in the mirror manifold, in the limit where t2 → 0, is simply proportional to e 1 . This seems to imply that, up to a normalization, the periods (in this limit for t2) do not depend on the Kähler class t1 of the original manifold. Whereas this is in contrast to the usual situation in mirror symmetry, it is consistent with the arguments of [10], where supermanifolds were proposed as candidates for mirrors of rigid Calabi–Yaus. That could only be possible if in these models the Kähler class decouples from the other observables. This issue deserves further study. 3|3 It is also worth pointing out that the discrete symmetry (t1 ↔ t2) which exchanges the two CP s of the quadric is not visible in the mirror geometric description (3.13) obtained from the Landau–Ginzburg dual (3.5). Of course, obtaining the geometric picture required us to integrate out certain fields which then broke the t1 ↔ t2 symmetry. Obviously we could follow a different route which would yield the same geometric mirror but with t1 and t2 interchanged in Eq. (3.12). It is tempting to speculate that this breaking of the t1 ↔ t2 symmetry is intrinsically related to the way these spaces are defined as twistor spaces. In particular, the bosonic part of CP3|3 × CP3|3,in which the quadric is embedded, is the space of all self-dual planes and anti-self-dual planes in C4. On the other 170 S.P. Kumar, G. Policastro / Physics Letters B 619 (2005) 163–170 hand, the bosonic part of CP3|4 is simply the space of self-dual (or anti-self-dual) planes in C4 and thus singles out states of a particular helicity. Finally, we consider the geometric mirror (3.12) in the general case ν = 0. One expects that it should be interpreted as a complex deformation of the twistor superspace. While CP3|4 itself may not have complex deformations, the twistor superspace PT which should actually be defined as CP3|4\CP1|4 [1], can have complex deformations. This is well known in the bosonic case [13,14]. It is interesting to note that if we ignore the fermions in the delta- functions in (3.12) and set ν = 0, after a simple variable change it is possible to interpret the resulting expressions as an O(1) ⊕ O(1) bundle over CP1. Turning on non-zero ν would be a deformation of this bundle. It would be interesting to pursue this interpretation in the presence of the fermions. These issues deserve further attention, particularly if we would like to interpret (3.12) (for generic Kähler parameters of the quadric) as a complex deformation of twistor superspace.

Acknowledgements

We thank C. Vafa and M. Mariño for reading and commenting on a draft of this Letter. We acknowledge PPARC for ﬁnancial support.

References

[1] E. Witten, hep-th/0312171. [2] A. Neitzke, C. Vafa, hep-th/0402128. [3] M. Aganagic, C. Vafa, hep-th/0403192. [4] E. Witten, Phys. Lett. B 77 (1978) 394. [5] N. Nekrasov, H. Ooguri, C. Vafa, hep-th/0403167. [6] A. Kapustin, hep-th/0404041. [7] A. Schwarz, Lett. Math. Phys. 38 (1996) 91, hep-th/9506070. [8] K. Hori, C. Vafa, hep-th/0002222. [9] E. Witten, Nucl. Phys. B 403 (1993) 159, hep-th/9301042. [10] S. Sethi, Nucl. Phys. B 430 (1994) 31, hep-th/9404186. [11] M. Rocek, E. Verlinde, Nucl. Phys. B 373 (1992) 630, hep-th/9110053. [12] K. Hori, A. Iqbal, C. Vafa, hep-th/0005247. [13] M.F. Atiyah, N.J. Hitchin, I.M. Singer, Proc. R. Soc. London, Ser. A 362 (1978) 425. [14] R.S. Ward, R.O. Wells, Twistor Geometry and Field Theory, Cambridge Univ. Press, Cambridge, UK, 1990. Physics Letters B 619 (2005) 171–176 www.elsevier.com/locate/physletb

On the SO(9) structure of supersymmetric Yang–Mills quantum mechanics

J. Wosiek

M. Smoluchowski Institute of Physics, Jagellonian University, Reymonta 4, 30-059 Kraków, Poland Received 8 April 2005; accepted 20 May 2005 Available online 31 May 2005 Editor: L. Alvarez-Gaumé

Abstract In ten space–time dimensions the number of Majorana–Weyl fermions is not conserved, not only during the time evolution, but also by rotations. As a consequence the empty Fock state is not rotationally symmetric. We construct explicitly the simplest singlet state which provides the starting point for building up invariant SO(9) subspaces. The state has non-zero fermion number and is a complicated combination of the 1360 elementary, gauge invariant, gluinoless Fock states with twelve fermions. Fermionic structure of higher irreps of SO(9) is also brieﬂy outlined.  2005 Elsevier B.V. All rights reserved.

PACS: 11.10.Kk; 04.60.Kz

Keywords: Supersymmetry; Quantum mechanics; (M)atrix theory

1. Introduction from the simple rescaling of a finite number of degrees of freedom. The whole family (for various D and N) Supersymmetric Yang–Mills quantum mechanics reveals a broad range of very interesting phenomena (SYMQM) emerge from the dimensional reduction and has many applications in seemingly distant areas of corresponding field theories to a single point in of theoretical physics [3–14]. Perhaps the most known = the (D − 1)-dimensional space [1,2]. Resulting sys- example is the conjectured relevance of the D 10 tems are characterized by two parameters: D—the di- system, at large N, to the M-theory [15–17]. mensionality of the space–time of the unreduced the- A series of results has been recently obtained for = = ory, and N—the number of colors specifying a gauge the D 2 and D 4, SU(2), models with the aid of group. Dependence on the gauge coupling follows the cut Fock space approach [18–20].InthisLetterwe address the D = 10, N = 2system[21] and construct explicitly the SO(9) singlet state which replaces the E-mail address: [email protected] (J. Wosiek). empty Fock state sufficient in lower dimensions. This

0370-2693/$ – see front matter  2005 Elsevier B.V. All rights reserved. doi:10.1016/j.physletb.2005.05.048 172 J. Wosiek / Physics Letters B 619 (2005) 171–176 state is highly non-trivial due to the non-conservation Ref. [22]. In chiral representation they are block di- of the Majorana–Weyl fermion number in ten dimen- agonal, hence we restrict ourselves to one chirality.1 It sions. is well known that both Majorana and Weyl conditions The Hamiltonian reads [1,21] can be simultaneously imposed only in D = 2 (mod 8) space–time dimensions. Consequently the Majorana H = HK + HP + HF , matrix turns out to be block diagonal as well, and we 1 can impose Majorana condition in one chirality sub- H = pi pi , K 2 a a block. Finally, a solution of the Majorana condition, g2 in chiral representation, has a simple form H = xi xj xi xj , P abc ade b c d e T = 1 2 3 4 5 6 7 8 8† − 7† 4 ψa f ,f ,f ,f ,f ,f ,f ,f ,f , f , ig H = ψ†Γ kψ xk. (1) f 6†, −f 5†, −f 4†,f3†, −f 2†,f1† (6) F 2 abc a b c a with f † and f being the standard, anticommuting, There are 27 bosonic coordinates xi , and their mo- a fermionic creation and annihilation operators. There- menta pi , i = 1,...,9, a = 1, 2, 3. Fermionic degrees a fore, the ten-dimensional supersymmetric Yang–Mills of freedom compose a Majorana–Weyl spinor in the quantum mechanics, with SU(2) gauge group has 24 adjoint representation of the SU(2) gauge group, ψα, a fermionic degrees of freedom. α = 1,...,16. Γ k are 16 × 16 subblocks of the big (32 × 32) Dirac αk matrices in chiral representation. In all explicit calculations we use the representation of 2. The cut Fock space approach Ref. [22]. The system has the internal Spin(9) rotational sym- There exists surprisingly powerful method to com- metry generated by the gauge invariant angular mo- pute the complete spectrum and eigenstates of polyno- mentum mial Hamiltonians with a “reasonably” large number [ ] 1 of degrees of freedom [18]. It was applied successfully J kl = x kpl + ψ†Σklψ , (2) a a 2 a a to the D = 2 and D = 4 SYMQM with 6 and 15 degrees of freedom respectively [19,20]. In the latter case with many new properties of this system were uncovered, i including identification of dynamical supermultiplets, Σkl =− Γ k,Γl . (3) 4 computation of their energies, wave functions, etc. Ten-dimensional system can also be attacked with this After the dimensional reduction, the local gauge in- approach. Presumably the high accuracy of Ref. [20] variance amounts to the global invariance under the could not be matched at the moment, but the recur- SU(2) rotations generated by the color angular mo- sive technique developed there offers a real possibil- mentum ity for some quantitative results. However, the ten- i dimensional system is more complex, also on the more G = xkpk − ψ†ψ . (4) a abc b c 2 b c fundamental level. Namely, the fermion number is not conserved by the Hamiltonian, and also by the SO(9) Furthermore, the Hamiltonian, Eq. (1), is invariant un- rotations, Eq. (2) [1]. In this work we address this diffi- N = der, 1, ten-dimensional supersymmetry with 16 culty in some detail and propose one possible solution. Majorana–Weyl generators In order to better illustrate the problem, we briefly sketch the method of Refs. [18–20]. Since the Hamil- = k k + jk j k Qα Γ ψa αpa igabc Σ ψa αxb xc . (5) tonian, Eq. (1) is a simple function of creation and an- Supersymmetry requires imposing both Weyl and Ma- nihilation operators, it is convenient use the eigenbasis jorana conditions on the 32-dimensional spinor. To identify explicitly fermionic degrees of freedom, we 1 Compared to Ref. [22] simple similarity transformation is reconstruct big (32 × 32) Dirac α matrices following quired to bring big α matrices to this form. J. Wosiek / Physics Letters B 619 (2005) 171–176 173 of the number operators associated with all individual just for the purpose of initial classification. In each degrees of freedom. Beginning with the empty (fermi- fermionic sector we then construct a complete basis onic and bosonic) state, |0=|0F , 0B , we construct of gauge invariant states allowing up to Bmax bosonic the physical, i.e., gauge invariant, basis of the Hilbert quanta. To this end one needs the empty Fock state space by acting on |0 with gauge invariant polynomi- |0=|0F , 0B mentioned earlier. In D = 2 and 4 this als of all creation operators.2 The basis is artificially is a simplest possible state, in particular in D = 4itis cut by limiting the total number of all bosonic quanta. rotationally symmetric, i.e., it is annihilated by the an- Then we calculate analytically matrix representation gular momentum. This is not the case for D = 10. We of the Hamiltonian and obtain numerically the spec- obtain trum. The procedure is then repeated for higher cut- 2| = | offs until the results converge. Originally the analytical J 0 78 0 . (7) part of the calculation was done in Mathematica by The empty Fock state is an eigenstate of J 2,butit defining the Mathematica representation of the Fock is not a singlet! It belongs to some higher represen- states and all relevant operators. In Ref. [20] this was tation of SO(9). Since SO(9) has rank 4, one needs replaced by the fast, recursive calculation of all ma- eigenvalues of other three Casimir operators to iden- trix elements. Where possible conservation laws were tify uniquely this representation. The precise answer used. For example, the fermionic number is conserved is not relevant here. The lowest candidate has dimen- in D = 2 and D = 4 models and the whole proce- sionality 132 132 and, in the Dynkin notation [24],is dure was carried out independently in each fermionic labeled by (1120). sector. In the four-dimensional model we reduced the Since the Hamiltonian (1) respects the SO(9) sym- problem even further by using (composite) creators metry, we would like to construct separate bases in with fixed fermion number and angular momentum. each channel of SO(9) angular momentum. To this This allowed to obtain the energies of the first 10–20 end however, we need to begin with the SO(9) sin- states, in every J(0 J ∼16) sector, with the four glet state. Evidently the empty state cannot be used digit precision. for this purpose. Where is the simplest singlet state? In ten dimensions fermion number is not conserved This question will be answered in the next section. [21], and things require more care. Even before that, it turns out that the empty Fock state—the state which is the root of the whole construction—has to be consid- 4. An answer erably modified. Table 1 gives sizes of the gauge invariant bases in all 25 fermionic sectors with none and one bosonic 3. A puzzle quantum. They were calculated with a fermionic variant of the TWS method introduced in Ref. [25].3 They = In dimensionally reduced theories the total number satisfy supersymmetric relations: # bosonic states of fermionic quanta is finite. Since all quanta occupy # fermionic states, for each cutoff. Such relations were the same point in space, one can only have as many also found in lower dimensions within our regular- fermionic quanta as there are different degrees of free- ization. It is clear that the brute force diagonalization = of the Hamiltonian is out of question, especially for dom. Hence in D 2, there are maximally three, 4 in D = 4—six, and in D = 10—24 fermions. Cor- higher B. As discussed earlier, one has to split these respondingly, there are four, seven and 25 fermionic sectors in the Hilbert spaces of these systems. In ten 3 In short, we generated all states built from elementary, gauge dimensions, fermion number is not conserved, never- i † j † invariant “bricks”: fa fa . Obvious linear dependences were ex- theless we shall use the concept of fermionic sectors cluded from the beginning, remaining ones were identified by Gram determinants. Many states mix, but in a regular pattern. TWS refers to a particular classification scheme, where the mixings are very 2 The BRST quantization of these systems has been recently dis- transparent and easy to control. cussedinRef.[23]. 4 For D = 4 we needed B ∼ 10–20 to reach convergent results. 174 J. Wosiek / Physics Letters B 619 (2005) 171–176

89 1 Table 1 J = (N1 + N2 + N3 + N4 + N5 + N6 + N7 Sizes of bases generated in each (B, F ) sector for the D = 10 sys- 2 tem. Ns is the number of basis vectors + N8 − 12), (9) F 024681012 = m † m BNs Ns Ns Ns Ns Ns Ns with Nm a(fa ) fa being the gauge invariant 0 1 28 406 4060 17 605 41 392 56 056 number operator of the mth Majorana–Weyl fermion, 1 – 324 9072 81 648 374 544 908 460 1 205 568 m = 1,...,8. Clearly, above Cartan generators are di- 2 45 3816 89 838 agonal in the occupation number representation. Their 3 84 23 652 eigenvalues can be just read off from the indices of 4 1035 5 2772 our basis. This substantially simplifies the search for 6 16 215 a spherically symmetric state. First, it is obvious from F 1357 9 11 Eq. (9) that a singlet can be only in the sector with F = 12. This is the most complex sector of the whole BN N N N N N s s s s s s theory with 56 056 basis states, cf. Table 1. Second, 0 – 120 1512 8856 29 512 51 520 using again Eqs. (9) one can readily identify the sub- 1 72 2016 29 232 192 528 626 040 1 126 944 B = 2 288 21 024 set 12(0, 0, 0, 0) of the F 12 states with all four 3 3240 magnetic quantum numbers equal to zero. It contains 4 12 960 “only” 1360 states. Singlet states, if any, must be linear combinations of these states. This problem is manage- able with our Mathematica representation of quantum bases into sectors with fixed SO(9) angular momen- mechanics. We have calculated matrix representation tum. Hence, again, one needs to construct the simplest 2 of J in the B12(0, 0, 0, 0) sub-basis and found the singlet state to begin with. spectrum. It turns out that there exists only one eigen- Since our cutoff state with zero eigenvalue. This is the desired singlet. ai †ai B , (8) All others 1359 eigenvalues belong to the known spec- b b max trum of the first Casimir operator of the SO(9), which b,i provides additional check on the whole procedure. As is invariant under SO(9) rotations, we can restrict the another test, we have reconstructed, from the numeri- search to the simplest, B = 0, sector. Even then the cal eigenvector, the singlet state in the Fock space, and problem would require calculation of the huge matrix checked that, indeed, it is annihilated by all 36 com- representation of the complicated J 2 operator, which ponents of the angular momentum, Eq. (2). is practically impossible. This is the main result of present Letter. The sim- Instead, we have analyzed the action of the 36 plest SO(9) invariant state is the linear combination of components of the angular momentum on all basis 1360 basis states from the sector with 12 Majorana– states with B = 0. Using the explicit representation of Weyl fermions and no bosonic quanta. Expansion co- Dirac matrices, discussed in Section 1, we have found efficients are known numerically. This state should that the four generators from the Cartan subalgebra of be used as the root when creating SO(9) invariant SO(9) are particularly simple for B = 0. subspaces. It replaces the empty state of the lower- 1 dimensional models. This is one more consequence of J 23 = (N − N + N − N + N − N the peculiar behavior of the Majorana–Weyl fermions 2 1 2 3 4 5 6 under SO(9) rotations. Had we not imposed Majorana + N − N ), 7 8 condition, the Weyl spinor in Eq. (6) would have con- 45 1 J = (N1 + N2 − N3 − N4 + N5 + N6 sisted of 16 annihilation operators. Then the empty 2 state would be symmetric again, cf. Eq. (2). − − N7 N8), Another way this peculiarity shows up is the 1 following. A fundamental fermionic representation, J 67 = (N + N + N + N − N − N 2 1 2 3 4 5 6 (0001), of SO(9) is 16-dimensional. On the other hand − N7 − N8), there are only 8 independent Majorana–Weyl creation J. Wosiek / Physics Letters B 619 (2005) 171–176 175

Table 2 of angular momentum are in general linear combina- SO(9) structure of the B = 0 sector of the D = 10 SYMQM. Irre- tions of states with different fermion number. ducible representations listed in the last two columns extend over fermionic sectors marked by × F 0 2 4 6 8 10 12 14 16 18 20 22 24 Irrep dim 5. Summary and outlook × (0, 0, 0, 0) 1 ××× (2, 0, 0, 0) 44 Number of Majorana–Weyl fermions is not con- ××××× (0, 0, 1, 0) 84 served by rotations in 9 space dimensions. This fact ××××××× (0, 1, 1, 0) 1650 has many unusual consequences for the D = 10 su- ×××××× × × ××××× (1, 1, 2, 0) 132 132 persymmetric Yang–Mills quantum mechanics. Irre- ducible representations of SO(9) cover many fermionic sectors of the theory. In particular, the empty operators.5 Therefore we need both creation and an- Fock state is not a singlet. It belongs to the compli- nihilation operators to form a covariant SO(9) spinor cated 132 132-dimensional representation which ex- as in Eq. (6). Such a spinor, when acting on a singlet tends over all sectors with even fermion number. The state, would create 16 states from (0001). With the simplest invariant state is unique and it is in the half- singlet in the F = 12 sector, everything is consistent: filled sector with 12 fermions. It is empty with respect 8 of above states are in the F = 11 sector and another to bosonic quanta, but has quite non-trivial structure 8haveF = 13. This would have not worked if a sin- in terms of the elementary, gauge invariant, fermionic glet was empty. Fock states. It is a linear combination of the 1360 basis The last example illustrates very well that irre- states which, out of a total of 56 056 states in this sec- ducible representations of SO(9) stretch across fermi- tor, are annihilated by the suitably chosen Cartan gen- onic sectors, which is another way to say that F is erators of SO(9). We have explicitly constructed this not conserved by rotations. In fact by diagonalizing expansion. This state replaces the empty state while 2 J in other sub-bases, BF (M23,M45,M67,M89), and building the SO(9) invariant subspaces of the theory. matching eigenvalues among different fermionic sec- Therefore one can proceed now with the diagonaliza- tors, one can construct higher representations and see tion of the Hamiltonian in channels with fixed SO(9) which F ’s they contain. Such a map is shown in Ta- angular momentum. ble 2 for few irreps. The singlet sits only in the mid- We conclude with some open questions, which dle, F = 12, sector. Higher irreps extend gradually might help to simplify present solution. Eq. (9) for towards the edges, i.e., towards the empty and filled the J 89 generator suggest that the normal ordering of fermionic states. Eventually, beginning with already Majorana–Weyl creation/annihilation operators might discovered (1120), representations span all fermionic help. However one has to check if this is consistent sectors. Remember that this structure holds only for with the whole SO(9) algebra and other symmetries Fock states without bosonic quanta. Since bosonic and of the model. Second, since the singlet is in the half- fermionic angular momenta add in a usual way, a sin- filled sector, one wonders if some variant of the Dirac glet with B = 1, say, would span through F = 10, 12 procedure might work. The trouble is that there are and F = 14 sectors, etc. In general, the higher B the many half-filled sates here and none of the simple re- wider are irreps in F . Beginning with B = 6evensim- definition of f ’s seems to work. Finally, one might plest irreps stretch across all fermionic sectors. look for the Bogoliubov transformation which makes Notice however, that each eigenstate of four Car- our singlet simple. One should keep in mind however, tan generators has well defined F if B = 0. In another that such a transformation, if exists, should also render words, eigenstates of Eqs. (9) never stretch across dif- simplicity of the Hamiltonian. ferent fermionic sectors, but irreducible representations do, even for B = 0. For B>0 the eigenstates Acknowledgements

5 Clearly the eight component object built only from creation I would like to thank W.A. Bardeen for the discus- operators does not form irreducible representation of SO(9) [26]. sion. This work is supported by the Polish Committee 176 J. Wosiek / Physics Letters B 619 (2005) 171–176 for Scientiﬁc Research under the grant No. 1 P03B 024 [13] M.V. Berry, J.P. Keating, in: I.V. Lerner, et al. (Eds.), Super- 27 (2004–2007). symmetry and Trace Formulae: Chaos and Disorder, Kluwer Academic, New York, 1999, p. 355. [14] S. Sethi, M. Stern, Commun. Math. Phys. 194 (1998) 675, hep- th/9705046. References [15] T. Banks, W. Fischler, S. Shenker, L. Susskind, Phys. Rev. D 55 (1997) 5112, hep-th/9610043. [1] M. Claudson, M.B. Halpern, Nucl. Phys. B 250 (1985) 689. [16] D. Bigatti, L. Susskind, in: Strings, Branes and Dualities, [2] L. Brink, J.H. Schwarz, J. Scherk, Nucl. Phys. B 121 (1977) Cargese, 1997, p. 277, hep-th/9712072. 77. [17] W. Taylor, Rev. Mod. Phys. 73 (2001) 419, hep-th/0101126. [3] E. Witten, Nucl. Phys. B 188 (1981) 513. [18] J. Wosiek, Nucl. Phys. B 644 (2002) 85, hep-th/0203116; [4] F. Cooper, A. Khare, U. Sukhatme, Phys. Rep. 251 (1995) 267, J. Kotanski, J. Wosiek, Nucl. Phys. B (Proc. Suppl.) 119 (2003) hep-th/9405029. 932, hep-lat/0208067; [5] B. de Wit, M. Lüscher, H. Nicolai, Nucl. Phys. B 320 (1989) See also J. Wosiek, hep-th/0204243. 135. [19] M. Campostrini, J. Wosiek, Phys. Lett. B 550 (2002) 121, hep- [6] M. Lüscher, Nucl. Phys. B 219 (1983) 233. th/0209140. [7] M. Lüscher, G. Münster, Nucl. Phys. B 232 (1984) 445. [20] M. Campostrini, J. Wosiek, Nucl. Phys. B 703 (2004) 454, hep- [8] G.K. Savvidy, Phys. Lett. B 159 (1985) 325. th/0407021. [9] P. van Baal, Acta Phys. Pol. B 20 (1989) 295. [21] M.B. Halpern, C. Schwartz, Int. J. Mod. Phys. A 13 (1998) [10] P. van Baal, The Witten index beyond the adiabatic approx- 4367, hep-th/9712133. imation, in: M. Olshanetsky, A. Vainshtein (Eds.), Michael [22] J. Polchinski, String Theory, Cambridge Univ. Press, Cam- Marinov Memorial Volume, Multiple Facets of Quantization bridge, 1998. and Supersymmetry, World Scientiﬁc, Singapore, 2002, p. 556, [23] A. Fuster, J.W. van Holten, hep-th/0504165. hep-th/0112072. [24] R. Slansky, Phys. Rep. 79 (1981) 1. [11] H. Nicolai, R. Helling, in: Nonperturbative Aspects of [25] J. Wosiek, Acta Phys. Pol. B (2003) 5103, hep-th/0309174; Strings, Branes and Supersymmetry, Trieste, 1998, p. 29, hep- See also J. Wosiek, hep-th/0410066. th/9809103. [26] V.G. Kac, A.V. Smilga, Nucl. Phys. B 571 (2000) 515, hep- [12] B. Simon, Ann. Phys. 146 (1983) 209. th/9908096. Physics Letters B 619 (2005) 177–183 www.elsevier.com/locate/physletb

Periodic monopoles

R.S. Ward

Department of Mathematical Sciences, University of Durham, Durham DH1 3LE, UK Received 14 May 2005; accepted 27 May 2005 Available online 6 June 2005 Editor: L. Alvarez-Gaumé

Abstract This Letter deals with static BPS monopoles in three dimensions which are periodic either in one direction (monopole chains) or two directions (monopole sheets). The Nahm construction of the simplest monopole chain is implemented numerically, and the resulting family of solutions described. For monopole sheets, the Nahm transform in the U(1) case is computed explicitly, and this leads to a description of the SU(2) monopole sheet which arises as a deformation of the embedded U(1) solution.  2005 Elsevier B.V. All rights reserved.

PACS: 11.27.+d; 11.10.Lm; 11.15.-q

1. Introduction Letter is to review what is known about the simplest (unit charge) monopole chains and monopole sheets, and to describe their appearance. In recent years, there has been interest in periodic In the chain case, we implement the Nahm con- BPS monopoles, namely solutions of the Bogomolny struction numerically, to obtain a one-parameter fam- equations on R3 which are periodic either in one direc- ily of 1-monopole chains; the parameter is the ratio tion (monopole chains) or two directions (monopole between the monopole size and the period. In the sheet sheets). This has arisen partly because of the interpre- case, there is a homogeneous U(1) monopole sheet tation and application of such solutions in the theory solution; we demonstrate that this is “self-reciprocal” of D-branes. For monopole chains, the details of the under the Nahm transform, and describe the appear- Nahm transform have been fully explored, and there ance of the SU(2) monopole sheet which arises as a are some partial existence results [1]; but for mono- deformation of this Abelian solution. pole sheets, much less is known [2]. In neither case are The ﬁelds we deal with are solutions of the Bogo- there any explicit solutions. The main purpose of this molny equations

1 E-mail address: [email protected] (R.S. Ward). D Φ =− ε F (1) j 2 jkl kl 0370-2693/$ – see front matter  2005 Elsevier B.V. All rights reserved. doi:10.1016/j.physletb.2005.05.070 178 R.S. Ward / Physics Letters B 619 (2005) 177–183 on R3. Here the coordinates are xj = (x,y,z),the has the asymptotic behaviour φ ∼ (log ρ)/(2π) for gauge field is Fjk = ∂j Ak − ∂kAj +[Aj ,Ak], and large ρ. Dj Φ = ∂j Φ +[Aj ,Φ]. We take the gauge group to be This U(1) example motivates the boundary condi- SU(2); except that in the section on monopole sheets, tions for the non-Abelian case [1]. In particular, we we start with U(1) fields. The norm-squared of the require that Higgs field Φ is defined by |Φ|2 =−1 tr(Φ2), and the 2 N energy density is |Φ|∼ log ρ, |DΦ|=O(1/ρ) (4) 2π 1 1 as ρ →∞, where N is a positive integer. In fact, N E =− tr (D Φ)2 + (F )2 . (2) 2 j 2 jk is a topological invariant: the eigenvector of Φ associated with its positive eigenvalue defines a line bundle E =∇2| |2 ∇2 If (1) is satisfied, then Φ , where is the over the 2-torus ρ = c 1, and the first Chern num- 3 Laplacian on R . ber of this line bundle is N. A smooth solution of (1) satisfying the boundary condition (4) may be thought of as an infinite chain of N-monopoles. 2. Monopole chains Through the Nahm transform [1], such monopole chains correspond to solutions of the U(N) Hitchin In this section, we are interested in BPS monopoles equations on the cylinder R × S1, with appropriate on R2 × S1; specifically, monopoles which are peri- boundary conditions. Let us concentrate on the N = 1 odic in z with period 2π. Let us begin with some case, and describe the Nahm construction of the N = 1 general remarks. In the case of periodic instantons monopole chain. (calorons), one may proceed by taking a finite chain Write s = r + it, where r ∈ R and t ∈[0, 1) are of instantons (m instantons strung along a line in R4 coordinates on the cylinder. Let be the first-order with equal spacing), and letting the number m tend to differential operator infinity—indeed, the first example of a caloron solu- 2∂s¯ − zP(s) tion was constructed in this way [3]. For monopoles, = , (5) there is a solution representing a string of m mono- P(s) 2∂s + z poles [4]: one can write down its Nahm data explicitly where P(s)= C cosh(2πs)− (x + iy), with C being a in terms of the m-dimensional irreducible representa- positive constant. For each spatial point xj = (x,y,z), tion of su(2). But this has no limit as m →∞, so one the L2 kernel of this operator is 2-dimensional. So does not get an infinite monopole chain in this way. there exists a 2 × 2matrixΨ(t,r; xj ) satisfying There is another way to understand why one ex- ∞ 1 pects something to go wrong in the m →∞ limit † [1,2]. In the asymptotic region ρ2 = x2 +y2 →∞,the Ψ = 0, Ψ Ψdtdr= I, (6) Higgs field Φ of a chain of single SU(2) monopoles −∞ 0 will behave like a chain of U(1) Dirac monopoles, where I denotes the 2 × 2 identity matrix. Then for which the Higgs field, by linear superposition, is =−1 [ 2 + − 2]−1/2 ∞ 1 φ 2 p∈Z ρ (z 2πp) . But this series diverges: the m-chain (which corresponds to a finite Φ = i rΨ†Ψdtdr, series) has no limit as m →∞. One may, instead, −∞ 0 define a chain of Dirac monopoles by subtracting an ∞ 1 infinite constant, to obtain † ∂ Aj = Ψ Ψdtdr (7) 1 1 1 1 ∂xj φ = α − − − , −∞ 0 2r 2 2 + − 2 2π|p| p=0 ρ (z 2πp) defines a 1-monopole chain satisfying (1) and (4). (3) The explicit solution of the boundary-value prob- where α is a constant. This field is smooth, except at lem (6) is not known. Part of the difficulty is the lack of the locations ρ = 0, z ∈ 2πZ of the monopoles, and symmetry—both the finite and the infinite monopole R.S. Ward / Physics Letters B 619 (2005) 177–183 179 chains seem to have only a D2 symmetry, correspond- f =|ξ|g/ξ and c = π|ξ|. In other words, one approx- ing to rotations by 180◦ about each of the x, y and z imate solution of (8) is axes. This is quite unlike the situation for the N = 1 g ξ instanton chain, where one has O(3) symmetry, and ≈ E(s − s0), an explicit caloron solution [3]. So to see what the f |ξ| monopole chain looks like, one has to solve (6) either which is strongly peaked at s = s0. The other (inde- approximately or numerically. pendent) solution is peaked at s =−s0, and is obtained This solution contains only one parameter, name- similarly. So we can take ly C. All the other moduli can (and have) been j − ξE(s − s0) −ξE(s + s0) removed by translations and rotations of the x . Ψ ≈|ξ| 1/2 , (9) (Of course, there will be far more parameters when |ξ|E(s − s0) |ξ|E(s + s0) N>1.) From (5), one might guess on dimensional where the normalization factor (ensuring that grounds that C determines the monopole size, and this Ψ †Ψ = I ) follows from is indeed the case; or rather, since the length-scale is already fixed by the period of z, the parameter C 1 E(r + it)2 dr dt ≈ . corresponds to the dimensionless ratio between the 2|ξ| monopole size and the z-period. If 0

Fig. 1. |Φ|2 and |DΦ|2 on the xy-plane, for C = 8. The upper figures use the approximate solution, and the lower figures use a numerical solution. z-axis). For large C, however, the energy density be- l = 1([1], discussed in the previous section) are well- comes approximately constant in the z-direction, and established; but not much is known about the l = 2 is peaked along two lines parallel to the z-axis, each case. In view of the general pattern, one would expect a distance C from it. The numerical results indicate that the Nahm transform of a monopole on R × T 2 that the zeros of the Higgs field Φ are located on the will be another monopole on R × T 2. It remains to z-axis, at z = 2πn for n ∈ Z; but for large C, Φ is be seen whether or not this is the case in general very close to zero on the whole of the planar segment (and under what circumstances), but we shall see now −C

(a)

(b)

Fig. 2. SU(2) monopole sheet: perturbation of homogeneous solution. (a) |Φ|2 on z = 0; (b) |DΦ|2 − π2 on z = 0. value B2/4 = π 2 in the unperturbed case, whereas the Acknowledgements perturbed version is non-constant near z = 0 and is peaked where Φ has its zero. Clearly much analysis remains to be done in this This work was supported by a research grant from case, to conﬁrm that solutions exist, understand their the UK Engineering and Physical Sciences Research moduli space, and work out the details of the Nahm Council, and by the grant “Classical Lattice Field The- transform. Work in this direction is currently under ory” from the UK Particle Physics and Astronomy Re- way. search Council. R.S. Ward / Physics Letters B 619 (2005) 177–183 183

References [4] N. Ercolani, A. Sinha, Commun. Math. Phys. 125 (1989) 385. [5] P. Rossi, Nucl. Phys. B 149 (1979) 170. [1] S. Cherkis, A. Kapustin, Commun. Math. Phys. 218 (2001) 333. [6] R.S. Ward, Phys. Lett. B 582 (2004) 203. [2] K. Lee, Phys. Lett. B 445 (1999) 387. [7] E.F. Corrigan, P. Goddard, Ann. Phys. 154 (1984) 253. [3] B.J. Harrington, H.K. Shepard, Phys. Rev. D 17 (1978) 2122. Physics Letters B 619 (2005) 184–191 www.elsevier.com/locate/physletb

Light quarks with twisted mass fermions

χLF Collaboration K. Jansen a, M. Papinutto a, A. Shindler a,C.Urbacha,b, I. Wetzorke a

a John von Neumann-Institut für Computing NIC, Platanenallee 6, D-15738 Zeuthen, Germany b Institut für Theoretische Physik, Freie Universität Berlin, Arnimallee 14, D-14195 Berlin, Germany Received 25 March 2005; accepted 20 May 2005 Available online 31 May 2005 Editor: L. Alvarez-Gaumé

Abstract We investigate Wilson twisted mass fermions in the quenched approximation using different definitions of the critical bare quark mass mc to realize maximal twist and, correspondingly, automatic O(a) improvement for physical observables. A particular definition of mc is given by extrapolating the value of mc obtained from the PCAC relation at non-vanishing bare twisted quark mass µ to µ = 0. Employing this improved definition of the critical mass the Wilson twisted mass formulation provides the possibility to perform reliable simulations down to very small quark masses with correspondingly small pion masses of 2 mπ 250 MeV, while keeping the cutoff effects of O(a ) under control.  2005 Elsevier B.V. All rights reserved.

1. Introduction calculations in chiral perturbation theory were done [9–11]. The combination of an automatic O(a)-improve- One of the problems that appeared in the discus- ment, the infrared regulation of small eigenvalues sion of twisted mass fermions is the exact deﬁnition and fast dynamical simulations render the so-called of the critical bare quark mass. In [12] it was shown twisted mass fermions [1–3] a most promising for- that using the critical value of the bare quark mass mulation of lattice QCD. There are already a number where the pion mass vanishes, leads to a quark mass of results for the quenched approximation [4–6].Also dependence of the pion decay constant fπ that devi- full QCD simulations with this approach have been ates strongly from the expected linear behavior. This performed and revealed a surprising phase structure of “bending phenomenon” was observed when the quark lattice QCD [7,8]. On the theoretical side, extensive mass mq obeys the inequality

2 2 E-mail address: [email protected] (I. Wetzorke). amq

The bending phenomenon was interpreted in a way We extract pseudoscalar and vector meson masses that for such small values of the quark mass the Wilson from the correlation functions at full twist (m0 = mc): term becomes the dominant term in the lattice Wilson– a = 3 a a = Dirac operator, thus leading to large lattice artefacts. CP (x0) a P (x)P (0) ,a1, 2, In Refs. [13,14] it has been suggested to take the x value of the critical mass where the PCAC quark mass 3 a3 vanishes. In chiral perturbation theory it was demon- Ca (x ) = Aa(x)Aa(0) ,a= 1, 2, A 0 3 k k strated that this definition leads to a stable situation k=1 x of “maximal twist” and hence to a reduction of lat- a3 3 tice artefacts that otherwise would be enhanced at Ca (x ) = T a(x)T a(0) ,a= 1, 2, T 0 3 k k very small values of the quark mass. In Refs. [15,16] k=1 x the theoretical background of such a definition is dis- a = cussed on a general ground. where we consider the usual local bilinears P ¯ τ a a = ¯ τ a a = ¯ τ a In this Letter, we employ the definition of the crit- ψγ5 2 ψ, Ak ψγkγ5 2 ψ and Tk ψσ0k 2 ψ. ical mass by computing at non-zero twisted mass the The untwisted PCAC quark mass mPCAC can be ex- critical hopping parameter as a functions of the twisted tracted from the ratio mass parameter and finally taking the limit to zero ∂ Aa(x)P a(0) x 0 0 twisted mass. Such a definition is expected to elimi- mPCAC = ,a= 1, 2. (4) 2 P a(x)P a(0) nate large O(a2) artifacts at small values of the pion x mass and should hence avoid the bending phenom- Using the PCAC relation we can also compute the enon. In the following we will shortly describe the pion decay constant at maximal twist without requir- encouraging results employing this improved defining any renormalization constant (see [4,18]): ition of the critical mass. A more detailed scaling 2µ analysis of mesonic quantities will be reported else- f = |P a|π ,a= , . π 2 0 1 2 (5) where [17]. (Mπ )

2. Wilson twisted mass fermions 3. Deﬁnition of the critical mass

The Wilson tmQCD action of Eq. (2) can be stud- In this Letter we will work with Wilson twisted ied in the full parameter space (m ,µ). A special case mass fermions that can be arranged to be O(a) 0 arises, however, when m is tuned towards a critical improved without employing specific improvement 0 bare quark mass m . In such, and only in such a situa- terms [1,2]. The Wilson tmQCD action in the twisted c tion, all physical quantities are, or can easily be, O(a) basis can be written as improved. The critical quark mass, or alternatively the 1 4 ¯ critical hopping parameter κc = + , has thus to S[U,ψ,ψ¯ ]=a ψ(x)(D + m + iµγ τ )ψ(x), 2amc 8 W 0 5 3 be fixed in the actual simulation to achieve automatic x (2) O(a)-improvement at maximal twist. In general, the definition of the critical mass has where the Wilson–Dirac operator D is given by W an intrinsic uncertainty that comes at O(a) for Wilson fermions. This could be improved to an O(a2) uncer- 3 1 ∗ ∗ tainty by using Clover fermions. If a definition of κc DW = γµ ∇ +∇µ − a∇ ∇µ (3) 2 µ µ from a linear extrapolation of (m a)2 → 0 is used, µ=0 π one has in addition an unrelated systematic error com- ∇ ∇∗ and µ and µ denote the usual forward and back- ing from the long way of extrapolation from large pion ward derivatives. We refer to [12] for further unex- masses of O(600 MeV). plained notations. The definition of the critical mass A better definition of κc can be obtained by the fol- mc will be discussed in detail in the next section. lowing procedure: 186 χLF Collaboration / Physics Letters B 619 (2005) 184–191

Fig. 1. Determination of the critical mass: 1/κc versus µa at β = 5.70, extrapolation to µa = 0, the open diamond indicates the 1/κc value 2 determined by (mπ a) → 0atµ = 0 for unimproved Wilson fermions.

Fig. 2. Comparison of results at β = 6.0 for the pion mass for different deﬁnitions of the critical mass.

• At fixed non-zero twisted mass parameter the hop- at small quark masses much below values indi- ping parameter κc(µa) is determined as the point cated in the inequality of Eq. (1). where the PCAC quark mass of Eq. (4) vanishes. The non-zero value of the twisted mass allows a In Fig. 1 we show an example of this extrapolation at safe interpolation in this case. β = 5.7, where we also indicate the κc value deter- • As a further step, an only short linear extrapola- mined by the vanishing pion mass squared. tion of κc(µa) from small values of µa to µa = 0 The possible choices for the determination of the yields a definition of κc which is expected [15,16] critical bare quark mass were also discussed in chiral to lead to small lattice artefacts, in particular also perturbation theory [13,14]. Recently, a definition of χLF Collaboration / Physics Letters B 619 (2005) 184–191 187

Fig. 3. Comparison of results at β = 5.70 for the pion decay constant (top) and vector meson mass (bottom) for different definitions of the critical mass. maximal twist from parity conservation has also been cay constant and the ρ-mass obtained with two differ- investigated in [6]. There, however, the critical masses ent definitions of κc. The first, “pion mass” definition, 2 mc(µa) were not extrapolated to µa = 0, but were refers to the limit (mπ a) → 0 taken at µa = 0. The used at the respective twisted mass parameter at which second, “PCAC quark mass” definition, corresponds to they were determined. the determination of κc(µa) at fixed µa = 0 from the vanishing of the PCAC quark mass, amPCAC(µa) and then taking the limit κ (µa → 0). The latter κ values 4. Numerical results c c were determined on a subset of O(150) configurations In this section we will provide a comparison of Wil- for three couplings, namely β = 5.70, 5.85 and 6.0. son twisted mass results for the pion mass, the pion de- The simulations were performed for a number of bare 188 χLF Collaboration / Physics Letters B 619 (2005) 184–191

Fig. 4. Comparison of overlap fermion and Wilson twisted mass results at β = 5.85 for the pion decay constant (top) and vector meson mass (bottom) for different deﬁnitions of the critical mass. quark masses in a corresponding pion mass range of [6] and is manifest in the smaller error bars displayed 250 MeV

Fig. 5. Comparison of results at β = 6.0 for the pion decay constant (top) and vector meson mass (bottom) for different deﬁnitions of the critical mass.

the pion mass definition of κc. This residual O(a) pion but non-zero value of the twisted mass. In Fig. 3 we mass at µa = 0 can be attributed to the O(a) error in show the results for the pion decay constant and the the critical mass determined by the vanishing of the vector meson mass. It is obvious that already this in- pion mass in the pure Wilson case. termediate definition of κc substantially diminishes the At β = 5.7 (lattice size 123 × 32, a ≈ 0.17 fm) we bending tendency of the data when approaching the computed correlation functions also for a third, inter- chiral limit. Such a definition of the critical hopping mediate definition of the critical mass: we chose the parameter κc(µ0a = 0) seems to be sufficient to elimi- point where the PCAC quark mass vanishes at small, nate the deviation from a linear behavior for all twisted 190 χLF Collaboration / Physics Letters B 619 (2005) 184–191 mass values µa µ0a. The results for κc(µa = 0) where maybe contact to chiral perturbation theory can from the PCAC quark mass definition show a straight be made. behavior for both observables down to pion masses of In Ref. [6], the same definition of the critical mass about 250 MeV. The bending phenomena reported in has been employed, but at fixed twisted mass parame- [12] can thus be attributed to the pion mass defini- ter separately at each simulation point. The results of tion of the critical mass in this simulation. Note that this reference taken together with the findings in the at β = 5.7 we are working at a rather coarse value of present Letter lead to the conclusion that the PCAC the lattice spacing. quark mass definition of the critical hopping parame- At β = 5.85 (lattice size 163 × 32, a ≈ 0.12 fm) ter is a crucial element of twisted mass simulations in we can now compare our previous and new Wil- order to keep O(a2) effects well under control at small son twisted mass results with the data from overlap quark masses. This conclusion is in accordance with fermion simulations [12] which were obtained on a theoretical considerations [13–16]. smaller lattice volume (123 × 24). Using the defini- Employing the PCAC quark mass definition of the tion of κc from the vanishing of the PCAC quark mass, critical mass the Wilson twisted mass formulation pro- the bending near the chiral limit vanishes almost com- vides the possibility to perform reliable simulations pletely both for the pion decay constant and the vector at very small quark masses (mπ 250 MeV). Thus, meson mass extracted from the tensor correlator (see Wilson twisted mass fermions can be used to explore Fig. 4). In both cases the overlap fermion results lie really light quarks on the lattice—as light as with over- slightly higher than the twisted mass fermion results. lap fermions, but at considerably lower cost (see [21] This fact might be explained by residual finite volume for a detailed cost comparison). This opens a very effects and/or different lattice artefacts. promising prospect for dynamical fermion simulations Finally we show in Fig. 5 the results at β = 6.0 (lat- and renders twisted mass fermions as a real alternative tice size 163 ×32, a ≈ 0.09 fm). Although the physical to staggered fermions without the conceptual difficul- volume is smaller than the one at β = 5.85 we are able ties of the latter. to confirm the absence of bending for small masses when using the PCAC quark mass definition of κc.For the pion decay constant we observe a very good agree- Acknowledgements ment with the overlap fermion results of Ref. [20] and with the twisted mass results using the parity conser- We thank R. Frezzotti, G.C. Rossi, L. Scorzato, vation definition of the critical mass from Ref. [6]. S. Sharpe and U. Wenger for many useful discussions. For the vector meson mass we refrain from compar- The computer centers at NIC/DESY Zeuthen, NIC ing with the parity conservation definition, since the at Forschungszentrum Jülich and HLRN provided the error bars would cover both our results with the pion necessary technical help and computer resources. This mass and PCAC quark mass definition of the critical work was supported by the DFG Sonderforschungs- mass. bereich/Transregio SFB/TR9-03.

5. Conclusions References

We have studied Wilson twisted mass fermions [1] R. Frezzotti, P.A. Grassi, S. Sint, P. Weisz, Nucl. Phys. B (Proc. Suppl.) 83 (2000) 941; with different deﬁnitions of the critical mass. The R. Frezzotti, P.A. Grassi, S. Sint, P. Weisz, JHEP 0108 (2001) “bending phenomena” in basic observables like the 058. vector meson mass and the pion decay constant using [2] R. Frezzotti, G.C. Rossi, JHEP 0408 (2004) 007; the critical mass from vanishing pion mass at µa = 0 R. Frezzotti, G.C. Rossi, Nucl. Phys. B (Proc. Suppl.) 128 (2004) 193. reported in [12] is absent for the choice of κc from [3] R. Frezzotti, Plenary talk at Lattice 2004, Nucl. Phys. B (Proc. the vanishing of the PCAC quark mass in the limit Suppl.) 140 (2005) 134. µa → 0. This is true even at small values of the [4] K. Jansen, A. Shindler, C. Urbach, I. Wetzorke, Phys. Lett. pion mass close to the experimentally measured value B 586 (2004) 432. χLF Collaboration / Physics Letters B 619 (2005) 184–191 191

[5] A.M. Abdel-Rehim, R. Lewis, Phys. Rev. D 71 (2005) 014503. [15] R. Frezzotti, G. Martinelli, M. Papinutto, G.C. Rossi, hep- [6] A.M. Abdel-Rehim, R. Lewis, R.M. Woloshyn, hep- lat/0503034. lat/0503007. [16] A. Shindler, private communication. [7] F. Farchioni, R. Frezzotti, K. Jansen, I. Montvay, G.C. Rossi, [17] χLF Collaboration, A. Shindler, et al., Nucl. Phys. B (Proc. E. Scholz, A. Shindler, N. Ukita, C. Urbach, I. Wetzorke, Eur. Suppl.) 140 (2005) 746; Phys. J. C 39 (2005) 421; χLF Collaboration, A. Shindler, et al., in preparation. F. Farchioni, R. Frezzotti, K. Jansen, I. Montvay, G.C. Rossi, [18] R. Frezzotti, S. Sint, Nucl. Phys. B (Proc. Suppl.) 106 (2002) E. Scholz, A. Shindler, N. Ukita, C. Urbach, I. Wetzorke, Nucl. 814; Phys. B (Proc. Suppl.) 140 (2005) 240. M. Della Morte, R. Frezzotti, J. Heitger, Nucl. Phys. B (Proc. [8] F. Farchioni, K. Jansen, I. Montvay, E. Scholz, L. Scorzato, A. Suppl.) 106 (2002) 260. Shindler, N. Ukita, C. Urbach, I. Wetzorke, hep-lat/0410031. [19] R.W. Freund, in: L. Reichel, A. Ruttan, R.S. Varga (Eds.), Nu- [9] S.R. Sharpe, R. Singleton, Phys. Rev. D 58 (1998) 074501. merical Linear Algebra, Berlin, 1993, p. 101; [10] G. Münster, JHEP 0409 (2004) 035. U. Gläsner, S. Güsken, Th. Lippert, G. Ritzenhöfer, K. [11] L. Scorzato, Eur. Phys. J. C 37 (2004) 445. Schilling, A. Frommer, hep-lat/9605008; [12] χLF Collaboration, W. Bietenholz, et al., JHEP 0412 (2004) B. Jegerlehner, Nucl. Phys. B (Proc. Suppl.) 63 (1998) 958. 044; [20] L. Giusti, et al., JHEP 0404 (2004) 013. χLF Collaboration, W. Bietenholz, et al., Nucl. Phys. B (Proc. [21] χLF Collaboration, T. Chiarappa, et al., Nucl. Phys. B (Proc. Suppl.) 140 (2005) 683. Suppl.) 140 (2005) 853; [13] S. Aoki, O. Bär, Phys. Rev. D 70 (2004) 116011. χLF Collaboration, T. Chiarappa, et al., in preparation. [14] S.R. Sharpe, J.M.S. Wu, Phys. Rev. D 70 (2004) 094029, hep- lat/0411021. Physics Letters B 619 (2005) 193–200 www.elsevier.com/locate/physletb

The fate of (phantom) dark energy universe with string curvature corrections

M. Sami a,1, Alexey Toporensky b, Peter V. Tretjakov b, Shinji Tsujikawa c

a IUCAA, Post Bag 4, Ganeshkhind, Pune 411 007, India b Sternberg Astronomical Institute, Moscow State University, Universitetsky Prospekt, 13, Moscow 119899, Russia c Department of Physics, Gunma National College of Technology, Gunma 371-8530, Japan Received 26 April 2005; received in revised form 7 June 2005; accepted 8 June 2005 Available online 16 June 2005 Editor: M. Cveticˇ

Abstract We study the evolution of (phantom) dark energy universe by taking into account the higher-order string corrections to Einstein–Hilbert action with fixed dilaton and modulus fields. While the presence of a cosmological constant gives stable de Sitter fixed points in the cases of heterotic and bosonic strings, no stable de Sitter solutions exist when a phantom fluid is present. We find that the universe can exhibit a Big Crunch singularity with a finite time for type II string, whereas it reaches a Big Rip singularity for heterotic and bosonic strings. Thus the fate of dark energy universe crucially depends upon the type of string theory under consideration.  2005 Published by Elsevier B.V.

PACS: 98.70.Vc

1. Introduction When w is less than −1, dubbed as phantom dark energy, the universe ends up with a Big Rip singularity [3,4] which is characterized by the divergence of cur- Recent observations suggest that the current uni- vature of the universe after a ﬁnite interval of time (see verse is dominated by dark energy responsible for an Refs. [5,6]). accelerated expansion [1]. The equation of state pa- The energy scale may grow up to the Planck scale rameter w for dark energy lies in a narrow region in the presence of phantom dark energy. This means around w =−1andmayevenbesmallerthan−1 [2]. that higher-order curvature or quantum corrections can be important around the Big Rip. For example, quan- E-mail address: [email protected] (S. Tsujikawa). tum corrections coming from conformal anomaly are 1 On leave from: Department of Physics, Jamia Millia, New taken into account in Refs. [7] for dark energy dynam- Delhi 110025. ics. It was found that such corrections can moderate

0370-2693/$ – see front matter  2005 Published by Elsevier B.V. doi:10.1016/j.physletb.2005.06.017 194 M. Sami et al. / Physics Letters B 619 (2005) 193–200 the singularity by providing a negative energy density where α is the string expansion parameter, φ is the [8]. Thus it is important to implement quantum effects dilaton field, and in order to predict the final fate of the universe. In low-energy effective string theory there exist L2 = Ω2, (3) higher-curvature corrections to the usual scalar cur- L = + µν αβ λρ 3 2Ω3 R R Rµν, (4) vature term. The leading quadratic correction corre- αβ λρ δB sponds to the product of dilaton/modulus and Gauss– L4 = L41 − δH L42 − L43, (5) Bonnet (GB) curvature invariant [9]. The GB term is 2 topologically invariant in four dimensions and hence with does not contribute to dynamical equations of motion 2 µν µναβ if the dilaton/modulus field is constant [10]. Mean- Ω2 = R − 4RµνR + Rµναβ R , (6) while it affects the cosmological dynamics in presence µν αβ µν Ω = R R Rλρ − 2R Rλβρ Rα of dynamically evolving dilaton and modulus fields. 3 αβ λρ µν αβ ν ρµλ The possible effects of the GB term for early universe 3 2 µναβ + RR + 6R RαµRβν cosmology and black hole physics were investigated 4 µναβ in Refs. [11,12]. Lately the GB correction was applied 3 + µν α − 2 + R to the study of cosmological dynamics of dark energy 4R RναRµ 6RRαβ , (7) 4 [13]. µγ µγ L = ζ(3)R Rανρβ R Rδσ − 2R Rδσ , (8) When the dilaton and modulus are fixed, it is impor- 41 µνρσ δβ αγ δα βγ tant to implement third and next-order string curvature 1 2 1 ρσ µν L = R Rµναβ + RγδR Rαβ R corrections [10]. This can change the resulting cos- 42 8 µναβ 4 µν γδ ρσ αβ mological dynamics drastically as it happens in the − 1 αβ ρσ µ νγδ − αβ ρν γδ µσ context of inflation [14] and black holes [15]. The goal Rµν Rαβ RσγδRρ Rµν Rαβ Rρσ Rγδ , 2 (9) of the present Letter is to study the effect of next-to- µναβ 2 µνασ βγδρ leading order string corrections to the cosmological L43 = Rµναβ R − 10Rµναβ R RσγδρR dynamics around the Big Rip singularity with an as- µνρ βσγδ α − Rµναβ R R R . (10) sumption that the dilaton and the modulus are stabi- σ δγρ lized. We would also investigate the existence and the Here one has δH(B) = 1 for heterotic (bosonic) string stability of de Sitter solutions in the presence of a cos- and zero otherwise. The Gauss–Bonnet term, Ω2, mological constant. We shall consider three types of as well as the Euler density, Ω3, does not con- string corrections and study the fate of the universe tribute to the background equation of motion for accordingly. D = 4 unless the dilaton is dynamically evolving. The coefficients (c1,c2,c3) are different depending on string theories [10].Wehave(c1,c2,c3) = 2. Evolution equations (0, 0, 1/8), (1/8, 0, 1/8), (1/4, 1/48, 1/8) for type II, heterotic, and bosonic strings, respectively. In the case of type II string with D = 4, for example, only the L41 Let us consider the Einstein–Hilbert action in low- term affects the dynamical evolution of the system. energy effective string theory: We shall consider the flat Friedmann–Robertson– Walker metric with a lapse function N(t): √ S = D − [ + L +···] d x g R c , (1) d ds2 =−N(t)2 dt2 + a(t)2 dxi 2, (11) where R is the scalar curvature and Lc is the string i=1 correction which is given by [10] where d = D − 1. The Ricci tensors under this metric are given in Appendix A. In what follows we shall −2φ 2 −4φ 3 −6φ Lc = c1α e L2 + c2α e L3 + c3α e L4, consider the case of D = 4 under the assumption that (2) the modulus field which corresponds to the radius of M. Sami et al. / Physics Letters B 619 (2005) 193–200 195 extra dimensions is stabilized after the compactifica- rise to another equation, but this can be derived from tion to four dimensions. Then we find Eqs. (16) and (18) by taking a derivative with respect 24 72N˙ to t.FromEq.(17) we find that the energy density ρc L = H 6 + I 3 − HI2, 3 6 7 (12) for type II and heterotic strings is N N 6ζ(3) = 8 + 4 + 4 2 + 2 3 + 6 L =− H 8 + H 4I 2 + H 2I 3 + I 4 ρc B a8H acI a4H I a2H I a6H I 41 8 3 4 4 N − 5 + 2 + 3 ˙ J a5H a1HI a3H I , (19) 6ζ(3)N 5 3 2 3 + 8H I + 12H I + 4HI , (13) ¨ ˙ 3 3 N 9 where J = H + 3HH + H . One has B = 6c3α × −6φ =− =− =− = 6 8 4 2 4 e ζ(3), a8 21, ac 3, a4 12, a2 4, L42 =− 5H + 2H I + 5I =− =− =− =− N 8 a6 24, a5 8, a1 12, a3 24 for type = 3 −6φ =− + 6N˙ II string, and B 6c3α e , a8 21ζ(3) 35, + H 5I + HI3 , =− + =− − = + 9 4 20 (14) ac 3ζ(3) 15, a4 12ζ(3) 6, a2 4ζ(3) N =− + =− + = 6 20, a6 24ζ(3) 12, a5 8ζ(3) 4, a1 L =− 60H 8 + 32H 4I 2 + 60I 4 −12ζ(3) + 60, a =−24ζ(3) for heterotic string. In 43 N 8 3 ˙ the case of bosonic strings we have 6N 5 3 + 64H I + 240HI , (15) 6 3 N 9 ρc = A 5H + 2I − 6HIJ where H ≡˙a/a is the Hubble rate and I is defined by + B −21ζ(3) + 210 H 8 + −3ζ(3) + 90 I 4 I ≡ H 2 + H˙ . It should be noted that, in the case of − 12ζ(3) + 48 H 4I 2 + 4ζ(3) + 120 H 2I 3 de Sitter spacetime, the expressions (12)–(15) reduce to their counterparts given by Eqs. (A.1)–(A.4) in Ap- + −24ζ(3) + 96 H 6I pendix A. + J 8ζ(3) − 32 H 5 + 12ζ(3) − 360 HI2 We shall implement the contribution of a barotropic perfect fluid to the action (1). The equation of state + 24ζ(3)H 3I , (20) parameter, w = p /ρ , is assumed to be constant. m m m where A = 24c α 2e−4φ and B = 6c α 3e−6φ. Our main interest is to study the final fate of universe 2 3 filled with a phantom-type fluid (wm < −1). In this case the universe eventually reaches a Big Rip singu- 3. The fate of dark energy universe larity [2] with a divergent Hubble rate in the absence of higher-curvature terms. We are interested in the effect of string curvature corrections to the cosmological In this section we study the cosmological evolution evolution around the Big Rip. in dark energy universe for the above three classes of Varying the action (1) with respect to N, we find string curvature corrections. Our main interest is the universe dominated by phantom dark energy, but we 2 6H = ρc + ρm, (16) consider the case of cosmological constant as well. We first study the effects of type II and heterotic correc- where tions and then proceed to the bosonic correction. L L L ≡ d ∂ c + ∂ c − ∂ c − L ρc ˙ 3H ˙ c . (17) dt ∂N ∂N ∂N N=1 3.1. Type II and heterotic strings

ρm is the energy density of the barotropic fluid, satisfying For the analysis of dynamics to follow, it would be convenient for us to cast Eqs. (16) and (18) with the ρ˙m + 3H(1 + wm)ρm = 0 . (18) correction term (19) in the form In what follows we shall consider the case with a fixed x˙ = y, (21) dilaton. Then we have two dynamical equations (16) y˙ = B a x8 + a I 4 + a x4I 2 + a x2I 3 + a x6I and (18) for our system. We note that the variation 8 c 4 2 6 of the action (1) in terms of the scale factor a gives − 3xy + x3 ξ + z − 6x2 /(Bξ), (22) 196 M. Sami et al. / Physics Letters B 619 (2005) 193–200 z˙ =−3(1 + wm)xz, (23) this stable solution disappears for the equation of state with w =−1. = = ˙ = = 2 + = m where x H , y H , z ρm, I x y and ξ In order to understand the fate of the universe 5 + 2 + 3 a5x a1xI a3x I . Here we shall consider the case which is dominated by a phantom-type fluid, we solve =− ˙ = ˙ = ˙ = with wm 1. By setting x 0, y 0 and z 0, we autonomous equations (21)–(23) numerically. Fig. 2 find the following de Sitter fixed point: shows the evolution of the Hubble rate for a phantom- =− 1/6 type fluid with wm 1.5 in the presence of string 6 curvature corrections. We find that in the type II case xc = ,yc = 0,zc = 0, (24) BD the Hubble rate begins to decrease because of the presence of string corrections and it eventually diverges where D = a + a + a + a + a − a − a − a . c 2 4 6 8 1 3 5 toward −∞ after crossing H = 0. Thus the fate of the Since D<0 for both type II and heterotic strings, we universe is characterized by a Big Crunch rather than do not have de Sitter fixed points. a Big Rip. Meanwhile in the heterotic case the Hubble When a cosmological constant Λ is present in- rate continues to increase and diverges after a finite instead of ρ , corresponding to the equation of state m terval of time, see Fig. 2. Thus the Big Rip singularity w =−1, we find from Eq. (16) that there exists one m is inevitable even when the heterotic string correction de Sitter solution which satisfies Λ = 6H 2 − BDH8. is present. One can study the stability of this solution by consid- We also studied cosmological evolution for several ering small perturbations δx and δy about the fixed different values of w and for different initial condi- point. We evaluate two eigenvalues for the matrix of m tions of the Hubble rate. For the type II correction we perturbations using the method in Ref. [16].Forthe find that the solutions approach the Big Crunch singu- type II correction we find that one eigenvalue is posi- larity for w −1.2, whereas they tend to approach tive while another is negative, thereby indicating that m the Big Rip singularity for −1.2 w < −1. Thus the de Sitter solution is not stable. Meanwhile in the m the fate of the universe depends upon the equation of heterotic case the de Sitter solution is either a stable state for phantom dark energy. For the heterotic case spiral (for smaller values of Λ,seeFig. 1)orastable node (for larger values of Λ). We note, however, that

Fig. 2. The evolution of the Hubble rate H in the presence of string Fig. 1. The phase portrait for heterotic string in the case of curvature corrections and a phantom fluid with an equation of state: ρm ≡ Λ = 1 for several different initial conditions. The stable fixed wm =−1.5. For the type II correction, the solution approaches point xc ≡ Hc = 0.408 corresponds to a de Sitter solution which is H =−∞by crossing H = 0, whereas in the heterotic and bosonic a stable spiral. cases the Hubble rate grows toward infinity. M. Sami et al. / Physics Letters B 619 (2005) 193–200 197 we find that the solutions reach to the Big Rip sin- the case of heterotic string. We also run our numerical gularity independent of the values of wm(< −1) and code for different values of wm and find that the solu- initial conditions of H . tions approach the Big Rip singularity for wm < −1. When a cosmological constant Λ is present instead 3.2. Bosonic string of ρm, de Sitter solutions satisfy 2 6 8 As demonstrated above, the type II and the het- Λ = 6H − AH − B 76 − 12ζ(3) H . (26) erotic string models do not exhibit de Sitter solutions There exist two solutions for this equation provided for w =−1. However, in the case of bosonic string, m that Λ ranges in the region 0 <Λ −1. Therefore m a stable spiral. As for the second critical point corre- the de Sitter solution characterized by Eq. (25) does sponding to H = 0.652, one of the eigenvalues turns not correspond to a stable attractor for w =−1. m out to be positive, which means that the de Sitter solu- In Fig. 2 we plot the evolution of the Hubble rate tion is unstable—a saddle in this case (see Fig. 3). with bosonic string corrections in the presence of a phantom-type fluid with wm =−1.5. The Hubble rate continues to grow and diverges with a finite time as in 4. Summary

In this Letter we studied the effect of higher- curvature corrections in low-energy effective string theory on the cosmological dynamics in the presence of dark energy fluid. Since the existence of a phantom fluid leads to the growth of the Hubble rate, the energy scale of the universe may reach the Planck scale in future. This means that string curvature corrections can be very important to determine the dynamical evolution of the universe. We have considered string corrections up to quartic in curvatures for three type of string theories—(i) type II, (ii) heterotic, and (iii) bosonic strings. In our analysis the contribution of the Gauss–Bonnet term does not affect the cosmological dynamics in D = 4 dimensions, since the dilaton and the modulus fields are fixed. For the fluid with an equation of state characterized by wm =−1, we find that de Sitter solutions Fig. 3. The phase portrait for bosonic string in the presence of a do not exist for type II and heterotic string corrections. cosmological constant (ρm ≡ Λ = 1) with A = 1/2andB = 3/4. There exist two de Sitter fixed points. The point A [x = 0.417] There is a de Sitter solution for the bosonic string even c =− corresponds to a stable spiral, whereas the point B [xc = 0.652] is a for wm 1, but this is found to be unstable. When saddle. the equation of state for fluid is that of a cosmologi- 198 M. Sami et al. / Physics Letters B 619 (2005) 193–200 cal constant (wm =−1), we find that stable de Sitter superstring whereas it is a Big Rip for bosonic and solutions exist for heterotic and bosonic strings. heterotic strings. This distinction mainly comes from We ran our numerical code to study the effect of the difference of the coefficients in Eqs. (19) and (20). string corrections around the Big Rip when a phantom In the type II case ρc becomes negative, which coun- fluid is present. In the type II case we found that the teracts the energy density of the phantom fluid. This solutions approach the Big Crunch singularity (H → property is crucially important to avoid the Big Rip −∞) after crossing H = 0 when the equation of state singularity as pointed out in Ref. [8]. We note that for dark energy is wm −1.2. Meanwhile the Hub- this behavior also appears in the presence of a dynami- ble rate diverges toward H →+∞with a finite time cal modulus field with second-order string corrections for heterotic and bosonic corrections, which implies [17]. It is really of interest to investigate how the final that the Big Rip singularity is difficult to be avoided in fate of the universe is changed when dynamical mod- these cases. The divergent behavior of the Hubble rate ulus/dilaton fields couple to third/fourth-order string for wm < −1 is associated with the fact that there are curvature corrections. We hope to address this issue in neither stable de Sitter nor stable Minkowski attractors future work. for the types of the corrections we considered. In addition, the higher-order curvature contribu- In the present Letter we restricted our attention tions used in our description have inbuilt ambiguities to purely geometrical effects assuming that non- related to particular metric redefinitions. It would be perturbative potentials may arise allowing to freeze important to investigate whether or not these ambigu- dilaton and modulus fields. Compactifications from ities can lead to different fate of cosmological evolu- higher dimensions to 4-dimensional spacetime result tion. in residual modulus fields which are related to the radii of internal space. In general a modulus field is dynamical and interacts with higher-order curvature Acknowledgements terms. The same is true for the dilaton field which is related to string coupling gs . These may give rise to We thank O. Bertolami, G. Calcagni, F. Fattoev, non-trivial effects, for instance, even the GB curva- T. Naskar, S. Nojiri and T. Padmanabhan for use- ture invariant which is purely topological in 4 dimen- ful discussions. A.T. acknowledges support from IU- sions, does not vanish in the presence of dynamical CAA’s “Program for enhanced interaction with the modulus and dilaton fields. Such a scenario would Africa–Asia–Pacific Region”. The work of S.T. was have important implications for future evolution of supported by JSPS (No. 30318802). the dark energy universe. Very recently, cosmological dynamics based upon effective string theory action was investigated with dynamically evolving modulus Appendix A. Calculation of curvature tensors and dilaton fields in the presence of second-order curvature corrections [17] (see Ref. [18] on the related For the metric (11) the non-zero components of theme). It was demonstrated that the second-order Christoffel symbols are curvature correction to Einstein–Hilbert action can ˙ ˙ ˙ 0 aa µ µ a 0 N significantly modify the structure of future singular- Γ = ,Γ= Γ = ,Γ= , µµ N 2 0µ µ0 a 00 N ities in dark energy universe. It is therefore important, µ = , ,...,d though technically cumbersome, to extend the analysis where 1 2 .Byusingtheformula of the present Letter to the case of dynamical modu- L = L − L + L R − L R RSMN ∂M ΓSN ∂N ΓSM ΓMRΓNS ΓNRΓMS, lus/dilaton fields. A comment is also in order about the distinct fea- we find that the non-zero components of Riemann ten- tures that different string models exhibit in cosmolog- sors are ical dynamics. With fixed modulus/dilaton, the evolu- aa¨ aa˙N˙ R0 =−R0 = − , tion of phantom dark energy universe is clearly dis- µ0µ µµ0 N 2 N 3 tinguished depending upon string models, namely, the ˙ µ µ a¨ a˙ N fate of such a universe is Big Crunch for the type II R =−R = − , 00µ 0µ0 a a N M. Sami et al. / Physics Letters B 619 (2005) 193–200 199

a˙2 Rµ =−Rµ = . J. Weller, A.M. Lewis, Mon. Not. R. Astron. Soc. 346 (2003) νµν ννµ N 2 987; These give U. Alam, V. Sahni, T.D. Saini, A.A. Starobinsky, Mon. Not. R. Astron. Soc. 354 (2004) 275; 1 a¨ a˙ N˙ P.S. Corasaniti, M. Kunz, D. Parkinson, E.J. Copeland, B.A. µ0 = 0µ =− µ0 =− 0µ = − Bassett, Phys. Rev. D 70 (2004) 083006. R0µ Rµ0 Rµ0 R0µ , N 2 a a N [2] R.R. Caldwell, Phys. Lett. B 545 (2002) 23. 1 a˙2 [3] A.A. Starobinsky, Grav. Cosmol. 6 (2000) 157. Rµν =−Rµν = . [4] R.R. Caldwell, M. Kamionkowski, N.N. Weinberg, Phys. Rev. νµ µν 2 2 N a Lett. 91 (2003) 071301. Noting the relation R = RL , we obtain [5] F. Hoyle, Mon. Not. R. Astron. Soc. 108 (1948) 372; MN MLN F. Hoyle, Mon. Not. R. Astron. Soc. 109 (1949) 365; a¨ a˙ N˙ F. Hoyle, J.V. Narlikar, Proc. R. Soc. A 282 (1964) 191; R00 =−d − , F. Hoyle, J.V. Narlikar, Mon. Not. R. Astron. Soc. 155 (1972) a a N 305; aa¨ aa˙N˙ a˙2 F. Hoyle, J.V. Narlikar, Mon. Not. R. Astron. Soc. 155 (1972) Rµµ = − + (d − 1) . 323; N 2 N 3 N 2 J.V. Narlikar, T. Padmanabhan, Phys. Rev. D 32 (1985) 1928. We also ﬁnd [6] S.M. Carroll, M. Hoffman, M. Trodden, Phys. Rev. D 68 (2003) 023509; N˙ S. Nojiri, S.D. Odintsov, Phys. Lett. B 562 (2003) 147; R = R =−R =−R = aa¨ − aa˙ , 0µµ0 µ00µ 0µ0µ µ0µ0 N S. Nojiri, S.D. Odintsov, Phys. Lett. B 565 (2003) 1; E. Elizalde, S. Nojiri, S.D. Odintsov, Phys. Rev. D 70 (2004) a2a˙2 R =−R = , 043539; µνµν µννµ N 2 P. Singh, M. Sami, N. Dadhich, Phys. Rev. D 68 (2003) R0µµ0 = Rµ00µ =−R0µ0µ =−Rµ0µ0 023522; M. Sami, A. Toporensky, Mod. Phys. Lett. A 19 (2004) 1509; 1 N˙ D.F. Torres, Phys. Rev. D 66 (2002) 043522; = a¨ −˙a , 4 3 V. Sahni, Y. Shtanov, JCAP 0311 (2003) 014; N a N P.F. Gonzalez-Diaz, Phys. Rev. D 68 (2003) 021303; a˙2 P.F. Gonzalez-Diaz, Phys. Lett. B 586 (2004) 1; Rµνµν =−Rµννµ = . N 2a6 P.F. Gonzalez-Diaz, Phys. Rev. D 69 (2004) 063522; P.F. Gonzalez-Diaz, C.L. Siguenza, Nucl. Phys. B 697 (2004) These relations are used to evaluate the correction term 363; Lc. B. Feng, X.L. Wang, X.M. Zhang, Phys. Lett. B 607 (2005) 35; We should note that the corrections can be eas- S. Tsujikawa, Class. Quantum. Grav. 20 (2003) 1991; = J.G. Hao, X.Z. Li, Phys. Rev. D 70 (2004) 043529; ily computed in case of de Sitter symmetry [Rabcd J.G. Hao, X.Z. Li, Phys. Lett. B 606 (2005) 7; Λ(gacgbd − gadgbc)], as M.P. Dabrowski, T. Stachowiak, M. Szydlowski, hep-th/ 0307128; 3 L3 = 4Λ d(d + 1), (A.1) L.P. Chimento, R. Lazkoz, Phys. Rev. Lett. 91 (2003) 211301, L =−3ζ(3)Λ4(d − 1)d(d + 1), (A.2) astro-ph/0405518; 41 V.K. Onemli, R.P. Woodard, Class. Quantum. Grav. 19 (2002) 1 4 2 4607, gr-qc/0406098; L42 = Λ (d − 7)d (d + 1), (A.3) 2 F. Piazza, S. Tsujikawa, JCAP 0407 (2004) 004; L =−4Λ4d(d + 1) 1 − d(d − 9) . (A.4) A. Feinstein, S. Jhingan, Mod. Phys. Lett. A 19 (2004) 457; 43 H. Stefancic, Phys. Lett. B 586 (2004) 5, astro-ph/0312484; These coincide with Eqs. (12)–(15) by setting N = 1 X. Meng, P. Wang, hep-ph/0311070; and H 2 = Λ. H.Q. Lu, hep-th/0312082; V.B. Johri, Phys. Rev. D 70 (2004) 041303, astro-ph/0409161; I. Brevik, S. Nojiri, S.D. Odintsov, L. Vanzo, hep-th/0401073; J. Lima, J.S. Alcaniz, astro-ph/0402265; References Z. Guo, Y. Piao, Y. Zhang, astro-ph/0404225; M. Bouhmadi-Lopez, J. Jimenez Madrid, astro-ph/0404540; [1] S. Hannestad, E. Mortsell, Phys. Rev. D 66 (2002) 063508; J. Aguirregabiria, L.P. Chimento, R. Lazkoz, astro-ph/ 0403157; A. Melchiorri, L. Mersini, C.J. Odman, M. Trodden, Phys. Rev. D 68 (2003) 043509; E. Babichev, V. Dokuchaev, Yu. Eroshenko, astro-ph/0407190; 200 M. Sami et al. / Physics Letters B 619 (2005) 193–200

Accelerating cosmologies from compactiﬁcation with a twist

Ishwaree P. Neupane a,b, David L. Wiltshire b

a Department of Physics and Astronomy, University of Canterbury, Private Bag 4800, Christchurch, New Zealand b Central Department of Physics, Tribhuvan University, Kirtipur, Kathmandu, Nepal Received 12 April 2005; accepted 2 June 2005 Available online 13 June 2005 Editor: M. Cveticˇ

Abstract It is demonstrated by explicit solutions of the (4 + n)-dimensional vacuum Einstein equations that accelerating cosmologies in the Einstein conformal frame can be obtained by a time-dependent compactiﬁcation of string/M-theory, even in the case that internal dimensions are Ricci-ﬂat, provided one includes one or more geometric twists. Such acceleration is transient. When both compact hyperbolic internal spaces and geometric twists are included, however, the period of accelerated expansion may be made arbitrarily large.  2005 Elsevier B.V. All rights reserved.

PACS: 98.80.Cq; 11.25.Mj; 11.25.Yb

The observation that the present expansion of the universe is accelerating [1] has proved a challenge to fundamental theories such as string/M-theory, and its low-energy supergravity limits. Fundamental scalar fields are abundant in such higher-dimensional gravity theories, and potentially provide a natural source for the gravitationally repulsive “dark energy” that could explain the present cosmic acceleration, or alternatively at much higher energy scales the very early period of cosmic inflation. However, the constraints imposed by string/M-theory on particular scalar fields, such as the moduli corresponding to 6 or 7 extra compactified dimensions, are such that realistic scenarios giving accelerating cosmologies are very difficult to arrive at. Indeed for many years, it was assumed that cosmic acceleration was ruled out for supergravity compactifications on the basis of a “no-go” theorem [2], which forbids accelerating cosmologies in the presence of static extra dimensions in supergravity compactifications, assuming that one wishes to stay within the realm of classical supergravity rather than resorting to the addition of “quantum correction” terms to the action. Recently Townsend and Wohlfarth [3] demonstrated, however, that it was possible to circumvent the no-go theorem in a time-dependent

E-mail addresses: [email protected] (I.P. Neupane), [email protected] (D.L. Wiltshire).

0370-2693/$ – see front matter  2005 Elsevier B.V. All rights reserved. doi:10.1016/j.physletb.2005.06.008 202 I.P. Neupane, D.L. Wiltshire / Physics Letters B 619 (2005) 201–207 supergravity compactification using compact extra dimensions with negative curvature. Many additional examples [4–10] were subsequently found. The advent of time-dependent compactifications is an important one, as it has the potential to offer a resolution to the dilemma posed by the observed cosmic acceleration within a natural theoretical framework. Nonetheless, a number of substantial problems remain in the models studied to date. In this Letter we will present new solutions arising from compactifications with a geometrical twist, which we believe overcome these problems to give time- dependent supergravity compactifications with compelling physical features. Let us turn to the open issues in time-dependent cosmological compactifications. Firstly, in the Townsend– Wohlfarth (henceforth TW) model [3] the no-go theorem was only circumvented by the choice of negatively curved internal spaces, which are made compact by periodic identification. While such spaces have particular features, such as the absence of massless Kaluza–Klein vectors which may have some phenomenological appeal, the restriction on the curvature is a severe one, especially given the view that Ricci-flat internal space compactifications are among the most natural in string theory. The second important issue is the question of the length of the period of accelerated expansion. The TW model and its successors incorporate scalar fields whose potential energy does not admit a local minimum with positive vacuum energy. A consequence of this is that the period of acceleration is transient, corresponding to just a few e-foldings. Again if one is to apply the model to describe the present cosmic evolution, then this is not a major difficulty but does place some constraints on the parameters involved. However, if rather than a model of the present accelerated cosmological evolution a model of very early universe inflation via time-dependent compactifications was desired, then this would be a significant problem. In this Letter we will present model cosmologies from time-dependent compactifications which overcome both of the problems above, while also satisfying a third property of theoretical naturalness. The limitations with the models studied to date arise, we believe, from an oversimplification of the effective potentials that arise from compactifications. In particular, when a number of scalars associated with different dimensions are present, their interactions may give rise to effects which are absent if the extra dimensions form a single space of constant curvature, as in the TW model. Product space compactifications have already been considered, sometimes also in the presence of higher-dimensional form-fields with non-zero fluxes [8]. However, while higher-dimensional fluxes go a little way towards altering the form of the compactification potential, for example, by allowing the presence of a local de Sitter minimum [8], in the examples studied to date there is not enough parameter freedom to allow significant periods of accelerated expansion. Here we will explicitly consider cosmologies that arise in models of gravity which correspond to the dimensional reduction to 4 dimensions of the Einstein equations in 10, 11 or generally 4 + n dimensions, where some of the extra dimensions form product spaces with a geometrical twist. In particular, consider a (4+n)-dimensional metric ansatz with n = p + 3 internal dimensions, − ds2 = e 2Φ ds2(M ) + r2e2φ1 ds2 Mp + r2e2φ2 ds2 M2 + r2e2φ3 (dz + )2, (1) 4+n 4 1 1 2 2 3 2 2 where φi = φi(u), the parameters ri define appropriate curvature radii, ds (M4) is the metric of the physical large dimensions in the form 2 =− 2δ 2 + 2 2 ds (M4) a (u) du a (u) dΩk,3, (2) and δ is a constant, the choice of which fixes the nature of the time coordinate, u. The internal space Mp is a p-dimensional space of constant curvature of sign 1 = 0, ±1, and the remaining 3 internal dimensions form a twisted product space, M2 M1, as follows 2 M2 = 2 + 2 2 = ds +1 dx sin x dy ,+1 f cos x dy, f ds2 M2 = dx2 + dy2,= (x dy − y dx), 0 0 2 2 M2 = 2 + 2 2 = (3) ds −1 dx sinh x dy ,−1 f cosh x dy, I.P. Neupane, D.L. Wiltshire / Physics Letters B 619 (2005) 201–207 203 when 2 =+1, 0, −1 respectively, f being the twist parameter. One chooses the Einstein conformal frame in four dimensions by setting Φ = pφ1/2 + φ2 + φ3/2, and so Newton’s constant is time-independent. To demonstrate the novel features introduced by a geometric twist, we will begin by presenting a special exact solution in the case that the physical universe is spatially flat, k = 0, and both Mp and M2 are Ricci-flat, i.e., 1 = 0 and 2 = 0, but with non-zero twist, f . This solution to the (p + 7)-dimensional vacuum Einstein equations is most readily written down in the gauge, δ = 3, used in Ref. [3]. In particular, we find [(3q+4)c+pc1]qu/8 q/4 a(u) = a0e cosh χ(u− u0) , 3 −φ 1 1 2 fr3a φ = 3 = χ(u− u ) + (pc + qc)u, φ = 0 − c u, 2 ln cosh 0 1 3 1 ln 2 1 (4) q 2 4 p χr2 where 1 χ ≡ 6pqcc + 3q(3q + 4)c2 − p (4 − q)p + 8 c2/q (5) 2 1 1 and u0, c and c1 are integration constants. We have ignored constants which can be absorbed into the ri .The parameter q denotes the number of twists, and for the metric (1)–(3) we take q√= 1in(4), (5). For reality of χ we require α1 −c0 which correspond to universes which expand as u increases. The value of u0 is merely a gauge choice and so we set u0 = 0. In terms of the physical cosmic time (i.e., proper time of co-moving observers), t =± u dua¯ 3(u)¯ , both the early and late time behaviour of the cosmic scale factor is a ∼ t1/3. This is similar to the TW solution with two minor differences. Firstly, here the t →∞limit corresponds to u →∞, whereas in the TW solution this limit corresponds to u → 0−. Secondly, the TW solution decelerates slightly less quickly at late times, with a ∼ tn/(n+2). Otherwise, the solution is physically very similar to the TW solution. In particular, since the acceleration parameter (for q = 1) is given by 1 3 1 a5a¨ = χ2 1 − tanh2 χu − (7c + pc )(4χ tanh χu+ 7c + pc ), (6) 4 2 32 1 1 where an overdot denotes differentiation w.r.t. cosmic time, t, it follows that√ solutions will exhibit a period of transient acceleration provided that α2 − c

The number of e-folds during the period of acceleration, Ne ≡ ln a(u+) − ln a(u−) is plotted in Fig. 1. It reaches a maximum Ne = 0.3041 at c1 = 0 independently of p, which corresponds to the special case in which the internal p space M is static with constant φ1.(Thep = 0 solution is formally equivalent to (4), (5) with c1 = 0 and no φ1.) In the general case φ2 slowly increases—giving a slow “decompactiﬁcation”—while φ3 decreases, and φ1 decreases (increases) if c1u>0(c1u<0). The number of e-folds in the examples is possibly too low to be consistent with observational bounds, especially when one notes that any additional matter obeying the strong energy condition would have a decelerating effect that may decrease the period of transient acceleration. However, the number of e-folds can be increased by increasing the dimension of the twisted space. In particular, consider the case n = p + 2q + 1 when the internal space is now p the product of a p-dimensional torus M (i.e., 1 = 0), and the (2q + 1)-dimensional twisted space q q 2 f ds2 = r2e2φ2 dx2 + dy2 + r2e2φ3 dz + (x dy − y dx ) , (7) T 1,...,1 2 i i 3 2 i i i i i=1 i=1 204 I.P. Neupane, D.L. Wiltshire / Physics Letters B 619 (2005) 201–207

= q [ 1+tanh χu+ κ−1 1−tanh χu− κ+1] ≡[ + + ] Fig. 1. The number of e-folds during acceleration epoch Ne 8 ln ( 1+tanh χu− ) ( 1−tanh χu+ ) where κ (3q 4)c pc1 /(2χ),as a function of the parameter α ≡ c1/c,forq = 1, p = 4, d = 11. which has topology (T 2 × ··· × T 2) S1.Eqs.(4), (5) with q arbitrary in fact represent the solution to the (4 + p + 2q + 1)-dimensional vacuum Einstein equations with this more general metric ansatz, where now Φ = pφ1/2 + qφ2 + φ3/2. For small values q>1 the qualitative features are the same as the q = 1 case, with transient acceleration and a maximum number of e-folds at c1 = 0, independently of p. For example, the maximum number of e-folds for q = 2, 3, 4 are respectively Ne = 0.3689, 0.4019, 0.4221, a marginal increase. Next we turn our attention to solutions with a period of acceleration which is not merely transient. In addition to a twist, we will take the curvature of the internal space to be non-zero. To this end, it is convenient to set δ = 0, so that the metric (2) takes the standard Friedmann–Robertson–Walker form, and u becomes the cosmic time, t. For the metric (1), with arbitrary i , the ﬁeld equations may be written in the form a¨ a˙ 2 a¨ 2k Φ¨ + 3H Φ˙ − 2K − 3 = 0, Φ¨ + 3HΦ˙ − 2 − − = 0, a a a a2 ¨ ˙ φi + 3Hφi − δ1Vi − δ2VF = 0,i= 1, 2, 3, (8) where δ1 = 2/p, δ2 = 0 (for i = 1), δ1 = 1, δ2 = 2 (for i = 2), δ1 = 0, δ2 =−2 (for i = 3), H ≡˙a/a is the Hubble parameter, k is the spatial curvature, and the kinetic and potential terms are respectively

p(p + 2) 3 p K = φ˙2 + 2φ˙2 + φ˙2 + pφ˙ φ˙ + φ˙ φ˙ + φ˙ φ˙ , 4 1 2 4 3 1 2 2 3 2 3 1 −(p+2)φ1−2φ2−φ3 −pφ1−4φ2−φ3 2 −pφ1−6φ2+φ3 V = V1 + V2 + VF = Λ1e + Λ2e + F e , (9) =− − 2 =− 2 ≡ 2 = where Λ1 p(p 1)1/(2r1 ), Λ2 2/r2 and F (f/2)(r3/r2 ). Even with i 0, the volume modulus field has a non-zero (positive) potential, and so the cosmological solutions with f>0 circumvent the no-go theorem [2], while retaining Ricci-flat internal spaces. Many additional examples are given in [11]. p+5 In terms of alternative canonically normalized scalars, which may be defined by ϕ1 ≡− + φ3, ϕ2 ≡ √ p 4 p(p+4) + 1 ≡ p + + 1 2 (φ1 p+4 φ3), and ϕ3 2 φ1 2φ2 2 φ3, the field equations are

dV ϕ¨i + 3Hϕ˙i + = 0, (10) dϕi − H˙ + K − ka 2 = 0, (11) I.P. Neupane, D.L. Wiltshire / Physics Letters B 619 (2005) 201–207 205 along with the Friedmann (constraint) equation

1 − H 2 = (K + V)− ka 2, (12) 3 = 1 3 ˙2 where K 2 i ϕi . The resulting scalar potential is √ 2 −ϕ2/β−ϕ3 −2ϕ3 2 − 5−β ϕ1+βϕ2−3ϕ3 V = Λ1e + Λ2e + F e , (13) √ √ where β ≡ p/ p + 4 < 1. As above we will confine our attention to k = 0 cosmologies. Solutions with no twist, 2 F = 0, and compact hyperbolic product spaces, i.e., Λi > 0, have been previously discussed [8], and give transient acceleration without a local minimum in the potential. However, if there is a non-trivial twist and at least one of Λi is positive, then it may be possible to find models with a large number of e-foldings. Indeed, if we choose Λi such that the potential (13) is strictly non-negative then it always has a minimum with respect to a subset of the ϕi directions. Even with k = 0 the general solution to (10)–(13) is highly non-trivial. To demonstrate the general physical 2 effects, we will take Λ2 = 0, so that M is a 2-torus, and specialize to the case in which d = 11 (or p = 4) and ϕ3 = b1= const. An explicit√ exact solution√ can then be found in terms of a new logarithmic time variable τ , defined t ¯ ¯ by τ = dt exp[−ϕ2(t)/ 2],orϕ2 = 2ln(dt/dτ). The explicit solution is then 1 − 1 − (ln a) = √ V ξ + ξ 1 ,ϕ= V ξ − ξ 1 , 0 2 2 0 √12 √ 2b0 6 − 1 5V0 ϕ1 = ϕ2 + ,ξ≡ √ tanh (τ − τ0), (14) 3 5 8 where ≡ d/dτ , τ0 is a constant,

−b1 2 −(b0+3b1) 2 −(b0+3b1) V0 ≡ Λ1e + F e = 3F e (15) 2 and b0 is fixed once b1 is chosen. At late times we find a ∝ t , with the corresponding value of ω<−2/3, where ω ≡ (K − V)/(K + V). In the limit ϕ1 → ϕ2 we find φ1 →−φ2/2 and φ3 →−2φ2. Thus two of the extra 2 dimensions associated with the space M may grow with time while other√ dimensions shrink√ (or vice versa). (0) = + 2 + The potential V has a minimum with respect to ϕ2 at ϕ2 ϕ1 ( 2/3) ln(2Λ1/F ) 2 2b0/3, with 1/3 √ 27 − − V(ϕ ) = Λ F 4e 7b0 e 2ϕ1 = V, (ϕ ). (16) 0 4 1 ϕ2ϕ2 0 = = Since the minimum has the curious feature, special to potentials with Λ2 0, that V,ϕ2ϕ2 (ϕ0) V(ϕ0), particle- 2 like states will have a mass m = V(ϕ0). If this value were to be assigned to the vacuum energy at the present −120 epoch, i.e., V(ϕ0) ∼ 10 in Planck units, then both the vacuum energy and scalar excitations about the vacuum are ultra-light. The degree to which the model would need to be fine-tuned to achieve such an outcome remains to be investigated. Another alternative would be to take V(ϕ0) to have the vastly higher energy scale associated with the epoch of = (0) inflation in the very early universe. Let us assume that ϕ1 starts initially at ϕ1 ϕ1 , and that Λ1 and F both are ∼ (0) + ∼ of order unity (in Planck units), then when ϕ1 ϕ1 47 the number of e-folds is Ne 65. The actual relation between√ Ne and the shift in ϕ1 can be different, however, depending upon the precise values of the compactification scale, Λ1, and the twist parameter, F . A point also worth emphasizing is that, especially in a flat universe, the de Sitter stage can be transient. If this is the case in our model, we must allow also ϕ3 to roll with t. Different choices of vacua could lead to different asymptotic expansion. More specifically, for the potential (13), when k = 0 and p = 4 the late time behaviour of 206 I.P. Neupane, D.L. Wiltshire / Physics Letters B 619 (2005) 201–207 the scale factor is characterized by a(t) ∝ tγ , where   = = 2 =  13/19 (Λ2 0,Λ1 0,F 0), = = 2 = = 3/5 (Λ1 0,Λ2 0,F 0), γ  2 = = (17)  3/4 (F 0,Λi 0), 2 7/9 (Λi = 0,F = 0). It is expected the contribution of dust or radiation could modify the above asymptotic behaviour. Let us summarize why we believe that exponential potentials arising from compactifications of supergravity models on a twisted product space of time-varying volume are a potentially significant source for dynamical dark energy at the present epoch. Firstly, Townsend and Wohlfarth [3] argued that the no-go theorem for accelerating cosmologies in supergravity compactifications could only be circumvented by including compact hyperbolic internal spaces, and would not work in the Ricci-flat cases. We have shown on the contrary that the no-go theorem is also circumvented by twisted Ricci-flat spaces by explicit construction of exact solutions with the internal space T p × (T 2 ×···×T 2) S1. Since Ricci-flat spaces are natural in the string/M-theory context, this is an important development. Secondly, we have demonstrated that it is possible to construct a metastable de Sitter vacuum in the general framework of [8] by incorporating one or more geometric twists in the internal space. As mentioned above, some of the extra dimensions may eventually grow large, though slowly, without bound and decompactify, a feature that our transient solutions share with those of the TW model. One might argue that provided that the relative size of the extra dimensions and their rate of change remain sufficiently small as to be consistent with observation over any epoch of physical relevance, then the eschatological consequences of such “decompactifications” are no more severe than those of a universe whose ordinary three dimensions accelerate forever. Nonetheless, given that the natural endpoint of the solutions is a fundamentally higher-dimensional run- away epoch [12], it is clear that we would have additional cosmic coincidence constraints in explaining why the universe appears to have three large spatial dimensions at the present epoch. However, before arriving at any such conclusions it is important to include additional matter, such as pressureless dust. Matter obeying the strong energy condition will certainly affect the length of any transient acceleration, and may also change the evolution of the scalar fields. Finally, we remark that geometric twists add richness to string/M-theory cosmology, and may potentially lead to the realization of new cosmological scenarios.

Acknowledgements

This work was supported in part by the Mardsen Fund of the Royal Society of New Zealand. I.P.N. thanks CERN-TH for hospitality while part of this work was done.

References

[1] C.L. Bennett, et al., Astrophys. J. Suppl. 148 (2003) 1. [2] G.W. Gibbons, in: F. del Aguila, J.A. de Azcarraga, L.E. Ibanez (Eds.), Supersymmetry, Supergravity and Related Topics, World Scientiﬁc, Singapore, 1985, p. 123; J.M. Maldacena, C. Nunez, Int. J. Mod. Phys. A 16 (2001) 822. [3] P.K. Townsend, M.N.R. Wohlfarth, Phys. Rev. Lett. 91 (2003) 061302. [4] C.-M. Chen, P.-M. Ho, I.P. Neupane, J.E. Wang, JHEP 0307 (2003) 017. [5] N. Ohta, Phys. Rev. Lett. 91 (2003) 061303. [6] M.N.R. Wohlfarth, Phys. Lett. B 563 (2003) 1; S. Roy, Phys. Lett. B 567 (2003) 322. [7] R. Emparan, J. Garriga, JHEP 0305 (2003) 028. [8] C.-M. Chen, P.-M. Ho, I.P. Neupane, N. Ohta, J.E. Wang, JHEP 0310 (2003) 058. I.P. Neupane, D.L. Wiltshire / Physics Letters B 619 (2005) 201–207 207

[9] I.P. Neupane, Class. Quantum Grav. 21 (2004) 4383; I.P. Neupane, Mod. Phys. Lett. A 19 (2004) 1093; L. Jarv, T. Mohaupt, F. Saueressig, JCAP 0408 (2004) 016. [10] E. Bergshoeff, A. Collinucci, U. Gran, M. Nielsen, D. Roest, Class. Quantum Grav. 21 (2004) 1947. [11] I.P. Neupane, D.L. Wiltshire, hep-th/0504135. [12] S.B. Giddings, R.C. Myers, Phys. Rev. D 70 (2004) 046005. Physics Letters B 619 (2005) 208–218 www.elsevier.com/locate/physletb

Noncompact KK theory of gravity: Stochastic treatment for a nonperturbative inﬂaton ﬁeld in a de Sitter expansion

José Edgar Madriz Aguilar a, Mauricio Bellini b

a Instituto de Física y Matemáticas, AP: 2-82, (58040) Universidad Michoacana de San Nicolás de Hidalgo, Morelia, Michoacán, Mexico b Departamento de Física, Facultad de Ciencias Exactas y Naturales, Universidad Nacional de Mar del Plata and Consejo Nacional de Ciencia y Tecnología (CONICET), Funes 3350, (7600) Mar del Plata, Argentina Received 11 March 2005; received in revised form 2 April 2005; accepted 23 May 2005 Available online 31 May 2005 Editor: N. Glover

Abstract We study a stochastic formalism for a nonperturbative treatment of the inflaton field in the framework of a noncompact Kaluza–Klein (KK) theory during an inflationary (de Sitter) expansion, without the slow-roll approximation.  2005 Elsevier B.V. All rights reserved.

PACS: 04.20.Jb; 11.10.Kk; 98.80.Cq

1. Introduction

Stochastic inflation model is one of the very few that solves almost all of the well-known cosmological problems. Since the differential microwave radiometer (DMR) mounted on the Cosmic Background Explorer satellite (COBE) first detected temperature anisotropies in the cosmic background radiation (CBR), we have the possibility δρ ≈ −5 to directly probe the initial density perturbation. The fact that the resulting energy density fluctuations ( ρ 10 ) fit the scaling spectrum predicted by the inflation model, suggests that they had indeed their origin in the quantum fluctuations of the “inflaton” scalar field during the inflationary era. Although in principle this problem is of a quantal nature, the fact that under certain conditions—which are made precise in [1–3]—the inflaton field can be considered as classical largely simplifies the approach, by allowing a Langevin-like stochastic treatment. The most widely accepted approach assumes that the inflationary phase is driving by a quantum scalar field ϕ with a potential V(ϕ). Within this perspective, the stochastic inflation proposes to describe the dynamics of this quantum field on the basis of a splitting of ϕ in a homogeneous and an inhomogeneous components. Usually the homogeneous

E-mail addresses: [email protected] (J.E. Madriz Aguilar), [email protected] (M. Bellini).

0370-2693/$ – see front matter  2005 Elsevier B.V. All rights reserved. doi:10.1016/j.physletb.2005.05.054 J.E. Madriz Aguilar, M. Bellini / Physics Letters B 619 (2005) 208–218 209 one φc(t) is interpreted as a classical field that arises from a coarsed-grained average over a volume larger than the observable universe, and plays the role of a global order parameter [4]. All information on scales smaller than this volume, such as the density fluctuations, is contained in the inhomogeneous component. Although this theory is ˙2 φc + widely used and accepted as general, one needs to make the approximation ρ 2 V(φc) to be able to make some calculations in a linear expansion for the scalar potential V(ϕ)around its classical background field φc(t) [3]. | It was Starobinsky the first one to derive a Fokker–Planck equation for the transition probabilities P(φL,t φL,t ) | in comoving coordinates [5] from the stochastic equation for the dynamics of the inflaton field. P(φL,t φL,t ) provides us with statistical information about the relative number of “domains” (metastable vacuum configurations) − that having a typical value φL of the coarse-grained inflaton field, evolve in a time interval (t t ) towards a new configuration with a typical value φL. The main aim of this work consists to make a consistent coarse-granning treatment for the scalar field dynamics on cosmological scales during the inflationary epoch. For simplicity, as an example, we shall study a de Sitter expansion for the universe, but the formalism here developed can be used to study other more realistic inflationary models. The strategy consists to starts from a 5D globally flat metric and an action for a purely kinetic quantum scalar field minimally coupled to gravity, which define a 5D vacuum state. The metric we consider can be mapped to a 5D generalized Friedmann–Robertson–Walker (FRW) on which we make a foliation by considering the fifth (spatial-like) coordinate as a constant. As a result of this foliation we obtain an effective 4D FRW metric and an effective 4D density Lagrangian in which appears an effective term which depends on the fifth coordinate and is not kinetic in 4D. This term is identified as a 4D scalar potential or source, and has an origin purely geometric. The idea that matter in four dimensions (4D) can be explained from a 5D Ricci-flat (RAB = 0) Riemannian manifold is a consequence of the Campbell’s theorem. It says that any analytic N-dimensional Riemannian manifold can be locally embedded in a (N + 1)-dimensional Ricci-flat manifold. This is of great importance for establishing the generality of the proposal that 4D field equations with sources can be locally embedded in 5D field equations without sources [6,7]. The advantage of this propose is that provides an exact (nonperturbative) treatment for the 4D dynamics of the inflaton field (the scalar field) with back-reaction effects included [8]. Furthermore, it is possible to make a consistent treatment for the effective 4D dynamics of the universe in other models governed by a single scalar field. In this Letter we consider a general version of the Kaluza—Klein theory in 5D, where the extra dimension is not assumed to be compactified. In other words, this means that the cylinder condition in the fifth coordinate of the original Kaluza—Klein theory is relaxed. From the mathematical viewpoint, this means that the 5D metric tensor is allowed to depend explicitly on the fifth coordinate. Without cylindricity, there is no reason to compactify the fifth dimension, so this approach is properly called noncompactified.

2. Review of the formalism

We consider an action (5)g (5)R I =− d4xdψ + (5)L(ϕ, ϕ ) , (1) (5) ,A g0 16πG for a scalar field ϕ, which is minimally coupled to gravity. Here, |(5)g|=ψ8e6N is the absolute value of the |(5) |= 8 6N0 determinant for the 5D metric tensor with components gAB (A, B take the values 0, 1, 2, 3, 4) and g0 ψ0 e (5) (5) is a constant of dimensionalization determined by | g| evaluated at ψ = ψ0 and N = N0. Furthermore, R is = (5) = 8 the 5D Ricci scalar and G is the gravitational constant. In this work we shall consider N0 0, so that g0 ψ0 . Here, the index “0” denotes the values at the end of inflation (i.e., when b¨ = 0). With the aim to describe a manifold in apparent vacuum the Lagrangian density L in (1) must be only kinetic in origin 1 (5)L(ϕ, ϕ ) = gAB ϕ ϕ , (2) ,A 2 ,A ,B 210 J.E. Madriz Aguilar, M. Bellini / Physics Letters B 619 (2005) 208–218 and the 5D metric should be globally flat [9–11]

dS2 = ψ2 dN2 − ψ2e2N dr2 − dψ2, (3) where dr2 = dx2 + dy2 + dz2. The coordinates (N,r) are dimensionless and the fifth coordinate ψ has spatial units. The metric (3) describes a flat 5D manifold in apparent vacuum (GAB = 0), so that the Ricci scalar must be zero: (5)R = 0. We consider a diagonal metric because we are dealing only with gravitational effects, which are the important ones for the global evolution for the universe [12]. The equation of motion for the scalar quantum field ϕ is 2 2 ∂ψ ∂ϕ ∂ ϕ − ∂ϕ ∂N ∂ϕ ∂ ϕ 2ψ + 3ψ2 + ψ2 − ψ2e 2N ∇2ϕ − 4ψ3 − 3ψ4 − ψ4 = 0, (4) ∂N ∂N ∂N2 r ∂ψ ∂ψ ∂ψ ∂ψ2 ∂N ∂ψ where ∂ψ and ∂N are zero because the coordinates (N, r,ψ) are independent. The Eq. (4) can be written as 2 − ∂ϕ ∂ ϕ ϕ + 3ϕ − e 2N ∇2ϕ − 4ψ + ψ2 = 0, (5) r ∂ψ ∂ψ2 where the overstar denotes the derivative with respect to N and ϕ ≡ ϕ(N,r,ψ) . The commutator between ϕ and L Π N = ∂ = gNNϕ is given by ∂ϕ,N ,N (5) N NN g0 ( ) ϕ(N,r,ψ),Π (N, r ,ψ ) = ig δ 3 (r −r )δ(ψ − ψ ), (6) (5)g

(5) where | g0 | is the inverse of the renormalized volume of the manifold (3) and gNN = ψ−2. Hence, the commutator (5)g between ϕ and ϕ will be (5) g0 ( ) ϕ(N,r,ψ), ϕ(N, r ,ψ ) = i δ 3 (r −r )δ(ψ − ψ ). (7) (5)g

= −3N/2 ψ0 2 By means of the transformation ϕ χe ( ψ ) we obtain the 5D generalized Klein–Gordon like equation for χ(N,r,ψ) and the commutator between χ and χ: 2 − ∂ 1 χ − e 2N ∇2 + ψ2 + χ = 0, (8) r ∂ψ2 4 χ(N,r,ψ), χ(N, r ,ψ ) = iδ(3)(r −r )δ(ψ − ψ ). (9) The redeﬁned ﬁeld χ can be written in terms of a Fourier expansion 1 3 i(kr .r+kψ ·ψ) † −i(kr .r+kψ ·ψ) ∗ χ(N,r,ψ) = d kr dkψ ak k e ξk k (N, ψ) + a e ξ (N, ψ) , (2π)3/2 r ψ r ψ kr kψ kr kψ (10) where the asterisk denotes the complex conjugate and (a ,a† ) are, respectively, the annihilation and creation kr kψ kr kψ operators which satisfy the following commutation expresions † = (3) − − akr kψ ,ak k δ (kr kr )δ(kψ kψ ), (11) r ψ † † =[ ]= ak k ,a akr kψ ,ak k 0. (12) r ψ kr kψ r ψ J.E. Madriz Aguilar, M. Bellini / Physics Letters B 619 (2005) 208–218 211

The expression (9) complies if the modes are renormalized by the following condition: ∗ − ∗ = ξkr kψ (ξ kr kψ ) (ξkr kψ ) ξ kr kψ i. (13)

This equation provides the renormalization for the complete set of solutions on all the spectrum (kr ,kψ ). On the other hand, the dynamics for the modes ξkr kψ (N, ψ) is well described by the equation 2 2 −2N 2 2 ∂ ∂ 1 ξ + k e ξk k + ψ k − 2ikψ − − ξk k = 0. (14) kr kψ r r ψ ψ ∂ψ ∂ψ2 4ψ2 r ψ To solve this equation we can make

ξ (N, ψ) = ξ (1)(N)ξ (2)(ψ), (15) kr kψ kr kψ where the dynamics for ξ (1)(N) and ξ (2)(ψ) are given by the following differential equations kr kψ (1) − ξ + k2e 2N − α ξ (1) = 0, (16) kr r kr d2ξ (2) dξ(2) − kψ kψ 2 (1/4 α) (2) + 2ikψ − k − ξ = 0 (17) dψ2 dψ ψ ψ2 kψ being α a dimensionless constant. Thesolutionof(14) can be written as [13] √ i π − − − ξ (N, ψ) = e ikψ .ψ H√(2) k e N = e ikψ .ψ ξ¯ (N), (18) kr kψ 2 α r kr H(1,2)[ ]=J [ ]± Y [ ] J [ ] Y [ ] where ν x(N) ν x(N) i ν√x(N) are the Hankel functions, ν x(N) and ν x(N) are the first and = = = −N ¯ second kind Bessel functions with ν α 1/2 and x(N) kr e . Furthermore the function ξkr (N) is given by √ i π − ξ¯ (N) = H(2) k e N . (19) kr 2 1/2 r − = ikψ .ψ ¯ ¯ In other words, ξkr kψ (N, ψ) e ξkr (N), where ξkr (N) is a solution of − 1 ξ¯ + k2e 2N − ξ¯ = 0, (20) kr r 4 kr ¯ such that the renormalization condition for ξkr (N) becomes ¯ ¯ ∗ − ¯ ∗ ¯ = ξkr (ξ kr ) (ξkr ) ξ kr i. (21) Note the discrepance of the result (18) with whole obtained in [8], which relies in the particular choice of the vacuum. The vacuum here used is general and solves the problem of the earlier work in the sense that now there is not dependence on ψ in χ. Hence, the field χ in Eq. (10) can be rewritten as 1 3 ikr .r ¯ † −ikr .r ¯ ∗ χ(N,r,ψ) = χ(N,r) = d kr dkψ ak k e ξk (N) + a e ξ (N) . (22) (2π)3/2 r ψ r kr kψ kr Finally, the field ϕ is given by 2 − 3N ψ0 ϕ(N,r,ψ) = e 2 χ(N,r), (23) ψ 212 J.E. Madriz Aguilar, M. Bellini / Physics Letters B 619 (2005) 208–218

± with χ(N,r) givenbyEq.(22). Note that exponentials e ikψ .ψ disappear in χ(N,r) and there is not dependence on the fifth coordinate ψ in this field. This is a very important fact that say us that the field ϕ(N,r,ψ) propagates only on the 3D spatially isotropic space r(x,y,z), but not on the additional space-like coordinate ψ.

3. Coarse-granning in 5D

To study the large scale evolution of the field ϕ on large 3D spatial scales, we can introduce the field χL 1 3 ikr .r ¯ χL(N, r,ψ) = d kr dkψ θ k (N) − kr ak k e ξk (N) + c.c. , (24) (2π)3/2 0 r ψ r √ N where c.c. denotes the complex conjugate of the first term inside the brackets and k0 = αe is the N-dependent wavenumber (related to the 3D spatially isotropic, homogeneous and flat space r2 = x2 + y2 + z2), which separates 2 2 2 2 the long (kr k0 ) and short (kr k0 ) sectors. Modes with kr /k0 <are referred to as outside the horizon. If the short wavelenght modes are described with the field χS 1 3 ikr .r ¯ χS(N, r,ψ) = d kr dkψ θ kr − k (N) ak k e ξk (N) + c.c. , (25) (2π)3/2 0 r ψ r such that χ = χL + χS , hence the equation of motion for χL will be approximately 2 k0(N) χ − χ = k η(N,r,ψ) + k κ(N,r,ψ) + 2 k γ(N,r,ψ) , (26) L a L 0 0 0 where the stochastic operators η, κ and γ are given, respectively, by 1 η(N,r,ψ) = d3k dk δ k (N) − k a eikr .r ξ¯ (N) + , 3/2 r ψ 0 r kr kψ kr c.c. (27) (2π) 1 3 ikr .r ¯ κ(N,r,ψ) = d kr dkψ δ k (N) − kr ak k e ξk (N) + c.c. , (28) (2π)3/2 0 r ψ r 1 3 ikr .r ¯ γ(N,r,ψ) = d kr dkψ δ k (N) − kr ak k e ξ (N) + c.c. . (29) (2π)3/2 0 r ψ kr Eq. (26) can be rewritten as d χ − αχ = k η(N,r,ψ) + k γ(N,r,ψ) . (30) L L dN 0 0 This is a second-order stochastic equation that can be written as two first order stochastic ones by introducing the auxiliar field u = χL − k0η u = αχL + k0γ, (31) χ L = u + k0η. (32) 2 2 2 2 In the system (31), (32) the role of the noise γ can be minimized if (k0) γ (k0) η , which holds if the following condition holds ¯ ¯ ∗ ξ k ξ r kr 1. (33) ξ¯ ξ¯ ∗ kr kr J.E. Madriz Aguilar, M. Bellini / Physics Letters B 619 (2005) 208–218 213

In such case the system (31), (32) can be approximated to

u = αχL, (34) χ L = u + k0η. (35) This system represents two Langevin equations with a noise η which is Gaussian and white in nature

η=0, (36) 2 (k ) ∗ η2 = 0 dk ξ¯ ξ¯ δ(N − N ). (37) ψ k0 k0 2 2π k0 [ (0) (0)| ] The equation that describes the dynamics of the transition probability P χL ,u χL,u from a conﬁguration (0) (0) (χL ,u )to(χL,u) is a Fokker–Planck one ∂P ∂P ∂P 1 ∂2P =−u − αχ + D , L 11 2 (38) ∂N ∂χL ∂u 2 ∂χL = 1 2[ 2] where D11 2 (k0) dN η is the diffusion coefﬁcient related to the variable χL due to the stochastic action of the noise η. Explicitely 3 2 (k0) ¯ ¯ ∗ D11 = k0 dkψ ξk ξ , (39) 4π 2 0 k0 which is divergent.

4. Ponce de Leon metric and 4D de Sitter expansion

In order to describe the metric (3) in physical coordinates we can make the following transformations:

t = ψ0N, R = ψ0r, ψ = ψ, (40) such that we obtain the 5D metric ψ 2 dS2 = dt2 − e2t/ψ0 dR2 − dψ2, (41) ψ0 where t is the cosmic time and R2 = X2 + Y 2 + Z2. This metric is the Ponce de Leon one [14], and describes a 3D spatially ﬂat, isotropic and homogeneous extended (to 5D) FRW metric in a de Sitter expansion [15]. To study the de Sitter evolution of the universe on the 4D spacetime we can take a foliation ψ = ψ0 in the metric (41), such that the effective 4D metric results

dS2 → ds2 = dt2 − e2t/ψ0 dR2, (42) which describes 4D globally isotropic and homogeneous expansion of a 3D spatially ﬂat, isotropic and homogeneous universe that expands with a Hubble parameter H = 1/ψ0 (in our case a constant) and has a 4D scalar curvature (4)R = 6(H˙ + 2H 2). Note that in this particular case the Hubble parameter is constant so that H˙ = 0. The 4D energy density ρ and the pressure p are 3 πGρ= , 8 2 (43) ψ0 214 J.E. Madriz Aguilar, M. Bellini / Physics Letters B 619 (2005) 208–218

3 πG =− , 8 p 2 (44) ψ0 = −2 = × 19 where G Mp is the gravitational constant and Mp 1.2 10 GeV is the Planckian mass. Furthermore, the universe describes a vacuum equation of state: p =−ρ, such that ϕ˙2 a2 ρ= + 0 (∇ ϕ)2 + V(ϕ) , (45) 2 2a2 where the brackets denote the 4D expectation vacuum and the cosmological constant Λ gives the vacuum energy = Λ = 2 density ρ 8πG. Thus, Λ is related with the ﬁfth coordinate by means of Λ 3/ψ0 [14]. Furthermore, the 4D Lagrangian is given by (4)g 1 (4)L(ϕ, ϕ ) =− gµνϕ ϕ + V(ϕ) , (46) ,µ (4) ,µ ,ν g0 2 where the effective potential for the 4D FRW metric [16],is 2 =−1 ψψ = 1 ∂ϕ V(ϕ) g ϕ,ψ ϕ,ψ = . (47) 2 ψ ψ0 2 ∂ψ ψ=ψ0 In our case this potential takes the form 2 V(ϕ)= ϕ2(t, R,ψ ), 2 0 (48) ψ0 = where kψ0 is the wavenumber for ψ ψ0. Notice this potential has a geometrical origin and assume different representations in different frames. In our case the observer is in a frame U ψ = 0, because we are taking a foliation ψ = ψ0 on the 5D metric (41). Furthermore, the effective 4D motion equation for ϕ is 2 2 3 − t/ψ ψ ∂ϕ ψ ∂ ϕ ϕ¨ + ϕ˙ − e 2 0 ∇2 ϕ − 4 + = 0, (49) ψ R 2 ∂ψ 2 ∂ψ2 0 ψ0 ψ0 ψ=ψ0 which means that the effective derivative (with respect to ϕ) for the potential, is 2 V (ϕ) = ϕ(R,t,ψ0). (50) ψ=ψ0 2 ψ0 Now we can make the following transformation: − 3t ϕ(R,t) = e 2ψ0 χ(R,t). (51) −3t/(2ψ ) Note that now ϕ ≡ ϕ(R = ψ0r, t = ψ0N,ψ = ψ0) = e 0 χ(R,t), where [see Eq. (22)] χ(t,R) = χ(t = ψ0N,R = ψ0r, ψ = ψ0): 1 3 ikR.R ¯ χ(R,t) = d kR dkψ ak k e ξk (t) + c.c. δ(kψ − kψ ). (52) (2π)3/2 R ψ R 0 Hence, we obtain the following 4D Klein–Gordon equation for χ 2t − 2 1 χ¨ − e ψ0 ∇ + χ = . R 2 0 (53) 4ψ0 ¯ The equation of motion for the time-dependent modes ξk (t) is R 2t ¯¨ 2 − 1 ¯ ξ + k e ψ0 − ξ = 0. (54) kR R 2 kR 4ψ0 J.E. Madriz Aguilar, M. Bellini / Physics Letters B 619 (2005) 208–218 215

It is important to notice that Eq. (54) is exactly Eq. (20) with the variables transformation (40), on the hypersurface ψ = ψ0.

4.1. 4D stochastic dynamics for χL in a de Sitter expansion As was made in 5D, we can define the fields χL(t, R) and χS(t, R), which describe respectively the long and short wavelenght sectors of the field χ. 1 χ (t, R) = d3k θ k (t) − k a eikR.Rξ¯ (t) + δ(k − k ), L 3/2 R 0 R kRkψ kR c.c. ψ ψ0 (55) (2π) 1 3 ikR.R ¯ χS(t, R) = d kR θ kR − k (t) ak k e ξk (t) + c.c. δ(kψ − kψ ), (56) (2π)3/2 0 R ψ R 0 1 t/ψ 2 2 where k (t) = e 0 . The field that describes the dynamics of χ on the infrared sector (k k )isχL.Its 0 2ψ0 R 0 dynamics obeys the Kramers-like stochastic equation

− 2t e ψ0 d χ¨ − χ = k˙ η(t,R) + k˙ γ(t,R) , L 2 L 0 0 (57) 4ψ0 dt where the stochastic operators η, κ and γ are 1 η = d3k δ(k − k ) a eikR.Rξ¯ (t) + c.c. , (58) 3/2 R 0 R kRkψ0 kR (2π) 1 3 ikR.R ¯˙ γ = d kR δ(k0 − kR) ak k e ξ (t) + c.c. . (59) (2π)3/2 R ψ0 kR This second-order stochastic equation can be rewritten as two Langevin stochastic equations

− 2t e ψ0 u˙ = χ + k˙ γ, 2 L 0 (60) 4ψ0 ˙ χ˙L = u + k0η, (61) ˙ where u =˙χL − k0γ . The condition to can neglect the noise γ with respect to η, now holds

¯˙ ¯˙ ∗ ¨ 2 ξ k ξ (k ) R kR 0 , (62) ξ¯ ξ¯ ∗ (k˙ )2 kR kR 0 on super Hubble scales. Notice this result is exactly the same in Eq. (33), with the transformation (40).Forade et/ψ0 Sitter expansion Eq. (62) becomes kR/k < 1. It means that the noise γ can be neglected on scales kR 0 ψ0 (0) (0)| (i.e., on super Hubble scales). The Fokker–Planck equation for the transition probability P(χL ,u χL,u)is − −2t ∂P ∂P e ψ0 ∂P ∂2P =−u − χ + D (t) , 2 L 11 2 (63) ∂t ∂χL 4ψ0 ∂u ∂χL 3k˙ k2 where D (t) = 0 0 |ξ¯ |2. Hence, the equation of motion for χ2= dχ duχ2P(χ ,u)is 11 4π2 k0 L L L L

3t d 2e ψ0 χ2 = D (t) . (64) dt L 11 2 3 32π ψ0 216 J.E. Madriz Aguilar, M. Bellini / Physics Letters B 619 (2005) 208–218

− 3t 2ψ In order to return to the original ﬁeld ϕL = e 0 χL Eq. (64) can be rewritten as d 3 2 ϕ2 =− ϕ2 + , (65) dt L ψ L 2 3 0 32π ψ0 which has the following solution

2 3t 2 − ϕ = 1 + Ce ψ0 , (66) L 2 2 96π ψ0 where C is a constant of integration. When 3t 1, one obtains [for 2(1 + C) = 24] ψ0 2 2 1 − 3t ≡ H [ − ] ϕL 3t 1 1 3Ht . (67) ψ 1 2 2 2 0 4π ψ0 ψ0 4π However, after the end of inflation, when 3t 1, it becomes ψ0 2 2 2 H ϕ t , L 3 1 2 (68) ψ0 96π which is valid only for 2 1. In order to understand better this result in the context of the inflaton field fluctuations φ(R,t) , we can make the following semiclassical approach:

ϕ(R,t) = ϕ(R,t) + φ(R,t), (69) where ϕ=φc(t) and φ=0. With this representation one obtains

2 = 2 + 2 ϕ φc φ , (70) where φc(t) is the solution of the equation 3 2 φ¨ + φ˙ + φ = . c c 2 c 0 (71) ψ0 ψ0 The general solution of the differential equation (71) is − t − t = (0) ψ + ψ φc(t) φc e 0 1 C1e 0 , (72) (0) where C1 is a constant of integration and φc = φc(t = ti), being ti the time when inflation starts. Note that after inflation ends φc(t →∞) → 0. Hence, after inflation one obtains the following result:

2 2 ϕ 3t 1 φ 3t 1, (73) ψ0 ψ0 which means that for 3t 1 the following approximation is fulfilled: ψ0 2 2 2 2 H φ t ϕ t . L 3 1 L 3 1 2 (74) ψ0 ψ0 96π This is an important result which says us that the expectation value for the second momenta of the field ϕL at the end of inflation is approximately given by the expectation value for the inflaton field fluctuations on cosmological scales (for 2 1). We can estimate the amplitude of density energy fluctuations on cosmological scales δρ V (ϕ) φ φ2 1/2 c φ2 1/2. (75) L 2 L ρ end V(ϕ) φL J.E. Madriz Aguilar, M. Bellini / Physics Letters B 619 (2005) 208–218 217

δρ | −5 = −3 In order to obtain ρ end 10 ,thevalueofφc at this moment should be (taking 10 ) × −10 0.66 10 = × −10 φc end 0.66 10 H. (76) ψ0 (0) 10 1/2 Finally, we can estimate the initial value for φc: φc . If we consider tend 10 G , we obtain (0) φc Mp, (77) −10 10 1/2 for H 10 Mp (or ψ0 10 G ). Hence, the value of φc when inﬂation starts assumes sub-Planckian values.

5. Conclusions

In this work we have developed a stochastic treatment for the effective 4D inflaton field from a KK theory of gravity without the hypothesis of a slow-roll regime. In this framework the long-wavelength modes of the inflaton field reduces to a quantum system subject to a quantum noise originated by the short-wavelength sector. In this approach, the effective 4D potential is quadratic in ϕ and has a geometrical origin. Hence, as in STM theory of gravity 4D source terms are induced from a 5D vacuum and the fifth dimension (here a space-like dimension) is noncompact. In our theory the 5D vacuum is represented by a 5D globally flat metric (related with a RAB = 0 manifold) and a purely kinetic density Lagrangian for a quantum scalar field minimally coupled to gravity. Since the treatment for the scalar field is nonperturbative a very important feature of this formalism is that back reaction effects are included in the calculations in a consistent manner. An important result here obtained is that the expectation value for the second momenta for the field ϕL at the end of inflation is approximately given by the expectation value for the inflaton field fluctuations on cosmological scales, being both determinated by the value of the fifth coordinate on which we take the foliation: ψ = ψ0. Furthermore, the initial value for the background (and spatially homogeneous) inflaton field take sub-Planckian values. This fact is very important because solves (0) the problem of initial conditions in other treatments of chaotic inflation, in which φc assumes trans-Planckian values.

Acknowledgements

J.E.M.A. acknowledges CONACyT and IFM of UMSNH (México) for ﬁnancial support. M.B. acknowledges CONICET and UNMdP (Argentina) for ﬁnancial support.

References

[1] M.D. Pollock, Nucl. Phys. B 298 (1998) 673; F.R. Graziani, K.A. Olive, Phys. Rev. D 38 (1988) 2386. [2] S. Habib, Phys. Rev. D 46 (1992) 2408. [3] M. Bellini, H. Casini, R. Montemayor, P. Sisterna, Phys. Rev. D 54 (1996) 7172. [4] A.S. Goncharov, A.D. Linde, Sov. J. Part. Nucl. 17 (1986) 369. [5] A.A. Starobinsky, Phys. Lett. B 117 (1982) 175. [6] P.S. Wesson, Phys. Lett. B 276 (1992) 299. [7] For a review of the space–time–matter theory the reader can see J.M. Overduin, P.S. Wesson, Phys. Rep. 283 (1997) 303, and references therein. [8] M. Bellini, Phys. Lett. B 609 (2005) 187. [9] D.J. Gross, M.J. Perry, Nucl. Phys. B 226 (1983) 29. [10] A. Davidson, D. Owen, Phys. Lett. B 177 (1986) 77. 218 J.E. Madriz Aguilar, M. Bellini / Physics Letters B 619 (2005) 208–218

[11] D.S. Ledesma, M. Bellini, Phys. Lett. B 581 (2004) 1. [12] J.W. van Holten, Phys. Rev. Lett. 89 (2002) 201301. [13] A. Raya Montaño, J.E. Madriz Aguilar, M. Bellini, gr-qc/0502027. [14] J. Ponce de Leon, Gen. Relativ. Gravit. 20 (1988) 539. [15] For a review of STM theory the reader can see J.M. Overduin, P.S. Wesson, Phys. Rep. 283 (1997) 303. [16] J.E. Madriz Aguilar, M. Bellini, Phys. Lett. B 596 (2004) 116; J.E. Madriz Aguilar, M. Bellini, Eur. Phys. J. C 38 (2004) 367. Physics Letters B 619 (2005) 219–225 www.elsevier.com/locate/physletb

Complementary constraints on non-standard cosmological models from CMB and BBN

Adam Krawiec a, Marek Szydłowski b,c, Włodzimierz Godłowski c

a Institute of Public Affairs, Jagiellonian University, Rynek Główny, 31-042 Kraków, Poland b Complex Systems Research Centre, Jagiellonian University, Reymonta 4, 30-059 Kraków, Poland c Astronomical Observatory, Jagiellonian University, Orla 171, 30-244 Kraków, Poland Received 28 April 2005; received in revised form 24 May 2005; accepted 25 May 2005 Available online 2 June 2005 Editor: N. Glover

Abstract We study metric-afﬁne gravity (MAG) inspired cosmological models. Those models were statistically estimated using the SNIa data. We also use the cosmic microwave background observations and the big-bang nucleosynthesis analysis to constrain the density parameter Ωψ,0 which is related to the non-Riemannian structure of the underlying spacetime. We argue that while the models are statistically admissible from the SNIa analysis, complementary stricter limits obtained from the CMB and BBN − indicate that the models with density parameters with a a 6 scaling behaviour are virtually ruled out. If we assume the validity of the particular MAG based cosmological model throughout all stages of the universe, the parameter estimates from the CMB and BBN put a stronger limit, in comparison to the SNIa data, on the presence of non-Riemannian structures at low redshifts.  2005 Elsevier B.V. All rights reserved.

1. Introduction

Astronomical observations brought important changes in modern cosmology [1]. While the type Ia supernovae (SNIa) data are most often employed, the cosmic microwave background (CMB) observations and big-bang nucleosynthesis (BBN) analysis can also be used. They allow the exotic physics of cosmological models to be checked against the observational data [2]. There is an increasing effort to develop some cosmological and astrophysical tools to search for new physics beyond the standard model. The metric-affine gravity (MAG) cosmological model with the Robertson–Walker symmetry was investigated by three different groups. First, it was considered the model with triplet ansatz in vacuum [3]. Second, it was considered the dilational hyperfluid model [4]. Third, it was shown that on the level of the field equations the

E-mail address: [email protected] (M. Szydłowski).

0370-2693/$ – see front matter  2005 Elsevier B.V. All rights reserved. doi:10.1016/j.physletb.2005.05.060 220 A. Krawiec et al. / Physics Letters B 619 (2005) 219–225 special case of the MAG model is equivalent to a model in the Weyl–Cartan spacetime if we choose a model parameter in the special form (a6 =−a4) [5]. Moreover after redefinition of some variables the second and third approach gives the same set of dynamical equations. The analysis of constraints on parameters in the MAG model can be addressed in all three approach, but we adopt the last one proposed by Puetzfeld and Chen [5]. Note that in the model with dust matter on the brane, apart from dark radiation which scales like a−4, there is a correction of the type a−6 to the Einstein equations on the brane which arise from the influence of a bulk geometry [6–8]. The term scaling like a−6 also appears in the Friedmann–Robertson–Walker model with spinning fluid [9]. It is possible to establish formally the one to one correspondence between the MAG model and either the Randall–Sundrum brane model when positive values of the non-Riemannian contribution to effective energy is admitted or the spinning fluid filled cosmology when this contribution is negative. However, if one takes the pure Randall–Sundrum type model then there is a constraint on the brane tension parameter coming from the theory itself. The brane tension parameter is not less than about (100 GeV)4. The MAG model is free from such a theoretical constraint. In our further discussion we examine the flat models which is motivated by the CMB WMAP observations [10] and consider the following formula for the Friedmann first integral 2 H − − − = Ω a 3 + Ω + Ω a 4 + Ω a 6, 2 m,0 Λ,0 r,0 ψ,0 (1) H0 where H = dlna/dt is the Hubble function, t is the cosmological time, a = a(t) is the scale factor, Ωm,0, ΩΛ,0 and Ωψ,0 are the density parameters for dust matter, the cosmological constant and fictitious fluid which mimics “non-Riemannian effects”, respectively. Their values in the present epoch are marked by the index “0”. All density parameters satisfy the constraint condition

Ωm,0 + ΩΛ,0 + Ωr,0 + Ωψ,0 = 1. (2) The density parameter for the fictitious fluid is defined as [5] υ ψ2 Ω = , (3) ψ H 2 a6 where κ2 3a υ = 1 − 0 . 144a0 b4 The sign of the parameter υ is undetermined and it can assume both positive and negative values.

2. Constraint from the SNIa

Let us start from the reestimation of the models parameters by using the latest sample of SNIa data [11].The motivation to study the SN constraint is to find the best estimation available from the latest data which gives the narrowest constraint for this method. In the next sections we compare it with constraints obtained from other methods. Riess et al.’s sample contains 157 type Ia supernovae [11]. We consider the flat model with and without priors on the Ωψ,0 and Ωm,0. We assume that the former can be of any value or only non-negative, and the latter is non- negative or equal 0.3 [12]. We estimate the best fits of the model parameters (Table 1). Additionally we find the maximum likelihood estimates with the errors at 2σ level (Table 2). We find that the estimates of the parameter Ωψ,0 are very close to zero although positive apart of one case when it is zero. We can conclude that the estimate of this parameter is order of magnitude of 0.01. To illustrate the results of the maximum likelihood analysis of the model we draw the levels of confidence on Fig. 1. A. Krawiec et al. / Physics Letters B 619 (2005) 219–225 221

Table 1 Best ﬁt estimation of the model parameters from the SNIa data 2 Priors Ωm,0 Ωr,0 Ωψ,0 ΩΛ,0 M χ Ωψ,0 0; Ωm,0 000.14 0.012 0.848 15.945 175.75 Ωψ,0 0; Ωm,0 = 0.3–00.005 0.695 15.965 177.30 Ωm,0 000.14 0.012 0.848 15.945 175.75 Ωm,0 = 0.3–00.005 0.695 15.965 177.30 Ωm,0 0; Ωr,0 = 0.0001 0.16 – 0.029 0.811 15.945 175.97 Ωm,0 = 0.3; Ωr,0 = 0.0001 – – 0.005 0.695 15.965 177.30

Table 2 Maximum likelihood estimation of the model parameters with 2σ errors from the SNIa data

Priors Ωm,0 Ωr,0 Ωψ,0 ΩΛ,0 +0.24 +0.15 +0.041 +0.100 Ωψ,0 0; Ωm,0 00−0.00 0−0.00 0.009−0.009 0.820−0.120 = +0.04 +0.017 +0.020 Ωψ,0 0; Ωm,0 0.3– 0−0.00 0−0.000 0.680−0.040 +0.27 +0.21 +0.042 +0.110 Ωm,0 00−0.00 0−0.00 0.009−0.036 0.800−0.140 = +0.09 +0.017 +0.020 Ωm,0 0.3–0−0.00 0.002−0.022 0.680−0.060 = +0.28 +0.038 +0.130 Ωm,0 0; Ωr,0 0.0001 0.14−0.14 –0.028−0.035 0.810−0.150 = = +0.013 +0.015 Ωm,0 0.3; Ωr,0 0.0001 – – 0.005−0.013 0.695−0.015

Fig. 1. The contours with 1σ and 2σ conﬁdence levels for Ωψ,0 versus H0, ΩΛ,0, Ωm,0,andΩr,0 from the SNIa data.

The MAG model fits well to SNIa data. We consider the model with any value of Ωr,0 we obtain the value of Ωm,0 equal zero as best fit and maximum likelihood estimator, while fixing the small amount radiation (Ωr,0 [13]) gives the low density matter universe. The estimation of the Hubble constant gives the value close to 65 km/s MPc. 222 A. Krawiec et al. / Physics Letters B 619 (2005) 219–225

3. CMB peaks in the MAG model

The hotter and colder spots in the CMB can interpreted as acoustic oscillation in the primeval plasma during the last scattering. Peaks in the power spectrum correspond to maximum density of the wave. In the Legendre multipole space these peaks correspond to the angle subtended by the sound horizon at the last scattering. Further peaks answer to higher harmonics of the principal oscillations. It is very interesting that locations of these peaks are very sensitive to the variations in the model parameters. Therefore, it can be used as another way to constrain the parameters of cosmological models. The acoustic scale A which puts the locations of the peaks is deﬁned as zdec dz = π 0 H(z ) , (4) A ∞ dz c zdec s H(z ) where 3 4 6 H(z)= H0 Ωm,0(1 + z) + Ωr,0(1 + z) + Ωψ,0(1 + z) + ΩΛ,0 (5) and cs is the speed of sound in the plasma given by 4 3 dp Ωγ,0(1 + z) + 6Ωψ,0(1 + z) c2 ≡ eff = 3 . s 3 (6) dρeff 3Ωb,0 + 4Ωγ,0(1 + z) + 6Ωψ,0(1 + z) Knowing the acoustic scale we can determine the location of mth peak

m ∼ A(m − φm), (7) where φm is the phase shift caused by the plasma driving effect. Assuming that Ωm,0 = 0.3, on the surface of last scattering zdec it is given by 0.1 0.1 0.1 r(zdec) 1 ρr(zdec) 1 Ωr,0(1 + zdec) φm ∼ 0.267 = 0.267 = 0.267 , (8) 0.3 0.3 ρm(zdec) 0.3 0.3 2 where Ωb,0h = 0.02, r(zdec) ≡ ρr(zdec)/ρm(zdec) = Ωr,0(1 + zdec)/Ωm,0 is the ratio of radiation to matter densities at the surface of last scattering. The CMB temperature angular power spectrum provides the locations of the first two peaks [14,15] and the BOOMERanG measurements give the location of the third peak [16]. They values with uncertainties on the level 1σ are the following = +0.8 = +10 = +12 1 220.1−0.8,2 546−10,3 845−25. Using the WMAP data only, Spergel et al. [14] obtained that the Hubble constant H0 = 72 km/s MPc (or the −2 −2 parameter h = 0.72), the baryonic matter density Ωb,0 = 0.024h , and the matter density Ωm,0 = 0.14h which give a good agreement with the observation of position of the first peak. To find whether cosmological models give these observational locations of peaks we fix some model parameters. Let the baryonic matter density Ωb,0 = 0.05, the spectral index for initial density perturbations n = 1, and the radiation density parameter [13] −2 −5 −2 −5 −2 −5 Ωr,0 = Ωγ,0 + Ων,0 = 2.48h × 10 + 1.7h × 10 = 4.18h × 10 , (9) which is a sum of the photon Ωγ,0 and neutrino Ων,0 densities. Assuming Ωm,0 = 0.3 and h = 0.72 we obtain for the standard CDM cosmological model the following positions of peaks

1 = 220,2 = 521,3 = 821 with the phase shift φm given by (8). A. Krawiec et al. / Physics Letters B 619 (2005) 219–225 223

Table 3 Values of Ωψ,0 and location of ﬁrst three peaks

Hubble constant Ωψ,0 1 2 3 −11 H0 = 65 km/sMPc 3× 10 220 522 825 − 7 × 10 14 220 523 826 − −1.4 × 10 10 223 530 847 −11 H0 = 72 km/sMPc 3.7 × 10 220 522 823 0 220 521 821 − −1.3 × 10 10 224 530 847

Fig. 2. The location of the ﬁrst peak in function of Ωψ,0.

From the SNIa data analysis it was found that the Hubble constant has lower value. Assuming that H0 = 65 km/s −5 MPc (or h = 0.65), we have Ωr,0 = 9.89 × 10 from Eq. (9). In further calculation we take Ωr,0 = 0.0001. If we consider the standard CDM model, with Ωm,0 = 0.3, Ωb,0 = 0.05, the spectral index for initial density perturbations n = 1, and h = 0.65, where sound can propagate in baryonic matter and photons we obtain the following locations of ﬁrst three peaks

1 = 225,2 = 535,3 = 847. We find some discrepancy between the observational and theoretical results in this case. Now it is interesting to check whether the presence of the fictitious fluid Ωψ,0 change the locations of the peaks. The properties of the fictitious fluid Ωψ,0 are unknown. In particular, we do not know whether the sound can or cannot propagate in this fluid. But we assume that sound can propagate in it as well as in baryonic matter and photons. We consider both values of the Hubble constant and assume that h = 0.65 or h = 0.72. The results of calculations of peak locations and the values of the parameter Ωψ,0 are presented in Table 3. If we choose the H0 = 65 km/s MPc then we obtain the agreement with the observation of the location of the first peak for three non-zero values of the parameter Ωψ,0.AsitisshownonFig. 2 there are two positive and one negative values of this parameter for which the MAG model is admissible. All these distinguished values of Ωψ,0 are in agreement with the result obtained from SNIa because the 2σ confidence interval for this parameter obtained from the SNIa data contains these three points. While the SNIa estimations give the possibility that Ωψ,0 is equal zero, the CMB calculations seem to exclude this case because the zero value of Ωψ,0 requires the first peak location at 225. If we choose the H0 = 72 km/s MPc than one of positive values of Ωψ,0 move to zero, while the second one move a little to the right. 224 A. Krawiec et al. / Physics Letters B 619 (2005) 219–225

We also calculated the age of the universe in the MAG model. We ﬁnd that the difference in the age of the universe is smaller than 1 mln years for all three values of Ωψ,0. Assuming that Ωm,0 = 0.3 the age of the universe is 14.496 Gyr for H0 = 65 km/s MPc, and 13.088 Gyr for H0 = 72 km/s MPc. The globular cluster analysis indicated that the age of the universe is 13.4 Gyr [17].

4. Constraint from the BBN

It is well known that the big-bang nucleosynthesis (BBN) is the very well tested area of cosmology and does not allow for any significant deviation from the standard expansion law apart from very early times (i.e., before the onset of BBN). The prediction of standard BBN is in well agreement with observations of abundance of light elements. Therefore, all non-standard terms added to the Friedmann equation should give only negligible small modifications during the BBN epoch to have the nucleosynthesis process unchanged. In our opinion the consistency with BBN is a crucial issue in the MAG models where the non-standard term a−6 in the Friedmann equation is added (see also discussion in [18]). This additional term scales like (1 + z)6. It is clear that such a term has either accelerated (Ωψ,0 > 0) or decelerated (Ωψ,0 < 0) impact on the universe expansion. Going backwards in time this term would become dominant at some redshift. If it would happen before the BBN epoch, the radiation domination would never occur and the all BBN predictions would be lost. The domination of the fictitious fluid Ωψ should end before the BBN epoch starts otherwise the nucleosynthesis process would be dramatically modified. If we assume that the BBN result are preserved in the MAG models we obtain another constraint on the amount of Ωψ,0. Let us assume that the model modification is negligible small during the BBN epoch and the nucleosynthesis process is unchanged. It means that the contribution of the MAG −4 8 term Ωψ,0 cannot dominate over the radiation term Ωr,0 ≈ 10 before the beginning of BBN (z 10 )

6 4 −20 Ωψ,0(1 + z) <Ωr,0(1 + z) ⇒ | Ωψ,0| < 10 . −2 The values of Ωψ,0 ∝ 10 obtained as best ﬁts in the SNIa data analysis as well as the smallest non-zero value −14 of Ωψ,0 = 7 × 10 calculated in the CMB analysis are unrealistic in the light of the above result. If we take into consideration the maximum likelihood analysis of SNIa data we have the possibility that the value of Ωψ,0 is lower than |10−20| in the 2σ conﬁdence interval. In the case of the CMB analysis only the value of the Hubble constant close to 72 km/s MPc gives the zero or close to zero value of Ωψ,0.

5. Conclusion

The Letter discusses observational constraint on “energy contributions” arising in certain cosmological models based on MAG. In particular it is focused on the non-standard term a−6. We test this model against the SNIa data, the location of the peaks of the CMB power spectrum, and constraints from the BBN. The MAG model fits well to SNIa data and the estimations give the amount of fluid Ωψ,0 to be order of magnitude 0.01, and the Hubble constant is close to 65 km/s MPc. Let us note that these results are compatible with constraints from FRIIb radio galaxies and X-ray gas mass fractions [19]. The CMB analysis gives that the Hubble constant is 72 km/s MPc which gives the too low age of the universe in comparison with the age of globular clusters. Taking lower value of the Hubble constant obtained from SNIa estimation resolves the problem of the age. However, the location of the first peak shifts to the right and is in conflict with the observed location. The introducing of the non-Riemannian structure of the underlying spacetime moves the location of the first peak back and this MAG model agrees with the CMB observations. The analysis of the CMB in this model cannot distinguish the character of the fictitious fluid and we do not know whether the parameter Ωψ,0 is positive or negative. A. Krawiec et al. / Physics Letters B 619 (2005) 219–225 225

The absolute values of Ωψ,0 obtained in the MAG model from the CMB analysis with h = 0.65 seems to be too large in comparison to the limit obtained from the BBN analysis. Using the BBN analysis we pointed out that the MAG part of the energy density to its present density parameter is of order 10−20. The limit of this order leads to the conclusion that the MAG model is virtually ruled out. However, we must remember that we insist that the MAG model does not change the physics during and after the BBN epoch. In this context, the merit of the SNIa analysis is its independency from any assumption on physical processes in the early universe.

Acknowledgements

M. Szydłowski acknowledges the support by KBN grant No. 1 P03D 003 26.

References

[1] S. Dodelson, Modern Cosmology, Academic Press, San Diego, 2003. [2] M. Kamionkowski, eConf C020805 (2002) TF04. [3] Y.N. Obukhov, E.J. Vlachynsky, W. Esser, F.W. Hehl, Phys. Rev. D 56 (1997) 7769. [4] O.V. Babourova, B.N. Frolov, Class. Quantum Grav. 20 (2003) 1423. [5] D. Puetzfeld, X.-L. Chen, Class. Quantum Grav. 21 (2004) 2703. [6] L. Randall, R. Sundrum, Phys. Rev. Lett. 83 (1999) 3370. [7] L. Randall, R. Sundrum, Phys. Rev. Lett. 83 (1999) 4690. [8] W. Godlowski, M. Szydlowski, Gen. Relativ. Gravit. 36 (2004) 767. [9] M. Szydlowski, A. Krawiec, Phys. Rev. D 70 (2004) 043510. [10] M. Kamionkowski, D.N. Spergel, N. Sugiyama, Astrophys. J. 426 (1994) L57. [11] A.G. Riess, et al., Astrophys. J. 607 (2004) 665. [12] P.J.E. Peebles, B. Ratra, Rev. Mod. Phys. 75 (2003) 559. [13] R.G. Vishwakarma, P. Singh, Class. Quantum Grav. 20 (2003) 2033. [14] D.N. Spergel, et al., Astrophys. J. Suppl. 148 (2003) 175. [15] L. Page, et al., Astrophys. J. Suppl. 148 (2003) 233. [16] P. de Bernardis, et al., Astrophys. J. 564 (2002) 559. [17] B. Chaboyer, L.M. Krauss, Science 299 (2003) 65. [18] D. Puetzfeld, Ph.D. dissertation, University of Cologne, 2003. [19] D. Puetzfeld, M. Pohl, Z.H. Zhu, Astrophys. J. 619 (2005) 657. Physics Letters B 619 (2005) 226–232 www.elsevier.com/locate/physletb

Constraint on B–L cosmic string from leptogenesis with degenerate neutrinos

Pei-Hong Gu, Hong Mao

Institute of High Energy Physics, Chinese Academy of Sciences, P.O. Box 918-4, Beijing 100049, People’s Republic of China Received 25 March 2005; received in revised form 10 May 2005; accepted 15 May 2005 Available online 1 June 2005 Editor: T. Yanagida

Abstract

In the early universe, as a consequence of U(1)B–L gauge symmetry-breaking, the so-called B–L cosmic strings are expected to be produced at the breaking scale ηB–L according to the Kibble mechanism. The decaying, collapsing closed loops of these strings can release the right-handed neutrinos, whose subsequent decay can contribute to the baryon asymmetry of the universe (BAU), through the “slow death” (SD) process and/or the “quick death” (QD) process. In this Letter, we assume that the decay of the lightest heavy Majorana neutrinos released from the B–L cosmic string loops can produce a baryon asymmetry consistent with the cosmic microwave background (CMB) observations. Considering the fact that both the neutrinoless double beta decay experiment and the cosmological data show a preference for degenerate neutrinos, we give the lower limits for the breaking ¯ = 2 + 2 + 2 1/2 scale ηB–L with the neutrino masses 0.06 eV m (m1 m2 m3) 1.0 eV, where the full possible cases of degenerate 15 15 15 neutrinos are included. We obtain ηB–L 3.3 × 10 GeV, 5.3 × 10 GeV and 9.5 × 10 GeV for m¯ = 0.2eV,0.4eVand 1.0 eV respectively in the SD process, and ﬁnd the B–L cosmic string has a very small contribution to the BAU in the QD process.  2005 Elsevier B.V. All rights reserved.

The baryon asymmetry of the universe (BAU) has there are several neutrino oscillation experiments [2,3] been determined precisely [1]: which have conﬁrmed the extremely small but nonzero neutrino masses. Then leptogenesis [4] is now an at- nB − ηCMB ≡ = (6.3 ± 0.3) × 10 10, (1) tractive scenario which can simultaneously explain B n γ the cosmological baryon asymmetry and the neutrino = − where nB nb nb¯ and nγ are the baryon and pho- properties by the seesaw mechanism [5]. ton number densities, respectively. At the same time, The simplest leptogenesis scenario is to extend the standard model (SM) by three generations of the right-handed neutrinos with Majorana mass. A more E-mail addresses: [email protected] (P.-H. Gu), [email protected] (H. Mao). appealing alternative is to consider this within the con-

0370-2693/$ – see front matter  2005 Elsevier B.V. All rights reserved. doi:10.1016/j.physletb.2005.05.053 P.-H. Gu, H. Mao / Physics Letters B 619 (2005) 226–232 227 text of unified models with an embedded U(1)B–L the transition, the infinite string network coarsens and gauge symmetry, which can be derived from the more loops form from the intercommuting of infinite SO(10) models. After spontaneous breaking of the strings. In the following discussion, we pay particu- U(1)B–L gauge symmetry, the right-handed neutri- lar attention to the formation of closed loops and their nos naturally acquire heavy Majorana mass and pro- subsequent evolution. duce a lepton asymmetry, which finally converted The formation and evolution of the cosmic string to the baryon asymmetry via the (B–L)-conserving loops have been studied extensively, both in analytical sphaleron process [6], by decaying into massless lep- and numerical methods, for details see [19,20].Af- tons and electroweak Higgs bosons. ter their formation, the evolution of the closed loops In the early universe, according to the Kibble mech- can be broadly categorized into two classes. One is the anism [7], the so-called B–L cosmic strings are ex- “slow death” (SD) process [14,19,20], where the loops pected to be produced at the U(1)B–L symmetry- born at time t have a longer lifetime (compared to the breaking scale during SO(10) breaking to the SM Hubble expansion time scale H −1(t)). In this scenario, gauge group [8]. The strings are formed by the gauge during the phase transition, the formed string loops os- field and Higgs field, and the Higgs field also gives cillate freely and loose their energy by emitting gravi- heavy Majorana mass to the right-handed neutrinos tational radiation. When the loop’s radius becomes of through Yukawa coupling [9,10]. As discussed in the order of the string width, the loop releases its final Refs. [11–16], the decaying, collapsing closed loops energy into massive particles. Among these particles of these strings can be a nonthermal source of the will be the massive gauge bosons, Higgs bosons and right-handed neutrinos whose subsequent decay can massive right-handed neutrinos which were trapped in contribute to the BAU. the string as fermion zero modes [19]. The other is the When the B–L cosmic string loops contribute sig- “quick death” (QD) process [21]—the loops die in- nificantly to the BAU, ones find that the U(1)B–L stantaneously as soon as they are formed due to the gauge symmetry-breaking scale ηB–L has a lower high probability of self intersecting [22]. Thus they 11 limit ηB–L 1.7 × 10 GeV [16] if the light neutrino would lose only a negligible amount of energy in grav- masses are hierarchical, especially for m3 0.05 eV, itational radiation and massive particle radiation rather where mi (i = 1, 2, 3) is the eigenvalue of the light than gravitational radiation plays the dominant role. neutrino mass matric. But once the evidence of neutri- For the above two cases, the number density of noless double beta decay with mee = (0.05–0.86) eV loops disappearing in the radiation dominated epoch (at 95% C.L.) [17] is confirmed, a degenerate neutrino at any time t can be described respectively as [16,19, spectrum is required. Recent studies on the cosmologi- 20]: cal data [1,18] with the neutrino oscillation experiment 3/2 dn 1 − (C + 1) − results [2,3] also showed a preference for degenerate SD = f (Γ Gµ) 1 t 4, (2) dt SD 2 C neutrinos with mi 0.23 eV or mi 0.56 eV. x In this Letter, we follow the discussions in Ref. [16] dnQD 1 1/2 −3 and assume that the lightest heavy Majorana neutri- = fQD µ t , (3) nos are released from the B–L cosmic string loops, dt x2 and hence a baryon asymmetry consistent with the ob- where x is approximately in the range ∼ 0.4–0.7 sup- servations can be induced by their decay modes. We ported by the extensive numerical simulations, Γ ∼ estimate the U(1)B–L symmetry-breaking scale ηB–L 100 is a geometrical factor that determines the average for 0.06 eV m¯ 1.0 eV, where m¯ is defined as loop length, µ is the mass per unit length of a cosmic ¯ 2 = 2 + 2 + 2 ¯ m m1 m2 m3. In this range of m, the full pos- string and related the symmetry-breaking scale η with sible cases of degenerate neutrinos are included. µ1/2 ∼ η, C is a numerical factor of order unity and = 2 In the generic picture, local B–L cosmic strings G 1/MPl is the Newton’s constant, while fSD and form at the phase transition associated with the spon- fQD denote the fraction of newly born loops which die taneous symmetry breaking of U(1)B–L. During the through the SD process and QD process, respectively. phase transition, a network of strings forms, consist- Noteworthy that the observations of the cosmic mi- ing of both infinite strings and cosmic loops. After crowave background (CMB) anisotropy give an upper 228 P.-H. Gu, H. Mao / Physics Letters B 619 (2005) 226–232 bound on the symmetry-breaking scale [23] which is finally converted to the baryon asymmetry, comes mainly from the decay of the lightest heavy Ma- η 1.0 × 1016 GeV. (4) jorana neutrino N1 and the contribution of the cosmic In addition, the measured flux of the cosmic gamma string loops to the BAU can be estimated as: ray background in the 10 MeV–100 GeV [24] energy tEW region puts a constraint on f [25] 28 1 dnN1 QD ηB = × 7.04 × ε1 dt, (10) 79 s dt 16 2 −6 t fQD η/10 GeV 9.6 × 10 , (5) F where 28/79 is the value of B/(B − L) for SM [26], but there is no equivalent constraint on f .Thisdif- SD 7.04 is the present density ratio of photon number ference can be understood easily, since the time de- and entropy, ε is the CP asymmetry of N decays pendence of the disappearing rate of loops is ∝ t−4 in 1 1 and s = (2π 2/45)g∗T 3 is the entropy density with theSDcase[seeEq.(2)] while ∝ t−3 in the QD case g∗ 106.75. t is the electroweak transition time, [see Eq. (3)], in other words, the SD process domi- EW while t denotes the epoch when the inverse decays nates at sufficiently early time, while the QD process F and L-nonconserving scatterings begin to freeze out dominates at relatively late time and can potentially dnN1 contribute to the nonthermal gamma ray background. and there is no washout effects any more. dt is the It is difficult to calculate exactly the total number releasing rate of N1 from the cosmic string loops. The of the heavy Majorana neutrinos from each loop, but temperature T and time t are related by as shown in Ref. [14] when a cosmic loop decays, 1 t = (11) it releases at least one heavy Majorana neutrino. For 2H(T) simplicity, we may expect that it would be a number of with Hubble constant order unity. Then the releasing rate of the heavy Ma- jorana neutrinos Ni (i = 1, 2, 3) from the SD process 8π 3g∗ H(T)= T 2/M . (12) and QD process at any time t can be written as: 90 Pl dnSD + 3/2 Therefore, we get Ni SD 1 −1 (C 1) −4 = N fSD (Γ Gµ) t , (6) dt Ni x2 C 1 dt =− dT (13) TH(T) QD dnN 1 − and can rewrite Eq. (10) as i = N QDf µ1/2t 3, (7) Ni QD 2 dt x TF 28 1 dn 1 with N SD and N QD ∼ O(1). = × × N1 Ni Ni ηB 7.04 ε1 (T ) dT. In the leptogenesis scenario, the decay of the heavy 79 s dt TH(T) TEW Majorana neutrinos Ni which produces the lepton (14) asymmetry is described by the following Lagrangian Now the key to calculate the baryon asymmetry in Eq. (14) is how to determine ε1 and TF . We note that −L = ¯ + 1 ¯ c + hij lLiφνRj MiνRiνRi H.c., (8) there is an upper bound on ε1 with m1

M T 1 In the SD case, using Eqs. (6), (11), (14), (15) and F 0.6 (16) 4.2(ln K) (23), we obtain with SD max max ηB ε1 ,TF ΓN ≡ 1 T max K . (17) F SD H(T) T =M 28 1 dn 1 1 = × 7.04 × εmax N1 (T ) dT 79 1 s dt TH(T) Here ΓN1 is the decay width of N1.Using TEW 1 = † 28 max −18 ΓN1 h h M1, (18) × 7.04 × ε × 5.33 × 10 8π 11 79 1 and Eq. (12), we get max 3 − 3 SD (TF /GeV) (TEW/GeV) × N fSD ˜ N1 (ηSD /GeV)2 = m1 B–L K (19) − m∗ 9.44 × 10 37 with (ηSD /GeV)2 × 4 SD B–L g1NN fSD . (25) † 2 1 [ ∗ ]1.8 (h h)11v (m3/eV) ln(m1/m ) m˜ 1 = , (20) M1 In the last step we have neglected the effect of TEW 100 GeV since the dominant contribution to the in- 1/2 5/2 2 tegral comes from TF TEW. Considering the con- 16π g∗ v −3 m∗ = √ 1.07 × 10 eV. (21) straint (24) from the CMB, we can get a lower limit 3 5 MPl for ηB–L Since m˜ 1 m1 [29], we replace m˜ 1 by m1 in Eq. (19) 13 1 and then give a lower limit for K ηB–L 2.39 × 10 2 SD 1/2 1/2 m g1(NN ) fSD K K ≡ 1 , (22) 1 min 1/2 0.9 m∗ × (m3/eV) ln(m1/m∗) GeV, (26) accordingly TF has an upper bound CMB = × −10 where the 3σ lower limit for (ηB )low 5.4 10 M1 has been adopted. Using the relations T T max . F F 0.6 (23) 4.2(ln Kmin) 1 m2 = m¯ 2 − m2 − 2m2 , (27) Assuming that the decay of the lightest heavy Ma- 1 3 atm sol jorana neutrinos, which are released from the cosmic string loops, can produce a baryon asymmetry con- 1 m2 = m¯ 2 + 2m2 − m2 , (28) sistent with the CMB observations (1), we obtain the 3 3 atm sol following restriction SD and ﬁxing N and fSD, we can get the lower limits N1 max max for ηB–L with m¯ . ηB ε1 ,TF In Fig. 1 we show ηB L as a function of m¯ for T max – F ¯ 28 1 dn 1 0.06 eV m 1.0 eV (corresponding to 10 k = × . × εmax N1 (T ) dT = SD = 7 04 1 300) with g1 1.0, 0.1, 0.01 by taking NN 1, 79 s dt TH(T) = 1 TEW fSD 1. We ﬁnd that in order to satisfy the CMB con- × 16 CMB straint ηB–L 1.0 10 GeV, the Yukawa coupling ηB . (24) g1 0.1. In the case of degenerate neutrino scenario 15 Then the U(1)B–L gauge symmetry-breaking scale with g1 = 0.1, we obtain ηB–L 3.3 × 10 GeV can be estimated by using the above equation in the SD for m¯ = 0.2 eV (corresponding to the upper bound process and QD process, respectively. In the following for the successful leptogenesis [29]), and ηB–L 15 calculations, we will take x = 0.5, Γ = 100, C = 1, 5.3 × 10 GeV for m¯ = 0.4eVorηB–L 9.5 × = 2 = 15 ¯ = µ ηB–L, and M1 g1ηB–L, where the Yukawa cou- 10 GeV for m 1.0 eV (corresponding to the up- pling g1 1 is natural for M1 M2,M3. per bounds from the cosmological data [1,18]). 230 P.-H. Gu, H. Mao / Physics Letters B 619 (2005) 226–232

Fig. 1. The evolution of the lower limits for η with 0.064 eV m¯ 1.0 eV in the SD process for NSD = 1, f = 1. B–L N1 SD

Fig. 2. The evolution of the lower limits for η with 0.064 eV m¯ 1.0 eV in the QD process for NQD = 1. B–L N1 QD QD dnN dnN max max 1 1 ηB ε1 ,TF Replacing dt in Eq. (14) by dt given by −27 Eq. (7), considering the additional constraint on fQD 4.05 × 10 from Eq. (5) and repeating the same steps in the SD 1 × g2N QD (η /GeV), case above, we can also obtain the baryon asymmetry 1 N1 0.6 B–L (m3/eV)[ln(m1/m∗] and the lower limit for ηB–L in the QD process (29) P.-H. Gu, H. Mao / Physics Letters B 619 (2005) 226–232 231

Fig. 3. The evolution of ηQD with 0.064 eV m¯ 1.0 eV in the QD process for NQD = 1andη = 1.0 × 1016 GeV. B N1 B–L

17 ηB–L 1.33 × 10 neutrinos are included. Especially we plot the lower limits for ηB–L with m¯ in the SD process and QD 1 0.6 × (m3/eV) ln(m1/m∗) GeV. process, respectively. g2N QD 1 N1 In the SD process, we find that the Yukawa cou- (30) pling g1 should be 0.1 due to the CMB constraint η 1.0 × 1016 GeV. In the case of degenerate In Fig. 2 we plot ηB–L as a function of m¯ for g1 = B–L × 15 1.0, 0.1, 0.01 by taking the parameters as N QD = 1. neutrino scenario, we obtain ηB–L 3.3 10 GeV, N1 × 15 × 15 We find that the lower limits for η with 0.06 eV 5.3 10 GeV and 9.5 10 GeV for the degener- B–L ¯ = m¯ 1.0 eV in the QD process are much higher than ate neutrino masses m 0.2eV,0.4 eV and 1.0eV, the values in the SD process, and higher than the up- respectively. And we also find that there is a very small per bound 1.0 × 1016 GeV. Furthermore by taking contribution from the B–L cosmic strings to the BAU in the QD process. η = 1.0 × 1016 GeV and N QD = 1inEq.(29), B–L N1 QD ¯ we plot the evolution of ηB as a function of m in Fig. 3. We can see that the contribution from the B–L Acknowledgements cosmic string loops to the BAU is small enough to be neglected with the above neutrino masses for the SD We thank Xiao-Jun Bi, Bo Feng, Zhi-Hai Lin and case. especially Xinmin Zhang for discussions. This work In summary, we study such leptogenesis scenario: is supported partly by the National Natural Science the lightest heavy Majorana neutrinos are released Foundation of China under the Grant No. 90303004. from the B–L cosmic string loops, and their decay can produce a baryon asymmetry consistent with the CMB observations. Considering the fact that both the neu- References trinoless double beta decay experiment and the cosmological data show a preference for the degenerate [1] M. Tegmark, et al., Phys. Rev. D 69 (2004) 103501. neutrinos, we give the lower limits for the U(1)B–L [2] G.L. Fogli, et al., hep-ph/0310012; symmetry-breaking scale ηB–L with 0.06 eV m¯ M.H. Ahn, et al., K2K Collaboration, Phys. Rev. Lett. 90 1.0 eV, where the full possible cases of degenerate (2003) 041801; 232 P.-H. Gu, H. Mao / Physics Letters B 619 (2005) 226–232

M. Shiozawa, et al., SK Collaboration, in: Proceedings of the [12] R.H. Brandenberger, A.C. Davis, M. Hindmarsh, Phys. Lett. Neutrino 2002, in press. B 263 (1991) 239. [3] Q.R. Ahmad, et al., SNO Collaboration, Phys. Rev. Lett. 92 [13] H. Lew, A. Riotto, Phys. Rev. D 49 (1994) 3837. (2004) 181301; [14] R. Jeannerot, Phys. Rev. Lett. 77 (1996) 3292. K. Eguchi, et al., KamLAND Collaboration, Phys. Rev. [15] P. Bhattacharjee, Phys. Rev. Lett. 81 (1998) 260. Lett. 90 (2003) 021802. [16] P. Bhattacharjee, N. Sahu, U.A. Yajnik, Phys. Rev. D 70 (2004) [4] M. Fukugita, T. Yanagida, Phys. Lett. B 174 (1986) 45; 083534. P. Langacker, P.D. Peccei, T. Yanagida, Mod. Phys. Lett. A 1 [17] H.V. Klapdor-Kleingrothaus, I.V. Krivosheina, A. Dietz, O. (1986) 541; Chkvorets, Phys. Lett. B 586 (2004) 198. M. Luty, Phys. Rev. D 45 (1992) 455; [18] D.N. Spergel, et al., Astrophys. J. Suppl. 148 (2003) 1. R.N. Mohapatra, X. Zhang, Phys. Rev. D 46 (1992) 5331. [19] A. Vilenkin, E.P.S. Shellard, Cosmic Strings and other Topo- [5] M. Gell-Mann, P. Ramond, R. Slansky, in: P. Van Nieuwen- logical Defects, Cambridge Univ. Press, Cambridge, 2000. huizen, D. Freedman (Eds.), Proceedings of the Supergravity [20] M.B. Hindmarsh, T.W.B. Kibble, Rep. Prog. Phys. 58 (1995) Stony Brook Workshop, New York, North-Holland, Amster- 477. dam, 1979; [21] G. Vincent, N. Antunes, M. Hindmarsh, Phys. Rev. Lett. 80 T. Yanagida, in: A. Sawada, A. Sugamoto (Eds.), Proceedings (1998) 2277. of the Workshop on Uniﬁed Theories and Baryon Number in [22] X.A. Siemens, T.W.B. Kibble, Nucl. Phys. B 438 (1995) 307. the Universe, Tsukuba, Japan, 1979, KEK Report No. 79-18; [23] M. Landriau, E.P.S. Shellard, Phys. Rev. D 69 (2004) 023003. R.N. Mohapatra, G. Senjanovic, Phys. Rev. Lett. 44 (1980) [24] P. Sreekumar, et al., Astrophys. J. 494 (1998) 523. 912; [25] P. Bhattacharjee, G. Sigl, Phys. Rep. 327 (2000) 109. S.L. Glashow, Caraese lectures, 1979, unpublished. [26] J.A. Harvey, M.S. Turner, Phys. Rev. D 42 (1990) 3344. [6] N.S. Manton, Phys. Rev. D 28 (1983) 2019; [27] S. Davidson, A. Ibarra, Phys. Lett. B 535 (2002) 25; F.R. Klinkhamer, N.S. Manton, Phys. Rev. D 30 (1984) 2212; W. Buchmüller, P. Di Bari, M. Plümacher, Nucl. Phys. B 643 V.A. Kuzmin, V.A. Rubakov, M.E. Shaposhnikov, Phys. Lett. (2002) 367. B 155 (1985) 36. [28] E.W. Kolb, M.S. Turner, The Early Universe, Addison– [7] T.W.B. Kibble, J. Phys. A 9 (1976) 1387. Wesley, Redwood City, 1990; [8] T.W.B. Kibble, G. Lazarides, Q. Shaﬁ, Phys. Lett. B 113 (1982) H.B. Nielsen, Y. Takanishi, Phys. Lett. B 507 (2001) 241. 237. [29] W. Buchmüller, P. Di Bari, M. Plümacher, Nucl. Phys. B 665 [9] R. Jackiw, P. Rossi, Nucl. Phys. B 190 (1981) 681. (2003) 445. [10] E.J. Weinberg, Phys. Rev. D 24 (1981) 2669. [11] P. Bhattacharjee, T.W.B. Kibble, N. Turok, Phys. Lett. B 119 (1982) 95. Physics Letters B 619 (2005) 233–239 www.elsevier.com/locate/physletb

Q-ball instability due to U(1) breaking

Masahiro Kawasaki, Kenichiro Konya, Fuminobu Takahashi

Institute for Cosmic Ray Research, University of Tokyo, Kashiwa 277-8582, Japan Received 19 April 2005; received in revised form 25 May 2005; accepted 31 May 2005 Available online 13 June 2005 Editor: T. Yanagida

Abstract Q-ball is a nontopological soliton whose stability is ensured by global U(1) symmetry. We study a Q-ball which arises in the Afﬂeck–Dine mechanism for baryogenesis and consider its possible instability due to U(1) breaking term (A-term) indispensable for successful baryogenesis. It is found that the instability destroys the Q-ball if its growth rate exceeds inverse of the typical relaxation time scale of the Q-ball. However, the instability is not so strong as it obstructs the cosmological formation of the Q-balls.  2005 Elsevier B.V. All rights reserved.

1. Introduction number is stable against decay into protons in theories based on the gauge-mediated SUSY breaking. The Q-ball is a nontopological soliton that arises in Therefore, such stable Q-ball can be a promising can- scalar field theory with some global U(1) symmetry didate for dark matter as well as the source of the [1]. The Q-ball solution naturally exists in the mini- baryon number of the universe, which makes the Q- mal supersymmetric (SUSY) standard model (MSSM) balls very attractive [9–12]. Q-balls in the gravity- [2–5], especially in the context of the Affleck–Dine mediated SUSY breaking models, on the other hand, (AD) baryogenesis [6] where the MSSM flat direc- are unstable and decay into the standard particles, pro- tions play important roles in baryon number genera- ducing the lightest SUSY particles (LSPs). Then it is tion. In this case the Q-ball consists of squarks and possible that the Q-balls account for the dark matter = sleptons, therefore carrying baryonic and/or leptonic ( LSPs) and the baryon asymmetry of the universe charges. It was shown that Q-balls with large baryon simultaneously [5]. For further applications of Q-balls number are actually produced in the early universe in and their variant [13], see, e.g., Refs. [14–17]. Refs. [7,8]. Furthermore, a Q-ball with large baryon In the AD mechanism we need the U(1)B(L) breaking terms for successful generation of the baryon (lepton) number. Here U(1)B(L) is the global symmetry E-mail address: [email protected] (F. Takahashi). associated with baryon (lepton) number, and in the fol-

0370-2693/$ – see front matter  2005 Elsevier B.V. All rights reserved. doi:10.1016/j.physletb.2005.05.082 234 M. Kawasaki et al. / Physics Letters B 619 (2005) 233–239 lowing we drop the subscript B(L), since the distinc- fluctuation: Φ = Φ¯ + δΦ. The equations of motion are tion makes no difference to the following discussion. | ¯ |2 If Q-balls are not formed, the U(1) breaking terms can ¨¯ + ˙¯ + 2 + + Φ ¯ Φ 3HΦ mΦ 1 K K log Φ be neglected soon after the baryon number is gener- M2 ated. This is because the cosmic expansion decreases Φ¯ ∗d−1 + Am = , the amplitude of the AD field and the U(1) breaking 3/2 d−3 0 (2) M∗ terms become much smaller than the U(1) conserving ones. However, once the Q-balls are formed, since the for the homogeneous mode, and amplitude of the AD field inside the Q-ball is fixed it k2 δΦ δΦ∗ becomes nontrivial whether the U(1) breaking terms δΦ¨ + 3HδΦ˙ + δΦ + Km2 Φ¯ + a2 Φ Φ¯ Φ¯ ∗ are truly negligible or not. Therefore, in the present Letter, we study the possible instability due to the |Φ¯ |2 + m2 1 + K + K log δΦ U(1) breaking A-terms and their effect on the Q-ball Φ M2 stability. It is found that the A-term induces instability ¯ ∗d−2 Φ ∗ similar to that of the parametric resonance and it up- + A(d − )m δΦ = , 1 3/2 d−3 0 (3) sets the stability of the Q-ball if the growth rate of the M∗ induced instability is larger than the inverse of the re- for the fluctuation with the wavenumber k in the mo- laxation time of the Q-ball configuration. However, in mentum space. Although we have introduced the Hub- the realistic cosmological situation, the A-term inside ble parameter H and the scale factor a in the above the Q-ball is not so strong to cause the strong insta- equations, we will neglect the cosmic expansion for bility, hence the previous studies which assumed the the moment. Q-ball stability remain valid. In order to parametrize the strength of the instability, let us define ξ as the ratio of the A-term to the mass term: 2. Linear analysis on instabilities d−2 |A|m3/2Φ ξ ≡ 2 0 , (4) 2 d−3 First let us consider the instabilities due to U(1) dmΦ M∗ breaking term in the homogeneous background. To where Φ0 is the maximal magnitude of Φ during the this end we perform linear analysis assuming small course of the oscillation in the limit of ξ → 0 and we perturbations. The potential of the AD field Φ is writ- will set M = Φ0. Note that the amplitude of the oscil- ten as lation in the limit of ξ → 0 is constant if we neglect |Φ|2 the cosmic expansion. We have numerically solved V(Φ)= m2 1 + K log |Φ|2 ¯ Φ M2 Eqs. (2) and (3) by decomposing Φ and δΦ into their real and imaginary components: Φd |Φ|2d−2 + Am + . + , 3/2 d−3 h.c 2d−6 (1) φ1 + iφ2 dM∗ M∗ Φ¯ = √ , 2 where mΦ is the mass of the AD field, K a numerical δφ1 + iδφ2 coefficient of the one-loop correction, m3/2 the grav- δΦ = √ . (5) itino mass, M the renormalization scale, and M∗ some 2 cut-off scale for the nonrenormalizable operator. Also For convenience we√ also define the polar decompo- A isassumedtobeaO(1) real parameter, and K is as- sition: Φ¯ = φeiθ/ 2. We have normalized the di- sumed to be negative. One can see that the second term mensionless variables as ϕi = φi/mΦ , δϕi = δφi/mΦ (called A-term) breaks the U(1) symmetry. In the fol- (i = 1, 2), κ = k/mΦ , τ = mΦ t, and χi = mΦ xi (i = lowing argument we can safely neglect the last term 1,...,D), where D is the spatial dimension. Further- since the amplitude of the AD filed is relatively small. more we set K = 0 in order to concentrate on the insta- In order to study the instabilities of the AD field we bility that originates from the A-term. Fig. 1 shows the first divide the AD field into homogeneous part and instability bands for d = 4 and ξ = 0.1 with the initial M. Kawasaki et al. / Physics Letters B 619 (2005) 233–239 235

Fig. 1. Instability bands due to the A-term for d = 4, ξ = 0.1and Fig. 3. Same as Fig. 1 but the elliptic orbit. circular orbit.

Fig. 2. Same as Fig. 1 but d = 6. Fig. 4. Same as Fig. 2 but the elliptic orbit. condition θ = π/8 and θ = 1, where the prime denotes the differentiation with respect to τ . Notice that with the above initial values the orbit of the homogeneous mode is close to a circle in the (φ1,φ2) plane. From the figure one can see that there exist instability bands similar to those of the parametric resonance; the strongest instability band is located at κ = 0–0.8 and several weaker ones at shorter wavelengths. We also show the result for d = 6 and ξ = 0.1 with the initial condition θ = π/12 and θ = 1inFig. 2. In this case the strongest instability is at larger κ around 1.7. In Figs. 3 and 4 the instability bands are shown Fig. 5. ξ-dependence of the growth rate of the most amplified mode, + × = = for the elliptic orbit of the AD field. We have cho- Γg.The ’s and ’s are the numerical results for d 4andd 6, respectively. The solid line denotes Γg/m 3.5ξ, while the dot- sen the initial conditions as θ = π/8 and θ = 0.01 Φ ted one denotes Γg/mΦ 2.0ξ. for d = 4 and θ = π/12 and θ = 0.01 for d = 6, respectively. One can see that the instability bands are ∝ located at different wavenumber, and that the instabil- strongest instability band grows as exp(Γgt).The ities are weaker and bands narrower, compared to the growth rate depends on ξ and we find that Γg is pro- cases of the circular orbit. portional to ξ as shown in Fig. 5. Thus we obtain In order to express the strength of the strongest in- Γ /m = γ ξ, (6) stability in the case of the circular orbit, we introduce g Φ d the growth rate Γg; the fluctuation at the peak of the with γ4 ≈ 2.0 and γ6 ≈ 3.5. 236 M. Kawasaki et al. / Physics Letters B 619 (2005) 233–239

Before closing this section, let us consider the ef- Table 1 fect of the cosmic expansion on the instabilities. When The critical values ξc for d = 4 the oscillation of the AD field starts, the Hubble para- Criterion K =−0.01 K =−0.03 K =−0.1 K =−0.3 − − − − meter H is comparable to mΦ . Hence the growth rate A4× 10 3 9 × 10 3 2.8 × 10 2 6.4 × 10 2 − − − − of the A-term instability is comparable to H only if B3× 10 3 9 × 10 3 2.6 × 10 2 5.5 × 10 2 ξ ∼ O(1). However, since the amplitude of the AD field decreases as a−3/2, ξ quickly becomes smaller − − (∝ a 3(d 2)/2) and the growth rate is soon overcome Table 2 by the cosmic expansion. Therefore, it seems that the The critical values ξc for d = 6 A-term instability does not play an important role Criterion K =−0.01 K =−0.03 K =−0.1 K =−0.3 − − − − at least in the Q-ball formation. However, it is still A2.2 × 10 2 3.5 × 10 2 2.3 × 10 2 1.2 × 10 2 − − − − nontrivial how the instability affects the evolution of B2.1 × 10 2 3.5 × 10 2 2.3 × 10 2 1.2 × 10 2 Q-balls after they are formed, since the amplitude inside them is fixed. In√ the numerical calculations we have set M = φ(0)/ 2 and investigated several values of K be- 3. Instabilities inside Q-ball tween −0.01 and −0.3 for both d = 4 and d = 6, varying ξ. For the Q-ball configuration, ξ is evaluated at In the previous section we have found that the the center of the Q-ball; Φ0 in Eq. (4) should be inter- A-term causes the instabilities in the homogeneous preted as Φc, the field value at the center. Also we have − motion of the AD field. Next let us investigate the in- added small initial seed for fluctuations of O(10 5). stabilities due to the A-term inside the Q-ball. Without The size of the lattices is 2048 and 200 × 200 for 1D the A-term, the Q-ball solution for the potential (1) is and 2D, respectively. A number of simulations with obtained with use of the Gaussian ansatz [5]: varying values of the lattice spacing and box size have been done to ensure that the exact values of the lattice 1 Φ(t,r) = √ eiωtφ(r), (7) parameters do not affect the results. 2 Then we have found that the Q-ball breaks up for − 2 2 − φ(r)= φ(0)e r /R , (8) ξ larger than O(10 2) (see Tables 1 and 2 for the precise values). Figs. 6–8 show how the Q-balls are where the Q-ball radius R and angular velocity ω are dispersed by the A-term instabilities. It is worth not- given by ing that the Q-ball is actually stable for small enough 2 values of ξ. The critical values (ξc) above which the R2 = , (9) m2 |K| Q-ball becomes unstable due to the A-term instabil- Φ ities for several values of K are shown in Tables 1 2 = 2 + | | ω mΦ 1 2 K . (10) and 2, for 1D lattice calculation. We have followed the = 4 To study the instabilities due to the A-term inside the evolution until τ 10 and then decided whether the Q-ball, we have numerically solved the evolution of Q-ball configuration is lost or not on the basis of the the AD field on one- and two-dimensional lattices. The following criterion; if the total charge of the Q-ball is equation of motion for the potential (1) is less than 10% (50%) of the initial value, we judge that the Q-ball is dispersed. We call this criterion A (B). |Φ|2 The critical value ξ is found to depend on the val- Φ¨ −∇2Φ + m2 1 + K + K log Φ c Φ M2 ues of both d and K to a certain degree. The depen- Φ∗d−1 dence of ξc on d comes from the fact the instability + Am = . 3/2 d−3 0 (11) bands have different structures between the cases of M∗ d = 4 and d = 6 (see Figs. 1 and 2). The dependence We have neglected the expansion of the universe since on the other variable, K, possibly comes from the fol- the Q-balls decouple from the cosmic expansion once lowing two effects; (i) the Q-ball is more stable for they are formed. larger |K|; (ii) the Q-ball is more tolerant to pertur- M. Kawasaki et al. / Physics Letters B 619 (2005) 233–239 237

Fig. 6. The typical breakdown of the Q-ball due to the A-term for Fig. 7. Same as Fig. 6 but d = 6. τ = 0, 3.0 × 103 and 5.0 × 103 d = 4on1+ 1 lattices. τ = 0, 103 and 104 from top to bottom. The −3 from top to bottom. The adopted parameters are K =−0.04 and adopted parameters are K =−0.01 and ξ = 8 × 10 . − ξ = 4.5 × 10 2. bations if its charge is larger, and the charge becomes the mass of the AD field, mΦ , possibly multiplied by larger for smaller |K| with the amplitude φ(0) fixed some powers of |K|. On the other hand, the growth because of the dependence of the radius on |K| (see rate calculated from Eq. (6) is about 0.1mΦ for the Eq. (9)). Such a naive reasoning might explain the be- critical values of ξ. So, the relaxation time scale of the −1 havior of ξc especially in Table 2. Q-ball should be around 10mφ . This relaxation time Apart from the detailed dependence, can we tell a scale is in an agreement with Ref. [18] where the ex- −2 rough value of ξc? Put another way, is ξc ∼ O(10 ) cited Q-balls were studied by numerical simulations. reasonable? It is conceivable that ξc is determined by Therefore it is probable that the instability due to the the competition between the growth of the instabilities A-term destroy the Q-ball if its growth rate exceeds and the relaxation of the Q-ball configuration. Since the inverse of the typical relaxation time scale of the the Q-ball configuration minimizes the energy of the Q-ball. system in the limit of ξ → 0, the Q-ball tends to keep Lastly let us comment on the realistic value of ξ its configuration for a certain range of ξ. It is expected at the Q-ball formation. When the AD field starts os- that the relaxation time scale of the Q-ball is set by cillating, ξ should be order unity if m3/2 mΦ and 238 M. Kawasaki et al. / Physics Letters B 619 (2005) 233–239

4. Conclusion

In this Letter we have investigated how the A-term affects the evolution of the AD ﬁeld, especially paying attention on the stability of the Q-balls. In the linear analysis we ﬁrst have found that there exist instability bands similar to those of parametric resonance. The growth rate of the instabilities, however, is not so large after the baryon asymmetry is generated. Thus the Q-ball formation in the expanding universe would not be disturbed by the presence of the instabilities. Next we have studied the stability of Q-balls and estimated the critical value of ξc, above which the Q-balls cannot stay stable, for several values of K. From our result it is conceivable that ξc is determined by the competition between the growth rate and the relaxation rate of the Q-balls. It should be also noted that the obtained ξc is rather large in the realistic cosmological situations. Therefore the extensive studies on Q-balls thus far should remain valid, since the realistic value of ξ should be smaller than ξc. On the other hand, the instability found in the present Letter may be important when Q-balls grow up by absorbing U(1) charge such as solitosynthesis [19], especially in the case of d = 4.

Acknowledgement

F.T. would like to thank the Japan Society for Pro- motion of Science for ﬁnancial support.

Fig. 8. The typical breakdown of the Q-ball due to the A-term for d = 6. τ = 0, 1.0 × 103 and 1.5 × 103 from top to bottom. The − parameters we adopted are K =−0.04 and ξ = 4.0 × 10 2. References

[1] S.R. Coleman, Nucl. Phys. B 262 (1985) 263; S.R. Coleman, Nucl. Phys. B 269 (1986) 744, Erratum. A O(1). Then the typical value of ξ is given by [2] A. Kusenko, Phys. Lett. B 405 (1997) 108, hep-ph/9704273. d−2 ξ ∼ (Φc/Φosc) , where Φosc is the amplitude of [3] G.R. Dvali, A. Kusenko, M.E. Shaposhnikov, Phys. Lett. B 417 the AD ﬁeld at the onset of oscillation. According (1998) 99, hep-ph/9707423. to the numerical calculations (see, e.g., Ref. [11]), [4] K. Enqvist, J. McDonald, Phys. Lett. B 425 (1998) 309, hep- × −3 | | 3/4 ph/9711514. Φc/Φosc 6 10 ( K /0.1) . Thus the realistic [5] K. Enqvist, J. McDonald, Nucl. Phys. B 538 (1999) 321, hep- value of ξ is much smaller than ξc: ph/9803380. [6] I. Afﬂeck, M. Dine, Nucl. Phys. B 249 (1985) 361; M. Dine, L. Randall, S. Thomas, Nucl. Phys. B 458 (1996) 291, −4 |K| 3/2 3 × 10 for d = 4, hep-ph/9507453. ξ ∼ 0.1 (12) −9 |K| 3 = [7] S. Kasuya, M. Kawasaki, Phys. Rev. D 61 (2000) 041301, hep- 10 0.1 for d 6. ph/9909509. [8] S. Kasuya, M. Kawasaki, Phys. Rev. D 62 (2000) 023512, hep- Note that ξ becomes smaller while d increases. ph/0002285. M. Kawasaki et al. / Physics Letters B 619 (2005) 233–239 239

[9] A. Kusenko, M.E. Shaposhnikov, Phys. Lett. B 418 (1998) 46, [15] M. Kawasaki, F. Takahashi, M. Yamaguchi, Phys. Rev. D 66 hep-ph/9709492. (2002) 043516, hep-ph/0205101. [10] S. Kasuya, M. Kawasaki, Phys. Rev. Lett. 85 (2000) 2677, hep- [16] K. Ichikawa, M. Kawasaki, F. Takahashi, Phys. Lett. B 597 ph/0006128. (2004) 1, astro-ph/0402522. [11] S. Kasuya, M. Kawasaki, Phys. Rev. D 64 (2001) 123515, hep- [17] M. Kawasaki, F. Takahashi, Phys. Rev. D 70 (2004) 043517, ph/0106119. hep-ph/0403199. [12] S. Kasuya, M. Kawasaki, F. Takahashi, Phys. Rev. D 68 (2003) [18] T. Multamaki, Phys. Lett. B 511 (2001) 92, hep-ph/0102339. 023501, hep-ph/0302154. [19] K. Griest, E.W. Kolb, Phys. Rev. D 40 (1989) 3231; [13] S. Kasuya, M. Kawasaki, F. Takahashi, Phys. Lett. B 559 A. Kusenko, Phys. Lett. B 406 (1997) 26, hep-ph/9705361; (2003) 99, hep-ph/0209358. M. Postma, Phys. Rev. D 65 (2002) 085035, hep-ph/0110199. [14] S. Kasuya, M. Kawasaki, F. Takahashi, Phys. Rev. D 65 (2002) 063509, hep-ph/0108171. Physics Letters B 619 (2005) 240–246 www.elsevier.com/locate/physletb

Strong suppression of nuclear-charge changing interactions for 18 TeV/c In ions channeled through a bent Si crystal

U.I. Uggerhøj a,H.D.Hansena,K.Jessena, H. Knudsen a, E. Uggerhøj a, C. Scheidenberger b, C. Biino c, M. Clément c,N.Doblec,K.Elsenerc, L. Gatignon c, P. Grafström c, P. Sona d,A.Mangiarottid, S. Ballestrero d

a Department of Physics and Astronomy, University of Aarhus, Denmark b GSI, Darmstadt, Germany c CERN, Geneva, Switzerland d University of Florence, Florence, Italy Received 18 February 2005; accepted 2 June 2005 Available online 13 June 2005 Editor: M. Doser

Abstract We present experimental results giving evidence for the strong reduction—a factor of more than 20—of nuclear-charge + changing interactions for 18 TeV In49 ions channeled through a silicon crystal bent to 7.5, 11.9 and 19.8 mrad. A very small fraction of the deﬂected ions suffer electromagnetic or nuclear interactions leading to proton loss while traversing the 60 mm long crystal, even though its thickness corresponds to about 0.13 nuclear interaction lengths for an amorphous material. By considering the deﬂected ions only, we show experimentally that the nuclear-charge pickup reaction believed to be induced by virtual photons is a short-range phenomenon.  2005 Elsevier B.V. All rights reserved.

PACS: 61.85.+p; 25.75.-q; 41.75.Ak; 41.85.Ja

Keywords: Channeling; Heavy ions; Nuclear interactions

1. Introduction heavy ions channeled and deﬂected in a bent crystal and (2) a nearly complete elimination under channel- In this Letter we report the observation of: (1) a ing conditions of the nuclear-charge pickup believed → − strong suppression of the proton-loss mechanism for to be a result of a γn pπ reaction by virtual photons [1,2]. Since its prediction [3] and the ﬁrst experiments in E-mail address: [email protected] (U.I. Uggerhøj). the late 1970s [4], the use of and knowledge about pro-

0370-2693/$ – see front matter  2005 Elsevier B.V. All rights reserved. doi:10.1016/j.physletb.2005.06.003 U.I. Uggerhøj et al. / Physics Letters B 619 (2005) 240–246 241 ton channeling in bent crystals has increased steadily tial in which the particle moves [13]. The particles to and is now at a well-established stage where almost all be channeled must be incident on the crystal within a aspects of the phenomenon have been explored. critical angle to the plane, ψp, which is of the order However, the investigation of effects pertaining to 10 µrad at (370 GeV/c)/Z. the use of high energy heavy ions in bent crystals is The guidance of channeled particles persists even if still evolving, with the first extensive studies appear- the crystal is slightly bent, such that the particle may ing at the turn of the millennium [5–7], following early be deviated from its original direction of motion as in experiments [8]. It turns out that, regarding deflection a dipole magnet. Since the fields that are responsible efficiencies and critical angles, heavy ions behave in for this deviation are the extremely strong (screened) bent crystals as protons of the same energy per charge, fields present near the lattice nuclei, the corresponding as expected. Nevertheless, since heavy, fully stripped bending power can reach a magnitude equivalent to a ions are composite particles of high charge a number magnetic field of several thousand Tesla. of additional effects may appear, such as electromag- In an earlier experiment [14] we investigated the re- netic dissociation and/or nuclear interactions. In the stricted energy loss of fully stripped Pb ions in silicon restframe of the incident ion with the Lorentz factor and showed that the well-channeled particles suffer an γ 160 the extremely strong, crystal electric fields energy loss of about 70% of the random value when E 1011 V/cm are boosted to very high values. It is channeled in a plane. thus not a priori excluded that electromagnetic disso- For a recent review of the deflection in bent crystals ciation for example through a giant dipole resonance as well as of its applications at high energy accelera- is significant [9–11]. The fundamental frequency in tors, see Ref. [15]. the ion restframe for interaction with the lattice is = ω0 2πγβc/d which is of the order 1 MeV for a 3. Experiment characteristic lattice spacing d 2Å. This Letter discusses the specific aspects of heavy The experiment was performed in the H2 beam line ion channeling in bent crystals and in particular the of the SPS accelerator at CERN, where In49+ ions of strong suppression of nuclear charge pick-up and loss momentum 370 GeV/c per charge unit are available reactions under channeling conditions. The main mo- with a small divergence. The ions were incident on a tivation is to demonstrate experimentally what can be silicon crystal, 60 mm long in the beam direction, bent the expected composition of a crystal-extracted heavy horizontally over 56 mm in a ‘three-point bender’. The ion beam in a scheme like the seemingly feasible one crystal was mounted ona2µradstepsizegoniometer proposed for the LHC [12]. and alignment of the crystalline planes to the beam direction was performed by observation of the deflected beam in a scintillator counter. 2. Ion channeling in bent crystals The experimental setup is shown schematically in Fig. 1. Here, S1 denotes a scintillator that was used In the so-called continuum approximation [13], as an event trigger in conjunction with SL1 which is charged particles incident on a single crystal with a scintillator with a 0.2 mm wide slit used as a veto. small angles to crystallographic directions, experience To detect the charge state of each ion before the inter- the collective atomic fields as if smeared along the action, a MUltiple Sampling Ionization Chamber [16], string or plane. If, further, a particle has sufficiently MUSIC1, was used. After the passage of the crystal, low transverse momentum with respect to the axis or the resulting charge state was detected in a down- plane of the crystal it can be restricted to areas away stream chamber, MUSIC2. At the downstream end, from the nuclei (positively charged particles) or close two scintillators, S2 and S3 (with a 5 mm overlap) to the nuclei (negatively charged particles). In this case that can be used in coincidence, S2·S3, were mounted the particle is channeled and is guided by the lattice on a movable table. These scintillators were used dur- such that a separation of the longitudinal and trans- ing alignment of the crystal which was performed by verse motions is present. The result is a conserved detection of the deflected particles (denoted as D in ‘transverse energy’ and therefore a transverse poten- Fig. 1) as a function of the turn angle of the goniome- 242 U.I. Uggerhøj et al. / Physics Letters B 619 (2005) 240–246

Fig. 1. A schematical drawing of the experimental setup.

Table 1 An advantage of working with fully stripped heavy Deflection angles, measured beam divergence of the incoming beam ions as compared to, e.g., protons, is the possibility of and deflection efficiency a large background rejection. Since the energy loss is θ [mrad] σ(ϑ)[µrad] ε [%] roughly proportional to the square of the charge, an + 7.5 ± 0.334± 13.0 ± 0.2 In49 ion gives a signal in a scintillator which by far 11.9 ± 0.235± 12.0 ± 0.1 exceeds that of a minimum ionizing particle. Thus, ± ± ± 19.8 0.22670.4 0.1 measurements can be made essentially background- free. This is demonstrated in Fig. 2(b) for angles between 11.9 and 19 mrad, where the dechanneling is ter. The distance between MUSIC1 and MUSIC2 was expected to be negligible (in a three-point bending 10.9 m with air at atmospheric pressure, for the 19.8 device, the local curvature of the crystal decreases be- mrad data reduced to 6.6 m. Advantage was taken of yond the central support pin). the horizontal position information of MUSIC2, by In Fig. 3 is shown the high-Z part of a typical which it is possible to sort the deflected particles (D) charge spectrum in the downstream MUSIC2. Here and those that go straight through the crystal (S). This it was required by the upstream ionization chamber identification can be performed on an event-by-event (MUSIC1) that the incoming particle had such a po- basis in conjunction with the charge-state identifica- sition that it was incident on the crystal endface and tion. Once the crystal was aligned, overlap between the could be identified as In49+. The spectrum is fitted slit of SL1 and the 1.5 mm wide endface of the crystal with Gaussians plus a constant background, and the was confirmed by a comparison of (1) the beam profile resulting sum is shown to fit the data. All charge states in MUSIC1 with SL1 in the trigger, looking at all par- (typically down to Z 20) can be identified (for clar- ticles and (2) the beam profile in MUSIC1 using only ity only the high-Z part is shown). In particular the deflected particles. The area thus corresponding to the nuclear-charge pick-up reaction leading to Z = 50 can crystal endface was used as a condition in the data be extracted. It was carefully checked that tighter event analysis. Furthermore, the beam divergence over the selections in the upstream MUSIC1 did not lead to a 1.5 mm frontface was derived from angular scans with significant change of the results, neither for position goniometer step-size 7 µrad and Gaussian fits to the nor charge selections. Furthermore, for all data sets number of deflected particles as a function of angle. the expected Z2 dependence of the signal on charge The resulting RMS values for the angular divergence, state was verified by a parabolic fit to the Gaussian σ(ϑ) , after deconvolution of the (110) planar critical centroids versus Z. For the deflected ions the Gaussian angle, 6 µrad, are given in Table 1. The errors stated fit to extract the pick-up channel Sn50+ was required to are statistical only and the comparatively large uncer- be centered at the same distance from the In49+ peak tainty for the angular divergence of the 19.8 mrad data and with the same width. set reflects the low deflection efficiency. The areas extracted from the Gaussian fits normalized to the number of incoming In49+ were used 4. Results to determine the fragmentation probability, pf ,asa function of charge number. These fragmentation prob- In Fig. 2 is shown a scan S2·S3 across the unde- abilities for the bent aligned crystal were then com- flected, dechanneled and fully deflected beams for the pared to the results for the randomly oriented crys- bend angles (a) 11.9 and (b) 19.8 mrad. All three parts tal by the fragmentation probability suppression η = random − bckg.,r crystal − bckg.,c of the beam are clearly identifiable. (pf pf )/(pf pf ) with subtrac- U.I. Uggerhøj et al. / Physics Letters B 619 (2005) 240–246 243

Fig. 2. Scan of a scintillator coincidence across the undeﬂected, dechanneled and fully deﬂected beams for the bend angles (a) 11.9 and (b) 19.8 mrad. Data points with zero counts have been put at 0.1 to enable their display on a logarithmic scale.

Fig. 3. The high-Z part of a typical charge spectrum observed in the downstream MUSIC2 for the randomly oriented crystal. Gaussian fits plus a constant and the resulting sum are shown. The different elements can be clearly identified, for details, see text. The error bars indicate a 1σ statistical uncertainty. tion of the measured values without target. For the the background is accounted for in the numerator of deflected beam, the background is difficult to mea- bckg.,r = η, i.e., pf 0). For the direct beam, background sure as there are no particles deviated to this region. subtraction amounts to 50%. Moreover, the method We have thus taken the conservative approach of not to determine the position of the crystal on MUSIC1 subtracting the background from the crystal signal for is not useful for the randomly oriented target. This bckg.,c = this case, i.e., pf 0 in the denominator (but still means that in the case of a varying position signal, 244 U.I. Uggerhøj et al. / Physics Letters B 619 (2005) 240–246

Fig. 4. The fragmentation probability suppression η for nuclear-charge changing probability versus charge number for the direct beam (open triangles) and the bent beam (filled squares), both for a bend angle of 7.5 mrad. non-interacting ions may be included in the event se- account of background is therefore not an explanation lection which leads to a value of η that is artificially of the tendency. too small. Thus, the suppression factor—in particu- Another effect may be part of the explanation for lar for Z = 50 where the counts in the aligned case is the large suppressions observed for proton-loss: If the consistent with zero—is a lower limit. Along the same ion dechannels it will not be fully deflected and there- crystal lines it should be mentioned, that for the large bend an- fore not contribute to pf . For the pick-up reaction, gles, ions emerging in the straight beam may not have however, this is unlikely to be significant since an in- sampled the whole 60 mm of the crystal which could crease of charge leads to a deeper transverse potential − also lead to a value of η that is artificially too small. and the momentum of the π is too low (typically As seen in Fig. 4 there is a strong suppression of a much smaller than 0.1 GeV/c [17,18]) to lead to an factor up to 25 for the channeled ions for fragmenta- appreciable change of angle of the In49+ ion. tion compared to the non-channeled ions. The suppres- As a cross-check, the direct beam data sets are all sion becomes progressively smaller for lower charge consistent with a suppression factor of 1, indicating number, but even for the lightest fragments detected it that systematic effects such as background subtraction is a factor 10. Furthermore, the nuclear-charge pickup or charge dependent angular distribution of fragments channel is suppressed significantly more—by a factor are insignificant. In the case of the bent beam, how- of 40—showing that this type of reaction must be ever, the suppression factor includes the deflection ef- a short-range phenomenon. The overall behaviour of ficiency to some extent as a particle has to be fully η with Z is partly explained by the limited number of deflected to contribute. counts for the channeled beam for Z 46 and the ap- In Fig. 5 the suppression factors for the fragments bckg.,c = proximation pf 0. Given that the particle must and the pick-up channel are shown for the three deflec- be fully deflected to be counted, one would expect a tion angles and for the 7.5 mrad data set also for an higher suppression for low Z than for high Z, con- incidence angle ψp/2. As expected, the suppression trary to observations. For the undeflected beam, the factor becomes larger for larger bend angles roughly background is about 50% for high Z-values, dropping speaking because those ions that are guided all the slightly to about 35% for low Z. Using these numbers way through the bent crystal are necessarily well chan- to estimate the background for the deflected beam, η neled from the beginning. However, the indication that increases by about a factor 3, but the overall tendency Z 40 has a stronger suppression than, e.g., Z 45 for η to rise with Z gets slightly stronger. Insufficient for the 11.9 mrad data set is not understood. For the U.I. Uggerhøj et al. / Physics Letters B 619 (2005) 240–246 245

Fig. 5. The fragmentation probability suppression η versus charge number in the upper region of Z for the bent beam for angles 7.5 mrad (ﬁlled squares: perfect alignment, open triangles: aligned at ψp/2), 11.9 mrad (open circles) and 19.8 mrad (ﬁlled triangles).

7.5 mrad deflection angle, η(Z) is slightly larger for 5. Conclusion ψ = ψp/2 than for ψ = 0 in the cases of Z = 47 and Z = 48. This is in contrast to smaller values of We have shown experimentally a large suppres- Z and is not expected. The scarcity of data points sion of the probability for proton-loss during de- for the 19.8 mrad deflection angle is due to the small flection of fully stripped relativistic heavy ions in deflection efficiency which precludes identification of a bent crystal. Furthermore, we have presented de- fragments below Z = 44. flection efficiencies for these ions for angles up to The fragmentation probability suppression η for 19.8 mrad. The results are promising for a potential + In49 to remain in its initial charge state is −12 and application of the deflection and extraction of rela- −19 for the 7.5 and 11.9 mrad data sets. Taking tivistic ions from an accelerator, as the large major- random bckg. pf 0.5pf this corresponds to an increase ity will exit in the same charge state as they enter of the effective interaction length by a factor 12 and the crystal. For such an application we note how- 19, respectively. Therefore, for the channeled ions the ever that the question of radiation damage to the crystal only presents 1.0% and 0.6% of an interaction crystal remains experimentally unaddressed in the length instead of the random value 13%, i.e., the large case of heavy ions. Finally, we have presented ev- majority of heavy ions exit in the same charge state idence for the almost complete absence of nuclear- as they enter the crystal even in a crystal as long as charge pickup under channeling conditions, corrob- 60 mm. We emphasize that the suppression factor is orating earlier results in amorphous targets which not the suppression for the channeling phenomenon as show that this is a close-encounter electromagnetic such, but the suppression under the condition that the process [1]. ions are deflected through the full bend angle. This is the figure of merit for the motivation discussed in the introduction, i.e., a potential extraction scheme based Acknowledgements on a bent crystal. We have obtained similar results for the fragmen- We acknowledge the strong support from P.B. tation probabilities in amorphous Pb, W, Sn and Ge Christensen and P. Aggerholm in setting up the data- targets and these will be published later [19]. taking system and the MUSIC detectors as well as 246 U.I. Uggerhøj et al. / Physics Letters B 619 (2005) 240–246 support from the Danish Natural Science Research [8] L.I. Bel’zer, et al., JETP Lett. 46 (1987) 382. Council. [9] R. Fusina, J.C. Kimball, Nucl. Instrum. Methods B 27 (1995) 368. [10] Yu.L. Pivovarov, A.A. Shirokov, S.A. Vorobiev, Nucl. Phys. A 09 (1990) 800. References [11] A.V. Stepanov, Phys. At. Nucl. 58 (1995) 2052. [12] E. Uggerhøj, U.I. Uggerhøj, Nucl. Instrum. Methods B 234 [1] C. Scheidenberger, et al., Phys. Rev. Lett. 88 (2002) 042301. (2005) 31. [2] C. Scheidenberger, et al., Phys. Rev. C 70 (2004) 014902. [13] J. Lindhard, Mat. Fys. Medd. Dan. Vid. Selsk. 34 (1965) 1. [3] E.N. Tsyganov, Fermilab TM-682, TM-684, Batavia (1976), [14] S.P. Møller, et al., Phys. Rev. A 64 (2001) 032902. unpublished. [15] A. Baurichter, et al., Nucl. Instrum. Methods B 164–165 (2000) [4] A.S. Vodop’yanov, et al., Sov. Phys. JETP Lett. 30 (1979) 442; 27. A.F. Elishev, et al., Phys. Lett. B 88 (1979) 387. [16] M. Pfützner, et al., Nucl. Instrum. Methods B 86 (1994) 213. [5] G. Arduini, et al., Phys. Rev. Lett. 79 (1997) 4182. [17] I.A. Pshenichnov, et al., Phys. Rev. C 60 (1999) 044901. [6] C. Biino, et al., Nucl. Instrum. Methods B 194 (2002) 417. [18] K.A. Chikin, et al., Eur. Phys. J. A 8 (2000) 537. [7] C. Biino, et al., Nucl. Instrum. Methods B 160 (2000) 536. [19] U.I. Uggerhøj, et al., in preparation. Physics Letters B 619 (2005) 247–254 www.elsevier.com/locate/physletb

Partial wave analysis of ψ → π +π −π 0 at BESII BES Collaboration M. Ablikim a,J.Z.Baia,Y.Banj, J.G. Bian a,X.Caia,J.F.Changa,H.F.Chenp, H.S. Chen a,H.X.Chena,J.C.Chena,JinChena,JunChenf,M.L.Chena,Y.B.Chena, S.P. Chi b,Y.P.Chua,X.Z.Cuia,H.L.Daia,Y.S.Dair,Z.Y.Denga, L.Y. Dong a, S.X. Du a,Z.Z.Dua,J.Fanga,S.S.Fangb,C.D.Fua,H.Y.Fua, C.S. Gao a,Y.N.Gaon, M.Y. Gong a,W.X.Gonga,S.D.Gua,Y.N.Guoa,Y.Q.Guoa,Z.J.Guoo, F.A. Harris o, K.L. He a,M.Hek,X.Hea, Y.K. Heng a,H.M.Hua,T.Hua,G.S.Huanga,1,L.Huangf, X.P. Huang a,X.B.Jia,Q.Y.Jiaj,C.H.Jianga,X.S.Jianga,D.P.Jina,S.Jina,Y.Jina, Y. F. L ai a,F.Lia,G.Lia,H.H.Lia,J.Lia,J.C.Lia,Q.J.Lia,R.B.Lia,R.Y.Lia, S.M. Li a,W.G.Lia,X.L.Lig,X.Q.Lii,X.S.Lin,Y.F.Liangm,H.B.Liaoe,C.X.Liua, F. Liu e,FangLiup,H.M.Liua,J.B.Liua,J.P.Liuq,R.G.Liua,Z.A.Liua,Z.X.Liua, F. Lu a,G.R.Lud,J.G.Lua,C.L.Luoh,X.L.Luoa,F.C.Mag,J.M.Maa,L.L.Mak, Q.M. Ma a,X.Y.Maa,Z.P.Maoa,X.H.Moa,J.Niea,Z.D.Niea,S.L.Olseno, H.P. Peng p,N.D.Qia,C.D.Qianl,H.Qinh,J.F.Qiua,Z.Y.Rena,G.Ronga, L.Y. Shan a, L. Shang a,D.L.Shena,X.Y.Shena, H.Y. Sheng a,F.Shia,X.Shij, H.S. Sun a,S.S.Sunp,Y.Z.Suna,Z.J.Suna,X.Tanga,N.Taop, Y.R. Tian n, G.L. Tong a,G.S.Varnero,D.Y.Wanga,J.Z.Wanga,K.Wangp,L.Wanga,L.S.Wanga, M. Wang a,P.Wanga,P.L.Wanga,S.Z.Wanga,W.F.Wanga,Y.F.Wanga, Zhe Wang a, Z. Wang a, Zheng Wang b,Z.Y.Wanga,C.L.Weia,D.H.Weic,N.Wua,Y.M.Wua, X.M. Xia a,X.X.Xiea,B.Xing,G.F.Xua,H.Xua,Y.Xua,S.T.Xuea,M.L.Yanp, F. Yang i,H.X.Yanga,J.Yangp,S.D.Yanga,Y.X.Yangc,M.Yea,M.H.Yeb,Y.X.Yep, L.H. Yi f,Z.Y.Yia,C.S.Yua,G.W.Yua,C.Z.Yuana,J.M.Yuana,Y.Yuana,Q.Yuea, S.L. Zang a,YuZenga,Y.Zengf, B.X. Zhang a,B.Y.Zhanga,C.C.Zhanga, D.H. Zhang a, H.Y. Zhang a, J. Zhang a, J.Y. Zhang a, J.W. Zhang a, L.S. Zhang a, Q.J. Zhang a,S.Q.Zhanga, X.M. Zhang a, X.Y. Zhang k,Y.J.Zhangj, Y.Y. Zhang a, Yiyun Zhang m, Z.P. Zhang p,Z.Q.Zhangd,D.X.Zhaoa,J.B.Zhaoa,J.W.Zhaoa, M.G. Zhao i,P.P.Zhaoa,W.R.Zhaoa,X.J.Zhaoa,Y.B.Zhaoa,Z.G.Zhaoa,2, H.Q. Zheng j, J.P. Zheng a, L.S. Zheng a,Z.P.Zhenga, X.C. Zhong a,B.Q.Zhoua,

G.M. Zhou a, L. Zhou a,N.F.Zhoua,K.J.Zhua,Q.M.Zhua,Y.C.Zhua,Y.S.Zhua, Yingchun Zhu a,Z.A.Zhua, B.A. Zhuang a,B.S.Zoua

Abstract + − The decay ψ → π π π0 is analyzed using a sample of 14 million ψ events taken with the BESII detector at the BEPC, + − − and the branching fraction is measured to be B(ψ → π π π0) = (18.1 ± 1.8 ± 1.9) × 10 5. A partial wave analysis is carried out using the helicity amplitude method. ψ → ρ(770)π is observed, and the branching fraction is measured to be − B(ψ → ρ(770)π) = (5.1 ± 0.7 ± 1.1) × 10 5, where the ﬁrst error is statistical and the second one is systematic. A high mass enhancement with mass around 2.15 GeV/c2 is also observed. Attributing this enhancement to the ρ(2150) resonance, B → → + − 0 = ± +11.5 × −5 the branching fraction is measured to be (ψ ρ(2150)π π π π ) (19.4 2.5−3.4 ) 10 . The results will help in the understanding of the longstanding “ρπ puzzle” between J/ψ and ψ hadronic decays.  2005 Elsevier B.V. All rights reserved.

From perturbative QCD (pQCD), it is expected that yields the pQCD “12% rule” both J/ψ and ψ decaying into light hadrons are dom- B B + − inated by the annihilation of cc¯ into three gluons or ψ →h ψ →e e Qh = = ≈ 12%. (1) one virtual photon, with a width proportional to the BJ/ψ→h BJ/ψ→e+e− square of the wave function at the origin [1].This A large violation of this rule was ﬁrst observed in decays to ρπ and K∗+K− + c.c. by Mark II [2], known as the ρπ puzzle, where only upper limits on the branching fractions were reported in ψ decays. Since E-mail address: [email protected] (Zheng Wang). then, many two-body decay modes of the ψ have been 1 Current address: Purdue University, West Lafayette, IN 47907, USA. measured by the BES Collaboration and recently by 2 Current address: University of Michigan, Ann Arbor, MI the CLEO Collaboration; some decays obey the rule 48109, USA. while others violate it [3,4]. BES Collaboration / Physics Letters B 619 (2005) 247–254 249

In the study of the ρπ puzzle, ψ → ρπ is a key de- (PWA). Monte Carlo samples of Bhabha, dimuon, cay mode and is of great interest to both theorists and and inclusive hadronic events generated with Lund- experimentalists. Many theoretical attempts, using, for charm [11] are used for background studies. The instance, intermediate vector glueballs, hadronic form simulation of the detector uses a Geant3 [12] based factors, final state interactions, etc., have been made program, which simulates the detector response, in- to solve the puzzle [5]. A recent calculation of the cluding the interactions of secondary particles with ψ → ρπ branching fraction, done in the framework the detector material. Reasonable agreement between of SU(3) symmetry and taking into consideration in- data and Monte Carlo simulation has been observed terference between ψ resonance decay and the con- in various channels tested [13], including e+e− → tinuum amplitude, predicts a branching fraction of (γ )e+e−, e+e− → (γ )µ+µ−, J/ψ → pp¯, and ψ → ψ → ρπ around 1 × 10−4 in Ref. [6] where the rela- J/ψπ+π −,J/ψ → +− ( = e,µ). tive phase between ψ strong and electromagnetic de- The final state of interest includes two charged pi- cay amplitudes is taken as −90◦. The measurement of ons and one neutral pion which is reconstructed from the ψ → ρπ mode is a direct test of the many models two photons. The candidate events must satisfy the fol- proposed to solve the ρπ puzzle [5,6]. lowing selection criteria: The data used for this analysis are taken with the Beijing Spectrometer (BESII) detector at the Beijing (1) A neutral cluster is considered to be a photon can- Electron Positron Collider (BEPC) storage ring op- didate when the deposited energy in the BSC is erating at the ψ energy. The number of ψ events greater than 80 MeV, the angle between the near- is 14 ± 0.6 million [7], determined from the number est track and the cluster is greater than 16◦,the of inclusive hadrons, and the luminosity is (19.72 ± first hit of the cluster is in the beginning six radi- 0.86) pb−1 [8] as measured using large angle Bhabha ation lengths of the BSC, and the angle between events. the cluster development direction in the BSC and BESII is a conventional solenoidal magnet detec- the photon emission direction is less than 37◦, tor that is described in detail in Refs. [9,10]. A 12- and the angle between two nearest photons is re- layer vertex chamber (VC) surrounding the beam pipe quired to be larger than 7◦. The number of pho- provides coordinate and trigger information. A forty- ton candidates after selection is required to be layer main drift chamber (MDC), located radially two. outside the VC, provides trajectory and energy loss (2) There are two tracks in the MDC with net charge (dE/dx) information for tracks over 85% of the to- zero. A track must have a good helix fit and sat- = | | tal solid angle. The momentum resolution is σp/p isfy cos θ < 0.80, where θ is the polar angle of 0.017 1 + p2 (p in GeV/c), and the dE/dx res- the track in the MDC. olution for hadron tracks is ∼ 8%.Anarrayof48 (3) For each track, the TOF and dE/dx measure- scintillation counters surrounding the MDC measures ments are used to calculate χ2 values and the cor- the time-of-flight (TOF) of tracks with a resolution of responding confidence levels for the hypotheses ∼ 200 ps for hadrons. Radially outside the TOF sys- that the particle is a pion, kaon, or proton (Probπ , tem is a 12 radiation length, lead-gas barrel shower ProbK , Probp). At least one track is required counter (BSC). This measures the energies of elec- to satisfy Probπ > ProbK and Probπ > Probp. trons and photons over ∼ 80% of the total solid√ angle Radiative Bhabha background is removed by re- with an energy resolution of σE/E = 22%/ E (E in quiring the tracks have small dE/dx or small GeV). Outside of the solenoidal coil, which provides energy deposited in the BSC. Dimuon background a 0.4 Tesla magnetic field over the tracking volume, is removed using the hit information in the muon is an iron flux return that is instrumented with three counter. double layers of counters that identify muons of mo- (4) A four-constraint kinematic fit is performed under + − mentum greater than 0.5 GeV/c. the hypothesis ψ → γγπ π , and the confi- A phase space Monte Carlo sample of 2 mil- dence level of the fit is required to be greater than lion ψ → π +π −π 0 events is generated for the ef- 1%. A four-constraint kinematic fit is also per- ficiency determination in the partial wave analysis formed under the hypothesis of ψ → γγK+K−, 250 BES Collaboration / Physics Letters B 619 (2005) 247–254

Fig. 1. Two photon invariant mass distribution after ﬁnal selection for (a) ψ data and (b) continuum data. The histograms are data, and the curves show the best ﬁts.

+ − Fig. 2. Dalitz plots of π π π0 for (a) ψ data and (b) continuum data after the ﬁnal selection.

2 2 2 and χγγππ <χγγKK is required to remove ties times a factor to account for the 1/s dependence K+K−π 0 events. of the cross section. This yields 229 ± 23 observed + − 0 (5) To remove background produced by ψ decays to ψ → π π π events. + − 0 γγJ/ψ and π 0π 0J/ψ with J/ψ → π +π − or Dalitz plots of the π π π system for the ψ and J/ψ → µ+µ−, where the muons are misiden- continuum data are shown in Fig. 2 after requiring tified as pions, the invariant mass of π +π − is that the invariant mass of the two photons lie within 2 0 required to be less than 2.95 GeV/c2. ±30 MeV/c of the nominal π mass. (The mass resolution from Monte Carlo simulation is 17.5 MeV/c2.) After applying the above selection criteria, the in- For the ψ sample, 250 events are obtained with 13% 0 variant mass distribution of the two photons is shown non-π background, while for the continuum sample, 0 in Fig. 1(a). A clear π 0 signal can be seen. A fit to the 11 events are obtained with 42% non-π background. 0 mass spectrum (shown in Fig. 1(a)) using a π 0 signal Here the fractions of non-π background are obtained from the π 0 mass sidebands as shown in Fig. 1.In shape determined from Monte Carlo simulation and a polynomial background yields 260 ± 19 π 0 events. ψ decays, besides clear ρ bands at the edges of the Dalitz plot, there is a prominent cluster of events in The contribution from the continuum is measured√ using (6.42 ± 0.24) pb−1 [8] of data taken at s = the center. This is very different than the Dalitz plot + − 0 3.65 GeV (“continuum data”). Fig. 1(b) shows the γγ for J/ψ → π π π decays [14], indicating different + − 0 invariant mass distribution and the fit. The number of decay dynamics between J/ψ and ψ → π π π . 0 ± The comparison of the ππ mass distribution be- π events from the fit (10.0 4.2) is subtracted incoherently from the number of π 0 events in the ψ data, tween the ψ data and the scaled continuum data is after normalizing by the ratio of the two luminosi- shown in Fig. 3(a). With the limited statistics, no BES Collaboration / Physics Letters B 619 (2005) 247–254 251

Fig. 3. (a) The comparison of the ππ mass distribution between the ψ data and the scaled continuum data (shaded histogram, including about 42% non-π0 background). (b) The comparison of the ππ mass distribution between the ψ data and the non-π0 background estimated by the Mπ0 sideband events (shaded histogram). Dots with error bars are ψ data. In these plots, the distributions for the three different dipion charge configurations are combined. clear structure can be seen for the continuum data in define the direction, and Figs. 3(a) or 2(b).Theππ mass distribution of the c 2 ±iφ 0 A± = B m sin θπ (cos φπ ± i cos θ sin φπ )e . non-π background, estimated using the Mπ0 side- 1 0 band events, is shown in Fig. 3(b). The non-π back- Here c = 0, +1, or −1 is the net charge of the dipion ground contribution for the ππ mass spectrum is ap- system, θ and φ are the polar and azimuthal angles proximately uniform. + − of the ρ in the e e system, θπ and φπ are the polar We now proceed to study the resonant substruc- and azimuthal angles of the designated pion in the ρ ture. Here, no continuum subtraction is made, and rest frame, and B(m2) describes the dependence of the the selected events are fitted in the helicity ampli- amplitude on the dipion mass m: tude formalism with an unbinned maximum likelihood method using MINUIT [15]. For the process BW (m2) + c eiβj BW (m2) 2 = ρ(770) j j j B m , + − ∗ − − 1 + c e e → γ → ρ 1 + π 0 j j − − 2 → π 0 + π 0 , where, BW(m ) is the Breit–Wigner form of the ρ(770) or its excited states. Here, the Gounaris– the intensity distribution dI for the final state is written Sakurai parameterization [16] is used; βj and cj are as the relative phase and the relative strength, respectively, between the excited ρ state j and the ρ(770). = | |2 +| |2 dI Ai Ci d(LIPS), Since the number of events is limited, the masses i=±1 and the widths of all states in the fit are fixed to their PDG values [17], and the number of background where C is an incoherent non-π 0 background term, i events is fixed to the number determined from the that is assumed to be either a constant or to have the γγ invariant mass fit. A fit with ρ(770), ρ(1450), same angular distribution as A . The differences be- i ρ(1700) and ρ(2150) results in insignificant ρ(1450) tween these two fits, 7.3% and 1.4% for ρ(770) and and ρ(1700) contributions. The fit after removing ρ(2150) respectively, are taken as the systematic er- these two components yields a likelihood decrease of ror on the background description. LIPS denotes the 10.7 with four less free parameters. The fit results are Lorentz-invariant phase space, and the amplitude shown in Fig. 4; the fit describes the data reasonably 0 − + + + 0 − 0 − well. It is noted that the data do not determine the mass Ai = A π ,π + A π ,π + A π ,π , i i i and width of the high mass ρ;theρ(2150) serves as where i =+1or−1 is the helicity of the γ ∗, the first an effective description of the high mass enhancement pion in each set of parentheses is the one designated to near 2.15 GeV/c2 in ππ mass. 252 BES Collaboration / Physics Letters B 619 (2005) 247–254

plitudes allows a test of isospin symmetry in the three ρ(770)π modes; the fit yields the relative numbers of ρ(770)0π 0, ρ(770)+π − and ρ(770)−π + events are 1 : 2.28±0.63 : 0.96±0.27, in fair agreement with the expectation of 1 : 1 : 1. A fit with the ρ(770), ρ(2150) and an additional P wave phase space shows that the contribution of the direct 3π process is small. The fit quality is checked using Pearson’s χ2 test by dividing the Dalitz plots into small areas with at least 20 events and comparing the number of events between data and normalized Monte Carlo simulation. A χ2/ndf = 14.6/7 = 2.1 is obtained, which corresponds to a confidence level of 4%. A fit with the ρ(2150) width or mass free; or a fit with ρ(770), ρ(1450), ρ(1700), and ρ(2150); or even with an extra excited ρ state does not improve the fit quality significantly. Considering these cases, the number of ρ(770)π events changes by less than 9.1%, which Fig. 4. Comparison between data (dots with error bars) and the final is included in the systematic error. The number of fit (solid histograms) for (a) two pion invariant mass, with a solid ρ(2150)π events increases by 57% when other excited line for the ρ(770)π, a dashed line for the ρ(2150)π, and a hatched histogram for background; (b) the ρ polar angle in the ψ rest frame; ρ states are added in the fit due to the large destructive and (c) and (d) for the polar and azimuthal angles of the designated interference between them; this is also included in the π in ρ helicity frame. systematic error. Using the parameters of the fit in the Monte Carlo → + − 0 Table 1 generator, the efficiency of ψ π π π is esti- + − ψ → π π π0 fitting parameters and the assumed values or fitting mated to be 9.02%, and the corresponding efficiencies results. For the assumed values (the numbers with no errors), the for ρ(770)π and ρ(2150)π are 10.54% and 8.70%, values are taken from PDG [17] andfixedinthefit respectively. The efficiency is considered in the PWA. + − Quantity Fit parameters Systematic errors in the ψ → π π π 0 branching 2 Mρ(770) (GeV/c ) 0.7758 fraction measurement come from the kinematic fit, the 2 Γ ρ(770) (GeV/c ) 0.1503 MDC tracking, particle identification, photon identifi- ◦ − ± βρ(2150) ( ) 102 10 cation, background estimation, continuum subtraction, 2 Mρ(2150) (GeV/c ) 2.149 etc. Most of the errors are measured using clean exclu- 2 Γ ρ(2150) (GeV/c ) 0.363 sive J/ψ and ψ decay samples [14,18], while some B(3π): B(ρ(770)π) 1.0:(0.28±0.03) : B(ρ(2150)π) : (1.07 ± 0.09) were described above. The uncertainty in the continuum subtraction listed in Table 2 is the error of the luminosity normalization factor between the contin- The fit parameters and results are given in Ta- uum and ψ data. The fluctuation of the continuum ble 1, where for results without errors, the parame- counts relative to the ψ yield is taken into considera- ter is fixed. The fit yields (28 ± 3)% ρ(770)π in all tion in the π +π −π 0 event subtraction, so this error is π +π −π 0 events (corrected for detection efficiency). included in the first error of the following branching By comparing the likelihood difference with and with- fraction calculation. + − + − 0 out the ρ(770√)π in the fit, the significance of e e → To determine B(ψ → ρ(770)π → π π π ) and ρ(770)π at s = 3.686 GeV is 7.4σ and varies be- B(ψ → ρ(2150)π → π +π −π 0), we assume that the tween 6.1σ to 7.7σ for the fit variations described ratios of branching fractions in Table 1 are the same below in the determination of systematic errors. The for the ψ data as for the continuum cross sections and significance of ρ(2150)π is larger than 10σ . Adding use these ratios and the continuum subtracted B(ψ → free parameters in front of the ρ+π − and ρ−π + am- π +π −π 0) to obtain the desired branching fractions. BES Collaboration / Physics Letters B 619 (2005) 247–254 253

Table 2 Summary of relative systematic errors (%). (The sources marked with a * were treated in common for all three modes) + − Source B(ψ → π π π0) B(ψ → ρ(770)π) B(ψ → ρ(2150)π) ∗ Trigger 0.5 0.5 0.5 ∗ MDC tracking 4.0 4.0 4.0 ∗ Kinematic fit 6.0 6.0 6.0 ∗ Photon efficiency 4.0 4.0 4.0 ∗ Number of photons 2.0 2.0 2.0 ∗ Background estimation 3.6 3.6 3.6 ∗ Particle ID negligible negligible negligible ∗ Total number of ψ 4.0 4.0 4.0 ∗ Continuum subtraction 3.0 3.0 3.0 Background shape in PWA no 7.3 1.4 +57 Different PWA fits no 9.1 −0.0 Continuum resonant structure no 16.0 13.7 ± ± +59.3 Total 10.5 22.4 −17.5

This subtracts the continuum with the stated assump- where the second errors are systematic, while the first tion. error of B(π +π −π 0) is the statistical error which con- For ρ(770)π and ρ(2150)π, the uncertainties asso- tains the error from the continuum 3π yield subtrac- ciated with the possibility of different resonant struction; and the first errors of B(ρ(770)π → π +π −π 0) ture between the continuum and the ψ data, 16.0% and B(ρ(2150)π → π +π −π 0) are the combinations and 13.7% respectively, and the uncertainties of fit- of the PWA fit errors (shown in Table 1) and the first ting with different high mass ρ states and with the error of B(π +π −π 0). ρ(2150) width or mass free, etc., are also included. Our B(ψ → π +π −π 0) agrees with the Mark Here, 16.0% is obtained from the difference of the II [2] result within 1.8σ and agrees well with the ψ → ρ(770)π events between the ρ(770)π subtrac- CLEO-c measurement [4].OurB(ψ → ρ(770)π) is tion using the component ratio in the PWA and the below the Mark II [2] upper limit and in agreement ρ(770)π subtraction estimated by CLEO-c’s contin- with the model prediction of B(ψ → ρ(770)π) = uum measurement [4], and the 13.7% is the difference (1.11 ± 0.87) × 10−4 [6]. This measurement is about of ψ → ρ(2150)π events between the ρ(2150)π sub- 2σ higher than the result of CLEO-c [4]; this is due traction using the component ratio in the PWA and to the different analysis procedure, namely the inter- CLEO-c’s zero subtraction of ρ(2150)π events. Ta- ference between ρ(770) and ρ(2150) considered in ble 2 summarizes the systematic errors for all chan- this analysis but not in the CLEO-c analysis and the nels. The total systematic error for ψ → π +π −π 0 is difference in the continuum subtractions in the two 10.5%, and those for ψ → ρ(770)π and ρ(2150)π analyses. The continuum amplitude, which is con- +59.3 are 22.4% and −17.5%, respectively. sidered incoherently in both analyses, could change Using the numbers obtained above, the branching the ρ(770)π branching fraction due to interference fractions of ψ → π +π −π 0, ρ(770)π and ρ(2150)π with the resonance [6]. This should be considered in a are higher statistics experiment [19]. Comparing with the corresponding J/ψ decay + − − + − B π π π 0 = (18.1 ± 1.8 ± 1.9) × 10 5, branching fractions, it is found that both π π π 0 and ρ(770)π are highly suppressed compared with B → + − 0 ρ(770)π π π π the “12% rule”, while for ρ(2150)π, there is no mea- − = (5.1 ± 0.7 ± 1.1) × 10 5, surement in J/ψ decays. It could be enhanced in ψ decays since the phase space in J/ψ decays is B → + − 0 ρ(2150)π π π π limited due to the large mass of the excited ρ state. = ± +11.5 × −5 Using the J/ψ and ψ → ρπ branching fractions, 19.4 2.5−3.4 10 , 254 BES Collaboration / Physics Letters B 619 (2005) 247–254

Measurement of single π 0 production in neutral current neutrino interactions with water by a 1.3 GeV wide band muon neutrino beam

K2K Collaboration S. Nakayama a, C. Mauger b,1,M.H.Ahnc, S. Aoki d,Y.Ashiea,H.Bhangc,S.Boyde,1, D. Casper f, J.H. Choi g, S. Fukuda a, Y. Fukuda h,R.Grane, T. Hara d, M. Hasegawa i, T. Hasegawa j, K. Hayashi i, Y. Hayato k, J. Hill b,1,A.K.Ichikawak,A.Ikedal, T. Inagaki i,1, T. Ishida k,T.Ishiik, M. Ishitsuka a,Y.Itowa, T. Iwashita k,H.I.Jangg,1, J.S. Jang g,E.J.Jeonc,K.K.Jooc,C.K.Jungb, T. Kajita a, J. Kameda a, K. Kaneyuki a, I. Kato i,1, E. Kearns m, A. Kibayashi n, D. Kielczewska o,p,B.J.Kimc,C.O.Kimq, J.Y. Kim g,S.B.Kimc, K. Kobayashi b, T. Kobayashi k,Y.Koshioa, W.R. Kropp f, J.G. Learned n,S.H.Limg,I.T.Limg, H. Maesaka i, T. Maruyama k,1, S. Matsuno n, C. Mcgrew b, A. Minamino a, S. Mine f,M.Miuraa,K.Miyanor, T. Morita i, S. Moriyama a, M. Nakahata a, K. Nakamura k, I. Nakano l, F. Nakata d, T. Nakaya i, T. Namba a,R.Nambua, K. Nishikawa i, S. Nishiyama d, K. Nitta k,S.Nodad, Y. Obayashi a,A.Okadaa,Y.Oyamak,M.Y.Pacs,H.Parkk,1,C.Sajia, M. Sakuda k,1, A. Sarrat b, T. Sasaki i,N.Sasaoi, K. Scholberg u,t, M. Sekiguchi d,E.Sharkeyb, M. Shiozawa a, K.K. Shiraishi e,M.Smyf,H.W.Sobelf, J.L. Stone m,Y.Sugad, L.R. Sulak m, A. Suzuki d, Y. Suzuki a, Y. Takeuchi a,N.Tamurar, M. Tanaka k, Y. Totsuka k, S. Ueda i, M.R. Vagins f, C.W. Walter u,W.Wangm, R.J. Wilkes e, S. Yamada a,1, S. Yamamoto i,C.Yanagisawab,H.Yokoyamav,J.Yooc, M. Yoshida w, J. Zalipska p

a Institute for Cosmic Ray Research, University of Tokyo, Kashiwa, Chiba 277-8582, Japan b Department of Physics and Astronomy, State University of New York, Stony Brook, NY 11794-3800, USA c Department of Physics, Seoul National University, Seoul 151-742, South Korea d Kobe University, Kobe, Hyogo 657-8501, Japan e Department of Physics, University of Washington, Seattle, WA 98195-1560, USA f Department of Physics and Astronomy, University of California, Irvine, Irvine, CA 92697-4575, USA g Department of Physics, Chonnam National University, Kwangju 500-757, South Korea h Department of Physics, Miyagi University of Education, Sendai 980-0845, Japan i Department of Physics, Kyoto University, Kyoto 606-8502, Japan

Abstract Neutral current single π0 production induced by neutrinos with a mean energy of 1.3 GeV is measured using a 1000 ton water Cherenkov detector in the K2K long baseline neutrino experiment. The cross section for this process relative to the total charged current cross section is measured to be 0.064 ± 0.001(stat.) ± 0.007(sys.). The momentum distribution of neutral current π0s from a water target is measured with high statistics for the ﬁrst time.  2005 Elsevier B.V. All rights reserved.

PACS: 13.15.+g; 14.60.Lm; 25.30.Pt

After the discovery of atmospheric neutrino oscil- showering Cherenkov rings. Understanding this rate lations by Super-Kamiokande in 1998 [1], the primary will make several significant contributions. The single aim of current and future long baseline (LBL) exper- π 0 production rate by atmospheric neutrinos could be iments using an accelerator-based neutrino beam is used to distinguish between the νµ ↔ ντ and νµ ↔ νs the accurate determination of oscillation parameters. oscillation hypotheses. The NC rate is attenuated in The uncertainties on the knowledge of the neutrino- the case of transitions of νµ’s into sterile neutrinos, nucleus cross sections and subsequent nuclear effects while it does not change in the νµ ↔ ντ scenario. In in the GeV neutrino energy region could become a addition, understanding single π 0 production is im- severe limitation in future oscillation studies. Near de- portant for electron neutrino appearance searches at tectors of LBL experiments can provide neutrino in- LBL experiments using a water Cherenkov far detec- teraction data with much higher statistics and better tor. In these experiments, the most serious background quality to reduce these systematic uncertainties. comes from events with a single π 0 where only one Single π 0 events are a good signature of neutral ring is found due to highly asymmetric energies, or current (NC) neutrino interactions in the GeV region, a small opening angle between the γ -rays of the de- and in a water Cherenkov detector a decay of the cay [2]. These single π 0s in the GeV neutrino energy π 0 can be clearly identified as two electromagnetic- region are mainly produced via the ∆ resonance as ν + N → ν + ∆, ∆ → N + π 0, where N and N are nucleons. Because of nuclear effects such as Fermi E-mail address: [email protected] (S. Nakayama). motion, Pauli blocking, nuclear potential and final 1 For current affiliations see http://neutrino.kek.jp/address05 state interactions, ∆ production and its decay could be may.pdf. different from a simple picture of neutrino-nucleon in- K2K Collaboration / Physics Letters B 619 (2005) 255–262 257 teractions. In addition, final state interactions of nucleons and mesons during propagation inside of nuclear matter could substantially modify the number, momenta, directions and charge states of produced particles. Though there are several theoretical approaches for modeling these processes, their uncertainties are large and there exist very few experimental data for NC single π 0 production with no measurements using a water target, the target material used in the far detector of some LBL experiments [3,4]. It is clear that good knowledge of the NC single π 0 production cross section and π 0 momentum distribution is required for the above studies. In this Letter, we present the first high statistics measurement of “NC1π 0 interactions in water” defined as a neutral current neutrino interaction where a single π 0 and no other mesons are emitted in the final state from the target nucleus. The KEK to Kamioka long baseline neutrino os- Fig. 1. The energy spectrum of the K2K neutrino beam at 300 m cillation experiment (K2K) uses an almost pure muon 20 ¯ downstream from the target with a 10 protons on target exposure neutrino beam (98.2% νµ,1.3%νe and 0.5% νµ) as predicted by a neutrino beam simulation. The spectrum is aver- with a mean neutrino energy of 1.3 GeV. This intense aged within 2 m from the beam center. wide-band beam is produced by the KEK proton synchrotron (KEK-PS). Protons extracted from the KEK- PS with 12 GeV kinetic energy are bent toward the The inner volume of the 1 kt detector is a cylinder of far detector, Super-Kamiokande, located 250 km away 8.6 m in diameter and 8.6 m in height. This volume is from KEK and interact with an aluminum target. Pos- viewed by 680 photomultiplier tubes (PMTs) of 50 cm itively charged secondary particles, mainly π +’s, are diameter, facing inward to detect Cherenkov light focused with a pair of magnetic horns and then decay from neutrino events. The PMTs and their arrange- to produce a neutrino beam. Fig. 1 shows the energy ment are identical to those of Super-Kamiokande, giv- spectrum of the K2K neutrino beam at 300 m down- ing a 40% photocathode coverage. The primary role of 20 stream from the target with a 10 protons on target the 1 kt detector is to measure the νµ interaction rate (p.o.t.) exposure as predicted by a beam simulation and the νµ energy spectrum. The 1 kt detector also pro- [3]. The beam simulation is validated by a pion moni- vides a high statistics measurement of neutrino-water tor, which measures the momentum and divergence of interactions. The analysis in this Letter is based on 1 kt pions just behind the second horn [3]. The flux shape data taken between January 2000 and July 2001, cor- is also measured through neutrino interactions by the responding to 3.2 × 1019 p.o.t. K2K near detectors [5]. The absolute neutrino flux is The 1 kt detector data acquisition (DAQ) system not estimated for the measurement of the NC1π 0 cross is also similar to that of Super-Kamiokande. The sig- section because it is difficult to verify the proton beam nal from each PMT is processed using custom elec- intensity, profile and targeting efficiency uncertainties. tronics modules called ATMs, which were developed Instead of deriving the absolute cross section, we mea- for the Super-Kamiokande experiment and are used sure the relative cross section of NC1π 0 interactions to to record digitized charge and timing information for the total charged current (CC) cross section. each PMT hit over a threshold of about 1/4 photoelec- As one of the near detectors for K2K, a 1000 ton tron [6]. The DAQ trigger threshold is about 40 PMT water Cherenkov detector (1 kt) is located about 300 m hits within a 200 ns time window in a 1.2 µs beam downstream of the pion production target. The 1 kt de- spill gate. This is roughly equivalent to the signal of a tector is a miniature of Super-Kamiokande using the 6 MeV electron. The analog sum of all 680 PMTs’ sig- same interaction target material and instrumentation. nals (PMTSUM) is also recorded for every beam spill 258 K2K Collaboration / Physics Letters B 619 (2005) 255–262 by a 500 MHz FADC to identify multiple interactions reconstructed in the 25 ton fiducial volume defined as in a spill gate. We determine the number of interac- a 4 m diameter and 2 m long cylinder along the beam tions in each spill by counting the peaks in PMTSUM axis. (v) The visible energy is larger than 30 MeV. greater than a threshold equivalent to a 100 MeV elec- A total of 60 545 events are selected by requirements tron signal. (i) to (v). The physical parameters of an event in the 1 kt Single π 0 events are extracted from the sample of detector such as the vertex position, the number of fully contained (FC) neutrino events, which deposit all Cherenkov rings, particle types and momenta are de- of their Cherenkov light inside the inner detector. An termined by using the same algorithms as in Super- exiting particle deposits a large amount of energy at Kamiokande [7,8]. First, the vertex position of an the PMT of the exiting position. Therefore, the FC event is determined from the PMT timing information. events are selected by requiring the maximum num- With knowledge of the vertex position, the number of ber of photoelectrons on a single PMT at the exit di- Cherenkov rings and their directions are determined rection of the most energetic particle to be less than by a maximum-likelihood procedure. Each ring is then 200. The events with the maximum number of photo- classified as e-like, representing a showering particle electrons greater than 200 are identified as a partially (e±,γ),orµ-like, representing a non-showering par- contained (PC) event. Because π 0s mostly decay into ticle (µ±,π±), using its ring pattern. On the basis two γ -rays, the following cut criteria are further ap- of this particle type information, the vertex position plied to select the single π 0 events in the FC sample: of a single-ring event is further refined. The momen- (1) The number of reconstructed rings is two. (2) Both tum corresponding to each ring is determined from the rings have a showering type (e-like) particle identifi- Cherenkov light intensity. cation. (3) The invariant mass is in the range of 85– In the 1 kt detector, the gain and timing of each 215 MeV/c2. Fig. 2 shows the invariant mass distribu- 0 PMT are calibrated using a Xe lamp and a N2 laser tion of the events which satisfy cuts (1) and (2). A π as light sources, respectively. The absorption and scat- mass peak is clearly observed. The peaks for both the tering coefficients of water are measured by a laser observed data and the neutrino Monte Carlo events are calibration, and the coefficients in the detector simula- slightly shifted towards higher values compared to the tion are further tuned to reproduce the observed charge nominal value of the π 0 mass, 135 MeV/c2. This shift patterns of cosmic ray muon events. The number of is due to ∼ 20 cm bias of vertex reconstruction for photoelectrons per unit track length is calibrated by γ -rays from π 0 decay and energy deposited by deexci- cosmic ray muons passing through the detector. The tation γ -rays from the oxygen nucleus. The difference energy scale is checked by cosmic ray muons stop- in peak position between the data distribution and the +2 ping inside the detector for which the track length is Monte Carlo prediction is within our quoted −3 %sys- determined by the vertex position of subsequent decay tematic uncertainty for the absolute energy scale. As electron events. The accuracy of the energy scale is es- shown in Table 1, a total of 2496 events are found as +2 0 timated to be −3 %. The performance of vertex recon- the single π sample by these criteria. struction is experimentally studied by special cosmic Fig. 3(a) shows the reconstructed π 0 momentum ray muon data utilizing a pipe inserted vertically into distribution for the single π 0 sample. The momentum the tank. Cosmic ray muons going through the pipe emulate the neutrino-induced muons whose vertex position is defined at the bottom end of the pipe. This Table 1 study demonstrates the vertex reconstruction works as The number of events after each selection to make the single π0 we expected from the Monte Carlo simulation. sample in 1 kt data. The Monte Carlo efficiencies are calculated for 0 Neutrino interaction candidates are selected by NC1π interactions whose real vertex is in the fiducial volume the following requirements: (i) An event is triggered Data NC1π0 efficiency (%) within a 1.2 µs beam spill gate. (ii) There is no de- FC 45 317 97 tector activity within a 1.2 µs window preceding the Two rings 11 117 57 beam spill. (iii) Only a single event is found by the Both e-like 3150 48 Invariant mass 2496 47 PMTSUM peak search in that spill. (iv) The vertex is K2K Collaboration / Physics Letters B 619 (2005) 255–262 259

Fig. 2. The invariant mass of two e-like ring events for the experimental data (black dots) and the neutrino Monte Carlo simulation (box histogram). The Monte Carlo histogram is normalized to the Fig. 3. (a) The reconstructed π0 momentum distribution for the sin- area of the data histogram. The error bars are statistical only. The gle π0 sample, comparing 1 kt data (black dots) and the neutrino hatched portion and the black portion in the Monte Carlo histogram Monte Carlo simulation (box histogram). The Monte Carlo his- 0 0 show the non-NC1π component and the non-π component, re- togram is normalized to the area of the data histogram. The error spectively. A pair of arrows shows the invariant mass cut (3) (see bars are statistical only. The hatched portion and the black portion text). in the Monte Carlo histogram show the non-NC1π0 component and the non-π0 component, respectively. (b) The detection efficiency for NC1π0 interactions as a function of real π0 momentum. The resolution is estimated to be 15 MeV/c for π 0s with a highest momentum bin in each figure integrates the events above 800 MeV/c. momentum of 200 MeV/c. As shown by the hatched portion and the black portion in Figs. 2 and 3(a), the single π 0 sample contains the non-NC1π 0 component ∆(1232) resonance is also determined by the Rein and the non-π 0 component, which are both the back- and Sehgal method. For the decays of the other reso- grounds to our signal, NC1π 0 interactions. The non- nances, the generated mesons are assumed to be emit- NC1π 0 component consists of events which include ted isotropically in the resonance rest frame. A 20% π 0s but do not satisfy our definition of NC1π 0 interac- suppression of pion production is adopted for simulat- tions. The composition of the non-NC1π 0 component ing pion-less ∆ decay, where the event contains only is described later. The non-π 0 component consists of a lepton and a nucleon in the final state [12]. The ax- 0 2 background events with no π . As described in de- ial vector mass, MA, is set to 1.1 GeV/c for both the tail later, the fraction of each component is estimated quasi-elastic scattering and the single meson produc- by the neutrino Monte Carlo simulation and the back- tion models. Coherent pion production is simulated ground components are subtracted from the single π 0 using the Rein and Sehgal model [13] with modified sample. cross section according to a description by Marteau The neutrino interactions with water are simu- et al. [14]. For deep inelastic scattering, GRV94 nu- lated by the NEUT program library. A detailed de- cleon structure function [15] with a correction by scription of NEUT can be found in Ref. [9]. Quasi- Bodek and Yang [16] is used to calculate the cross sec- elastic scattering is simulated based on the Llewellyn tion. In order to generate the final state hadrons, the Smith’s model [10]. The Rein and Sehgal model [11] PYTHIA/JETSET [17] package is used for hadronic is used to simulate single meson production mediated invariant masses W larger than 2.0 GeV/c2. A cus- by a baryon resonance. The decay kinematics of the tom program [18] is used for W in the range of 1.3– 260 K2K Collaboration / Physics Letters B 619 (2005) 255–262

Table 2 Summary of the systematic errors on the measurement of the number of NC1π0 interactions Sources Systematic errors (%) (A) Background subtraction MA in quasi-elastic and single meson (±10%) 0.2 Quasi-elastic scattering (total cross section, ±10%) 0.0 Single meson production (total cross section, ±10%) 0.9 Coherent pion production (model dependence) 1.6 Deep inelastic scattering (model dependence) 5.1 Deep inelastic scattering (total cross section, ±5%) 0.5 CC/NC cross section ratio (±20%) 3.2 Nuclear effects in 16O (pion absorption, ±30%) 1.5 Nuclear effects in 16O (pion inelastic scattering, ±30%) 0.7 Pion interaction outside the target nucleus (±20%) 2.3 (B) Fiducial volume correction Fiducial cut 1.6 (C) Efﬁciency correction Ring counting 5.4 Particle identiﬁcation 4.2 Energy scale 0.3

2.0 GeV/c2. Nuclear effects for ν–16O scattering are rest of the mesons are absorbed inside the nucleus). also taken into account. Fermi motion of nucleons The non-NC1π 0 component accounts for 26% of the is simulated using the relativistic Fermi gas model. single π 0 sample as shown in Figs. 2 and 3. The non- Pauli blocking effect is considered in the simulation NC1π 0 component contains NCπ 0 production where of quasi-elastic scattering and single meson produc- outgoing mesons except a single π 0 have low motion. Pions generated in 16O are tracked taking into menta (7%), CCπ 0 production where accompanying account their inelastic scattering, charge exchange and muon and mesons have low momenta (9%), and π 0 absorption. The pion generation point in the nucleus is production by a recoil nucleon or mesons outside the set according to Woods and Saxon’s nucleon density target nucleus (10%). The non-π 0 component caused distribution [19]. The mean free path of each pion in- by mis-reconstruction accounts for the remaining 3% teraction is calculated using the model by Salcedo et of the single π 0 sample. al. [20]. The direction and momentum of pions after The NC1π 0 fraction and the background fraction inelastic scattering or charge exchange are determined are estimated for each of the nine π 0 momentum bins based on the results of a phase shift analysis obtained shown in Fig. 3. The number of NC1π 0 events is ex- from π–N scattering experiments [21]. Emission of tracted from the number of observed single π 0 events deexcitation low energy γ -rays from a hole state of multiplied by the NC1π 0 fraction bin by bin. The residual nuclei is also taken into account [22]. Outside systematic errors of this background subtraction are the nucleus, particles are tracked using GEANT and estimated based on the uncertainties of neutrino inter- CALOR packages except pions with momenta below actions as listed in Table 2(A). The error due to the 500 MeV/c, which are tracked with a custom program uncertainty on the deep inelastic scattering cross sec- [18] including the effect of the ∆ resonance. tion is evaluated by comparing the two models in Refs. The NC1π 0 fraction in the single π 0 sample is esti- [15,16]. For coherent pion production, the difference mated to be 71% using the neutrino Monte Carlo sim- between the two models in Refs. [13,14] is assumed ulation (52% from single π 0 production via resonance to be the uncertainty. The systematic error in the CC decay, 3% from single π ± production via resonance component subtraction is estimated by considering the decay and subsequent charge exchange into π 0 in the uncertainty on the CC cross section relative to the NC target nucleus, 10% from coherent π 0 production and cross section [23]. The systematic errors due to uncer- 4% from neutrino deep inelastic scattering where the tainties on pion nuclear effects in the target nucleus K2K Collaboration / Physics Letters B 619 (2005) 255–262 261 are evaluated by varying the probabilities of pion absorption and inelastic scattering (including charge exchange) in the 16O nucleus independently by ±30% [9]. The uncertainty on the cross section of π–16O interactions outside the target nucleus is taken from Refs. [24,25]. For the number of NC1π 0 events, we apply a 2% fiducial correction, estimated using the Monte Carlo simulation, to account for the difference between the number of events in the true 25 ton fiducial volume and the number in the reconstructed fiducial volume. The vertex distribution of the π 0 events is in good agreement with that of the Monte Carlo events. The systematic error due to the uncertainty of the fiducial volume is estimated to be 1.6% as shown in Table 2(B) by varying the boundary of the fiducial volume by ±30 cm along the beam direction, which is the ver- 0 tex resolution of the π events. Fig. 4. The momentum distribution of NC1π0 events in the 25 ton 0 Finally, the number of NC1π interactions in the fiducial volume (black dots). The inner and outer error bars attached true 25 ton fiducial volume is corrected for the detec- to data points show statistical errors and total errors including sys- tion efficiency bin by bin. Fig. 3(b) shows the detection tematic errors, respectively. The distribution predicted by the neu- efficiency for NC1π 0 interactions as a function of real trino Monte Carlo simulation is also shown as a box histogram for 0 comparison. The size of inner boxes represents the statistical errors. π momentum. The inefficiency for higher momentum The size of outer boxes includes uncertainties on nuclear effects of 0 π s results from the reconstruction of only one ring of pions in 16O. a π 0 decay with highly asymmetric energies or a small opening angle between the two γ -rays. The overall detection efficiency for NC1π 0 interactions, estimated π 0 momentum distribution due to the uncertainties of by the Monte Carlo simulation, is 47% as shown in nuclear effects, where the largest source is inelastic Table 1. The systematic errors from the efficiency cor- scattering of pions in the target 16O nucleus. Consider- rection are due to uncertainties of reconstruction al- ing these uncertainties, the discrepancy in the momen- gorithms such as ring counting, particle identification tum distribution can be explained. The effect from this and energy scale as listed in Table 2(C). discrepancy is not significant in the cross section mea- After corrections for background subtraction, the surement. true/reconstruction fiducial difference and the detec- As previously described, we use CC interactions for tion efficiency, the true number of NC1π 0 interactions normalization in order to derive the relative cross sec- is measured to be (3.61 ± 0.07 ± 0.36) × 103 in the tion for NC1π 0 interactions. To make a CC enriched 25 ton fiducial volume. Fig. 4 shows the measured mo- sample, FC µ-like events and PC events are selected mentum distribution of NC1π 0 after all corrections, in the same period. By using the CC enriched sam- compared with the distribution predicted by the neu- ple as a normalization, the uncertainty of the neutrino trino Monte Carlo simulation. The Monte Carlo his- energy spectrum [5] is almost canceled in the mea- togram is normalized by the total number of neutrino surement since the expected mean energy of neutrinos events in the fiducial volume after all the cuts. The producing the CC sample, 1.45 GeV, is almost same as size of inner boxes for the Monte Carlo histogram rep- that of neutrinos producing the π 0 sample, 1.50 GeV. resents the statistical errors of Monte Carlo events. The sample consists of 22 612 FC single-ring µ-like There exists small discrepancy between the measured events, 12 386 FC multi-ring events with the most en- distribution and the Monte Carlo expectation in the ergetic ring identified as µ-like and 15 228 PC events, 0 π momentum around 300 and 600 MeV/c. The size resulting in a total of 50 226 events. The νµCC fraction of outer boxes includes the error in the shape of the in this sample is estimated to be 96% by the Monte 262 K2K Collaboration / Physics Letters B 619 (2005) 255–262

Carlo simulation (96.5% for the FC single-ring µ-like Acknowledgements sample, 91.2% for the FC multi-ring µ-like sample and 98.5% for the PC sample). The remaining 4% We thank the KEK and ICRR Directorates for of the sample is mostly composed of NC interactions their strong support and encouragement. K2K is made with an outgoing charged pion above its Cherenkov possible by the inventiveness and the diligent efforts threshold. The fiducial volume correction factor is es- of the KEK-PS machine and beam channel groups. timated to be 1.02. The detection efficiency for νµCC We gratefully acknowledge the cooperation of the interactions by this selection is estimated to be 85%. Kamioka Mining and Smelting Company. This work The inefficiency mainly comes from mis-identification has been supported by the Ministry of Education, Cul- of the ring type in multi-ring events and ∼ 100 MeV ture, Sports, Science and Technology, Government of visible energy threshold by peak counting of the PMT- Japan and its grants for Scientific Research, the Japan SUM signal. Society for Promotion of Science, the US Department By applying non-νµCC background subtraction, of Energy, the Korea Research Foundation, the Ko- fiducial volume correction and detection efficiency rea Science and Engineering Foundation, the CHEP correction, the number of νµCC neutrino interactions in Korea, and Polish KBN grant 1P03B03826 and during the analysis period is measured to be (5.65 ± 1P03B08227. 0.03 ± 0.26) × 104 in the 25 ton fiducial volume. The systematic errors are estimated to be 4% from the uncertainty of vertex reconstruction, 1% from the un- References certainty of neutrino interaction models, 1% from the uncertainty of particle identification and 1% from the [1] Y. Fukuda, et al., Phys. Rev. Lett. 81 (1998) 1562. uncertainty of absolute energy scale. [2] M.H. Ahn, et al., Phys. Rev. Lett. 93 (2004) 051801. [3] S.H. Ahn, et al., Phys. Lett. B 511 (2001) 178. By taking the ratio, the relative cross section for [4] Y. Itow, et al., hep-ex/0106019, http://neutrino.kek.jp/jhfnu/. 0 NC1π interactions to the total νµCC cross section [5] M.H. Ahn, et al., Phys. Rev. Lett. 90 (2003) 041801. is measured to be 0.064 ± 0.001(stat.) ± 0.007(sys.). [6] Y. Fukuda, et al., Nucl. Instrum. Methods A 501 (2003) 418. Since the systematic errors on the NC1π 0 measure- [7] Y. Fukuda, et al., Phys. Lett. B 433 (1998) 9. [8] M. Shiozawa, Nucl. Instrum. Methods A 433 (1999) 240. ment and the νµCC measurement have almost no [9] Y. Hayato, Nucl. Phys. B (Proc. Suppl.) 112 (2002) 171. correlation, errors are added in quadrature. Our neu- [10] C.H. Llewellyn Smith, Phys. Rep. 3 (1972) 261. trino interaction model predicts the ratio to be 0.065, [11] D. Rein, L.M. Sehgal, Ann. Phys. 133 (1981) 79. which is in good agreement. For reference, the to- [12] S.K. Singh, M.J. Vicente-Vacas, E. Oset, Phys. Lett. B 416 tal νµCC cross section is calculated to be 1.1 × (1998) 23. 10−38 cm2/nucleon in the neutrino Monte Carlo sim- [13] D. Rein, L.M. Sehgal, Nucl. Phys. B 223 (1983) 29. [14] J. Marteau, J. Delorme, M. Ericson, Nucl. Instrum. Methods ulation averaged over the K2K neutrino beam energy. A 451 (2000) 76. In summary, we have measured the cross sec- [15] M. Gluck, E. Reya, A. Vogt, Z. Phys. C 67 (1995) 433. tion and the π 0 momentum distribution of NC single [16] A. Bodek, U.K. Yang, Nucl. Phys. B (Proc. Suppl.) 112 (2002) π 0 production for neutrinos with a mean energy of 70. 1.3 GeV using the K2K 1 kt water Cherenkov de- [17] T. Sjostrand, Comput. Phys. Commun. 82 (1994) 74. 0 [18] M. Nakahata, et al., J. Phys. Soc. Jpn. 55 (1986) 3786. tector. The ratio of cross sections for NC1π to the [19] R.D. Woods, D.S. Saxon, Phys. Rev. 95 (1954) 577. total νµCC interaction is measured to be 0.064 ± [20] L.L. Salcedo, E. Oset, M.J. Vicente-Vacas, C. Garcia-Recio, 0.001(stat.) ± 0.007(sys.), showing good agreement Nucl. Phys. A 484 (1988) 557. with the prediction by our neutrino interaction model. [21] G. Rowe, M. Salomon, R.H. Landau, Phys. Rev. C 18 (1978) This measurement provides essential information to 584. 0 [22] H. Ejiri, Phys. Rev. C 48 (1993) 1442. understand π production in water in the 1 GeV neu- [23] E.H. Monsay, Phys. Rev. Lett. 41 (1978) 728. trino energy region, and is relevant to present and [24] J.P. Albanese, et al., Nucl. Phys. A 350 (1980) 301. future neutrino oscillation experiments. [25] A.S. Clough, et al., Nucl. Phys. B 76 (1974) 15. Physics Letters B 619 (2005) 263–270 www.elsevier.com/locate/physletb

A measurement of the antineutrino asymmetry B in free neutron decay

M. Kreuz a,b, T. Soldner b,S.Baeßlerc,B.Branda,F.Glückd,e,U.Mayera,D.Munda, V. Nesvizhevsky b, A. Petoukhov b, C. Plonka a,J.Reicha, C. Vogel a,H.Abelea

a Physikalisches Institut, Universität Heidelberg, Philosophenweg 12, D-69120 Heidelberg, Germany b Institut Laue Langevin, 6 rue Jules Horowitz, BP 156, F-38042 Grenoble cedex 9, France c Institut für Physik, Universität Mainz, Staudinger Weg 7, D-55099 Mainz, Germany d Institut für Kernphysik, Forschungszentrum Karlsruhe, Pf. 3640, D-76021 Karlsruhe, Germany e Theory Department, Research Institute for Nuclear and Particle Physics, PO Box 49, H-1525 Budapest 114, Hungary Received 11 April 2005; received in revised form 30 May 2005; accepted 31 May 2005 Available online 8 June 2005 Editor: L. Rolandi

Abstract We have measured the antineutrino asymmetry B in neutron beta decay, i.e., the correlation of the neutron spin and the antineutrino momentum, with a new method. Our result is B = 0.967 ± 0.006stat ± 0.010syst = 0.967 ± 0.012. Statistical and systematic uncertainty can be considerably reduced in future experiments.  2005 Elsevier B.V. All rights reserved.

PACS: 11.30.Er; 24.80.+y; 23.40.-s; 13.30.Ce

Keywords: Beta decay; Weak interaction; P violation; Right handed currents

1. Introduction action. In contrast, left–right symmetric models have been proposed (e.g., [1,2]) seeking to provide a spon- Weak interaction is the only fundamental interac- taneous origin of parity violation. tion known to violate discrete symmetries like parity In neutron beta decay, such models can be tested or charge conjugation. In the Standard Model of par- by measuring the beta and the antineutrino asymmetry ticle physics, parity violation for weak interaction is A and B and the neutron lifetime [3,4]. The present maximal, corresponding to a purely left-handed inter- limits [5] for manifest left–right symmetric models are less stringent than those from other direct and indirect searches [6]. An improvement of the precision E-mail address: [email protected] (M. Kreuz). of the neutron decay data is required. In the Standard

Model, the antineutrino is completely polarized in the 2. Principle of the measurement direction of its momentum. Additional, right-handed interactions will reduce this polarization. The antineu- For the determination of the antineutrino asymme- trino asymmetry B is very sensitive to this reduction. try B, the projection of the antineutrino momentum to The momentum distribution for electron and an- the neutron spin has to be derived from the electron tineutrino in the decay of free polarized neutrons is and the proton. described by [7] A difficulty in absolute measurements of correlation coefficients in neutron decay is the correct inte- dW gration of Eq. (1) over the detectors and the neutron beam distribution, i.e., the precise determination of the dEedΩedΩν¯ detector’s solid angle. The approach of the PERKEO pepν¯ me II spectrometer [13] is the use of a high magnetic field ∝ F(Ee) 1 + a + b EeEν¯ Ee (1 T) perpendicular to the neutron beam. The neutron spin is aligned parallel or antiparallel to the magnetic σ p p ¯ p × p ¯ + A e + B ν + D e ν . (1) field vector. Due to the high magnetic field, all charged σ Ee Eν¯ EeEν¯ particles emitted into one of the two hemispheres defined by this field are collected and guided to one of Here pi , Ei , and Ωi are the momentum, the energy, two detectors. Apart from magnetic mirror effects, the and the solid angle of the electron e and the antineu- × π ¯ detectors’ solid angle is truly 2 2 . Under these trino ν respectively, F(Ee) is the Fermi spectrum, σ conditions Eq. (1) can be integrated analytically for the neutron spin, me the electron mass and a, b, A, B, all combinations of electron and proton detectors in- and D are correlation coefficients. These coefficients stalled in the two hemispheres [14]. To avoid trapping are related to the coupling constants of the interac- of particles, the magnetic field has a maximum in the tion. In the Standard Model, the weak interaction is center of the spectrometer and drops monotonously a pure V–A one, corresponding to maximal parity vi- towards the detectors. Electron backscattering effects olation. Therefore, the correlation coefficients depend = are effectively suppressed by this field configuration. only on the ratio λ gA/gV of axial vector and vector If electron and proton are detected in the same coupling constant (apart from radiative and recoil or- hemisphere, the antineutrino was unambiguously emit- der corrections). With the present experimental status, =− ted into the opposite one, independent of the energy λ 1.2695(29) [6] is obtained most precisely from of the observed particles. Therefore the correspond- the beta asymmetry A. ing detector geometry is rather insensitive to energy In V–A theory, B can be calculated from λ: calibration and resolution of the detectors and particularly clean from the systematic point of view. This |λ|2 − Re(λ) is shown in Fig. 1 and compared to the sensitivity of B = 2 = 0.9876(2). (2) SM 1 + 3|λ|2 a geometry detecting electron and proton in different hemispheres, where the reconstruction of the antineu- Remnants of right handedness, e.g., a (V + A)-type trino direction is very sensitive to the energy of the interaction, will deviate B from BSM. The general ex- detected particles. With similar arguments the statisti- pression for B, including left and right handed vector cal sensitivity of the detection in the same hemisphere and axial vector as well as scalar and tensor couplings, is much higher than that of opposite hemispheres, even can be found in [8]. though—due to kinematics—in only about 22% of Previously, measurements of B have been per- the decays electron and proton are emitted into the formed by [9–12]. We have measured B with a new same hemisphere. In this paper we consider therefore and—in principle—systematically clean method. In only this geometry (in the experiment, data were taken the first measurement we could not fully exploit its for both geometries simultaneously). The statistical potential due to unexpected high background. In this one standard√ deviation error of B for this geometry Letter we present the results of this measurement and is σB = 2.6/ N [14] where N is the total number discuss ways for future improvements. of neutron decays inside the decay volume (i.e., inde- M. Kreuz et al. / Physics Letters B 619 (2005) 263–270 265

Table 1 The corrections (corr.) and errors of the two detectors. Corrections ∗ marked by have been included in the fit function Systematic effect Detector 1 Detector 2 corr. [%] error [%] corr. [%] error [%] ∗ ∗ polarization: +1.5 +1.8 polarization 0.50.5 flip efficiency 0.10.3 data set: statistics 0.80.8 ∗ ∗ random coincidences +3.0 0.5 +3.5 0.6 proton suppression −0.80.4 −0.90.5 detector function: ∗ ∗ gain 0.00 0.01 0.00 0.01 ∗ ∗ offset 0.00 0.03 0.00 0.05 Fig. 1. The expected experimental asymmetries α (Eq. (3))forthe ∗ ∗ ep resolution 0.00 < 0.01 0.00 < 0.01 detection of electron and proton in the same and in opposite hemi- solid angle effects: spheres as function of the electron energy E . Corrections specific ∗ ∗ e edge effect −0.1 0.05 −0.1 0.05 to the set-up (see Table 1) are not included. ∗ ∗ el.-magn. mirror +0.5 0.05 +0.5 0.05 ∗ ∗ grid effect −0.05 0.05 −0.05 0.05 pendent on the emission direction of the decay prod- physical constants ucts but assuming a detection efficiency of 1 and no A 0.00 0.03 0.00 0.03 a 0.00 0.06 0.00 0.06 background). For systematic reasons, in data analysis only the electron energy range between 250 and sum 4.05 1.15 4.75 1.4 450 keV is√ used, reducing the statistical sensitivity to = × 7 σB 4.0/ N. Still, only about 1.6 10 neutron de- by the small radiative and recoil order corrections [15] cays are needed in the decay volume in order to obtain and the systematic effects marked in Table 1, are used a statistical accuracy of 0.1%. to fit the experimental asymmetry Eq. (3). The corre- If N ↑(E ) and N ↓(E ) are the coincident electron- e e lation coefficients a and A enter in αep and thus in the proton count rates in one detector for the neutron spin measurement of B, but for the detection of electron pointing away from and towards the detector, respec- and proton in the same hemisphere the systematic er- tively, the experimental asymmetry αep(Ee) is ror caused by the uncertainties of these coefficients is ↑ − ↓ not important (see Table 1). = N (Ee) N (Ee) αep(Ee) ↑ ↓ . (3) N (Ee) + N (Ee) With β = v/c, the velocity of the electron as a fraction 3. The experimental setup of the speed of light, one finds by integrating Eq. (1) over the hemisphere [14] Fig. 2 shows a cross section of the spectrometer  PERKEO II. The detection of electron and proton in  Aβ(2r−3)+B(3−r2) 4 − + 2− for r<1, the same detector is challenging since the energies α (E ) = P 8 4r aβ(r 2) (4) ep e  − + of these particles are small and about three orders of 3 Aβ 2Br for r>1 4r−aβ magnitude different from each other (maximal energy with 780 keV for electrons and 750 eV for protons). The electron detection was done via plastic scin- E E r = β e ≈ β e = r(E ). tillators with photomultiplier readout. The light out- − e (5) Eν E0 Ee put was optimized by the following means: six high E0 is the total decay energy and P the degree of neu- magnetic field resistant mesh photomultipliers (Hama- tron polarization. The kinetic energy of the proton matsu R5504) were used to read out the scintillator (Ep 750 eV) can be neglected. The energy depen- from the back (local field ≈ 0.6 T). Between the scin- dence of αep is plotted in Fig. 1.Eqs.(4)–(5), extended tillator (thickness 5 mm) and the photomultipliers a 266 M. Kreuz et al. / Physics Letters B 619 (2005) 263–270

Fig. 2. Scheme of the PERKEO II detector. The transversally polarized neutron beam enters the decay volume from the left side. The length of the decay volume is defined by the aluminum baffles. Electron and proton are guided to the detectors via the magnetic field.

20 mm layer of acrylic glass was used to improve The decay volume, limited by aluminum baffles light distribution and homogeneity. The front side of outside the neutron beam (see Fig. 2), was electri- the scintillator was coated with a 50 nm light reflect- cally separated from the potential of the foil by four ing aluminum layer to ground the detector, increase grounded grids of carbon wires with 8 µm diameter the light collection efficiency, and optically decouple and 6 mm distance between the wires. The residual the two detectors from each other. An energy resolu- potential in the decay volume was calculated by 2 tion of ≈ 14% (FWHM) at 1 MeV and a low trigger different methods: the MAFIA software package [17, threshold (50% efficiency at 75 keV, 90% efficiency at 18] and a C-program package developed by one of us 95 keV) was obtained. (F.G.). The latter uses the charge density or boundary The protons were detected indirectly by generat- element method [19,20], which is especially appropri- ing and detecting secondary electrons [16] as fol- ate for potential calculations of an electrode system lows: a very thin carbon foil (≈ 20 µg/cm2)cov- with thin wires [21]. These computations show that the ered with MgO on the detector side (≈ 5µg/cm2; change of the potential inside the decay volume along this layer increases the electron yield) was put be- the magnetic field lines is smaller than 100 mV, thus tween the decay volume and the electron detector. the correction on the antineutrino asymmetry due to A potential of −20 kV was applied to the foil. The the electric field is less than 0.1%. protons were accelerated towards the foil and pro- The spacing in the spectrometer was generous to duced in average 4 to 5 secondary electrons in for- avoid beam related background and discharges of the ward direction when traversing it. These electrons high voltage of the carbon foils. The first grounded were then accelerated by the same electric poten- grid was placed at about 100 mm distance from the tial towards the electron detector on ground, de- center of the decay volume, the carbon foil at 300 mm posing totally in average 80 to 100 keV. With this and the plastic scintillator at 400 mm. method a proton detection efficiency of 50–60% was With this set-up, the proton signal appears in aver- achieved. age about 4 µs after the electron signal. In about 90% M. Kreuz et al. / Physics Letters B 619 (2005) 263–270 267

Fig. 3. Set-up of the beam-line. The horizontal cut through the detector chamber is shown in Fig. 2. of the decays, the proton signals arrives within 9 µs tron beam was 0.987(5). The uncertainty is composed (all numbers for electron and proton in the same hemi- of contributions from the uncertainty of the analyzer sphere). The measurement was carried out in delayed efficiency (0.3%), the neutron wavelength spectrum coincidence. Data acquisition was triggered by simul- (0.3%), the inhomogeneity of the polarization across taneous signals of at least two photomultipliers of the the beam (0.3%), and statistics (0.1%). The average same detector. For the stop signal, only one photomul- flipping efficiency was 0.997(2) [26]. tiplier was requested in order to increase the proton Behind the flipper, the beam was collimated by a efficiency. An event consisted of the energy signals of system of five 6LiF diaphragms. The thermal equiva- the photomultipliers for both detectors and for the start lent flux in the last diaphragm (32 × 25 mm2,1.5m and the stop signal and the time between start and stop in front of the decay volume) was 3 × 108 cm−2 s−1, signal. Only the first stop signal within a coincidence leading to a total decay rate of ≈111 s−1 in the decay window of 58 µs was accepted. All triggered events volume (50 × 58 × 90 mm3). were stored independent of the appearance of a stop Behind the decay volume a free flight zone of 4 m signal. To determine the background of random coin- up to the 6LiF beam stop was installed, allowing a gen- cidences, the stop branch of the data acquisition was erous shielding of the latter. The volume between the reopened for every 10th event 100 µs after the start first diaphragm and the beam stop was evacuated to signal. about 2 × 10−6 mbar. Two shutters in front of the de- The experiment was carried out in 2001 at the cay volume, one right behind the polarizer and one cold neutron beam facility PF1b of the Institut Laue- right after the last diaphragm, allowed to close the Langevin (ILL) in Grenoble, France. The ILL operates neutron beam and to measure the background signal a 58 MW high flux reactor. The instrument PF1b is separately. The trigger rates with opened and closed located at the end position of a 78 m long m = 2su- shutters were ≈ 360 s−1 and ≈ 180 s−1, the coinci- per mirror guide [22] using the horizontal cold source. dence rates were ≈ 60 s−1 and ≈ 30 s−1, respectively. The neutron thermal equivalent flux at the guide exit For the flipper and the shutters, the usual drift com- was 1.6 × 1010 cm−2 s−1. pensating switching schemes (on, off, off, on, off, on, The set-up of the beam-line is shown in Fig. 3.The on, off) were used (80 s period for the flipper, 1 h pe- neutrons traversed a neutron wavelength cut-off filter riod for the shutters). Twice per hour, a 207Bi source [23] in order to suppress neutrons with a wavelength was measured for detector re-calibration. of more than 13 Å. They were spin polarized by a super mirror polarizer (3×4.5cm2) [24]. A current sheet spin flipper allowed the inversion of the spin direction 4. Data analysis and results every 10 s. Polarization and flipping efficiency were measured behind the PERKEO spectrometer using the new experimental method of two crossed super mirror The analyzed data were taken over a period of 10 analyzers [25]. The average polarization of the neu- days. 268 M. Kreuz et al. / Physics Letters B 619 (2005) 263–270

First, data sets with high voltage discharges were coincidences with analytically known self-suppression identified and removed by a semi automatic procedure. appeared. The procedure was based on the smallest time unit of data taking available provided by the spin flipper state The steps (i) and (ii) were applied to the data sets (10 s). Intervals with count rates deviating more than taken with opened and with closed shutters. 3 sigma from the respective median and neighbor intervals were removed. Medians were used in order to (iii) The coincident background (a) consisted of avoid a bias caused by strongly deviating count rates. a beam-independent and a beam-dependent compo- A similar procedure was applied to the intervals de- nent. The beam-independent one was measured with fined by the shutter state (about 6 min). The stability the closed shutters and subtracted. Since it is inde- of these cuts was tested by means of a raw asymme- pendent from the electron-proton signals, there is no try. Between 10% and 30% of the measured data were mutual suppression with the decay signal and the sub- discarded per run. In the remaining data no influence traction can be done without further corrections to of the instabilities could be seen within the given sta- the spectra. As was proven by measurements with ra- tistics. In total the cleaned data set comprised around dioactive sources, electrons and gamma quanta did not 5 million events with valid start and stop signal. create coincident background (apart from the already Then, the background in the electron energy spec- removed background type (b)). Therefore the beam- tra of the events with valid stop signal was determined dependent component of the coincident background and subtracted. The background consisted of different was caused by decay protons registered as start signal. components superposing each other and the electron– The corresponding energy is below the fitting range. proton signal: (a) a coincident beam-independent background caused by the high voltage, (b) a coin- With the steps (i) to (iii), all background was re- cident background due to metastable scintillator exci- moved or subtracted in the fitting range. The resulting tations or photomultiplier after-pulses, and (c) random electron energy spectra contain in total 1.3 × 106 coin- coincidences. The two latter ones suppress the proton cident electron–proton events. They deviate from the signal from neutron decay because only the first stop theoretical energy spectra for two reasons: first, for signal of an event was registered. The background was a decay, the probability of a coincident background determined and removed in three steps: signal of type (b) depends on the electron energy, resulting in an energy dependent suppression of decays (i) The scintillator excitations (causing delayed by removing this background (step (i)). Second, proton single photon events after a scintillator signal) or pho- signals with a larger TOF have an increased probabil- tomultiplier after-pulses (b) could be identified since ity to be suppressed by faster stop signals (from back- they deposed energy in only one single photomulti- ground (b) or (c)). The influence of these effects on plier. They had a typical time constant of about 1 µs the measured antineutrino asymmetry was estimated and an energy dependent probability of about 15% via Monte Carlo simulations. The effects lead to a cor- (averaged over the fitting range). This software cut rection of less than 1% (called proton suppression in removes also proton signals, reducing the proton effi- Table 1). ciency to 30%. (Note that the proton efficiency is only All other systematic effects are small. The correc- important for the statistics and does not cause system- tions and errors are listed in Table 1. The edge ef- atic effects.) fect describes the absorption of decay particles in the (ii) The rate of random coincidences (c) can be vicinity of the aluminum baffles. It depends on the gy- obtained from the measurements with the reopened ration radius and thus on the energy of the particle. stop branch. In the first coincidence window, later The electro-magnetic mirror effect corresponds to a random stop signals are suppressed by earlier ones reflection of some particles by the electric or the mag- and by real coincidences. This suppression was cal- netic field and their detection in the wrong hemisphere. culated from channel to channel from the measured Both effects and the influence of the correlation coeffi- TOF spectra and using the rate of random coinci- cients a and A were determined by Monte Carlo sim- dences [26]. In the reopened stop branch, only random ulations. The grid effect describes the influence of the M. Kreuz et al. / Physics Letters B 619 (2005) 263–270 269

Fig. 4. Fit to the experimental asymmetry, calculated from the spectra after background subtraction, for the two detectors. The fit function includes all corrections and is extrapolated outside the fitting range (250–450 keV, indicated by the dashed lines). The deviation of the data for low energies is caused by remaining background (see step (iii)). grounded carbon wires on the particle trajectories and Within its precision, it is in agreement with the Stan- was estimated using the program PENELOPE [27]. dard Model prediction Eq. (2) and the present world The effects of polarization and flipping efficiency were average Bworld = 0.983 ± 0.004 [6]. included in the fit function. Due to the small dependence of the asymmetry on the electron energy (same hemisphere in Fig. 1) the final result is not sensitive to 5. Outlook the detector function. The highest corrections are due to background and polarization. The systematic errors The systematic error of the measurement is domi- are treated as independent from each other and added nated by the contributions of 0.5% from polarization quadratically. and of 0.7% from background (random coincidences The statistical error results from the fit of B to the and proton suppression). The effects related to the de- experimental asymmetry α (Eqs. (3)–(5))shownin ep tector itself sum up to only 0.1%. Fig. 4. The used fitting range was 250 to 450 keV. For The main systematic errors can be addressed by dif- smaller electron energies background is limiting and ferent improvements: the influence of the scintillator the electrons can be reflected by the electric potential excitations or photomultiplier after-pulses can be sup- of the proton detectors, for higher energies the correc- pressed by requiring coincident triggers of at least two tion due to the magnetic mirror effect becomes large. photomultipliers for the stop signal. This is more ef- The results of the two detectors are fective than the software cut (i) used in the analysis stat syst as the full time resolution of the photomultipliers can BDet1 = 0.976 ± 0.008 ± 0.008 , be exploited. The random coincidences are easier to = ± stat ± syst BDet2 0.957 0.008 0.012 . calculate if all stop signals instead of only the first The systematic uncertainties do not include effects that one are registered. In this case the suppression of pro- cancel out by averaging the two detectors (e.g., shift ton signals by faster coincident signals no longer ex- of the beam relative to the maximum of the magnetic ists. Furthermore the new polarization analysis method field leading to opposite contributions to the magnetic provides a perfect means to polarize the beam, rais- mirror effect for the two detectors). The final result is ing polarization to a value of almost 1 and measurable with a precision of 10−3 [25]. Thus all major uncer- B = 0.967 ± 0.006stat ± 0.010syst = 0.967 ± 0.012. tainties can be reduced and a significant improvement (6) in the accuracy is achievable. 270 M. Kreuz et al. / Physics Letters B 619 (2005) 263–270

Acknowledgements [12] A.P. Serebrov, I.A. Kuznetsov, I.V. Stepanenko, A.V. Al- dushchenkov, M.S. Lasakov, Y.A. Mostovoi, B.G. Erozolim- We particularly acknowledge D. Dubbers who di- skii, M.S. Dewey, F.E. Wietfeldt, O. Zimmer, H. Börner, JETP 86 (1998) 1074. rectly motivated much of the work covered in this let- [13] H. Abele, S. Baeßler, D. Dubbers, J. Last, U. Mayerhofer, ter and who has given advice and support. We thank C. Metz, T.M. Müller, V. Nesvizhevsky, C. Raven, O. Schärpf, the target laboratories of the Technical University of O. Zimmer, Phys. Lett. B 407 (1997) 212. Munich and of the GSI Darmstadt who have provided [14] F. Glück, J. Joó, J. Last, Nucl. Phys. A 593 (1995) 125. the magnesium coated carbon foils pushing the tech- [15] F. Glück, Phys. Lett. B 436 (1998) 25. [16] D.E. Kraus, F.A. White, IEEE Trans. Nucl. Sci. NS-13 (1966) nical possibilities to the limit. We acknowledge the 765. support from all services of the ILL and from the Uni- [17] Computer Simulation Technology, MAFIA software package, versity of Heidelberg whose efforts have made this http://www.cst.de. work possible. [18] B. Brand, Ein Detektor für die Messung der Neutrinoasymme- trie beim Zerfall freier Neutronen, Diploma thesis, Universität Heidelberg, 2000. [19] A. Renau, F.H. Read, J.N.H. Brunt, J. Phys. E 15 (1982) 347. References [20] CPO (Charged Particle Optics Program), http://www. electronoptics.com. [1] J.C. Pati, A. Salam, Phys. Rev. D 10 (1974) 275. [21] F.H. Read, N.J. Bowring, P.D. Bullivant, R.R.A. Ward, Rev. [2] R.N. Mohapatra, J.C. Pati, Phys. Rev. D 11 (1975) 2558. Sci. Instrum. 69 (1998) 2000. [3] A.-S. Carnoy, J. Deutsch, B.R. Holstein, Phys. Rev. D 38 [22] H. Häse, A. Knöpﬂer, K. Fiederer, U. Schmidt, D. Dubbers, (1988) 1636. W. Kaiser, Nucl. Instrum. Methods A 485 (2002) 453. [4] B.G. Erozolimskii, Nucl. Instrum. Methods A 284 (1989) 89. [23] P. Høghøj, H. Abele, M. Astruc Hoffmann, S. Baeßler, [5] H. Abele, Nucl. Instrum. Methods A 440 (2000) 499. J. Reich, J. Nesvizhevsky, O. Zimmer, Nucl. Instrum. Meth- [6] S. Eidelman, et al., Review of Particle Physics, Phys. Lett. ods B 160 (2000) 431. B 592 (2004) 1, http://pdg.lbl.gov. [24] O. Schaerpf, N. Stuesser, Nucl. Instrum. Methods A 284 [7] J.D. Jackson, S.B. Treiman, H.W. Wyld, Phys. Rev. 106 (1957) (1989) 208. 517. [25] M. Kreuz, V. Nesvizhevsky, A. Petoukhov, T. Soldner, Nucl. [8] J. Deutsch, P. Quin, Symmetry-tests in semileptonic weak in- Instrum. Methods, in press. teractions: A search for new physics, in: P. Langacker (Ed.), [26] M. Kreuz, Messung von Winkelkorrelationen im Zerfall po- Precision Tests of the Standard Model, World Scientiﬁc, Sin- larisierter Neutronen mit dem Spektrometer PERKEO II, gapore, 1995, p. 706. PhD thesis, Universität Heidelberg, 2004, http://www.ub. [9] B.G. Erozolimsky, L.N. Bondarenko, Y.A. Mostovoy, uni-heidelberg.de/archiv/4799. B.A. Obinyakov, V.A. Titov, V.P. Zacharova, A.I. Frank, Phys. [27] J. Baró, J. Sempau, J.M. Fernández-Varea, F. Salvat, Nucl. In- Lett. B 33 (1970) 351. strum. Methods B 100 (1995) 31. [10] C.J. Christensen, V.E. Krohn, G.R. Ringo, Phys. Rev. C 1 (1970) 1693. [11] I.A. Kuznetsov, A.P. Serebrov, I.V. Stepanenko, A.V. Al- duschenkov, M.S. Lasakov, A.A. Kokin, Y.A. Mostovoi, B.G. Yerozolimsky, M.S. Dewey, Phys. Rev. Lett. 75 (1995) 794. Physics Letters B 619 (2005) 271–280 www.elsevier.com/locate/physletb

Observation of the ankle and evidence for a high-energy break in the cosmic ray spectrum

High Resolution Fly’s Eye Collaboration R.U. Abbasi a, T. Abu-Zayyad a, J.F. Amman b, G. Archbold a, R. Atkins a, J.A. Bellido c, K. Belov a,J.W.Belzd,S.Y.BenZvie,D.R.Bergmanf,∗,G.W.Burta,Z.Caoa, R.W. Clay c, B.C. Connolly e, W. Deng a,B.R.Dawsonc, Y. Fedorova a, J. Findlay a, C.B. Finley e,W.F.Hanlona,C.M.Hoffmanb, G.A. Hughes f, M.H. Holzscheiter b, P. Hüntemeyer a, C.C.H. Jui a,K.Kima,M.A.Kirnd,E.G.Loha,M.M.Maestasa, N. Manago h, L.J. Marek b,K.Martensa, J.A.J. Matthews g,J.N.Matthewsa, A. O’Neill e, C.A. Painter b, L. Perera f,K.Reila,R.Riehlea, M. Roberts g, M. Sasaki h, S.R. Schnetzer f, K.M. Simpson c,G.Sinnisb,J.D.Smitha,R.Snowa,P.Sokolskya, C. Song e, R.W. Springer a, B.T. Stokes a, J.R. Thomas a, S.B. Thomas a, G.B. Thomson f,D.Tupab,S.Westerhoffe,L.R.Wienckea,A.Zechf

a University of Utah, Department of Physics and High Energy Astrophysics Institute, Salt Lake City, UT, USA b Los Alamos National Laboratory, Los Alamos, NM, USA c University of Adelaide, Department of Physics, Adelaide, South Australia, Australia d University of Montana, Department of Physics and Astronomy, Missoula, MT, USA e Columbia University, Department of Physics and Nevis Laboratory, New York, NY, USA f Rutgers, The State University of New Jersey, Department of Physics and Astronomy, Piscataway, NJ, USA g University of New Mexico, Department of Physics and Astronomy, Albuquerque, NM, USA h University of Tokyo, Institute for Cosmic Ray Research, Kashiwa, Japan Received 17 January 2005; received in revised form 16 May 2005; accepted 27 May 2005 Available online 8 June 2005 Editor: H. Weerts

Abstract We have measured the cosmic ray spectrum at energies above 1017 eV using the two air ﬂuorescence detectors of the High Resolution Fly’s Eye experiment operating in monocular mode. We describe the detector, PMT and atmospheric calibrations,

* Corresponding author. E-mail address: [email protected] (D.R. Bergman).

0370-2693/$ – see front matter  2005 Elsevier B.V. All rights reserved. doi:10.1016/j.physletb.2005.05.064 272 High Resolution Fly’s Eye Collaboration / Physics Letters B 619 (2005) 271–280 and the analysis techniques for the two detectors. We ﬁt the spectrum to models describing galactic and extragalactic sources. Our measured spectrum gives an observation of a feature known as the “ankle” near 3 × 1018 eV, and strong evidence for a suppression near 6 × 1019 eV.  2005 Elsevier B.V. All rights reserved.

1. Introduction ter the atmosphere. The HiRes detector collects the fluorescence light emitted by EAS as they propa- The highest energy cosmic rays yet detected, of gate through the atmosphere. Charged particles in the 20 energies up to and above 10 eV, are interesting in shower excite nitrogen molecules which fluoresce in that they shed light on two important questions: how the ultraviolet (300 to 400 nm). The fluorescence yield are cosmic rays accelerated in astrophysical sources, is about five photons per minimum ionizing particle and how do they propagate to us through the cosmic per meter of path length [10]. As an EAS propagates microwave background radiation (CMBR) [1]?The through the atmosphere, the detector measures the acceleration of cosmic rays to ultra high energies is number of photons seen as a function of time and an- thought to occur in extensive regions of high magnetic gle. From this information, we reconstruct the geom- fields, regions which are expanding at relativistic ve- etry of the shower and the solid angle subtended by locities [2]. Such regions are rare and are to be counted the detector from each point of the shower. From the among the most violent and interesting objects in the number of photons collected, we reconstruct the num- Universe. ber of charged particles in the shower as a function Once accelerated, interactions between the ultra of the depth of the atmosphere traversed. We integrate high energy cosmic rays (UHECR) and the CMBR the energy deposited in the atmosphere [11] to find the cause the cosmic rays to lose energy. The strongest energy of the primary cosmic ray. energy loss mechanism comes from the production of UHECRs are thought to be protons or heavier nu- pions in these CMBR interactions at UHECR energies clei up to iron. While nucleus–nucleus collisions are × 19 above about 6 10 eV. This energy loss mecha- complex, the general features of the interaction can be nism produces the Greisen–Zatsepin–Kuz’min (GZK) + − understood in terms of a simple superposition model. suppression [3,4]. In addition, e e production in In this model each nucleon generates an independent these same interactions provides a somewhat weaker EAS. The superposition of many, lower energy show- energy loss mechanism above a threshold of about ers will result in an EAS with different statistical prop- × 17 5 10 eV. A third important energy-loss mechanism erties than an EAS produced by one high energy pro- at all energies comes from universal expansion. ton. This allows one to measure the composition of the In previous publications [5,6], we have reported on primary cosmic rays on a statistical basis. our measurements of the cosmic ray spectrum using data collected independently, in monocular mode, by the two detectors of the High Resolution Fly’s Eye experiment (HiRes). We here report on an updated mea- 2. The HiRes detectors surement of the flux of UHECR, covering an energy range from 2.5 × 1017 eV to over 1020 eV, using a The HiRes detectors have been described exten- significantly larger data set for the HiRes-II detector. sively elsewhere [12,13]. In brief, they consist of With the improved statistical power available in this spherical mirrors, of area 5.1 m2, which collect the data, we study two features in this spectrum: a break fluorescence light and focus it onto a cluster of 256 in the spectral slope at 3 × 1018 eV, called the “an- photomultiplier tubes arranged in a 16×16 array. Each kle” [7–9], and a steepening of the spectrum near the tube in the cluster views about one square degree of threshold for pion production. the sky. Time and pulse height information are col- The HiRes experiment performs a calorimetric lected from each tube. The HiRes detectors trigger on measurement of the energy of cosmic rays. UHECR and reconstruct showers that occur within a radius of produce extensive air showers (EAS) when they en- about 35 km. High Resolution Fly’s Eye Collaboration / Physics Letters B 619 (2005) 271–280 273

The HiRes-I detector is located atop Little Granite this intensity is traceable to NIST-calibrated photodi- Mountain on the US Army Dugway Proving Ground odes and is stable to about 2%, flash-to-flash. Sep- in west-central Utah. It consists of 21 mirrors, and arate calibrations of PMT gains using photoelectron their associated phototube arrays, arranged in one ring, statistics and using the absolute light intensity of the observing from 3 to 17 degrees in elevation and pro- xenon flash lamp agree within uncertainties. Xenon viding almost complete coverage in azimuthal angle. flash lamp data are collected about once a month. The detector uses a sample and hold readout system A second calibration system, using a frequency-tripled which integrates phototube pulses for 5.6 µs. This is YAG laser, is used to monitor phototube gains on a long enough to collect the signal from all cosmic ray night-to-night basis. We estimate that the relative cal- showers of interest. ibration techniques are accurate to about 3% with an The HiRes-II detector is located on Camel’s Back absolute calibration uncertainty of about ±10%. Ridge, also on Dugway Proving Ground, about The atmosphere is our calorimeter, but it is also 12.6 km SW of HiRes-I. It consists of 42 mirrors, the medium through which fluorescence light prop- arranged in two rings, covering from 3 to 31 degrees in agates to the detectors. To calculate the number of elevation and almost the whole azimuthal angle range. fluorescence photons emitted by a cosmic ray shower, This detector uses a flash ADC (FADC) readout sys- we must understand the way in which the atmosphere tem with a 100 ns sampling time. scatters this light between the EAS and the detector. In this Letter, we present data collected from June The molecular component of the atmosphere is quite 1997 to February 2003 for HiRes-I, and from Decem- constant, with only small seasonal variations, and the ber 1999 through September 2001 for HiRes-II. For Rayleigh scattering it produces is well understood. HiRes-II, this is about four times the data that was re- The aerosol content of the atmosphere can vary con- ported on previously. We collect data on nights when siderably over time, and with it, the amount of light the moon is down for three hours or more. In a typical scattered and its angular distribution. year each detector collects up to about 1000 hours of To measure these quantities, we perform an at- data. mospheric calibration using YAG lasers operating at The weather is clear about 2/3 of the time at the wavelength λ = 355 nm. At each of our two sites, we HiRes sites. Since clouds can reduce the experiment’s have a steerable beam laser which is fired in a pattern aperture, we record the existence of clouds by opera- of shots that covers the detector’s aperture, and which tor observations, infrared cameras, and evidence from is repeated every hour. The scattered light from the data collected by the detector (this consists primar- laser at one site is collected by the detector at the other ily of the upward going laser and flasher pulses, used site. The amount of detected light is then analyzed to to measure the atmospheric conditions, which have a determine the scattering properties of the atmosphere. distinct signature upon encountering a cloud; actual The properties that we measure are the vertical aerosol cosmic rays also appear emerging from clouds). Only optical depth (VAOD), the horizontal aerosol extinc- data from those nights in which the aperture is not tion length, and the aerosol scattering phase function reduced by cloud cover are used in our spectrum mea- (the angular distribution of the differential scattering surements. cross section). Because about half of the data from HiRes-I were collected before the lasers were installed, we use aver- 3. Calibration age values of the measured parameters in this analysis: a horizontal aerosol extinction length of 25 km (the av- The two most important calibrations we perform erage horizontal molecular extinction length is 17 km), are of the photomultiplier tube (PMT) gains [14,15], an average phase function, and a VAOD of 0.04 [5,16, and of the clarity of the atmosphere [16].Weuseasta- 17]. The atmosphere at our sites is quite clear: the av- ble xenon flash lamp, carried to each detector and used erage atmospheric correction to an event’s energy is to illuminate the photomultiplier array, to find PMT about 10% (see below for the effect on flux measure- gains. The xenon lamp produces a light intensity of ments). We are most sensitive to the value of VAOD. about 10 photons per mm2 at the face of the PMTs; The RMS of the VAOD distribution is 0.02, and we 274 High Resolution Fly’s Eye Collaboration / Physics Letters B 619 (2005) 271–280 use this RMS value as a conservative estimate of the lated from the phototube pulse heights. Corrections systematic uncertainty in this parameter. are made for atmospheric scattering of the light, and for other effects such as mirror reflectivity, phototube quantum efficiency, etc. A correction is made for the 4. HiRes-II data analysis Cerenkovˇ light produced by charged particles in the The analysis of the HiRes-II monocular data has shower. Both direct and scattered Cerenkovˇ light con- been described previously [6]. The data presented here tributions to the light seen by the PMTs are calculated were collected during 540 hours of good weather run- and subtracted. The number of charged particles is cal- ning, and consists of 21 million triggers, mostly of ran- culated from the fluorescence light at the shower using dom sky noise and events generated by atmospheric the fluorescence yield and its pressure and temperature lasers and other man-made light sources. Events were variation as given by Kakimoto et al. [10]. The result- selected that satisfied the following criteria: ing shower development profile, expressed as a function of slant depth, is fit to the Gaisser–Hillas parame- • angular speed 11◦ µs−1; terization [18] (this has been seen to fit UHE cosmic • selected tubes 6; ray showers quite well [11,19]). We integrate over the • photoelectrons/degree 25; Gaisser–Hillas function and multiply by the average • track length 7◦,or 10◦ for events extending energy loss rate of 2.19 MeV/(gcm−2) to calculate above 17◦ elevation; the energy of the primary cosmic ray. We then correct • zenith angle 80◦; for unobserved energy, mostly neutrinos and muons • in-plane angle 130◦; which hit the ground. This correction [17], which • in-plane angle uncertainty 30◦; varies from 10% at 3 × 1017 eV to 5% at 1019 eV, is 2 • 150 Xmax 1200 g/cm , and is within determined during the Monte Carlo calculation of the 50 g/cm2 of begin visible in detector; aperture. It is similar to the calculation in Ref. [11]. • average Cerenkovˇ correction 70%; A fraction of the HiRes-II events are also observed • geometry fit χ2/d.o.f. 10; by HiRes-I. In this case we perform a cross-check on • profile fit χ2/d.o.f. 10; our monocular determination of the shower geometry. • minimal trigger from signal tubes required after Fig. 1 shows a scatter plot of the energy using monocu- March 2001.

These cuts remove events in which the monocular geometric reconstruction is poor or in which the longitudinal profile cannot be determined accurately. The final event sample consisted of 2685 events covering × 17 = an energy range from 1.6 10 eV (log10 E 17.2) to 1020 eV. The geometry of each event is reconstructed using the time and angle information from the hit PMTs. First a pattern recognition step is performed to choose phototubes that lie on a line both in angle and in time. Next the plane that contains both the shower and the detector is determined from the azimuth and elevation of hit tubes; the angle of the shower in this plane is determined from a fit to phototube time and angle information. The resolution of shower-detector plane determination is about 0.6◦, and the in-plane angle un- ◦ certainty is 5 on average. Fig. 1. A scatter plot of the HiRes-II energy calculated using monoc- With the geometry determined, the profile of the ular geometry versus the energy calculated using the stereo geome- number of charged particles in the shower is calcu- try for those events observed in stereo. High Resolution Fly’s Eye Collaboration / Physics Letters B 619 (2005) 271–280 275 lar geometry versus the energy using stereo geometry, for those events in which such a comparison is possible. The energy resolution, including statistical and systematic effects and the uncertainty in reconstructing the shower geometry, has been calculated in the Monte Carlo simulation [17]. The overall resolution is ±17%. It improves from ±18% below 1018 eV to ±12% above 1019 eV.

5. Monte Carlo simulation

To calculate the aperture as a function of cosmic ray energy, a very accurate Monte Carlo (MC) simulation of the experiment was performed [6]. Two libraries of cosmic ray showers, one for proton primaries and one for iron primaries, were generated using the Corsika Fig. 2. Comparison of HiRes-II data and MC for the photoelectrons 5.61 [20] EAS simulation program and the QGSJet per degree of track. In the upper frame, the filled squares with the 01 [21] hadronic event generator. Events from these histogram are the data, the open squares are the MC. The lower libraries were placed by a detector simulation in the frame shows the ratio of data to MC for each bin. vicinity of the HiRes-II detector. This program also simulated the fluorescence and Cerenkovˇ light generated by the showers, and calculated how much of this light would have been collected by the detectors. A complete simulation of the optical path, trigger, and readout electronics was performed. This simulation followed the experimental conditions that pertained over the data-collection period. The results were written out in the same format as the data and analyzed by the same data analysis program described above. The stereoscopic energy spectrum of the Fly’s Eye experiment [7], in the form of a broken power law fit, and the composition measurements made by the HiRes/MIA hybrid experiment [22] and by HiRes in stereo [23] were used as inputs. To convince ourselves that the MC simulation is accurate, we compare many MC distributions of geometrical and kinematic variables to the data. The agreement in these comparisons is excellent and indicates 2 that we understand our detector. Fig. 2 shows the Fig. 3. Comparison of HiRes-II data and MC for the χ of a fit to the time vs. angle plot assuming a vertical shower. In the upper frame, brightness of showers: the number of photoelectrons the filled squares with the histogram are the data, the open squares per degree of track. The agreement between the data are the MC. The lower frame shows the ratio of data to MC for each and MC simulation indicates that the same amount of bin. light is collected in the MC as in the data. Fig. 3 shows the χ2 of a fit to the time vs. angle plot from which we that of the data. Fig. 4 shows a histogram of the num- determine shower geometry. The agreement here in- ber of events vs. the logarithm of their energy in eV. dicates that the resolution of the MC is the same as The agreement here shows that, when we use previ- 276 High Resolution Fly’s Eye Collaboration / Physics Letters B 619 (2005) 271–280

Fig. 5. Energy resolution using PCF, after bias correction. The histogram shows MC resolution, the data points show the data monocular resolution in stereo events. For the MC, EREC refers to the reconstructed, monocular energy, while EMC refers to the gener- Fig. 4. Comparison of HiRes-II data and MC for the reconstructed ated energy in the same events. For the data, EMONO refers to the energy. In the upper frame, the filled squares with the histogram are energy reconstructed using the monocular geometry (corresponding the data, the open squares are the MC. The lower frame shows the to EREC in the MC), while EST refers to the energy reconstructed ratio of data to MC for each bin. using the stereo geometry. ous measurements of the spectrum and composition in the MC, and a complete simulation of the accep- works poorly for events close to the detector (within tance, we reproduce the experiment’s energy depen- about 5 km), and for lower energy events (below dence. 3 × 1018 eV), where less of the shower profile is seen. These events are excluded from the HiRes-I monocular sample. The PCF also works poorly if too much 6. HiRes-I analysis Cerenkovˇ light contaminates the fluorescence signal; these events are cut also. In reconstructing MC events, The analysis of the HiRes-I monocular data has it is found that, even with these cuts, the resolution is also been described previously [5,6]. The main differ- somewhat worse than for HiRes-II, and that there is an ence from the HiRes-II analysis is that, with only one energy bias. ring of mirrors, most tracks are too short to reliably Since stereo events are seen in both detectors, they determine the geometry from timing alone. Although have excellent geometrical determination using the in- the determination of the shower-detector plane is still tersection of the two shower-detector planes. For these excellent, correlations between the fit distance to the events, comparison of the PCF reconstruction to the shower and the fit in-plane angle become large for stereo reconstruction shows the same energy resolu- short tracks. tion and bias as seen in the MC sample. Having confi- A reconstruction procedure using the pulse height dence that we understand the PCF, we correct for the information in addition to the tube angles and tim- bias. Fig. 5 shows the energy resolution of the PCF re- ing information has been developed: the profile con- construction for MC events and for stereo events after strained fit (PCF). The PCF uses the one-to-one corre- the correction. The agreement is excellent. lation between in-plane angle and shower profile: the Fig. 6 shows comparisons between the HiRes-I data in-plane angle with the best fit shower profile is cho- and the MC simulation for the distance to the shower sen as the in-plane angle of the shower. The Gaisser– core of showers in three energy bands. Again the Hillas function is used in this profile fit. The PCF agreement is excellent. High Resolution Fly’s Eye Collaboration / Physics Letters B 619 (2005) 271–280 277

Fig. 7. The calculated apertures of HiRes-I (red squares) and HiRes-II (black circles) as a function of energy.

studied [17,24]. The changes to the energy scale and flux are negligible. The limited elevation coverage of the HiRes-II Fig. 6. Shower core distance distribution using PCF, in HiRes-I data detector makes the aperture calculation sensitive to and in MC. The squares with error bars are the data, the histogram the composition assumptions used in the MC simula- is the MC. tion. This and other sources of systematic uncertainty (the given input spectrum and using an average at- 7. Systematic uncertainties mosphere) are considered in Ref. [24]. The composition assumptions have a negligible effect on the aper- The largest systematic uncertainties in the calcula- ture above an energy of 1018 eV, and give a systematic tions of energy are the absolute calibration of the pho- uncertainty of order the statistical uncertainty only at totubes (±10%) [15], the fluorescence yield (±10%) 3 × 1017 eV. [10], and the correction for unobserved energy in the shower (±5%) [11,17]. These three uncertainties, added in quadrature, give an uncertainty in the energy 8. Results of ±15%. This effect of this energy uncertainty in calculating the flux is ±27% [5]. Fig. 7 shows the calculated aperture of the two To test the sensitivity of the flux measurement to HiRes detectors. At an energy of 1020 eV the aperture atmospheric uncertainties, we generated new MC sam- is nearly 10 000 km2 sr. ples with VAOD values of 0.02 and 0.06, i.e., with the Fig. 8 shows the measured spectrum of cosmic rays average plus and minus one RMS value, and analyzed [25]. The spectrum has been multiplied by E3 for clar- them (and the data) using the same VAOD values. This ity. The closed squares (open circles) are the HiRes-I provides a conservative estimate of the flux uncer- (HiRes-II) measurements. For comparison to previous tainty since the systematic uncertainty in the average experiments, the up-triangles are the stereo Fly’s Eye VAOD is less than the RMS. The result was a change spectrum [7], and the down-triangles are the result of in the flux of ±15%. Adding this in quadrature with the Akeno Giant Air Shower Array (AGASA) [8].The the sources of systematic uncertainty described above HiRes-I and HiRes-II monocular measurements agree results in a net uncertainty of ±31%. This uncertainty with each other very well in the overlap region, and is common to the flux measurements from HiRes-I and are also in good agreement with the Fly’s Eye stereo HiRes-II. spectrum. The effect of using an average VAOD value, rather In this plot the ankle shows up clearly at 3 × 18 = than the changing but measured values has also been 10 eV (log10 E 18.5). The spectrum steepens 278 High Resolution Fly’s Eye Collaboration / Physics Letters B 619 (2005) 271–280

Fig. 8. E3 times the UHECR Flux. Results from the HiRes-I (red squares) and HiRes-II (black circles) detectors, the AGASA experiment (blue down-triangles) and the Fly’s Eye experiment (in stereo mode; magenta up-triangles) are shown. The Fly’s Eye points have been shifted to the = right by one quarter bin ( log10 E 0.025) for clarity. Also shown are two spectral law ﬁts to the HiRes-I and HiRes-II spectra as described in the text. The 1σ upper limits for two empty bins of each HiRes spectra are also shown.

× 19 = = again at 6 10 eV (log10 E 19.8). The AGASA point energy of log10 E 19.79, where we really have spectrum appears to continue unabated above this en- 11. The Poisson probability for 11 or fewer events with ergy while the HiRes spectrum falls above this point. a mean of 28 is 2.4×10−4. We therefore conclude that We test whether our data are consistent with this our data is not consistent with a continuation of the interpretation of the AGASA spectrum by fitting our spectrum unabated above the pion production thresh- data to a broken power law. This fit is also shown old. It is worth emphasizing that we have considerable on Fig. 8. This fit had two floating break points sep- sensitivity to such a continuation, but the data do not arating three regions of constant spectral slope. The support it. fit was performed using the normalized, binned max- A similar fit with only one break point has a χ2 imum likelihood method [26], which allows us to in- of 46.0 for 35 degrees of freedom, worse by nearly 16 clude sparsely populated and empty bins. The fitted than the fit above (a significance of ∼ 3.7σ ). The fitted = ± = ± break points are at log10 E (in eV) 18.47 0.06 break point is at log10 E 18.45 0.03, and the fitted and 19.79 ± 0.09. The fitted spectral slopes are γ = spectral slopes are γ = 3.32 ± 0.03 and 2.85 ± 0.05. 3.32± 0.04, 2.86± 0.04, and 5.2 ± 1.3. The χ2 for the A fit with no break points at all has a bad χ2 of 114 for fit is 30.1 for 33 degrees of freedom. If we extend the 37 degrees of freedom, demonstrating that the ankle is middle section of the fit (as shown by the red/gray line clearly observed in our data. The spectral slope is γ = in Fig. 8) to higher energies, our aperture predicts that 3.12 ± 0.01. This fit is shown on Fig. 8 as a cyan/light we should have 28.0 events above the second break gray line. High Resolution Fly’s Eye Collaboration / Physics Letters B 619 (2005) 271–280 279

9. Fitting the spectrum 10. Summary

We have measured the flux of ultrahigh energy cos- The implications of our spectrum measurement can mic rays from 1.6 × 1017 eV to over 1020 eV. Our ex- be explored using a toy model of UHECR. In this periment detects atmospheric fluorescence light from model, there are two types of sources, galactic and ex- cosmic ray showers and performs a calorimetric mea- tragalactic. We choose the galactic sources to be con- surement of cosmic ray energies. We perform calibra- sistent with the HiRes/MIA and HiRes stereo compo- tions of our detector and measure the light-scattering sition measurements [22,23]: we assign the iron com- properties of the atmosphere. The total systematic un- ponent of the cosmic ray flux to be galactic [27].This certainty in our spectrum measurement averages 31%. assignment is consistent with the expectation that the In our energy range we observe two features in the highest energy galactic cosmic rays should be those of UHECR spectrum visible through changes in the spec- the highest charge. The proton component we take to tral power law. We observe the ankle at 3 × 1018 eV. be extragalactic. We also have evidence for a suppression at a higher To describe the extragalactic cosmic rays, we as- energies, above 6 × 1019 eV. sume that all sources have the same power law spectrum, and that cosmic rays lose energy in propagating to the earth by pion and e+e− production from Acknowledgements the CMBR photons, and by the cosmological red shift [28]. The sources are assumed to be uniformly distrib- This work is supported by US NSF grants PHY- uted and to evolve in density by (1 + z)m. Fig. 9 shows 9321949, PHY-9322298, PHY-0098826, PHY- our spectrum result with the best fit superimposed on 0245428, PHY-0305516, PHY-0307098, by the DOE it. The fitted values m and of −γ , the spectral slope grant FG03-92ER40732, and by the Australian Re- of the spectrum at the source, are m = 2.6 ± 0.4 and search Council. We gratefully acknowledge the con- −γ = 2.38 ± 0.05. tributions from the technical staffs of our home in- stitutions and the Utah Center for High Performance Computing. The cooperation of Colonels E. Fischer and G. Harter, the US Army, and the Dugway Proving Ground staff is greatly appreciated.

References

[1] A.A. Penzias, R. Wilson, Astrophys. J. 142 (1965) 419; R.H. Dicke, et al., Astrophys. J. 142 (1965) 414; G.F. Smoot, et al., Astrophys. J. 396 (1992) L1. [2] R.J. Protheroe, in: M.A. DuVernois (Ed.), Topics in Cosmic Ray Astrophysics, Nova Science Publishing, New York, 2000, p. 258, astro-ph/9812055. [3] K. Greisen, Phys. Rev. Lett. 16 (1966) 748. [4] G.T. Zatsepin, V.A. Kuz’min, Pis’ma Zh. Eksp. Teor. Fiz. 4 (1966) 114, JETP Lett. 4 (1966) 78. [5] R. Abbasi, et al., Phys. Rev. Lett. 92 (2004) 151101. [6] R. Abbasi, et al., astro-ph/0208301, Astropart. Phys., in press. [7] D.J. Bird, et al., Phys. Rev. Lett. 71 (1993) 3401. [8] M. Takeda, et al., Astropart. Phys. 19 (2003) 447. [9] M. Ave, et al., Astropart. Phys. 19 (2003) 47. 3 Fig. 9. E times the UHECR Flux. Results from the HiRes-I (red [10] F. Kakimoto, et al., Nucl. Instrum. Methods Phys. Res. A 372 squares) and HiRes-II (black circles) detectors are shown. Also (1996) 527; shown is a ﬁt to a model described in the text. The 1σ upper limits M. Nagano, et al., Astropart. Phys. 20 (2004) 293; for two empty bins of each HiRes spectra are also shown. J. Belz, et al., FLASH Collaboration, in preparation. 280 High Resolution Fly’s Eye Collaboration / Physics Letters B 619 (2005) 271–280

[11] C. Song, Z. Cao, et al., Astropart. Phys. 14 (2000) 7. [20] D. Heck, J. Knapp, J.N. Capdevielle, G. Schatz, T. Thouw, [12] T. Abu-Zayyad et al., in: Proceedings of the 26th International CORSIKA: A Monte Carlo Code to Simulate Extensive Cosmic Ray Conference, Salt Lake City, vol. 5, 1999, p. 349. Air Showers, Report FZKA 6019 (1998), Forschungszentrum [13] J. Boyer, B. Knapp, E. Mannel, M. Seman, Nucl. Instrum. Karlsruhe. Methods Phys. Res. A 482 (2002) 457. [21] N.N. Kalmykov, S.S. Ostapchenko, A.I. Pavlov, Nucl. Phys. B [14] J.H.V. Girard, et al., Nucl. Instrum. Methods Phys. Res. A 460 (Proc. Suppl.) 52B (1997) 17. (2001) 278. [22] T. Abu-Zayyad, et al., Phys. Rev. Lett. 84 (2000) 4276. [15] J.N. Matthews, S.B. Thomas for the HiRes Collaboration, in: [23] R. Abbasi, et al., Astrophys. J. 622 (2005) 910. Proceedings of the 28th International Cosmic Ray Conference, [24] A. Zech, in: Proceedings of the Cosmic Ray International Sem. Tsukuba, 2003, p. 911. (CRIS ’04), Nucl. Phys. B (Proc. Suppl.) 136 (2004) 34. [16] K. Martens, L. Wiencke for the HiRes Collaboration, in: Pro- [25] The spectrum is available in tabular form at http://www. ceedings of the 28th International Cosmic Ray Conference, physics.rutgers.edu/~dbergman/HiRes-Monocular-Spectra. Tsukuba, 2003, p. 485; html. L. Wiencke for the HiRes Collaboration, in: Proceedings of the [26] S. Baker, R. Cousins, Nucl. Instrum. Methods 221 (1984) 437. 27th International Cosmic Ray Conference, Hamburg, 2001, [27] E. Waxman, Astrophys. J. 452 (1995) L1; p. 635. J.N. Bahcall, E. Waxman, Phys. Lett. B 556 (2003) 1. [17] A. Zech, PhD thesis, Rutgers University, 2004, http://www. [28] V. Berezinsky, A.Z. Gazizov, S.I. Grigorieva, hep-ph/0204357; physics.rutgers.edu/~aszech/thesis.html. S.T. Scully, F.W. Stecker, Astropart. Phys. 16 (2002) 271; [18] T. Gaisser, A.M. Hillas, in: Proceedings of the 15th Interna- D. De Marco, P. Blasi, A.V. Olinto, Astropart. Phys. 20 (2003) tional Cosmic Ray Conference, Plovdiv, vol. 8, 1977, p. 353. 53. [19] T. Abu-Zayyad, et al., Astropart. Phys. 16 (2001) 1. Physics Letters B 619 (2005) 281–287 www.elsevier.com/locate/physletb

A precision determination of the mass of the η meson

GEM Collaboration M. Abdel-Bary a,A.Budzanowskib,A.Chatterjeec,J.Ernstd,P.Hawraneka,e, R. Jahn d,V.Jhac, K. Kilian a, S. Kliczewski b, D. Kirillov f,D.Kolevg, M. Kravcikova h,T.Kutsarovai, M. Lesiak e,J.Liebj,H.Machnera,∗,A.Magierae, R. Maier a,G.Martinskak,S.Nedevl, N. Piskunov f, D. Prasuhn a,D.Protic´ a, P. von Rossen a,B.J.Roya,c, I. Sitnik f, R. Siudak b,d, M. Smiechowicz e,H.J.Steina, R. Tsenov g, M. Ulicny a,k,J.Urbana,d,G.Vankovaa,g, C. Wilkin m

a Institut für Kernphysik, Forschungszentrum Jülich, Jülich, Germany b Institute of Nuclear Physics, Polish Academy of Sciences, Krakow, Poland c Nuclear Physics Division, BARC, Bombay, India d Helmholtz-Institut für Strahlen- und Kernphysik der Universität Bonn, Bonn, Germany e Institute of Physics, Jagellonian University, Krakow, Poland f Laboratory for High Energies, JINR, Dubna, Russia g Physics Faculty, University of Sofia, Sofia, Bulgaria h Technical University, Kosice, Slovakia i Institute of Nuclear Physics and Nuclear Energy, Sofia, Bulgaria j Physics Department, George Mason University, Fairfax, VA, USA k P.J. Safarik University, Kosice, Slovakia l University of Chemical Technology and Metallurgy, Sofia, Bulgaria m Department of Physics and Astronomy, UCL, London, UK

Received 12 April 2005; received in revised form 31 May 2005; accepted 1 June 2005

Available online 14 June 2005

Editor: D.F. Geesaman

Abstract Several processes of meson production in proton–deuteron collisions have been measured simultaneously using a calibrated magnetic spectrograph. Among these processes, the η meson is seen clearly as a sharp missing-mass peak on a slowly varying background in the p + d → 3He + X reaction. Knowing the kinematics of the other reactions with well determined masses, it is possible to deduce a precise mass for the η meson. The ﬁnal result, m(η) = 547.311 ± 0.028(stat) ± 0.032(syst) MeV/c2,

* Corresponding author. E-mail address: [email protected] (H. Machner).

0370-2693/$ – see front matter  2005 Elsevier B.V. All rights reserved. doi:10.1016/j.physletb.2005.06.004 282 GEM Collaboration / Physics Letters B 619 (2005) 281–287 is signiﬁcantly lower than that found by the recent NA48 measurement, though it is consistent with values obtained in earlier counter experiments.  2005 Elsevier B.V. All rights reserved.

PACS: 13.75.-n; 14.40.Aq

Keywords: Eta meson production; Meson mass

Compared to other light mesons, the mass of the tor beam and the detector and hence to measure the η is surprisingly poorly known. Though the Particle η-mass with potentially a very small systematic er- Data Group (PDG) quote a value of mη = 547.75 ± ror. The crucial difference from SATURNE is that the 0.12 MeV/c2 in their 2004 review [1], this error hides spectrograph (Big Karl) that we have used has a large differences of up to 0.7MeV/c2 between the results acceptance. Therefore all the reactions could be stud- of some of the modern counter experiments quoted. ied simultaneously. The PDG average is in fact dominated by the re- Following ideas already developed in Ref. [6],we sult of the CERN NA48 experiment, mη = 547.843 ± have measured simultaneously the following three re- 0.051 MeV/c2, which is based upon the study of the actions: 0 kinematics of the six photons from the 3π decay of + p + d → π + t, (1) 110 GeV η-mesons [2]. In the other experiments em- + ploying electronic detectors, which typically suggest a p + d → t + π , (2) 2 mass ≈ 0.5MeV/c lighter, the η was produced much p + d → 3He + η, (3) closer to threshold and its mass primarily determined where in each case it was the first particle on the right through a missing-mass technique where, unlike the that was measured. NA48 experiment, precise knowledge of the beam mo- Charged particles were detected with the help of mentum plays an essential part. In the Rutherford Lab- Big Karl, a focussing 3Q2DQ magnetic spectrograph oratory experiment the momentum of the pion beam − whose principal elements are indicated in Fig. 1.It in the π + p → n + η reaction was fixed macro- scopically using the floating wire technique [3].Inthe measurement making use of the photoproduction reaction γ + p → p + η, the energy of the electrons that were the source of the bremsstrahlung photons was fixed to a relative precision of 2 × 10−4 by measuring the distance of the beam paths in the third race track microtron of the MAMI accelerator [4].Inthe Saclay SATURNE experiment a high resolution, but small acceptance, spectrometer was used and, through an ingenious series of measurements on different nuclear reactions, the beam energy and spectrograph settings were both calibrated. The value of the η-mass was then extracted from the missing-mass peak in the p + d → 3He + X reaction [5]. In an attempt to clarify the situation, we have performed an experiment at COSY in Jülich specifically designed to determine the η-mass with high precision. The methodology is very similar in spirit to that used at SATURNE [5] in that several reactions were Fig. 1. Sketch of the Big Karl magnetic spectrograph and the focal measured, thus allowing one to calibrate the accelera- plane detector arrangement. GEM Collaboration / Physics Letters B 619 (2005) 281–287 283 should be noted that the final quadrupole magnet Q3 We start by discussing the principles of a momen- was not actually used in this study. We define the beam tum measurement with the magnetic spectrograph. to be incident in the z-direction, with the y-direction At any specified position in the system, an arbitrary being vertical, and the x-direction horizontal and per- charged particle is represented by a column vector V , pendicular to the beam. The optics with respect to the whose components are the positions, angles, and mo- horizontal and vertical motions are almost decoupled mentum of the particle with respect to the reference in Big Karl [7]. In the horizontal direction the spectro- trajectory, which is chosen to be the z-axis. This vec- graph has a point-to-point imaging from the target to tor then reads: the focal plane with dispersion whereas, in the vertical   x direction, it operates in the point-to-parallel mode.    x  Tracks were measured with two packs of multiwire    y  drift chambers (MWDCs). Each pack consists of six V =   , (4)  y  layers, allowing a precise determination of the posi-   tion of a charged particle. The drift time measurement l was started by a signal from the hodoscope layer P δ and an individual drift time calibration was performed where we have used the following definitions: for each particle type. Signals from hodoscope layers P and R, approximately 3.5 m apart, were used for a x is the horizontal displacement of an arbitrary ray time-of-flight measurement. Together with specific en- (or particle track) with respect to the assumed central ergy loss in the scintillators and the momentum vector, trajectory. this allowed particle identification and hence a deter- x is the tangent of the angle that this ray makes in mination of the energy of the particle. It is important the horizontal plane with respect to the assumed cen- to note that at a beam momentum around 1640 MeV/c tral trajectory. all three reactions can be observed simultaneously y is the vertical displacement of the ray with re- with a single setting of the Big Karl magnetic fields. spect to the assumed central trajectory. The first two reactions were used to calibrate the beam y is the tangent of the vertical angle of the ray and the spectrograph with the third determining mη. with respect to the assumed central trajectory. The precision of a missing-mass measurement de- l is the path length difference between the actual pends on the accuracy with which the four-momentum ray and the central trajectory. vectors of the incident particles in the entrance chan- δ = p/p is the fractional momentum deviation nel and of the detected particle in the exit channel of the ray from the assumed central trajectory which are known. In order to define well the reaction ver- corresponds to the assumed Big Karl central momen- tex, a liquid deuterium target only 2 mm thick was tum pBK. employed [8]. The Mylar windows were only 1 µm thick, thus making background reactions in the win- For any two different positions in the overall system, dow material negligible. The target was operated at a such as the target (t) and the focal plane (f ), the corre- temperature of 18.7 K which can lead to freezing out sponding vectors are connected through the transport of residual gas on the windows. The target was there- matrix R fore cleaned by warming it up periodically. The proton beam at COSY was electron-cooled at injection energy V t = RVf . (5) and then stochastically extracted, which resulted in the following beam properties (uncooled beam prop- The matrix elements were not necessary constant −5 −4 erties in brackets): p/p0 = 7.5 × 10 (3.2 × 10 ), and, where needed, they were expanded in powers px/p0 = 0.9 (1.8) mrad, py/p0 = 0.8 (5.8) mrad. of δ. The most important parameter in the transport It can be seen that electron cooling gave an impor- matrix for the reconstruction of the momentum in the tant improvement for this experiment. Another benefit present case is the element R16, which is the disper- from the cooling was the reduction in the beam halo sion. Some of the other elements are either zero or and hence in the associated background. small and can be neglected, whereas others, including 284 GEM Collaboration / Physics Letters B 619 (2005) 281–287

R16, must be determined using data from calibration Although the target was rather thin, corrections measurements, which will be discussed now. were made for the energy losses of the particles. While A low intensity 793 MeV/c proton beam and empty for pions this correction is negligible, and for tritons it target were used to investigate the dependence of R is modest, for 3He ions it is significant. We then pro- on δ. For this purpose the central momentum pBK ceeded as follows. The 804.5MeV/c setting was, in a was changed and the proton tracks reconstructed in first approximation, assumed to be exact and constant the drift chambers. Before each such measurement the with time. The beam momentum, target thickness, and magnetic fields in the dipoles were set and measured η-mass were then free parameters to be fit to the data. with nuclear magnetic resonance probes; the differ- In a second step, we checked the approximations made ences between the predicted and measured values of by inverting the calibration method. The spectrograph the field was of order 10−5. We measured in this way was assumed to be at its nominal value and the missing twice at 17 values of the central momentum. masses of the particles were derived. This comparison The next calibration was based upon detecting the yields a measure of the precision to which our method deuterons from the p + p → d + π + reaction, also at works and give us an estimate of the systematic errors. 793 MeV/c. At this momentum the spectrograph has In Fig. 2 coincident events from reactions (1) and full acceptance for this reaction and this momentum is (2) are shown in terms of their longitudinal and trans- close to 804.4MeV/c, the central spectrograph mo- verse momentum components. The expected kine- mentum, where all three reactions (1), (2), and (3) fit matic loci are rotational ellipsoids in three dimensions. into the acceptance. In the time-of-flight part of the Projections of their surfaces are shown as curves. set-up shown in Fig. 1 we used an additional scintil- The beam momentum was deduced from the mea- lator layer, S, in the veto mode. A 5 cm aluminum surement of the pion four-momenta, which are almost absorber was placed between this and the R layer. unaffected by the target thickness (see Fig. 2). The This thickness was sufficient to stop deuterons, but target thickness was then deduced from the measure- not protons with the same momentum. This reduced ment of the four-momenta of the tritons. The results the background originating from the direct beam proof these two measurements are shown in Fig. 3 as tons that have a momentum close to those of detected functions of the time of measurement. The beam mo- deuterons. mentum was found to be quite stable, with a variation Finally we studied the p + p → π + + d reaction at a beam momentum of 1642 MeV/c. The central momentum of the spectrograph was again varied and the pions measured for 12 different field settings. From each calibration experiment the values of the possible parameters were extracted and these were used as start values to fix all the elements of the transport matrix Rij in one least-squares fit to all of the calibration measurements. For the production runs measuring reactions (1), (2) and (3) simultaneously, the spectrograph was set to the nominal momentum pBK = 804.5MeV/c. The experiment was performed in a series of runs that were analyzed separately. This analysis yielded an unexpected result, indicating a change in some parameters with time. One possible cause could have been a variation of the magnetic fields in the dipoles. However, this is Fig. 2. Plot of (px ,pz) and (py ,pz) for coincident events from reactions (1) and (2). The solid curves give the predictions for the contrary to the very high precision measurements of mean loci for such coincidences. The points in a restricted range of the fields with NMR probes and so the variation must ordinate values with a larger scatter on the abscissa are the (px ,pz) have another origin. This was found when inspecting data points. The different scatter result from the different projections the target thickness as a function of time. of a rotational ellipsoid onto the different planes. GEM Collaboration / Physics Letters B 619 (2005) 281–287 285

Fig. 4. The massing mass spectrum from the p + d → 3He + X reaction. The solid curve shows a ﬁt with a Gaussian peak superimposed on a linear background.

from the beginning to the end of the experiment of only 3 × 10−5! However, the target thickness showed a steady increase with time. Sixty hours after the beginning of the experiment there was an interruption during which the target was warmed up and any possible freeze-out on the windows was removed. It should be noted that the increase of effective target thickness corresponds to a freeze-out of ≈ 100 µm frozen air within ≈ 70 hours. As shown in Fig. 3(b), after the interruption at 88 hours the effective target thickness may have started to increase once again. We are now in a position to extract the value of mη from the missing-mass distribution in the p + d → 3He + X reaction using the four-momenta of the 3He- ions measured simultaneously with reactions (1) and (2). Our extracted values of the η-mass are shown as a function of measuring time in Fig. 3(c) where, because of the limited count rate, several runs have been grouped together. No correlation is visible with the other two reconstructed observables. Also shown is the mean value and the uncertainty. The combined missing-mass distribution for all events is shown in Fig. 4, together with a ﬁt in terms of a Gaussian peak on top of an almost constant linear background corresponding to multipion production. The width of the p + d → 3He + η peak is in accord with Monte Carlo Fig. 3. (a) The reconstructed beam momentum pbeam from mea- simulations of this reaction. sured pions in reaction (1). (b) The target thickness deduced from To get an estimate of some of the systematic er- the tritons in reaction (2), and (c) the η-mass as functions of the time of measurement. The error bars shown are purely statistical. rors, we investigated the inﬂuence of the assumption The thick solid line denotes the mean and the two thin lines the that the mean momentum setting of the spectrograph is ±1σ error band. known. For this purpose we applied the deduced para- 286 GEM Collaboration / Physics Letters B 619 (2005) 281–287

Fig. 5. The deviation of the measured missing-mass mm from the rest mass m(x) [1] for the particle types x as function of the deviation of the mean momentum setting δ of the spectrograph. The horizontal lines indicate the 1σ band of ±28 keV/c2. Fig. 6. The results of the η-mass measurements, in order of publication date, taken from the Rutherford Laboratory (RL) [3],SAT- meters and kept the mass of the measured particle as a URNE [5],MAMI[4],NA48[2], and GEM. When two error bars are shown, the smaller is statistical and the larger total. variable. In the case of the direct beam this is of course zero whereas for p + p → d + π + at 793 MeV/c it is + + agreement with the other results. This is very puzzling the π and for p + p → π + d at 1642 MeV/c it + in that the NA48 experiment yields an excellent value is the deuteron. The deviations from the p, π and d for the K0 mass, also through the 3π 0 decay, though masses are shown in Fig. 5. the statistics were then much higher and the systemat- The mean mass differences are ±20 keV/c2 for ics not completely identical. protons, ±32 keV/c2 for pions, and ±21 keV/c2 for deuterons. The average of these, which is one measure of the systematic error, is ±28 keV/c2, and this Acknowledgements interval is shown in Fig. 5. Now there seems to be a stronger deviation of the missing mass from the true The quality of the beam necessary for the success value for larger positive values of δ but it is important of this work is due mainly to the efforts of the COSY to note that the η-mass was determined at the position operator crew. Support by Internationales Büro des δ ≈−2.8% where the deviation is minimal. BMBF (IND 01/022), SGA, Slovakia (1/1020/04), and Another systematic error arises from the uncer- the Forschungszentrum Jülich is gratefully acknowl- tainty in the liquid deuterium density depending on the edged. We acknowledge European Community— target temperature. This uncertainty was studied with Access to Research Infrastructure action of the Im- the help of the codes GEANT3 [9] and SRIM [10], proving Human Potential Programme. gives only an additional 0.004 MeV/c2 to the systematic error. The ﬁnal result of our measurement is References 2 mη = 547.311 ± 0.028(stat) ± 0.032(syst) MeV/c . [1] Particle Data Group, S. Eidelman, et al., Phys. Lett. B 592 (6) (2004) 1. Our value of the mass of the η meson is compared [2] A. Lai, et al., Phys. Lett. B 533 (2002) 196. in Fig. 6 with the results of all other measurements [3] A. Duane, et al., Phys. Rev. Lett. 32 (1972) 425. [4] B. Krusche, et al., Z. Phys. A 351 (1995) 237; taken into consideration in the current PDG compi- B. Krusche, Habilitationsschrift, Justus-Liebig-Universität, lation [1]. Though signiﬁcantly smaller than that re- Gießen, 1995. ported in the NA48 experiment [2], it is in excellent [5] F. Plouin, et al., Phys. Lett. B 276 (1992) 526. GEM Collaboration / Physics Letters B 619 (2005) 281–287 287

[6] GEM Collaboration, M.G. Betigeri, et al., Nucl. Instrum. [8] V. Jaeckle, et al., Nucl. Instrum. Methods A 349 (1994) 15. Methods A 426 (1999) 249. [9] GEANT3 manual, CERN Program Library, W5013. [7] GEM Collaboration, M. Drochner, et al., Nucl. Phys. A 643 [10] J.F. Ziegler, J.P. Biersack, Program SRIM2003, http://www. (1998) 55. srim.org. Physics Letters B 619 (2005) 288–292 www.elsevier.com/locate/physletb

Remark on double diffractive χ meson production

Adam Bzdak

M. Smoluchowski Institute of Physics, Jagiellonian University, Reymonta 4, 30-059 Kraków, Poland Received 25 April 2005; received in revised form 10 June 2005; accepted 10 June 2005 Available online 17 June 2005 Editor: N. Glover

Abstract 0 0 The double pomeron exchange contributions to the central inclusive and exclusive χc and χb mesons production in the Bialas–Landshoff approach are calculated. We ﬁnd the model to be consistent with the preliminary CDF upper limit on double 0 diffractive exclusive χc production cross section.  2005 Elsevier B.V. All rights reserved.

PACS: 12.40.Nn; 13.85.Ni; 14.40.Gx

1. Introduction [11–13], see also [14,15]. In the exclusive DPE event the central object χ is produced alone, separated from The study of the double pomeron exchange (DPE) the outgoing hadrons by rapidity gaps: production processes is interesting in its own right. It is an ideal way to improve our understanding of dif- pp¯ → p + gap + χ + gap +¯p. fractive processes and the dynamics of the pomeron exchange. In the central inclusive DPE event an additional radia- However, the great interest in such reactions is tion accompanying the central object is allowed. caused by the possibility of the DPE processes to be The basis for our considerations is the Bialas– one of the main mechanisms leading to Higgs boson Landshoff model for central inclusive double diffrac- production [1–10] within a very clean experimental tive Higgs boson production [2]. We showed that the environment. Bialas–Landshoff model and its exclusive extension In the present Letter we are particularly interested [10] give satisfactory description of the DPE central in the exclusive and central inclusive (central inelastic) inclusive and exclusive dijet cross sections [10,16]. DPE production of heavy quarkonium states χc and χb In this Letter we show that the exclusive extension of the Bialas–Landshoff model is consistent with the preliminary CDF upper limit on double diffractive ex- 0 E-mail address: [email protected] (A. Bzdak). clusive χc production cross section [17].

2. Central inclusive χ meson production

In the Bialas–Landshoff approach pomeron exchange corresponds to the exchange of a pair of non- perturbative gluons which takes place between a pair of colliding quarks [18]. Thus, the process of χ meson production is described by a sum of the four (reggeized) diagrams shown in Fig. 1 where dashed lines correspond to the non-perturbative gluons. The χ coupling is taken to be through a c-quark and b- quark loop for χc and χb, respectively. Fig. 1. Four diagrams contributing to the amplitude of the process Our calculation follows closely that of Ref. [2] of χ meson production by double pomeron exchange. The dashed where the DPE contribution to the central inclusive lines represent the exchange of the non-perturbative gluons. The χ coupling is taken to be through a c-quark and b-quark loop for χ Higgs boson production is calculated. c and χb, respectively. First, one calculates the (non-reggeized) production amplitude in the forward direction i.e. for vanishing transverse momenta of the produced χ meson and of the final hadrons. In this case the sum of the diagrams of Fig. 1 can be approximately replaced by the s-channel discontinuity of the first one [2],shownin Fig. 2. In the second step one introduces phenomenologically the effects of reggeization, transverse momentum dependence and the gap survival so that the full ampli- Fig. 2. Putting the inner quark lines “on-shell” is equivalent to cal- tude becomes:1 culating the amplitude for χ meson production by double pomeron exchange in the forward scattering limit i.e. t1 = t2 = 0. − − s α(t2) 1 s α(t1) 1 M = M0 momentum carried by the non-perturbative gluon):2 s1 s2 D p2 = D exp p2/τ 2 , (2) ×F(t1)F (t2) exp β(t1 + t2) Sgap. (1) 0 with τ = 1 GeV and D G2τ = 30 GeV−1 [2] where G M 0 Here 0 is the amplitude in the forward scattering is the scale of the process independent non-perturba- = + limit given by the diagram shown in Fig. 2. α(t) 1 tive quark gluon coupling. + ≈ α t is the pomeron Regge trajectory with 0.08, The factor S takes the gap survival effect into = −2 = + 2 = + 2 gap α 0.25 GeV . s (p1 p2) , s1 (k1 P) , account i.e. the probability (S2 ) of the gaps not to be = + 2 = − 2 = − 2 gap s2 (k2 P) , t1 (p1 k1) , t2 (p2 k2) where populated by secondaries produced in the soft rescat- = p1, p2, k1, k2 and P are defined in Fig. 2. F(t) tering. It is not a universal number but it depends on = −2 exp(λt) is the nucleon form-factor with λ 2GeV . the initial energy and the particular final state. Theo- + = The phenomenological factor exp(β(t1 t2)) with β S2 −2 retical predictions of the gap survival factor gap can 1GeV takes into account the effect of the momen- be found in Ref. [19]. In our calculations, follow- tum transfer dependence of the non-perturbative gluon ing [10,16], we take for the Tevatron (LHC) energy 2 propagator given by (p is the Lorentz square of the 2 2 2 = Sgap/(G /4π) 0.6 (0.25).

1 2 For DPE central inclusive χc and χb mesons production we can As was stated in Ref. [2] there is no reason to believe that the neglect the additional gap spoiling effect i.e. the Sudakov effect. It true form of D is as simple as this. We hope it is not a serious ob- is justified by the relatively small masses of the produced mesons. jection to our model. 290 A. Bzdak / Physics Letters B 619 (2005) 288–292 √ √ max max Before we present the details of our calculation Here y,ln(Mχ0 /(δ2 s)) y ln(δ1 s/Mχ0 ), 0 one point must be emphasized. Here we calculate in a is a rapidity of the produced χ where δ1 and δ2 are model dependent way the non-perturbative part of the defined as δ1,2 ≡ 1 − k1,2/p1,2 (k1, k2, p1, p2,are whole amplitude. The contribution coming from small max = defined in Fig. 2). In the following we take δ1 distances, being another part of the whole amplitude, max = = δ2 δ 0.1. is beyond our approach. However, for the case of χ The factors C and R are defined as: meson production the main part of the cross section 2 Γ 0 S may has a non-perturbative origin. 18 χ 1 2 6 2 gap C = D0G τ τ , (7) (12π)6 M3 α 2 (G2/4π)2 Following the calculation presented in Ref. [2] we χ0 M 3 4 find 0 for colliding hadrons to be in the form: 4 3 = 2 2 − 2 = 2G D0 2 λ J ν 2 2 R 9 dQ Q exp 3Q 1. (8) M = d Q p V p exp −3Q/τ . 0 π 2 1 λν 2 (3) In the expression (6) R2 reflects the structure of the Here Q is the transverse momentum carried by each loop integral and is shown explicitly for the reason J → J of the three gluons. Vλν is the gg χ vertex de- which will become clear in the following. pending on the polarization J of the χJ meson state. Performing integration over the whole range of ra- It was shown [13] that the DPE contribution to χ1 and pidity we obtain the following result for the central χ2 production in the forward scattering limit is vanish- inclusive total cross section: ing (either perturbative or non-perturbative two gluon s 2 CR2 exchange models). σ = √ 2 M (λ + β)/α + ln[ s/M 0 ] It turns out that for J = 0 we have the following χ0 χ simple result [2,11,12]: (λ + β)/α + ln(δs/M2 ) × χ0 2 ln . (9) sQ (λ + β)/α − ln δ pλV 0 pν = A, (4) 1 λν 2 2M2 χ0 This completes the calculation of the cross section. where A is expressed by the mass Mχ0 and the width 0 Γχ0 of the χ meson through the relation: 3. Sudakov factor A2 Γ 0 = . The calculation presented in the previous section, χ ! (5) 2 4πMχ0 based on the original Bialas–Landshoff model, is a Substituting (3) with (4) and (5) to the full ampli- central inclusive one, i.e. the QCD radiation accom- tude (1) and then performing the appropriate calcula- panying the produced object is allowed.5 Thus, to de- tions [16] we find the differential cross section dσ/dy scribe the exclusive processes where the central object for DPE central inclusive χ0 meson production to be is produced alone one has to forbid this radiation. To in the form: this end we shall include the Sudakov survival factor T(Q,µ) [5] inside the loop integral over Q (8). dσ s 2 CR2 = √ . The Sudakov factor T(Q,µ) is the survival proba- 2 2 2 dy M ((λ + β)/α + ln[ s/M 0 ]) − y χ0 χ bility that a gluon with transverse momentum Q re- (6) mains untouched in the evolution up to the hard scale µ = M/2. 3 Naturally a question of internal consistency arises. The calculation of M0 for colliding hadrons is performed in two steps. First, we calculate the diagram presented in Fig. 2 for Namely, the Sudakov factor uses perturbative gluons quark–quark scattering. Next, we multiply by a factor 32 to take the whilst in our calculations of the Born amplitude (3) presence of three quarks in each (anti)proton into account. 4 This formula is only valid in the limit of δ1,2 1whereδ1,2 ≡ 5 1 − k1,2/p1,2, and for small momentum transfer between initial and Let us notice that the central inclusive process contains exclu- final quarks. sive process. It implies that the relation σc.incl. >σexcl. holds. A. Bzdak / Physics Letters B 619 (2005) 288–292 291

µ2 we used non-perturbative gluons. We hope however ˜ 2 2 2 that taking the Sudakov factor in the loop integral into R(µ) = 9 dQ Q exp −3Q T(Q,µ). (13) account we obtain an approximate insight into exclu- 0 sive processes. Moreover, it is shown [10] that such approach leads to the satisfactory description of the DPE exclusive dijet production cross sections [20,21]. 4. Exclusive χ meson production. The CDF result We take the function T(Q,µ) to be in the form [5]: Now we are ready to give our predictions for DPE exclusive χ0 and χ0 mesons production at the Teva- µ2 c b 2 2 αs(k) dk tron and the LHC energies. The masses and widths of T(Q,µ)= exp − 0 0 = 2 the χc and χb are taken as follows: Mχ0 3.4GeV, 2π k c 2 M 0 = 9.8GeV,Γ 0 = 15 MeV, Γ 0 = 2.15 MeV. Q χb χc χb Taking into account (13) we find the effective Sudakov 1−∆ suppression of the cross section to be (µ = M/2): × zPgg(z) + Pqg(z) dz . ˜2 0 = q R χc 0.3, 0 (10) ˜2 0 = R χb 0.07. (14) Here ∆ =|k|/(µ +|k|), Pgg(z) and Pqg(z) (we take q = u, d, s, u,¯ d,¯ s¯) are the GLAP spitting func- In Table 1 the predictions for the Tevatron and the 6 tions. αs is the strong coupling constant. Taking into LHC energies are shown. As was discussed earlier we = 2 2 2 account the leading-order contributions [22] to the take δ 0.1 and assume Sgap/(G /4π) to be 0.6 GLAP splitting functions: (0.25) for the Tevatron (LHC) energy. As can be seen from Table 1 the obtained results for P (z) = 6 z/(1 − z) + (1 − z)/z + z(1 − z) gg the LHC energy are comparable or even smaller than + δ(1 − z)(11/2 − nf /3) , those for the Tevatron energy. It is due to the faster decrease of the rapidity gap survival factor S2 with = 2 + − 2 gap Pqg(z) z (1 z) /2, (11) increasing energy than the increase of our “soft” en- 2 we obtain: ergy dependence s . It should be noted, however, that the s dependence is the most serious uncertainty of our 1−∆ approach and the results presented in Table 1 should 2 3 zPgg(z) dz =−11/2 + 12∆ − 9∆ + 4∆ be regarded only as an order of magnitude estimates. Bearing in mind this uncertainty our results seem to be 0 comparable with those obtained in Refs. [11–13]. − 4 − 3∆ /2 6ln∆, At this point one comment is to be in order. At first 1−∆ sight it seems that the calculation presented in this Let- P (z) dz = 1/3 − ∆/2 + ∆2/2 − ∆3/3. (12) ter based on the model with the effective gluon prop- qg 2 agator of the exponential form D0 exp(−Q) can be 0 only used to estimate the contribution coming from the Now to describe the exclusive processes we use the formulas (6), (9) with R2 = 1 replaced by R˜2(µ) ˜ 7 Table 1 where R(µ) is defined as: 0 0 Our results for DPE exclusive χc , χb mesons production cross section for the Tevatron and the LHC energies √ 6 2 sdσ/dy(y= 0)σ In the following we take αs at one loop accuracy i.e. αs (q ) = 2 2 0 (4π/β0)(1/ ln(q /Λ )) with β0 = 9andΛ = 200 MeV. Below q = χc 2 TeV 45 nb 370 nb 0.8 GeV we freeze αs to be 0.5. 14 TeV 30 nb 350 nb 2 0 7 µ 2 2 × χb 2 TeV 0.05 nb 0.3 nb Notice that µ>1.5 GeV is required so that 9 0 dQ Q 2 14 TeV 0.03 nb 0.3 nb exp(−3Q) = 1. 292 A. Bzdak / Physics Letters B 619 (2005) 288–292

Table 2 are hold. The recently proposed exclusive extension 0 Comparison of the CDF upper limit for DPE exclusive χc meson of the Bialas–Landshoff model was found to be con- production with the result obtained in the presented model. The sat- sistent with the preliminary CDF upper limit on double isfactory consistency is observed 0 √ diffractive exclusive χc production cross section. s = 2TeV,|y| < 0.6 CDF upper limit Model 0 ± χc 80 100 nb 55 nb Acknowledgements relatively low Q. However, it is not the case. The rea- I would like to thank Dr. Leszek Motyka for helpful son is following. As was mentioned before we take for discussions. This investigation was supported by the 2 2 2 = the Tevatron energy Sgap/(G /4π) 0.6 [10,16].If Polish State Committee for Scientiﬁc Research (KBN) 2 we assume the gap survival factor Sgap to be 0.05 [12, under grant 2 P03B 043 24. 19] we obtain the non-perturbative coupling G2/4π to be 0.3. From the magnitude of the total cross section 2 −2 References we conclude that D0G = 30 GeV [2] what allows −2 us to extract D0 = 8GeV . Now if we compare the numerical values of our non-perturbative gluon propa- [1] A. Schafer, O. Nachtmann, R. Schopf, Phys. Lett. B 249 (1990) − 2 2 331. gator 8 exp( Q) with the perturbative one 1/Q we [2] A. Bialas, P.V. Landshoff, Phys. Lett. B 256 (1991) 540. 2 2 2 observe that 8 exp(−Q)>1/Q for 0.15 < Q < [3] J.R. Cudell, O.F. Hernandez, Nucl. Phys. B 471 (1996) 471. . 2 [4] D. Kharzeev, E. Levin, Phys. Rev. D 63 (2001) 073004. 3 3GeV . It means that the results presented in Table 1 [5] V.A. Khoze, A.D. Martin, M.G. Ryskin, Eur. Phys. J. C 23 include contribution coming not only from the non- (2002) 311. 2 2 perturbative region, say Q < 1GeV , but also from [6] B. Cox, J. Forshaw, B. Heinemann, Phys. Lett. B 540 (2002) 2 2 263. the quasi-perturbative region 1 < Q < 3.3GeV. [7] M. Boonekamp, R. Peschanski, C. Royon, Phys. Rev. Lett. 87 Moreover, we have checked that the contribution com- (2001) 251806. 2 2 ing from the perturbative region Q > 3.3GeV with [8] R. Enberg, G. Ingelman, A. Kissavos, N. Timneanu, Phys. Rev. the non-perturbative gluon propagators replaced by Lett. 89 (2002) 081801. perturbative ones is negligible. [9] V.A. Petrov, R.A. Ryutin, Eur. Phys. J. C 36 (2004) 509. At the end, let us notice that the result dσ/dy(y = [10] A. Bzdak, Phys. Lett. B 615 (2005) 240. = 0 [11] V.A. Khoze, A.D. Martin, M.G. Ryskin, Eur. Phys. J. C 19 0) 45 nb for DPE exclusive χc production is con- (2001) 477; sistent with the preliminary CDF Run II upper limit V.A. Khoze, A.D. Martin, M.G. Ryskin, Eur. Phys. J. C 20 [17],seeTable 2. Indeed, if we integrate the differen- (2001) 599, Erratum. tial cross section (6) over the rapidity range |y| < 0.6 [12] V.A. Khoze, A.D. Martin, M.G. Ryskin, W.J. Stirling, Eur. [17] we obtain the value8 55 nb to be compared with Phys. J. C 35 (2004) 211. ± ± [13] F. Yuan, Phys. Lett. B 510 (2001) 155. the CDF upper limit 80 30(stat) 70(syst) nb [17]. [14] E. Stein, A. Schafer, Phys. Lett. B 300 (1993) 400. [15] H.A. Peng, Z.M. He, C.S. Ju, Phys. Lett. B 351 (1995) 349. [16] A. Bzdak, Phys. Lett. B 608 (2005) 64. 5. Conclusions [17] CDF Collaboration, K. Terashi, in: Deep Inelastic Scattering, Strbske Pleso, 2004, pp. 546–553. [18] P.V. Landshoff, O. Nachtmann, Z. Phys. C 35 (1987) 405; In conclusion, in this Letter we investigated the A. Donnachie, P.V. Landshoff, Nucl. Phys. B 311 (1988) 509. 0 0 [19] V.A. Khoze, A.D. Martin, M.G. Ryskin, Eur. Phys. J. C 18 central inclusive and exclusive DPE process of χc , χb mesons production in pp ( pp¯) collisions. We observed (2000) 167; ≈ 0 0 A.B. Kaidalov, V.A. Khoze, A.D. Martin, M.G. Ryskin, Eur. that the relation σexcl./σc.incl. 0.3(χc ), 0.07 (χb ) be- Phys. J. C 21 (2001) 521. tween exclusive and central inclusive cross sections [20] CDF Collaboration, T. Affolder, et al., Phys. Rev. Lett. 85 (2000) 4215. [21] K. Goulianos, AIP Conf. Proc. 698 (2004) 110; 8 Since the differential cross section (6) weekly depends on the M. Gallinaro, Acta Phys. Pol. B 35 (2004) 465. rapidity of the produced meson it is enough to multiply dσ/dy(y = [22] G. Altarelli, G. Parisi, Nucl. Phys. B 126 (1977) 298. 0) = 45 nb by a factor 1.2. Physics Letters B 619 (2005) 293–304 www.elsevier.com/locate/physletb

Strong interaction and bound states in the deconﬁnement phase of QCD

Yu.A. Simonov

State Research Center, Institute of Theoretical and Experimental Physics, Moscow 117218, Russia Received 10 March 2005; received in revised form 20 May 2005; accepted 3 June 2005 Available online 14 June 2005 Editor: N. Glover

Abstract ¯ Recent striking lattice results on strong interaction and bound states above Tc can be explained by the nonperturbative QQ potential, predicted more than a decade ago in the framework of the ﬁeld correlator method. Explicit expressions and quantitative estimates are given for Polyakov loop correlators in comparison with lattice data. New theoretical predictions for glueballs and baryons above Tc are given.  2005 Elsevier B.V. All rights reserved.

1. Introduction

There is a growing understanding nowadays that nonperturbative dynamics plays important role in the deconfinement phase, for reviews and references see [1]. An additional part of this understanding, not contained in [1], is the realization of the fact, that at Tc Tc [4]. In 1991 the author has found [5] that colorelectric fields also survive the deconfinement transition in the form ¯ of potential V1(r), which can support QQ bound states in some temperature interval Tc

E-mail address: [email protected] (Yu.A. Simonov).

1 ∞ At the same time the evidence for V1(r) and selfenergies 2 V1( ) has been also obtained on the lattice in the form of Polyakov loop averages and of free and internal energies above Tc [10–12]. The light quark (mq ∼ ms)qq¯ bound states have also been observed in [13]. The theory used in [5] and below is based on the powerful method of field correlators MFC [14] (for a review ··· 1 see [15]), where the basic dynamic ingredients are the field correlators tr Fµ1ν1 (x1) Fµnνn (xn) . It was shown later [16] that the lowest quadratic (so-called Gaussian) correlator explains more than 90% of all dynamics and it will be considered in what follows. The quadratic correlator consists of two scalar form-factors, D and D1: g2 Dµν,λσ (x, 0) ≡ tr Fµν(x)Φ(x, 0)Fλσ (0) Nc 1 ∂ = (δµλδνσ − δµσ δνλ)D(x) + (xλδνσ − xσ δνλ) + (µλ ↔ νσ) D1(x) (1) 2 ∂xµ which produce the following static potential [17] between heavy quarks at zero temperature (obtained from the Wilson loop r × t with t →∞) r ∞ r ∞ 2 2 2 2 2 2 V(r)= 2r dλ dτ D λ + τ + λdλ dτ −2D λ + τ + D1 λ + τ 0 0 0 0 = V (r) + V (r). (2) D 1 = = 1 2 One can notice that linear confinement part of potential, VD σR, is due to correlator D(x), σ 2 D(x)d x. E E H H At T>0 one should distinguish between electric and magnetic correlators, D (x), D1 (x), and D (x), D1 (x) and correspondingly between σ (E) and σ (H ). It was argued in [2,3] that the principle of minimality of free energy E (E) H H requires D and electric confinement, σ , to vanish, while colormagnetic correlators, D (x), D1 (x) should stay roughly unchanged at least up to 2Tc. Several years later in detailed studies on the lattice in [18] these statements have been confirmed, and indeed E magnetic correlators do not change at 1.5Tc >T >Tc while D (x) vanishes in vicinity of Tc. E Not much was said about the second electric correlator D1 (x), since in the parametrization of [18] it was found E to be smaller than D (x) and hence not so important at TTc. In the latter = case F1(r, T ) was found to saturate at large r at the√ values of the order of several hundred MeV (e.g., for T 1.2Tc the value of F1(∞,T)found in [12] is around 0.7 σ , while the internal energy is around 3Tc) and this fact cannot be explained by perturbative contributions alone—we consider it as the most striking revelation of nonperturbative electric fields above Tc. At the same time several groups have calculated the so-called spectral function of heavy [6–10] and light quarkonia [13] at T>Tc. In all cases sharp peaks have been observed, corresponding to the ground state levels of cc¯ at L = 0, 1 and of light (mq ≈ ms) quarkonia in V,A,S,PS channels. In both heavy and light cases the peaks are possibly displaced as compared to T = 0 positions and apparently almost degenerate in different nn¯ channels. All these facts cannot be explained in the framework of the commonly accepted perturbative quark–gluon plasma and call for a new understanding of the nonperturbative physics at T>Tc. In what follows we shall argue following [5] that at T>Tc not only nonperturbative magnetic fields, but also strong nonperturbative electric fields are present, which can be calculated in MFC and explain the observed data.

1 We omit for simplicity parallel transporters Φ(xi ,xj ) necessary to maintain gauge invariance, see [15] for details. Yu.A. Simonov / Physics Letters B 619 (2005) 293–304 295

2. Dynamics of Polyakov loops and the correlator D1

In this section we consider the Polyakov loop and apply to it the non-Abelian Stokes theorem and Gaussian approximation, taking first the loop as a circle on the plane and making limiting process with the cone surface S inside loop and finally transforming cone into the cylinder by tending the vertex of the cone to infinity.2 In doing so we are writing the non-Abelian Stokes theorem and cluster expansion for the surface S which is transformed from the cone to the (half) cylinder surface. As a result one has for the Polyakov loop average 1 1 L = tr P exp − dσµν(u) dσρλ(v) Dµν,ρλ(u, v) Nc 2 S S 1/T 1/T ∞ 1 = exp − dτ dτ ξdξD ξ 2 + (τ − τ )2 . (3) 4 1 0 0 0

In obtaining (3) we have omitted the contribution of D(x) in Dµν,ρλ, since this would cause vanishing of L in the limiting process described above due to the infinite cone surface S. This exactly corresponds to vanishing of L in the confinement region, observed on the lattice. Therefore the result (3) refers to the deconfinement phase, T>Tc. As it is known from lattice [18] and analytic calculations, D1(x) [19] exponentially falls off at large x as 3 exp(−M1x), with M1 1 GeV and for T M1 one can approximate (3) as follows 1/T ∞ 1 1 1 L = exp − dν − ν ξdξD ξ 2 + ν2 ≈ exp − V (∞) , 2 T 1 2T 1 0 0 1/T r 2 2 V1(r, T ) = dν(1 − νT ) ξdξD1 ξ + ν . (4) 0 0 We turn now to the correlator of Polyakov loops following notations from [11]. Using the same limiting procedure as for one Polyakov loop, one can apply it to the correlator tr˜ L tr˜ L+≡P(x − y), tr˜ = 1 tr, representing x y Nc the loops Lx and Ly as two concentric loops on the cylinder separated by the distance |x − y| along its axis, the cylinder obtained in the limiting procedure from the cone with the vertex tending to infinity. One can apply in this situation the same formalism as was used in [20] for the case of the vacuum average of two Wilson loops. For opposite orientation of loops using Eqs. (21)–(28) from [20] one arrives at the familiar form found in [21] 1 F˜ (r, T ) N 2 − 1 F˜ (r, t) P( − ) = − 1 + c − 8 ,r≡| − |. x y 2 exp 2 exp x y (5) Nc T Nc T In Appendix A two different ways of derivation of Eq. (5) are given, with the result ˜ = + F1(r, T ) V1(r, T ) VD(r, T ), (6) 1 exp −F˜ (r, T )/T = L (T ) exp − V (r, T ) − V (r, T ) T . (7) 8 adj D 8 1

2 In doing so one is changing topology of the surface and as a result loses the Z(N) subgroup of SU(N). This however does not inﬂuence our results as long as one is remaining in the j = 0 sector of Z(N) broken vacua (see last reference of [1] for more discussion of Z(N)). 3 The correlators D,D1 in (3) in principle should be taken in the periodic form, as was suggested in [22].HoweverforT 2Tc this modiﬁcation brings additional terms of the order of exp(−M1/T) which are neglected below. 296 Yu.A. Simonov / Physics Letters B 619 (2005) 293–304

Here 9 ∗ 9 L = exp − V r + V (∞,T) T (8) adj 4 D 8 1 is the vacuum average of the adjoint Polyakov loop, which vanishes in the leading approximation in the confinement phase, as it is explained in Appendix A, and nonzero when gluon loops are taken into account, in which case 9 ∗ ∼ ∼ − 4 VD(r ,T) M1 and Ladj exp( M1/T) 1. ˜ The suppression of exp(−F8/T) in our approach in the confinement phase has thus the same origin as the strong damping of the adjoint Polyakov loop in that phase [23] and the persistence of the Casimir scaling for adjoint static potential in the interval 0 r<1.2fm(see[24] for discussion and references). ˜ ˜ It is clear that in the deconfinement phase with D ≡ 0,VD ≡ 0 one has only V1(r) in both F1 and F8, and all these quantities are finite (after the renormalization of the perturbative divergencies specific for the fixed contours, which are discussed in Section 3). Thus in the deconfined phase one can write

−F˜ ¯ /T 1 −V (r)/T 8 −( 9 V (∞)− 1 V (r))/T P(x − y) ≡ e qq = e 1 + e 8 1 8 1 , (9) 9 9 ˜ where V1(r) and V1(∞) are renormalized. It is clear from (9) that at small r one has limr→0(Fqq¯ (r) − V1(r)) → 2 T ln Nc as was noticed and measured in [11]. At this point one should stress the difference between the genuine free energy Fi(r, T ), i = 1, 8, which is ˜ ˜ measured with some accuracy on the lattice, and the calculated above Fi(r, T ). It is clear that Fi do not contain the contribution due to excitation of QQ¯ and gluon degrees of freedom existing at ﬁnite T . The latter is contained in the free energy F1(r, T ) and in the internal energy, which we denote Ui(r, T ) = Fi + ST to distinguish from our Vi(r, T ), since they are not equal. In general for nonzero temperature and comparing to the lattice data on heavy-quark potential one should have in mind, that temperature effects might be of two kinds. First, the intrinsic temperature dependence due to changing of the vacuum structure and the vacuum correlators and hence of our potentials F(r,T)˜ . Second, the physical quantities like Fi(r, T ), Ui(r, T ) are thermal averages over all excited states, e.g., − − e F1(r,T )/T = e En(r,T )/T , (10) n −En(r,T )/T U1(r, T ) = Ene . (11) n ˜ One can associate F1(r, T ) = E0(r, T ), while the structure of excited spectrum can be traced in the temperature dependence of F1 and U1. E.g., assuming in the conﬁnement phase the string-like spectrum and multiplicity for the multihybrid spectrum with two static quarks, En = σr + πn/r, n = 1, 2,..., and multiplicity ρ(m)= exp(m/m0)θ(m − m1), m = πn/r, one arrives at 1 1 F1(r, T ) = σr + m1(1 − T/m0) − T ln − , (12) T m0 U1(r, T ) = σr + m1 + T/(1 − T/m0). (13)

The increase of U1(r, T ) below Tc in the quenched case was indeed observed in lattice calculations (see Fig. 3 of [12]). Above Tc one can see in lattice data [10] the striking drop of entropy S1(∞,T) and U1(∞,T) in the region Tc T 1.2Tc which can be possibly explained again by the multihybrid states occurring due to the potential ˜ V1(r, T ) = F1(r, T ) connecting quarks and gluons, and assuming that the magnitude of V1(r, T ) decreases with temperature passing at T ≈ 1.05Tc the critical value enabling to bind those states of high multiplicity. In this way one assumes that both below and above Tc in quenched and unquenched cases the dominant (in entropy) Yu.A. Simonov / Physics Letters B 619 (2005) 293–304 297

¯ conﬁguration is the gluon chain connecting Q and Q with gluons bound together by conﬁning string (below Tc) and potential V1 (above Tc). Thus the comparison to the lattice data on F1,U1 needs the exact knowledge of the ˜ spectrum. In what follows we shall associate our F1(r, τ) with the free energy F1(r, T ), since its temperature dependence is not so steep as that of U1(r, T ) in this region and this discussion will be of qualitative character, leading detailed discussion of the spectrum to future publications.

3. Properties of D1(x) and F1(r, T )

The correlator D1(x) was measured on the lattice [18] both below and above Tc, and decays exponentially with M1 ≈ 1–1.5 GeV (in the quenched case). At the same time D1(x) can be connected to the gluelump Green’s E ≈ function, and the corresponding M1 for the electric correlator D1 (x) is M1 1.5 GeV at zero T [26]. Moreover in E a recent paper [19] D1 (x) was found analytically for T Tc, and can be represented symbolically as a sum, with perturbative part acting at small x, 4C α D (x) = D(pert)(x) + D(np)(x), Dpert = 2 s + O α2 , (14) 1 1 1 1 πx4 s and the nonperturbative part having the asymptotic form A1 − | | D(np)(x) = e M1 x + O α2 ,A= 2C α σ M ,x 1/M . (15) 1 |x| s 1 2 s adj 1 1

As will be argued below, the form of D1(x) (14) does not change for T>Tc, however the mass M1 and A1 may be there different. Using the asymptotics (15) in the whole x region for a qualitative estimate, one has √ 1/T r −M ξ 2+ν2 (np) ξdξe 1 V (r, T ) = A1 (1 − νT )dν 1 ξ 2 + ν2 0 0 1/T √ A1 − − 2+ 2 = (1 − νT )dν e νM1 − e r ν M1 M1 0 A1 T − − = V np(∞) − K (M r)M r − e M1r ( + M r)+ O e M1/T . 1 2 1 1 1 1 1 (16) M1 M1 Finally the Polyakov loop exponent is (np) V (∞) A1 T − L = − 1 ,V(np)(∞) = − − e M1/T . exp 1 2 1 1 (17) 2T M1 M1 (np) ∞ One can see from (17) that V1 ( ) is ﬁnite and is of the order of few hundred MeV in the interval 0 T 1.5Tc. (np) ≈ × 2 = (np) + (pert) At small r from (16) V1 const r . The total V1 V1 V1 , contains also perturbative contribution at small r, which to the order O(αs) is 1/T 2C α 1 1 V (pert)(r) = 2 s dν(1 − νT ) − = V (pert)(∞) + V (C)(r, T ), 1 π ν2 ν2 + r2 1 1 0

C α 2 rT − V (C)(r, T ) =− 2 s f(r,T), f(r,T)= 1 − arctan(rT ) − ln 1 + (rT ) 2 . (18) 1 r π π 298 Yu.A. Simonov / Physics Letters B 619 (2005) 293–304

= (np) + (C) = ; ; Fig. 1. A comparison of behavior of V1(r, T ) V1 (r, T ) V1 (r, T ) Eqs. (16), (18), (19) (solid lines, T/Tc 1.05 1.2 1.5 from above), with the singlet free energy F1(r, T ) measured in Ref. [12] (ﬁlled circles).

(pert) ∞ (pert) ∞ ≈ 2C2αs 1 − From (18) it is clear that V1 ( ) is divergent and should be renormalized, V1 ( ) π ( a → (np) ∼ 2 T ln a), a 0. Since the dominant divergent part is T -independent and V1 (r) r at small r, one can renor- malize matching V1(r, T ) with the Coulomb interaction at small r, as it was done in [9,11] for Fi(r, T ). (pert) (pert) ∞ As a result in the renormalized V1 (r, T ) the term V1 ( ) can be put equal to zero, and we shall use it in what follows. At this point we are able to compare V1(r, T ) with the lattice data for F1(r, T ) at T Tc.InFig. 1 we compare the lattice data for F1(r, T ) taken from [12] for T = 1.05Tc, 1.2Tc and 1.5Tc with the potential V1(r, T ) in the form (16) parametrizing M and a(T ) ≡ A1 in it as 1 M1

T − Tc M1 = const,a(T)= a0 − c ,αs = 0.3 (19) Tc ∼ 2 and ﬁnd that M1 = 0.69 GeV and a0 = a(conf) = 2C2(f )αsσadj = 0.432 GeV , c = 0.36 provides a good agreement with the data points at 1.5Tc T Tc, while a(T ) in (19) smoothly matches at T = Tc the amplitude of = C + (np) the gluelump Green’s function [19]. One can see that the behaviour of the total V1(r, T ) V1 (r, T ) V1 (r, T ) = (np) ∞ which has a Coulomb part at smaller r and saturates at V1 V1 ( ) is qualitatively very similar to the behavior of F1(r, T ) as a function of r. We also compare in Fig. 2 our results with lattice data [11] for the Polyakov loop (17) and ﬁnd reasonable agreement. It is clear that both Figs. 1 and 2 are qualitative illustrations, and for quantitative comparison one needs knowledge of excitation spectrum and analytic or lattice predictions for a(T ),M1(T ) which will be given elsewhere [25,37].

4. Bound states of qq¯ in the deconﬁnement region

In the recent lattice studies sharp peaks have been found in the spectral function of cc¯ system [6–10], which can be associated with the quark–antiquark bound states surviving at T Tc. Yu.A. Simonov / Physics Letters B 619 (2005) 293–304 299

Fig. 2. The Polyakov loop exponent V1(∞,T) as a function of T (GeV) from Eqs. (17), (19) (solid line), in comparison with lattice data for F1(∞,T)in Ref. [11] (ﬁlled circles).

To understand qualitatively whether the interaction V1(r, T ) can support bound states, one can use the Bargmann condition [27] for monotonic attractive potentials ∞ 2m¯ rdrU(r,T) > 1, (20) 0 where 2m¯ = mc, and U(r,T) = V1(r, T ) − V1(∞,T) which yields the condition for the bound S-states. Taking = = (np) mc 1.4GeV,M1 0.6 GeV, one can deduce that V1 (r, T ) can support bound states in some interval of −M /T temperature Tc

3 1 V (r , r , r ; T)= V (r ,T)+ V (r − r ,T), (21) 3q 1 2 3 D i 2 1 i j i=1 i>j 300 Yu.A. Simonov / Physics Letters B 619 (2005) 293–304

= = = where√V1(r, T ) is given in (16), (18). In the deconﬁnement phase when ri R, i 1, 2, 3, one obtains V3q 3 (C) 1 (C) = | = 2 V1( 3R). For the perturbative part one has from (21) V3q 2 i>j V1 (rij ,T) rij R. For the nonperturbative part from Eq. (21) it follows that V (np)(R →∞) = 3 V (np)(∞). One can check that this √ 3q 2 1 = 3 prediction and the general form of V3q 2 V1( 3R) as function of R is supported by the recent measurement of singlet free energy of the 3Q system in [33] at T>Tc. Thus it is of interest to measure the spectral functions of baryons at T>Tc in the same way as it was done for mesons.

5. Summary and conclusions

Citing the 1991 paper [5] when the magnitude of D1 was not exactly known “...Using an exponential parame- E trization for D1, we can find D1 with parameter values which satisfy the condition for the appearance of levels. In (np) ∼ 2 → → →∞ this case ε(r) (our V1 (r, T )) is a well with a behaviour ε(r) r as r 0 and ε(r) const > 0asr .The quark and antiquark are thus bound but there exists a threshold ε(∞) above which quarks fly apart, each acquiring = 1 ∞ a nonperturbative mass increment δm 2 ε( )...”. In the present Letter this picture was further substantiated and quantified using lattice and analytic knowledge on D1. Comparing to recent lattice data in [6–13] it was shown that this picture is qualitatively supported by data, and new proposals have been done for searching the glueball and baryon systems at T>Tc. The results obtained on the lattice [6–13] and in the present approach establish a new picture of the QCD thermodynamics at 1.5Tc >T Tc widely discussed in [28]. As a new feature compared to the works [28] the 1 9 main emphasis in this Letter is done on the selfenergies ( V1(∞) ≡ εq for quarks and V1(∞) ≡ εg for gluons) ∼ 2 8 which are large (V1(∞,Tc)>F1(∞,Tc) = 600 MeV for nf = 2 [34]) and cancel each other at small distances for white bound states, like qq¯, (gg)1,(qgq)¯ 1,(qg...gq)¯ 1, etc. In contrast to that colored states are higher in potential and mass by several units of εq and εg and are suppressed by the corresponding Boltzmann factors. As a result in this region white bound states of quarks and gluons are energetically preferable, while individual quarks and gluons acquire selfenergies, so that the thermodynamics of the system resembles that of the neutral gas, and for higher temperature T>1.5–2Tc a smooth transition to the “ionized” plasma of colored quarks and qluons possibly occurs. This new state of the quark–gluon matter should be taken into account when considering ion–ion collisions. For more discussion of the thermodynamics above Tc see [28] and references therein. It was noted before [35] that the behaviour of the free and internal energies above Tc, with a bump around ∼ ε−3P T 1.1–1.2Tc in T can be explained if gluons are supplied with the nonperturbative mass term of the order of 0.6 GeV, while for higher T this mass is less important. This can be easily understood now taking into account the 1 ∞ value of 2 V1( ,T) and its decreasing with growing T . In this way the nonperturbative dynamics in the form of correlator D1 can explain the observed dynamics of the deconfined QCD. A more detailed analysis of bound states requires explicit calculation of QQ¯ and 3Q bound states taking into account spin splitting in the mass of P -wave charmonia and quasi-degeneration of spectra of light qq¯ V,A,S,PS states observed in [13]. Here spin-dependent forces are different from the confining case, since only the correlator D1 contributes, and one can list the corresponding terms in [17,36]. It is interesting to note, that due to the vector character of V1(r, T ), not violating chiral symmetry, bound states of massless quarks should exhibit parity doubling. All this analysis is now in progress [37].

Acknowledgements

The author is grateful to N.O. Agasian, S.M. Fedorov and V.I. Shevchenko for discussions and D.V. Antonov, N. Brambilla, E.-M. Ilgenfritz, A. Vairo and C.-Y. Wong for useful correspondence, and P. Petreczky for valuable remarks and comments. Yu.A. Simonov / Physics Letters B 619 (2005) 293–304 301

The work is supported by the Federal Program of the Russian Ministry of Industry, Science and Technology No. 40.052.1.1.1112, and by the grant for scientiﬁc schools NS-1774.2003.2.

Appendix A. Derivation of the Polyakov loop correlator ˜ We give here two different derivations of Fi(r, T ), i = 1, 8. The ﬁrst is based on the correlator of two concentric ˜ Wilson loops, derived in [20], in which case F1,8 are expressed in terms of surface integrals of ﬁeld correlators Dµν,ρλ(u, v)

I(Si,Sk) ≡ dσµν(u) dσρλ(v) Dµν,ρλ(u, v). (A.1)

Si Sk In this way one obtains for two oppositely directed Polyakov loops from Eqs. (29), (23), (20) of [20] 1 F˜ (r, T )/T = I(S ,S ), (A.2) 1 2 12 12 1 1 1 F˜ (r, T )/T = I(S ,S ) + I(S ,S ) + I(S ,S ). (A.3) 8 1 1 2 2 2 − 1 2 2 2 Nc 1 Here S1 is the surface on the cylinder with circumference 1/T extending from the loop 1 at coordinate x in the direction y to infinity, the surface S2 is also infinite surface from the loop 2 at coordinate y in the same direction (the answer does not depend on the choice of this direction). The surface S12 lies on the cylinder between the loops 1 and 2. Note that surface orientation in (A.1) is fixed to ˜ be same. Calculation of F1 according to (A.2) reduces to that of the Wilson loop and yields

F1(r, T ) = V1(r, T ) + VD(r, T ), (A.4) where V1(r, T ) is given in (4) and VD is 1/T r 2 2 VD(r, T ) = 2 dν(1 − νT ) (r − ξ)dξ D ξ + ν . (A.5) 0 0 ˜ Calculation of F8 is more subtle. To this end one can use connection of D1(x) to the gluelump Green’s func- 2 2 tion G (x) = δ N (N 2 − 1)f (x2) [19], D (x) =−2g (N 2 − 1) df (x ) , f(x2 → 0) ∼ 1 and inserting this µν µν c c 1 Nc c dx2 4π2x2 into (4), one has 1/T 2 2 − g (Nc 1) 2 2 2 V1(r, T ) = dν(1 − νT ) f ν − f r + ν = V1(∞,T)+ vex(r, T ). (A.6) Nc 0 ∞ ≡ + ¯ One can see that V1( ,T) VQ VQ¯ is the sum of equal selfenergy parts of Q and Q, while vex(r, T ) describes interaction due to one gluelump exchange between Q and Q¯ . Note that V1(0,T)= V1(∞,T)+ vex(0,T)= 0 and vex(∞,T)= 0. Therefore vex appears only in I(S1,S2) in + (A.3) and one should restore there the original (opposite) orientation of loops Lx and Ly to get the correct sign of vex (the same sign and factor appears in the second derivation below). From (A.1) one obtains for I(Si,Si) 1 ∗ 1 1 I(S ,S ) = V r + V ¯ , I(S ,S ) = I(S ,S ) + V (r), 2 2 2 D Q 2 1 1 2 2 2 D I(S1,S2) =−vex(r, T ) + I(S2,S2). (A.7) 302 Yu.A. Simonov / Physics Letters B 619 (2005) 293–304

As a result one can write using (A.3) for Nc = 3 ˜ 9 ∗ 1 F (r, T ) = V r ,T + V + V ¯ − v (r, T ) + V (r, T ) 8 4 D Q Q 8 ex D 9 1 9 ∗ = V (∞,T)− V (r, T ) + V r ,T + V (r, T ). (A.8) 8 1 8 1 4 D D In (A.8) the value of r∗ is infinitely large, when one neglects the valence gluon loops, as it is done everywhere ∗ ˜ above. In this case VD(r ,T)→∞and the term exp(−F8/T) vanishes in the confinement region. This is in line with the strong damping of the adjoint Polyakov loop in this region observed on the lattice [23], and with the persistence of Casimir scaling for adjoint static potential for 0 r 1.2 fm found on the lattice (see discussion and references in [24]). The correction due to the gluon determinant, producing additional gluon loops becomes important for r 1.2fm ∗ ∗ ∗ (see discussion in the second reference in [24]) and makes finite the value of VD(r ,T)≈ σr ,r ≈ 1.2fm/2 = 0.6fm. 1 VD(r, T ) − V1(r, T ) exp −F˜ (r, T )/T = L (T ) exp − 8 , 8 adj T 9 9 ∗ L (T ) = exp − V (∞,T)− V r ,T . (A.9) adj 8 1 4 D In (A.9) the effects of loop–loop interaction and of the total (adjoint) loop are separated. Physically the result (A.9) can be easily understood: in absence of the internal interaction one has do with the adjoint Polyakov loop, which strongly changes around Tc, namely Ladj(T < Tc) is much smaller than Ladj(T > Tc). The behaviour similar to (A.9) was observed on the lattice, see Fig. 3 of second reference in [10] and Ref. [34]. ˜ 1 = αs Here one observes linear growth in r of F8(r, T ) below Tc, and repulsive Coulomb behaviour from 8 V1(r, T ) 6r . In the second derivation one is connecting two loops using parallel transporters and using the completeness relation 1 δ δ = δ δ + 2ta ta (A.10) α1β1 α2β2 α1β2 β1α2 β2α1 β1α2 Nc so that the second term produces the adjoint Wilson loop on the cylinder surface ˜ a a + exp(−F8/T) = 2trU(x, X; 0)t U(X, y, 0)L(y)U(y, X,t)t U(X, x,t)L (x) . (A.11) This can be compared to the approach, suggested in [38]. Our results however are different from those of [38] ˜ ˜ in that both perturbative and nonperturbative interactions in F1 and F8 are different (and calculable through D1). Now (A.11) can be rewritten using cluster expansion and non-Abelian Stokes theorem, which finally results in the same equations as in (A.9). Alternatively one can use the technical exploited in [39] to separate the contributions of perturbative exchanges from the nonperturbative confining terms. For the first ones one commutes as in [39] the color generators tc of exchanged gluon (gluelump) with ta (A.11) according to the equality tctatc =−ta/2N which finally gives in (A.11) the adjoint Coulomb interaction c αs →−1 c c a → a 6r 8 V1(r), while the selfenergy parts and confining terms arise from sequences t t t δcc t and do not change sign. In this way one arrives at the same answer as given in (A.9).

References

[1] P. Arnold, L.G. Yaffe, Phys. Rev. D 52 (1995) 7208, hep-ph/9508280; L.G. Yaffe, Nucl. Phys. B (Proc. Suppl.) 106 (2002) 117, hep-th/0111058; R.D. Pisarski, hep-ph/0203271. Yu.A. Simonov / Physics Letters B 619 (2005) 293–304 303

[36] A.M. Badalian, Yu.A. Simonov, Phys. At. Nucl. 59 (1996) 2164. [37] N.O. Agasian, S.M. Fedorov, Yu.A. Simonov, in preparation. [38] O. Philipsen, Phys. Lett. B 535 (2002) 138; O. Jahn, O. Philipsen, hep-lat/0407042. [39] Yu.A. Simonov, S. Titard, F.J. Yndurain, Phys. Lett. B 354 (1995) 435. Physics Letters B 619 (2005) 305–312 www.elsevier.com/locate/physletb

Study of B → ΛΛK¯ and B → ΛΛπ¯ C.Q. Geng, Y.K. Hsiao

Department of Physics, National Tsing Hua University, Hsinchu 300, Taiwan Received 30 March 2005; received in revised form 20 May 2005; accepted 24 May 2005 Available online 13 June 2005 Editor: T. Yanagida

Abstract We study three-body charmless baryonic B decays of B → ΛΛP¯ with P = π and K in the standard model. We ﬁnd that the branching ratios of the K modes are about one order of magnitude larger than those of the corresponding π modes unlike − − − the cases of B → ppP¯ . Explicitly, we obtain that Br(B → ΛΛK¯ ) = (2.8 ± 0.2) × 10 6 and Br(B¯ 0 → ΛΛ¯K¯ 0) = (2.5 ± × −6 +0.90 ± × −6 0.3) 10 . The former agrees well with the Belle experimental measurement of (2.91−0.70 0.38) 10 , while the latter should be seen at the ongoing B factories soon.  2005 Elsevier B.V. All rights reserved.

There have been lots of attentions recently on charmless three-body baryonic B decays due to the several new experimental measurements by Belle and BaBar [1–5]. It has been realized that baryonic B decays could happen since the B-meson mass can be heavier than the invariant baryon masses. In particular, the reduced energy release can make the decays as signiﬁcant as the two-body mesonic decay modes [6]. As the main characteristic of the three-body baryonic decays, the baryon pair threshold effect [6,7] results in the decay modes being accessible at the current B factories. Indeed, some of the modes have been seen and limited recently by the Belle and BaBar Collaborations, with the data given by − − +0.73 −6 Br B → ppπ¯ = 3.06− ± 0.37 × 10 (Belle) [1], 0.62 0 +0.32 −6 Br B → ppK¯ S = 1.20− ± 0.14 × 10 (Belle) [2], 0.22 +0.45 −6 − − 5.3− ± 0.58 × 10 (Belle) [2], Br B → ppK¯ = 0.39 6.7 ± 0.9 ± 0.6 × 10−6 (BaBar) [3], ¯ 0 → ¯ + = +0.62 ± × −6 Br B Λpπ 3.27−0.51 0.39 10 (Belle) [2],

E-mail address: [email protected] (Y.K. Hsiao).

Fig. 1. Diagrams for B → BB¯ P with B = p and Λ, P = π(K) for q = d(s) and q = u(d) for charged (neutral) modes, and where (a) and (b) represent C and T terms, respectively. − + − Br B → Λpγ¯ = 2.16 0.58 ± 0.20 × 10 6 (Belle) [4], −0.53 − → ¯ − = +0.90 ± × −6 Br B ΛΛK 2.91−0.70 0.38 10 (Belle) [5], (1) and − − − Br B → ΛΛπ¯ < 2.8 × 10 6 (Belle) [5], (2) respectively. Note that the three-body decays in Eq. (1) are much larger than the corresponding two-body baryonic modes, for which only upper bounds of O(10−7) have been reported [8]. There are mainly two kinds of approaches to study the baryonic B decays in the literature. One is the pole model, presented in Refs. [9–11], where the intermediate particles couple dominantly to the final states. The other, proposed in Refs. [7,12,13], is based on the QCD counting rules [14,15], which deal with the baryonic form factors by power expansions. We note that global χ2 fits in Ref. [16] for Br(B− → ppπ¯ −), Br(B0 → ppK¯ 0) and Br(B− → ppK¯ −) [1] with the QCD counting rules have been performed and consistent results with data have been derived. Furthermore, various radiative three-body baryonic B decays [16] have been studied. In this report, we will concentrate on the three-body charmless baryonic decays of B → ΛΛP¯ (P = π and K) (3) based on the QCD counting rules. We note that so far there has been no theoretical study on the modes in Eq. (3) in the literature. In particular, we do not know the reason why the decay branching ratio of B− → ΛΛπ¯ − is smaller than that of B− → ΛΛK¯ −, whereas the corresponding pp¯ modes are comparable, in terms of the data shown in Eqs. (1) and (2). In addition, these decays are of great interest since they provide opportunities for probing T violating effects [17] due to the measurable polarization of Λ [18]. There are two types of diagrams which contribute to the decays in Eq. (3) under the factorization approximation [7], with Figs. 1(a) and (b) representing typical diagrams of a current-produced baryon pair with B → P and a current-produced P with B → BB¯ , named as C and T terms, respectively. The amplitude of B → BB¯ P with B = p or Λ and P = π or K in the factorization approximation is given by [7] A(B → BB¯ P)= C(B → BB¯ P)+ T (B → BB¯ P), (4) where [19,20] C → ¯ = G√F ∗ ¯ | ¯ | | ¯ | (B BBP) VubVuq a2 BB (uu)V −A 0 P (qb)V −A B 2 − ∗ ¯ | ¯ + ¯ +¯ | + ¯ | ¯ | VtbVtq a3 BB (uu dd ss)V −A 0 a4 BB (qq)V −A 0 C.Q. Geng, Y.K. Hsiao / Physics Letters B 619 (2005) 305–312 307 ¯ ¯ 3 ¯ ¯ + a5BB|(uu¯ + dd +¯ss)V +A|0+ a9BB|(euuu¯ + ed dd + esss)¯ V −A|0 2 | ¯ | + ∗ ¯ | ¯ | | ¯ | P (qb)V −A B VtbVtq2a6 BB (qq)S+P 0 P (qb)S−P B , (5) ¯ with q = d(s) for P = π(K), while T (B → BBP) is decomposed with the P -meson decay constant fP induced from the P creation and B → BB¯ through scalar and pseudoscalar currents as ¯ GF ¯ ¯ T (B → BBP)= i √ fP mb αP BB|¯q b|B+βP BB|¯q γ5b|B , (6) 2 with q = u(d) for charged (neutral) modes, and 2m2 = ∗ − ∗ ± P αP (βP ) VubVuq aP VtbVtq aP aP , (7) mb(mq + mq ) where a − = a − = a , a 0 = a , a 0 = 0, a − = a − = a = a , a =−a + 3a /2, a = a and π K 1 π √ 2 K π K K0 4 π0 4 9 π−,K−,K0 6 a =−a . Here, f 0 = f / 2, V denote the CKM mixing matrix elements [21], a (i = 1,...,10) are com- π0 6 π π ij i (eff) = posed with (effective) Wilson coefficients ci (i 1, 2,...,10), defined in Refs. [19,20], and Nc is the effective (eff) color number. The coefficients ai and ci are related by ceff ceff = eff + i+1 = = eff + i−1 = ai ci (i odd), ai ci (i even). (8) Nc Nc To calculate the decay rate, we need to know the hadronic transition matrix elements in Eqs. (5) and (6).The B → P transition matrix element can be parameterized as 2 2 m − m → P |¯qγµ( − γ )b|B= (p + p )µ − B P (p − p )µ F B P (t) 1 5 B P 2 B P 1 (pB − pP ) 2 2 m − m → + B P (p − p )µF B P (t), 2 B P 0 (9) (pB − pP ) ≡ + 2 B→P where t (pB pB¯ ) and F1,0 (t) are defined by [22] F B→P (0) F B→P (0) B→P = 1 B→P = 0 F1 (t) 2 ,F0 (t) 2 , (10) − t − σ11t + σ12t − σ01t + σ02t (1 2 )(1 2 4 ) 1 2 4 MV MV MV MV MV B→π = B→π = = = = = with the input parameter values of F1 (0) F0 (0) 0.29, σ11 0.48, σ12 0, σ01 0.76, σ02 0.28 and = → B→K = B→K = = = = = MV 5.32 GeV for B π and F1 (0) F0 (0) 0.36, σ11 0.43, σ12 0, σ01 0.70, σ02 0.27 and MV = 5.42 GeV for B → K, respectively. For those of the baryon pair involving the vector, axial-vector, scalar and pseudoscalar currents in Eq. (5),wehave ¯ F2(t) BB|Vµ|0=u(p ¯ B) F1(t)γµ + σµν(p ¯ + pB)µ v(p¯ ) m + m ¯ B B B B F2(t) =¯u(pB) [F1(t) + F2(t)]γµ + (p ¯ − pB)µ v(p¯ ), m + m ¯ B B B B ¯ hA(t) |A | =u(p ¯ ) g (t)γ + (p ¯ + p ) γ v(p¯ ), BB µ 0 B A µ + B B µ 5 B mB mB¯ ¯ | | = ¯ ¯ | | = ¯ BB S 0 fS(t)u(pB)v(pB¯ ), BB P 0 gP (t)u(pB)γ5v(pB¯ ), (11) 308 C.Q. Geng, Y.K. Hsiao / Physics Letters B 619 (2005) 305–312

Table 1 Relations of form factors between different sets of parameterizations, where the form factors [F1(t)+F2(t)], gA(t), fS (t) and gP (t) correspond to the notations V , A, S and P in the ﬁrst column, respectively

where Vµ =¯qiγµqj , Aµ =¯qiγµγ5qj , S =¯qiqj and P =¯qiγ5qj with qi = u, d and s. We note that F2(t) alone cannot be determined by the present experimental data. However, F2(t) can be ignored since it acquires one more 1/t than F1(t) according to the power expansion in a perturbative QCD re-analysis [23,24]. By using equation of motion and adopting zero quark mass limits, we have

2 (mB + m ¯ ) h (t) =−g (t) B . (12) A A t

In terms of the SU(3) ﬂavor symmetry, the form factors [F1(t) + F2(t)], gA(t), fS(t) and gP (t) in Eq. (11) can be related to another set of form factors DX(t), FX(t) and SX(t) where X = V,A,S and P denote vector, axial- vector, scalar, pseudoscalar currents, deﬁned in Table 1, respectively. It is noted that the zero value of pp¯|(ss)¯ X|0 ¯ ¯ in the second column is due to the OZI suppression rule. As an illustration, we take ΛΛ|(uu¯ + dd +¯ss)V,A|0 in Eq. (5) and we have ΛΛ¯ + ΛΛ¯ = + ΛΛ¯ = + F1 (t) F2 (t) 2DV (t) 3SV (t), gA (t) 2DA(t) 3SA(t), ¯ ¯ (m + m ¯ )2 hΛΛ(t) =−gΛΛ(t) Λ Λ . (13) A A t p(n) For DX(t), FX(t) and SX(t), since they are related to the nucleon magnetic (Sachs) form factors GM (t),we p(n) adopt the results for GM (t) in Ref. [25], which are extracted from experiments. p(n) The functions of GM (t), DX(t), FX(t), and SX(t) are parameterized as 5 −γ 2 −γ 2 −γ p xi t n yi t si t G (t) = ln ,G(t) = ln ,SV (t) = ln , M ti+1 Λ2 M ti+1 Λ2 ti+1 Λ2 i=1 0 i=1 0 i=1 0 2 ˜ −γ 2 ˜ −γ 2 −γ di t fi t s˜i t DA(t) = ln ,FA(t) = ln ,SA(t) = ln , ti+1 Λ2 ti+1 Λ2 ti+1 Λ2 i=1 0 i=1 0 i=1 0 2 ¯ −γ 2 ¯ −γ 2 −γ di t fi t s¯i t DP (t) = ln ,FP (t) = ln ,SP (t) = ln , ti+1 Λ2 ti+1 Λ2 ti+1 Λ2 i=1 0 i=1 0 i=1 0 (14) with

=−3 n = p + 1 n =−3 n DV (t) GM (t), FV (t) GM (t) GM (t), DS(t) nq GM (t), 2 2 2 1 F (t) = n Gp (t) + Gn (t) ,S(t) = n S (t). (15) S q M 2 M S q V C.Q. Geng, Y.K. Hsiao / Physics Letters B 619 (2005) 305–312 309

4 The input numbers can be found in Refs. [7,12,13]. Explicitly, we take γ = 2.148, x1 = 420.96 GeV , x2 = 6 8 10 12 4 −10485.50 GeV , x3 = 106390.97 GeV , x4 =−433916.61 GeV , x5 = 613780.15 GeV , y1 = 292.62 GeV , =− 6 = − = 6 ˜ = − 3 ˜ = 2 + 1 ˜ =− + ˜ = y2 579.51 GeV , s1 x1 2y1, s2 500 GeV , d1 x1 2 y1, f1 3 x1 2 y1, s1 x1/3 2y1, d2 ˜ 6 ¯ 3 ¯ 3 ¯ ¯ 6 f2 =−478 GeV , s˜2 = 0, d1 = nq y1, f1 = nq (x1 − y1), s¯1 = nq (x1 − 2y1), s¯2 = nq s2, d2 = f2 =−952 GeV 2 2 and Λ = 0.3 GeV. We remark that the parameter n in Eq. (15) corresponds to the (m − m )/(m − m ¯ ) term 0 q B B¯ q q in connecting scalar form factors to vector ones in the case of B = B¯ .InB → BB¯ P decays, this parameter is not well defined. However, by taking m ¯ → m and m → m , it has been shown [12] that n is around 1.3–1.4. Here q q B¯ B q we fix nq 1.4. For the B → BB¯ transition in Eq. (6), we first discuss the case with B = p.ForB− → pp¯, one has [7] − − pp¯|¯ub B = iu(p¯ p)[FApγ/ 5 + FP γ5]v(pp¯ ), pp¯|¯uγ5b B = iu(p¯ p)[FV p/ + FS]v(pp¯ ), (16) ¯ 0 where p = pB − (pp¯ + pp). Note that FS = FP asshowninRef.[12].ForB → pp¯, one gets ¯| ¯ ¯ 0 = ¯ pp¯ + pp¯ ¯| ¯ ¯ 0 = ¯ pp¯ + pp¯ pp db B iu(pp) FA pγ/ 5 FP γ5 v(pp¯ ), pp dγ5b B iu(pp) FV p/ FS v(pp¯ ). (17) The form factors in Eqs. (16) and (17) can be related [7,16] by the SU(3) symmetry and the helicity conservation [14,15] and one obtains that ¯ 1 ¯ 1 ¯ 1 F pp = (11F + 9F ), F pp = (9F + 11F ), F pp =− F . (18) A 10 A V V 10 A V P(S) 4 P Moreover, the three form factors FA, FV and FP in Eq. (16) can be simply presented by [7] C C F = A,V ,F= P , (19) A,V t3 P t4 where Ci (i = A,V,P) are new parameterized form factors, which are taken to be real. By following the approach of Refs. [7,15], the form factors in B−,0 → ΛΛ¯ transitions are given by

ΛΛ¯ ΛΛ¯ ΛΛ¯ ¯ C ¯ C ¯ C F ΛΛ = A ,FΛΛ = V ,FΛΛ = P , (20) A t3 V t3 P(S) t4 where ¯ 1 ¯ 1 ¯ CΛΛ = (9C + 6C ), CΛΛ = (6C + 9C ), CΛΛ = 0. (21) A 10 A V V 10 A V P ΛΛ¯ = 1 ΛΛ¯ It is interesting to note that FP(S) 0 from Eqs. (20) and (21). The reason for this result is that FP(S) correspond to a helicity (chirality) flipped terms in the B → ΛΛ¯ matrix elements in the large momentum transfer. It is well known that the spin of Λ is carried by the s-quark component. Hence, such a term requires a chirality flip in the s-quark, which is absent in the decay amplitude of B → ΛΛ¯. To obtain these unknown form factors, we use the χ2 fit with the experimental data in Eq. (1).Herewehave neglected [16] the CP term since it has one more 1/t over CA and CV as shown in Eq. (19). Therefore, we keep the numbers of the degree of freedom (ndf) to be 2. In our fit, we also include the uncertainties from the Wolfenstein 2 parameters [26] in the CKM matrix. Explicitly, we use λ = 0.2200 ± 0.0026, A =|Vcb|/λ = 0.853 ± 0.037, ρ = 0.20 ± 0.09 and η = 0.33 ± 0.05 [27]. We follow the Refs. [20,28] to deal with ai (i = 1,...,10) and we take Wilson coefficients from Ref. [29]. The experimental inputs and the fitted results are shown in Table 2. With the fitted values in Table 2, the theoretical predictions with 1σ error for B− → ΛΛP¯ are given by − − − − − − Br B → ΛΛK¯ = (2.8 ± 0.2) × 10 6, Br B → ΛΛπ¯ = (1.7 ± 0.7) × 10 7 , (22)

1 We thank the argument given by the referee. 310 C.Q. Geng, Y.K. Hsiao / Physics Letters B 619 (2005) 305–312

Table 2 4 Fits of (CA, CV ) in units of GeV Input Experimental data Fit result Best ﬁt (with 1σ error) − − Br(B → ppπ¯ ) [1] Eq. (1) CA −71.0 ± 5.3 0 0 Br(B → ppK¯ ) [2] CV 42.0 ± 9.2 − − Br(B → ppK¯ ) [2] χ2/ndf 1.9 − Br(B → Λpγ)¯ [4]

− → ¯ − → ¯ = − = − Fig. 2. dBr/dmBB¯ as a function of mBB¯ for (a) B ppP and (b) B ΛΛP , where the solid and dashed lines for P K and P π , respectively. which agree very well with the recent Belle data [5] in Eq. (1). We note that the spectrum of B− → ΛΛK¯ − shown in Fig. 2(b) is consistent with that in Ref. [5] by Belle. Our predicted values in Eq. (22) show that Br(B− → ΛΛπ¯ −) Br(B− → ΛΛK¯ −) unlike Br(B− → ppπ¯ −) ∼ Br(B− → ppK¯ −) as indicated in Eq. (1). This result can be easily understood from the theoretical point of view. As B− → ppπ¯ − cannot obtain a large contribution from C in Eq. (5) since a2 is color suppressed, its main contribution is from a1 term in T as seen in Eqs. (6) and (7), whereas B− → ppK¯ − is due to the penguin part which gives contributions to the terms in both C and T . − ¯ − ¯ However, it is not the case for B → ΛΛK since ΛΛ|(ss)¯ X|0 in Eq. (5) escapes from the OZI suppression. Its contribution is mainly from C which is enhanced by a6 with the chiral enhancement, whereas it is suppressed − ¯ − in T .ForB → ΛΛπ , on the other hand, because of the a2 color suppression in C and the small contribution in T , its branching ratio is small. Moreover, to explicitly see the T -suppression in the ΛΛ¯ case, we show the spectra of B− → ppπ¯ − and B− → ΛΛπ¯ − in Fig. 2. We note that the two modes contain the same T -amplitudes in Eq. (6).AsshowninFig. 2 the main contributions to the rates are due to the threshold effect, resulting from the = 3 ΛΛ¯ = ΛΛ¯ 3 − → ¯ − − → ¯ − form factors of FA,V CA,V /t and FA,V CA,V /t in B ppπ and B ΛΛπ , respectively. When −3 −3 t = (p + p ¯ )2 → t = (m + m ¯ )2, t t /3. Furthermore, the relations from the numerator of the form B B min B B ΛΛ¯ pp¯ ΛΛ¯ ΛΛ¯ − factor are CA CA/2 and CV CV /10. Once we combine the relations above and integrate them through the phase space, we obtain that Br(B− → ΛΛπ¯ −) O(10−1)Br(B− → ppπ¯ −). Similarly, we can also study the neutral decay modes of B¯ 0 → ΛΛ¯K¯ 0(π 0), which have not been measured yet. The decay branching ratios are found to be − − Br B¯ 0 → ΛΛ¯K¯ 0 = (2.5 ± 0.3) × 10 6, Br B¯ 0 → ΛΛπ¯ 0 = (0.4 ± 0.4) × 10 7. (23) As seen from Eq. (23), the decay branching ratio of B¯ 0 → ΛΛ¯K¯ 0 is almost the same as that of B− → ΛΛK¯ −, whereas B¯ 0 → ΛΛπ¯ 0 is still suppressed as B− → ΛΛπ¯ −. We note that the errors of Br(B− → ΛΛK¯ −) and ¯ 0 → ¯ ¯ 0 ΛΛ¯ Br(B ΛΛK ) in Eqs. (22) and (23) are small since the main contributions are not from CA,V which receive almost all uncertainties from the data. C.Q. Geng, Y.K. Hsiao / Physics Letters B 619 (2005) 305–312 311

To describe the possible non-factorizable effects, we also fit the data with Nc = 2 and ∞. As expected, we find that the branching ratios of the K modes are slightly changed, while the central values of B− → ΛΛπ¯ − and ¯ 0 ¯ 0 B → ΛΛπ shift to 2.3 and 1.3 (0.7 and 0.1) for Nc = 2 (∞), respectively. We may conclude that the two π modes remain small even with including all possible non-factorizable effects. However, as pointed in Ref. [30], a2 can only be determined by the experimental data in the two-body B decays, since the experimental data of Br(B¯ 0 → π 0π 0) and Br(B¯ 0 → D0π 0) are much larger than the theoretical values, which means the failure of the factorization approximation. On the other hand, in the three-body baryonic B decays, due to the complicated topology of Feynman diagrams, it is not as easy as those of the two-body decays to influence a2 by the annihilations [31] as well as the final state interactions [32]. Therefore, the value of a2 may not change much. Nevertheless, we leave the surprise to the experimentalists if the factorization does not work well in these two modes as those in the two-body decays. In sum, we have studied three-body charmless baryonic B decays of B → ΛΛπ¯ and B → ΛΛK¯ based on the QCD counting rules in the standard model. We have shown that Br(B−,0 → ΛΛK¯ −,0) Br(B−,0 → ΛΛπ¯ −,0) unlike the cases of B → ppπ¯ (K). Explicitly, we have found that Br(B− → ΛΛK¯ −) = (2.8 ± 0.2) × 10−6, Br(B¯ 0 → ΛΛ¯K¯ 0) = (2.5 ± 0.3) × 10−6, Br(B− → ΛΛπ¯ −) = (1.7 ± 0.7) × 10−7 and Br(B¯ 0 → ΛΛπ¯ 0) = (0.4 ± 0.4) × 10−7. It is interesting to note that the decay of B¯ 0 → ΛΛ¯K¯ 0 should be seen at the ongoing B factories soon.

Acknowledgements

This work was supported in part by the National Science Council of the Republic of China under Contracts Nos. NSC-93-2112-M-007-014 and NSC-93-2112-M-007-025.

References

[1] Belle Collaboration, M.Z. Wang, et al., Phys. Rev. Lett. 92 (2004) 131801. [2] J. Schümann, Talk presented at QCD and Hadronic Interactions, La Thuile, Italy, 12–19 March 2005, hep-ex/0505098. [3] BaBar Collaboration, B. Aubert, et al., hep-ex/0408037. [4] Belle Collaboration, K. Abe, et al., hep-ex/0409009. [5] Belle Collaboration, Y.J. Lee, et al., Phys. Rev. Lett. 93 (2004) 211801. [6] W.S. Hou, A. Soni, Phys. Rev. Lett. 86 (2001) 4247. [7] C.K. Chua, W.S. Hou, S.Y. Tsai, Phys. Rev. D 66 (2002) 054004. [8] Belle Collaboration, K. Abe, et al., Phys. Rev. D 65 (2002) 091103; BaBar Collaboration, B. Aubert, et al., Phys. Rev. D 69 (2004) 091503. [9] H.Y. Cheng, K.C. Yang, Phys. Rev. D 65 (2002) 054028; H.Y. Cheng, K.C. Yang, Phys. Rev. D 65 (2002) 099901, Erratum. [10] H.Y. Cheng, K.C. Yang, Phys. Rev. D 66 (2002) 014020. [11] H.Y. Cheng, K.C. Yang, Phys. Lett. B 533 (2002) 271; See also H.Y. Cheng, hep-ph/0311035. [12] C.K. Chua, W.S. Hou, Eur. Phys. J. C 29 (2003) 27. [13] C.K. Chua, W.S. Hou, S.Y. Tsai, Phys. Rev. D 65 (2002) 034003. [14] S.J. Brodsky, G.R. Farrar, Phys. Rev. D 11 (1975) 1309. [15] S.J. Brodsky, G.P. Lepage, S.A. Zaidi, Phys. Rev. D 23 (1981) 1152. [16] C.Q. Geng, Y.K. Hsiao, Phys. Lett. B 610 (2005) 67. [17] C.Q. Geng, Y.K. Hsiao, in preparation. [18] M. Suzuki, J. Phys. G 29 (2003) B15. [19] A. Ali, G. Kramer, C.-D. Lu, Phys. Rev. D 58 (1998) 094009. [20] Y.H. Chen, et al., Phys. Rev. D 60 (1999) 094014. [21] N. Cabibbo, Phys. Rev. Lett. 10 (1963) 531; M. Kobayashi, K. Maskawa, Prog. Theor. Phys. 49 (1973) 652. [22] D. Melikhov, B. Stech, Phys. Rev. D 62 (2000) 014006. 312 C.Q. Geng, Y.K. Hsiao / Physics Letters B 619 (2005) 305–312

[23] A.V. Belitsky, X. Ji, F. Yuan, Phys. Rev. Lett. 91 (2003) 092003. [24] S.J. Brodsky, Phys. Rev. D 69 (2004) 076001. [25] C.K. Chua, W.S. Hou, S.Y. Tsai, Phys. Lett. B 528 (2002) 233. [26] L. Wolfenstein, Phys. Rev. Lett. 51 (1983) 1945. [27] Particle Data Group, Phys. Lett. B 592 (2004) 1. [28] H.Y. Cheng, K.C. Yang, Phys. Rev. D 62 (2000) 054029. [29] G. Buchalla, A.J. Buras, M.E. Lautenbacher, Rev. Mod. Phys. 68 (1996) 1125. [30] M. Neubert, A.A. Petrov, Phys. Lett. B 519 (2001) 50. [31] H.Y. Cheng, Eur. Phys. J. C 26 (2003) 551. [32] H.Y. Cheng, C.K. Chua, A. Soni, Phys. Rev. D 71 (2005) 014030. Physics Letters B 619 (2005) 313–321 www.elsevier.com/locate/physletb

B → χc0,2K decays: A model estimation

T.N. Pham a, Guohuai Zhu b,1

a Centre de Physique Theorique, Centre National de la Recherche Scientiﬁque, UMR 7644, Ecole Polytechnique, 91128 Palaiseau cedex, France b Institut für Theoretische Physik, Universität Hamburg, Luruper Chaussee 149, 22761 Hamburg, Germany Received 18 April 2005; received in revised form 3 June 2005; accepted 3 June 2005 Available online 13 June 2005 Editor: N. Glover

Abstract

In this Letter, we investigate the vertex corrections and spectator hard scattering contributions to B → χc0,2K decays, which has no leading contribution from naive factorization scheme. A non-zero binding energy b = 2mc −M is introduced to regularize the infrared divergence of the vertex part. The spectator diagrams also contain logarithmic and linear infrared divergences, for which we adopt a model dependent parametrization. If we neglect possible strong phases in the hard spectator contributions, we obtain a too small branching ratio for χc0K while too large one for χc2K, as can be seen from the ratio of the branching + → + + → + +0.63 ratio of B χc2K to that of B χc0K , which is predicted to be 2.15−0.76 in our model, while experimentally it should be about 0.1 or even smaller. But a closer examination shows that, assuming large strong phases difference between the twist-2 and twist-3 spectator terms, together with a slightly larger spectator infrared cutoff parameter Λh, it is possible to accommodate the experimental data. This shows that, for B → χc0,2K decays with no factorizable contributions, QCDF seems capable of producing decay rates close to experiments, in contrast to the B → J/ψK decay which is dominated by the factorizable contributions.  2005 Elsevier B.V. All rights reserved.

PACS: 13.25.Hw; 12.38.Bx; 14.40.Gx

1. Introduction

Hadronic B decays attract a lot of attention because of its role in determining the Cabibbo–Kobayashi–Maskawa (CKM) matrix elements, extracting CP-violating angles and even revealing physics beyond the Standard Model (SM). However in most cases, a deep understanding on the strong dynamics in hadronic B decays is prerequisite for the above purposes.

E-mail addresses: [email protected] (T.N. Pham), [email protected], [email protected] (G. Zhu). 1 Alexander-von-Humboldt Fellow.

Phenomenologically the naive factorization ansatz (NF) [1], supported by color transparency argument [2],is widely used in hadronic two-body B decays. However the unphysical dependence of the decay amplitude on renormalization scale indicates a prominent role of QCD corrections to NF. In this respect, B → χc0,2K decays are of special interest as these channels vanish in the approximation of NF, due to the spin-parity and vector current conservation. Therefore they provide a good opportunity to study the QCD corrections to NF. It was generally believed that the branching ratios of these channels should be quite small as the QCD corrections are either suppressed by strong coupling αs or ΛQCD/mb. But BaBar [3] and Belle [4] have found a surprisingly large branching ratio of + + B → χc0K decay,

+2.1 −4 + + (6.0− ± 1.1) × 10 (Belle), B B → χc0K = 1.8 (1) (2.7 ± 0.7) × 10−4 (BaBar).

Actually this large branching ratio is even comparable, for example, to that of B → χc1K decay which is not forbidden in NF. Another surprising observation is that, the upper limit of B → χc2K decay is roughly an order of + + magnitude smaller than the observed branching ratio of B → χc0K decay [5],

+ + −5 B B → χc2K < 3.0 × 10 (BaBar), (2) while naively the branching ratios of B → χc0,2K decays are expected to be at the same order. In the following we shall discuss these decay channels using the QCD factorization (QCDF) approach [6].In this framework, the final state light meson is described by the light-cone distribution amplitude(LCDA), while for the P -wave charmonium χc0,2, we shall adopt the covariant projection method of non-relativistic QCD [7]. It is well known that, for the inclusive decay and production of P -wave charmonia, the color-octet mechanism must be introduced to guarantee the infrared safety. However it is still unclear how to incorporate this mechanism in a model-independent way into exclusive processes. Thus the decay amplitudes A(B → χc0,2K) would be in- evitably infrared divergent when only the color-singlet picture is adopted for χc, which is shown explicitly in [8]. Thus strictly speaking, the QCDF approach is not applicable for B → χc0,2K decays due to the breakdown of factorization. In this Letter, to get a model estimation, we will introduce the binding energy b = 2mc − M [9] as an effective cutoff to regularize the infrared divergence appearing in the diagrams of vertex corrections (see Fig. 1). In fact, the logarithmic divergence ln(b) term in the limit b → 0 for the vertex corrections in B → χc0,2K decays is similar to the ln(b) term found in Ref. [9] for the production of P -wave charmonium in e+e− collisions. As for the spectator scattering contributions, there appears logarithmic divergence at twist-2 level and linear divergence at twist-3 level. Phenomenologically we shall parameterize these divergence as ln[mB /Λh] and mB /Λh respectively, where the non perturbative parameter Λh = 500 MeV again acts as an effective cutoff to regularize the endpoint divergence [10]. According to the QCDF approach, all other contributions are power suppressed by ΛQCD/mb. + + −4 We find that, with the above method, the branching ratio of B → χc0K decay is about 0.78 × 10 , which is several times smaller than the experimental measurements. At the same time, we also get the branching ratio of + + −4 −5 B → χc2K decay at about 1.68 × 10 , which is significantly larger than the upper limit 3 × 10 observed by BaBar [5]. But the above estimation is very crude in that the strong phases effects are completely ignored. Notice further that for the spectator contributions, there contains only logarithmic divergence at twist-2 level, while linear divergence appears at the twist-3 level, the strong phases of the twist-2 and twist-3 spectator terms could be quite different. We then briefly discuss the potential strong phases effects and argue that very different strong phases between twist-2 and twist-3 spectator terms together with a slightly larger Λh seems to be able to reproduce the + + + + experimental hierarchy B(B → χc0K ) B(B → χc2K ). T.N. Pham, G. Zhu / Physics Letters B 619 (2005) 313Ð321 315

Fig. 1. Order of αs contributions to B → χcJ K decay. (a)–(d) and (e)–(f) are called vertex corrections and spectator scattering diagrams, respectively.

2. Vertex and spectator corrections

In the QCDF approach, K meson is described by the following light-cone projection operator in momentum space [6] if /l /l MK = K /lγ Φ(x)− µ γ 2 1 Φ (x) , αβ 5 K 5 · P (3) 4 l2 l1 αβ where l is the momentum of K meson and l1 (l2) is the momentum of quark (antiquark) in K meson. Φ(x) and ΦP (x) are leading twist and twist-3 distribution amplitudes of K meson, respectively. It is understood that only after the factor l2 · l1 in the denominator is canceled, may we take the collinear limit l1 = xl, l2 = (1 − x)l. Notice that in principle we could also start directly from the original light-cone projector of K meson in coordinate space [11], and the physical results should be the same. But in this case care must be taken that, only with a proper regularization, can one do the relevant convolution integrals correctly. The readers may refer to the appendix of [12] for further details. Since P -wave charmonium χcJ is involved, we shall use covariant projection method [7,9] to calculate the decay amplitude A → = E(0,2) ∂ αC A (B χc0,2K) αβ Tr Π1 1 . (4) ∂qβ q=0 √ Here A is the standard QCD amplitude for cc¯ production, amputated of the heavy quark spinors, C1 = δij / 3is the color singlet projector. While Π α is the S = 1 heavy quark spinor projector 1 1 P/ P/ Π α = − q/ − m γ α + q/ + m , (5) 1 3 8mc 2 2 ¯ E(0,2) where P is the momentum of charmonium and 2q is the relative momentum between the cc pair in χcJ . αβ is the polarization tensor of χc0,2 which satisﬁes the following sum over polarization relation [7]

(0) (0) 1 (2) (2) 1 1 E E = Π Π , E E = (Π Π + Π Π ) − Π Π , (6) αβ α β D − 1 αβ α β αβ α β 2 αα ββ αβ βα D − 1 αβ α β 316 T.N. Pham, G. Zhu / Physics Letters B 619 (2005) 313Ð321 with

PαPβ Π =−g + . (7) αβ αβ M2

Here M is the mass of χcJ . For charmonium B decays, we shall start with the effective Hamiltonian [13]

6 H = G√F ∗ c + c − ∗ eff VcbVcs C1(µ)Q1(µ) C2(µ)Q2(µ) VtbVts Ci(µ)Qi(µ) , (8) 2 i=3 where Ci are Wilson coefﬁcients which are perturbatively calculable and Q1,2 (Q3–6) are the effective tree (QCD penguin) operators. Notice that we have dropped the electroweak penguin contributions here which are numerically negligible. The four-quark effective operators are deﬁned as

c = ¯ ¯ c = ¯ ¯ Q1 (qαbα)V −A(cβ cβ )V −A,Q2 (sαbβ )V −A(cβ cα)V −A,

Q3,5 = (s¯αbα)V −A (q¯β qβ )V ∓A,Q4,6 = (s¯β bα)V −A (q¯αqβ )V ∓A. (9) q q

Here q denotes all the active quarks at the scale µ = O(mb), i.e., q = u, d, s, c, b. While α and β are color indices. It is then straightforward to get the decay amplitude of B → χc0,2K decay by considering the vertex and spectator corrections drawn in Fig. 1,

6|R (0)| A → = iG√F √ 1 αs CF ∗ − ∗ + (B χc0,2K) VcbVcsC1 VtbVts(C4 C6) 2 πM 4π Nc

2 × B→K I + 4π fB fK II2 + II3 F0 f(0,2) B→K f(0,2) f(0,2) , (10) Nc F0

B→K → where R1(0) is the derivative of the χcJ wave function at the origin and F0 the form factor of B K.The function f I represents the contributions from vertex corrections while f II2 (f II3) arising from the twist-2 (twist-3) spectator contributions. The vertex function f I is actually infrared divergent and therefore depends on the binding energy b = 2mc − M. In the following we shall keep ln(b/M) term and drop the terms suppressed by b/M.The I explicit expressions of f(0,2) are as follows

2m ((1 + 12a)(1 − 4a) + 16a ln [4a]) −b f I= B √ ln + f I + O(b/M), 0 (1 − 4a)2 3a M 0ﬁn √ (2)∗ β 32E pα p a((1 + 12a)(1 − 4a) + 16a ln [4a]) −b f I= αβ B B + f I + O(b/M), 2 3 ln 2ﬁn (11) mB (1 − 4a) M T.N. Pham, G. Zhu / Physics Letters B 619 (2005) 313Ð321 317

= 2 2 I I → where a mc/mb, and ffin is the finite part of the function f in the limit b/M 0. The explicit expressions of I ffin are as follows, −m f I = B √ −6 − 22 ln 2 + 4a 26 + 15 − 56 ln 2 ln 2 0fin − 2 − 3 2(1 4a) (1 2a) 3a + 8a2 65 + 52 ln 2 − 84 ln2 2 + 384a4(1 + 2ln2) + 2a −85 + 28 ln 2(−1 + 6ln2) −1 + 4a + 32a3 −23 − 32 ln 2 + 14 ln2 2 − 8lna + 4a −(1 − 4a)2 5 − 24(1 − a)a ln a + 9lna + 2 a(−3 + 4a) 13 − 46a + 56a2 − 4(1 − 2a)3 ln [64a] ln a

+ 16(1 − 2a)3 ln2ln[−1 + 4a] 2 − 4a 1 − 2a − 64a(1 − 2a)3 Li 2 + Li 2[1 − 4a]−Li 2 , 1 − 4a 1 − 4a −E(2)∗ α β αβ p p 32a f I = √B B 4ln2(1 − 2a)2 + (1 − 4a) 4a2(1 + 2ln2) − 1 2ﬁn − 3 − 4mB a (1 2a) (1 4a) − 8a(3a − 1) ln a − ln [4a − 1] + VAB [a] , (12) where the function VAB [a] denotes the ﬁnite part of vertex corrections from Fig. 1(a)–(b). The analytical form of VAB [a] is too complicated to be shown here, but numerically it has a very mild dependence on the parameter a, for example,

VAB [0.1]=11.3,VAB [0.15]=11.9.

As for the spectator functions, we have

1 1 1 φ (ξ) φ (y) f II2 = √ dξ B dy K −8a + (1 − 4a)y¯ , 0 ¯2 mb(1 − 4a) 3a ξ y 0 0 1 1 2 µ φ (ξ) φ (y) f II3 = √ K dξ B dy P 8a − (1 − 4a)y¯ , 0 2 ¯2 mb(1 − 4a) 3a mb ξ y 0 0 ∗ √ 1 1 E(2) α β 16 αβ p p a φ (ξ) φ (y) f II2 = B B dξ B dy K 4a + (1 − 4a)y¯ , 2 3 − 3 ¯2 mb(1 4a) ξ y 0 0 ∗ √ 1 1 E(2) α β 32 αβ p p a µ φ (ξ) φ (y) f II3 = B B K dξ B dy P 8a − (1 + 8a)y¯ . (13) 2 3 − 4 ¯2 mb(1 4a) mb ξ y 0 0 Here ξ is the momentum fraction of the light spectator quark in the B meson, and y¯ = 1 − y the light-cone momentum fraction of the quark in the K meson which is from the spectator quark of B meson. Notice that our II2 expressions for twist-2 spectator function f(0,2) are consistent with those of [8]. 318 T.N. Pham, G. Zhu / Physics Letters B 619 (2005) 313Ð321

3. Numerical results and discussion

To get a numerical estimation on the branching ratios of B → χc0,2K decays, several parameters appearing in Eqs. (10)–(13) should be ﬁrst decided on. The derivative of χcJ wave function at the origin |R (0)| may be either 2 estimated by QCD-motivated potential models [14], or extracted from χcJ decays [15]. |R (0)| varies from 0.075 5 to 0.131 GeV in different potential models [14] while using χcJ decays, for instance [16],

36 |R(0)|2 16 α Γ(χ → γγ)= e4 α2 − s , c2 Q em 4 1 (14) 5 mc 3 π 2 5 it is easy to get |R (0)| = (0.062 ± 0.007) GeV if we take mc = 1.5 GeV. This result is a little bit lower than, but still consistent with the potential model calculations, especially considering that it is very sensitive to the choice of charm quark mass. In this Letter, we shall take |R(0)|2 = (0.10 ± 0.03) GeV5 as input. = − − = For the binding energy, if we take mc 1.5 GeV,√ the ratio b/M is about 0.11 ( 0.16) for χc0 (χc2), while a 2 2 mc/mb 0.1. The QCD scale µ should be order of mbΛ, as in charmless B decays, which is about (1–1.5) GeV. In the following we shall ﬁx the scale µ = 1.3 GeV with αs = 0.36. Notice also that the Wilson coefﬁcients should be evaluated at leading order, to be consistent with the leading order formula of Eq. (10),

C1 = 1.26,C4 =−0.049,C6 =−0.074. (15) The relevant CKM parameters are chosen to be A = 0.83 and λ = 0.224. As for the spectator contributions, we adopt the following LCDAs for the ﬁnal K meson,

(3/2) φK (y) = 6y(1 − y) 1 + anCn (2y − 1) ,φP (y) = 1, (16) n1 (3/2) where Cn (x) are Gegenbauer polynomials. The parameters an are set to be [17]

a1 = 0.17,a2 = 0.115,a4 = 0.015,a3 = an>4 = 0. (17) Then logarithmic and linear divergences appear in Eq. (13), which may be phenomenologically parameterized as [6] dy m dy m = B , = B , ln 2 (18) y Λh y Λh with Λh = 500 MeV. Notice that the above parametrization of linear divergence would violate the power counting of QCDF, but we do not have better way yet to deal with it. This is clearly a very rough estimation, for example, we do not consider here the strong phase effect. We also know little about B wave function, but fortunately only the following integral is involved which may be parameterized as φ (ξ) m dξ B = B (19) ξ λB and we shall simply ﬁx λB = 350 MeV in our calculation. The chirally enhanced ratio rK = µK /mb is chosen to +0.11 = ± + = be 0.43−0.08, which corresponds to taking ms(2GeV) (90 20) MeV and (mu md )(2GeV) 9 MeV. The B→K 2 2 = form factor F0 (mχc ) may be read from [17], in which as stated, the uncertainty of form factor at q 0is 2 = B→K 2 = ± likely to be smaller than that of q 0, which is about 12%. Therefore we will cite F0 (mχc ) 0.48 0.06 as our input. The decay constants are set as fK = 160 MeV and fB = (210 ± 25) MeV. With the above input, we get B + → + = +0.46 × −4 B + → + = +0.78 × −4 B χc0K 0.78−0.35 10 , B χc2K 1.68−0.69 10 . (20) We also show separately the contributions from vertex corrections and hard spectator scattering diagrams in Ta- ble 1, with all the input parameters taken at their central values. For the case of χc0K channel, our results are T.N. Pham, G. Zhu / Physics Letters B 619 (2005) 313Ð321 319

Table 1 The numerical estimations of vertex corrections and hard spectator scattering contributions, with all the parameters taken at their central values. 2 The constant C ≡ 4π fB fK Nc B→K F0 Decay channels f I C ∗ f II2 C ∗ f II3

χc0K 46.3 − 33.6i −43.180.7 χc2K 1.7 + 14.1i 69.368.3 approximately four times smaller than the average of BaBar and Belle measurements, (3.0 ± 0.7) × 10−4, while −5 our prediction on B → χc2K decay is obviously too large compared with the experimental upper limit, 3.0 × 10 . This is a little bit surprising, because for charmonium B decays, the theoretical results are normally a few times smaller than the experimental measurements. The careful reader may have noticed that in the above analysis we did not consider the uncertainty related to = 2 2 → the parameter a mc/mb. In fact a larger a could enhance the branching ratio of B χc0K decay significantly, but unfortunately it would also enhance that of B → χc2K decay with similar magnitude. Notice that B → χc0,2K share many common inputs, the ratio of branching ratios of these two channels should have mild dependence on the input parameters, for example it is independent on the parameter |R(0)|. Our numerical analysis shows that this is indeed the case, with a = 0.10 ± 0.03: B + → + R = (B χc2K ) = +0.26 +0.33 +0.36 +0.30 + + 2.15−0.55 −0.31 −0.28 −0.31, (21) B(B → χc0K ) B→K where the uncertainties arise from the parameters a, rK , F0 and fB , respectively. The above ratio is clearly in strong contradiction with the experimental hierarchy R 0.1 1. Notice that the chirally enhanced power corrections, namely twist-3 spectator contributions in this case, have been included in the above estimation. For the rest part of power corrections, there is no systematic way to estimate them yet. But since the power corrections are suppressed by ΛQCD/mb, intuitively they might lead to an uncertainty of about 20% to the decay amplitude, which is unlikely to be able to change our estimation Eq. (21) dramatically. In our model, the parameters Λh and λB will introduce additional uncertainties to B → χc0,2K decays. It is very unlikely that we could reproduce the experimental observations by fine tuning λB , because although a larger λB would lead to a smaller branching ratio for χc2K decay, it would also make the already too small branching ratio of χc0K decay even smaller. However a larger Λh does help to close the gap between our predictions and the experimental data, due to the fact that a larger Λh will lead to a significantly smaller branching ratio for χc2K decay while χc0K decay does not change much. Of course we cannot choose a too large Λh, say larger than 1 GeV, because it is anyway a non-perturbative parameter. As an illustration, we take Λh = 700 MeV and get + + −4 + + −4 B B → χc0K = 0.78 × 10 , B B → χc2K = 0.74 × 10 . (22) Although it seems to be on the right way, this effort alone is still not enough to accommodate the experimental data. Let us take a closer look at the decay amplitudes. From Table 1, it is clear that the spectator hard scattering mechanism is dominant in B → χc2K decay and also very important for χc0K channel. Furthermore there is significant destructive (constructive) interference between the twist-2 spectator term and the twist-3 one for χc0K (χc2K) mode. This is probably the reason that we get too small χc0K decay as well as too large χc2K decay in our model. Notice that there are logarithmic and linear divergences appear in the spectator contributions, which are parameterized by Eq. (18). It is obviously a very rough model estimation and for example, strong phases effects are completely ignored. It is also reasonable to assume that the strong phase of twist-2 spectator term could be different from that of twist-3 part. As an illustration, the endpoint divergences could be parameterized as [6]: dy mB dy mB = + ρ eiθ2,3 , = + ρ eiθ3 , ln 1 2,3 2 1 3 (23) y Λh y Λh 320 T.N. Pham, G. Zhu / Physics Letters B 619 (2005) 313Ð321 with 0 ρ 1 and the phase θ completely free. In the above equations, (ρ2,θ2) denotes the parameters for twist-2 spectator term and (ρ3,θ3) for twist-3 one. In this case, the interference effects and therefore the predictions of the branching ratios, could be changed dramatically. For example, if we take a somewhat extreme case

ρ2 = 0.6,θ2 = π, ρ3 = 0,θ3 = 0, with Λh = 600 MeV while keep all other input parameters fixed at their central values, we will get + + −4 + + −5 B B → χc0K = 3.3 × 10 , B B → χc2K = 1.7 × 10 , (24) which are in good agreement with the experimental observations. Certainly, due to the non-perturbative nature of the above strong phases, there is strong model dependence of our predictions. Therefore it is not so meaningful to fine tune the parameters to get the best fit of the experimental data. The key point here is that, different strong phases between twist-2 and twist-3 spectator terms might be able to account for the experimental hierarchy that B(B → χc0K) is at least an order of magnitude larger than B(B → χc2K). The authors of Ref. [18] also studied the B → χc0K decay with the same QCDF method. But they used the gluon mass and gluon momentum cutoff, instead of binding energy adopted in this paper, to regularize the infrared divergences of the vertex corrections. Another difference is that they calculated the spectator contributions directly from the original light-cone projector of K meson in coordinate space and got a different result from this Letter. They claimed that the difference was due to the light-cone projector adopted in this paper which is inappropriate for χcJ K channels: to get the projector Eq. (3) from the original one in coordinate space [11], the integration by parts has been used and the boundary terms were dropped. However because of the linear singularities appeared in the above calculations, the boundary terms seem to be divergent and thus the justification of using the integration by parts is in doubt in this case. But Beneke has elaborated on this subtle point in the appendix of Ref. [12] and it is shown there that the boundary terms are indeed zero provided the propagators are regularized carefully when they go close to the mass-shell. Therefore the integration by parts can be used here and the light-cone projector adopted in this paper is justified. Certainly, with a proper regularization, the calculation starting directly from the coordinate space projector should give the same results as this Letter. Most recently, the B → χc0K decay was discussed by using the PQCD method [22]. Notice that the vertex corrections were not included in their calculations, and the spectator contributions alone are enough in their paper to account for the experimental data. It would be very interesting to see whether they could also reproduce the very small branching ratio of χc2K channel observed by BaBar, which has not been done yet. The B → χc0K decay was also analyzed with light-cone sum rules [19,20]. Although there are some discrepancies in their papers, they agreed on the point that their results were too small to accommodate the experimental (∗) (∗) data. A large charmed meson rescattering effects B → Ds D → χc0,2K could account for the surprisingly large B → χc0K decay [21], but generally it will also lead to a large branching ratio for χc2K mode. In summary, we discuss in this Letter the vertex corrections and spectator hard scattering contributions to B → χc0,2K decays. Since there is no model independent way yet to estimate the color-octet contribution to exclusive processes, it is no wonder that the vertex corrections here are infrared divergent. The non-zero binding = − energy b 2mc MχcJ makes the charm quark slightly off-shell inside χcJ , and effectively acts as a cutoff to regularize the vertex part. There are also less serious logarithmic and linear endpoint divergences which appears in the spectator contributions and are parameterized in a model-dependent way as usually done in charmless B decays. This means that the spectator diagrams are actually dominated by soft gluon exchange, which in a sense could be viewed as a model estimation of color-octet contributions. Then our numerical analysis predicts the + + −4 branching ratio of B → χc0K decay to be about 0.78 × 10 , about four times smaller than the experimental + + −4 observations, while for B → χc2K decay, we get 1.68 × 10 , which is about five times larger than the experimental upper limit. But concerning the large theoretical uncertainties, it is more interesting to consider the ratio + + + + R = B(B → χc2K )/B(B → χc0K ), in which a large part of the theoretical uncertainty can be eliminated. = +0.63 Numerically we find the ratio to be R 2.15−0.76, in sharp contrast to the experimental observation that this ratio T.N. Pham, G. Zhu / Physics Letters B 619 (2005) 313Ð321 321

+ + should be about 0.1 or even smaller, if the BaBar analysis of the upper limit of B(B → χc2K ) decay will be conﬁrmed by further measurements. We then have a closer look at the decay amplitudes. One observation is that, χc0K channel is not very sensitive to the spectator infrared cutoff parameter Λh while a larger Λh could reduce the branching ratio of χc2K decay signiﬁcantly. Another observation is that, in our model there is large destructive (constructive) interference between the twist-2 and twist-3 spectator terms for χc0K (χc2K) mode. But notice that the twist-2 spectator contributions contain only logarithmic endpoint divergence while twist-3 ones contain more severe linear endpoint divergence, it should be reasonable to assume that their strong phases could be quite different. Since the interference effects are very sensitive to the strong phases difference, this might change our model predictions dramatically. As an illustration, we then show in an explicit case that, with a slightly larger Λh and large strong phases difference between twist-2 and twist-3 spectator terms, our predictions are in good agreement with the experimental data. In conclusion, what we have shown in this Letter, is that by adjusting the parameters for the spectator hard scattering contributions, as with the annihilation terms for charmless B decays, QCDF is able to produce appreciable + + + + non-factorizable contributions to B → χc0,2K decays close to experiments, in contrast with the B → J/ψK decay which needs a large factorizable contribution in addition to the small non-factorizable one obtained in QCDF [23].

Acknowledgement

G.Z. acknowledges the support from Alexander-von-Humboldt Stiftung.

References

[1] M. Bauer, B. Stech, M. Wirbel, Z. Phys. C 34 (1987) 103. [2] J.D. Bjorken, Nucl. Phys. B (Proc. Suppl.) 11 (1989) 325. [3] BaBar Collaboration, B. Aubert, et al., Phys. Rev. D 69 (2004) 071103. [4] Belle Collaboration, K. Abe, et al., Phys. Rev. Lett. 88 (2002) 031802. [5] BaBar Collaboration, B. Aubert, et al., Phys. Rev. Lett. 94 (2005) 171801, hep-ex/0501061. [6] M. Beneke, G. Buchalla, M. Neubert, C.T. Sachrajda, Phys. Rev. Lett. 83 (1999) 1914; M. Beneke, G. Buchalla, M. Neubert, C.T. Sachrajda, Nucl. Phys. B 591 (2000) 313; M. Beneke, G. Buchalla, M. Neubert, C.T. Sachrajda, Nucl. Phys. B 606 (2001) 245. [7] A. Petrelli, et al., Nucl. Phys. B 514 (1998) 245. [8] Z. Song, C. Meng, J.Y. Gao, K.T. Chao, Phys. Rev. D 69 (2004) 054009; Z. Song, K.T. Chao, Phys. Lett. B 568 (2003) 127. [9] J. Kuhn, J. Kaplan, E. Saﬁani, Nucl. Phys. B 157 (1979) 125. [10] D.S. Du, D.S. Yang, G.H. Zhu, Phys. Lett. B 509 (2001) 263. [11] V.M. Braun, I.E. Filyanov, Z. Phys. C 44 (1989) 157; V.M. Braun, I.E. Filyanov, Z. Phys. C 48 (1990) 239. [12] M. Beneke, Nucl. Phys. B (Proc. Suppl.) 111 (2002) 62. [13] G. Buchalla, A.J. Buras, M.E. Lautenbacher, Rev. Mod. Phys. 68 (1996) 1125. [14] E. Eichten, C. Quigg, Phys. Rev. D 52 (1995) 1726. [15] M. Mangano, A. Petrelli, Phys. Lett. B 352 (1995) 445. [16] R. Barbieri, et al., Nucl. Phys. B 192 (1981) 61. [17] P. Ball, R. Zwicky, Phys. Rev. D 71 (2005) 014015. [18] C. Meng, Y.J. Gao, K.T. Chao, hep-ph/0502240. [19] Z.G. Wang, L. Li, T. Huang, Phys. Rev. D 70 (2004) 074006. [20] B. Melic, Phys. Lett. B 591 (2004) 91. [21] P. Colangelo, F. De Fazio, T.N. Pham, Phys. Lett. B 542 (2002) 71. [22] C.H. Chen, H-N. Li, hep-ph/0504020. [23] H.Y. Cheng, K.C. Yang, Phys. Rev. D 63 (2001) 074011. Physics Letters B 619 (2005) 322–332 www.elsevier.com/locate/physletb

¯ Reduction of charm quark mass scheme dependence in B → Xsγ at the NNLL level

H.M. Asatrian a,C.Greubb, A. Hovhannisyan a,T.Hurthc,d,1, V. Poghosyan a

a Yerevan Physics Institute, 375036 Yerevan, Armenia b Institute for Theoretical Physics, University Berne, CH-3012 Berne, Switzerland c Theoretical Physics Division, CERN, CH-1211 Geneva 23, Switzerland d SLAC, Stanford University, Stanford, CA 94309, USA Received 16 May 2005; accepted 29 May 2005 Available online 13 June 2005 Editor: G.F. Giudice

Abstract ¯ The uncertainty of the theoretical prediction of the B → Xsγ branching ratio at NLL level is dominated by the charm mass renormalization scheme ambiguity. In this Letter we calculate those NNLL terms which are related to the renormalization of mc, in order to get an estimate of the corresponding uncertainty at the NNLL level. We find that these terms significantly reduce ¯ (by typically a factor of two) the error on BR(B → Xsγ)induced by the definition of mc. Taking into account the experimental accuracy of around 10% and the future prospects of the B factories, we conclude that a NNLL calculation would increase the ¯ sensitivity of the observable B → Xsγ to possible new degrees of freedom beyond the SM significantly.  2005 Elsevier B.V. All rights reserved.

1. Introduction

¯ The branching ratio of B → Xsγ is a very sensitive probe for new degrees of freedom beyond the standard model (SM) (for a review, see [1]). Within supersymmetric extensions of the SM, for example, one can derive stringent bounds on the parameter space of these models [2–8]. Clearly, such bounds will be most valuable when the general nature of the new physics beyond the SM will be identiﬁed at the forthcoming LHC experiments. ¯ Because of the heavy mass expansion that is valid for inclusive decay modes, the decay rate of B → Xsγ is dominated by the perturbatively calculable partonic decay rate Γ(b→ Xsγ ). QCD corrections to the latter, due to hard-gluon exchange, are the most important perturbative contributions; they were calculated in the past up to

E-mail address: [email protected] (C. Greub). 1 Heisenberg Fellow.

0370-2693/$ – see front matter  2005 Elsevier B.V. All rights reserved. doi:10.1016/j.physletb.2005.05.080 H.M. Asatrian et al. / Physics Letters B 619 (2005) 322–332 323 the next-to-leading logarithmic (NLL) level [9–18]. Subsequently, also electroweak corrections were calculated [19–22]. After completion of these computations, it was generally believed that the theoretical uncertainty of the branching ratio is below 10%. However, as first pointed out in 2001 in [23], there is an additional uncertainty in the NLL results for Γ(b→ Xsγ)which is related to the definition (renormalization scheme) of the charm quark mass. Technically, the charm quark mass dependence enters through the matrix elements sγ|O1,2|b which in the context of a NLL have to be calculated up to O(αs). As these matrix elements vanish at the lowest order, the charm quark mc only enters (through the ratio mc/mb)atO(αs). As a consequence, the charm quark mass does not get renormalized in a NLL calculation, which means that the symbol mc can be identified with mc,pole or with the MS mass m¯ c(µc) at some scale µc or with some other definition of mc. Formally, all these assignments are equivalent, as they lead to 2 differences which are of order αs . Note that in contrast to the c-quark mass the b-quark mass does get renormalized in a NLL calculation and we choose to express all the following results in terms of mb,pole. In this respect we do not follow Ref. [23], where the mb,1S mass was used. Unless stated otherwise, the symbol mb stands for mb,pole in all the formulas in this Letter. Numerically, we use mb = 4.8 GeV throughout. Numerically, it turns out that the NLL result for Γ(b→ Xsγ)strongly depends on which mass definition of the charm quark mass is used in the NLL expressions. To illustrate this, we first identify mc with mc,pole as it was done in all analyses before the paper of Gambino and Misiak [23]. Numerically, we use mc,pole/mb,pole = 0.29 which is based on the mass difference mb,pole − mc,pole = 3.4 GeV fixed through the heavy mass expansion of mB and mD and mb,pole = 4.8 GeV. The corresponding branching ratio then reads [23] [ ¯ → ] = × −4 BR B Xsγ Eγ >mb/20 3.35 10 . (1)

As the charm quarks which are propagating in a loop have a typical virtuality of mb/2, the authors of Ref. [23] suggested to use m¯ c(µc) with µc ∈[mc,mb] instead of mc,pole. A typical value for the corresponding ratio is m¯ c(µc)/mb,pole = 0.22. Using this value, the branching ratio gets increased w.r.t. (1) by about 11% [23] [ ¯ → ] = × −4 BR B Xsγ Eγ >mb/20 3.73 10 . (2) In a recent theoretical update of the NLL prediction of this branching ratio, the uncertainty related to the deﬁnition of mc was taken into account by varying mc/mb in the conservative range 0.18 mc/mb 0.31 which covers both, the pole mass (with its numerical error) value and the running mass m¯ c(µc) value with µc ∈[mc,mb] [24] [ ¯ → ]= ± | ± | ± | ± | × −4 BR B Xsγ (3.70 0.35 mc/mb 0.02 CKM 0.25 param. 0.15 scale) 10 . (3) ¯ There exists a large number of measurements of the inclusive decay B → Xsγ [25–30] and the present experimental accuracy has reached the 10% level [31]

¯ −4 BR[B → Xsγ ]=(3.52 ± 0.30) × 10 . (4) In the near future, more precise data on this mode are expected from the B factories. Thus, it is mandatory to reduce the present theoretical uncertainty accordingly. A systematic improvement certainly consists in performing a complete NNLL calculation. This is, however, a very complicated task (for discussion and some results see [32– 35]) and a certain motivation is needed to enter such an enterprise. In the present Letter we try to give such a motivation: by calculating those NNLL terms which are induced by renormalizing the charm quark mass in the NLL expressions, i.e., those terms which are sensitive to the definition of the charm quark mass, we show that the large error at the NLL level related to the mc definition gets significantly reduced. As this error is the dominant one at the NLL level (see Eq. (3)), we conclude that a complete NNLL calculation will drastically improve the theoretical prediction of the branching ratio. We stress here that in the present Letter we only make a statement about the reduction of the error at the NNLL level, and not about the central value of the branching ratio; this remains the topic of a complete NNLL calculation! 324 H.M. Asatrian et al. / Physics Letters B 619 (2005) 322–332

The remainder of this Letter is organized as follows. In Section 2 we discuss in some detail how to calculate the NNLL terms induced by renormalizing mc in the NLL results. In order to make the Letter self-contained, we ﬁrst list in Section 3 the structure of the NNL results and then we present the analytical results for the new terms discussed in Section 2. Finally, in Section 4, we numerically investigate by how much the error related to the deﬁnition of mc gets reduced at the NNLL level.

2. NNLL terms related to mc renormalization

virt = | | As already explained in the introduction, the matrix elements M1,2 (mc) sγ O1,2(µb) b only start at or- 1 2 der O(αs ), or, in other words at the NLL order. As a consequence, the deﬁnition of mc is not ﬁxed at this brems = order, because mc does not get renormalized. This is also true for the bremsstrahlung contributions M1,2 (mc) sγg|O1,2(µb)|b, which are needed up to O(gs) for a NLL calculation. In this section we concentrate on the virt brems virtual terms M1,2 (mc), as the extension to the bremsstrahlung contributions M1,2 (mc) is straightforward. virt 2 When going to NNLL precision, the matrix elements M1,2 (mc) are needed to O(αs ). At this level, there are— among many other diagrams—counterterm contributions to these matrix elements, induced by the renormalization of mc (see the left frame of Fig. 1). The complete set of such diagrams is generated by inserting the operator − ¯ virt() iδmcψcψc in the O(αs) diagrams of O1,2 in all possible ways. The sum δM1,2 (mc)δmc of all these insertions → + 1 virt() can be obtained by replacing mc mc δmc in the O(αs ) results M1,2 (mc), followed by expanding in δmc up to linear order virt() + = virt() + virt() + 2 M1,2 (mc δmc) M1,2 (mc) δM1,2 (mc)δmc O (δmc) . (5)

virt() 1 As δmc is ultraviolet divergent, the matrix elements M1,2 (mc) are needed in our application up to order ,as indicated by the notation in Eq. (5). The explicit shift δmc depends of course on the renormalization scheme. When aiming at expressing the results virt() ¯ = for M1,2 (mc) in terms of mc(µb), the shift reads (CF 4/3)

α (µ ) 3 δm¯ (µ ) =− s b C m¯ (µ ). c b 4π F c b

Fig. 1. Left frame: Typical δmc insertion diagram (see text). Right frame: Typical diagram with a charm quark self-energy insertion (see text).

2 In the present Letter we use the operator basis as ﬁrst introduced in Ref. [11]. µb denotes the renormalization scale of O(mb). H.M. Asatrian et al. / Physics Letters B 619 (2005) 322–332 325

On the other hand, when the result is expressed in terms of mc,pole, the shift reads α (µ ) 3 µ2 δm =− s b C + b + m . c,pole F 3ln 2 4 c,pole 4π mc The inﬁnities induced by the 1/ terms in δmc get canceled in a full NNLL calculation, in particular by self-energy diagrams depicted in the right frame of Fig. 1. As we do not perform a full NNLL calculation, we suggest to 2 2 = 2 consider self-energy insertions, where the self-energy Σ(p ) is replaced by Σ1(p mc). 2 The Σ1-part of the self-energy Σ1(p ) is deﬁned through the decomposition of the full unrenormalized self- energy Σ(p2) as 2 2 2 Σ p ≡ Σ2 p (/p − mc) + Σ1 p . (6) R R At the one-loop level, the corresponding pieces Σ1 and Σ2 of the renormalized self-energy are R 2 = 2 + R 2 = 2 + Σ2 p Σ2 p δZc,Σ1 p Σ1 p δmc, (7) where Zc = 1 + δZc denotes the wave function renormalization constant of the charm quark. Eq. (7) implies that virt() 2 the singularities in δM1,2 (mc)δmc cancel when combined with the diagrams with Σ1(p ) insertions. However, 2 2 for general p , the function Σ1(p ) depends on the gauge parameter ξ: α (µ ) 1 µ2 m2 m2 p2 Σ p2 = s b C m + b + + − c ξ − − ξ c − . 1 F c 3 ln 2 4 1 2 3 2 ln 1 2 4π mc p p mc 2 = 2 2 = 2 2 = 2 As for p mc the self-energy piece Σ1(p mc) is gauge-independent, we add Σ1(p mc) insertions to virt() δM1,2 (mc)δmc. 2 = 2 eff These momentum independent Σ1(p mc) insertions can be straightforwardly absorbed into δmc insertions: eff = 2 = 2 + = δmc,pole Σ1 p mc δmc,pole 0, α (µ ) µ2 δm¯ eff(µ ) = Σ p2 = m2 + δm¯ (µ ) = s b C b + m¯ (µ ). c b 1 c c b F 3ln 2 4 c b (8) 4π mc virt() ¯ Finally, if we wish to express the matrix elements M1,2 (mc) in terms of mc(µc), the shift reads α (µ ) µ2 δm¯ eff(µ ) = s b C c + m¯ (µ ). c c F 3ln 2 4 c c (9) 4π mc

3. Analytical results

Before turning to the contributions induced through the renormalization of the charm quark mass, which are NNLL terms, we ﬁrst summarize the structure of the NLL result for the branching ratio for b → Xsγ . We write the → = mb − = − decay width for b Xsγ using a photon energy cut E0 2 (1 δ) Emax(1 δ) as → = → + → Γ(b Xsγ)Eγ E0 Γ(b sγ) Γ(b sγg)Eγ E0 , (10) where the two parts are deﬁned as follows: 2 GF ∗ 2 5 2 Γ(b→ sγ) = V Vtb α m |D| , 32π 4 ts em b,pole 2 GF ∗ 2 5 Γ(b→ sγg) = V Vtb α m A, Eγ E0 32π 4 ts em b,pole 326 H.M. Asatrian et al. / Physics Letters B 619 (2005) 322–332 8 = (0)eff + αs(µb) (1)eff + (0)eff + (0)eff mb − 16 (0)eff D C7 (µb) C7 (µb) Ci (µb) ri γi7 ln C7 (µb) , (11) 4π µb 3 i=1

8 − [ + ] 2 αs(µb) A = e αs (µb) ln(δ) 7 2ln(δ) /(3π) − 1 C(0)eff(µ ) + C(0)eff(µ )C(0)eff(µ )f (δ). (12) 7 b π i b j b ij i,j=1,ij

The expressions for the Wilson coefﬁcients Ci(µb) can be found in [36]. Their numerical values we take from Table 5.1 in Ref. [37]. Writing the results in this speciﬁc form, the functions fij (δ) and ri are understood to be taken from [11] and not from the original paper [10] where the results were parametrized differently. Following common practice, we write the branching ratio (without taking into account nonperturbative corrections) as

Γ(b→ X γ) → = s Eγ E0 exp BR(b Xsγ)Eγ E0 BRsl , (13) Γ(b→ Xceν)¯ where the semileptonic decay rate is given by 2 5 2 2 − GF mb,pole m m Γ(b→ X e ν)¯ = |V |2g c K c . c 3 cb 2 2 (14) 192π mb mb g(z) = 1 − 8z + 8z3 − z4 − 12z2 ln(z) is the phase-space factor and the function 2α (m ) f(z) K(z) = 1 − s b , 3π g(z) with 25 239 25 4 17 f(z)=− 1 − z2 − z + z2 + z ln(z) 20 + 90z − z2 + z3 4 3 4 3 3 17 64 17 + z2 ln2(z) 36 + z2 + 1 − z2 − z + z2 ln(1 − z) 3 3 3 − 4 1 + 30z2 + z4 ln(z) ln(1 − z) − 1 + 16z2 + z4 6Li(z) − π 2 √ √ √ 1 − z − 32z3/2(1 + z) π 2 − 4Li( z)+ 4Li(− z)− 2ln(z) ln √ 1 + z accounts for O(αs) QCD corrections. We note that mc is understood to be the pole mass in Eq. (14). We now turn to that part of NNLL corrections which is responsible for the reduction of the charm quark mass virt() renormalization scheme dependence, as explained in Section 2.WefirstturntotermsδM1,2 induced by mc virt() virt() 1 renormalization in the matrix elements M1,2 . To this end, we need M1,2 up to oder .In[10] have calculated these matrix elements up to terms 0, using Mellin–Barnes representations for generalized propagator to obtain 2 analytic results in the form of the series in z = (mc/mb) and L = ln(z). As in these calculations the expansion in virt() 1 was the last step, it is straightforward to calculate M1,2 up to order . In order to get finite results for these matrix elements, we add counterterms related to operator mixing as in virt,ren Ref. [10], adapted, however, to the operator basis defined in Ref. [11]. This step leads to M1,2 , which we decompose as in Ref. [10] virt,ren = | | αs 416 mb − 784 2 mb − mb (0) + (0) + (1) M2 sγ O7 b ln ln 4 ln r2 r2 r2 . (15) 4π 81 µb 81 µb µb H.M. Asatrian et al. / Physics Letters B 619 (2005) 322–332 327

= (0) + (1) =−1 We obtain for r2 r2 r2 (note that r1 6 r2): 1666 8 32 r(0) =− − −48 − 3L2 − L3 + 5π 2 + 9L −4 + π 2 + 36ζ(3) z + π 2z3/2 2 243 27 27 8 4 + L3 − 6L −2 + π 2 + 18 + 2π 2 z2 − 9 − 182L + 126L2 + 14π 2 z3 27 81 8iπ 10 8 − + 2 −15 − 3L − 3L2 + π 2 z + 2 −3L2 + π 2 z2 + (−7 + 3L)z3 , (16) 27 3 3

19577 184 2 r(1) =− + π 2 − −18180 + 75L4 + 3240π 2 + 46π 4 − 30L2 −24 + 7π 2 2 729 243 405 32 + 9000ζ(3) + 120L −66 + 14π 2 + 27ζ(3) z − π 2 −49 + 6L + 24 ln(2) z3/2 81 2 + 48L3 − 15L4 + 24L2 −3 + π 2 − 24L 3 + 5π 2 + 1116 + 36π 2 + 40π 4 + 432ζ(3) z2 81 1120 1 − π 2z5/2 + 22705 − 2484L2 + 4536L3 − 6036π 2 + 6L −1783 + 192π 2 + 8208ζ(3) z3 81 729 8iπ 221 + − + 15L2 − 6L3 − 4L −9 + π 2 + 186 − 10π 2 − 36ζ(3) z 27 9 4 − 2 −3 − 6L2 + 3L3 + 2π 2 + 2Lπ 2 + 18ζ(3) z2 + −67 + 66L + 9L2 + 12π 2 z3 . (17) 9 In these formulas we retained all terms up to order z3, as higher order terms contribute much less than 1%. Never- theless, in the numerical evaluations in Section 4 all terms up to z6 were included. At the level of the decay width, the implementation of the contribution coming from renormalization of the c-quark mass in the virtual contributions is (according to Eq. (5)) most easily achieved by replacing r1,2 in Eq. (11) (0) + by r1,2 r1,2, where = d (0) + (1) r1,2 δmc r1,2 r1,2 . (18) dmc

At the NLL order, the bremsstrahlung corrections to the decay width are encoded in the quantities fij (δ) (see Eq. (12)), which correspond to the interference terms (Oi,Oj ). In the following, we calculate the shifts f ij to these quantities induced by the renormalization of the charm quark mass. In principle, we calculate the decay width using a photon energy cut δ = 0.9(seeEq.(10)). However, as all bremsstrahlung contributions which contain charm quark loops are finite for δ → 1, we can approximate these terms by putting δ = 1. Numerically the relative error is of order 10−4. We first calculate the shift f 27. To this end, we shift the charm quark mass in the matrix element of sγg|O2|b as in Eq. (5) and then work out the interference with sγg|O7|b. Because of the 1/ term in δmc, the result is ultraviolet singular. In a full NNLL calculation this singularity gets canceled when combined with self-energy insertions in the charm quark lines in the matrix element of O2. We therefore do the phase space integrations involved in the derivation of f27 (or f 27)ind = 4 dimensions. As only the matrix element of O2 depends on mc, the shift f 27 can be constructed by first considering the quantity f27 itself. Using the integral representation for the building block for photon and gluon emission from the c-quark loop [10], one obtains after integration over all but one of the phase space parameters 8 µ 2 [x(1 − x)]− (2 + )eγ+iπ f =− b dxdyduu2(1 − u)(1 − x)y . (19) 27 [ − z + ](1+) 9 mb uy x(1−x) iη 328 H.M. Asatrian et al. / Physics Letters B 619 (2005) 322–332

Here x,y are Feynman parameters and u is the remaining phase space parameter, 0 x,y,u 1. To solve the integrals, we use the Mellin–Barnes representation for the generalized propagator

− − z 1 1 eiπs (−s) (1 + + s) z s uy − + iη = ds x(1 − x) 2iπ (1 + ) (uy)1+ uyx(1 − x) γ appearing in Eq. (19). γ denotes the integration path parallel to imaginary axes which hits the real axes somewhere between (−1 − ) and 0. Closing the integration path in the right s-half plane, one gets an expansion for f27 in 2 z = (mc/mb) and L = ln(z). The shift f 27 is then obtained as = dz df27 = mc δmc d(0) + d(1) f 27 δmc 2 f27 f27 . (20) dmc dz mb mb

To summarize, the NNLL contributions due to renormalization of mc in the (O2,O7) interference are taken into → (0) + account by replacing f27 f27 f 27 in Eq. (12). Explicitly, we ﬁnd (0) =−8 1 + 1 − 2 + + 2 + 2 − − 2 2 + 1 − − 2 − + 2 3 f27 7 2π 6L 2L z π 2L L z 11 6π 4L 6L z 9 12 8 4 1 1 1 + (−6 + 10L)z4 + (13 + 70L)z5 + (32 + 63L)z6 , (21) 3 24 15 d(0) =−8 1 − 2 + + 2 + − + 2 − − 2 + 1 − − 2 + 2 2 f27 13 2π 10L 2L 2 1 π 3L L z 37 18π 18L z 9 8 4 2 5 1 + (−7 + 20L)z3 + (27 + 70L)z4 + (85 + 126L)z5 , (22) 3 24 5 8 1 f d(1) =− 165 − 52π 2 − 4 −15 + π 2 L − 18L2 − 8L3 − 72ζ(3) 27 9 48 1 + 2 −12 + π 2 L − 3L2 + 4L3 + 6 −5 + 4π 2 + 6ζ(3) z 3 3 − −7 + 15π 2 + 15 + 2π 2 L − 15L2 + 4L3 + 36ζ(3) z2 4 5 1 − 235 + 48π 2 − 204L + 36L2 z3 + −10076 − 4200π 2 + 19425L − 3150L2 z4 27 432 1 + −3554 − 4200π 2 + 20685L − 3150L2 z5 . (23) 250

0 Note that f27 in Eq. (21) is an expanded version in z of the integral expression for f27 in Ref. [11]. We further =−1 =−1 = 1 note that f28 3 f27, f17 6 f27, f18 18 f27; the same relations also hold for the respective f ij (see, for instance, [38]). Finally, we turn to the shift f 22 related to the (O2,O2)-interference. To derive this quantity, one has to perform → + brems = | | the shift mc mc δmc only in one of the two interfering one-loop amplitudes M2 sγg O2 b .Tothis brems brems end, one writes integral representations for both, M2 and dM2 /dmc. f 22 is then represented as a ﬁve- dimensional integral (4 Feynman parameters and one phase space parameter), which can be solved by double Mellin–Barnes techniques (see, for instance, [39]). Omitting the detail of this calculation, the terms induced by → (0) + renormalizing mc in the (O2,O2) bremsstrahlung terms are implemented in Eq. (12) by replacing f22 f22 H.M. Asatrian et al. / Physics Letters B 619 (2005) 322–332 329

f 22, where = mc δmc d(0) + d(1) f 22 2 f22 f22 . (24) mb mb Explicitly, we get f (0) = 0.04938272 + 16.64197 + 1.887290L − 0.4444444L2 − 0.09876543L3 z 22 + 57.92026 + 47.67037L + 1.185185L2 + 3.134737L3 + 0.05925926L5 z2 + −93.12628 + 32.36078L − 12.95977L2 + 1.777778L3 − 0.2962963L4 z3 + 11.92082 − 11.21491L + 2.074074L2 − 0.5925926L3 z4 + 0.6482797 − 4.160089L + 0.1810700L2 − 0.3292181L3 z5 + −1.125313 − 4.320604L − 0.2444444L2 − 0.3456790L3 z6, (25) f d(0) = 18.52926 + 0.9984013L − 0.7407407L2 − 0.09876543L3 22 + 163.5109 + 97.71112L + 11.77458L2 + 6.269473L3 + 0.2962963L4 + 0.1185185L5 z + −247.0180 + 71.16280L − 33.54596L2 + 4.148148L3 − 0.8888889L4 z2 + 36.46839 − 40.71149L + 6.518519L2 − 2.370370L3 z3 + −0.9186906 − 20.43831L − 0.08230453L2 − 1.646091L3 z4 + −11.07248 − 26.41251L − 2.503704L2 − 2.074074L3 z5, (26)

f d(1) = 41.24600 − 7.794263L − 0.7525535L2 + 0.3950617L3 + 0.07407407L4 22 + 234.4505 + 44.95451L − 64.68047L2 − 0.5200208L3 − 4.498135L4 − 0.05925926L5 − 0.08559671L6 z + −368.8104 + 245.2526L − 95.81857L2 + 28.84099L3 − 3.851852L4 + 0.5728395L5 z2 + −3.708986 − 93.16023L + 34.50854L2 − 7.670782L3 + 1.283951L4 z3 + −45.10910 − 54.73258L + 13.39517L2 − 4.073160L3 + 0.8779150L4 z4 + −92.60340 − 74.33726L + 10.32328L2 − 4.786008L3 + 1.094650L4 z5. (27) We decided to give the expansion coefﬁcients in these equations in numerical form, because the exact results are 0 somewhat lengthy. We note that f22 in Eq. (25) is an expanded version in z of the integral expression for f22 in = 1 =−1 Ref. [11]. We further note that f11 36 f22 and f12 3 f22; the same relations also hold for the respective f ij . These analytical results are deﬁned parts of the complete NNLL contribution which can be used within a future NNLL calculation.

4. Numerical results

In the following analysis we show that the NNLL terms, induced through the renormalization of mc, drastically reduce the error related to the definition of the charm quark mass in BR(b → Xsγ). To illustrate this feature as clearly as possible, we take the fixed values shown in Table 1 for the input parameters. In particular, we use the fixed ratio mc,pole/mb,pole = 0.29. Furthermore, we always leave the semileptonic decay width, which enters the branching ratio for b → Xsγ through Eq. (13), expressed in terms of mc,pole as given in Eq. (14). In this way the mc definition dependence of the BR(b → Xsγ)only comes from the numerator in Eq. (13). For our studies, we neglect 330 H.M. Asatrian et al. / Physics Letters B 619 (2005) 322–332

Table 1 Input parameters used in the numerical analysis mb = 4.8GeV mc,pole/mb = 0.29 mZ = 91.187 GeV = = | ∗ |2 = = αs (mZ) 0.119 αem 1/137.036 VtsVtb/Vcb 0.95 BRsl 10.49%

Table 2 m¯ c(µc)/mb for µc = 1.25, 2.5, 5GeVusingmc,pole/mb = 0.29 as input m¯ c(1.25)/mb = 0.257 m¯ c(2.5)/mb = 0.214 m¯ c(5.0)/mb = 0.187

Fig. 2. BR(b → Xs γ)for three values of µb. For each value of µb the left string shows the NLL results for mc,pole (solid dot) and for m¯ c(µc) with µc = 1.25, 2.5, 5.0 GeV (open symbols). The right strings show the corresponding NLL results supplemented by the δmc mass insertions 2 = 2 and the Σ1(p mc) insertions (see text for more details). electroweak corrections and nonperturbative effects. As already mentioned, in the bremsstrahlung contribution we use δ = 0.9 for the lower cut in the photon energy (see Eq. (10)). Starting from mc,pole = 0.29 · 4.8GeV= 1.392 GeV, we ﬁrst calculate m¯ c(mc,pole), using the one-loop expression αs(mc,pole) m¯ (m ) = m 1 − C . (28) c c,pole c,pole π F

To get m¯ c(µc) for an arbitrary scale (typically between 1.25 and 5 GeV), we use two-loop running (with 5 ﬂavours) according to (0) α (µ ) γm /2β0 γ (1) β γ (0) α (µ ) − α (µ ) m¯ (µ ) =¯m (µ ) s c + m − 1 m s c s 0 c c c 0 1 2 (29) αs(µ0) 2β0 2β0 4π with µ0 = mc,pole. Numerically, we get the values shown in Table 2.InFig. 2 our results are given for three different values of µb, where µb represents the usual renormalization scale of the effective ﬁeld theory. We compare the branching ratio for b → Xsγ within the pole and the MS scheme for the charm quark mass. Within each vertical string the solid dot represents the branching ratio using mc,pole, while the open symbols correspond to m¯ c(µc) for µc = 1.25 GeV (triangle), µc = 2.5 GeV (quadrangle) and µc = 5.0 GeV (pentagon), respectively. H.M. Asatrian et al. / Physics Letters B 619 (2005) 322–332 331

For each µb the left string shows the value of the branching ratio at the NLL level, while the right string 2 = 2 shows the corresponding value where in addition δmc mass insertions and Σ1(p mc) insertions were taken into account, as explained in detail in Section 2. Because the combination of these insertions is zero by construction for the pole scheme (see Eq. (8)), the solid dots are at the same place in the left and the right string for a given value of µb. From Fig. 2 we see that the error related to the charm quark mass definition gets significantly reduced when taking into account NNLL terms connected with mass insertions. Taking as an example the results for µb = 5GeV, we find that at the NLL level the branching ratio evaluated for m¯ c(2.5GeV) is 12.6% higher than the one based on mc,pole, in agreement with Ref. [23]. Including the new contributions, these 12.6% get reduced to 5.1%. A remark concerning the remaining NNLL terms is in order: as these terms give contributions to the branching 3 ratio which (up to terms of order αs ) do not depend on charm quark mass definition, the error related to mc in the full NNLL result is expected to stay essentially the same as estimated in the present Letter. However, to obtain a NNLL prediction for the central value of the branching ratio, it is of course necessary to calculate all NNLL terms. Summing up, we have shown that the relatively large error related to the definition of the charm quark mass in the NLL result for BR(b → Xsγ)gets significantly reduced (typically by a factor of 2) at the NNLL level. Taking into account the present experimental accuracy of around 10% and the future prospects of the B factories and also of possible super-B factories [40,41], we conclude that a future NNLL QCD calculation of the b → Xsγ branching ratio will significantly increase the sensitivity of this observable to possible new physics.

Acknowledgements

The team from Yerevan is partially supported by the ANSEF-05-PS-hepth-813-100 and the NFSAT PH 095-02 (CRDF 12050) programs. C.G. is partially supported by the Swiss National Foundation; RTN, BBW-Contract No. 01.0357 and EC-Contract HPRN-CT-2002-00311 (EURIDICE).

References

[1] T. Hurth, Rev. Mod. Phys. 75 (2003) 1159, hep-ph/0212304. [2] S. Bertolini, F. Borzumati, A. Masiero, Phys. Lett. B 192 (1987) 437; S. Bertolini, F. Borzumati, A. Masiero, G. Ridolﬁ, Nucl. Phys. B 353 (1991) 591. [3] G. Degrassi, P. Gambino, G.F. Giudice, JHEP 0012 (2000) 009, hep-ph/0009337. [4] M. Carena, D. Garcia, U. Nierste, C.E.M. Wagner, Phys. Lett. B 499 (2001) 141, hep-ph/0010003. [5] G. D’Ambrosio, G.F. Giudice, G. Isidori, A. Strumia, Nucl. Phys. B 645 (2002) 155, hep-ph/0207036. [6] F. Borzumati, C. Greub, T. Hurth, D. Wyler, Phys. Rev. D 62 (2000) 075005, hep-ph/9911245. [7] T. Besmer, C. Greub, T. Hurth, Nucl. Phys. B 609 (2001) 359, hep-ph/0105292. [8] M. Ciuchini, E. Franco, A. Masiero, L. Silvestrini, Phys. Rev. D 67 (2003) 075016, hep-ph/0212397; M. Ciuchini, E. Franco, A. Masiero, L. Silvestrini, Phys. Rev. D 68 (2003) 079901, Erratum. [9] K. Adel, Y. Yao, Phys. Rev. D 49 (1994) 4945, hep-ph/9308349. [10] C. Greub, T. Hurth, D. Wyler, Phys. Lett. B 380 (1996) 385, hep-ph/9602281; C. Greub, T. Hurth, D. Wyler, Phys. Rev. D 54 (1996) 3350, hep-ph/9603404. [11] K. Chetyrkin, M. Misiak, M. Münz, Phys. Lett. B 400 (1997) 206, hep-ph/9612313. [12] C. Greub, T. Hurth, Phys. Rev. D 56 (1997) 2934, hep-ph/9703349. [13] A.J. Buras, A. Czarnecki, M. Misiak, J. Urban, Nucl. Phys. B 611 (2001) 488, hep-ph/0105160. [14] P. Gambino, M. Gorbahn, U. Haisch, hep-ph/0306079. [15] A.J. Buras, A. Czarnecki, M. Misiak, J. Urban, Nucl. Phys. B 631 (2002) 219, hep-ph/0203135. [16] H.M. Asatrian, H.H. Asatryan, A. Hovhannisyan, Phys. Lett. B 585 (2004) 263. [17] A. Ali, C. Greub, Z. Phys. C 49 (1991) 431. [18] N. Pott, Phys. Rev. D 54 (1996) 938, hep-ph/9512252. 332 H.M. Asatrian et al. / Physics Letters B 619 (2005) 322–332

[19] A. Czarnecki, W.J. Marciano, Phys. Rev. Lett. 81 (1998) 277, hep-ph/9804252. [20] A.L. Kagan, M. Neubert, Eur. Phys. J. C 7 (1999) 5, hep-ph/9805303. [21] K. Baranowski, M. Misiak, Phys. Lett. B 483 (2000) 410, hep-ph/9907427. [22] P. Gambino, U. Haisch, JHEP 0009 (2000) 001, hep-ph/0007259; P. Gambino, U. Haisch, JHEP 0110 (2001) 020, hep-ph/0109058. [23] P. Gambino, M. Misiak, Nucl. Phys. B 611 (2001) 338, hep-ph/0104034. [24] T. Hurth, E. Lunghi, W. Porod, Nucl. Phys. B 704 (2005) 56, hep-ph/0312260. [25] ALEPH Collaboration, R. Barate, et al., Phys. Lett. B 429 (1998) 169. [26] Belle Collaboration, K. Abe, et al., Phys. Lett. B 511 (2001) 151, hep-ex/0103042. [27] CLEO Collaboration, S. Chen, et al., Phys. Rev. Lett. 87 (2001) 251807, hep-ex/0108032. [28] BaBar Collaboration, B. Aubert, et al., hep-ex/0207074. [29] BaBar Collaboration, B. Aubert, et al., hep-ex/0207076. [30] Belle Collaboration, P. Koppenburg, et al., Phys. Rev. Lett. 93 (2004) 061803, hep-ex/0403004. [31] Heavy Flavor Averaging Group (HFAG), J. Alexander, et al., hep-ex/0412073. [32] M. Misiak, Nucl. Phys. B (Proc. Suppl.) 116 (2003) 279. [33] K. Bieri, C. Greub, M. Steinhauser, Phys. Rev. D 67 (2003) 114019, hep-ph/0302051. [34] M. Gorbahn, U. Haisch, Nucl. Phys. B 713 (2005) 291, hep-ph/0411071. [35] M. Gorbahn, U. Haisch, M. Misiak, hep-ph/0504194. [36] C. Bobeth, M. Misiak, J. Urban, Nucl. Phys. B 574 (2000) 291, hep-ph/9910220. [37] H.M. Asatrian, K. Bieri, C. Greub, M. Walker, Phys. Rev. D 69 (2004) 074007, hep-ph/0312063. [38] H.H. Asatryan, H.M. Asatrian, C. Greub, M. Walker, Phys. Rev. D 66 (2002) 034009, hep-ph/0204341. [39] H.H. Asatryan, H.M. Asatrian, C. Greub, M. Walker, Phys. Rev. D 65 (2002) 074004, hep-ph/0109140. [40] SuperKEKB Physics Working Group, A.G. Akeroyd, et al., hep-ex/0406071. [41] J. Hewett, et al., in: SLAC Workshops, Stanford, USA, 2003, hep-ph/0503261. Physics Letters B 619 (2005) 333–339 www.elsevier.com/locate/physletb

Discrete light-cone quantization in pp-wave background

Kunihito Uzawa a, Kentaroh Yoshida b

a Yukawa Institute for Theoretical Physics, Kyoto University, Kyoto 606-8502, Japan b Theory Division, High Energy Accelerator Research Organization (KEK), Tsukuba, Ibaraki 305-0801, Japan Received 3 March 2005; received in revised form 20 April 2005; accepted 5 June 2005 Available online 17 June 2005 Editor: T. Yanagida

Abstract We discuss the discrete light-cone quantization (DLCQ) of a scalar ﬁeld theory on the maximally supersymmetric pp-wave background in ten dimensions. It has been shown that the DLCQ can be carried out in the same way as in the two-dimensional Minkowski spacetime. Then, the vacuum energy is computed by evaluating the vacuum expectation value of the light-cone Hamiltonian. The results are consistent with the effective potential obtained in our previous work [hep-th/0402028].  2005 Elsevier B.V. All rights reserved.

Keywords: Discrete light-cone quantization; Effective potential; pp-wave

1. Introduction

Recently, the discrete light-cone quantization (DLCQ) method, which has been originally developed by [1,2] (for reviews, see [3,4]), has a renewed interest related to the M-theory formulation [5] (for a review related to the DLCQ, see [6]). The M-theory formulation [5] has been extended to the pp-wave background [7–9] by Berenstein– Maldacena–Nastase [10], and so it is also interesting to study the DLCQ method on the pp-wave background. On the other hand, for the type IIB string theory on the pp-wave [11,12], an interesting work [13] has been done related to the DLCQ in the pp-wave background. Scalar ﬁeld theories on pp-wave backgrounds are nice laboratories for studies of the DLCQ method in the pp- wave case. Furthermore these may give interesting cosmological models (for studies of cosmological structure of pp-wave background, see [14–18]). In fact, scalar ﬁeld theories on the maximally supersymmetric pp-wave background in ten dimensions are fairly studied. The propagator in this theory was computed in [19], and then the effective potential was discussed in [20] in the path integral formulation [21,22].

E-mail addresses: [email protected] (K. Uzawa), [email protected] (K. Yoshida).

In this Letter, we discuss the DLCQ of scalar ﬁeld theories in the maximally supersymmetric pp-wave background in ten dimensions. The vacuum energies of them are computed by evaluating the vacuum expectation value (VEV) of the light-cone Hamiltonian. The results completely agree with the effective potential calculated in our previous paper [20].

2. DLCQ and vacuum energy of a scalar ﬁeld in pp-wave

We shall consider a scalar field theory on the maximally supersymmetric pp-wave background: 8 8 + − + ds2 =−2dx dx − µ2 xi 2(dx )2 + dxi 2, (2.1) i=1 i=1 F+1234 = F+5678 = 4µ. (2.2) The constant five-form flux√F is a field strength of Ramond–Ramond four-form. The light-cone coordinates are defined as x± = (x0 ± x9)/ 2. The action of a scalar field theory on this background is given by √ 1 I = d10x −g − gµν∂ φ∂ φ − V(φ) s 2 µ ν 10 1 2 2 1 = d x ∂+φ∂−φ − µ r ∂−φ∂−φ − ∂ φ∂ φ − V(φ) , (2.3) 2 2 i i 2 = 8 i 2 = 2 2 where r i=1(x ) and V(φ)is a potential of a scalar field φ. We will concentrate on the case V(φ) m φ /2 below, in order to discuss the DLCQ method for a free scalar field. The classical equation of motion is 8 2 + 2 = 2 ≡ − 2 − 2 2 2 (pp) m φ 0, (pp) 2∂−∂+ ∂i µ r ∂−. (2.4) i=1 By using the identity xδ(x) = 0, we can easily find the solution of (2.4):

8 + − dk− dk+ 2 i(k+x +k−x ) i φ(x)= δ 2k+k− +|k−|E − m χ{ }(k−,k+)e ψ x . (2.5) 4π n ni ni {ni } i=1 Here we have introduced the following notations:

8 8 1 E ≡ E = µ n + (i = 1,...,8,n = 0,...,∞), n ni i 2 i i=1 i=1

| | 1/4 i − 1 µ|k−|(xi )2 i µ k− 1 ψ x ≡ N e 2 H µ|k−| x ,N≡ √ . ni ni ni ni n π 2 i ni! The orthonormal and complete conditions: ∞ ∞ dxψm(x)ψn(x) = δmn, ψn(x)ψn(y) = δ(x − y), (2.6) −∞ n=0 are available to check the normalization of the Poisson bracket. K. Uzawa, K. Yoshida / Physics Letters B 619 (2005) 333–339 335

After some algebra, the classical solution (2.5) can be rewritten as ∞ 8 1 dk− ˆ + − ˆ + − = −i(k+x +k−x ) + ∗ i(k+x +k−x ) i φ(x) a{ni }(k−)e a{n }(k−)e ψni x , (2.7) − i 4π { } k = ni 0 i 1

ˆ 2 where we have defined the on-shell energy as k+ ≡ (|k−|En + m )/(2k−) and redefined the coefficients as follows: ˆ ∗ ˆ a{ }(k−) ≡ χ{ }(k−, k+), a (k−) ≡ χ{ }(−k−, −k+). (2.8) ni ni {ni } ni The classical Poisson bracket at the simultaneous light-cone time x+ = y+ is given by 1 − − φ(x),π(y) = δ x − y δ(8) xi − yi , (2.9) P 2 where the π(y) is the light-cone canonical momentum: ∂L π ≡ = ∂−φ. (2.10) ∂(∂+φ) Here the factor 1/2 in the r.h.s. of (2.9) is determined by the Schwinger’s action principle and it depends on the convention of the light-cone coordinates1 (for the detail, see [4]). In terms of the coefficients a and a∗, the classical Poisson bracket (2.9) is described as

8 ∗ =− − a{ni }(k−), a{ }(k−) 4πik−δ(k− k−) δn n . (2.11) ni P i i i=1 It is possible to obtain the usual commutation relation after an appropriate rescaling as we will see later. In the next we will consider to quantize the theory in the canonical formulation.

2.1. DLCQ method in pp-wave background

Now let us consider the canonical quantization of classical scalar field theories by replacing the classical Poisson bracket of scalar field φ(x) and the light-cone canonical momentum π(y) with the commutator at the simultaneous light-cone time x+ = y+ (for detail, see [4]). However, we should be careful for the constraint conditions before the replacement of the commutator. In order to carry out the canonical quantization in the light-cone frame, we follow the DLCQ procedure as a standard manner. In this method, the physical system is enclosed in a finite volume with − a periodic boundary conditions in x and the longitudinal momentum k− is discretized. At first, we compactify the light-cone coordinate x− as −L x− L, and impose periodic boundary condition for the field; + − + − φ x ,x =−L,xi = φ x ,x = L,xi . (2.12)

The corresponding longitudinal momentum is thus discretized as k− = πq/L (q ∈ Z) and then the Fourier expansion of the ﬁeld becomes 8 + 1 + −i πq x− ∗ + i πq x− i φ(x)= a0(x ) + √ a{n }q (x )e L + a{ } (x )e L ψn ,q x . (2.13) 4πq i ni q i {ni } q>0 i=1

± − 1 If we take the light-cone coordinates as x = x0 ± xD 1, then the factor 1/2 is removed. 336 K. Uzawa, K. Yoshida / Physics Letters B 619 (2005) 333–339

Here after the descritization as ∞ 1 dk− 1 1 −→ ,k−δ(k− − k−) −→ qδq,q , 4π k− 4π q 0 q>0 the annihilation and creation operators have been redeﬁned by rescaling as

−→ a{ni }q 4πqa{ni }q , + ∗ and the plane-wave with x is included in the deﬁnition of a and a . We also have written the zero mode a0 separately. Though one may think that the zero-mode naively vanish because of the normalization factors of the Hermite polynomials, the k− = 0 implies a plane-wave expansion rather than a harmonic oscillator one. Hence we still need to treat carefully the zero-mode part. Substituting the expansion (2.13) into the Lagrangian (2.3),we obtain the following expression (discarding a total time derivative) ∗ L πq 2 ∗ 2 2 L[a ,a{ } ]= −ia{ } a˙ − E + m a{ } a − m La 0 ni q ni q {ni }q n ni q {ni }q 0 { } 2πq L ni q>0 ∗ =−i a{ } a˙ − H, (2.14) ni q {ni }q {ni } q>0 where the Hamiltonian H is given by 2 2 L πq 2 ∗ H = m La + E + m a{ } a{ } , (2.15) 0 2πq L n ni q ni q {ni } q>0 · + ˙ ≡ + and the symbol “ ” implies the derivative with respect to “x ”, such as a{ni }q da{ni }q /dx . From the above, the light-cone Lagrangian is linear with respect to the velocity, i.e., a ﬁrst order system. In this case we can determine the Poisson bracket by comparing the Euler–Lagrange equation with the canonical equation as discussed in [4]. On the one hand, the Euler–Lagrange equations are

L πq 2 −ia˙{ } + E + m a{ } = 0, (2.16) ni q 2πq L n ni q 2 2m La0 = 0. (2.17)

The second equation (2.17) is non-dynamical and gives a constraint implying the absence of zero mode, i.e., a0 = 0. On the other hand, using the Hamiltonian (2.15), the canonical equations are

L πl 2 ∗ ˙{ } ={ { } } = + { } a ni q a ni q ,H P En m a{n }l a ni q ,a{n }l . (2.18) 2πl L i i P { } l>0 ni These are identical with the Euler–Lagrange equations (2.16) if we identify the canonical bracket as follows: 8 ∗ =− · a{ni }q ,a{ } iδql δn ,n . (2.19) ni l P i i i=1 The constraint condition (2.17) can be also derived by differentiating the Hamiltonian as

∂H 2 = 2m La0 = 0. (2.20) ∂a0 The quantization of a free scalar field is now performed as usual by replacing the classical Poisson bracket ˆ ˆ with the commutator as [A, B]=i{A,B}P. As the result, the commutation relation of the creation and annihilation K. Uzawa, K. Yoshida / Physics Letters B 619 (2005) 333–339 337 operators is given by 8 † ˆ{ } ˆ = · a ni q , a{ } δql δni ,n . (2.21) ni l i i=1 Finally, the Fock space expansion of the scalar field φ becomes 8 + − i 1 + −i πq x− † + i πq x− i φ x ,x ,x = √ aˆ{n }q (x )e L +â{ } (x )e L ψn ,q x . (2.22) 4πq i ni q i {ni } q>0 i=1 As one can see from the expansion (2.22), the Fock space contains only particles with positive longitudinal momenta. Operators with negative longitudinal momenta are annihilation operators. If longitudinal momenta are conserved, positive and discrete, then states with k− = qπ/L can have at most q particles among them. Thus, in the sector with q units of momentum, the theory reduces to non-relativistic quantum mechanics including eight harmonic oscillators, with a fixed number of particles. Finally, it should be remarked that the structure of Fock space in the case of pp-wave background is quite similar to that in two-dimensional Minkowski spacetime. Namely, it is the product of the Fock space of two-dimensional Minkowski and Hermite polynomials. From this result, one can guess that this structure may be extended to other pp-wave cases. In the standard pp-wave cases, it is only the difference that the Hermite polynomials are modified in terms of the oscillation numbers or the number of harmonic oscillators. Remarkably speaking, this structure may be expected even for the pp-waves which lead to harmonic oscillators with negative mass terms. We encounter this type of background when Penrose limits are taken for black hole geometries. In these pp-waves, hyperbolic cylinder functions, which imply an instability of the system, would appear as well as Hermite polynomials. However, the hyperbolic cylinder functions do not mean instabilities as already discussed in [15]. On the other hand, our result is also compatible with this fact because the Fock vacuum structure is determined by the light-cone directions only. Namely, it is the two-dimensional Minkowski one. In the next subsection we will evaluate the vacuum energy of a scalar field by using this operator representation.

2.2. Computation of the vacuum energy

Next we shall compute the vacuum energy of a free scalar field in ten-dimensional pp-wave background. In the previous subsection we have discussed the DLCQ of a scalar field and derived the operator expression of the field. We shall compute the vacuum energy density by using this quantized scalar field. For the Fock vacuum |0 of the light-cone Hamiltonian defined by a(k−)|0=0, the vacuum energy density E is given by2 1 E = 0|P+|0,V9: 9-dim. volume, (2.23) V9 where the light-cone Hamiltonian is ∞ ∞ − 8 P+ = dx d x π∂−φ − L[φ,∂µφ] . (2.24) −∞ −∞ If we use the commutation relation (2.21) and the expression (2.22), the light-cone Hamiltonian is rewritten as

L πq 2 † P+ = En + m 2aˆ{ } aˆ{n }q + 1 . (2.25) 4πq L ni q i {ni } q>0

2 For computations of vacuum energy of light-cone Hamiltonian in some other models such as Gross–Neveu, SU(N) Thirring, O(N) vector models with large N limit, see [23]. 338 K. Uzawa, K. Yoshida / Physics Letters B 619 (2005) 333–339

We can evaluate the vacuum energy density E as ∞ + 1 L πq 2 b 1 2 1 E = n+7C7 |µ|(n + 4) + m = m , (2.26) V9 4πq L 8πV8 q n=0 q>0 q>0 where we have used the V9 = 2L · V8. The combination factor n+7C7 denotes the degeneracy of the sum of ni , and the coefﬁcient b + 1 is given by

1 2497 b + 1 = ζ (−7) − 14ζ (−5) + 49ζ (−3) − 36ζ (−1) = . (2.27) 7! R R R R 3628800

Here ζR(s) is the Riemann’s zeta function. Since the coefﬁcient b + 1 is computed by the zeta function regularization, there may be the ambiguity for the method of mode sum of n. When the continuum limit is considered by taking L →∞, we can recover the expression: ∞ + b 1 2 1 E = m dk− . (2.28) 8πV8 k− 0

This can be evaluated by introducing the cut-off for the longitudinal momentum as m2/Λ |k−| Λ. The resulting vacuum energy density is

b + 1 m2 E = m2 . ln 2 (2.29) 8πV8 Λ Note that the contribution of the factor b + 1 does not appear in the tree level calculation [24,25]. This result agrees with the 1-loop effective potential with V(φ)= m2φ2/2 that is obtained by using the path integral method [20]. On the other hand, we may consider the limit L →∞before the concrete computation, and introduce the cut-off for the longitudinal momentum as m2/Λ |k−| Λ.Thek− = 0 seems to have a subtlety, but this mode decouple from the theory as already discussed and so this point may have no problem. After taking the decompactiﬁcation limit, we use the expansion of scalar ﬁeld (2.7) with the creation and annihilation operators whose commutation relations are

8 † ˆ{ } − ˆ = − − − a ni (k ), a{ }(k−) 4πk δ(k k−) δni n . (2.30) ni i i=1

Thus, the light-cone Hamiltonian P+ is given by using the expressions (2.7) and (2.30)

Λ 8 1 dk− 1 2 P+ = δ(0) µk− ni + + m . (2.31) 2 k− 2 {n } i=1 i m2/Λ

By using the relation δ(0)/V9 = (2π)V1/V9 = (2π)/V8, we can rederive the vacuum energy density (2.29) again. Notably, the effective potential is independent of the parameter µ. We can see this fact by noting that the effect of µ (i.e., En) can be absorbed by shifting the light-cone momentum k+. But physically, as we discussed in [20], the vacuum energies produced by the quantum fluctuations flow from transverse space to the k+-direction due to the flux equipped with the pp-wave geometry. As the result of the energy flow, the effect of the pp-wave background would be realized only as the numerical coefficients, and thus the vacuum energy may not explicitly depend on the parameter µ. K. Uzawa, K. Yoshida / Physics Letters B 619 (2005) 333–339 339

3. A conclusion and discussions

We have discussed the DLCQ of a scalar field theory in the maximally supersymmetric pp-wave background in ten dimensions. The DLCQ method in the pp-wave background has been shown to work well in the same way as in two-dimensional case, even if we consider higher-dimensional theory. It is because transverse momenta are discretized and so the treatment in the pp-wave case is quite similar to the two-dimensional Minkowski case. We also calculated the vacuum energy of a free scalar field in the pp-wave and it has been shown that the resulting vacuum energy surely agrees with the effective potential obtained in our previous paper. The DLCQ method may play an important role in the M-theory formulation [5], where the discrete light-cone quantized M-theory can be described by a matrix model. It would be an important key ingredient in studies of the pp-wave matrix model [10]. We believe that our study of DLCQ in a scalar field theory on the pp-wave background should be a clue to shed light on some features of DLCQ method in the pp-wave case.

Acknowledgements

The work of K.U. is partially supported by Yukawa fellowship. The work of K.Y. is supported in part by JSPS Research Fellowships for Young Scientists.

References

[1] T. Maskawa, K. Yamawaki, Prog. Theor. Phys. 56 (1976) 270. [2] H.C. Pauli, S.J. Brodsky, Phys. Rev. D 32 (1985) 1993; H.C. Pauli, S.J. Brodsky, Phys. Rev. D 32 (1985) 2001. [3] J.R. Hiller, Nucl. Phys. B (Proc. Suppl.) 90 (2000) 170, hep-ph/0007309. [4] T. Heinzl, in: Lecture Notes in Physics, vol. 572, Springer, Berlin, 2001, p. 55, hep-th/0008096, and references therein. [5] T. Banks, W. Fischler, S.H. Shenker, L. Susskind, Phys. Rev. D 55 (1997) 5112, hep-th/9610043. [6] T. Banks, hep-th/9911068. [7] M. Blau, J. Figueroa-O’Farrill, C. Hull, G. Papadopoulos, JHEP 0201 (2002) 047, hep-th/0110242. [8] M. Blau, J. Figueroa-O’Farrill, C. Hull, G. Papadopoulos, Class. Quantum Grav. 19 (2002) L87, hep-th/0201081. [9] M. Blau, J. Figueroa-O’Farrill, G. Papadopoulos, Class. Quantum Grav. 19 (2002) 4753, hep-th/0202111. [10] D. Berenstein, J. Maldacena, H. Nastase, JHEP 0204 (2002) 013, hep-th/0202021. [11] R.R. Metsaev, Nucl. Phys. B 625 (2002) 70, hep-th/0112044. [12] R.R. Metsaev, A.A. Tseytlin, Phys. Rev. D 65 (2002) 126004, hep-th/0202109. [13] M.M. Sheikh-Jabbari, JHEP 0409 (2004) 017, hep-th/0406214. [14] D. Marolf, S.F. Ross, Class. Quantum Grav. 19 (2002) 6289, hep-th/0208197; V.E. Hubeny, M. Rangamani, JHEP 0211 (2002) 021, hep-th/0210234. [15] D. Brecher, J.P. Gregory, P.M. Safﬁn, Phys. Rev. D 67 (2003) 045014, hep-th/0210308; D. Marolf, L.A. Pando Zayas, JHEP 0301 (2003) 076, hep-th/0210309. [16] G. Papadopoulos, J.G. Russo, A.A. Tseytlin, Class. Quantum Grav. 20 (2003) 969, hep-th/0211289. [17] M. Blau, M. O’Loughlin, Nucl. Phys. B 654 (2003) 135, hep-th/0212135; M. Blau, M. O’Loughlin, G. Papadopoulos, A.A. Tseytlin, Nucl. Phys. B 673 (2003) 57, hep-th/0304198; M. Blau, P. Meessen, M. O’Loughlin, JHEP 0309 (2003) 072, hep-th/0306161. [18] M. Sakaguchi, K. Yoshida, JHEP 0311 (2003) 030, hep-th/0309025. [19] S.D. Mathur, A. Saxena, Y.K. Srivastava, Nucl. Phys. B 640 (2002) 367, hep-th/0205136. [20] K. Uzawa, K. Yoshida, Nucl. Phys. B 683 (2004) 122, hep-th/0402028. [21] T.M. Yan, Phys. Rev. D 7 (1973) 1780. [22] T. Heinzl, hep-th/0212202. [23] S. Kojima, N. Sakai, T. Sakai, Prog. Theor. Phys. 95 (1996) 621, hep-th/9509143. [24] D.S. Bak, M.M. Sheikh-Jabbari, JHEP 0302 (2003) 019, hep-th/0211073. [25] D. Sadri, M.M. Sheikh-Jabbari, Rev. Mod. Phys. 76 (2004) 853, hep-th/0310119. Physics Letters B 619 (2005) 340–346 www.elsevier.com/locate/physletb

Quasinormal modes in Schwarzschild black holes due to arbitrary spin ﬁelds

Fu-Wen Shu a,d, You-Gen Shen a,b,c

a Shanghai Astronomical Observatory, Chinese Academy of Sciences, Shanghai 200030, People’s Republic of China b National Astronomical Observatories, Chinese Academy of Sciences, Beijing 100012, People’s Republic of China c Institute of Theoretical Physics, Chinese Academy of Sciences, Beijing 100080, People’s Republic of China d Graduate School of Chinese Academy of Sciences, Beijing 100039, People’s Republic of China Received 22 April 2005; received in revised form 27 May 2005; accepted 31 May 2005 Available online 9 June 2005 Editor: M. Cveticˇ

Abstract The Newman–Penrose formalism is used to deal with the massless scalar, neutrino, electromagnetic, gravitino and gravitational quasinormal modes (QNMs) in Schwarzschild black holes in a united form. The quasinormal mode frequencies evaluated by using the 3rd-order WKB potential approximation show that the boson perturbations and the fermion perturbations behave in a contrary way for the variation of the oscillation frequencies with spin, while this is no longer true for the damping’s, which variate with s in a same way both for boson and fermion perturbations.  2005 Elsevier B.V. All rights reserved.

PACS: 04.70.Dy; 04.70.Bw; 97.60.Lf

Keywords: Schwarzschild black hole; QNMs; Arbitrary spin ﬁelds; WKB approximation

Ever since Chandrasekhar [1] and Vishveshwara such as its mass, charge, and angular momentum, and [2] discovered the quasinormal modes of black holes, thus, help uniquely identify a black hole. much effort has been devoted to investigating the QNMs are described as the pure tones of black QNMs of various black hole cases [3–7]. By obtaining hole. They are deﬁned as solutions of the perturbation quasinormal mode (QN) frequencies, we cannot only equations belonging to certain complex characteristic test the stability of the spacetime against small pertur- frequencies which satisfy the boundary conditions ap- bations, but also probe the parameters of black hole, propriate for purely ingoing waves at the event horizon and purely outgoing waves at inﬁnity [8]. QNMs are excited by the external perturbations (may be induced, E-mail address: [email protected] (Y.-G. Shen). for example, by the falling matter). They appear as

0370-2693/$ – see front matter  2005 Elsevier B.V. All rights reserved. doi:10.1016/j.physletb.2005.05.077 F.-W. Shu, Y.-G. Shen / Physics Letters B 619 (2005) 340–346 341 damped oscillations described by the complex char- where M is the mass of the black hole. acteristic frequencies which are entirely fixed by the The Teukolsky’s master equations [15,16] for mass- properties of background geometry, and independent less arbitrary spin fields (s = 0, 1/2, 1, 3/2, 2) in of the initial perturbation. These frequencies can be Newman–Penrose formalism can be written as [17] detected by observing the gravitational wave signal ∗ ∗ D − (2s − 1) + − 2sρ − ρ (∆¯ − 2sγ + µ) [9]: this makes QNMs be of particular relevance in ∗ ∗ gravitational wave astronomy. − δ − (2s − 1)β − α − 2sτ + π (δ¯ − 2sα + π) QNMs were firstly used to study the stability of a − (s − 1)(2s − 1)Ψ2 Φs = 0, (3) black hole. Detweiler and Leaver found the relations between the parameters of a black hole and QNMs. and Latest studies show that QNMs play an important role ∗ ∗ ∆¯ + (2s − 1)γ − γ + 2sµ + µ (D + 2s − ρ) in the quantum theory of gravity. This is related to ∗ ∗ the quantization of black hole area [10]. For example, − δ¯ + (2s − 1)α + β + 2sπ − τ (δ + 2sβ − τ) there exit some possible relations between the classical − (s − 1)(2s − 1)Ψ2 Φ−s = 0. (4) vibrations of black holes and various quantum aspects, such as the relation between the real part of the qua- Assume that the wave-functions in Eqs. (3) and (4) sinormal mode frequencies and the Barbero–Immirzi have a t- and a ϕ-dependence specified in the form i(ωt+mϕ) parameter, a factor introduced by hand in order that e , i.e., loop quantum gravity reproduces correctly entropy of i(ωt+mϕ) Φ = R+ (r)A+ (θ)e , the black hole [11–13]. All these works deal with as- s s s −i(ωt+mϕ) ymptotically flat spacetimes. The recently proposed Φ−s = R−s(r)A−s(θ)e (5) AdS/CFT correspondence makes the QNMs more ap- and define pealing, due to its argument that string theory in anti- s 2s de Sitter (AdS) space is equivalent to conformal field P+s = ∆ R+s,P−s = r R−s (6) theory (CFT) in one less dimension [14]. In addition (where R± and A± are, respectively, functions of r to this context, many studies also have been done on s s and θ only, and ∆ = r2 − 2M) one can decouple equa- QNMs of various spin-fields [3,7,8]. Chandrasekhar = tions (3) and (4) as two pairs of equations, [8] investigated the QNMs of fields with spin s 1/2, 1, 2 in Kerr black hole, and s = 2 in Schwarz- † − − = ∆D1−sD0 2(2s 1)iωr P+s λP+s, (7) schild and Reissner–Nordström black hole. Cardoso † =− and Lemos [3] studied the QNMs of Schwarzschild– L1−sLsA+s λA+s, (8) anti-de Sitter for fields with spin s = 1, 2. Cho [7] and calculated the massive Dirac quasinormal mode fre- † + − = quencies of Schwarzschild black hole. However, prob- ∆D1−sD0 2(2s 1)iωr P−s λP−s, (9) lem on how to deal with QNMs due to arbitrary spin † =− L1−sLs A−s λA−s, (10) field (s = 0, 1/2, 1, 3/2, 2) in a united form has never been discussed in these previous works. The main pur- where λ is a separation constant. The reason we have pose of this article is to give a possible way to deal not distinguished the separation constants which de- with QNMs in Schwarzschild black hole of arbitrary rived from Eqs. (3) and (4) is that λ is a parameter that spin field in a united form. is to be determined by the fact that A+s should be reg- We start with the line element in standard coordi- ular at θ = 0 and θ = π, and thus the operator acting nates for a Schwarzschild space–time on A−s on the left-hand side of Eq. (10) is the same as the one on A+s in Eq. (8) if we replace θ by π − θ. − ds2 =−e2U dt2 + e 2U dr2 + r2 dθ2 + sin2 θdϕ2 , In Schwarzschild black hole, the separation con- (1) stant can be determined analytically [16,18] for boson with λ = l +|s| l −|s|+1 , 2M e2U = 1 − , (2) l =|s|, |s|+1,..., (11) r 342 F.-W. Shu, Y.-G. Shen / Physics Letters B 619 (2005) 340–346 for fermion where ξ and W are certain functions of r∗ to be determined. One can then deduce the following equa- = +| | −| |+ λ j s j s 1 , tions [8] =| | | |+ = ±| | j s , s 1,..., and j l s , (12) dT χ = ξV + , (20) where l and j represent angular quantum number and dr∗ total quantum number, respectively. Since P+s and 4s 4s d r = r − + P− satisfy complex-conjugate equations (7) and (9), χ (QT 2iωχ) β, (21) s dr∗ ∆s ∆s it will suffice to consider Eq. (7) only. dT ∆s ∆s By introducing a tortoise coordinate transformation χ χ − + βT = K, (22) = r2 † dr∗ r4s r4s dr∗ ∆ dr, one can rewrite the operators D0 and D0 ∆s dβ as χV − QξV = , 4s (23) 2 2 r dr∗ r † r D = Λ+, and D = Λ−, (13) 0 ∆ 0 ∆ where K is a constant, and χ, T , and β are certain functions of r∗. d ± = ± where we have defined Λ dr∗ iω. The following work is to look for solutions of 1−2s With the definition Y = r P+s ,Eq.(7) can be Eqs. (20)–(23). These equations provide four equa- written as tions for five functions ξ, β, χ, T , and V . As a result, 2s+3 4s there is considerable difficulty in seeking useful solu- r 2 d r Λ Y + ln Λ−Y tions of these equations. An obvious fact is that χ and ∆ dr∗ ∆s V are independent of ω (i.e., they do not contain any d 1 d − − + ∆s r2s 1 − λr2s 1 Y = 0, term linear in iω). Under these considerations, we can, dr ∆s−1 dr without loss of generality, suppose that T , β, K are of (14) the forms [8] 2 where Λ2 = d + ω2. On further simplification, d(r∗)2 T = T1(r∗) + 2iωf(s), β = β1(r∗) + 2iωβ2, Eq. (14) can be brought to the form K = κ1 + 2iωκ2, (24) 2 Λ Y + PΛ−Y − QY = 0, (15) where β2, κ1, κ2 are constants and f(s)is function of where s. In this article, we take the choice d r4s 1 P = ln , (16) f(s)= s(2s − 1) 6s2 − 23s + 23 for dr∗ ∆s 6 1 3 and s = 0, , 1, , 2. (25) 2 2 ∆ 2∆ ∆ Q = λ − ( s − )(s − ) − . Making use of the fact that, for a equation contains real 4 2 1 1 2 (17) r r r and imaginary parts, the real parts and imaginary parts Taking the purpose of this article into account, we in two sides of the equation are equal, respectively, we should seek to transform Eq. (15) to a one-dimensional can separate each of Eqs. (21), (22) into two equations wave-equation of the form by substituting Eq. (24), i.e., 2 = ∆s Λ Z VZ, (18) χ = fQ+ β , (26) r4s 2 where V represents potential. d r4s r4s The transformation theory introduced in Ref. [8] is = + s χ s QT1 β1, (27) applicable to solve this problem. We assume that Y is dr∗ ∆ ∆ related to Z in the manner and = + Y ξΛ+Λ+Z WΛ+Z, (19) β1f + β2T1 = κ2, (28) F.-W. Shu, Y.-G. Shen / Physics Letters B 619 (2005) 340–346 343

s s 2 − dT + ∆ = ∆ obtain a simpler form of V (±) by defining a new func- χ χ s β1T1 s κ, (29) dr∗ r4 r4 tion F˜ 2 where κ = κ1 + 4ω fβ2. ˜ (±) =± dF + 2 ˜ 2 + ˜ Substituting Eqs. (26) and (28) into Eq. (27),we V κ2 κ2 F κF. (34) dr∗ obtain The expression of this new function is 2 f F,r∗ − κ2 T = . | − | 1 − (30) ∆ s 1 fF β2 F˜ = . (35) r4|s−1|[λ + 2(2s − 1)(s − 1) M ] = r4s Q r Here we have defined F ∆s , and ‘,r∗’ denotes the (+) (−) differential with respect to r∗.Eq.(29) can then be Refs. [8,19] have shown that potentials V and V written as related in the way Eq. (34) shows are equivalent and hence possess the same spectra of quasinormal mode ∆s (f F + β )F + (f F + β )2 − f 2 2 ,r∗,r∗ frequencies. We shall therefore concentrate on V ( ) 4s 2 − r fF β2 only in evaluating the quasinormal mode frequen- (f 4F 2 − κ2)F cies. + ,r∗ 2 = κ. 2 (31) So we can simplify the radial Eq. (7) to a one- (f F − β2) dimensional wave-equation of the form It is obvious that Eq. (31) is a condition on F if solutions of the chosen form are to exist, and hence the d2Z + ω2Z = VZ, 2 (36) work to seek available β2, κ, and κ2 whose values sat- dr∗ isfy Eq. (31) is a key step to obtain the function of 2 with smooth real potentials, independent of ω, i.e., potential V . Since κ2 occurs as κ2 in Eq. (31),two + − choices which associated with κ2 and κ2 are pos- ∆s sible to satisfy the equation. Further study finds, an V = F r4s available choice is the following 2 2 (f F − β2)f F,r∗,r∗ − f F + κ2F,r∗ − ,r∗ , (37) 1 − 2 β =− (2s − 3)(s − 2)(4s − 1)λ, (f F β2) 2 3 where F is a known function of r∗. Note that we have 1 (+) κ = (2s − 1)(s − 1)(2s − 3)(5s − 8)λ(λ + s), written V as V because we shall not work with 6 (−) V , which will give the same quasinormal mode fre- 8 1 quencies. κ = s(s − 1) (2 − s)λ s − + λ 2 3 2 Fig. 1 demonstrates the variation of the effective potential V with spin s. From this we can see that + (2s − 1)(2s − 3)M . (32) peak values of the effective potential V decrease with s for boson perturbations, while they increase with s The solution for V can then be obtained by substitut- for fermion perturbations. This phenomena is closely ing the expressions of β2, κ, and κ2 into Eq. (23), i.e., related to the value of the separation constant λ. Analytic expressions for the quasinormal mode fre- s − − 2 2 ± ∆ (f F β2)f F,r∗,r∗ f F quencies are usually very difficult to obtain. One hence V ( ) = F − ,r∗ 4s 2 r (f F − β2) should appeal to approximation schemes to evaluate these frequencies. Many methods are available for our κ2F,r∗ ∓ , (33) purpose. One often used is WKB approximation, a nu- (f F − β )2 2 merical method first proposed by Mashhoon [20], de- where we have distinguished the transformations asso- vised by Schutz and Will [21], and was subsequently ciated with +κ2 and −κ2 by superscripts (±). extended to higher orders in [22,23]. An obvious char- It is obvious that V (+) equals to V (−) for s = 0, 1 acter of this scheme is that it is very accurate for low- = 1 3 from Eq. (33). For the case of s 2 , 2 , 2, one can lying modes (n

Fig. 1. Variation of the effective potential V with spin s.

Table 1 Quasinormal mode frequencies for boson perturbations. ωi (i = 0, 1, 2) represent scalar, electromagnetic and gravitational perturbations, respectively. ωCD represent Chandrasekhar’s results for gravitational perturbations lnω0 ω1 ω2 ωCD 000.1046+0.1152i 100.2911+0.0980i 0.2459+0.0931i 200.4832+0.0968i 0.4571+0.0951i 0.3730+0.0891i 0.3737+0.0890i 10.4632+0.2958i 0.4358+0.2910i 0.3452+0.2746i 0.3484+0.2747i 300.6752+0.0965i 0.6567+0.0956i 0.5993+0.0927i 0.5994+0.0927i 10.6604+0.2923i 0.6415+0.2898i 0.5824+0.2814i 0.5820+0.2812i 20.6348+0.4941i 0.6151+0.4901i 0.5532+0.4767i 0.4263+0.3727i 400.8673+0.0964i 0.8530+0.0959i 0.8091+0.0942i 0.8092+0.0941i 10.8557+0.2909i 0.8411+0.2893i 0.7965+0.2844i 0.7965+0.2844i 20.8345+0.4895i 0.8196+0.4870i 0.7736+0.4790i 0.5061+0.4232i 30.8064+0.6926i 0.7909+0.6892i 0.7433+0.6783i

Table 2 Quasinormal mode frequencies for fermion perturbations. ωi (i = 1/2, 3/2) represent Dirac, Rarita–Schwinger perturbations, respectively. ωCho represent Cho’s results for Dirac perturbations. Notice that κ = l + 1forj = l + 1/2, according to Cho’s deﬁnition lκnω1/2 ωCho ω3/2 1200.3786+0.0965i 0.379 + 0.097i 2300.5737+0.0963i 0.574 + 0.096i 0.7346+0.0949i 10.5562+0.2930i 0.556 + 0.293i 0.7206+0.2870i 3400.7672+0.0963i 0.767 + 0.096i 0.9343+0.0954i 10.7540+0.2910i 0.754 + 0.291i 0.9233+0.2876i 20.7304+0.4909i 0.730 + 0.491i 0.9031+0.4835i 4500.9602+0.0963i 0.960 + 0.096i 1.1315+0.0956i 10.9496+0.2902i 0.950 + 0.290i 1.1224+0.2879i 20.9300+0.4876i 0.930 + 0.488i 1.1053+0.4828i 30.9036+0.6892i 0.904 + 0.689i 1.0817+0.6812i F.-W. Shu, Y.-G. Shen / Physics Letters B 619 (2005) 340–346 345

Fig. 2. Variation of the QN frequencies with spin s, the left for boson perturbations, the right for fermion perturbations. we may use WKB approximation to evaluate the quasinormal mode frequencies within low-lying modes. The values for n

Acknowledgements [10] J. Bekenstein, Lett. Nuovo Cimento 11 (1974) 467; J. Bekenstein, in: M. Novello (Ed.), Cosmology and Gravita- One of the authors (F.-W. Shu) wishes to thank tion, Atlasciences, France, 2000, pp. 1–85, gr-qc/9808028; J. Bekenstein, V.F. Mukhanov, Phys. Lett. B 360 (1995) 7. Doctor Xian-Hui Ge for his valuable discussions. The [11] S. Hod, Phys. Rev. Lett. 81 (1998) 4293. work was supported by the National Natural Science [12] O. Dreyer, Phys. Rev. Lett. 90 (2003) 081301. Foundation of China under Grant No. 10273017. [13] A. Corichi, Phys. Rev. D 67 (2003) 087502. [14] J. Maldacena, Adv. Theor. Math. Phys. 2 (1998) 253. [15] S.A. Teukolsky, Phys. Rev. Lett. 29 (1972) 1114; S.A. Teukolsky, Astrophys. J. 185 (1973) 635; References S.A. Teukolsky, W.H. Press, Astrophys. J. 193 (1974) 443. [16] W.H. Press, S.A. Teukolsky, Astrophys. J. 185 (1973) 649. [1] S. Chandrasekhar, S. Detweiler, Proc. R. Soc. London, [17] G.F. Torres del Castillo, J. Math. Phys. 29 (1988) 2078; Ser. A 344 (1975) 441. G.F. Torres del Castillo, J. Math. Phys. 30 (1989) 446. [2] C.V. Vishveshwara, Nature 227 (1970) 936. [18] E.T. Newman, R. Penrose, J. Math. Phys. 7 (1966) 863; [3] V. Cardoso, J.P.S. Lemos, Phys. Rev. D 64 (2001) 084017. J.N. Goldberg, A.J. Macfarlane, E.T. Newman, F. Rohrlich, [4] C. Molina, Phys. Rev. D 68 (2003) 064007. E.C.G. Sudarshan, J. Math. Phys. 8 (1967) 2155. [5] F.-W. Shu, Y.-G. Shen, Phys. Rev. D 70 (2004) 084046; [19] A. Anderson, Phys. Rev. D 37 (1988) 536; F.-W. Shu, Y.-G. Shen, Phys. Lett. B 614 (2005) 195. A. Anderson, R.H. Price, Phys. Rev. D 43 (1991) 3147. [6] E. Berti, V. Cardoso, S. Yoshida, Phys. Rev. D 69 (2004) [20] B. Mashhoon, in: H. Ning (Ed.), Proceedings of the Third Mar- 124018. cel Grossmann Meeting on Recent Developments of General [7] H.T. Cho, Phys. Rev. D 68 (2003) 024003. Relativity, Shanghai, 1982, North-Holland, Amsterdam, 1983. [8] S. Chandrasekhar, The Mathematical Theory of Black Holes, [21] B.F. Schutz, C.M. Will, Astrophys. J. Lett. 291 (1985) L33. Oxford Univ. Press, Oxford, 1983. [22] S. Iyer, C.M. Will, Phys. Rev. D 35 (1987) 3621. [9] F. Echeverria, Phys. Rev. D 40 (1989) 3194; [23] R.A. Konoplya, Phys. Rev. D 68 (2003) 024018. L.S. Finn, Phys. Rev. D 46 (1992) 5236; [24] S. Iyer, Phys. Rev. D 35 (1987) 3632. O. Dreyer, B. Kelly, B. Krishnan, L.S. Finn, D. Garrison, R. Lopez-Aleman, Class. Quantum Grav. 21 (2004) 787. Physics Letters B 619 (2005) 347–351 www.elsevier.com/locate/physletb

Condition for the superradiance modes in higher-dimensional rotating black holes with multiple angular momentum parameters

Eylee Jung, Sung Hoon Kim, D.K. Park

Department of Physics, Kyungnam University, Masan 631-701, South Korea Received 6 June 2005; accepted 7 June 2005 Available online 13 June 2005 Editor: N. Glover

Abstract The condition for the existence of the superradiance modes is derived for the incident scalar, electromagnetic and gravitational waves when the spacetime background is a higher-dimensional rotating black hole with multiple angular momentum parameters. The ﬁnal expression of the condition is 0 <ω< i miΩi,whereΩi is an angular frequency of the black hole and, ω and mi are the energy of the incident wave and the ith azimuthal quantum number. The physical implication of this condition in the context of the brane-world scenarios is discussed.  2005 Elsevier B.V. All rights reserved.

Recently, much attention is paid to the various and suppresses the emission rate while the presence of properties of the absorption and emission problems in n reduces the absorptivity and increases the emission the higher-dimensional black holes. This is mainly due rate regardless of the brane-localized and bulk fields. to the fact that the brane-world scenarios [1–3] opens In fact, this fact can be deduced by considering the the possibility to make tiny black holes in the future Hawking temperature or by computing the effective colliders [4–7] by high-energy scattering. For the non- potential generated by the horizon structure. rotating black holes the complete absorption and emis- Also, the ratio of the low-energy absorption cross sion spectra are calculated numerically in Ref. [8], section for the Dirac fermion to that for the scalar field where the effect of the number of extra dimensions is derived analytically [9] in the charged black hole n and the inner horizon radius r− on the spectra is background. For the case of the bulk fields this ratio carefully examined. It has been found in Ref. [8] that factor becomes the presence of r− generally enhances the absorptivity σ BL γ BL ≡ F σ BL S E-mail addresses: [email protected] (E. Jung), n+1 (n+2)/(n+1) − + + r− [email protected] (S.H. Kim), [email protected] = 2 (n 3)/(n 1) 1 − (1) (D.K. Park). r+

0370-2693/$ – see front matter  2005 Elsevier B.V. All rights reserved. doi:10.1016/j.physletb.2005.06.012 348 E. Jung et al. / Physics Letters B 619 (2005) 347–351 and for the case of the brane-localized ﬁelds this factor Here, we would like to extend Ref. [23] to the ro- becomes tating black holes which have multiple angular mo- BR n+1 2/(n+1) mentum parameters. This is important because the tiny σ − + r− γ BR ≡ F = 2(n 3)/(n 1) 1 − . rotating black holes that will be produced in the fu- σ BR r+ S ture colliders due to the non-zero impact parameter (2) can have multiple components of the angular momen- In the Schwarzschild limit (r− ∼ 0)γBL ∼ tum since the brane thickness is of order of 1/TeV. − + + − + 2 (n 3)/(n 1) and γ BR ∼ 2(n 3)/(n 1) which reduces We start with a spacetime of the (N + 1)-dimen- to 1/8 when n = 0, which was derived by Unruh long sional rotating black hole derived by Myers and Perry BL = BL ago [10]. It is interesting to note that σF σS /2 in Ref. [25] BR = BR =∞ and σF 2σS when n . In the extremal limit (r− ∼ r+) the low-energy absorption cross sections N/2 2 =− 2 + 2 + 2 2 + 2 2 for the brane-localized and bulk fermions goes to zero. ds dt r ai dµi µi dφi The explicit n-dependence of these ratio factors may i=1 play important role in the experimental proof on the 2 2 N/2 existence of the extra dimensions. µr 2 + dt + aiµ dφi Recently, there is a controversy in the question ΠF i i=1 of whether the higher-dimensional black holes radi- ΠF ate mainly on the brane or in the bulk. Refs. [11,12] + dr2, (3) argued that the Hawking radiation into the bulk is Π − µr2 dominant compared to the brane-emission. The main where reason for this is because of the fact that for the tiny black holes the Hawking temperature is much larger N/2 2 2 than the mass of the light Kaluza–Klein modes. How- a µ F = 1 − i i , ever, Ref. [13] argued that the Hawking radiation on r2 + a2 i=1 i the brane is dominant because the radiation into the bulk by the light Kaluza–Klein modes is strongly sup- N/ 2 = 2 + 2 pressed by the geometrical factor. This argument is Π r ai . (4) supported numerically by Ref. [8] when n is not too i=1 large in the charged black holes. In Eq. (3) we assumed that N is even. The odd N case However, the situation can be completely differ- will be discussed later. The µ are not all independent ent when the black hole background has an angu- i but obeys lar momentum. For the rotating black holes the incident waves can be scattered backward with extrac- 2 + 2 +···+ 2 = tion of the black hole’s rotating energy, which is µ1 µ2 µN/2 1. (5) called superradiance. The effect of the superradiance The mass M and angular momenta J of the black hole in the 4-dimensional Kerr black hole was discussed i (3) are in Refs. [14–17]. In this context Refs. [18,19] argued that the conventional claim that the black holes radi- (N − 1)Ω − ate mainly on the brane can be changed if the effect M = N 1 µ, 16πG of the superradiance is involved. In fact, the existence the superradiance was proved analytically [20] and nu- 2 N J = Ma i = 1, 2,..., , (6) merically [21,22]. Also the general condition for the i N − 1 i 2 existence of the superradiance modes for the scalar, N/2 electromagnetic and gravitational waves was derived where ΩN−1 = 2π /[N/2] is the area of a unit in Ref. [23] using the Bekenstein argument [24] when (N − 1)-sphere and G is a (N + 1)-dimensional New- the black hole has a single angular momentum para- ton constant, which will be assumed to be unity from meter. now on. E. Jung et al. / Physics Letters B 619 (2005) 347–351 349

Now, we would like to calculate the horizon area A Since the surface gravity is proportional to the Hawk- of the spacetime (3), which is given by ing temperature, we can assume B > 0. Inserting Eq. (12) into (11) and using Eq. (6) yields 1−µ2 2π 1 1 ∂A = 8πrH A = dφ1 ···dφN dµ1 dµ2 ··· . (16) 2 ∂M B 0 0 0 Next, let us compute ∂A/∂Jj , which is 1−µ2−···−µ2 1 N −2 2 √ ∂A ∂rH = ΩN−1µ . (17) × dµN − det M, (7) ∂Jj ∂Jj 2 1 0 Differentiating Eq. (8) with respect to Jj , one can = | where det M det(gµi ,µj ) det(gφi ,φj ) r=rH . The hori- show easily zon radius r is deﬁned by solving H ∂r (N − 1)r H =− H Ω , (18) N/ 2 ∂J 2MB j 2 + 2 = 2 j rH ai µrH . (8) i=1 where

The factorization of det M comes from gµ ,φ = 0. It is aj i j Ωj = (19) = 2 2 2 2 ··· 2 r2 + a2 not difﬁcult to show det M µ rH µ1µ2 µN/2−1, H j which makes A in the following simple form is a frequency of the black hole arising due to the jth A = ΩN−1µrH . (9) angular momentum Jj . Inserting Eq. (18) into (17) yields Now, we regard M and Ji (i = 1, 2,...,N/2) are independent variables. Then one can write ∂A 8πrH =− Ωj . (20) N/2 ∂J B ∂A ∂A j dA = dM + dJi. (10) ∂M ∂Ji Thus inserting (16) and (20) into (10) simply yields i=1 N/2 Firstly, let us compute ∂A/∂M, which is 8πrH dA= dM − Ωi dJi . (21) ∂A µ ∂rH B = Ω − r + µ . (11) i=1 ∂M N 1 M H ∂M Bekenstein has shown in Ref. [24] that for scalar, To compute ∂r /∂M we use Eq. (8). Differentiating H electromagnetic, and gravitational waves dJi/dM is Eq. (8) with respect to M, one can show easily expressed in terms of the stress-energy tensor T as µν N/2 following a2 ∂rH = 1 rH + rH i , (12) r ∂M B 2M M r2 + a2 dJ T i=1 H i i =− φi r , (22) where dM Tt N/2 where φ are the azimuthal angles associated with J . r2 i i B = H − 1. (13) Since the incident waves should have the factorization 2 2 − r + a imi φi iωt i=1 H i factors e e , one can show easily that Eq. (22) It is important to note that B can be expressed as reduces to dJi mi B = r κ, (14) = , (23) H dM ω where κ is a surface gravity deﬁned where mi and ω are the azimuthal quantum numbers − ∂r Π 2µr corresponding to φi and energy of the incident waves, κ = . (15) 2µr2 respectively. Inserting (23) into (21), one can show r=rH 350 E. Jung et al. / Physics Letters B 619 (2005) 347–351 easily The mass M and the angular momenta Ji of the black hole (27) are same with Eq. (6). The horizon area is N/2 8πrH 1 slightly different from Eq. (9): dA = dM 1 − m Ω . (24) B i i ω = i 1 1−µ2 2π 1 1 Since A/4 is a black hole entropy, Eq. (24) gives a A = dφ1 ···dφN−1 dµ1 dµ2 ··· condition 2 N/2 0 0 0 1 − − 2−···− 2 dM 1 miΩi > 0. (25) 1 µ1 µ N−3 ω 2 i=1 × dµN−1 Since the existence of the superradiance modes im- 2 plies dM < 0, it is easy to show that the condition for 0 the existence of the superradiance is × | | det(gµi µj ) r=rH det(gφi φj ) r=rH N/2 ΩN−1 0 <ω< m Ω . (26) = µrH . (30) i i 2 i=1 Then it is straightforward to show In Ref. [20] the condition for the superradiance for the incident scalar wave was shown to be 0 <ω

Summing up non-anticommutative Kähler potential ✩

Tomoya Hatanaka, Sergei V. Ketov, Shin Sasaki

Department of Physics, Tokyo Metropolitan University, 1-1 Minami-osawa, Hachioji-shi, Tokyo 192-0397, Japan Received 1 May 2005; received in revised form 24 May 2005; accepted 6 June 2005 Available online 15 June 2005 Editor: T. Yanagida

Abstract We offer a simple non-perturbative formula for the component action of a generic N = 1/2 supersymmetric chiral model in terms of an arbitrary number of chiral superfields in four dimensions, which is obtained by the non-anticommutative (NAC) deformation of a generic four-dimensional N = 1 supersymmetric non-linear sigma-model (NLSM) described by arbitrary Kähler superpotential and scalar superpotential. The auxiliary integrations responsible for fuzziness are eliminated in the case of a single chiral superfield. The scalar potential in components is derived by eliminating the auxiliary fields. The NAC-deformation of the CP1 Kähler NLSM with an arbitrary scalar superpotential is calculated as an example.  2005 Elsevier B.V. All rights reserved.

1. Introduction

There was a lot of recent activity in investigating various aspects of non-anticommutative (NAC) superspace and related deformations of supersymmetric ﬁeld theories (see, e.g., the most recent references [1–7] directly related to our title, and the references therein for the earlier work in the NAC-deformed N = 1 superspace). It is supposed to enhance our understanding of the role of spacetime in supersymmetry, while keeping globally supersymmetric ﬁeld theory under control. . We work in four-dimensional Euclidean1 N = 1 superspace (xµ,θα, θ¯α), and use the standard notation [8].The NAC deformation is given by α β = αβ θ ,θ ∗ C , (1.1)

✩ Supported in part by the Japanese Society for Promotion of Science (JSPS). E-mail addresses: [email protected] (T. Hatanaka), [email protected], [email protected] (S.V. Ketov), [email protected] (S. Sasaki). 1 The use of Atiyah–Ward spacetime of the signature (+, +, −, −) is another possibility [9].

0370-2693/$ – see front matter  2005 Elsevier B.V. All rights reserved. doi:10.1016/j.physletb.2005.06.014 T. Hatanaka et al. / Physics Letters B 619 (2005) 352–358 353 where Cαβ are some constants. The remaining superspace coordinates in the chiral basis (yµ = xµ + iθσµθ¯, µ,ν = 1, 2, 3, 4 and α,β,...= 1, 2) still (anti)commute, . . . . yµ,yν = θ¯α, θ¯β = θ α, θ¯β = yµ,θα = yµ, θ¯α = 0. (1.2) The Cαβ = 0 explicitly break the four-dimensional ‘Lorentz’ invariance at the fundamental level. The NAC nature of θ’s can be fully taken into account by using the Moyal–Weyl-type (star) product of superfields [1], ←− −→ Cαβ ∂ ∂ f(θ)∗ g(θ) = f(θ)exp − g(θ), (1.3) 2 ∂θα ∂θβ which respects the N = 1 superspace chirality. The star product (1.3) is polynomial in the deformation parameter, Cαβ ∂f ∂g ∂2f ∂2g f(θ)∗ g(θ) = fg + (−1)degf − det C , (1.4) 2 ∂θα ∂θβ ∂θ2 ∂θ2 where we have used the identity 1 det C = ε ε Cαβ Cγδ, (1.5) 2 αγ βδ and the notation ∂2 1 ∂ ∂ = εαβ . (1.6) ∂θ2 4 ∂θα ∂θβ We also use the following book-keeping notation for 2-component spinors: . . = α ¯ ¯ = ¯ ¯ α 2 = α ¯2 = ¯ ¯α θχ θ χα, θχ θα.χ ,θθ θα, θ θα.θ . (1.7) The spinorial indices are raised and lowered by the use of two-dimensional Levi-Civita symbols [8]. Grassmann integration amounts. to Grassmann differentiation. The antichiral covariant derivative in the chiral superspace basis ¯ =− ¯α is Dα. ∂/∂θ . The field component expansion of a chiral superfield Φ reads √ Φ(y,θ) = φ(y)+ 2θχ(y)+ θ 2M(y). (1.8) An antichiral superfield Φ¯ in the chiral basis is given by √ √ √ ¯ µ µ ¯ ¯ ¯ ¯ ¯2 ¯ µ ¯2 µ ¯ ¯ Φ y − 2iθσ θ,θ = φ(y) + 2θχ(y)¯ + θ M(y)+ 2θ iσ ∂µχ(y)¯ θ − i 2σ θ∂µφ(y) + θ 2θ¯22φ(y),¯ (1.9) µ where 2 = ∂µ∂ . The bars over fields serve to distinguish between the ‘left’ and ‘right’ components that are truly independent in Euclidean spacetime. Our major concern in this Letter is a derivation of the NAC deformation of a generic four-dimensional N = 1 supersymmetric action

¯ ¯ S = d4x d2θd2θK¯ Φi, Φ¯ j + d2θW Φi + d2θ¯ W¯ Φ¯ j (1.10) specified by a Kähler superpotential K(Φ,Φ)¯ and a scalar superpotential W(Φ), in terms of an arbitrary number n of chiral and antichiral superfields, i, j¯ = 1, 2,...,n. This problem in four dimensions was addressed in Refs. [4,7], where the perturbative answers (in terms of infinite sums) were found. A similar problem in two dimensions was solved perturbatively in Ref. [3], while the non-perturbative summation (in terms of finite functions) was done in Ref. [6]. In this Letter we give simple non-perturbative formulas in four dimensions and offer a simple way of their derivation. Our results presumably amount to a full summation of the infinite sums in Refs. [4,7]. Summing up is 354 T. Hatanaka et al. / Physics Letters B 619 (2005) 352–358 crucial for further non-perturbative physical applications of the NAC-deformation and its geometrical interpretation. We also made progress in eliminating the auxiliary integrations and solving the auxiliary field equations, as well as in investigating some concrete examples (see below). We use the chiral basis, which is most suitable for investigating NAC-deformation, and reduce the most non- trivial problem of calculation of the NAC-deformed Kähler superpotential to that for the NAC-deformed scalar superpotential. The remarkably simple non-trivial results about the NAC-deformed scalar superpotential are already available in Refs. [5,6]. In Section 2 we describe our idea in the undeformed case (it is not really new there). In Section 3 we present the results of our calculation for the most general NAC-deformed action (1.10). In Section 4 we specialize our results to the case of a single chiral superfield and the CP1 (Kähler) superpotential, as the simplest non-trivial examples. Section 5 is our conclusion.

2. Chiral reduction of Kähler superpotential

Let us use the identity

4 2 2 ¯ ¯ 1 4 2 ¯ 2 ¯ 4 d xd θd θK(Φ,Φ) =− d yd θ D K(Φ,Φ) ≡ d yL, (2.1) 4 | ¯ ¯ i = where denotes the θ-independent part of a superﬁeld, and the constraint Dα.Φ 0. Having performed Grassmann differentiation in D¯ 2K, we arrive at the spacetime (NAC-undeformed) component Lagrangian in the chiral form

2 I L = d θV(K) Φ , (2.2)

I whose effective scalar superpotential V(K)(Φ ) is given by 1 ¯ p¯q¯ ¯ p¯ V =− K ¯ ¯ Φ(y,θ),φ(y) Φ + K ¯ Φ(y,θ),φ(y) Φ (2.3) (K) 2 ,pq n+1 ,p n+2 I ={ i p¯q¯ p¯ } in terms of the extended set of the chiral superfields Φ Φ ,Φn+1,Φn+2 . We use the notation (valid for any function F(φ,φ)¯ ) s+t = ∂ F F,i1i2...is p¯1p¯2...p¯t ¯ ¯ ¯ , (2.4) ∂φi1 ∂φi2 ···φis ∂φ¯p1 ∂φ¯p2 ···∂φ¯pt ¯ j¯|= ¯j¯ + p¯q¯ and the fact that Φ φ (y). The additional (composite) n(n 1)/2 chiral superfields Φn+1 and n chiral super- p¯ fields Φ + are given by n 2 √ p¯q¯ 1 ¯ ¯ p¯ ¯ α. ¯ q¯ p¯ q¯ µ (p¯ ¯q)¯ 2 µ ¯(p¯ ¯q)¯ Φ + (y, θ) = D.Φ D Φ =¯χ χ¯ − 2 2i θσ χ¯ ∂µφ − 2θ ∂ φ ∂µφ (2.5a) n 1 2 α and √ p¯ 1 ¯ 2 ¯ p¯ ¯ p¯ µ p¯ 2 ¯p¯ Φ + (y, θ) =− D Φ = M + 2i θσ ∂µχ¯ + θ 2φ . (2.5b) n 2 4 Therefore, the NAC deformation of the Kähler superpotential is not an independent problem, since it is deriv- able from the NAC deformation of the effective scalar superpotential (2.3) having the extended number of chiral superfields. The simple non-perturbative form of an arbitrary NAC-deformed scalar superpotential V , depending upon a single chiral superfield Φ of Eq. (1.8), was first calculated in Ref. [5],

2 2 1 χ d θV∗(Φ) = V(φ+ cM) − V(φ− cM) − V (φ + cM) − V (φ − cM) , (2.6) 2c 4cM ,φ ,φ T. Hatanaka et al. / Physics Letters B 619 (2005) 352–358 355 where we have introduced the (finite) deformation parameter √ c = − det C. (2.7) As is clear from Eq. (2.6), the NAC-deformation in the single superfield case gives rise to the scalar potential split controlled by the auxiliary field M. When using an identity

+1 ∂ f(x+ a) − f(x− a) = a dξ f(x + ξa), (2.8) ∂x −1 valid for any function f , we can rewrite Eq. (2.6) to the equivalent form,

+1 +1 2 2 1 ∂ 1 2 ∂ d θV∗(Φ) = M dξ V(φ + ξcM)− χ dξ V(φ + ξcM), (2.9a) 2 ∂φ 4 ∂φ2 −1 −1 which is very suitable for an immediate generalization to the case of several chiral superﬁelds (cf. Ref. [6]),

2 2 I 1 I ∂ ˜ 1 I J ∂ ˜ d θV∗ Φ = M V(φ,M)− χ χ V(φ,M), (2.9b) 2 ∂φI 4 ∂φI ∂φJ where the auxiliary prepotential V˜ has been introduced [6],

+1 V(φ,M)˜ = dξ V φI + ξcMI . (2.10) −1 Therefore, the NAC-deformation of a generic scalar superpotential V results in its smearing or fuzziness controlled by the auxiliary ﬁelds MI [6].

3. The NAC-deformed Kähler superpotential

Our general result in components is just given by Eq. (2.9b) after a substitution of Eq. (2.3) in terms of the deﬁnitions of Section 2. We ﬁnd (cf. Refs. [3,4,6,7]) 1 i 1 µ ¯p¯ ¯q¯ 1 ¯p¯ 1 i j 1 i µ p¯ ¯q¯ L = M Y + ∂ φ ∂ φ Z ¯ ¯ + 2φ Z ¯ − χ χ Y − i χ σ χ¯ ∂ φ Z ¯ ¯ 2 ,i 2 µ ,pq 2 ,p 4 ,ij 2 µ ,ipq 1 i µ p¯ − i χ σ ∂ χ¯ Z ¯ , (3.1) 2 µ ,ip where we have introduced the (component) smeared Kähler prepotential

+1 ¯ Z(φ,φ,M)¯ = dξ Kξ with Kξ ≡ K φi + ξcMi, φ¯j , (3.2a) −1 as well as the extra (auxiliary) prepotential

+1 ¯ ¯ ¯ p¯ 1 p¯ q¯ µ ¯p¯ ¯q¯ ξ ¯p¯ ξ Y(φ,φ,M,M)= M Z ¯ − χ¯ χ¯ Z ¯ ¯ + c dξ ξ ∂ φ ∂ φ K + 2φ K . (3.2b) ,p 2 ,pq µ ,p¯q¯ ,p¯ −1 356 T. Hatanaka et al. / Physics Letters B 619 (2005) 352–358

We veriﬁed that Eq. (3.1) reduces to the standard (Kähler) N = 1 supersymmetric non-linear sigma-model in the → = i ¯ j¯ limit c 0. Also, in the case of a free (bilinear) Kähler potential K δij¯Φ Φ , there is no deformation at all. Given, in addition, an independent scalar superpotential W(Φ),asinEq.(1.10), the following component terms are to be added to Eq. (3.1): 1 i ˜ 1 i j ˜ ¯ p¯ ¯ 1 p¯ q¯ ¯ L = M W − χ χ W + M W ¯ − χ¯ χ¯ W ¯ ¯ , (3.3) potential 2 ,i 4 ,ij ,p 2 ,pq where we have used the fact that the antichiral scalar superpotential terms are inert under the NAC-deformation, and we have introduced the smeared scalar prepotential [6]

+1 W(φ,M)˜ = dξ W φi + ξcMi . (3.4) −1 Though our general results of this section are quite explicit and non-perturbative, as regards their applications, it is desirable to perform all integrations over ξ (thus evaluating the smearing effects) and eliminate the auxiliary fields (M, M)¯ by using their algebraic equations of motion. It is clear that the ξ-integrals can be evaluated once the Kähler and scalar functions, K and W , are specified. The anti-chiral auxiliary fields M¯ p¯ enter the action linearly (as Lagrange multipliers), so that their algebraic equations of motion determine the auxiliary fields Mi = Mi(φ, φ)¯ ,

1 i ¯ M Z ¯ + W ¯ = 0. (3.5) 2 ,ip ,p The bosonic scalar potential in components is thus given by

¯ 1 i ˜ Vscalar(φ, φ) = M W,i , (3.6) 2 M=M(φ,φ)¯ in agreement with Ref. [6]. In Section 4 we study some examples, by restricting our general results to the case of a single chiral superﬁeld, and then to the case of the CP1 Kähler potential.

4. Examples

It is quite natural to begin with an example of a single chiral superﬁeld, while keeping arbitrary both Kähler and scalar functions. In this case all the ξ-integrations can be easily performed, e.g., by using Eq. (2.8) and the related identity obtained by differentiating Eq. (2.8) with respect to the parameter a,

+1 +1 ∂ f(x+ a) + f(x− a) = dξ f(x + ξa)+ a dξ ξf(x + ξa). (4.1) ∂x −1 −1 Of course, one would get the same results by using the chiral reduction of Section 2 and the crucial identity (2.6) from the very beginning. We found some cancellations amongst the bosonic terms, with the result 1 µ L =+ ∂ φ∂¯ φ¯ K ¯ ¯ (φ + cM,φ)¯ + K ¯ ¯ (φ − cM,φ)¯ bosonic 2 µ ,φφ ,φφ 1 M¯ + 2φ¯ K ¯ (φ + cM,φ)¯ + K ¯ (φ − cM,φ)¯ + K ¯ (φ + cM,φ)¯ − K ¯ (φ − cM,φ)¯ 2 ,φ ,φ 2c ,φ ,φ 1 ∂W¯ + W(φ+ cM) − W(φ− cM) + M¯ . (4.2) 2c ∂φ¯ T. Hatanaka et al. / Physics Letters B 619 (2005) 352–358 357

The bosonic terms are to be supplemented by the following fermionic terms: 1 2 L =− χ¯ K ¯ ¯ (φ + cM,φ)¯ − K ¯ ¯ (φ − cM,φ)¯ fermionic 4c ,φφ ,φφ i µ − χσ χ¯ ∂ φ¯ K ¯ ¯ (φ + cM,φ)¯ − K ¯ ¯ (φ − cM,φ)¯ 2cM µ ,φφ ,φφ i µ − χσ ∂ χ¯ K ¯ (φ + cM,φ)¯ − K ¯ (φ − cM,φ)¯ 2cM µ ,φ ,φ ¯ M 2 − χ K ¯ (φ + cM,φ)¯ − K ¯ (φ − cM,φ)¯ 4cM ,φφ ,φφ 1 2 µ − χ ∂ φ∂¯ φ¯ K ¯ ¯ (φ + cM,φ)¯ + K ¯ ¯ (φ − cM,φ)¯ 4M µ ,φφφ ,φφφ 1 2 µ ¯ ¯ ¯ ¯ + χ ∂ φ∂µφ K ¯ ¯ (φ + cM,φ) − K ¯ ¯ (φ − cM,φ) 4cM2 ,φφ ,φφ 1 2 − χ 2φ¯ K ¯ (φ + cM,φ)¯ + K ¯ (φ − cM,φ)¯ 4M ,φφ ,φφ 1 2 + χ 2φ¯ K ¯ (φ + cM,φ)¯ − K ¯ (φ − cM,φ)¯ 4cM2 ,φ ,φ 1 2 2 + χ χ¯ K ¯ ¯ (φ + cM,φ)¯ − K ¯ ¯ (φ − cM,φ)¯ 8cM ,φφφ ,φφφ 1 2 1 2 ¯ − χ W (φ + cM) − W (φ − cM) − χ¯ W ¯ ¯ . (4.3) 4cM ,φ ,φ 2 ,φφ In the case of the CP1 Kähler potential

K(φ,φ)¯ = ln(1 + φφ),¯ (4.4) the auxiliary ﬁeld equation (3.5) gives rise to a quadratic equation on M, whose roots are given by

1 ± 1 + (2cφ(¯ 1 + φφ)¯ W¯ )2 M = , (4.5) 2c2φ¯2W¯ where we have used the notation W¯ = ∂W/∂¯ φ¯. Taking the anticommutative limit c → 0 implies that we should choose minus in Eq. (4.5). The scalar potential is given by Eq. (3.6), after substituting Eqs. (3.4) and (4.5) overthere. More examples and applications will be considered elsewhere [10].

5. Conclusion

A comparison to the NAC-deformed N = 2NLSMintwo dimensions [3,6] is possible after dimensional reduction of our results in Sections 3 and 4 by assuming ∂3 = ∂4 = 0. As was already demonstrated in Ref. [6],the inﬁnite series found in Ref. [3] can be resummed in terms of the ‘minimally’ deformed Kähler potential and superpotential in the sense of Eqs. (3.2a) and (3.4), plus some additional ‘non-minimal’ terms with deformed coupling as in Eq. (3.2b),inprecise agreement with our basic formulae (2.6) and (2.9).2 As regards the inﬁnite series found in Refs. [4,7] in four dimensions, those results seem to be very complicated to allow us a direct comparison. An explicit resummation is necessary not only (i) for comparison, but also (ii) checking the locality of the deformed

2 The split (2.6) of a NAC-deformed superpotential in the case of a single chiral superfield was found in Ref. [5]. The smearing (2.9) and (2.10) of NAC-deformed Kähler potential and superpotential in the case of several chiral superfields was found in Ref. [6]. 358 T. Hatanaka et al. / Physics Letters B 619 (2005) 352–358 action in spacetime, (iii) verifying the auxiliary fields to be still non-propagating, and (iv) ultimately solving the auxiliary field equations. We believe that our results in this Letter are useful for all those purposes, because we addressed the most general case, formulated our results in a compact and transparent form, and offered a clear ‘short cut’ for their easy derivation. As is well known in the theory of NLSM (see, e.g., Ref. [11]), the so-called quotient construction (or gauging isometries of the NLSM target space) can be used to represent some NLSM with homogeneous target spaces as the gauge theories. It was used in Ref. [12] to construct the NAC-deformed supersymmetric NLSM in four dimensions with the CPn target space, by combining the quotient construction with the results of Ref. [1] about the NAC- deformed supersymmetric gauge theories. As is clear from our results about generic NAC-deformed NLSM in Section 3, the NAC deformation of a Kähler potential is controlled by the auxiliary fields entering the deformed Kähler potential in the highly non-linear way. The auxiliary fields are determined by their algebraic equations of motion that are also dependent upon a superpotential, even when all fermions are ignored. As a result, the NAC deformation of Kähler geometry is controlled by a scalar superpotential! It is in drastic contrast with the standard supersymmetric (Kähler) NLSM in undeformed superspace, whose target space geometry is unaffected by a scalar superpotential. In the absence of a scalar superpotential, W = W¯ = 0, we found that the NAC deformation c = 0 does not affect the supersymmetric CP1 NLSM action at all, in agreement with Ref. [12]. A detailed analysis of the actions (3.1), (4.2) and (4.3), including the deformed supersymmetric CPn NLSM, will be given elsewhere [10].

Acknowledgements

The authors are grateful to Y. Kobayashi for discussions, B. Chandrasekhar, A. Kumar and A. Lerda for correspondence, and the referee for his careful reading of our manuscript and useful suggestions.

References

[1] N. Seiberg, JHEP 0306 (2003) 010, hep-th/0305248. [2] M. Billo, M. Frau, F. Lonegro, A. Lerda, hep-th/0502084. [3] B. Chandrasekhar, A. Kumar, JHEP 0403 (2004) 013, hep-th/0310137; B. Chandrasekhar, Phys. Rev. D 70 (2004) 125003, hep-th/0408184; B. Chandrasekhar, hep-th/0503116. [4] O.D. Azorkina, A.T. Banin, I.L. Buchbinder, N.G. Pletnev, hep-th/0502008. [5] T. Hatanaka, S.V. Ketov, Y. Kobayashi, S. Sasaki, Nucl. Phys. B 716 (2005) 88, hep-th/0502026. [6] L. Alvarez-Gaume, M. Vazquez-Mozo, hep-th/0503016. [7] T.A. Ryttov, F. Sannino, hep-th/0504104. [8] J. Wess, J. Bagger, Supersymmetry and Supergravity, Princeton Univ. Press, Princeton, 1992. [9] S.J. Gates Jr., S.V. Ketov, H. Nishino, Nucl. Phys. B 393 (1993) 149, hep-th/9207042. [10] T. Hatanaka, S.V. Ketov, Y. Kobayashi, S. Sasaki, in preparation. [11] S.V. Ketov, Quantum Non-Linear Sigma-Models, Springer-Verlag, Heidelberg, 2000. [12] T. Inami, H. Nakajima, Prog. Theor. Phys. 111 (2004) 961, hep-th/0402137. Physics Letters B 619 (2005) 359–366 www.elsevier.com/locate/physletb

Fundamental strings in SFT

L. Bonora, C. Maccaferri, R.J. Scherer Santos, D.D. Tolla

International School for Advanced Studies (SISSA/ISAS), Via Beirut 2-4, 34014 Trieste, Italy INFN, Sezione di Trieste, Trieste, Italy Received 26 January 2005; accepted 27 May 2005 Available online 6 June 2005 Editor: L. Alvarez-Gaumé

Abstract In this Letter we show that vacuum string ﬁeld theory contains exact solutions that we propose to interpret as macroscopic fundamental strings. They are formed by a condensate of inﬁnitely many completely space-localized solutions (D0-branes).  2005 Elsevier B.V. All rights reserved.

PACS: 11.25.-w; 11.25.Sq; 11.25.Uv

Keywords: String ﬁeld theory; Solitonic lumps; Fundamental strings

1. Introduction The existence of such solutions confirms the conjecture at the basis of VSFT. The tachyon condensation Vacuum string field theory (VSFT) is a version of vacuum physics can only represent closed string the- Witten’s open string field theory which is conjectured ory and thus, if VSFT is to represent string theory at to represent string theory at the tachyon condensation the tachyon condensation vacuum, it should be able to vacuum [1]. Its action is formally the same as the orig- describe closed string theory in the sense of [2].The inal Witten theory except that the BRST charge takes above D-brane solutions, expressed in the open string a simplified form: it has been argued that it can be language of VSFT, correspond precisely to objects that expressed simply in terms of the ghost creation and in closed string language appear either as boundary annihilation operators. By virtue of this simplification state or as solutions of low energy effective actions. it has been possible to determine exact classical solu- Recently it has been possible to find in VSFT an tions which have been shown to represent D-branes. exact time-dependent solution, [3], with the characteristics of a rolling tachyon [4]. A rolling tachyon describes in various languages (effective field theory, E-mail addresses: [email protected], [email protected] (L. Bonora), [email protected] (C. Maccaferri), [email protected] BCFT, SFT) the decay of unstable D-branes. It is by (R.J. Scherer Santos), [email protected] (D.D. Tolla). now clear that the final product of a brane decay is

0370-2693/$ – see front matter  2005 Elsevier B.V. All rights reserved. doi:10.1016/j.physletb.2005.05.067 360 L. Bonora et al. / Physics Letters B 619 (2005) 359–366 formed by massive closed string states. However, it Ψm|Ψm is the ordinary inner product. We will con- has been shown that, in the presence of a background centrate on the matter part, Eq. (5), assuming the ex- electric field also (macroscopic) fundamental strings istence of a universal ghost solution. The solutions are appear as final products of a brane decay. Now, since projectors of the ∗m algebra. The ∗m product is defined our aim is to be able to describe a brane decay in the as follows framework of VSFT we must show first of all that such | | = ∗ | fundamental strings exist as solutions of VSFT. In this 123 V3 Ψ1 1 Ψ2 2 3 Ψ1 m Ψ2 , (7) Letter we want to present some evidence that such so- see [5–8] for the definition of the three string ver- lutions exist. tex 123V3|. In the following we need both transla- The Letter is organized as follows. In the next sec- tionally invariant and nontranslationally invariant so- tion we collect some well-known formulas which are lutions. Although there is a great variety of such solu- needed in the sequel. In Section 3 we show in a rather tions we will stick to those introduced in [9], i.e., the informal way how to construct new one-dimensional sliver and the lump. The former is translationally in- solutions as condensate of D0-branes. In Section 4 we variant and is defined by give a more motivated account of the same construc- − 1 a†·S·a† tion by introducing a background B field. In the last |Ξ=N e 2 |0, section we provide evidence that the new solutions ∞ † · · † = µ† ν† represent fundamental strings. a S a an Snmam ηµν, (8) n,m=1 where S = CT and 2. A reminder 1 T = 1 + X − (1 + 3X)(1 − X) (9) 2X In this section we recall the notation and some use- 11 ful formulas. The VSFT action is with X = CV . In proving that this is a solution it is crucial that the matrices X = CV 11, X+ = CV 12 1 1 21 S(Ψ ) =− Ψ |Q|Ψ + Ψ |Ψ ∗ Ψ , (1) and X− = CV , are mutually commuting and com- n 2 3 mute with T = CS = SC, where Cnm = (−1) δnm. where The normalization constant N needs being regularized and is formally vanishing. It has been showed in other n Q = c0 + (−1) (c2n + c−2n). (2) papers how this problem could be dealt with, [10,11]. n>0 The lump solution is engineered to represent a The ansatz for nonperturbative solutions is in the fac- lower-dimensional brane (Dk-branes), therefore it will − torized form have (25 k) transverse space directions along which translational invariance is broken. Accordingly we Ψ = Ψm ⊗ Ψg, (3) split the three string vertex into the tensor product of the perpendicular part and the parallel part where Ψg and Ψm depend purely on ghost and matter degrees of freedom, respectively. The equation of |V3=|V3,⊥⊗|V3,. (10) motion splits into The parallel part is the same as in the sliver case while the perpendicular part is modified as follows. Follow- QΨg =−Ψg ∗g Ψg, (4) ing [9], we denote by xα, pα, α = 1,...,k the coor- = ∗ Ψm Ψm m Ψm, (5) dinates and momenta in the transverse directions and where ∗ and ∗ refers to the star product involving introduce the canonical zero modes oscillators g m √ only the ghost and matter part. The action for this type 1 1 a(r)α = bpˆ(r)α − i √ xˆ(r)α, of solution is 0 2 b √ 1 (r)α† = 1 ˆ(r)α + √1 ˆ(r)α S(Ψ ) =− Ψ |Q|Ψ Ψ |Ψ , (6) a0 bp i x , (11) 6 g g m m 2 b L. Bonora et al. / Physics Letters B 619 (2005) 359–366 361 where b is a free parameter. Denoting by |Ωb the os- solution displaced by an amount s in the positive x α| = ˆ cillator vacuum (a0 Ωb 0), in this new basis the direction (x being the eigenvalue of x). The appropri- three string vertex is given by ate solution has been constructed by Rastelli, Sen and Zwiebach, [15]: | = −E | V3,⊥ Ke Ωb (12) − ˆ Ξ (s) = e isp|Ξ . (14) with 0 0 √ k/2 It satisfies |Ξ (s)∗|Ξ (s)=|Ξ (s).Eq.(14) can 2πb3 0 0 0 K = , 2 be written explicitly as 3(V00 + b/2) 2 3 − s − = N 2b (1 S00) 1 (r)α† rs (s)β† Ξ0(s) e E = a V a ηαβ , (13) 2 M MN N = is r,s 1 M,N 0 × exp −√ (1 − S ) · a† b 0 where M, N denote the couple of indices {0,m} and

{0,n}, respectively. The coefficients V rs are given 1 MN × exp − a† · S · a† |Ω , (15) in Appendix B of [9]. The new Neumann coefficients 2 b matrices V rs satisfy the same relations as the V rs − · † = ∞ − † ones. In particular one can introduce the matrices where ((1 S ) a )0 N=0((1 S )0N aN ) and rs = rs = − N † · · † = ∞ † † N | X CV , where CNM ( 1) δNM, which turn a S a N,M=0 aN SNMaM ; is the Ξ0 nor- out to commute with one another. The lump solution malization constant. Moreover one can show that | Ξk has the form (8) with S along the parallel direc- = | tions and S replaced by S along the perpendicular Ξ0(s) Ξ0(s) Ξ0 Ξ0 . (16) ones. In turn S = CT and T has the same form as T Eq. (9) with X replaced by X . The normalization The meaning of this solution is better understood constant N is defined in a way analogous to N and if we make its space profile explicit by contracting it the same remarks hold for it. It can be verified that with the coordinate eigenfunction the ratio of tensions for such solutions is the appropri- 2 1/4 ate one for Dk-branes. Moreover, the space profile of |ˆx= πb these solutions in the transverse direction is given by a Gaussian (see [12,14]). This reinforces the interpreta- 2 × −x − √2 † + 1 † † | tion of these solutions as branes. exp i a0x a0a0 Ωb . (17) b b 2 The result is

3. Constructing new solutions −S 2 1/4 1 00 (x−s) 2 N − ˆ| = 1+S b x Ξ0(s) e 00 In this section we would like to show how qual- πb + 1 S00 itatively new solutions to (5) can be constructed by √2i x−s † − S a 1 † † accretion of infinite many lumps. Let us start from a b 1+S 0m m − a W a × e 00 e 2 n nm m |0, (18) lump solution representing a D0-brane as introduced S S in the previous section: it has a Gaussian profile in = − n0 0m where Wnm Snm 1+S . It is clear that (18) rep- all space directions, the form of the string field—let 00 | resents the same Gaussian profile as |Ξ =|Ξ (0) us denote it Ξ0 —will be the same as (9) with S re- 0 0 placed by S , while the ∗-product will be determined shifted away from the origin by s. by the primed three strings vertex (12). Let us pick It is important to remark now that two such states | | ∗ one particular space direction, say the αth. For sim- Ξ0(s) and Ξ0(s ) are -orthogonal and bpz-or- = plicity in the following we will drop the corresponding thogonal provided that s s . For we have α α label from the coordinate xˆ , momenta pˆ and oscil- ∗ = −C(s,s ) lators aα along this direction. Next we need the same Ξ0(s) Ξ0(s ) e Ξ0(s, s ) , (19) 362 L. Bonora et al. / Physics Letters B 619 (2005) 359–366

| where the state Ξ0(s, s ) becomes proportional to In conclusion we can write |Ξ (s) when s = s and needs not be explicitly writ- 0 ∗ = ˆ ten down otherwise; while Ξ0(s) Ξ0(s ) δ(s,s ) Ξ0(s) , (22) where δˆ is the Kronecker (not the Dirac) delta func- 1 T (1 − T ) C(s, s ) =− s2 + s 2 tion. 2b 1 + T 00 Similarly one can prove that 2 (1 − T ) + ss . (20) + Ξ0(s ) Ξ0(s) 1 T 00 N 2 2 − s (1−S ) T (1−T ) = e b 00 The quantity ( + )00 can be evaluated by using 1 T det(1 − S 2) the basis of eigenvectors of X and T , [3,16,17]: 1 2 2 S (1−S ) 1−S (s +s ) +2ss × e 2b 1+S 00 1+S 00 . (23) T (1 − T ) + We can repeat the same argument as above and con- 1 T 00 ∞ clude that − 2 t(k)(1 t(k)) = 2 dk V0(k) Ξ (s ) Ξ (s) = δ(s,sˆ )Ξ |Ξ . (24) 1 + t(k) 0 0 0 0 0 After the above preliminaries, let us consider a se- −| | −| | ¯ ¯ e η (1 − e η ) quence s ,s ,... of distinct real numbers and the cor- + V (ξ)V (ξ) + V (ξ)V (ξ) . (21) 1 2 0 0 0 0 + −|η| | 1 e responding sequence of displaced D0-branes Ξ0(sn) . The variable k parametrizes the continuous spectrum Due to the property (22) also the string state and V (k) is the relevant component of the continuous ∞ 0 | = basis. The modulus 1 numbers ξ and ξ¯ parametrize Λ Ξ0(sn) (25) ¯ n=1 the discrete spectrum and V (ξ), V (ξ) are the relevant 0 0 | ∗| =| components of the discrete basis (see [17] for explicit is a solution to (5): Λ Λ Λ . To figure out what expressions of the eigenvectors and for the connection it represents let us study its space profile. To this end between ξ, η and b). The discrete spectrum part of the we must sum all the profiles like (18) and then pro- RHS of (21) is just a number. Let us concentrate on ceed to a numerical evaluation. In order to get a one- dimensional object, we render the sequence s ,s ,... the continuous spectrum contribution. We have t(k)= 1 2 | | dense, say, in the positive x-axis so that we can replace − exp(− π k ). Near k = 0, V (k) ∼ 1 b and the in- 2 0 2 2π the summation with an integral. The relevant integral tegrand ∼− b 1 , therefore the integral diverges log- 2π2 k is arithmically, a singularity we can regularize with an ∞ infrared cutoff . Taking the signs into account we find dsexp −α(x − s)2 − iβ(x − s) that the RHS of (21) goes like b log as a function 2π2 (1−T )2 0 √ of the cutoff. Similarly one can show that ( + )00 2 1 T π − β iβ √ goes like − b log . Since for s = s , s2 + s 2 > 2ss , = √ e 4α 1 + Erf √ + αx , (26) π2 2 α 2 α we can conclude that C(s, s ) ∼−c log , where c is a positive number. Therefore, when we remove the cut- where Erf is the error function and −C off, the factor e (s,s ) vanishes, so that (19) becomes 1 − S S a† = 1 00 = √2 0m m a ∗-orthogonality relation. Notice that the above log- α ,β . b 1 + S b 1 + S arithmic singularities in the two pieces in the RHS of 00 00 = Of course (26) is a purely formal expression, but it be- (21) neatly cancel each other when s s and we get the finite number comes meaningful in the α → 0 limit. As usual, [12], we parametrize this limit with a dimensionless para- 2 s → C(s, s) =− (1 − S ). meter and take 0. Using the√ results of [3,12]√ , 2b 00 one can see that α ∼ 1/ , β ∼ 1/ , so that β/ α L. Bonora et al. / Physics Letters B 619 (2005) 359–366 363 tends to a finite limit. Therefore, in this limit, we can 4. An improved construction disregard the first addend in the argument of Erf. Then, up to normalization, the space profile of |Λ is deter- In this section we would like to justify some of the mined by passages utilized in discussing the space profile of the 1 √ fundamental string solution in Section 3. The prob- 1 + Erf( αx) . (27) 2 lems in Section 3 are linked to the well-known singu- In the limit → 0 this factor tends to a step func- larity of the lump space profile, [12], which arises in → tion valued 1 in the positive real x-axis and 0 in the the low energy limit ( 0) and renders some of the negative one. Of course a similar result can be ob- manipulations rather slippery. The origin of this singu- + S tained numerically to any degree of accuracy by using larity is the denominator 1 00 that appears in many exponentials. Since, when → 0, S →−1 the expo- a dense enough discrete {sn} sequence. 00 Another way of getting the same result is to use nentials are ill-defined because the series expansions the recipe of [12] first on (18). In this way the mid- in 1/ are. The best way to regularize them is to in- dle exponential disappears, while the first exponential troduce a constant background B-field, [30–32].The →− relevant formulas can be found in [13]. For the pur- is regularized by hand (remember that S00 1as → 0), so we replace S by a parameter s and keep it pose of this Letter we introduce a B field along two 00 space directions, say x and y (our aim is to regularize =−1. Now it is easy to sum over sn. Again we replace the summation by an integration and see immediately the solution in the x direction, but, of course, there is that the space profile becomes the same as (27). no way to avoid involving in the process another space Let us stress that the derivation of the space pro- direction). α = file in the low energy regime we have given above is Let us use the notation x with α 1, 2 to denote far from rigorous. This is due to the very singular na- x, y and let us denote ture of the lump in this limit, first pointed out by [12]. G = ∆δ ,∆= 1 + (2πB)2 (29) A more satisfactory derivation will be provided in the αβ αβ next section after introducing a background B field. the open string metric. As is well known, as far as In summary, the state |Λ is a solution to (5), which lump solutions are concerned, there is an isomorphism represents, in the low energy limit, a one-dimensional of formulas with the ordinary case by which X , S , object with a constant profile that extends from the ori- T are replaced, respectively, by X , S, T , which ex- gin to infinity in the x-direction. Actually the initial plicitly depend on B. One should never forget that the point could be any finite point of the x-axis, and it is latter matrices involve two space directions. We will ˆ not hard to figure out how to construct a configuration denote by |Ξ0 the D0-brane solution in the presence that extend from −∞ to +∞. How should we inter- of the B field. pret these condensate of D0-branes? In the absence of Without writing down all the details, let us see the supersymmetry it is not easy to distinguish between D- significant changes. Let us replace formula (14) by strings and F-strings (see, for instance, [18] for a com- α ˆ α −is pˆα ˆ parison), however in the last section we will provide Ξ0 s = e |Ξ0. (30) some evidence that the one-dimensional solutions of ˆ ˆ ˆ the type |Λ can be interpreted as fundamental strings. It satisfies |Ξ0(s)∗|Ξ0(s)=|Ξ0(s) and ˆ ˆ ˆ ˆ This kind of objects are very well known in string the- Ξ0(s)|Ξ0(s)=Ξ0|Ξ0. Instead of (17) we have ory as classical solutions, [19–23], see also [24–26]. 2∆ 1/2 For the time being let us notice that, due to (24) xˆα = ∞ ∞ πb Λ|Λ= Ξ (sn) Ξ (sm) = Ξ |Ξ . (28) α β 0 0 0 0 x x 2 α† β n,m=1 n=1 × exp − − i √ a x b b 0 It follows that the energy of the solution is infinite. Such an (unnormalized) infinity is a typical property + 1 α† β† | a0 a0 Gαβ Ωb . (31) of fundamental string solutions, see [19]. 2 364 L. Bonora et al. / Physics Letters B 619 (2005) 359–366

Next we have see [14]. Therefore the 1 + S00 denominators in (32) are not dangerous any more. Similarly one can prove ˆα ˆ x Ξ0(s) that in the same limit S → 0. Moreover the - 0n 1/2 Nˆ expansions about these values are well-defined. There- = 2∆ √ πb + S fore the space profile we are interested in is det(1 00) µ µ 1 1 − S00 ∼ − + 2 − 2 × exp − xα − sα xβ − sβ exp (x s) (y) b 1 + S b b 00 αβ × −1 α†S β† | 2i α α β γ † exp an nm,αβ am 0 (35) − √ x − s (1 + S00)αβ S0m γ am 2 b 2|a|−1 with a finite normalization factor and µ = | |+ ∆. 1 α† β† 2 a 1 × exp − a Wnm,αβ a |0, (32) Now one can safely integrate s and obtain the result 2 n m illustrated in Section 3. This also shed light on how + S × where det(1 00) means the determinant of the 2 2 the resulting state couples to the Bµν field. Indeed matrix (1 + S00)αβ and the length of this one-dimensional objects is measured with the open string metric (29), in other words the 1 W = S − S γ S δβ. B-field couples to the string by “stretching” it. nm,αβ nm,αβ n0,α + S 0m 1 00,γ δ (33)

The state we start from, i.e., |Ξ0(s), and the relevant 5. Fundamental strings space profile, are obtained by setting s1 = s and s2 = 0 in the previous formulas. In this section we would like to discuss the prop- Next we have an analog of (19) with C(s, s ) re- erties of the Λ solutions we found in the previous placed by sections. In order to justify the claim we made that they represent fundamental strings, we note that they 1 T (1 − T ) Cˆ(s, s ) =− s2 + s 2 are naturally attached to D-branes on both ends. These 2b 1 + T 00,11 are the two extremal D0-branes in the sn sequence (see ss (1 − T )2 also below). It easy to envisage systems in which such − . + T (34) strings are attached to other D-branes as well. For in- 2b 1 00,11 stance let us pick |Λ as given by (25) with sn > 0 Proceeding in the same way as in Section 3 we can for all n’s. Now let us consider a D24-brane with the prove the analog of Eq. (22). By using the spectral only transverse direction coinciding with the x-axis representation worked out in [29] one can show that and centered at x = 0. The corresponding lump solu- ˆ C picks up a logarithmic singularity unless s = s .Ina tion has been introduced at the end of Section 2 (case similar way one can prove the analog of (24). = | k 24). Let us call it Ξ24 . Due to the particular con- Now let us discuss the properties of | +| figuration chosen, it is easy to prove that Ξ24 Λ is |Ξ ∞ still a solution to (5). This is due to the fact that 24 ˆ ˆ is ∗-orthogonal to the states |Ξ (s ) for all n’s. To |Λ= Ξ0(sn) 0 n = be even more explicit we can study the space profile n 1 of |Ξ +|Λ, assuming the sequence sn to become 1 = 24 in the low energy limit. We refer to (32) with s s dense in the positive x-axis. Using the previous results 2 = and s 0. The fundamental difference between this it is not hard to see that the overall configuration is S formula and (18) is that in the low energy limit 00,αβ a Gaussian centered at x = 0inthex direction (the becomes diagonal and takes on a value different from D24-brane) with an infinite prong attached to it and − 1. More precisely extending along the positive x-axis. The latter has a Gaussian profile in all space directions except x. 2|a|−1 π 2 S → G ,a=− B We remark that the condition s > 0 for all n’s is 00,αβ | |+ αβ b n 2 a 1 V00 + | +| 2 important because Ξ24 Λ is not anymore a pro- L. Bonora et al. / Physics Letters B 619 (2005) 359–366 365

{ } | jector if the sn sequence contains 0, since Ξ0(0) is This is still a solution to (5) and can be interpreted in ∗ | | +++| −− | +−+ not -orthogonal to Ξ24 . This remark tells us that it two ways: either as Λ Λ or as Λ | −+ | is not possible to have solutions representing configu- Λ , up to addition to both of Ξ0 (a bit removed rations in which the string crosses the brane and/or is from the origin). This addition costs the same amount not attached to D-branes by the endpoints: the string of energy in the two cases, an amount that vanishes in has to stop at a brane.1 the continuous limit. Therefore the solution (37) repre- Needless to say it is trivial to generalize the solution sents precisely the exchange property of fundamental | +| of the type Ξ24 Λ to lower-dimensional branes. strings. It is worth pointing out that it is also possible to So far we have considered only straight one- construct string solutions of finite length. It is enough dimensional solutions (in terms of space profiles), or to choose the sequence {sn} to lie between two fixed at most solutions represented by straight lines at right values, say a and b in the x-axis, and then ‘condense’ angles. However, this is an unnecessary limitation. It the sequence between these two points. In the low en- is easy to generalize our construction to any curve in ergy limit the resulting solution shows precisely a flat space. For instance, let us consider two directions in profile for a

1 corresponding charge. When the D-brane decays there It should be noted that, according to Section 5 of [15], one is nothing that prevents the (fundamental) strings at- ∗ | →| ˇ | ˇ ∗| = could -rotate Ξ24 Ξ24 so that Ξ24 Ξ0(0) 0; however this simply means that |Λ has to be rotated too in a similar way into tached to it from decaying themselves. However in the | ˇ | ˇ presence of a background E-ﬁeld, the latter are ex- Λ in order for the corresponding string to end on Ξ24 .Ifwedo | ˇ cited by the coupling with the E-ﬁeld and persist (or, not do so, we simply obtain a string that may cross the Ξ24 brane, but still ends on the two terminal D0-branes. at least, persist longer than the other unstable objects). 366 L. Bonora et al. / Physics Letters B 619 (2005) 359–366

This phenomenon is described in [23] in effective ﬁeld [12] G. Moore, W. Taylor, JHEP 0201 (2002) 004, hep-th/0111069. theory and BCFT language (see also [29]). [13] L. Bonora, D. Mamone, M. Salizzoni, Nucl. Phys. B 630 (2002) 163, hep-th/0201060. [14] L. Bonora, D. Mamone, M. Salizzoni, JHEP 0204 (2002) 020, hep-th/0203188; Acknowledgements L. Bonora, D. Mamone, M. Salizzoni, JHEP 0301 (2003) 013, hep-th/0207044. This research was supported by the Italian MIUR [15] L. Rastelli, A. Sen, B. Zwiebach, JHEP 0111 (2001) 035, hep- under the program “Teoria dei Campi, Superstringhe th/0105058. e Gravità” and by CAPES-Brasil as far as R.J.S.S. is [16] L. Rastelli, A. Sen, B. Zwiebach, JHEP 0203 (2002) 029, hep- th/0111281. concerned. [17] D.M. Belov, hep-th/0204164. [18] J.H. Schwarz, Phys. Lett. B 360 (1995) 13, hep-th/9508143; J.H. Schwarz, Phys. Lett. B 364 (1995) 252, Erratum. References [19] C.G. Callan, J.M. Maldacena, Nucl. Phys. B 513 (1998) 198, hep-th/9708147. [1] L. Rastelli, A. Sen, B. Zwiebach, hep-th/0106010. [20] G.W. Gibbons, Nucl. Phys. B 514 (1998) 603, hep-th/9709027. [2] A. Sen, hep-th/0308068. [21] P.S. Howe, N.D. Lambert, P.C. West, Nucl. Phys. B 515 (1998) [3] L. Bonora, C. Maccaferri, R.J. Scherer Santos, D.D. Tolla, hep- 203, hep-th/9709014. th/0409063. [22] K. Dasgupta, S. Mukhi, Phys. Lett. B 423 (1998) 261, hep- [4] A. Sen, JHEP 0204 (2002) 048, hep-th/0203211; th/9711094. A. Sen, JHEP 0207 (2002) 065, hep-th/0203265; [23] P. Mukhopadhyay, A. Sen, JHEP 0211 (2002) 047, hep- A. Sen, JHEP 0210 (2002) 003, hep-th/0207105; th/0208142; A. Sen, Int. J. Mod. Phys. A 18 (2003) 4869, hep-th/0209122; A. Sen, hep-th/0305011. A. Sen, JHEP 0405 (2004) 076, hep-th/0402157; [24] A. Dabholkar, G. Gibbons, J.A. Harvey, F. Ruiz Ruiz, Nucl. A. Sen, hep-th/0410103. Phys. B 340 (1990) 33. [5] D.J. Gross, A. Jevicki, Nucl. Phys. B 283 (1987) 1; [25] J.A. Harvey, A. Strominger, Nucl. Phys. B 449 (1995) 535, E. Cremmer, A. Schwimmer, C. Thorn, Phys. Lett. B 179 hep-th/9504047; (1986) 57. J.A. Harvey, A. Strominger, Nucl. Phys. B 458 (1996) 456, [6] N. Ohta, Phys. Rev. D 34 (1986) 3785; Erratum. N. Ohta, Phys. Rev. D 35 (1987) 2627, Erratum. [26] A. Dabholkar, Phys. Lett. B 357 (1995) 307, hep-th/9506160. [7] L. Bonora, C. Maccaferri, D. Mamone, M. Salizzoni, hep- [27] A. Strominger, Phys. Lett. B 383 (1996) 44, hep-th/9512059. th/0304270. [28] E. Witten, Nucl. Phys. B 460 (1996) 335, hep-th/9510135. [8] A. Leclair, M.E. Peskin, C.R. Preitschopf, Nucl. Phys. B 317 [29] C. Maccaferri, R.J. Scherer Santos, D.D. Tolla, hep-th/ (1989) 411. 0501011. [9] L. Rastelli, A. Sen, B. Zwiebach, Adv. Theor. Math. Phys. 5 [30] E. Witten, hep-th/0006071. (2002) 393, hep-th/0102112. [31] M. Schnabl, JHEP 0011 (2000) 031, hep-th/0010034. [10] L. Bonora, C. Maccaferri, P. Prester, JHEP 0401 (2004) 038, [32] F. Sugino, JHEP 0003 (2000) 017, hep-th/9912254; hep-th/0311198. T. Kawano, T. Takahashi, Prog. Theor. Phys. 104 (2000) 459, [11] L. Bonora, C. Maccaferri, P. Prester, Phys. Rev. D 71 (2005) hep-th/9912274. 026003, hep-th/0404154. Physics Letters B 619 (2005) 367–376 www.elsevier.com/locate/physletb

An adiabatic approximation to the path integral for relativistic fermionic ﬁelds

J.L. Cortés a, J. Gamboa b,S.Lepeb,c, J. Lopez-Sarrión a,b

a Departamento de Física Teórica, Universidad de Zaragoza, Zaragoza 50009, Spain b Departamento de Física, Universidad de Santiago de Chile, Casilla 307, Santiago 2, Chile c Instituto Física, Universidad Católica de Valparaíso, Casilla 4059, Valparaíso, Chile Received 22 March 2005; accepted 20 April 2005 Available online 28 April 2005 Editor: M. Cveticˇ

Abstract A new approach to the path integral over fermionic fields, based on the extension of a reformulation of the adiabatic approximation to some quantum-mechanical systems, is presented. A novel non-analytic contribution to the effective fermionic action for a fermion field coupled to a non-Abelian vector field is identified. The possible interpretation of this contribution as a violation of the decoupling theorem in quantum field theory (QFT) is discussed. The generalization of the approach to the case of finite temperature and density suggests the possibility to apply it to the understanding of non-perturbative properties in QFT and their dependence on temperature and density.  2005 Elsevier B.V. All rights reserved.

1. Introduction

The adiabatic approximation is one of the most important methods going beyond perturbation theory in quantum mechanics. In QFT, the necessity of non-perturbative methods is clear in many cases (low energy limit of asymptotically free theories, high energy limit of infrared safe theories). Unfortunately, the attempts to translate the adiabatic approximation to QFT have been very limited and the main results are the identification of Wess– Zumino terms and anomalies as geometric phases [1]. The complexity of QFT (a quantum-mechanical system with infinite degrees of freedom) has been an obstacle to the possible use of the adiabatic approximation as the starting point to an alternative to the perturbative expansion. In this Letter we attempt to give a first step in this direction by considering the path integral over a relativistic fermionic field system. In the next section we take as a starting point

E-mail addresses: [email protected] (J.L. Cortés), [email protected] (J. Gamboa), [email protected] (S. Lepe), [email protected] (J. Lopez-Sarrión).

0370-2693/$ – see front matter  2005 Elsevier B.V. All rights reserved. doi:10.1016/j.physletb.2005.04.044 368 J.L. Cortés et al. / Physics Letters B 619 (2005) 367–376 a very simple quantum-mechanical system, a spin coupled to a time-dependent magnetic field, and the adiabatic approximation is reformulated in an appropriate way to be generalized to other systems. In Section 3, we point out the difficulties found when one tries a generalization of this reformulation in QFT. In Section 4, a possible way to circumvent these problems is presented and our proposal is applied to the path integral over a fermion field in the fundamental representation of SU(2) coupled to a vector field in the adjoint representation. The leading term in the adiabatic approximation is determined and its possible relation with non-perturbative properties of the theory are discussed at a qualitative level. In Section 5, a generalization of the results at finite density and temperature is obtained and finally in Section 6 a discussion of possible physical realizations of the adiabatic approximation in QFT is presented.

2. A quantum-mechanical example

In order to formulate the adiabatic approximation in QFT, let us first discuss it at the level of quantum mechanics in the most simple non-trivial example, namely, a half-integer spin (j) coupled to an external magnetic field (B(t) ) varying periodically in time (B(T) = B( −T)). Let us review the well-known solution of this problem [2]. One has a Hamiltonian = · † H B(t) Jαβ aαaβ , (1) α,β where α, β =−j,−j + 1,...,j − 1,j and a, a† are operators satisfying the (anti)commutation relations † = aα,aβ δαβ . (2) We want to determine the probability amplitude that the system remains in the ground state (|0) during time evolution, i.e., the matrix element 0(T )|0(−T). In order to do this calculation it is convenient to use at each time, the direction of the magnetic field as the spin quantization axis to rewrite the Hamiltonian as j = † H(t) mB(t)am(t)am(t). (3) m=−j The ground state of the system, |0(t) is the state satisfying the conditions | = † | = am(t) 0(t) 0form>0,am(t) 0(t) 0form<0. (4) In the adiabatic approximation one has two contributions to the matrix element 0(T )|0(−T), one due to the energy of the ground state (j + 1/2)2 E (t) = mB(t) =− B(t), (5) 0 2 m<0 and a contribution due to the time evolution of the phase of the ground state |0(t)

T 0(t)|∇ H |k(t)∧k(t)|∇ H |0(t) dt 0(t)|i∂ |0(t)=−Im dS B B . (6) t − 2 = (Ek E0) −T k 0 Stoke’s theorem has been applied to rewrite the time integral as a surface integral in magnetic ﬁeld space and the matrix elements of the gradient of the Hamiltonian can be directly read from the matrix representation of the angular momentum operator. J.L. Cortés et al. / Physics Letters B 619 (2005) 367–376 369

After some straightforward algebra one ﬁnds

T (j + 1/2)2 dt 0(t)|i∂ |0(t)= Ω[Bˆ ], (7) t 2 −T where B Ω[Bˆ ]= dS · , (8) B3 is the solid angle that B subtends. This is the standard derivation of the adiabatic approximation including Berry’s phase (7) [2]. Let us now see how these results can be rederived in an alternative formulation based on a Grassmann path integral representation of the evolution, which will be useful in the discussion of QFT in Section 4. One can represent the operators a, a† by Grassmann variables ψ, ψ¯ and the probability amplitude that the system remains in the ground state by a Euclidean Grassmann path integral, ¯ −S dψ dψe − [ ] lim 0(T )|0(−T)= = e Γ B , (9) T →∞ ¯ −S| dψ dψe B=0 where the Euclidean action for the fermionic system is ¯ S = dτ ψ(∂τ + J · B)ψ. (10)

In order to simplify the action, one introduces new Grassmann variables cm, c¯m through the expansion j j = ¯ = † ¯ ψ cmfm, ψ fmcm, (11) m=−j m=−j where fm(t) are the eigenspinors deﬁned by J · Bfm = mBfm. (12) The action as a function of the new variables is given by, S = dτ c¯m(∂τ + mB)cm + c¯miAm,m cm , (13) m m,m † where iAm,m = fm∂τ fm . In the adiabatic approximation, one neglects off-diagonal terms (m = m ) and the path integral becomes a product of independent integrals

j β/2 −Γ (ad)[B] e = dcm dc¯m exp − dτ c¯m(∂τ + iAm + mB)cm , (14) =− m j −β/2 with Am = Am,m and the limits on the Euclidean time (τ = it) incorporate finite temperature (1/β) effects in the imaginary time formalism [3]. The integral on each pair of variables (cm, c¯m), is a standard quantum-mechanical determinant [4] β det ∂ + iA (τ) + mB(τ) = N cosh (mB˜ + iA˜ ) , (15) τ m 2 m 370 J.L. Cortés et al. / Physics Letters B 619 (2005) 367–376 where N is an infinite constant that will cancel in the ratio of Grassmann integrals in (9) and we have introduced the notation β/2 1 f˜ = dτ f(τ). (16) β −β/2 In the zero temperature limit the quantum-mechanical determinants (15) become exponentials and the effective fermionic action in the adiabatic approximation takes the simple form i m −Γ (ad)[B]=− |m| dt B(t)+ dt A (t) , (17) 2 |m| m m where the first term can be recognized as the dynamical phase −i dtE0(t) with E0 the energy of the ground state of the quantum-mechanical system (5). The second term in (17) reproduces Berry’s phase as one can show by using once more Stokes theorem and the definition of fm in (12), i.e., A = † =− [ ˆ ] dt i m(t) dt fm∂t fm imΩ B , (18) with Ω[Bˆ ] the solid angle that the magnetic field subtends in its evolution. The adiabatic approximation to this simple quantum-mechanical system in the Grassmann path integral representation, will reappear as an ingredient in some approximation to a QFT with fermionic fields as we will see later. Also this reformulation of the adiabatic approximation is interesting because it allows to go beyond the zero temperature limit by using the quantum-mechanical determinants in (15).

3. Formal direct approach and its difﬁculties

The purpose of this section is to introduce a direct extension to QFT of the reformulation of the previous section. Before doing that we will show the problems of a direct implementation of the adiabatic approximation. The most natural way to formulate the adiabatic approximation and the related Born–Oppenheimer approach in QFT is based on the use of the Schrödinger representation, where the wave function of quantum mechanics is replaced by a functional in the space of field configurations. After this replacement, all the standard results of the adiabatic quantum-mechanical expansion can be applied directly in QFT [5]. This formulation gives a new perspective of the anomaly in chiral gauge theories which appears as a geometric phase in the space of gauge field configurations related to the gauge non-invariance of the phase of the fermionic Fock states [6]. Unfortunately, a quantum-mechanical system with infinite degrees of freedom is too complicated to go beyond the study of a few topological properties and the adiabatic approximation remains as a reformulation of the theory at a formal level. An alternative way to implement the adiabatic formulation is based on a direct use of the reformulation of the quantum-mechanical example of the previous section in a QFT system. For definiteness, let us consider the action in Euclidean space for a fermionic field Ψ coupled to a vector field Aµ, D ¯ S = d x Ψγµ(∂µ + ieAµ)Ψ . (19)

Using the decomposition = ΨR ¯ = † † Ψ , Ψ ΨL ΨR , (20) ΨL J.L. Cortés et al. / Physics Letters B 619 (2005) 367–376 371 for the fermionic ﬁeld in a representation where all gamma matrices are off-diagonal, one has S = 4 † + + † + + † · − ∇+ − † · − ∇+ d x ΨL(∂4 ieA4)ΨL ΨR(∂4 ieA4)ΨR ΨLσ ( i eA)ΨL ΨRσ ( i eA)ΨR . (21) Following the steps of the previous section, we introduce the eigenfunctions Φn(x) σ · (−i∇+eA) Φn(x) = nΦn(x). (22)

These eigenfunctions and the eigenvalues n are, in fact, functionals of the vector ﬁeld at a given time and a more precise notation for them is Φn[A(τ)](x) and n[A(τ)]. Next step is to use the decomposition of the fermionic ﬁelds in terms of the eigenfunctions in (22), = = ΨL(x,τ) cLn (τ)Φn A(τ) (x), ΨR(x,τ) cRn (τ)Φn A(τ) (x), (23) n n Ψ †(x,τ) = c† (τ)Φ† A(τ) (x), Ψ †(x,τ) = c† (τ)Φ† A(τ) (x). (24) L Ln n R Rn n n n

Using the orthogonality of the eigenfunctions Φn, the action takes a compact form in terms of the (Grassmann) coefﬁcients cL, cR S = dτ c† (τ)∂ c (τ) + c† (τ) c (τ) + ic† (τ)A c (τ) Ln τ Ln Ln n Ln Ln n Ln n + c† (τ)∂ c (τ) − c† (τ) c (τ) + ic† (τ)A c (τ) Rn τ Rn Rn n Rn Rn n Rn n + i c† (τ)A c (τ) + c† (τ)A c (τ) , (25) Ln nm Lm Rn nm Rm n=m where we have introduced the connection An, A = † − + n A(τ) dxΦn A(τ) (x)( i∂τ eA4)Φn A(τ) (x), (26) and Anm for n = m, A = † − + nm A(τ) dxΦn A(τ) (x)( i∂τ eA4)Φm A(τ) (x). (27)

In the adiabatic approximation one neglects the off-diagonal terms (n = m) and then one has infinite copies of quantum-mechanical systems each of them similar to the one discussed in Section 2. However, for a general vector field configuration the spectrum ( n) will be continuous and the difference of energy levels can be arbitrarily small rendering the adiabatic expansion out of control. Besides that, the eigenvalues n and eigenfunctionals Φn are not known, except for very special choices of the vector field, and the formulation remains once more at a formal level.

4. A new approach and its application to an SU(2) gauge theory

The only way we have found to use the reformulation of the adiabatic approximation to get a useful expansion in QFT, is based on the introduction of variables independently at each point. In order to do that, one has to select an operator at each point and use its eigenfunctions in the expansion of some of the fields at this point. We can then identify two ingredients in the formulation of the new approach. The first one is a separation of the fields in two sets, one of them corresponding to the spin degrees of freedom of the quantum-mechanical example. The second 372 J.L. Cortés et al. / Physics Letters B 619 (2005) 367–376 one is the choice of the operator at each point whose eigenfunctions are used to introduce new variables for the spin-like fields. Several requirements constraint the ambiguities in this two ingredients. The action should be quadratic in the spin-like fields either directly or after the introduction of appropriate auxiliary fields. The expression for some of the terms in the action as a function of the new variables should be as simple as possible. The contribution from the remaining terms in the action (including the space derivatives) as well as the corrections to the “adiabatic” approximation (off-diagonal contributions in the new variables) should be small. The search of a good set of fields and local operators defining the new variables has to be done, however, case by case. The usefulness of the approach will be established if one finds examples where all these requirements are satisfied yielding to a dominant contribution with interesting results. In order to illustrate our approach let us consider the Lagrangean of a fermionic system coupled to a vector field L = ¯ µ − ¯ µ a a − ¯ Ψiγ ∂µΨ gΨγ AµT Ψ mΨΨ, (28) where T a are the generators of SU(2) acting on the fermionic fields in the fundamental representation. It is convenient to use the Dirac representation for the γ matrices with I 0 γ 0 = (29) 0 −I and the decomposition in bispinors of the Dirac field ϕ Ψ = . (30) χ

We neglect for a moment, the terms proportional to the space components of the vector field (Aa) and to space derivatives of the fermionic field. The remaining terms take a simple form if we use the eigenvectors f± of the â a operator A0T â a 1 A T f± =± f±, (31) 0 2 a = â â â = where we have used the parametrization A0 A0A0 with a A0A0 1. With these eigenvectors, one can introduce the new fermionic variables ϕn,i , χn,i ϕ = ϕn,ifn,i, (32) n=± = i1,2 χ = χn,i fn,i, (33) n i=1,2 where the bispinors fn,i are given by fn 0 fn,1 = ,fn,2 = . (34) 0 fn Note that the new fermionic variables have been introduced independently at each point in space. We then have Grassmann variables ϕn,i , χn,i at each space–time point. In the new representation for the fermionic variables, one has † = † + † − † + † A Ψ ∂τ Ψ ϕn,i∂τ ϕn,i χn,i ∂τ χn,i ϕn,iϕn ,i χn,i χn ,i i n,n (35) n,i n,n ,i † with iAn,n = fn ∂τ fn and τ = it is the Euclidean time. J.L. Cortés et al. / Physics Letters B 619 (2005) 367–376 373

For the interaction term one has − † a a =− † + † gΨ A0T Ψ gnA0 ϕn,iϕn,i χn,i χn,i , (36) n i =±g with g± 2 . The Lagrangian density before including space derivatives and the space components of the vector field as a function of the new variables is L a = † − − + A + † − + + A A0 ϕn,i(∂τ gnA0 m i n)ϕn,i χn,i(∂τ gnA0 m i n)χn,i n,i + † + † A ϕn,iϕn ,i χn,iχn ,i i n,n . (37) n=n ,i a We then have at each point, a generalization of the quantum-mechanical example in Section 2 with A0 playing the role of the magnetic field B (the direction in the internal SU(2) space is the analog to the orientation of the magnetic field) and four instead of one spin variable (corresponding to the four components of the Dirac spinor). Then, the energies associated to the new variables are

E(ϕn,i) = gnA0 + m, E(χn,i) = gnA0 − m. (38) The corrections to the adiabatic approximation due to the off-diagonal terms (n = n )in(37) involve levels separated by a gap gA0. Then, the adiabatic approximation will be justified when the product of the coupling and ˙â the time component of the vector field is large compared to the time derivative of its orientation (A0). With respect to the consistency of treating the space derivatives and space components of the vector field as a correction, we have a sum of two contributions † · − ∇+ a a = † − ∇ + † − ∇ † Ψ α i gA T Ψ ϕn,i( i )χn,i χn,i( i )ϕn,i fn,iσfn,i n,i,i + † + † † · − ∇+ a a ϕn,iχn ,i χn,iϕn ,i fn,iσ i gA T fn ,i . (39) n,n ,i,i

The first term is a non-diagonal contribution between energy levels separated by 2m. Since the eigenvectors fn depend only on the direction in the internal space of A0, then the corrections due to these terms will be proportional ∇ â + to A0/2m. The second term has non-diagonal contributions between energy levels separated by 2m,2m gA0 | − | ∇ â a or 2m gA0 and there are two types of terms, ones proportional to A0 and the others proportional to gA . From these simple arguments one can see what are the conditions on the vector field and the fermion mass in order to treat (39) as a small correction to (37). It should be noted that considering spacial derivatives as corrections does not mean that we are making use of the usual derivative expansion method [7]. Then, in the approximation where we keep only the terms diagonal in (37) and using the result (17) and (18) we have a new approximation to the effective fermionic action, g Γ = d3x sgn(g A + m) dτ (g A + m) + n iΩ Aâ ad n 0 n 0 g 0 n g + d3x sgn(g A − m) dτ (g A − m) + n iΩ Aâ , (40) n 0 n 0 g 0 n [ â] â where Ω A0 is the solid angle that A0 subtends in internal space in its time evolution. We can distinguish two different regions. In the weak coupling region (gA0 2m) there is a cancellation of contributions from the two sets of new variables ϕn,i and χn,i and Γad reduces to a constant. On the other hand, in 374 J.L. Cortés et al. / Physics Letters B 619 (2005) 367–376 the strong coupling region (gA0 2m) the effective fermionic action in the adiabatic approximation is (s) = 3 + â Γad 2g dτ d xA0 2iΩ A0 . (41)

The presence of a contribution in the effective fermionic action which disappears in the weak coupling region, suggest a possible relation between the presence of such contribution and the non-pertubative properties of the theory. One should note that the limit m →∞for fixed gA0 corresponds to the weak coupling region, where the new non-perturbative contribution disappears as expected from the decoupling theorem [8]. On the other hand, for an arbitrarily large mass, one can be in the strong coupling limit if one has a sufficiently large coupling g and/or vector field A0 and then the non-perturbative trace of the fermionic system remains.

5. Finite density and temperature effects

The generalization of the adiabatic approximation to the SU(2) gauge theory in the case of ﬁnite density is trivial. All one has to do is to include the chemical potential (µ) through a term µΨγ¯ 0Ψ in the Lagrangian which modiﬁes the energies associated to the new variables

E(ϕn,i) = gnA0 + m + µ, E(χn,i) = gnA0 − m + µ. (42)

The energy differences are not modified and all the estimates of the corrections due to space derivatives and the space components of the vector field are not changed. The effective fermionic action in the adiabatic approximation is now g Γ (µ) = dx sgn(g A + m + µ) dt (g A + m + µ) + n iΩ Aâ ad n 0 n 0 g 0 n g + dx sgn(g A − m + µ) dt (g A − m + µ) + n iΩ Aâ . (43) n 0 n 0 g 0 n We consider µ>0 for definiteness. In this case, we can consider three different regions. The first one corresponds to gA0 < 2|m − µ|, where once more there is a cancellation of contributions and one has a trivial adiabatic effective action as in the weak coupling region of the case without chemical potential. There is also an analog of the strong coupling region where gA0 > 2|m + µ| and the adiabatic effective action is (41). Finally, there is an intermediate region, 2|m − µ|

Once more, we can discuss the decoupling of the fermion degrees of freedom. There are once more cases with arbitrarily large mass and density where for sufficiently large gA0 a fermionic signal remains. On the other hand, for fixed mass and gA0 the fermion decouples in the limit µ →∞. This shows that the non-perturbative properties of the non-Abelian gauge theory related to the presence of a non-trivial adiabatic effective action disappear in the infinite density limit. It is also very easy to generalize the adiabatic approximation to the case of finite temperature. As we have seen in Section 2, the modification of the fermionic integral for each quantum-mechanical system is very simple and J.L. Cortés et al. / Physics Letters B 619 (2005) 367–376 375 the effective fermionic action in the adiabatic approximation at finite temperature (and also including a chemical potential) is given by

β/2 − = + + + â Γad(µ, β) dx ln cosh dτ (gnA0 m µ) iΩ A0 n −β/2 β/2 + − + + â dx ln cosh dτ (gnA0 m µ) iΩ A0 , (45) n −β/2 where β is the inverse of the temperature. If one takes the high temperature limit, β → 0 for fixed (µ, m, gA0), then the adiabatic effective action is proportional to β2 and then the new non-perturbative contribution disappears.

6. Discussion

The result for the effective fermionic action in the adiabatic approximation (40) is not gauge-invariant neither Lorentz-invariant. This should not be a surprise. A similar situation happens in the simplest quantum-mechanical example of the spin coupled to a magnetic field [9], where each term in the adiabatic expansion is not separately rotational invariant, the variation of each term being canceled by that of subsequent terms in the expansion. This makes difficult to understand how one can find a physical situation where the adiabatic approximation to the effective fermionic action (40) can give a consistent first-order approximation to a relativistic QFT with fermionic fields. a One possibility is to consider a gauge field theory, identifying the vector field Aµ with the gauge field of the SU(2) gauge theory. In that case one should add a term in the action depending on the dynamical gauge field. The Lorentz and gauge invariance of the theory is inconsistent with the adiabatic approximation; in fact, this could have been anticipated because by applying an arbitrary gauge transformation on a vector field whose space components are sufficiently small and whose time component orientation in internal space varies sufficiently slowly to justify the approximation one gets a gauge field configuration where the approximation is not justified. The only way to look for a consistent realization of the adiabatic approximation in a gauge field theory is by considering the vector field as the gauge field satisfying an appropriate non-covariant gauge fixing condition. It is not possible with our present understanding of gauge field theories at the non-perturbative level to check whether there is a situation were a gauge fixing condition can be found such that the relevant gauge field configurations in that gauge satisfy all the conditions to justify the adiabatic approximation to the effective fermionic action considered in this work. All one can do is to explore the consequences of the assumption that this is the case. In all cases, where the effective action 3 in the adiabatic approximation is non-trivial, one has a term proportional to dτ d xA 0 and then, in order to have a finite action, the gauge field A0 should be concentrated in a finite region in space–time. This fact, together with the disappearance of the new contribution in the high temperature or high density limits suggests a relation of the presence of a non-trivial adiabatic approximation and the confinement in the non-Abelian gauge theory. Note that if this relation holds, then the confinement at low energies would be due to the presence of heavy quarks which do not decouple as one would naively expect due to non-perturbative effects. Other possibilities for a physical realization of the adiabatic approximation to the effective fermionic action could correspond to a situation where the vector field is not a dynamical field but a background field which parametrizes some of the non-perturbative properties of the vacuum of QFT or an auxiliary field introduced to linearize a fermion self-coupling. In those cases the Lorentz and gauge non-invariance of the result inherent to the adiabatic approximation should be related with the details of the introduction of the background or auxiliary field. 376 J.L. Cortés et al. / Physics Letters B 619 (2005) 367–376

Acknowledgements

This work has been partially supported by the grants 1050114, 7010516 Fondecyt, Chile, DI-PUCV No. 123.771/2004 (S.L.) by AECI (Programa de Cooperación con Iberoamérica) and by MCYT (Spain), grant FPA2003-02948. J.L.S. thanks MECESUP-0108 and the Spanish Ministerio de Educación y Cultura for support.

References

[1] The literature is very extensive and some important references are: P. Nelson, L. Alvarez-Gaumé, Commun. Math. Phys. 99 (1985) 103; E. Witten, Nucl. Phys. B 223 (1983) 422; M. Stone, Phys. Rev. D 33 (1986) 1191; Others important references are reprinted in: A. Shapere, F. Wilczek, in: Geometric Phases in Physics, World Scientiﬁc, Singapore, 1989. [2] M.V. Berry, Proc. R. Soc. London A 392 (1984) 45. [3] See, e.g., J. Kapusta, Finite-Temperature Field Theory, Cambridge Univ. Press, Cambridge, 1989. [4] G. Dunne, K. Lee, C. Lu, Phys. Rev. Lett. 78 (1997) 3434. [5] For a ﬁrst attempt to formulate gauge theory along these lines see: J.P. Ralston, Phys. Rev. D 51 (1995) 2018. [6] A. Niemi, G. Semenoff, Phys. Rev. Lett. 55 (1985) 927; A. Niemi, G. Semenoff, Phys. Rev. Lett. 56 (1986) 1019; H. Sonoda, Phys. Lett. B 156 (1985) 220; H. Sonoda, Nucl. Phys. B 266 (1986) 410. [7] J. Goldstone, F. Wilczek, Phys. Rev. Lett. 47 (1981) 986; R. Mackenzie, F. Wilczek, Phys. Rev. D 30 (1984) 2194; R. Mackenzie, F. Wilczek, Phys. Rev. D 30 (1984) 2260. [8] T. Appelquist, J. Carazzone, Phys. Rev. D 11 (1975) 2856. [9] R. Jackiw, Int. J. Mod. Phys. A 3 (1988) 285. Physics Letters B 619 (2005) 377–386 www.elsevier.com/locate/physletb

Weak gauge-invariance of dimension two condensate in Yang–Mills theory

Kei-Ichi Kondo

Department of Physics, Faculty of Science, Chiba University, Chiba 263-8522, Japan Received 13 April 2005; received in revised form 24 May 2005; accepted 2 June 2005 Available online 13 June 2005 Editor: T. Yanagida

Abstract We give a formal proof that the spacetime average of the vacuum condensate of mass dimension two, i.e., the vacuum expec- Ꮽ2 tation value of the squared potential µ, is gauge invariant in the weak sense that it is independent of the gauge-fixing condition adopted in quantizing the Yang–Mills theory. This is shown at least for the small deformation from the generalized Lorentz and the modified maximal Abelian gauge in the naive continuum formulation neglecting Gribov copies. This suggests that the numerical value of the condensate could be the same no matter what gauge-fixing conditions for choosing the representative from the gauge orbit are adopted to measure it. Finally, we discuss how this argument should be modified when the Gribov copies exist.  2005 Elsevier B.V. All rights reserved.

PACS: 12.38.Aw; 12.38.Lg

Keywords: Vacuum condensation; Gauge invariance; Gauge ﬁxing; Yang–Mills theory; Gribov copy

1. Introduction

In the gauge theory, only the gauge invariant quantities are observable, since gauge noninvariant quantities change their values by the gauge transformation and cannot take deﬁnite values. The dimension-three quark– ¯ µν antiquark bilinear operator ψ(x)ψ(x) and the dimension-four gluon operator tr(Ᏺµν(x)Ᏺ (x)) are typical gauge- invariant operators in QCD. They are local operators deﬁned on a spacetime point. We wish to discuss how the gauge noninvariant operator may be gauge invariant. If we do not restrict to the local operator, we can enlarge the candidates of the gauge invariant operator. In this Letter, we focus on composite operators of mass dimension two, since they were taken up by many papers from the analytical and numerical view-

E-mail address: [email protected] (K.-I. Kondo).

2 points, see, e.g., [1–10]. For example, we know that the squared potential tr(Ꮽµ(x) ) of Yang–Mills theory is gauge 4 Ꮽω 2 variant. However, the minimized squared potential minω d x tr( µ(x) ) with respect to the gauge transforma- 2 tion ω in Euclidean space is clearly gauge invariant [5], since tr(Ꮽµ(x) ) cannot be negative and has a nonnegative 4 Ꮽω 2 minimum and hence minω d x tr( µ(x) ) no longer change the value under the gauge transformation. Thus, the composite operator of mass dimension two can be gauge invariant by this procedure in classical gauge theory. In quantum theory, however, we cannot measure the value of the operator itself and can measure only the vacuum expectation value of the operator. In this sense, the gauge invariance of the operator stated in the above (strong gauge invariance in the operator level) is too strong. If the gauge theory is written in terms of the gauge potential, we must fix the gauge in quantizing the gauge theory and the Becchi–Rouet–Stora–Tyutin (BRST) symmetry in quantum gauge theory plays a role of the gauge symmetry of classical gauge theory. Therefore, we can check whether the (elementary or composite) operator is gauge invariant or not, respectively, by examining whether or not the vacuum expectation value of the operator keeps the value under deforming gauge fixing condition. It is the weak gauge invariance (the gauge-fixing-condition independence) that we wish to discuss in this Letter. A first step in this direction shown in [11] is a fact that the spacetime average of the mixed gluon–ghost condensate of mass dimension two,

− 1 ᏻ =V 1 d4x tr Ꮽ (x) · Ꮽµ(x) − λiᏯ(x) · Ꮿ¯ (x) , (1) K G/H 2 µ V is BRST invariant in the generalized Lorentz gauge and the modiﬁed maximal Abelian gauge, although the operator 1 Ꮽ 2 ᏻ tr( 2 µ(x) ) is gauge variant. This claim is based on the on-shell BRST invariance of the operator K .Hereitis essential to take both the spacetime average and the vacuum expectation value. Recently, Slavnov [12] has shown that the spacetime average of the dimension two condensate,

− 1 1 V 1 d4x tr Ꮽ (x)2 := tr Ꮽ2 , (2) 2 µ 2 µ V is gauge invariant by using a noncommutative field theory technique under three hypotheses. Subsequently, he 1 2 has shown [13] that the averaged condensate 2 Aµ in the Abelian gauge theory is gauge independent, i.e., gauge-fixing-parameter independent without using the noncommutative technique and in the standard commutative formalism, i.e., the averaged condensate does not change at least in the Abelian gauge theory when the gauge- fixing-parameter is changed. 1 Ꮽ 2 In view of these, we expect that the tr( 2 µ(x) ) can become gauge invariant only after taking both the spacetime V−1 4 1 Ꮽ 2 average and the vacuum expectation value, i.e., V d x tr( 2 µ(x) ) . Note that either the spacetime average, V−1 4 1 Ꮽ 2 1 Ꮽ 2 V d x tr( 2 µ(x) ), or the vacuum expectation value, tr( 2 µ(x) ) , is not expected to be gauge invariant. What we wish to discuss in this paper is the gauge-fixing-condition independence of the dimension two condensate, rather than the gauge-fixing-parameter independence, in the non-Abelian gauge theory in the standard commutative Yang–Mills theory formalism. In this Letter, we give a formal proof that the spacetime average of a vacuum condensate of mass dimension two ᏻK is gauge invariant in the weak sense that it is independent of the gauge-fixing condition adopted in quantizing the Yang–Mills theory, at least for the small deformation from the generalized Lorentz and the modified maximal Abelian gauge when we neglect Gribov copies. In the final section, we examine if the statement holds by restricting the functional integral to the Gribov region or the fundamental modular region, when the Gribov copies exist. K.-I. Kondo / Physics Letters B 619 (2005) 377–386 379

2. Gauge-ﬁxing-condition independence

For our purpose, it is convenient to adopt the quantization method based on the functional integration. The vacuum expectation value of the operator ᏻ is defined by the functional integral − + |ᏻ| = 1 iSYM iSGF+FP ᏻ 0 0 ZYM dµYM e , (1) with the integration measure dµYM given by ¯ dµYM =[dᏭµ][dᏺ][dᏯ][dᏯ], (2) ¯ where Ꮽµ, ᏺ, Ꮿ, Ꮿ are respectively the gauge field, the Nakanishi–Lautrup (NL) auxiliary field, the Faddeev– Popov (FP) ghost field and antighost field. ZYM guarantees the normalization 0|1|0=1. Here, SYM is the gauge- invariant Yang–Mills action, while SGF+FP is the sum of the gauge-fixing (GF) and the associated FP ghost terms. The BRST transformation δB is defined so that SYM is BRST invariant. Then SGF+FP is also BRST invariant, 2 = since it is written in the BRST-exact form using the nilpotent BRST transformation, i.e., δB 0. Note that the measure [dᏭµ] is gauge invariant and dµYM is BRST invariant. For the gauge-fixing condition F [Ꮽ]=0, we consider the BRST-exact SGF+FP term which has the form, 4 −1 ¯ λ ¯ S + = d xi δ Ꮿ(x) · F Ꮽ(x) + G ᏺ(x), Ꮿ(x), Ꮿ(x) , (3) GF FP B 2 where λ is the gauge-fixing parameter. In the usual covariant Lorentz gauge, we see

µ ¯ F [Ꮽ]=∂ Ꮽµ,G[ᏺ, Ꮿ, Ꮿ]=ᏺ. (4)

Now we introduce an operation δF of changing the gauge ﬁxing condition F [Ꮽ]=0(λ-independent part) inﬁnitesimally. We apply this operation to the vacuum expectation value 0|ᏻ|0 of an elementary or composite operator ᏻ. Then we obtain 4 ¯ 4 ¯ δF 0|ᏻ|0=0|ᏻ d y δB Ꮿ(y) · δF F Ꮽ(y) |0−0|ᏻ|00| d y δB Ꮿ(y) · δF F Ꮽ(y) |0 4 ¯ =0|ᏻ d y δB Ꮿ(y) · δF F Ꮽ(y) |0, (5) since the BRST transformation is generated by the BRST charge QB according to

δB (∗) =[iQB , ∗]∓, (6) and the vacuum is annihilated by the BRST charge (Kugo–Ojima subsidiary condition)

QB |0=0. (7) Taking into account an identity 4 ¯ 0|δB ᏻ d y Ꮿ(y) · δF F Ꮽ(y) |0=0, (8) following from (6) and (7), we obtain a basic relationship, 4 ¯ δF 0|ᏻ|0=∓0|δB ᏻ d y Ꮿ(y) · δF F Ꮽ(y) |0, (9) where the − (+) in the right-hand side should be understood for an operator ᏻ with zero and even (odd) ghost number. 380 K.-I. Kondo / Physics Letters B 619 (2005) 377–386

Eq. (9) implies that, if an operator ᏻ is BRST-invariant δB ᏻ = 0, the vacuum expectation value of the operator ᏻ does not depend on the gauge-fixing condition (procedure). Hence, the BRST-invariant operator is gauge invariant in this restricted sense. This corresponds to the opposite of the statement in the operator level that the gauge- invariant operator is BRST-invariant. In general, BRST-invariant operator is not necessarily gauge invariant. In our approach, it is essential to consider the vacuum expectation value as a criterion of gauge invariance of the operator (weak gauge invariance), which is weaker than the gauge invariance in the operator level (strong gauge invariance). In this Letter, we check the gauge invariance of the operator in the weak sense, by first breaking the gauge through the gauge fixing procedure and then seeing the stability of the vacuum expectation value under the arbitrary deformation of the gauge fixing.

2.1. Generalized Lorentz gauge

In what follows, we focus on the mixed gluon-ghost operator of mass dimension two defined by [11] − 1 ᏻ := V 1 d4x Ꮽ · Ꮽµ − λiᏯ · Ꮿ¯ , (10) K 2 µ V in the generalized Lorentz gauge [11], 4 ¯ 1 µ λ ¯ S + = d xiδ δ Ꮽ · Ꮽ − iᏯ · Ꮿ , (11) GF FP B B 2 µ 2 ¯ with the anti-BRST transformation δB . The generalized Lorentz gauge is equivalent to take g F [Ꮽ]=∂µᏭ ,G[ᏺ, Ꮿ, Ꮿ¯ ]=ᏺ − i (Ꮿ¯ × Ꮿ). (12) µ 2 At λ = 0, the generalized Lorentz gauge coincides with the usual Lorentz gauge, i.e., the Landau gauge. It has been shown [11] that the operator ᏻK is on-shell BRST invariant, but it is not (off-shell) BRST invariant, ¯ δB ᏻK = 0. The on-shell BRST transformation is defined for Ꮽµ, Ꮿ, Ꮿ and is obtained by eliminating the NL field ᏺ from the off-shell BRST transformation through the equation of motion for ᏺ.TheGF+ FP term can be rewritten ¯ in terms only of Ꮽµ, Ꮿ, Ꮿ and the resulting term is checked to be on-shell BRST invariant, but it is not written in the on-shell BRST-exact form. Moreover, the on-shell BRST transformation is not nilpotent, unless the equation of motion for the ghost is used. In this Letter, we do not use the on-shell BRST transformation. The key identity we use in the following is − δS + δ ᏻ =−V 1 d4x Ꮿ(x) · GF FP . (13) B K δᏺ(x) V In fact, the BRST transform of the operator reads −1 4 µ ¯ ¯ δB ᏻK = V d x Ꮽµ · δB Ꮽ − λiδB Ꮿ · Ꮿ + λiᏯ · δB Ꮿ V − −g = V 1 d4x Ꮽ · Dµ[Ꮽ]Ꮿ − λi (Ꮿ × Ꮿ) · Ꮿ¯ + λiᏯ · iᏺ µ 2 V − g = V 1 d4x Ꮽ · ∂µᏯ + λi Ꮿ · (Ꮿ × Ꮿ¯ ) − λᏯ · ᏺ µ 2 V − g = V 1 d4x Ꮿ · −∂µᏭ + λi (Ꮿ × Ꮿ¯ ) − λᏺ . (14) µ 2 V K.-I. Kondo / Physics Letters B 619 (2005) 377–386 381

If this is combined with the fact, δS + g GF FP = ∂µᏭ (x) − λi (Ꮿ × Ꮿ¯ )(x) + λᏺ(x), (15) δᏺ(x) µ 2 the desired equality (13) is obtained. Therefore, by substituting (13) into (9), we obtain −1 4 δ A 4 ¯ δ 0|ᏻ |0=V d x 0| Ꮿ (x)S + d y Ꮿ(y) · δ F Ꮽ(y) |0 F K δᏺA(x) GF FP F V −1 4 −1 iS +iS + δ A 4 ¯ = V d xZ dµ e YM GF FP Ꮿ (x)S + d y Ꮿ(y) · δ F Ꮽ(y) YM YM δᏺA(x) GF FP F V − − δ + = V 1 d4xZ 1 dµ −iᏯA(x)eiSYM iSGF+FP d4y Ꮿ¯ (y) · δ F Ꮽ(y) . (16) YM YM δᏺA(x) F V Note that the right-hand side is zero.1 Therefore, we conclude

δF 0|ᏻK |0=0, (17) which means that the spacetime average of the vacuum expectation value of mass dimension two: − 1 ᏻ =V 1 d4x Ꮽ (x) · Ꮽµ(x) − λiᏯ(x) · Ꮿ¯ (x) , (18) K 2 µ V is unchanged even if we adopt the gauge-fixing condition which is slightly deformed from the original one. In particular for λ = 0, starting from the Landau gauge in the usual Lorentz gauge fixing, we see the invariance of the vacuum condensate − 1 ᏻ =V 1 d4x Ꮽ (x) · Ꮽµ(x) (19) K 2 µ V µ for the deformation of the Landau gauge-fixing condition ∂ Ꮽµ = 0. The gauge fixing is performed by choosing a representative from a gauge orbit (gauge equivalent configurations of the given gauge potential) as an intersecting point of a gauge orbit with a gauge-fixing hypersurface specified by the gauge-fixing condition. This is clear for λ = 0 case. Any gauge-fixing hypersurface is obtained by repeating the infinitesimal continuous change from an initial gauge-fixing hypersurface. Therefore, the gauge-fixing independence of ᏻ implies that the value of ᏻ has a unique value and does not change no matter how we choose one representative from each gauge orbit. In this sense, the gauge-fixing independence leads to the gauge invariance. See Fig. 1.

2.2. Modiﬁed maximal Abelian gauge

For the gauge group G = SU(N), we consider the Cartan decomposition of Ꮽ into diagonal (maximal torus group H = U(1)N−1) and off-diagonal (G/H ) pieces, Ꮽ(x) = ᏭA(x)T A = ai(x)H i + Aa(x)T a A = 1,...,N2 − 1 , (20)

1 Dᏺ δ ᏺ = The integration of the differentiation is identically zero, δᏺ f( ) 0. This follows only from the shift invariance of the measure, i.e., D + = D D = D + + = D + (ϕ a) ϕ for arbitrary a, since for an arbitrary functional f(ϕ)of ϕ it implies that ϕf(ϕ) (ϕ a)f(ϕ a) ϕf(ϕ) a Dϕ δ f(ϕ)+··· should hold for arbitrary a, and hence Dϕ δ f(ϕ)= 0 is obtained. This also follows if the boundary values f(ϕ ), δϕ δϕ max D δ = − f(ϕmin) of the functional vanish, since ϕ δϕ f(ϕ) f(ϕmax) f(ϕmin). 382 K.-I. Kondo / Physics Letters B 619 (2005) 377–386

Fig. 1. Gauge orbits and the gauge-fixing hypersuface (solid line) specified by the gauge-fixing condition and its infinitesimal deformation (broken line). where i = 1, 2,...,N − 1. In particular, for G = SU(3), the generators are given by the Gell-Mann matrices: λ3 λ8 λa H 1 = ,H2 = ,Ta = (a = 1, 2, 4, 5, 6, 7). (21) 2 2 2 The maximal Abelian (MA) gauge-fixing condition plays a role of the partial gauge fixing G → H and is given by a[ ]:= [ ] µ a = µa + aib i µb = F a,A Dµ a A ∂µA (x) gf aµ(x)A (x) 0. (22) The naive MA gauge is given by a 4 ¯ a µ α S + =− d xiδ C D [a]A + N . (23) GF FP B µ 2 In order to fix the residual Abelian gauge group H , we add an additional GF + FP term for the diagonal part, e.g., β i − d4xi δ C¯ i ∂µa + N . (24) B µ 2 We consider the operator − 1 ᏻ := V 1 d4x tr Ꮽ Ꮽµ − αiᏯᏯ¯ , (25) K G/H 2 µ V in the modified MA gauge defined by [14] 4 ¯ 1 µ α ¯ S + := d xiδ δ tr Ꮽ Ꮽ − iᏯᏯ . (26) GF FP B B G/H 2 µ 2 Note that (26) is obtained from (23) by adding the ghost self-interaction terms and by adjusting the parameter for the ghost self-interaction term, g F a[Ꮽ]= Dµ[a]A a,Ga[ᏺ, Ꮿ, Ꮿ¯ ]=N a − igf abiC¯ bCi − i f abcCbC¯ c, (27) µ 2 since a ¯ 1 a µa − α a ¯ a =−¯ a µ[ ] + α + α abi ¯ a ¯ b i + α abc a ¯ b ¯ c δB AµA iC C C D a Aµ N i f C C C i f C C C 2 2 2 2 4 α g =−C¯ a Dµ[a]A a + N a − igf abiC¯ bCi − i f abcCbC¯ c , (28) µ 2 2 K.-I. Kondo / Physics Letters B 619 (2005) 377–386 383 where the structure constants f ABC are completely antisymmetric in the indices and we have used a fact f aij = 0 (T i and T j commute). It is easy to check that the similar identity holds in the modified MA gauge, − δS + δ ᏻ =−V 1 d4xCa(x) GF FP , (29) B K δNa(x) V where − g δ ᏻ =−V 1 d4xCa DµA a − αgf abiiCiC¯ b + α f abciCbC¯ c + αNa . (30) B K µ 2 V In the same way as in the Lorentz gauge, therefore, we conclude |ᏻ | = δF 0 K 0 0, (31) which means that the spacetime average of the vacuum expectation value of mass dimension two: − 1 α ᏻ =V 1 d4x Aa (x)Aµa(x) − iCa(x)C¯ a(x) , (32) K 2 µ 2 V is unchanged even if we adopt the gauge-fixing condition which is slightly deformed from the original one. In particular for α = 0, starting from the Landau gauge in the naive MA gauge, we see the invariance of the vacuum condensate − 1 ᏻ =V 1 d4x Aa (x)Aµa(x) (33) K 2 µ V for the deformation of the gauge-fixing condition.

3. Conclusion and discussion

In this Letter, we have given a formal proof that the spacetime average of a vacuum condensate of mass dimension two ᏻK is gauge invariant in the weak sense that it is independent of the gauge-fixing condition adopted in quantizing the Yang–Mills theory, at least for the small deformation from the generalized Lorentz and the modified MA gauge in the naive continuum formulation without Gribov copies. This suggests that the numerical value of the condensate must be the same no matter what gauge-fixing conditions for choosing the representative from the gauge orbit are adopted to measure it. We discuss this point for a while. The Lorentz gauge and the MA gauge can be interpolated by introducing a parameter ξ [15] for the off-diagonal gauge-fixing condition a := µ a + aib µi b = Fξ ∂ Aµ(x) ξgf a (x)Aµ(x) 0, (34) and by taking the same Landau gauge condition for the diagonal part i := µ i = Fξ ∂ aµ(x) 0, (35) = = A = Aib µi b where ξ 0 is the Lorentz gauge and ξ 1 is the MA gauge. Then δF F ξgf a Aµ. Therefore, for small change of ξ from 0 and 1, the vacuum condensation of mass dimension two must not change the value, i.e., 2 δF 0|ᏻK |0=O(ξ ). 384 K.-I. Kondo / Physics Letters B 619 (2005) 377–386

Is it possible to show the above independence also for a finite deformation of the gauge-fixing condition? This is trivial for a finite deformation, if it is obtained by performing the infinitesimal transformation successively, as far as the obstruction for the deformation does not exist. In order to treat the finite deformation, we denote the vacuum expectation value calculated under the gauge-fixing condition F = 0by0|ᏻK |0F . If we consider the change of gauge-fixing condition F by δF F , the relationship is obtained:

iδ S + 0|ᏻK e F GF FP |0F 0|ᏻ |0 + = . (36) K F δF F iδ S + 0|e F GF FP |0F The denominator reads ∞ ∞ iδ S + 1 n 1 0|e F GF FP |0 = 1 + 0|(iδ S + ) |0 = 1 + 0|δ (∗)δ (∗) ···δ (∗) |0 = 1, (37) F n! F GF FP F n! B B B F n=1 n=1 4 ¯ where we deﬁned ∗:= d x Ꮿ(x) · δF F [Ꮽ(x)] and have used the nilpotency of the BRST transformation. On the other hand, the numerator reads

|ᏻ | = |ᏻ iδF SGF+FP | 0 K 0 F +δF F 0 K e 0 F ∞ 1 n =0|ᏻ |0 + 0|ᏻ (iδ S + ) |0 K F n! K F GF FP F n=1 ∞ 1 = |ᏻ | + |ᏻ δ (∗) δ (∗) ···δ (∗) | 0 K 0 F ! 0 K B B B 0 F = n n 1 n−1 ∞ 1 = |ᏻ | − |δ ᏻ (∗) δ (∗) ···δ (∗) | , 0 K 0 F ! 0 B K B B 0 F (38) = n n 1 n−1 where we have used the nilpotency of the BRST transformation in the third equality. The term ∗ does not include the NL field ᏺ. This is also the case for the n = 1 piece of this expansion. In the n = 2 piece, however, the NL field ¯ ᏺ appears from δB Ꮿ in δB (∗). This invalidates the argument given in this Letter. Therefore, our proof is too naive to extend the independence proof to a finite deformation beyond the infinitesimal deformation of the gauge-fixing condition. In the above, we implicitly assumed that the gauge-fixing hypersurface intersects the gauge orbit only once. It is known, however, that this is not necessarily achieved by the usual gauge-fixing condition, e.g., Landau gauge. In fact, there are many intersection points called Gribov copies in a gauge orbit with the given gauge-fixing hypersurface. This difficulty is known as the Gribov problem. For the generalized Lorentz gauge λ = 0, however, the GF + FP term includes the quartic ghost self-interaction term and hence the Gribov problem is not manifest and could be circumvented. For the Landau gauge λ = 0, the Gribov copies are shown to exist. A simple way (a proposal due to Gribov) to avoid the Gribov copies is to restrict the functional integral into the interior Ω of the Gribov horizon where the µ Faddeev–Popov determinant keeps the positive value. In the Landau gauge λ = 0forF [Ꮽ]:=∂ Ꮽµ, the ghost and antighost field can be integrated out to obtain − − δ δ 0|ᏻ |0=V 1 d4xZ 1 [dᏭ ] [dᏺ] F K YM µ δᏺA(x) V Ω 4 −1 B iSYM ¯ A ¯ B iSGF+FP × d yi δF F Ꮽ(y) e [dᏯ][dᏯ]Ꮿ (x)Ꮿ (y)e K.-I. Kondo / Physics Letters B 619 (2005) 377–386 385 − − δ = V 1 d4xZ 1 [dᏭ ] [dᏺ] YM µ δᏺA(x) V Ω − 4 ᏺ· [Ꮽ] −1 × 4 1 B Ꮽ iSYM i d x F − · [Ꮽ] − · [Ꮽ] d yi δF F (y) e e ∂ D AB (x, y) det ∂ D , (39) where we have used the identity (16) and the ghost and antighost field are integrated out. The right-hand side is zero, since the integration of the total derivative with respect to ᏺ vanishes. Within the Gribov proposal, therefore, the gauge-fixing independence is shown for arbitrary gauge-fixing hypersurface which is deformed from the µ hypersurface ∂ Ꮽµ = 0 inside the Gribov horizon of the Landau gauge fixing. However, this reasoning is clearly insufficient, since it is proved that there is no Gribov copies only in a subset of the Gribov region, called the fundamental modular region H . In the rigorous sense, therefore, we must consider the functional integration restricted to the fundamental modular region H . In fact, in gauge-fixed lattice simulations, the fundamental modular region is selected and used to measure the averaged condensate, V−1 4 1 Ꮽ 2 V d x trG/H ( 2 µ(x) ) . In light of these facts, our analyses in this paper is still insufficient, although there is a claim that the Gribov copies inside the Gribov region have no influence on expectation-values of gauge invariant operators [16], in other words, the fundamental modular gauge fixing would be possible by the Langevin simulation, which does not seem to be confirmed, e.g., by the lattice simulations. It is not clear whether the dependence on the gauge-fixing subset H disappears. Therefore, it is an open question whether the expectation value of the gauge dependent operator in the Landau gauge after fundamental modular gauge fixing, and that of the maximal Abelian gauge should be the same after spacetime averaging, i.e., sample averaging. The gauge-fixing independence could be checked by perturbation theory up to Gribov problem and by the numerical simulations where we need to remove the Gribov copies.

Acknowledgements

The author would like to thank Yukinari Sumino and Kazuo Fujikawa for valuable comments on the relationship between gauge fixing and BRST invariance. This work is supported by Grant-in-Aid for Scientific Research (C)14540243 from Japan Society for the Promotion of Science (JSPS), and in part by Grant-in-Aid for Scientific Research on Priority Areas (B)13135203 from the Ministry of Education, Culture, Sports, Science and Technology (MEXT).

References

[1] J. Greensite, M. Halpern, Nucl. Phys. B 271 (1986) 379. [2] M.J. Lavelle, M. Schaden, Phys. Lett. B 208 (1988) 297; M. Lavelle, M. Oleszczuk, Mod. Phys. Lett. A 7 (1992) 3617. [3] M. Schaden, hep-th/9909011; K.-I. Kondo, T. Shinohara, Phys. Lett. B 491 (2000) 263, hep-th/0004158; D. Dudal, H. Verschelde, J. Phys. A 36 (2003) 8507, hep-th/0209025. [4] Ph. Boucaud, A. Le Yaouanc, J.P. Leroy, J. Micheli, O. Pene, J. Rodriguez-Quintero, Phys. Lett. B 493 (2000) 315, hep-ph/0008043; Ph. Boucaud, A. Le Yaouanc, J.P. Leroy, J. Micheli, O. Pene, J. Rodriguez-Quintero, Phys. Rev. D 63 (2001) 114003, hep-ph/0101302. [5] F.V. Gubarev, L. Stodolsky, V.I. Zakharov, Phys. Rev. Lett. 86 (2001) 2220, hep-th/0010057; F.V. Gubarev, V.I. Zakharov, Phys. Lett. B 501 (2001) 28, hep-ph/0010096. [6] D. Dudal, J.A. Gracey, V.E.R. Lemes, M.S. Sarandy, R.F. Sobreiro, S.P. Sorella, H. Verschelde, Phys. Rev. D 70 (2004) 114038, hep- th/0406132; D. Dudal, H. Verschelde, J.A. Gracey, V.E.R. Lemes, M.S. Sarandy, R.F. Sobreiro, S.P. Sorella, JHEP 0401 (2004) 044, hep-th/0311194. [7] E.R. Arriola, P.O. Bowman, W. Broniowski, hep-ph/0408309. [8] S. Furui, H. Nakajima, hep-lat/0503029. 386 K.-I. Kondo / Physics Letters B 619 (2005) 377–386

[9] A.J. Niemi, N.R. Walet, hep-ph/0504034. [10] M. Chaichian, K. Nishijima, hep-th/0504050. [11] K.-I. Kondo, Phys. Lett. B 514 (2001) 335, hep-th/0105299; K.-I. Kondo, Phys. Lett. B 572 (2003) 210, hep-th/0306195; K.-I. Kondo, T. Murakami, T. Shinohara, T. Imai, Phys. Rev. D 65 (2002) 085034, hep-th/0111256. [12] A.A. Slavnov, hep-th/0407194. [13] A.A. Slavnov, Phys. Lett. B 608 (2005) 171. [14] K.-I. Kondo, Phys. Rev. D 58 (1998) 105019, hep-th/9801024. [15] K.-I. Kondo, Phys. Rev. D 57 (1998) 7467, hep-th/9709109. [16] D. Zwanziger, Phys. Rev. D 69 (2004) 01600, hep-ph/0303028. Physics Letters B 619 (2005) 387–388 www.elsevier.com/locate/physletb Erratum

Erratum to: “Majorana phase in minimal S3 invariant extension of the standard model” [Phys. Lett. B 578 (2004) 156]

Jisuke Kubo 1

Max-Planck-Institut für Physik, Werner-Heisenberg-Institut, D-80805 Munich, Germany Received 7 June 2005 Available online 13 June 2005

† 1. The first sentence of the last paragraph of p. 160 (just before Eq. (26)) “Now the product UeLUν defines...” should be changed to † = ν “Now the product UeLPUν with P diag(1, 1, exp(i arg(Y4 ))) defines...”. 2. This new phase introduces a Dirac CP phase that is responsible for CP violations in neutrino oscillations. Consequently, Eq. (28) should be changed to: = ν − δ arg Y4 φν. (28) As a consequence, CP violations can exist in neutrino oscillations as well as in form of Majorana phases. Therefore, the corresponding statements in the abstract and also in the text should be corrected. 3. Furthermore, Eqs. (29), (30) and (31), respectively, should be changed to: m sin φ sin 2α = sin(φ − φ ) =± ν3 ν m2 − m2 sin2 φ + m2 − m2 sin2 φ (29) 1 2 m m ν2 ν3 ν ν1 ν3 ν ν1 ν2 ± − 2 2 2sinφν(mν3 /mν2 ) 1 (mν3 /mν2 ) sin φν, (30) sin φ sin 2β = sin(φ − φ ) =± ν m 1 − sin2 φ + m2 − m2 sin2 φ (31) 1 ν ν3 ν ν1 ν3 ν mν1

for φ1 + φ2 ∼±π, where φ1,φ2 and φν are deﬁned in (15). 4. Because of the changes above, the Figs. 3 and 4 should be changed to:

DOI of original article: 10.1016/j.physletb.2003.10.048. E-mail address: [email protected] (J. Kubo). 1 Permanent address: Institute for Theoretical Physics, Kanazawa University.

= 2 = × −5 2 2 = × −3 2 Fig. 3. sin 2α (solid) and sin 2β (dotted) versus sin φν for tan θ12 0.68,m21 6.9 10 eV and m23 2.3 10 eV in the case of φ1 + φ2 ∼ π.

2 = 2 = × −5 2 Fig. 4. The effective Majorana mass mee as a function of sin φν with sin θ12 0.3andm21 6.9 10 eV . The dashed, solid and 2 = × −3 2 2 dot-dashed lines stand for m23 1.4, 2.3and3.0 10 eV , respectively. The m21 dependence is very small. Physics Letters B 619 (2005) 389–427 www.elsevier.com/locate/physletb

Cumulative author index to volumes 611–619

Abazov, V.M., 617,1 Aichelin, J., 612, 201 Abbaneo, D., 611, 66; 614,7 Aihara, H., 613, 20; 614, 27; 615, 39; 617, 141, 198; 618,34 Abbasi, R.U., 619, 271 Ajaltouni, Z.J., 614, 165 Abbott, B., 617,1 Akaishi, Y., 613, 140 Abdalla, E., 611,21 Akatsu, M., 614, 27; 615,39 Abdalla, M.C.B., 613, 213 Akhmetshin, R.R., 613,29 Abdel-Bary, M., 619, 281 Aktas, A., 616,31 Abdesselam, A., 617,1 Alamanos, N., 619,82 Abe, K., 613, 20, 20; 614, 27, 27; 615, 39, 39; Alberg, M., 611, 111 617, 141, 141, 198, 198; 618, 34, 34 Alcaniz, J.S., 619,11 Abel, S., 618, 201 Alcaraz, J., 613, 118; 615, 19; 616, 145, 159; 619,71 Abele, H., 619, 263 Alemanni, G., 613, 118; 615, 19; 616, 145, 159; 619,71 Ablikim, M., 614, 37; 619, 247 ALEPH Collaboration, 611, 66; 614,7 Abolins, M., 617,1 Alexakhin, V.Yu., 612, 154 Abramov, V., 617,1 Alexandrov, Yu., 612, 154 Abu-Zayyad, T., 619, 271 Alexandru, A., 612, 21; 617,49 Abuki, H., 615, 102 Alexeev, G.D., 612, 154; 617,1 Achard, P., 613, 118; 615, 19; 616, 145, 159; 619,71 Alimonti, G., 618,23 Acharya, B.S., 617,1 Alkofer, R., 611, 279 Ackerman, L., 611,53 Allaby, J., 613, 118; 615, 19; 616, 145, 159; 619,71 Adachi, I., 614, 27; 617, 198 Alner, G.J., 616,17 Adams, D.L., 617,1 Aloisio, A., 613, 118; 615, 19; 616, 145, 159; 619,71 Adams, J., 612, 181; 616,8 ALPHA Collaboration, 612, 313 Adams, M., 617,1 Altmann, M., 616, 174 Adeva, B., 619,50 Alton, A., 617,1 Adler, C., 612, 181 Alves, G.A., 617,1 Adriani, O., 613, 118; 615, 19; 616, 145, 159; 619,71 Alviggi, M.G., 613, 118; 615, 19; 616, 145, 159; 619,71 Afanasyev, L., 619,50 Ambrosino, F., 619,61 Ageev, E.S., 612, 154 Amir-Ahmadi, H.R., 617,18 Aggarwal, M.M., 612, 181; 616,8 Amman, J.F., 619, 271 Agostino, L., 618,23 Amonett, J., 612, 181; 616,8 Aguilar-Benitez, M., 613, 118; 615, 19; 616, 145, 159; 619,71 Amoroso, A., 612, 154 Aguilar-Saavedra, J.A., 613, 170 Amsler, C., 615, 153 Aguirre, R., 611, 248 Anderhub, H., 613, 118; 615, 19; 616, 145, 159; 619,71 Ahammed, Z., 612, 181; 616,8 Anderson, B.D., 612, 181; 616,8 Ahmad, I., 618,51 Anderson, J.D., 613,11 Ahmed, S.N., 617,1 Andreev, V., 616,31 Ahn, M.H., 619, 255 Andreev, V.P., 613, 118; 615, 19; 616, 145, 159; 619,71

0370-2693/2005 Published by Elsevier B.V. doi:10.1016/S0370-2693(05)00891-9 390 Cumulative author index to volumes 611–619 (2005) 389–427

Andronic, A., 612, 173 Badełek, B., 612, 154 Anjos, J.C., 618,23 Baden, A., 617,1 Anselmo, F., 613, 118; 615, 19; 616, 145, 159; 619,71 Badyal, S.K., 612, 181; 616,8 Anthonis, T., 616,31 Baeßler, S., 619, 263 Antonelli, A., 611, 66; 614,7;619,61 Bafﬁoni, S., 617,1 Antonelli, M., 611, 66; 614,7;619,61 Bagchi, M., 618, 115 Antusch, S., 618, 150 Bagliesi, G., 611, 66; 614,7 Anzivino, G., 615,31 Bagnaia, P., 613, 118; 615, 19; 616, 145, 159; 619,71 Aoki, S., 613, 105; 614, 155; 619, 255 Bahinipati, S., 613, 20; 615, 39; 617, 141, 198; 618,34 Araújo, H.M., 616,17 Bähr, J., 616,31 Archbold, G., 619, 271 Bai, J.Z., 614, 37; 619, 247 Arcidiacono, R., 615,31 Bai, Y., 616,8 Areﬁev, A., 613, 118; 615, 19; 616, 145, 159; 619,71 Bajo, A., 613, 118; 615, 19; 616, 145, 159; 619,71 Arena, V., 618,23 Bakich, A.M., 613, 20; 614, 27; 615, 39; 617, 141, 198; 618,34 Arhrib, A., 612, 263 Baksay, G., 613, 118; 615, 19; 616, 145, 159; 619,71 Arkhipkin, D., 612, 181; 616,8 Baksay, L., 613, 118; 615, 19; 616, 145, 159; 619,71 Arleo, F., 614,44 Balagura, V., 618,34 Armstrong, S.R., 611, 66; 614,7 Balantekin, A.B., 613,61 Arnison, G.J., 616,17 Balata, M., 616, 174 Arnoud, Y., 617,1 Baldew, S.V., 613, 118; 615, 19; 616, 145, 159; 619,71 Arnowitt, R., 618, 182 Baldin, B., 617,1 Artamonov, A., 613, 105; 614, 155 Balestra, F., 612, 154 Asahi, K., 615, 186 Balewski, J., 612, 181; 616,8 Asano, Y., 613, 20; 614, 27; 615, 39; 617, 141, 198; 618,34 Ball, J., 612, 154 Asatrian, H.M., 619, 322 Ballestrero, S., 619, 240 Balm, P.W., 617,1 Ashie, Y., 619, 255 Ban, Y., 614, 27, 37; 615, 39; 618, 34; 619, 247 Asmone, A., 616,31 Bandos, I.A., 615, 127 Assunção, M., 615, 167 Banerjee, S., 611, 27; 613, 118; 614, 27; 615, 19, 39; Atkins, R., 619, 271 616, 145, 159; 617,1;618, 34; 619,71 Auger, F., 619,82 Banerjee, Sw., 613, 118; 615, 19; 616, 145, 159; 619,71 Aulchenko, V., 615, 39; 617, 141; 618,34 Banzarov, V.Sh., 613,29 Aulchenko, V.M., 613,29 Barannikova, O., 612, 181; 616,8 Aushev, T., 613, 20; 614, 27; 615, 39; 617, 141; 618,34 Baranov, P., 616,31 Avelino, P.P., 611,15 Barate, R., 611, 66; 614,7 Avenier, M., 615, 153 Baratt, A., 613,29 Averett, T., 613, 148 Barberis, E., 617,1 Averichev, G.S., 612, 181; 616,8 Barberis, S., 618,23 Avila, C., 617,1 Barbuto, E., 613, 105; 614, 155 Awunor, O., 611, 66; 614,7 Barczyk, A., 613, 118; 615, 19; 616, 145, 159; 619,71 Axiotis, M., 619,88 Barea, J., 613, 134 Azemoon, T., 613, 118; 615, 19; 616, 145, 159; 619,71 Barger, V., 613, 61; 614, 67; 617, 78, 167 Aziz, T., 613, 118; 615, 19; 616, 145, 159; 618, 34; 619,71 Barillère, R., 613, 118; 615, 19; 616, 145, 159; 619,71 Azzurri, P., 611, 66; 614,7 Baringer, P., 617,1 Barklow, T., 611, 66; 614,7 Baaquie, B.E., 615, 134 Barkov, L.M., 613,29 Baba, H., 614, 174 Barnby, L.S., 612, 181; 616,8 Babaev, A., 616,31 Barr, G., 615,31 Babichev, E., 614,1 Barrelet, E., 616,31 Babintsev, V.V., 617,1 Barret, V., 612, 173 Babukhadia, L., 617,1 Barreto, J., 617,1 Bacci, C., 619,61 Bartalini, P., 613, 118; 615, 19; 616, 145, 159; 619,71 Bacelar, J.C.S., 617,18 Bartel, W., 616,31 Backovic, S., 616,31 Bartlett, J.F., 617,1 Bacon, T.C., 617,1 Barton, J.C., 616,17 Badaud, F., 611, 66; 614,7 Bashtovoy, N.S., 613,29 Cumulative author index to volumes 611–619 (2005) 389–427 391

Basile, M., 613, 118; 615, 19; 616, 145, 159; 619,71 Bender, C.M., 613,97 Basrak, Z., 612, 173 Benelli, A., 619,50 Bass, S.A., 618,77 Benoit, A., 616,25 Bassler, U., 617,1 Benussi, L., 618,23 Bastid, N., 612, 173 Benzoni, G., 615, 160 Batalova, N., 613, 118; 615, 19; 616, 145, 159; 619,71 Ben Zvi, S.Y., 619, 271 Batist, L., 619,88 Berbeco, R., 613, 118; 615, 19; 616, 145, 159; 619,71 Battiston, R., 613, 118; 615, 19; 616, 145, 159; 619,71 Berdugo, J., 613, 118; 615, 19; 616, 145, 159; 619,71 Battistoni, G., 615,14 Berek, G., 612, 173 Baudot, J., 612, 181; 616,8 Berezinsky, V., 612, 147 Bauer, C.W., 611,53 Bergé, L., 616,25 Bauer, D., 617,1 Berger, Ch., 616,31 Baum, G., 612, 154 Berger, J., 612, 181; 616,8 Bäumer, C., 612, 165 Berger, N., 616,31 Baumgartner, S., 616,31 Berges, P., 613, 118; 615, 19; 616, 145, 159; 619,71 Bay, A., 613, 118; 614, 27; 615, 19; 616, 145, 159; 617, 198; Berglund, P., 612, 154 618, 34; 619,71 Bergman, D.R., 619, 271 Bean, A., 617,1 Beri, S.B., 617,1 Beaudette, F., 617,1 Berka, Z., 619,50 Beaumel, D., 619,82 Berkelman, K., 611, 66; 614,7 Becattini, F., 613, 118; 615, 19; 616, 145, 159; 619,71 Bernabei, R., 616, 174 Becherini, Y., 615,14 Bernabéu, J., 613, 162 Becheva, E., 619,82 Bernardi, G., 617,1 Becker, H.G., 615,31 Berndt, T., 616,31 Becker, J., 616,31 Bernet, C., 612, 154 Becker, U., 613, 118; 615, 19; 616, 145, 159; 619,71 Bertani, M., 618,23 Beckingham, M., 616,31 Bertini, R., 612, 154 Bedaque, P.F., 616, 208 Bertolucci, S., 619,61 Bedfer, Y., 612, 154 Bertram, I., 617,1 Bediaga, I., 618,23 Bertucci, B., 613, 118; 615, 19; 616, 145, 159; 619,71 Bednarczyk, P., 615, 160 BES Collaboration, 614, 37; 619, 247 Bedny, I., 613, 20; 614, 27; 615, 39; 617, 141; 618,34 Besson, A., 617,1 Begel, M., 617,1 Betev, B.L., 613, 118; 615, 19; 616, 145, 159; 619,71 Behner, F., 613, 118; 615, 19; 616, 145, 159; 619,71 Beuselinck, R., 611, 66; 614,7;617,1 Behnke, O., 616,31 Bevan, A., 615,31 Behrendt, O., 616,31 Bewick, A., 616,17 Bekele, S., 612, 181; 616,8 Beylin, A., 616, 228 Belaga, V.V., 612, 181; 616,8 Bezverkhny, B.I., 612, 181; 616,8 Belle Collaboration, 613, 20; 614, 27; 615, 39; 617, 141, 198; Bezzubov, V.A., 617,1 618,34 Bhang, H., 619, 255 Belli, P., 616, 174 Bharadwaj, S., 616,8 Bellido, J.A., 619, 271 Bhardwaj, S., 612, 181 Bellini, M., 619, 208 Bhasin, A., 616,8 Bellorín, J., 616, 118 Bhat, P.C., 617,1 Bellotti, E., 616, 174 Bhati, A.K., 612, 181; 616,8 Bellucci, L., 613, 118; 615, 19; 616, 145, 159; 619,71 Bhatia, V.S., 616,8 Bellucci, S., 612, 283; 616, 228 Bhatnagar, V., 617,1 Bellwied, R., 612, 181; 616,8 Bhattacharjee, M., 617,1 Belousov, A., 616,31 Bhattacharyya, A., 611,27 Belov, K., 619, 271 Bian, J.G., 614, 37; 619, 247 Beltrame, P., 619,61 Bianco, S., 618,23 Belyaev, A., 617,1 Biasini, M., 613, 118; 615, 19; 616, 145, 159; 619,71 Belz, J.W., 619, 271 Bichsel, H., 612, 181; 616,8 Benabderrahmane, L., 612, 173 Bidder, S.J., 612,75 Benayoun, M., 619,50 Bielcik, J., 616,8 Bencivenni, G., 611, 66; 614,7;619,61 Bielcikova, J., 616,8 392 Cumulative author index to volumes 611–619 (2005) 389–427

Biglietti, M., 613, 118; 615, 19; 616, 145, 159; 619,71 Booth, C.N., 611, 66; 614,7 Biino, C., 615, 31; 619, 240 Borcea, R., 619,88 Biland, A., 613, 118; 615, 19; 616, 145, 159; 619,71 Borcherding, F., 617,1 Billmeier, A., 612, 181; 616,8 Bordalo, P., 612, 154 Binétruy, P., 611,39 Borean, C., 611, 66; 614,7 Bini, C., 619,61 Borge, M.J.G., 618,43 Birsa, R., 612, 154 Borgia, B., 613, 118; 615, 19; 616, 145, 159; 619,71 Bisplinghoff, J., 612, 154 Bos, K., 617,1 Bissegger, M., 613,57 Boschini, M., 618,23 Biswas, A., 613, 208 Bose, T., 617,1 Bitenc, U., 613, 20; 614, 27; 615, 39; 617, 141, 198; 618,34 Bossi, F., 611, 66; 614,7;619,61 Bizjak, I., 613, 20; 614, 27; 615, 39; 617, 141, 198; 618,34 Botje, M., 612, 181; 616,8 Bizot, J.C., 616,31 Bottai, S., 613, 118; 615, 19; 616, 145, 159; 619,71 Bizzeti, A., 615,31 Boucham, A., 612, 181; 616,8 Bjerrum-Bohr, N.E.J., 612,75 Boucrot, J., 611, 66; 614,7 Blair, G.A., 611, 66; 614,7 Boudry, V., 616,31 Blaising, J.J., 613, 118; 615, 19; 616, 145, 159; 619,71 Bouhova-Thacker, E., 611, 66; 614,7 Blaizot, J.-P., 615, 221 Boumediene, D., 611, 66; 614,7 Bland, L.C., 612, 181; 616,8 Bourilkov, D., 613, 118; 615, 19; 616, 145, 159; 619,71 Blasi, N., 615, 160 Bourquin, M., 613, 118; 615, 19; 616, 145, 159; 619,71 Blazey, G., 617,1 Bowdery, C.K., 611, 66; 614,7 Bleicher, M., 612, 201 Bowring, D., 619,61 Blekman, F., 617,1 Boyd, S., 619, 255 Blessing, S., 617,1 Bozek, A., 613, 20; 614, 27; 615, 39; 617, 141, 198; 618,34 Bloch-Devaux, B., 611, 66; 614,7 Bozza, C., 613, 105; 614, 155 Bloise, C., 619,61 Braccini, S., 613, 118; 615, 19; 616, 145, 159; 619,71 Blondel, A., 611, 66; 614,7 Bracco, A., 615, 160 Blumenfeld, Y., 613, 128; 619,82 Bracinik, J., 616,31 Blumenschein, U., 611, 66; 614,7 Bracko,ˇ M., 613, 20; 614, 27; 615, 39; 617, 141, 198; 618,34 Blümer, J., 616,25 Bradamante, F., 612, 154 Blümlein, J., 614,53 Bragin, A.V., 613,29 Blyth, C.O., 612, 181; 616,8 Branchini, P., 619,61 Blyth, S., 613, 20; 614, 27; 615, 39; 618,34 Branco, G.C., 614, 187 Blyth, S.C., 613, 118; 615, 19; 616, 145, 159; 619,71 Brand, B., 619, 263 Bobbink, G.J., 613, 118; 615, 19; 616, 145, 159; 619,71 Brandenburg, A., 615, 68; 617,99 Boca, G., 618,23 Brandin, A., 612, 181 Boccali, T., 611, 66; 614,7 Brandin, A.V., 616,8 Bocci, V., 619,61 Brandolini, F., 619,88 Bocquet, G., 615,31 Brandt, A., 617,1 Boehnlein, A., 617,1 Brandt, S., 611, 66; 614,7 Boenig, M.-O., 616,31 Branson, J.G., 613, 118; 615, 19; 616, 145, 159; 619,71 Böhm, A., 613, 118; 615, 19; 616, 145, 159; 619,71 Braun, V., 618, 252 Böhme, J., 616,31 Bravar, A., 612, 154, 181; 616,8 Böhrer, A., 611, 66; 614,7 Bravo, S., 611, 66; 614,7 Boinepalli, S., 616, 196 Brekhovskikh, V., 619,50 Bojko, N.I., 617,1 Bressan, A., 612, 154 Boldizsar, L., 613, 118; 615, 19; 616, 145, 159; 619,71 Brient, J.-C., 611, 66; 614,7 Bolton, T.A., 617,1 Brihaye, Y., 615,1 Bondar, A., 613, 20; 614, 27; 615, 39; 617, 198; 618,34 Briskin, G., 617,1 Bondar, A.E., 612, 215; 613,29 Brisson, V., 616,31 Bondarev, D.V., 613,29 Britto, R., 611, 167 Bonissent, A., 611, 66; 614,7 Brochu, F., 613, 118; 615, 19; 616, 145, 159; 619,71 Bonner, B.E., 612, 181; 616,8 Brock, R., 617,1 Bonomi, G., 618,23 Brodzicka, J., 613, 20; 614, 27; 615, 39; 617, 141, 198; 618,34 Bonora, L., 619, 359 Broggini, C., 615, 153 Cumulative author index to volumes 611–619 (2005) 389–427 393

Bröker, H.-B., 616,31 Cano-Ott, D., 619,88 Broniatowski, A., 616,25 Cao, Z., 619, 271 Brooijmans, G., 617,1 Capell, M., 613, 118; 615, 19; 616, 145, 159; 619,71 Bross, A., 617,1 Caplar,ˇ R., 612, 173 Browder, T.E., 613, 20; 614, 27; 617, 141, 198; 618,34 Capon, G., 611, 66; 614,7;619,61 Brown, D.P., 616,31 Capussella, T., 619,61 Bruncko, D., 616,31 Caragheorgheopol, G., 619,50 Brunelière, R., 611, 66; 614,7 Cara Romeo, G., 613, 118; 615, 19; 616, 145, 159; 619,71 Bruski, N., 613, 105; 614, 155 Carlino, G., 613, 118; 615, 19; 616, 145, 159; 619,71 Buchholz, D., 617,1 Caron, S., 616,31 Buchmüller, O., 611, 66; 614,7 Carpenter, M.P., 618,51 Budzanowski, A., 619, 281 Carrillo, S., 618,23 Buehler, M., 617,1 Carroll, J., 612, 181 Buescher, V., 617,1 Carson, M.J., 616,17 Bungau, C., 616,17 Cartacci, A., 613, 118; 615, 19; 616, 145, 159; 619,71 Bunyatyan, A., 616,31 Cartiglia, N., 615,31 Buontempo, S., 613, 105; 614, 155 Cartwright, S., 611, 66; 614,7 Burger, J.D., 613, 118; 615, 19; 616, 145, 159; 619,71 Carvalho, D.F., 618, 162 Burger, W.J., 613, 118; 615, 19; 616, 145, 159; 619,71 Carvalho, W., 617,1 Burkert, E., 616, 174 Casado, M.P., 611, 66; 614,7 Burnstein, R.A., 617,11 Casalbuoni, R., 615, 297 Burt, G.W., 619, 271 Casali, R., 615,31 Burtin, E., 612, 154 Casaus, J., 613, 118; 615, 19; 616, 145, 159; 619,71 Burtovoi, V.S., 617,1 Casey, D., 617,1 Buschhorn, G., 616,31 Casimiro, E., 618,23 Bussa, M.P., 612, 154 Casper, D., 619, 255 Büsser, F.W., 616,31 Cassol-Brunner, F., 616,31 Busto, J., 615, 153 Castelijns, R., 617,18 Butler, J.M., 617,1 Castilla-Valdez, H., 617,1 Butler, J.N., 618,23 Castillo, J., 612, 181; 616,8 Butter, D., 612, 304 Castoldi, M., 615, 160 Byrne, A.P., 618,51 Cataldo, M., 619,5 Bystersky, M., 616,8 Catanesi, M.G., 613, 105; 614, 155 Bystritskaya, L., 616,31 Cattadori, C., 616, 174 Bytchkov, V.N., 612, 154 Cattaneo, M., 611, 66; 614,7 Bzdak, A., 615, 240; 619, 288 Catu, O., 616,8 Caurier, E., 619,88 Cachazo, F., 611, 167 Cavallari, F., 613, 118; 615, 19; 616, 145, 159; 619,71 Cadman, R.V., 612, 181; 616,8 Cavallo, N., 613, 118; 615, 19; 616, 145, 159; 619,71 Cai, X., 614, 37; 619, 247 Cavanaugh, R., 611, 66; 614,7 Cai, X.D., 613, 118; 615, 19; 616, 145, 159; 619,71 Cavero-Pelaez, I., 613,97 Cai, X.Z., 612, 181; 616,8 Cawlﬁeld, C., 618,23 Caines, H., 612, 181; 616,8 Cebra, D., 612, 181; 616,8 Calderón de la Barca Sánchez, M., 612, 181; 616,8 Cecchi, C., 613, 118; 615, 19; 616, 145, 159; 619,71 Callot, O., 611, 66; 614,7 Cecchini, S., 615,14 Caloi, R., 619,61 Ceccucci, A., 615,31 Calvetti, M., 615,31 Cechak, T., 619,50 Camacho, A., 617, 118 Cenci, P., 615,31 Camanzi, B., 616,17 Censier, B., 616,25 Camera, F., 615, 160 Ceradini, F., 619,61 Cameron, W., 611, 66; 614,7 Cerini, L., 612, 154 Campana, P., 619,61 Cerna, C., 615, 153 Campbell, A.J., 616,31 Cerny, K., 616,31 Canelli, F., 617,1 Cerrada, M., 613, 118; 615, 19; 616, 145, 159; 619,71 Canfora, F., 614, 131 Cerri, C., 615,31 394 Cumulative author index to volumes 611–619 (2005) 389–427

Cerulli, R., 616, 174 Chi, S.P., 614, 37; 619, 247 Cerutti, A., 618,23 Chiarella, V., 611, 66; 614,7 Cerutti, F., 611, 66; 614,7 Chiarini, M., 616, 174 Chabert, L., 616,25 Chiba, M., 619,50 Chajecki, Z., 616,8 Chiefari, G., 613, 118; 615, 19; 616, 145, 159; 619, 61, 71 Chakrabarti, D., 617,92 Chikanian, A., 612, 181; 616,8 Chakraborty, D., 617,1 Chikawa, M., 613, 105; 614, 155 Chakravorty, A., 617,11 Chimento, L.P., 615, 146 Chaloupka, P., 612, 181; 616,8 Chiodini, G., 618,23 Chamizo, M., 613, 118; 615, 19; 616, 145, 159; 619,71 Chistov, R., 613, 20; 614, 27; 615, 39; 617, 141, 198; 618,34 Chan, C.-T., 611, 193 Chmeissani, M., 611, 66; 614,7 Chan, K.M., 617,1 Cho, D.K., 617,1 Chandrasekhar, B., 614, 207 Cho, K., 618,23 Chang, J.F., 614, 37; 619, 247 Cho, Y.M., 616, 101 Chang, M.-C., 613, 20; 617, 141 Choi, J.H., 619, 255 Chang, P., 613, 20; 617, 141 Choi, S., 613, 148; 617,1 Chang, Y.H., 613, 118; 615, 19; 616, 145, 159; 619,71 Choi, S.-K., 614, 27; 617, 141, 198; 618,34 Chao, Y., 614, 27; 617, 141, 198; 618,34 Choi, Y., 613, 20; 614, 27; 615, 39; 617, 198; 618,34 Chapellier, M., 616,25 Choi, Y.K., 613, 20; 614, 27; 615, 39; 618,34 Chapiro, A., 612, 154 Chollet, J.C., 615,31 Chardin, G., 616,25 Chomaz, P., 613, 128 Chatterjee, A., 619, 281 Chong, Z.-W., 614,96 Chattopadhyay, S., 612, 181; 616,8 Choong, W.-S., 617,11 Chbihi, A., 613, 128 Chopra, S., 617,1 Chekelian, V., 616,31 CHORUS Collaboration, 613, 105; 614, 155 Chekulaev, S.V., 617,1 Chowdhury, P., 618,51 Chemarin, M., 613, 118; 615, 19; 616, 145, 159; 619,71 Christie, W., 612, 181; 616,8 Chen, A., 613, 20, 118; 614, 27; 615, 19, 39; 616, 145, 159; Chu, Y.P., 614, 37; 619, 247 617, 141, 198; 618, 34; 619,71 Chung, Y.S., 618,23 Chen, C.-M., 611, 156 Chuvikov, A., 613, 20; 614, 27; 615, 39; 617, 141, 198; 618,34 Chen, G., 613, 118; 615, 19; 616, 145, 159; 619,71 Ciambrone, P., 619,61 Chen, G.M., 613, 118; 615, 19; 616, 145, 159; 619,71 Cicuttin, A., 612, 154 Chen, H.F., 612, 181; 613, 118; 614, 37; 615, 19; 616, 8, 145, 159; Cifarelli, L., 613, 118; 615, 19; 616, 145, 159; 619,71 619, 71, 247 Cindolo, F., 613, 118; 615, 19; 616, 145, 159; 619,71 Chen, H.S., 613, 118; 614, 37; 615, 19; 616, 145, 159; 619, 71, 247 Cinquini, L., 618,23 Chen, H.X., 614, 37; 619, 247 Cirilli, M., 615,31 Chen, J., 614, 37, 37; 619, 247, 247 Ciulli, V., 611, 66; 614,7 Chen, J.-P., 613, 148 Claes, D., 617,1 Chen, J.-W., 616, 208 Clare, I., 613, 118; 615, 19; 616, 145, 159; 619,71 Chen, J.C., 614, 37; 619, 247 Clare, R., 613, 118; 615, 19; 616, 145, 159; 619,71 Chen, K.-F., 614, 27; 617, 141; 618,34 Clark, A.R., 617,1 Chen, M.L., 614, 37; 619, 247 Clark, K., 617,11 Chen, S.-L., 612,29 Clarke, D.P., 611, 66; 614,7 Chen, W.T., 613, 20; 614, 27; 617, 141 Clay, R.W., 619, 271 Chen, Y., 612, 21, 181; 616,8;617,49 Clément, M., 619, 240 Chen, Y.B., 614, 37; 619, 247 Clerbaux, B., 611, 66; 614,7 Chen, Y.C., 617,11 Cleymans, J., 615,50 Cheng, J., 616,8 Clifft, R.W., 611, 66; 614,7 Cheon, B.G., 613, 20; 614, 27; 615, 39; 617, 141, 198; 618,34 Cline, D., 618,51 Chernenko, S.P., 612, 181 Cocco, A.G., 613, 105; 614, 155 Cherney, M., 612, 181; 616,8 Cofﬁn, J.P., 612, 181; 616,8 Chernyak, V.L., 612, 215 Cogan, J., 615,31 Cheshkov, C., 615,31 Cohen, T.D., 619, 115 Cheung, H.W.K., 618,23 Coignet, G., 613, 118; 615, 19; 616, 145, 159; 619,71 Cheze, J.B., 615,31 Colaleo, A., 611, 66; 614,7 Chi, S., 619,61 Colantoni, M., 612, 154 Cumulative author index to volumes 611–619 (2005) 389–427 395

Colas, P., 611, 66; 614,7 Cvach, J., 616,31 Colavita, A.A., 612, 154 Czosnyka, P., 615,55 Cole, S., 614, 27; 617, 141, 198 Czosnyka, T., 615,55 Colino, N., 613, 118; 615, 19; 616, 145, 159; 619,71 Czy˙z, H., 611, 116 Collazuol, G., 615,31 Collin, S., 616,25 D’Agostini, G., 615,31 Combley, F., 611, 66; 614,7 Dai, H.L., 614, 37; 619, 247 COMPASS Collaboration, 612, 154 Dai, Y.S., 614, 37; 619, 247 Conetti, S., 619,61 Dainton, J.B., 616,31 Connolly, B., 617,1 Dall’Agata, G., 619, 149 Connolly, B.C., 619, 271 Dalla Torre, S., 612, 154 Constantinescu, S., 619,50 Dalpiaz, P., 615,31 Contalbrigo, M., 615,31 Dalseno, J., 613, 20; 614, 27; 615, 39; 617, 141, 198; 618,34 Conte, E., 614, 165 D’Ambrosio, N., 613, 105; 614, 155 Contreras, J.G., 616,31 Da Motta, H., 617,1 Controzzi, D., 617, 133 D’Angelo, P., 618,23 Cooper, W.E., 617,1 D’Angelo, S., 616, 174 Coppage, D., 617,1 Danielewicz, P., 618,60 Coppens, Y.R., 616,31 Danilov, M., 614, 27; 615, 39; 617, 141, 198; 618,34 Coraggio, L., 616,43 Dappiaggi, C., 615, 291 Cordier, E., 612, 173 Darabi, F., 615, 141 Cormier, T.M., 612, 181; 616,8 Daraktchieva, Z., 615, 153 Cortés, J.L., 619, 367 Darwish, E.M., 615,61 Costa, S., 612, 154 Das, D., 612, 181; 616,8 Costantini, F., 615,31 Das, P.K., 618, 221 Costantini, S., 613, 118; 615, 19; 616, 145, 159; 619,71 Das, S., 612, 181; 616,8 Coughlan, J.A., 616,31 Dasgupta, S.S., 612, 154 Covi, L., 617,99 Dash, M., 613, 20; 614, 27; 615, 39; 617, 141, 198; 618,34 Cowan, G., 611, 66; 614,7 Dau, W.D., 616,31 Cox, B.E., 616,31 Daum, K., 616,31 Coyle, P., 611, 66; 614,7 D’Auria, R., 619, 149 Cozzi, M., 615,14 Davenport III, T.F., 618,23 Cozzika, G., 616,31 David, A., 611, 66; 614,7 Cramer, J.G., 612, 181; 616,8 Davidge, D., 616,17 Cranmer, K., 611, 66; 614,7 Davids, B., 615, 167 Crawford, H.J., 612, 181; 616,8 Davidson, P.M., 611,81 Creanza, D., 611, 66; 614,7 Davier, M., 611, 66; 614,7 Crépé-Renaudin, S., 617,1 Davies, A.D., 611,81 Crespo, J.M., 611, 66; 614,7 Davies, G., 611, 66; 614,7 Crespo, M.L., 612, 154 Davies, G.J., 616,17 Cribier, M., 616, 174 Davies, J.C., 616,17 Crochet, P., 612, 173 Davis, A.-C., 611,39 Cruz, N., 619,5 Davis, G.A., 617,1 Csatlós, M., 615, 175 Davis, S.C., 611,39 Cuautle, E., 618,23 Daw, E., 616,17 Cucciarelli, S., 613, 118; 615, 19; 616, 145, 159; 619,71 Dawson, B.R., 619, 271 Cugnon, J., 614,44 Dawson, J.V., 616,17 Cuhadar, T., 615,31 De, K., 617,1 Cui, X.Z., 614, 37; 619, 247 Deacon, A.N., 618,51 Cumalat, J.P., 618,23 De Angelis, G., 615, 160; 619,88 Cummings, M.A.C., 617,1 De Asmundis, R., 613, 118; 615, 19; 616, 145, 159; 619,71 Cundy, D., 615,31 D’Eath, P.D., 613, 181 Curien, D., 615, 160 De Azcárraga, J.A., 615, 127 Curtil, C., 611, 66; 614,7 De Beer, M., 615,31 Cutts, D., 617,1 De Bonis, I., 611, 66; 614,7 396 Cumulative author index to volumes 611–619 (2005) 389–427

Debreczeni, J., 613, 118; 615, 19; 616, 145, 159; 619,71 Deschamps, H., 616,25 Debu, P., 615,31 De Schauenburg, B., 612, 173 Decamp, D., 611, 66; 614,7 De Séréville, N., 615, 167 Dedek, N., 612, 154 Desesquelles, P., 613, 128 De Filippis, N., 611, 66; 614,7 Deshpande, N.G., 615, 111 Déglon, P., 613, 118; 615, 19; 616, 145, 159; 619,71 De Simone, P., 619,61 Degré, A., 613, 118; 615, 19; 616, 145, 159; 619,71 Dessagne, S., 611, 66; 614,7 Dehmelt, K., 613, 118; 615, 19; 616, 145, 159; 619,71 Detraz, C., 619,50 De Huu, M.A., 612, 165 Deur, A., 613, 148 Deiters, K., 613, 118; 615, 19; 616, 145, 159; 619,71 De Wolf, E.A., 616,31 De Jager, C.W., 613, 148 Dey, J., 618, 115 De Jésus, M., 616,25 Dey, M., 618, 115 De Jong, M., 613, 105; 614, 155 Dey, T.K., 613, 208 De Jong, P., 613, 118; 615, 19; 616, 145, 159; 619,71 De Zorzi, G., 619,61 De Jong, S.J., 617,1 Dhamotharan, S., 611, 66; 614,7 Dekhissi, H., 615,14 Dhara, L., 612, 154 De la Cruz, B., 613, 118; 615, 19; 616, 145, 159; 619,71 D’Hose, N., 612, 154 Delaere, C., 611, 66; 614,7 Diaconu, C., 616,31 Del Aguila, F., 613, 170 Diaz Kavka, V., 612, 154 Delaunay, F., 619,82 Dibon, H., 615,31 Delbar, T., 613, 105; 614, 155 Di Capua, E., 613, 105; 614, 155 Delcourt, B., 616,31 Di Capua, F., 613, 105; 614, 155 De Lellis, G., 613, 105; 614, 155 DiCorato, M., 618,23 De Lesquen, A., 616,25 Didenko, L., 612, 181; 616,8 Dell’Agnello, S., 619,61 Di Domenico, A., 619,61 Della Morte, M., 612, 313 Di Donato, C., 619,61 Della Volpe, D., 613, 118; 615, 19; 616, 145, 159; 619,71 Diehl, H.T., 617,1 Delmeire, E., 613, 118; 615, 19; 616, 145, 159; 619,71 Diemoz, M., 613, 118; 615, 19; 616, 145, 159; 619,71 Del Re, G., 616, 174 Dierckxsens, M., 613, 118; 615, 19; 616, 145, 159; 619,71 Deltuva, A., 617,18 Diesburg, M., 617,1 De Lucia, E., 619,61 Dietel, T., 612, 181; 616,8 Demarteau, M., 617,1 Dietl, H., 611, 66; 614,7 De Masi, R., 612, 154 Di Falco, S., 619,61 Demina, R., 617,1 Diget, C.Aa., 618,43 Demine, P., 617,1 Di Micco, B., 619,61 De Miranda, J.M., 618,23 Dingfelder, J., 616,31 Demirchyan, R., 616,31 Dini, P., 618,23 De Moura, M.M., 612, 181; 616,8 Dinkelbach, A.M., 612, 154 Denes, P., 613, 118; 615, 19; 616, 145, 159; 619,71 Dionisi, C., 613, 118; 615, 19; 616, 145, 159; 619,71 Deng, W., 619, 271 Dissertori, G., 611, 66; 614,7 Deng, Z.Y., 614, 37; 619, 247 Di Stefano, P., 616,25 Denig, A., 619,61 Dittmaier, S., 612, 223 Denisov, D., 617,1 Dittmar, M., 613, 118; 615, 19; 616, 145, 159; 619,71 Denisov, O.Yu., 612, 154 Doble, N., 615, 31; 619, 240 Denisov, S.P., 617,1 DØCollaboration, 617,1 Denner, A., 612, 223 Dodonov, V., 616,31 DeNotaristefani, F., 613, 118; 615, 19; 616, 145, 159; 619,71 Dogra, S.M., 616,8 De Palma, M., 611, 66; 614,7 Dolgopolov, A.V., 612, 154 Derevschikov, A.A., 612, 181; 616,8 Dolgorouky, Y., 616,25 De Roeck, A., 616,31 Dombrádi, Zs., 614, 174 De Rosa, G., 613, 105; 614, 155 Donagi, R., 618, 259 Desai, S., 617,1 Dong, L.Y., 614, 37; 618, 34; 619, 247 De Salvo, A., 613, 118; 615, 19; 616, 145, 159; 619,71 Dong, S.J., 612, 21; 617,49 De Santis, A., 619,61 Dong, W.J., 612, 181; 616,8 Desch, K., 616,31 Dong, X., 612, 181; 616,8 Cumulative author index to volumes 611–619 (2005) 389–427 397

Donnachie, A., 611, 255 Dutta, B., 618, 182 Donoghue, J.F., 612, 311 Dutta Majumdar, M.R., 612, 181 Donskov, S.V., 612, 154 Dutta Mazumdar, M.R., 616,8 Dore, U., 613, 105; 614, 155 Dyshkant, A., 617,1 Doria, A., 613, 118; 615, 19; 616, 145, 159; 619, 61, 71 Dželalija, M., 612, 173 Döring, J., 619,88 Dornan, P.J., 611, 66; 614,7 Ealet, A., 611, 66; 614,7 Dorofeev, V.A., 612, 154 Eberl, H., 618, 171 Dosanjh, R.S., 615,31 Ebert, K.H., 616, 174 Doshita, N., 612, 154 Echenard, B., 613, 118; 615, 19; 616, 145, 159; 619,71 Dos Reis, A.C., 618,23 Eckardt, V., 612, 181; 616,8 Døssing, T., 615, 160 Eckerlin, G., 616,31 Doté, A., 613, 140 EDELWEISS Collaboration, 616,25 Dotsenko, V.S., 611, 189 Edera, L., 618,23 Doulas, S., 617,1 Edgecock, T.R., 611, 66; 614,7 Dova, M.T., 613, 118; 615, 19; 616, 145, 159; 619,71 Edmunds, D., 617,1 Dracoulis, G.D., 618,51 Edwards, W.R., 616,8 Dragic, J., 618,34 Eﬁmov, L.G., 612, 181; 616,8 Drain, D., 616,25 Efremenko, V., 616,31 Draper, J.E., 612, 181; 616,8 Efremov, A.V., 612, 233 Draper, T., 612, 21; 617,49 Egli, S., 616,31 Dreossi, D., 619,50 Ehlers, J., 612, 154 Dreucci, M., 619,61 Eichler, R., 616,31 Drevermann, H., 611, 66; 614,7 Eidelman, S., 613, 20; 614, 27; 615, 39; 617, 141, 198; 618,34 Drijard, D., 619,50 Eidelman, S.I., 613,29 Drouart, A., 619,82 Eiges, V., 614, 27; 617, 198; 618,34 Drutskoy, A., 614, 27; 615, 39; 617, 141; 618,34 Eisele, F., 616,31 Du, D.-S., 619, 105 Eisermann, Y., 615, 175 Du, F., 612, 181; 616,8 Eitel, K., 616,25 Du, S.X., 614, 37; 619, 247 El-Nabulsi, R.A., 619,26 Du, Z.Z., 614, 37; 619, 247 Elekes, Z., 614, 174 Dubak, A., 616,31 El Hage, A., 613, 118; 615, 19; 616, 145, 159; 619,71 Dubey, A.K., 612, 181; 616,8 Eline, A., 613, 118; 615, 19; 616, 145, 159; 619,71 Duchesneau, D., 613, 118; 615, 19; 616, 145, 159; 619,71 Ellerbrock, M., 616,31 Duclos, J., 615,31 Ellis, G., 611, 66; 614,7 Duda, M., 613, 118; 615, 19; 616, 145, 159; 619,71 Ellis, J., 618, 162; 619, 17, 30 Dudarev, A., 619,50 Ellison, J., 617,1 Dudko, L.V., 617,1 El Mamouni, H., 613, 118; 615, 19; 616, 145, 159; 619,71 Duffy, C., 616,17 Elsen, E., 616,31 Duﬂot, L., 611, 66; 614,7;617,1 Elsener, K., 619, 240 Dugad, S.R., 617,1 Eltzroth, J.T., 617,1 Duic, V., 612, 154 Elvira, V.D., 617,1 Dukes, E.C., 617,11 Emelianov, V., 612, 181; 616,8 Dumoulin, L., 616,25 Emori, S., 615, 186 Dunbar, D.C., 612,75 Enari, Y., 613, 20; 614, 27; 615, 39; 617, 141; 618,34 Dunin, V.B., 612, 181; 616,8 Engelage, J., 612, 181; 616,8 Dunlop, J.C., 612, 181; 616,8 Engelmann, R., 617,1 Dünnweber, W., 612, 154 Engh, D., 618,23 Duperrin, A., 617,1 Engler, A., 613, 118; 615, 19; 616, 145, 159; 619,71 Dupieux, P., 612, 173 Eno, S., 617,1 Duprel, C., 616,31 Epifanov, D., 613, 20; 615,39 Durandet, C., 617,11 Epifanov, D.A., 613,29 Durkin, T.J., 616,17 Eppard, K., 615,31 Dürr, S., 612, 313 Eppard, M., 615,31 Durrer, R., 614, 125 Eppley, G., 612, 181; 616,8 398 Cumulative author index to volumes 611–619 (2005) 389–427

Eppling, F.J., 613, 118; 615, 19; 616, 145, 159; 619,71 Feng, B., 611, 167 Erazmus, B., 612, 181; 616,8 Ferbel, T., 617,1 Erba, S., 618,23 Ferencei, J., 616,31 Erdmann, W., 616,31 Ferguson, D.P.S., 611, 66; 614,7 Ermolov, P., 617,1 Ferguson, T., 613, 118; 615, 19; 616, 145, 159; 619,71 Ernst, J., 619, 281 Fernandez, E., 611, 66; 614,7 Eroshin, O.V., 617,1 Fernandez-Bosman, M., 611, 66; 614,7 Esposito, L.S., 615,14 Ferrara, S., 619, 149 Estienne, M., 612, 181; 616,8 Ferrari, A., 619,61 Estrada, J., 617,1 Ferrari, N., 616, 174 Evangelou, I., 619,50 Ferrari, R., 611, 215 Evans, H., 617,1 Ferrer, M.L., 619,61 Evdokimov, V.N., 617,1 Ferrero, A., 612, 154 Eversheim, P.D., 612, 154 Ferrero, L., 612, 154 Extermann, P., 613, 118; 615, 19; 616, 145, 159; 619,71 Ferro-Luzzi, M., 619,50 Eyrich, W., 612, 154 Fesefeldt, H., 613, 118; 615, 19; 616, 145, 159; 619,71 Fesquet, M., 616,25 Fabbri, F.L., 618,23 Fiandrini, E., 613, 118; 615, 19; 616, 145, 159; 619,71 Fabbro, B., 611, 66; 614,7 Field, J.H., 613, 118; 615, 19; 616, 145, 159; 619,71 Fabro, M., 612, 154 Filimonov, K., 612, 181; 616,8 Fachini, P., 612, 181; 616,8 Filip, P., 612, 181; 616,8 Faessler, M., 612, 154 Filthaut, F., 613, 118; 615, 19; 616, 145, 159; 617,1;619,71 Faestermann, T., 615, 175 Finch, A.J., 611, 66; 614,7 Fahlander, C., 619,88 Finch, E., 612, 181; 616,8 Faine, V., 612, 181 Findlay, J., 619, 271 Faivre, J., 612, 181; 616,8 Fine, V., 616,8 Falagan, M.A., 613, 118; 615, 19; 616, 145, 159; 619,71 Finger, M., 612, 154 Falaleev, V., 612, 154; 615,31 Finger Jr., M., 612, 154 Falciano, S., 613, 118; 615, 19; 616, 145, 159; 619,71 Finley, C.B., 619, 271 Fallot, M., 613, 128 Finocchiaro, G., 619,61 Falvard, A., 611, 66; 614,7 Fiorillo, G., 613, 105; 614, 155 Fang, F., 617, 141 Fiorini, L., 615,31 Fang, J., 614, 37; 619, 247 Fiorucci, S., 616,25 Fang, S.S., 614, 37; 619, 247 Fischer, C.S., 611, 279 Fantechi, R., 615,31 Fischer, G., 615,31 Farley, A.N.St.J., 613, 181 Fischer, H., 612, 154 Farnea, E., 619,88 Fisher, P.H., 613, 118; 615, 19; 616, 145, 159; 619,71 Fatemi, R., 612, 181; 616,8 Fisher, W., 613, 118; 615, 19; 616, 145, 159; 619,71 Fauland, P., 612, 154 Fisk, H.E., 617,1 Faulkner, P.J.W., 616,31 Fisk, I., 613, 118; 615, 19; 616, 145, 159; 619,71 Favara, A., 613, 118; 615, 19; 616, 145, 159; 619,71 Fisyak, Y., 612, 181; 616,8 Favart, D., 613, 105; 614, 155 Fleischer, M., 616,31 Favart, L., 616,31 Fleischmann, P., 616,31 Fay, J., 613, 118; 615, 19; 616, 145, 159; 619,71 Fleming, Y.H., 616,31 Fayard, L., 615,31 Fleurot, F., 615, 167 Fayolle, D., 611, 66; 614,7 Flierl, D., 612, 181 Fedin, O., 613, 118; 615, 19; 616, 145, 159; 619,71 Flucke, G., 616,31 Fedorisin, J., 616,8 Flügge, G., 616,31 Fedorova, Y., 619, 271 Foà, L., 611, 66; 614,7 Fedotov, A., 616,31 Focardi, E., 611, 66; 614,7 Fedotovitch, G.V., 613,29 FOCUS Collaboration, 618,23 Felcini, M., 613, 118; 615, 19; 616, 145, 159; 619,71 Fodor, Z., 612, 173 Felici, G., 619,61 Foley, K.J., 612, 181 Felix, J., 617,11 Fomenko, A., 616,31 Felst, R., 616,31 Fomenko, K., 616,8 Cumulative author index to volumes 611–619 (2005) 389–427 399

FOPI Collaboration, 612, 173 Gadelha, A.L., 613, 213 Forconi, G., 613, 118; 615, 19; 616, 145, 159; 619,71 Gagliardi, C.A., 612, 181; 616,8 Foresti, I., 616,31 Gagunashvili, N., 612, 181 Formánek, J., 616,31 Gaillard, L., 616,8 Formica, A., 615,31 Gaillard, M.K., 612, 304 Forti, C., 619,61 Gaines, I., 618,23 Fortner, M., 617,1 Gajdosik, T., 618, 171 Forty, R.W., 611, 66; 614,7 Galaktionov, Yu., 613, 118; 615, 19; 616, 145, 159; 619,71 Foster, F., 611, 66; 614,7 Gallas, E., 617,1 Fouchez, D., 611, 66; 614,7 Gallas, M.V., 619,50 Fox, H., 615, 31; 617,1 Galyaev, A.N., 617,1 Frabetti, P.L., 615,31 Gamble, T., 616,17 Fraga, E.S., 614, 181 Gamboa, J., 619, 367 Fraile, L.M., 618,43 Ganguli, S.N., 613, 118; 615, 19; 616, 145, 159; 619,71 Frank, A., 613, 134 Ganis, G., 611, 66; 614,7 Frank, M., 611, 66; 614,7 Gans, J., 612, 181; 616,8 Franke, G., 616,31 Ganti, M.S., 612, 181; 616,8 Frankfurt, L., 616, 59; 619,95 Gao, C.J., 612, 127 Frankland, J., 613, 128 Gao, C.S., 614, 37; 619, 247 Franz, J., 612, 154 Gao, H., 613, 148 Franzini, P., 619,61 Gao, M., 617,1 Frascaria, N., 613, 128 Gao, Y., 611, 66; 614,7 Fratina, S., 613, 20; 614, 27; 615, 39; 617, 141, 198; 618,34 Gao, Y.N., 614, 37; 619, 247 Freeman, S.J., 618,51 Garbincius, P.H., 618,23 Frekers, D., 612, 165; 613, 105; 614, 155 Garbrecht, B., 612, 311 Freudenreich, K., 613, 118; 615, 19; 616, 145, 159; 619,71 Garcia-Abia, P., 613, 118; 615, 19; 616, 145, 159; 619,71 Friar, J.L., 618,68 Garcia-Bellido, A., 611, 66; 614,7 Friedrich, J.M., 612, 154 Gardner, R., 618,23 Fries, R.J., 618,77 Garfagnini, R., 612, 154 Frigerio, M., 612,29 Garibaldi, F., 613, 148 Frising, G., 616,31 Garmash, A., 613, 20; 614, 27; 615, 39; 617, 198; 618,34 Frolov, V., 612, 154 Garren, L.A., 618,23 Fry, J.N., 612, 122 Garrido, Ll., 611, 66; 614,7 Fu, C.D., 614, 37; 619, 247 Garutti, E., 616,31 Fu, H.Y., 614, 37; 619, 247 Garvey, J., 616,31 Fu, J., 612, 181; 616,8 Gascon, J., 616,25 Fu, S., 617,1 Gasparic, I., 612, 173 Fu, Y., 617,11 Gataullin, M., 613, 118; 615, 19; 616, 145, 159; 619,71 Fuchs, U., 612, 154 Gates, G., 613, 148 Fuess, S., 617,1 Gatignon, L., 615, 31; 619, 240 Fujii, K., 611, 223 Gatti, C., 619,61 Fujii, Y., 616, 141 Gatto, R., 615, 297 Fukuda, S., 619, 255 Gaudichet, L., 612, 181; 616,8 Fukuda, Y., 619, 255 Gautheron, F., 612, 154 Fülöp, Zs., 614, 174 Gauzzi, P., 619,61 Fulton, B.R., 618,43 Gavrichtchouk, O.P., 612, 154 Furetta, C., 613, 118; 615, 19; 616, 145, 159; 619,71 Gavrilov, V., 617,1 Fynbo, H.O.U., 618,43 Gay, P., 611, 66; 614,7 Gayler, J., 616,31 Gabadadze, G., 617, 124 Gazizov, A.Z., 612, 147 Gabathuler, E., 616,31 Ge, X.-H., 612,61 Gabathuler, K., 616,31 Gehrmann, T., 612, 36, 49 Gabyshev, N., 613, 20; 614, 27; 615, 39; 617, 141, 198; 618,34 Gehrmann-De Ridder, A., 612, 36, 49 Gabyshev, N.I., 613,29 GEM Collaboration, 619, 281 Gadea, A., 619,88 Geng, C.Q., 619, 305 400 Cumulative author index to volumes 611–619 (2005) 389–427

Genser, K., 617,1 Göbel, C., 618,23 Gentile, S., 613, 118; 615, 19; 616, 145, 159; 619,71 Godbole, R.M., 618, 193 Gérard, J.-M., 616,85 Godłowski, W., 619, 219 Gerassimov, S., 612, 154 Goeke, K., 612, 233; 618,90 Gerber, C.E., 617,1 Goerlich, L., 616,31 Gerbier, G., 616,25 Goertz, S., 612, 154 Gerhards, R., 616,31 Gogitidze, N., 616,31 Gerland, L., 619,95 Gogohia, V., 611, 129; 618, 103 Gerlich, C., 616,31 Gokhroo, G., 613, 20; 614, 27; 615, 39; 617, 141, 198; 618,34 Germain, M., 612, 181 Golak, J., 617,18 Gerndt, J., 619,50 Goldbach, C., 616,25 Gershon, T., 613, 20; 614, 27; 615, 39; 617, 141, 198; 618,34 Goldberg, J., 613, 105; 614, 155 Gershon, T.J., 615,31 Goldin, D., 619,50 Gershtein, Y., 617,1 Golec-Biernat, K., 613, 154 Geurts, F., 612, 181; 616,8 Golob, B., 614, 27; 617, 141; 618,34 Geweniger, C., 611, 66; 614,7 Gómez, B., 617,1 Geyer, R., 612, 154 Gómez, F., 619,50 Gharibyan, V., 611, 231 Gómez, M.E., 618, 162 Ghazaryan, S., 616,31 Gomis, J., 617, 182 Ghazikhanian, V., 612, 181; 616,8 Goncharov, P.I., 617,1 Ghete, V.M., 611, 66; 614,7 Goncharov, Yu.P., 617,67 Ghosh, A., 616, 114 Gong, M.Y., 614, 37; 619, 247 Ghosh, P., 612, 181; 616,8 Gong, W.X., 614, 37; 619, 247 Ghosh, S., 618, 243 Gong, Z.F., 613, 118; 615, 19; 616, 145, 159; 619,71 Ghosh, S.K., 611,27 Gonidec, A., 615,31 Giacomelli, G., 615,14 Gonzalez, J.E., 612, 181; 616,8 Giacomich, R., 619,50 González, S., 611, 66; 614,7 Giagu, S., 613, 118; 615, 19; 616, 145, 159; 619,71 González Felipe, R., 618,7 Giammanco, A., 611, 66; 614,7 Gorbachev, D.A., 613,29 Gianini, G., 618,23 Gorbar, E.V., 611, 207 Giannakis, I., 611, 137 Gorbounov, S., 616,31 Giannini, G., 611, 66; 614,7 Gorbunov, P., 613, 105; 614, 155 Gianoli, A., 615,31 Gorchakov, O., 619,50 Gianotti, F., 611, 66; 614,7 Gorin, A., 619,50 Gianotti, P., 619,50 Gorin, A.M., 612, 154 Giassi, A., 611, 66; 614,7 Gorini, B., 615,31 Gibelin, J., 614, 174 Gorini, E., 619,61 Gidal, G., 617,11 Gorišek, A., 617, 141 Gillibert, A., 619,82 Górska, M., 619,88 Gilman, R., 613, 148 Gottschalk, E., 618,23 Ginther, G., 617,1 Gounder, K., 617,1 Ginzburgskaya, S., 616,31 Goussiou, A., 617,1 Giorgi, M., 612, 154 Govi, G., 615,31 Giorgini, M., 615,14 Goy, C., 611, 66; 614,7 Giot, L., 619,82 Grab, C., 616,31 Giovannella, S., 619,61 Grachov, O., 612, 181; 616,8 Girone, M., 611, 66; 614,7 Graesser, M.L., 611, 53; 613,5 Girtler, P., 611, 66; 614,7 Grafström, P., 615, 31; 619, 240 Giudici, S., 615,31 Grajek, O.A., 612, 154 Glöckle, W., 617,18 Gran, R., 619, 255 Glover, E.W.N., 612, 36, 49 Grange, P., 616, 135 Glück, F., 619, 263 Granier de Cassagnac, R., 615,31 GNO Collaboration, 616, 174 Grannis, P.D., 617,1 Go, A., 614, 27; 617, 198 Grassi, C., 615, 160 Gobbo, B., 612, 154 Grässler, H., 616,31 Cumulative author index to volumes 611–619 (2005) 389–427 401

Grasso, A., 612, 154 Gupta, V.K., 613, 118; 615, 19; 616, 145, 159; 619,71 Graw, G., 615, 175 Gurtu, A., 613, 118; 615, 19; 616, 145, 159; 619,71 Grawe, H., 619,88 Gurzhiev, S.N., 617,1 Graziani, E., 619,61 Gustafson, H.R., 617,11 Graziani, G., 615,31 Gutay, L.J., 613, 118; 615, 19; 616, 145, 159; 619,71 Grebeniuk, A.A., 613,29 Gutierrez, G., 617,1 Grebenyuk, O., 612, 181; 616,8 Gutierrez, P., 617,1 Green, M.G., 611, 66; 614,7 Gutierrez, T.D., 612, 181; 616,8 Greenﬁeld, M.B., 615, 193 Gwilliam, C., 616,31 Greenlee, H., 617,1 Greenshaw, T., 616,31 H1 Collaboration, 616,31 Greenwood, Z.D., 617,1 Haas, D., 613, 118; 615, 19; 616, 145, 159; 619,71 Grégoire, G., 613, 105; 614, 155 Haba, J., 613, 20; 614, 27; 617, 141, 198; 618,34 Gregori, M., 616,31 Haba, N., 615, 247 Grella, G., 613, 105; 614, 155 Habs, D., 615, 175 Grenier, G., 613, 118; 615, 19; 616, 145, 159; 619,71 Hadley, N.J., 617,1 Greub, C., 616, 93; 619, 322 Hagemann, G.B., 615, 160 Grigoriev, D.N., 613,29 Haggerty, H., 617,1 Grigorieva, S.I., 612, 147 Hagino, K., 615,55 Grimm, O., 613, 118; 615, 19; 616, 145, 159; 619,71 Hagopian, S., 617,1 Grindhammer, G., 616,31 Hagopian, V., 617,1 Grinstein, B., 615, 213 Haidt, D., 616,31 Grinstein, S., 617,1 Hajduk, L., 616,31 Grisa, L., 617, 124 Hall, R.E., 617,1 Grishkin, Yu., 612, 173 Haller, J., 616,31 Grivaz, J.-F., 611, 66; 614,7 Hallman, T.J., 612, 181; 616,8 Groer, L., 617,1 Hamaguchi, K., 617,99 Gronstal, S., 612, 181 Hamed, A., 612, 181; 616,8 Groot Nibbelink, S., 616, 125 Hammond, N.J., 618,51 Gros, M., 616,25 Hampel, W., 616, 174 Grosnick, D., 612, 181; 616,8 Han, C., 617,1 Grube, B., 612, 154 Han, L., 618, 209 Gruenewald, M.W., 613, 118; 615, 19; 616, 145, 159; 619,71 Han, T., 616, 215 Grünemaier, A., 612, 154 Handler, T., 618,23 Grünendahl, S., 617,1 Hanke, P., 611, 66; 614,7 Grupen, C., 611, 66; 614,7 Hanlon, W.F., 619, 271 Grzelinska,´ A., 611, 116 Hannappel, J., 612, 154 Gu, P.-H., 619, 226 Hannen, V.M., 612, 165 Gu, S.D., 614, 37; 619, 247 Hansen, H.D., 619, 240 Guaraldo, C., 619,50 Hansen, J.B., 611, 66; 614,7 Guchait, M., 618, 193 Hansen, J.D., 611, 66; 614,7 Guedon, M., 612, 181 Hansen, J.R., 611, 66; 614,7 Guertin, S.M., 612, 181; 616,8 Hansen, P.H., 611, 66; 614,7 Guida, M., 613, 118; 615, 19; 616, 145, 159; 619,71 Hansen, S., 617,1 Guillot, J., 615, 167 Hansroul, M., 619,50 Güler, M., 613, 105; 614, 155 Hansson, J., 616,1 Gulyás, J., 615, 175 Hansson, M., 616,31 Gumbsheimer, R., 616,25 Hao, H., 618,97 Guo, W.J., 617,24 Hara, K., 618,34 Guo, Y., 616,8 Hara, T., 613, 105; 614, 27, 155; 618, 34; 619, 255 Guo, Y.N., 614, 37; 619, 247 Harakeh, M.N., 612, 165; 615, 167, 175 Guo, Y.Q., 614, 37; 619, 247 Hardtke, D., 612, 181; 616,8 Guo, Z.J., 614, 37; 619, 247 Harindranath, A., 617,92 Gupta, A., 612, 181; 616,8 Harris, F.A., 614, 37; 619, 247 Gupta, K.S., 618, 237 Harris, J.W., 612, 181; 616,8 402 Cumulative author index to volumes 611–619 (2005) 389–427

Hart, S.P., 616,17 Herrmann, N., 612, 173 Hartmann, F.X., 616, 174 Herskind, B., 615, 160 Hartmann, O.N., 612, 173 Hertenberger, R., 615, 175 Harvey, J., 611, 66; 614,7 Hervé, A., 613, 118; 615, 19; 616, 145, 159; 619,71 Hasegawa, M., 617, 150; 619, 255 Herynek, I., 616,31 Hasegawa, T., 612, 154; 619, 255 Heß, C., 612, 154 Hasenfratz, P., 613,57 Hess, J., 611, 66; 614,7 Haseyama, T., 615, 186 Heuer, R.-D., 616,31 Hashimoto, M., 611, 207 Heusse, Ph., 611, 66; 614,7 Hastings, N.C., 618,34 Heusser, G., 616, 174 Hatakeyama, A., 611, 239 High Resolution Fly’s Eye Collaboration, 619, 271 Hatanaka, K., 615, 193 Higuchi, T., 614,27 Hatanaka, T., 619, 352 Hildebrandt, M., 616,31 Hatano, M., 615, 193 Hildenbrand, K.D., 612, 173 Hatzifotiadou, D., 613, 118; 615, 19; 616, 145, 159; 619,71 Hildreth, M.D., 617,1 Hauptman, J.M., 617,1 Hill, J., 619, 255 Hawranek, P., 619, 281 Hillenbach, M., 616, 125 Hay, B., 615,31 Hiller, K.H., 616,31 Hayasaka, K., 613, 20; 614, 27; 615, 39; 617, 141, 198; 618,34 Hinterberger, F., 612, 154 Hayashi, K., 619, 255 Hinz, L., 613, 20; 614, 27; 617, 141, 198; 618,34 Hayashii, H., 613, 20; 614, 27; 615, 39; 618,34 Hippolyte, B., 612, 181; 616,8 Hayato, Y., 619, 255 Hirayama, Y., 611, 239 Hayes, O.J., 611, 66; 614,7 Hirosky, R., 617,1 Hazumi, M., 613, 20; 614, 27; 617, 141, 198; 618,34 Hirsch, A., 612, 181; 616,8 He, H., 611, 66; 614,7 Hirsch, J.G., 613, 134 He, K.L., 614, 37; 619, 247 Hirschfelder, J., 613, 118; 615, 19; 616, 145, 159; 619,71 He, L., 615,93 Hjort, E., 612, 181; 616,8 He, M., 614, 37; 619, 247 Hobbs, J.D., 617,1 He, X., 614, 37; 619, 247 Hodgson, P.N., 611, 66; 614,7 He, Y.-H., 618, 252, 259 Hoeneisen, B., 617,1 Hebbeker, T., 613, 118; 615, 19; 616, 145, 159; 619,71 Hofer, H., 613, 118; 615, 19; 616, 145, 159; 619,71 Hebert, C., 617,1 Hoffman, C.M., 619, 271 Hedicke, S., 612, 154 Hoffmann, D., 616,31 Hedin, D., 617,1 Hoffmann, G.W., 612, 181; 616,8 Heinmiller, J.M., 617,1 Hofmann, F., 612, 165 Heinsius, F.H., 612, 154 Hohlmann, M., 613, 118; 615, 19; 616, 145, 159; 619,71 Heinson, A.P., 617,1 Hokuue, T., 613, 20; 614, 27; 617, 141; 618,34 Heintz, U., 617,1 Holder, M., 615,31 Heinz, M., 612, 181; 616,8 Hölldorfer, F., 611, 66; 614,7 Heinzelmann, G., 616,31 Hollingworth, R., 616,17 Heitger, J., 612, 313 Holmstrom, T., 617,11 Hellström, M., 619,88 Holstein, B.R., 612, 311 Henderson, R.C.W., 616,31 Holzner, G., 613, 118; 615, 19; 616, 145, 159; 619,71 Heng, Y.K., 614, 37; 619, 247 Holzscheiter, M.H., 619, 271 Henley, E.M., 611, 111 Homer, G.J., 616,17 Henrich, E., 616, 174 Hong, B., 612, 173 Henry, T.W., 612, 181; 616,8 Horikawa, N., 612, 154 Henschel, H., 616,31 Horikawa, S., 612, 154 Henshaw, O., 616,31 Horisberger, R., 616,31 Hepp, V., 611, 66; 614,7 Horn, M., 616,25 Heppelmann, S., 612, 181 Horns, D., 611, 297 Hepplemann, S., 616,8 Horsley, M., 612, 181 Hermann, R., 612, 154 Horváth, Á., 614, 174 Hernandez, H., 618,23 Horváth, I., 612, 21; 617,49 Herrera, G., 616,31 Horváthy, P.A., 615,87 Cumulative author index to volumes 611–619 (2005) 389–427 403

Hosack, M., 618,23 Iijima, T., 613, 20; 614, 27; 615, 39; 617, 141, 198; 618,34 Hosek, R., 619,50 Ijaduola, R.B., 612, 154 Hoshi, Y., 613, 20; 614, 27; 615, 39; 617, 141, 198; 618,34 Ikeda, A., 619, 255 Hoshino, K., 613, 105; 614, 155 Ilgenfritz, E.-M., 611,27 Hosotani, Y., 615, 257 Ilgner, C., 612, 154 Höting, P., 616,31 Iliescu, M., 619,50 Hou, H.-S., 618, 209 Illingworth, R., 617,1 Hou, S., 613, 20; 614, 27; 615, 39; 617, 141, 198; 618,34 Imbergamo, E., 615,31 Hou, S.R., 613, 118; 615, 19; 616, 145, 159; 619,71 Imoto, A., 613, 20; 614, 27; 615, 39; 617, 141, 198; 618,34 Hou, W.-S., 613, 20; 614, 27; 615, 39; 617, 141, 198; 618,34 Inagaki, T., 619, 255 Hovhannisyan, A., 616, 31; 619, 322 Inami, K., 613, 20; 614, 27; 615, 39; 617, 141, 198; 618,34 Howard, A.S., 616,17 Incagli, M., 619,61 Hristov, P., 615,31 Inzani, P., 618,23 Hristova, I.R., 613, 105; 614, 155 Ioukaev, A.I., 612, 154 Hsiao, Y.K., 619, 305 Ishida, T., 619, 255 Hsiung, Y.B., 617, 141 Ishihara, A., 612, 181; 616,8 Hu, H., 611, 66; 614,7 Ishii, T., 619, 255 Hu, H.M., 614, 37; 619, 247 Ishikawa, A., 613, 20; 614, 27; 615, 39; 617, 141, 198; 618,34 Hu, T., 614, 37; 619, 247 Ishimoto, S., 612, 154 Huang, G.S., 614, 37; 619, 247 Ishitsuka, M., 619, 255 Huang, H.Z., 612, 181; 616,8 Ismail, M., 616,31 Huang, J., 617,1 Itaco, N., 616,43 Huang, L., 614, 37; 619, 247 Itakura, K., 615, 221 Huang, M., 617,11 Ito, A.S., 617,1 Huang, S.L., 612, 181; 616,8 Itoh, R., 614, 27; 615, 39; 617, 141, 198; 618,34 Huang, T., 611, 260 Itow, Y., 619, 255 Huang, W.-H., 615, 266 Ivaniouchenkov, I., 616,17 Huang, X., 611, 66; 614,7 Ivanov, O., 612, 154 Huang, X.P., 614, 37; 619, 247 Ivanov, R.I., 611,34 Huang, Y., 617,1 Ivanova, T.A., 612,65 Huber, P., 617, 167 Iwasa, N., 614, 174 Hughes, E., 612, 181 Iwasaki, H., 614, 174 Hughes, E.W., 613, 148; 616,8 Iwasaki, M., 613, 20; 615, 39; 617, 141, 198; 618,34 Hughes, G., 611, 66; 614,7 Iwasaki, Y., 613, 20; 614, 27; 615, 39; 617, 141; 618,34 Hughes, G.A., 619, 271 Iwashita, T., 619, 255 Humanic, T.J., 612, 181; 616,8 Iwata, T., 612, 154 Hüntemeyer, P., 619, 271 Izumi, H., 611, 239 Hunyadi, M., 615, 175 Hurth, T., 619, 322 Jacholkowska, A., 611, 66; 614,7 Hutchcroft, D.E., 611, 66; 614,7 Jack, I., 611, 199 Hüttmann, K., 611, 66; 614,7 Jackson, K.P., 611, 239 HyperCP Collaboration, 617,11 Jacobs, P., 612, 181; 616,8 Jacobs, W.W., 612, 181; 616,8 Iacopini, E., 615,31 Jacquet, M., 616,31 Iancu, E., 615, 221 Jaffré, M., 617,1 Iaselli, G., 611, 66; 614,7 Jahn, R., 612, 154; 619, 281 Iashvili, I., 617,1 Jain, S., 617,1 Ibbotson, M., 616,31 Jain, V., 617,1 Ibe, M., 615, 120 Jakobs, K., 611, 66; 614,7 Ichikawa, A.K., 619, 255 James, C., 617,11 Ichikawa, Y., 614, 174 Janas, Z., 619,88 Iconomidou-Fayard, L., 615,31 Janata, A., 612, 154 Ideguchi, E., 614, 174 Janauschek, L., 616,31 Ignatov, F.V., 613,29 Jang, H.I., 619, 255 Igo, G., 612, 181; 616,8 Jang, J.S., 619, 255 404 Cumulative author index to volumes 611–619 (2005) 389–427

Janik, M., 612, 181; 616,8 Juget, F., 615, 153 Janot, P., 611, 66; 614,7 Jui, C.C.H., 619, 271 Jansen, K., 619, 184 Juillard, A., 616,25 Janssen, X., 616,31 Jung, C.K., 619, 255 Janssens, R.V.F., 618,51 Jung, E., 614, 78; 615, 273; 619, 347 Jarlskog, C., 615, 207 Jung, H., 616,31 Jastrz˛ebski, J., 615,55 Juste, A., 617,1 Jeitler, M., 615,31 Jemanov, V., 616,31 K2K Collaboration, 619, 255 Jenkins, A., 613,5 Kabana, S., 612, 181; 616,8 Jenkins, C.M., 617,11 Kabuß, E., 612, 154 Jeon, E.J., 619, 255 Kachelrieß, M., 614,1 Jeppesen, H.B., 618,43 Kado, M., 611, 66; 614,7 Jesik, R., 617,1 Kaether, F., 616, 174 Jessen, K., 619, 240 Kahl, W., 617,1 Jézéquel, S., 611, 66; 614,7 Kahn, S., 617,1 Jha, V., 619, 281 Kajfasz, E., 617,1 Ji, X.B., 614, 37; 619, 247 Kajita, T., 619, 255 Jia, Q.Y., 614, 37; 619, 247 Kalantar-Nayestanaki, N., 617,18 Jiang, C.H., 614, 37; 619, 247 Kalinin, A.M., 617,1 Jiang, H., 612, 181; 616,8 Kalinin, S., 613, 105; 614, 155 Jiang, J., 615, 111 Kalinnikov, V., 612, 154 Jiang, W.Z., 617,33 Kalinovsky, Y., 614,44 Jiang, X., 613, 148 Kalmus, G.E., 615,31 Jiang, X.S., 614, 37; 619, 247 Kalmykov, Y., 612, 165 Jiang, Y., 618, 209 Kalter, A., 615,31 Jin, B.N., 613, 118; 615, 19; 616, 145, 159; 619,71 Kamada, H., 617,18 Jin, D.P., 614, 37; 619, 247 Kameda, D., 615, 186 Jin, S., 611, 66; 614,7,37;619, 247 Kameda, J., 619, 255 Jin, Y., 614, 37; 619, 247 Kamiya, J., 615, 193 Jindal, P., 613, 118; 615, 19; 616, 145, 159; 619,71 Kamleh, W., 616, 196 Johns, K., 617,1 Kamon, T., 611, 223; 618, 182 Johns, W.E., 618,23 Kaneko, K., 617, 150 Johnson, D.P., 616,31 Kaneyuki, K., 619, 255 Johnson, I., 612, 181 Kang, D., 612, 154 Johnson, M., 617,1 Kang, J.H., 613, 20; 614, 27; 615, 39; 617, 141, 198; 618,34 Jonckheere, A., 617,1 Kang, J.S., 613, 20; 614, 27; 615, 39; 617, 141, 198; 618, 23, 34 Jones, D.R.T., 611, 199 Kang, K., 616,8 Jones, G.D., 618,51 Kang, S.K., 619, 129 Jones, L.W., 613, 118; 615, 19; 616, 145, 159; 619,71 Kanno, S., 614, 174 Jones, P.G., 612, 181; 616,8 Kant, D., 616,31 Jones, R.W.L., 611, 66; 614,7 Kanungo, R., 614, 174 Jones, T.D., 617,11 Kao, C., 614,67 Jones, W.G., 616,17 Kapichine, M., 616,31 Jonson, B., 618,43 Kaplan, D.M., 617,11 Jönsson, L., 616,31 Kaplan, M., 612, 181; 616,8 Joo, K.K., 619, 255 Kapusta, P., 614, 27; 618,34 Joosten, R., 612, 154 Karliner, M., 612, 197; 616, 191 Josa-Mutuberría, I., 613, 118; 615, 19; 616, 145, 159; 619,71 Karlsson, M., 616,31 Joshi, M.K., 616,17 Karmanov, D., 617,1 Jost, B., 611, 66; 614,7 Karmgard, D., 617,1 Jöstlein, H., 617,1 Karpov, S.V., 613,29 Jouravlev, N.I., 612, 154 Karpukhin, V., 619,50 Jousset, J., 611, 66; 614,7 Karstens, F., 612, 154 Judd, E.G., 612, 181; 616,8 Kasper, P.H., 618,23 Cumulative author index to volumes 611–619 (2005) 389–427 405

Kastaun, W., 612, 154 Kiko, J., 616, 174 Kataoka, S.U., 613, 20; 615, 39; 618,34 Kile, J., 611, 66; 614,7 Katayama, N., 613, 20; 614, 27; 615, 39; 617, 141, 198; 618,34 Kilian, K., 619, 281 Kato, G., 615, 186 Kim, B.J., 619, 255 Kato, H., 615, 193 Kim, C.O., 619, 255 Kato, I., 619, 255 Kim, C.S., 619, 129 Kato, Y., 611, 223 Kim, D.Y., 618,23 Katzy, J., 616,31 Kim, H.D., 616, 108 Kaur, M., 613, 118; 615, 19; 616, 145, 159; 619,71 Kim, H.J., 613, 20; 614, 27; 615, 39; 617, 141; 618,34 Kaus, P., 611, 147 Kim, J.E., 612, 293; 616, 108 Kawada, J., 613, 105; 614, 155 Kim, J.H., 613, 20; 614, 27; 615, 39; 618,34 Kawai, H., 611, 269; 613, 20; 614, 27; 615, 39; 617, 141, 198; Kim, J.K., 613, 118; 615, 19; 616, 145, 159; 619,71 618,34 Kim, J.Y., 619, 255 Kawai, S., 614, 174 Kim, K., 619, 271 Kawamura, T., 613, 105; 614, 155 Kim, S.B., 619, 255 Kawasaki, M., 618,1;619, 233 Kim, S.H., 614, 78; 615, 273; 619, 347 Kawasaki, T., 613, 20; 614, 27; 615, 39; 617, 141, 198; 618,34 Kim, S.K., 613, 20; 615, 39; 617, 141; 618,34 Kayis-Topaksu, A., 613, 105; 614, 155 Kim, S.M., 613, 20; 614, 27; 615, 39; 617, 141; 618,34 Kayser, F., 611, 66; 614,7 Kim, Y.J., 612, 173 Kazanin, V.F., 613,29 King, S.F., 618, 150 Keane, D., 612, 181; 616,8 Kinoshita, K., 614, 27; 617, 141, 198; 618,34 Kearns, E., 619, 255 Kirchner, R., 619,88 Kecskemeti, J., 612, 173 Kirejczyk, M., 612, 173 Keeley, N., 619,82 Kirillov, D., 619, 281 Kehoe, R., 617,1 Kirkby, J., 613, 118; 615, 19; 616, 145, 159; 619,71 Kekelidze, V., 615,31 Kirkpatrick, J., 616,17 Kelic,´ A., 616,48 Kirn, M.A., 619, 271 Keller, N., 616,31 Kirsten, T., 616, 174 Kemper, K.W., 619,82 Kiryluk, J., 612, 181; 616,8 Kennedy, J., 611, 66; 614,7 Kiš, M., 617,18 Kenyon, I.R., 616,31 Kisiel, A., 612, 181; 616,8 Kerler, W., 617,60 Kisielinski,´ M., 615,55 Kesisoglou, S., 617,1 Ketov, S.V., 619, 352 Kislov, E.M., 616,8 Ketzer, B., 612, 154 Kisselev, Yu., 612, 154 Khalil, S., 618, 201 Kitamura, Y., 615, 193 Khan, E., 613, 128 Kitazawa, M., 615, 102 Khan, H.R., 613, 20; 615, 39; 617, 141, 198; 618,34 Kittel, W., 613, 118; 615, 19; 616, 145, 159; 619,71 Khanov, A., 617,1 Klay, J., 612, 181; 616,8 Kharchilava, A., 617,1 Klein, F., 612, 154 Khaustov, G.V., 612, 154 Klein, M., 616,31 Khazin, B.I., 613,29 Klein, S.R., 612, 181; 616,8 Khlebnikov, S., 615,55 Kleinert, H., 611, 182 Khodyrev, V.Yu., 612, 181; 616,8 Kleinknecht, K., 611, 66; 615,31 Khokhlov, Yu.A., 612, 154 Kleinwort, C., 616,31 Khomutov, N.V., 612, 154 Kliczewski, S., 619, 281 Khoo, T.L., 618,51 Klima, B., 617,1 Khotilovich, V., 611, 223; 618, 182 Klimentov, A., 613, 118; 615, 19; 616, 145, 159; 619,71 Khovansky, V., 613, 105; 614, 155 Klimkovich, T., 616,31 Kibayashi, A., 619, 255 KLOE Collaboration, 619,61 Kichimi, H., 613, 20; 614, 27; 615, 39; 617, 141; 618,34 Kluge, E.E., 611, 66; 614,7 Kielczewska, D., 619, 255 Kluge, T., 616,31 Kiener, J., 615, 167 Kluge, W., 619,61 Kienzle-Focacci, M.N., 613, 118; 615, 19; 616, 145, 159; 619,71 Kluson, J., 619,50 Kiesling, C., 616,31 Klyachko, A., 612, 181 Kijima, G., 615, 186 Kmiecik, M., 615, 160 406 Cumulative author index to volumes 611–619 (2005) 389–427

Kneringer, E., 611, 66; 614,7 Kotchenda, L., 612, 181; 616,8 Knies, G., 616,31 Kotcher, J., 617,1 Knowles, I., 615,31 Kothari, B., 617,1 Knudsen, H., 619, 240 Kotte, R., 612, 173 Knutsson, A., 616,31 Kotzinian, A.M., 612, 154 Ko, B.R., 618,23 Kou, E., 616,85 Ko, P., 611,87 Koutchinski, N.A., 612, 154 Koang, D.H., 615, 153 Koutouev, R., 616,31 Kobayashi, K., 619, 255 Koutsenko, V., 613, 118; 615, 19; 616, 145, 159; 619,71 Kobayashi, M., 619,50 Kovalenko, A.D., 612, 181 Kobayashi, T., 619, 255 Kovalenko, A.V., 613,52 Koblitz, B., 616,31 Kowalczyk, M., 615,55 Koblitz, S., 612, 154 Kowalik, K., 612, 154 Koch, U., 615,31 Kozelov, A.V., 617,1 Koczon, P., 612, 173 Kozlovsky, E.A., 617,1 Kodama, K., 613, 105; 614, 155 Kraan, A.C., 611, 66; 614,7 Koetke, D.D., 612, 181; 616,8 Kräber, M., 613, 118; 615, 19; 616, 145, 159; 619,71 Kohli, J.M., 617,1 Kraemer, R.W., 613, 118; 615, 19; 616, 145, 159; 619,71 Koivuniemi, J.H., 612, 154 Kramer, K., 613, 148 Kokkas, P., 619,50 Kramer, M., 612, 181; 616,8 Kolbe, E., 616,48 Krane, J., 617,1 Kolev, D., 613, 105; 614, 155; 619, 281 Kraniotis, G.V., 611, 156 Kollegger, T., 612, 181; 616,8 Krasznahorkay, A., 615, 175 Kolosov, V.N., 612, 154 Kravchuk, N.P., 612, 154 Komarov, V., 619,50 Kravchuk, V.L., 615, 167 Komatsu, M., 613, 105; 614, 155 Kravcikova, M., 619, 281 Komissarov, E.V., 612, 154 Kravtsov, P., 612, 181; 616,8 Kondev, F.G., 618,51 Kravtsov, V.I., 612, 181; 616,8 Kondo, K., 612, 154 Krawiec, A., 619, 219 Kondo, K.-I., 619, 377 Krein, G., 614, 181 Kondo, Y., 611, 93; 614, 174 Kress, T., 612, 173 König, A.C., 613, 118; 615, 19; 616, 145, 159; 619,71 Kreuz, M., 619, 263 Königsmann, K., 612, 154 Kreymer, A.E., 618,23 Konoplyannikov, A.K., 612, 154 Krishnaswamy, M.R., 617,1 Konorov, I., 612, 154 Krivkova, P., 617,1 Konstandin, T., 612, 311 Krivokhizhin, G.V., 612, 154 Konstantinov, V.F., 612, 154 Krivonos, S., 612, 283; 616, 228 Konya, K., 619, 233 Križan, P., 615, 39; 617, 141; 618,34 Koohi-Fayegh-Dehkordi, R., 617,18 Krogulski, T., 615,55 Koop, I.A., 613,29 Krokovny, P., 613, 20; 614, 27; 615, 39; 617, 141, 198; 618,34 Kopal, M., 613, 118; 615, 19; 616, 145, 159; 619,71 Krokovny, P.P., 613,29 Köpke, L., 615,31 Kropivnitskaya, A., 616,31 Koppenburg, P., 614, 27; 618,34 Kropp, W.R., 619, 255 Kopytine, M., 612, 181; 616,8 Kroseberg, J., 616,31 Korbel, V., 616,31 Kroumchtein, Z.V., 612, 154 Kordyasz, A., 615,55 Krueger, K., 612, 181; 616,8 Korentchenko, A.S., 612, 154 Krüger, A., 613, 118; 615, 19; 616, 145, 159; 619,71 Korolija, M., 612, 173 Krüger, K., 616,31 Korpar, S., 614, 27; 615, 39; 617, 141, 198; 618,34 Kruglov, V., 619,50 Korsch, W., 613, 148 Kruglova, L., 619,50 Korzenev, A., 612, 154 Kryemadhi, A., 618,23 Köse, U., 613, 105; 614, 155 Krzywdzinski, S., 617,1 Koshio, Y., 619, 255 Kubantsev, M., 617,1 Kostka, P., 616,31 Kubischta, W., 615,31 Kostritskiy, A.V., 617,1 Kubo, J., 619, 387 Cumulative author index to volumes 611–619 (2005) 389–427 407

Kückens, J., 616,31 Lamanna, M., 612, 154 Kudryavtsev, V.A., 616,17 Lamberto, A., 619,50 Kuhn, C., 612, 181; 616,8 Lamblin, J., 615, 153 Kuhn, D., 611, 66; 614,7 Lamont, M.A.C., 612, 181; 616,8 Kühn, J.H., 611, 116 Lanaro, A., 619,50 Kuhn, R., 612, 154 Lançon, E., 611, 66; 614,7 Kulasiri, R., 615, 39; 618,34 Landgraf, J.M., 612, 181; 616,8 Kuleshov, S., 617,1 Landi, G., 613, 118; 615, 19; 616, 145, 159; 619,71 Kulik, Y., 617,1 Landon, M.P.J., 616,31 Kulikov, A., 619,50 Landsberg, G., 617,1 Kulikov, A.I., 612, 181; 616,8 Lane, G.J., 618,51 Kulikov, V., 619,61 Lanfranchi, G., 619,61 Kumar, A., 612, 181; 616,8 Lanfranchi, J., 616, 174 Kumar, S., 617, 141, 198 Langacker, P., 614,67 Kumar, S.P., 619, 163 Langanke, K., 616,48 Kunde, G.J., 612, 181 Lange, J.S., 614, 27; 617, 198 Kundrát, V., 611, 102 Lange, S., 612, 181; 616,8 Kunihiro, T., 615, 102 Lange, W., 616,31 Kunin, A., 613, 118; 615, 19; 616, 145, 159; 619,71 Lapoux, V., 619,82 Kunne, F., 612, 154 Lapshin, V., 619,50 Kunori, S., 617,1 Lasiuk, B., 612, 181 Kunz, C.L., 612, 181 Laštovicka,ˇ T., 616,31 Kunz, J., 614, 104 Laubenstein, M., 616, 174 Kunz, M., 614, 125 Laue, F., 612, 181; 616,8 Kuo, C.C., 613, 20; 614, 27; 615, 39; 617, 141, 198; 618,34 Laurelli, P., 611, 66; 614,7 Laurent, H., 615, 167 Kuo, T.-L., 617, 141 Lauret, J., 612, 181; 616,8 Kupco, A., 617,1 Lauritsen, T., 618,51 Kuptsov, A., 619,50 Laville, J.L., 613, 128 Kurek, K., 612, 154 Lawson, T.B., 616,17 Kurochkin, I., 619,50 Laycock, P., 616,31 Kuroda, K.-I., 619,50 Lazkoz, R., 615, 146 Kuroda, M., 618,84 Lazzeroni, C., 615,31 Kuroki, T., 611, 269 Learned, J.G., 619, 255 Kutsarova, T., 619, 281 Lebeau, M., 613, 118; 615, 19; 616, 145, 159; 619,71 Kutschke, R., 618,23 Lebed, R.F., 619, 115 Kutuev, R.Kh., 612, 181; 616,8 Lebedenko, V., 616,17 Kuzmin, A., 613, 20; 615, 39; 617, 141, 198; 618,34 Lebedev, A., 612, 173, 181; 613, 118; 615, 19; Kuzmin, A.S., 613,29 616, 8, 31, 145, 159; 619,71 Kuznetsov, A.A., 612, 181; 616,8 Leberig, M., 612, 154 Kuznetsov, V.E., 617,1 Lebrun, D., 615, 153 Kwak, J.W., 618,23 Lebrun, P., 613, 118; 615, 19; 616, 145, 159; 619,71 Kwon, Y.-J., 613, 20; 614, 27; 615, 39; 617, 141, 198; 618,34 Lechtenfeld, O., 612,65 Kyriakis, A., 611, 66; 614,7 Lecomte, P., 613, 118; 615, 19; 616, 145, 159; 619,71 Lecoq, P., 613, 118; 615, 19; 616, 145, 159; 619,71 L3 Collaboration, 613, 118; 615, 19; 616, 145, 159; 619,71 Le Coultre, P., 613, 118; 615, 19; 616, 145, 159; 619,71 Lacava, F., 619,61 Leder, G., 613, 20; 615, 39; 617, 141, 198; 618,34 Lachenmaier, T., 616, 174 Lednický, R., 612, 181 La Commara, M., 619,88 Lednicky, R., 616,8;619,50 Lacourt, A., 615,31 Lee, F.X., 612,21 Ladron de Guevara, P., 613, 118; 615, 19; 616, 145, 159; 619,71 Lee, H.-S., 614,67 Ladygin, M.E., 612, 154 Lee, J., 611, 87; 619, 129 Lai, A., 615,31 Lee, J.-C., 611, 193 Lai, Y.F., 614, 37; 619, 247 Lee, K.B., 618,23 Laktineh, I., 613, 118; 615, 19; 616, 145, 159; 619,71 Lee, S., 614, 113 Lamanna, G., 615,31 Lee, S.E., 613, 20; 614, 27; 615, 39; 617, 141; 618,34 408 Cumulative author index to volumes 611–619 (2005) 389–427

Lee, S.H., 613, 20; 614,27 Li, T., 617, 112 Lee, T., 611,87 Li, W.G., 614, 37; 619, 247 Lee, W.M., 617,1 Li, X.-Z., 611,8 Lee, X.G., 617,24 Li, X.L., 614, 37; 619, 247 Lee, Y.-J., 617, 141 Li, X.Q., 614, 37; 619, 247 Lee-Franzini, J., 619,61 Li, X.S., 614, 37; 619, 247 Lees, J.-P., 611, 66; 614,7 Li, Y., 616,8 Lefebvre, A., 615, 167 Liang, Y.F., 614, 37; 619, 247 Leﬂat, A., 617,1 Liao, H.B., 614, 37; 619, 247 Le Goff, J.M., 612, 154; 613, 118; 615, 19; 616, 145, 159; 619,71 Lichtenstadt, J., 612, 154 Lehner, F., 617,1 Liddick, S.N., 611,81 Lehner, M.J., 616,17 Lieb, J., 619, 281 Lehocka, S., 616,8 Ligabue, F., 611, 66; 614,7 Lehto, M., 611, 66; 614,7 Lightfoot, P.K., 616,17 Leibenguth, G., 611, 66; 614,7 Liguori, G., 618,23 Leifels, Y., 612, 173 Likhoded, S., 613, 118; 615, 19; 616, 145, 159; 619,71 Leinweber, D.B., 616, 196 Liko, T., 617, 193 Leißner, B., 616,31 Lim, I.T., 619, 255 Leiste, R., 613, 118; 615, 19; 616, 145, 159; 619,71 Lim, S.H., 619, 255 Leitner, O., 614, 165 Lima, J.G.R., 617,1 Lemaire, M.-C., 611, 66; 614,7 Limosani, A., 614,27 Lemaitre, V., 611, 66; 614,7 Lin, C.H., 613, 118; 615, 19; 616, 145, 159; 619,71 Lemrani, R., 616,31 Lin, G., 616,8 Lendermann, V., 616,31 Lin, J., 611, 66; 614,7 Lenti, M., 615,31 Lin, S.-W., 613, 20; 614, 27; 615, 39; 617, 141, 198; 618,34 Lenzi, S.M., 619,88 Lin, W.T., 613, 118; 615, 19; 616, 145, 159; 619,71 Leone, D., 619,61 Lincoln, D., 617,1 Leoni, S., 615, 160 Linde, F.L., 613, 118; 615, 19; 616, 145, 159; 619,71 Leonidopoulos, C., 617,1 Lindenbaum, S.J., 612, 181; 616,8 Lepe, S., 617, 174; 619, 5, 367 Lindfeld, L., 616,31 Leruste, P., 619,50 Link, J.M., 618,23 Lesiak, M., 619, 281 Link, O., 615, 153 Lesiak, T., 613, 20; 614, 27; 615, 39; 617, 141, 198; 618,34 Linn, S.L., 617,1 Leupold, S., 616, 203 Linnemann, J., 617,1 Leveraro, F., 618,23 Lipka, K., 616,31 LeVine, M.J., 612, 181; 616,8 Lipkin, H.J., 612, 197; 616, 191 Levi Sandri, P., 619,50 Lipton, R., 617,1 Levonian, S., 616,31 Lisa, M.A., 612, 181; 616,8 Levtchenko, M., 613, 118; 615, 19; 616, 145, 159; 619,71 Liska, T., 612, 154 Levtchenko, P., 613, 118; 615, 19; 616, 145, 159; 619,71 List, B., 616,31 Levy, C.D.P., 611, 239 Lista, L., 613, 118; 615, 19; 616, 145, 159; 619,71 Lewin, J.D., 616,17 Lister, C.J., 618,51 Li, C., 612, 181; 613, 118; 615, 19; 616, 8, 145, 159; 619,71 Litke, A.M., 611, 66; 614,7 Li, F., 614, 37; 619, 247 Liu, C.X., 614, 37; 619, 247 Li, G., 614, 37; 619, 247 Liu, D.-J., 611,8 Li, H.H., 614, 37; 619, 247 Liu, F., 612, 181; 614, 37, 37; 616,8;619, 247, 247 Li, J., 613, 20; 614, 27, 37; 615, 39; 617, 1, 141; 618, 34; 619, 247 Liu, H.M., 614, 37; 619, 247 Li, J.-W., 619, 105 Liu, J.B., 614, 37; 619, 247 Li, J.C., 614, 37; 619, 247 Liu, J.P., 614, 37; 619, 247 Li, Q., 612, 181; 616,8 Liu, J.Y., 617,24 Li, Q.J., 614, 37; 619, 247 Liu, K.F., 612, 21; 617,49 Li, Q.Z., 617,1 Liu, L., 612, 181; 616,8 Li, R.B., 614, 37; 619, 247 Liu, Q.J., 612, 181; 616,8 Li, R.Y., 614, 37; 619, 247 Liu, R.G., 614, 37; 619, 247 Li, S.M., 614, 37; 619, 247 Liu, Z., 612, 181; 616,8 Cumulative author index to volumes 611–619 (2005) 389–427 409

Liu, Z.A., 613, 118; 614, 37; 615, 19; 616, 145, 159; 619, 71, 247 Luk, K.-B., 617,11 Liu, Z.X., 614, 37; 619, 247 Lüke, D., 616,31 Liubarsky, I., 616,17 Lukin, P.A., 613,29 Liventsev, D., 614, 27; 615, 39; 617, 141; 618,34 Luminari, L., 613, 118; 615, 19; 616, 145, 159; 619,71 Liyanage, N., 613, 148 Lundstedt, C., 617,1 Ljubicic, T., 612, 181; 616,8 Luo, C., 617,1 Llanes-Estrada, F.J., 611, 279 Luo, C.L., 614, 37; 619, 247 Llope, W.J., 612, 181; 616,8 Luo, X.L., 614, 37; 619, 247 Lobanov, A.E., 619, 136 Lüscher, R., 616,17 Lo Bianco, G., 615, 160 Lustermann, W., 613, 118; 615, 19; 616, 145, 159; 619,71 Lobodzinska, E., 616,31 Lütjens, G., 611, 66; 614,7 Locci, E., 611, 66; 614,7 Lützenkirchen, K., 616, 174 Loewe, M., 617,87 Lux, T., 616,31 Logashenko, I.B., 613,29 Lynch, J.G., 611, 66; 614,7 Loginov, E.K., 618, 265 Lynn, D., 612, 181; 616,8 Loh, E.G., 619, 271 Lysenko, A.P., 613,29 Lohmann, W., 613, 118; 615, 19; 616, 145, 159; 619,71 Lytkin, L., 616,31 Löhner, H., 617,18 Lokajícek,ˇ M., 611, 102 Ma, B.-Q., 615, 200 Loktionova, N., 616,31 Ma, E., 612,29 Lola, S., 618, 162 Ma, F.C., 614, 37; 619, 247 Long, H., 612, 181; 616,8 Ma, G.L., 616,8 Longacre, R.S., 612, 181; 616,8 Ma, J., 612, 181 Longo, E., 613, 118; 615, 19; 616, 145, 159; 619,71 Ma, J.G., 616,8 Longo, M.J., 617,11 Ma, J.M., 614, 37; 619, 247 Lopes da Silva, P., 615,31 Ma, J.P., 613,39 Lopes Pegna, D., 618,23 Ma, L.L., 614, 37; 619, 247 Lopez, A.M., 618,23 Ma, Q.M., 614, 37; 619, 247 Lopez, X., 612, 173 Ma, W.-G., 618, 209 Lopez-Fernandez, R., 616,31 Ma, W.G., 613, 118; 615, 19; 616, 145, 159; 619,71 Lopez-Noriega, M., 612, 181; 616,8 Ma, X.Y., 614, 37; 619, 247 Lopez-Sarrión, J., 619, 367 Ma, Y.G., 612, 181; 616,8 Lopez Aguera, A., 619,50 Mabe, M., 615, 257 Love, W.A., 612, 181; 616,8 Maccaferri, C., 619, 359 Loverre, P.F., 613, 105; 614, 155 Macchiavelli, A.O., 618,51 Lu, F., 614, 37; 619, 247 Machado, A.A., 618,23 Lu, G.R., 614, 37; 619, 247 Machefert, F., 611, 66; 614,7 Lü, H., 614,96 Machner, H., 619, 281 Lu, J.G., 614, 37; 619, 247 Macías, A., 617, 118 Lu, L.C., 617,11 Maciel, A.K.A., 617,1 Lu, Y., 616,8 Mackintosh, R.S., 619,82 Lu, Y.S., 613, 118; 615, 19; 616, 145, 159; 619,71 MacNaughton, J., 614, 27; 617, 141, 198 Lu, Z., 615, 200 Madaras, R.J., 617,1 Lubimov, V., 616,31 Madigojine, D., 615,31 Lubrano, P., 615,31 Madriz Aguilar, J.E., 619, 208 Luca, M., 616,25 Maeda, Y., 615, 193 Lucherini, V., 619,50 Maesaka, H., 619, 255 Luci, C., 613, 118; 615, 19; 616, 145, 159; 619,71 Maestas, M.M., 619, 271 Ludlam, T., 612, 181; 616,8 Magestro, D., 612, 181; 616,8 Ludovici, L., 613, 105; 614, 155 Maggi, G., 611, 66; 614,7 Ludwig, I., 612, 154 Maggi, M., 611, 66; 614,7 Luebke, W., 617,11 Maggiora, A., 612, 154 Lueders, H., 616,31 Maggiora, M., 612, 154 Lueking, L., 617,1 Magiera, A., 619, 281 Luiggi, E., 618,23 Magnin, J., 618,23 410 Cumulative author index to volumes 611–619 (2005) 389–427

Magnon, A., 612, 154 Markert, C., 612, 181; 616,8 Mahajan, S., 612, 181; 616,8 Markou, C., 611, 66; 614,7 Mahapatra, D.P., 612, 181; 616,8 Marks, J., 616,31 Mahjour-Shaﬁei, M., 617,18 Markytan, M., 615,31 Maier, A., 615,31 Marnieros, S., 616,25 Maier, H.J., 615, 175 Marotta, A., 613, 105; 614, 155 Maier, R., 619, 281 Marouelli, P., 615,31 Maina, E., 614, 216 Marras, D., 615,31 Maj, A., 615, 160 Marroncle, J., 612, 154 Majerotto, W., 618, 171 Marshall, R., 616,31 Majka, R., 612, 181; 616,8 Marshall, T., 617,1 Majumder, G., 614, 27; 615, 39; 617, 198; 618,34 Martens, K., 619, 271 Makankine, A., 616,31 Martin, A., 612, 154 Makhlioueva, I., 613, 105; 614, 155 Martin, F., 611, 66; 614,7 Maki, T., 619,50 Martin, J.P., 613, 118; 615, 19; 616, 145, 159; 619,71 Malden, N., 616,31 Martin, L., 612, 181; 616,8 Malgeri, L., 613, 118; 615, 19; 616, 145, 159; 619,71 Martin, M.I., 617,1 Malinin, A., 613, 118; 615, 19; 616, 145, 159; 619,71 Martin, V., 615,31 Malinovski, E., 616,31 Martina, L., 615,87 Mallet, J., 616,25 Martinez, M., 611, 66; 614,7 Mallot, G.K., 612, 154 Martínez-Pinedo, G., 619,88 Malvezzi, S., 618,23 Martínez de la Ossa, A., 613, 170 Malyshev, V.L., 617,1 Martini, M., 615, 31; 619,61 Maña, C., 613, 118; 615, 19; 616, 145, 159; 619,71 Martinovic,ˇ L., 617,92 Manago, N., 619, 271 Martinska, G., 619, 281 Manankov, V., 617,1 Martisikova, M., 616,31 Mandl, F., 613, 20; 614, 27; 615, 39; 617, 141, 198; 618,34 Martyn, H.-U., 616,31 Mandrioli, G., 615,14 Maruyama, T., 619, 255 Mangano, S., 616,31 Marx, J., 612, 181 Mangiarotti, A., 612, 173; 619, 240 Marx, J.N., 616,8 Mangotra, L.K., 612, 181; 616,8 Marzano, F., 613, 118; 615, 19; 616, 145, 159; 619,71 Mannarelli, M., 615, 297 Marzec, J., 612, 154 Mannelli, I., 615,31 Massafferri, A., 618,23 Männer, W., 611, 66; 614,7 Massarotti, P., 619,61 Mannocchi, G., 611, 66; 614,7 Máté, Z., 615, 175 Mans, J., 613, 118; 615, 19; 616, 145, 159; 619,71 Mathur, N., 612, 21; 617,49 Manthos, N., 619,50 Matis, H.S., 612, 181; 616,8 Mantica, P.F., 611,81 Mato, P., 611, 66; 614,7 Manuilov, I., 619,50 Matsuda, T., 612, 154 Manuilov, I.V., 612, 154 Matsumoto, T., 613, 20; 614, 27; 615, 39; 617, 141, 198; 618,34 Manvelyan, R., 613, 197 Matsuno, S., 619, 255 Manweiler, R., 612, 181; 616,8 Matsuo, M., 615, 160 Manzoor, S., 615,14 Matthews, J.A.J., 619, 271 Mao, H., 619, 226 Matthews, J.N., 619, 271 Mao, H.S., 617,1 Mattingly, S.E.K., 617,1 Mao, Z.P., 614, 37; 619, 247 Matulenko, Yu.A., 612, 181; 616,8 Marage, P., 616,31 Matyja, A., 613, 20, 118; 614, 27; 615, 39; 617, 141, 198; 618,34 Marchand, C., 612, 154 Mauger, C., 619, 255 Marchetto, F., 615,31 Mavromatos, N.E., 619,17 Mardanpour, H., 617,18 Maxﬁeld, S.J., 616,31 Marek, L.J., 619, 271 Maximov, A.N., 612, 154 Marfatia, D., 613, 61; 617, 78, 167 Mayer, K., 616, 174 Margetis, S., 612, 181; 616,8 Mayer, U., 619, 263 Margiotta, A., 615,14 Mayes, V.E., 611, 156 Marinelli, N., 611, 66; 614,7 Mayet, F., 614, 143 Cumulative author index to volumes 611–619 (2005) 389–427 411

Mayorov, A.A., 617,1 Meyer, A.B., 616,31 Mazumdar, K., 613, 118; 615, 19; 616, 145, 159; 619,71 Meyer, H., 616,31 Mazzocchi, C., 619,88 Meyer, J., 616,31 Mazzucato, E., 615,31 Meyer, W., 612, 154 McCarthy, R., 617,1 Meziani, Z.-E., 613, 148 McClain, C.J., 612, 181; 616,8 Mezzadri, M., 618,23 Mcgrew, C., 619, 255 Miao, C., 617,1 McMahon, T., 617,1 Michel, B., 611, 66; 614,7 McMillan, J.E., 616,17 Middleton, C., 613, 189 McNamara III, P.A., 611, 66; 614,7 Mielech, A., 612, 154 McNeil, R.R., 613, 118; 615, 19; 616, 145, 159; 619,71 Miettinen, H., 617,1 McNeile, C., 619, 124 Migliozzi, P., 613, 105; 614, 155 McShane, T.S., 612, 181; 616,8 Mihalcea, D., 617,1 Medcalf, T., 611, 66; 614,7 Mihul, A., 613, 118; 615, 19; 616, 145, 159; 619,71 Medved, K.S., 612, 154 Mikhailov, K.Yu., 613,29 Meer, D., 616,31 Mikhailov, Yu.V., 612, 154 Mehmandoost-Khajeh-Dad, A.A., 617,18 Mikocki, S., 616,31 Mehta, A., 616,31 Mikulec, I., 615,31 Mei, W., 619,61 Milcent, H., 613, 118; 615, 19; 616, 145, 159; 619,71 Meier, K., 616,31 Milcewicz-Mika, I., 616,31 Meinhard, H., 613, 105; 614, 155 Miller, M.L., 612, 181; 616,8 Meissner, F., 612, 181; 616,8 Million, B., 615, 160 Meister, M., 618,43 Milosevich, Z., 612, 181 Melanson, H.L., 617,1 Milstead, D., 616,31 Mele, S., 613, 118; 615, 19; 616, 145, 159; 619,71 Milstein, A.I., 613,29 Meljanac, S., 613, 221 Milton, K.A., 613,97 Mellado, B., 611,60 Minaev, N.G., 612, 181; 616,8 Melnick, Yu., 612, 181; 616,8 Minamino, A., 619, 255 Melnitchouk, A., 617,1 Minard, M.-N., 611, 66; 614,7 Melnitchouk, W., 613, 148 Mine, S., 619, 255 Meloni, D., 613, 170 Mirabelli, G., 613, 118; 615, 19; 616, 145, 159; 619,71 Melzer-Pellmann, I., 615,31 Miransky, V.A., 611, 207 Menasce, D., 618,23 Mironov, C., 612, 181; 616,8 Mendez, H., 618,23 Miscetti, S., 619,61 Mendizabal, S., 617,87 Mischke, A., 612, 181; 616,8 Menichetti, E., 615,31 Mishra, D., 612, 181 Menzel, S., 612, 233 Mishra, D.K., 616,8 Meola, S., 619,61 Misiejuk, A., 611, 66; 614,7 Merkin, M., 617,1 Miškovic,´ O., 615, 277 Merle, E., 611, 66; 614,7 Mitaroff, W., 613, 20; 614, 27; 615, 39; 617, 141, 198; 618,34 Merlo, M.M., 618,23 Mitchell, J., 612, 181; 616,8 Merola, L., 613, 118; 615, 19; 616, 145, 159; 619,71 Mitchell, R., 618,23 Merritt, K.W., 617,1 Mitra, I., 611, 289 Merschmeyer, M., 612, 173 Mitra, P., 616, 114 Mertzimekis, T.J., 611,81 Miura, M., 619, 255 Meschanin, A., 612, 181; 616,8 Miyabayashi, K., 613, 20; 618,34 Meschini, M., 613, 118; 615, 19; 616, 145, 159; 619,71 Miyake, H., 613, 20; 614, 27; 615, 39; 617, 141; 618,34 Meshkov, S., 611, 147 Miyanishi, M., 613, 105; 614, 155 Messchendorp, J.G., 617,18 Miyano, K., 619, 255 Messina, M., 613, 105; 614, 155 Miyata, H., 613, 20; 614, 27; 615, 39; 617, 141, 198; 618,34 Messineo, A., 611, 66; 614,7 Miyatake, H., 611, 239 Mestvirishvili, A., 615,31 Miyoshi, H., 615, 186 Metlitski, M.A., 612, 137 Mizuk, R., 614, 27; 615, 39; 617, 141; 618,34 Metz, A., 612, 233; 618,90 Mizusaki, T., 617, 150 Metzger, W.J., 613, 118; 615, 19; 616, 145, 159; 619,71 Mnich, J., 613, 118; 615, 19; 616, 145, 159; 619,71 412 Cumulative author index to volumes 611–619 (2005) 389–427

Mo, X.H., 614, 37; 619, 247 Munday, D.J., 615,31 Moch, S.-O., 614,53 Munhoz, M.G., 612, 181; 616,8 Moegel, P., 616, 174 MUNU Collaboration, 615, 153 Mohamed, A., 616,31 Murín, P., 616,31 Mohanty, B., 612, 181; 616,8 Murphy, A., 616,17 Mohanty, G.B., 613, 118; 615, 19; 616, 145, 159; 619,71 Murtas, F., 619,61 Mohapatra, D., 614, 27; 615, 39; 618,34 Murtas, G.P., 611, 66; 614,7 Mohapatra, R.N., 615, 231; 618, 150 Musicar, B., 613, 118; 615, 19; 616, 145, 159; 619,71 Moinester, M.A., 612, 154 Mussardo, G., 617, 133 Mokhov, N., 617,1 Musy, M., 613, 118; 615, 19; 616, 145, 159; 619,71 Molke, H., 612, 313 Mutaf, Y.D., 617,1 Molnar, L., 612, 181; 616,8 Mutterer, M., 615,55 Molokanova, N., 615,31 Moloney, G.R., 614, 27; 617, 141, 198 Nabi, J.-U., 612, 190 Mondal, N.K., 617,1 Nácher, E., 619,88 Montanet, L., 619,50 Nagae, D., 615, 186 Monteil, S., 611, 66; 614,7 Nagamine, T., 613, 20; 614, 27; 615, 39; 617, 141, 198; 618,34 Montgomery, H.E., 617,1 Nagasaka, Y., 613, 20; 614, 27; 615, 39; 617, 141, 198; 618,34 Moore, C.F., 612, 181; 616,8 Nagovizin, V., 616,31 Moore, E.F., 618,51 Nagy, E., 617,1 Moore, J.E., 618,23 Nagy, S., 613, 118; 615, 19; 616, 145, 159; 619,71 Moore, R.W., 617,1 Nähle, O., 612, 154 Mora-Corral, M.J., 612, 181 Nakadaira, T., 617, 198 Moradi, S., 613,74 Nakahata, M., 619, 255 Moreau, F., 616,31 Nakamura, K., 619, 255 Moretti, S., 614, 216 Nakamura, M., 613, 105; 614, 155 Morgan, B., 616,17 Nakano, E., 613, 20; 614, 27; 615, 39; 617, 141, 198; 618,34 Mori, T., 615, 39; 618,34 Nakano, I., 619, 255 Morimatsu, O., 611,93 Nakano, T., 613, 105; 614, 155 Morita, T., 611, 269; 619, 255 Nakao, M., 613, 20; 614, 27; 615, 39; 617, 141, 198; 618,34 Moriyama, S., 619, 255 Nakata, F., 619, 255 Moroni, L., 618,23 Nakaya, T., 619, 255 Morozov, A., 616,31 Nakayama, S., 619, 255 Morozov, D.A., 612, 181; 616,8 Nakazawa, H., 615, 39; 617, 141; 618,34 Morozov, V., 612, 181 Nalpas, L., 619,82 Morozumi, T., 616, 108 Namba, T., 619, 255 Morris, C.L., 615, 193 Nambu, R., 619, 255 Morris, J.V., 616,31 Nandi, B.K., 612, 181; 616,8 Mosca, L., 616,25 Nandi, S., 617, 112 Moser, H.-G., 611, 66; 614,7 Nankov, K., 616,31 Motobayashi, T., 614, 174 Nanopoulos, D.V., 611, 156; 619,17 Motta, D., 616, 174 Napoli, D., 615, 160 Moulin, E., 614, 143 Napoli, D.R., 619,88 Moulson, M., 619,61 Napolitano, M., 613, 118; 615, 19; 616, 145, 159; 619, 61, 71 Moutoussi, A., 611, 66; 614,7 Nappi, A., 615,31 Mozer, M.U., 616,31 Narain, M., 617,1 Muanza, G.S., 613, 118; 615, 19; 616, 145, 159; 619,71 Narasimham, V.S., 617,1 Muciaccia, M.T., 613, 105; 614, 155 Narayanan, R., 616,76 Muijs, A.J.M., 613, 118; 615, 19; 616, 145, 159; 619,71 Nardulli, G., 615, 297 Mukherji, S., 613, 208 Narita, K., 613, 105; 614, 155 Müller, A.-S., 611, 66; 614,7 Narjoux, J.-L., 619,50 Müller, B., 618, 77, 123 Naroska, B., 616,31 Müller, K., 616,31 Nasri, S., 615, 231 Müller, S., 619,61 Nassalski, J., 612, 154; 615,31 Mund, D., 619, 263 Nasseri, F., 614, 140; 618, 229 Cumulative author index to volumes 611–619 (2005) 389–427 413

Natale, S., 613, 118; 615, 19; 616, 145, 159; 619,71 Nishikawa, T., 611,93 Natkaniec, Z., 613, 20; 614, 27; 615, 39; 617, 141, 198; 618,34 Nishiyama, S., 619, 255 Naumann, J., 616,31 Nisi, S., 616, 174 Naumann, N.A., 617,1 Nitoh, O., 613, 20; 614, 27; 615, 39; 617, 141, 198; 618,34 Naumann, Th., 616,31 Nitta, K., 619, 255 Navarro-Lérida, F., 614, 104 Niu, K., 613, 105; 614, 155 Navick, X.-F., 616,25 Niwa, K., 613, 105; 614, 155 Nayak, G.C., 613,45 Noda, S., 619, 255 Nayak, S.K., 612, 181; 616,8 Nogach, L.V., 612, 181; 616,8 Nayak, T.K., 612, 181; 616,8 Nogga, A., 617,18 Neal, H.A., 617,1 Nojiri, M.M., 611, 223 Nedel, D.L., 613, 213 Nollez, G., 616,25 Nedev, S., 619, 281 Nomerotski, A., 617,1 Needham, M.D., 615,31 Nonaka, N., 613, 105; 614, 155 Negret, J.P., 617,1 Norman, B., 612, 181 Negus, P., 611, 66; 614,7 Norton, A., 615,31 Nehring, M., 618,23 Norton, P.R., 611, 66; 614,7 Neliba, S., 612, 154 Notani, M., 614, 174 Nelson, J.M., 612, 181; 616,8 Novak, T., 613, 118; 615, 19; 616, 145, 159; 619,71 Nelson, K.S., 617,11 Nowacki, F., 619,88 Nelson, S., 617,1 Nowak, G., 616,31 Nemenov, L., 619,50 Nowak, H., 613, 118; 615, 19; 616, 145, 159; 619,71 Nersessian, A., 616, 228 Nowell, J., 611, 66; 614,7 Nessi-Tedaldi, F., 613, 118; 615, 19; 616, 145, 159; 619,71 Nozaki, T., 614, 27; 615,39 Nesterenko, I.N., 613,29 Nozdrin, A.A., 612, 154 Nesvizhevsky, V., 619, 263 Nozicka, M., 616,31 Netrakanti, P.K., 612, 181; 616,8 Núñez Pardo, T., 619,50 Neuberger, H., 616,76 Nunnemann, T., 617,1 Neubert, M., 612,13 Nurushev, S.B., 612, 181; 616,8 Neubert, W., 612, 173 Nuzzo, S., 611, 66; 614,7 Neuhofer, G., 615,31 Nyman, G., 618,43 Neupane, I.P., 619, 201 Newman, H., 613, 118; 615, 19; 616, 145, 159; 619,71 Obayashi, Y., 619, 255 Newman, P.R., 616,31 Obraztsov, V.F., 612, 154 Neyret, D.P., 612, 154 Ocariz, J., 615,31 Ngac, A., 611, 66; 614,7 Odyniec, G., 612, 181; 616,8 Nguyen, F., 619,61 Oehm, J., 616, 174 Nicklin, G., 616,17 Oeschler, H., 615,50 Nickolls, A., 616,17 Oﬁerzynski, R., 613, 118; 615, 19; 616, 145, 159; 619,71 Nie, J., 614, 37; 619, 247 Oganezov, R., 616,31 Nie, Z.D., 614, 37; 619, 247 Ogawa, A., 612, 181; 616,8 Niebuhr, C., 616,31 Ogawa, S., 613, 20, 105; 614, 27, 155; 615, 39; 617, 141, 198; Nielsen, J., 611, 66; 614,7 618,34 Nieto, M.M., 613,11 Oguri, V., 617,1 Nikiforov, A., 616,31 Ohnishi, T., 614, 174 Nikitin, D., 616,31 Ohshima, T., 613, 20; 614, 27; 615, 39; 617, 141, 198; 618,34 Nikitin, M., 619,50 Okabe, T., 613, 20; 614, 27; 615, 39; 617, 141, 198; 618,34 Nikitin, V.A., 612, 181; 616,8 Okada, A., 619, 255 Nikolaenko, V.I., 612, 154 Okada, K., 619,50 Nikulin, M.A., 613,29 Okamura, H., 615, 193 Nilsson, B.S., 611, 66; 614,7 Okhapkin, V.S., 613,29 Nilsson, T., 618,43 Okorokov, V., 612, 181; 616,8 Nisati, A., 613, 118; 615, 19; 616, 145, 159; 619,71 Okuno, S., 613, 20; 614, 27; 615, 39; 617, 198; 618,34 Nishida, S., 613, 20; 614, 27; 615, 39; 617, 141, 198; 618,34 Okusawa, T., 613, 105; 614, 155 Nishikawa, K., 619, 255 Olaiya, E., 615,31 414 Cumulative author index to volumes 611–619 (2005) 389–427

Olchevskii, V., 619,50 Panebratsev, Y., 612, 181; 616,8 Oldeman, R.G.C., 613, 105; 614, 155 Panitkin, S.Y., 612, 181; 616,8 Oldenburg, M., 612, 181; 616,8 Panman, J., 613, 105; 614, 155 Olive, K.A., 619,30 Pantea, D., 618,23 Olivier, B., 616,31 Pantev, T., 618, 252 Olsen, S.L., 613, 20; 614, 27, 37; 615, 39; 617, 141, 198; 618, 34; Panzer-Steindel, B., 615,31 619, 247 Panzieri, D., 612, 154 Olshevsky, A.G., 612, 154 Paolucci, P., 613, 118; 615, 19; 616, 145, 159; 619,71 Olson, D., 612, 181; 616,8 Papavassiliou, J., 613, 162 Olsson, J.E., 616,31 Papinutto, M., 619, 184 O’Neil, D., 617,1 Paramatti, R., 613, 118; 615, 19; 616, 145, 159; 619,71 O’Neill, A., 619, 271 Paramonov, A., 616,31 Önengüt, G., 613, 105; 614, 155 Parashar, N., 617,1 Orazi, E., 616, 228 Pari, P., 616,25 O’Reilly, B., 618,23 Paris, A., 618,23 Organtini, G., 613, 118; 615, 19; 616, 145, 159; 619,71 Park, C.W., 613, 20; 614, 27; 615, 39; 617, 141 Ortín, T., 616, 118 Park, D.K., 614, 78; 615, 273; 619, 347 O’Shea, V., 611, 66; 614,7 Park, H., 614, 27; 615, 39; 617, 198; 618, 23, 34; 619, 255 Oshima, N., 617,1 Park, H.K., 617,11 Ostrick, M., 612, 154 Park, J.-H., 611,87 Ostrowicz, W., 613, 20; 614, 27; 615, 39; 617, 141, 198; 618,34 Park, K.S., 614, 27; 617, 198 Otboev, A.V., 613,29 Parker, M.A., 615,31 Ouyang, Q., 611, 66; 614,7 Parrini, G., 611, 66; 614,7 Ovrut, B.A., 618, 252, 259 Parslow, N., 613, 20; 614, 27; 615, 39; 617, 141, 198; 618,34 Oyama, Y., 619, 255 Partridge, R., 617,1 Ozaki, H., 617, 141, 198; 618,34 Parua, N., 617,1 Ozawa, A., 614, 174 Pascaud, C., 616,31 Ozerov, D., 616,31 Pascolo, J.M., 611, 66; 614,7 Pasqualucci, E., 619,61 Pac, M.Y., 619, 255 Passalacqua, L., 611, 66; 614,7 Paccetti Correia, F., 613,83 Passaleva, G., 613, 118; 615, 19; 616, 145, 159; 619,71 Pacheco, A., 611, 66; 614,7 Passera, M., 613, 162 Padee, A., 612, 154 Passeri, A., 619,61 Padley, P., 617,1 Passerini, F., 617, 182 Pagano, P., 612, 154 Pastrone, N., 615,31 Paic, G., 612, 181 Patel, G.D., 616,31 Painter, C.A., 619, 271 Patera, V., 619,61 Pakhlov, P., 613, 20; 615, 39; 618,34 Patricelli, S., 613, 118; 615, 19; 616, 145, 159; 619,71 Pakou, A., 619,82 Patrizii, L., 615,14 Pakvasa, S., 613,61 Patwa, A., 617,1 Pal, I., 613, 118; 615, 19; 616, 145, 159; 619,71 Paul, S., 612, 154 Pal, P., 618, 243 Paul, T., 613, 118; 615, 19; 616, 145, 159; 619,71 Pal, S.K., 612, 181; 616,8 Pauluzzi, M., 613, 118; 615, 19; 616, 145, 159; 619,71 Pal, S.S., 614, 201 Paus, C., 613, 118; 615, 19; 616, 145, 159; 619,71 Paleni, A., 615, 160 Pauss, F., 613, 118; 615, 19; 616, 145, 159; 619,71 Palestini, S., 615,31 Pavlinov, A.I., 612, 181; 616,8 Paling, S.M., 616,17 Pavšic,ˇ M., 614,85 Palka, H., 613, 20; 614, 27; 615, 39; 617, 141; 618,34 Pawlak, T., 612, 181; 616,8 Palla, F., 611, 66; 614,7 Payne, G.L., 618,68 Pallin, D., 611, 66; 614,7 Payre, P., 611, 66; 614,7 Palomares, C., 613, 118; 615, 19; 616, 145, 159; 619,71 Pazos, A., 619,50 Palutan, M., 619,61 Peak, L.S., 615, 39; 617, 141, 198; 618,34 Pan, Y.B., 611, 66; 614,7 Pearson, M.R., 611, 66; 614,7 Pandola, L., 616, 174 Pedace, M., 613, 118; 615, 19; 616, 145, 159; 619,71 Panebianco, S., 612, 154 Pedrini, D., 618,23 Cumulative author index to volumes 611–619 (2005) 389–427 415

Peez, M., 616,31 Pignanelli, M., 615, 160 Peitzmann, T., 612, 181; 616,8 Piilonen, L.E., 613, 20; 614, 27; 615, 39; 617, 141, 198; 618,34 Pelte, D., 612, 173 Pilkuhn, H., 614,62 Peng, H.P., 614, 37; 619, 247 Ping, R.G., 611, 123 Pensotti, S., 613, 118; 615, 19; 616, 145, 159; 619,71 Pioppi, M., 613, 118; 615, 19; 616, 145, 159; 619,71 Pentia, M., 619,50 Piragino, G., 612, 154 Penzo, A., 619,50 Pirjol, D., 615, 213 Pepe, I.M., 618,23 Piroué, P.A., 613, 118; 615, 19; 616, 145, 159; 619,71 Pepe, M., 615,31 Piskunov, N., 619, 281 Pereira, H.D., 612, 154 Pistolesi, E., 613, 118; 615, 19; 616, 145, 159; 619,71 Perera, L., 619, 271 Pitzl, D., 616,31 Perevedentsev, E.A., 613,29 Pivovarov, G.B., 617,92 Perevoztchikov, V., 612, 181; 616,8 Placakytˇ e,˙ R., 616,31 Perez, E., 616,31 Plagnol, E., 613, 128 Perez, P., 611, 66; 614,7 Planinic, M., 612, 181; 616,8 Perfetto, F., 619,61 Platchkov, S., 612, 154 Perieanu, A., 616,31 Platzer, K., 612, 154 Perkins, C., 612, 181; 616,8 Plettner, C., 619,88 Perkins, W.B., 612,75 Pló, M., 619,50 Pernicka, M., 615,31 Płochocki, A., 619,88 Perreau, J.-M., 619,50 Plonka, C., 619, 263 Perret, P., 611, 66; 614,7 Pluta, J., 612, 181; 616,8 Perret-Gallix, D., 613, 118; 615, 19; 616, 145, 159; 619,71 Plyaskin, V., 613, 118; 615, 19; 616, 145, 159; 619,71 Perroud, J.-P., 617,11 Pochodzalla, J., 612, 154 Peryt, W., 612, 181; 616,8 Poghosyan, V., 619, 322 Peshekhonov, D.V., 612, 154 Pohl, M., 613, 118; 615, 19; 616, 145, 159; 619,71 Peshekhonov, V.D., 612, 154 Pojidaev, V., 613, 118; 615, 19; 616, 145, 159; 619,71 Pestotnik, R., 613, 20; 614, 27; 615, 39; 617, 141; 618,34 Policastro, G., 619, 163 Peters, A., 615,31 Polikarpov, M.I., 613,52 Peters, O., 617,1 Pollacco, E.C., 613, 128; 619,82 Petersen, A.K., 614, 104 Polunin, A.A., 613,29 Petit, F., 612, 105 Polyakov, D., 611, 173 Petoukhov, A., 619, 263 Polyakov, V.A., 612, 154 Petrache, C.M., 615, 160 Polycarpo, E., 618,23 Petrascu, C., 619,50 Ponomarev, V.Yu., 612, 165 Petricca, F., 616, 174 Ponta, T., 619,50 Pétroff, P., 617,1 Pontecorvo, L., 619,61 Petrov, V.A., 612, 181; 616,8 Pontoglio, C., 618,23 Petrovici, M., 612, 173 Pop, D., 619,50 Petrucci, F., 615,31 Popa, V., 615,14 Petrukhin, A., 616,31 Pope, B.G., 617,1 Peyaud, B., 615,31 Pope, C.N., 614,96 Pham, T.N., 619, 313 Popov, A.A., 612, 154 Phatak, S.C., 612, 181; 616,8 Popov, A.S., 613,29 Piasecki, E., 615,55 Porile, N., 612, 181; 616,8 Piasecki, K., 615,55 Porter, J., 612, 181; 616,8 Picariello, M., 611, 215 Portheault, B., 616,31 Piccini, M., 615,31 Pöschl, R., 616,31 Piccolo, D., 613, 118; 615, 19; 616, 145, 159; 619,71 Poskanzer, A.M., 612, 181; 616,8 Picha, R., 612, 181; 616,8 Potekhin, M., 612, 181; 616,8 Picón, M., 615, 127 Pothier, J., 613, 118; 615, 19; 616, 145, 159; 619,71 Piegaia, R., 617,1 Potrebenikov, Yu., 615,31 Pierazzini, G., 615,31 Potrebenikova, E., 612, 181; 616,8 Pierella, F., 613, 118; 615, 19; 616, 145, 159; 619,71 Potukuchi, B.V.K.S., 612, 181; 616,8 Pietrzyk, B., 611, 66; 614,7 Potzel, W., 616, 174 416 Cumulative author index to volumes 611–619 (2005) 389–427

Povh, B., 616,31 Ramos, S., 612, 154 Prange, G., 611, 66; 614,7 Rancoita, P.G., 613, 118; 615, 19; 616, 145, 159; 619,71 Prasuhn, D., 619, 281 Rander, J., 611, 66; 614,7 Pratt, S., 618,60 Ranieri, A., 611, 66; 614,7 Preece, R.M., 616,17 Ranieri, R., 613, 118; 615, 19; 616, 145, 159; 619,71 Prelz, F., 618,23 Raniwala, R., 612, 181; 616,8 Pretz, J., 612, 154 Raniwala, S., 612, 181; 616,8 Prezado, Y., 618,43 Ranjard, F., 611, 66; 614,7 Primavera, M., 619,61 Rapaport, J., 615, 193 Prindle, D., 612, 181; 616,8 Rapidis, P.A., 617,1 Prodanov, E.M., 611,34 Rappazzo, G.F., 619,50 Prokoﬁev, D., 613, 118; 615, 19; 616, 145, 159; 619,71 Raso, G., 611, 66; 614,7 Prosper, H.B., 617,1 Raspereza, A., 613, 118; 615, 19; 616, 145, 159; 619,71 Protic,´ D., 619, 281 Rastkar, A.R., 615, 141 Protopopescu, S., 617,1 Ratabole, R., 611, 289 Pruneau, C., 612, 181; 616,8 Ratti, S.P., 618,23 Przybycien, M.B., 617,1 Ravel, O., 612, 181; 616,8 Puglierin, G., 615, 153 Ray, R.L., 612, 181; 616,8 Putschke, J., 612, 181; 616,8 Raychaudhuri, S., 618, 221 Putz, J., 611, 66; 614,7 Razeti, M., 616,25 Putzer, A., 611, 66; 614,7 Razin, S.V., 612, 181; 616,8 Razis, P., 613, 118; 615, 19; 616, 145, 159; 619,71 Qi, N.D., 614, 37; 619, 247 Reay, N.W., 617,1 Qian, C.D., 614, 37; 619, 247 Rebelo, M.N., 614, 187 Qian, J., 617,1 Rebourgeard, P.C., 612, 154 Qin, H., 614, 37; 619, 247 Redin, S.I., 613,29 Qiu, J.-W., 613,45 Redlich, K., 615,50 Qiu, J.F., 614, 37; 619, 247 Reich, J., 619, 263 Quadri, A., 611, 215 Reichenbach, T., 612, 275 Quadt, A., 617,1 Reicherz, G., 612, 154 Quayle, W., 611,60 Reichhold, D., 612, 181; 616,8 Quenby, J.J., 616,17 Reid, J.G., 612, 181; 616,8 Quinones, J., 618,23 Reil, K., 619, 271 Quintans, C., 612, 154 Reimer, P., 616,31 Reinbacher, R., 618, 259 Raabe, R., 619,82 Reisdorf, W., 612, 173 Radu, E., 615,1 Reisert, B., 616,31 Ragusa, F., 611, 66; 614,7 Reitz, B., 612, 165 Raha, S., 611,27 Ren, D., 613, 118; 615, 19; 616, 145, 159; 619,71 Rahal-Callot, G., 613, 118; 615, 19; 616, 145, 159; 619,71 Ren, H.-C., 611, 137 Rahaman, M.A., 613, 118; 615, 19; 616, 145, 159; 619,71 Ren, Z.Y., 614, 37; 619, 247 Rahimi, A., 618,23 Ren, Z.Z., 617,24 Rahman, M.-U., 612, 190 Renault, G., 612, 181; 616,8 Rai, G., 612, 181 Renk, B., 611, 66; 614,7;615,31 Rai, S.K., 618, 221 Rescigno, M., 613, 118; 615, 19; 616, 145, 159; 619,71 Raicevic, N., 616,31 Retiere, F., 612, 181; 616,8 Raics, P., 613, 118; 615, 19; 616, 145, 159; 619,71 Reucroft, S., 613, 118; 615, 19; 616, 145, 159; 617,1;619,71 Raja, N., 613, 118; 615, 19; 616, 145, 159; 619,71 Reyes, M., 618,23 Rajagopalan, S., 617,1 Reymann, J., 612, 154 Rajaram, D., 617,11 Rezaei-Aghdam, A., 615, 141 Rakers, S., 612, 165 Riccardi, C., 618,23 Rakness, G., 612, 181; 616,8 Richter, A., 612, 165 Ramelli, R., 613, 118; 615, 19; 616, 145, 159; 619,71 Richter, H., 616, 174 Rami, F., 612, 173 Ridel, M., 617,1 Ramirez, J.E., 618,23 Ridiger, A., 612, 181; 616,8 Cumulative author index to volumes 611–619 (2005) 389–427 417

Riehle, R., 619, 271 Roy, B.J., 619, 281 Riemann, S., 613, 118; 615, 19; 616, 145, 159; 619,71 Roy, C., 612, 181; 616,8 Riisager, K., 618,43 Roy, D.P., 618, 193 Rijssenbeek, M., 617,1 Roynette, J.C., 613, 128 Riles, K., 613, 118; 615, 19; 616, 145, 159; 619,71 Royon, C., 617,1 Rimmer, A., 616,31 Rozanov, A., 613, 105; 614, 155 Risler, C., 616,31 Rozanska, M., 614, 27; 617, 198; 618,34 Rith, K., 612, 154 Rozhdestvensky, A.M., 612, 154 Ritter, H.G., 612, 181; 616,8 Ruan, L., 616,8 Rizatdinova, F., 617,1 Ruan, L.J., 612, 181 Rizvi, E., 616,31 Ruban, A.A., 613,29 Roberts, J.B., 612, 181; 616,8 Rubin, H.A., 617,11 Roberts, J.W., 616,17 Rubinov, P., 617,1 Roberts, M., 619, 271 Rubio, B., 619,88 Robertson, N.A., 611, 66; 614,7 Rubio, J.A., 613, 118; 615, 19; 616, 145, 159; 619,71 Robinson, M., 616,17 Ruchti, R., 617,1 Robmann, P., 616,31 Rudolph, G., 611, 66; 614,7 Rockwell, T., 617,1 Ruggieri, F., 611, 66; 614,7 Rodriguez Fernandez, A., 619,50 Ruggieri, M., 615, 297 Roe, B.P., 613, 118; 615, 19; 616, 145, 159; 619,71 Ruggiero, G., 613, 118; 615, 19, 31; 616, 145, 159; 619,71 Roeckl, E., 619,88 Rühl, W., 613, 197 Rogachevski, O.V., 612, 181 Ruiz, H., 611, 66; 614,7 Rogachevskiy, O.V., 616,8 Ruppert, J., 618, 123 Rojas, J.C., 617,87 Rurikova, Z., 616,31 Roland, B., 616,31 Rusakov, S., 616,31 Rolandi, L., 611, 66; 614,7 Rutherford, S.A., 611, 66; 614,7 Rolf, J., 612, 313 Ryazantsev, A., 619,50 Romano, G., 613, 105; 614, 155 Rybicki, K., 616,31 Romão, J.C., 618, 162 Rykaczewski, H., 613, 118; 615, 19; 616, 145, 159; 619,71 Romero, A., 619,50 Rykalin, V., 619,50 Romero, J.L., 612, 181; 616,8 Ryskulov, N.M., 613,29 Romero, L., 613, 118; 615, 19; 616, 145, 159; 619,71 Rondio, E., 612, 154; 615,31 Saavedra, J., 617, 174 Rong, G., 614, 37; 619, 247 Sabirov, B.M., 617,1 Roosen, R., 616,31 Saborido, J., 619,50 Root, N., 613, 20; 617, 141 Sacco, R., 615,31 Root, N.I., 613,29 Sadovski, A.B., 612, 154 Rosa, G., 613, 105; 614, 155 Sagawa, H., 613, 20; 614, 27; 615, 39; 617, 141, 198; 618,34 Rosca, A., 613, 118; 615, 19; 616, 145, 159; 619,71 Sahoo, R., 612, 181; 616,8 Rose, A., 612, 181; 616,8 Saito, K., 612,5 Rosemann, C., 613, 118; 615, 19; 616, 145, 159; 619,71 Saito, T., 615, 193 Rosenbleck, C., 613, 118; 615, 19; 616, 145, 159; 619,71 Saitta, B., 613, 105; 614, 155 Rosier-Lees, S., 613, 118; 615, 19; 616, 145, 159; 619,71 Saji, C., 619, 255 Ross, D.A., 614, 216 Sajot, G., 617,1 Rostovtsev, A., 616,31 Sakai, H., 615, 193 Roszkowski, L., 617,99 Sakai, Y., 613, 20; 614, 27; 615, 39; 617, 141, 198; 618,34 Roth, M., 612, 223 Sakellariadou, M., 614, 125 Roth, S., 613, 118; 615, 19; 616, 145, 159; 619,71 Sakemi, Y., 615, 193 Rothberg, J., 611, 66; 614,7 Sakharov, A., 613, 118; 615, 19; 616, 145, 159; 619,71 Rougé, A., 611, 66; 614,7;619,43 Sakrejda, I., 612, 181; 616,8 Rouhani, S., 613,74 Sakuda, M., 619, 255 Roussel-Chomaz, P., 613, 128; 619,82 Sakurai, H., 614, 174 Roux, B., 616, 159 Sala, S., 618,23 Rovere, M., 618,23 Salicio, J., 613, 118; 615, 19; 616, 145, 159; 619,71 Rowley, N., 615,55 Saller, E., 612, 154 418 Cumulative author index to volumes 611–619 (2005) 389–427

Salur, S., 612, 181; 616,8 Schambach, J., 612, 181; 616,8 Salzmann, N., 616,93 Schamberger, R.D., 617,1 Sami, M., 619, 193 Scharenberg, R.P., 612, 181; 616,8 Samoylenko, V.D., 612, 154 Schätzel, S., 616,31 Samsarov, A., 613, 221 Schegelsky, V., 613, 118; 615, 19; 616, 145, 159; 619,71 Sanchez, E., 613, 118; 615, 19; 616, 145, 159; 619,71 Scheidenberger, C., 619, 240 Sánchez-Hernández, A., 618,23 Scheins, J., 616,31 Sandacz, A., 612, 154 Schellman, H., 617,1 Sander, H.-G., 611, 66; 614,7 Scherer Santos, R.J., 619, 359 Sandin, F., 616,1 Schiavon, P., 612, 154 Sandweiss, J., 612, 181; 616,8 Schietinger, T., 613, 20; 614, 27; 615, 39; 617, 141, 198; 618,34 Sanglard, V., 616,25 Schildknecht, D., 618,84 Sanguinetti, G., 611, 66; 614,7 Schill, C., 612, 154 Sankey, D.P.C., 616,31 Schilling, F.-P., 616,31 Sans, M., 612, 154 Schilling, S., 616,93 Santacesaria, R., 613, 105; 614, 155 Schlatter, D., 611, 66; 614,7 Santachiara, R., 611, 189 Schlegel, M., 618,90 Santamarina, C., 619,50 Schleper, P., 616,31 Santangelo, P., 619,61 Schmeling, S., 611, 66; 614,7 Santoro, A., 617,1 Schmidt, I., 612, 258 Santos, D., 614, 143 Schmidt, K., 619,88 Santos, J., 619,11 Schmidt, K.-H., 616,48 Santovetti, E., 619,61 Schmidt, M.G., 613,83 Sapeta, S., 613, 154 Schmidt, S., 616,31 Sapozhnikov, M.G., 612, 154 Schmidt, S.A., 615,31 Saracino, G., 619,61 Schmidt, T., 612, 154 Sarangi, T.R., 614,27 Schmidt-Kaerst, S., 616, 145 Saremi, S., 613, 118; 615, 19; 616, 145, 159; 619,71 Schmitt, H., 612, 154 Sarkar, S., 613, 118; 615, 19; 616, 145, 159; 619,71 Schmitt, L., 612, 154 Sarrat, A., 619, 255 Schmitt, S., 616,31 Sarrazin, M., 612, 105 Schmitz, N., 612, 181; 616,8 Sarsour, M., 616,8 Schneider, M., 616,31 Sasaki, M., 619, 271 Schneider, O., 613, 20; 614, 27; 615, 39; 617, 141, 198; 618,34 Sasaki, S., 619, 352 Schnetzer, S.R., 619, 271 Sasaki, T., 619, 255 Schoeffel, L., 616, 25, 31 Sasao, N., 619, 255 Schoenert, S., 616, 174 Sato, N., 613, 20; 615, 39; 617, 141, 198; 618,34 Scholberg, K., 619, 255 Sato, O., 613, 105; 614, 155 Schönharting, V., 615,31 Sato, Y., 613, 105; 614, 155 Schöning, A., 616,31 Satta, A., 613, 105; 614, 155 Schönmeier, P., 618,34 Sauer, P., 617,18 Schopper, H., 613, 118; 615, 19; 616, 145, 159; 619,71 Sauvan, E., 616,31 Schotanus, D.J., 613, 118; 615, 19; 616, 145, 159; 619,71 Savin, I., 612, 181; 616,8 Schraußer, B., 618, 171 Savin, I.A., 612, 154 Schröder, V., 616,31 Savrié, M., 615,31 Schroeder, L.S., 612, 181 Savvidy, G., 615, 285 Schué, Y., 615,31 Sawyer, L., 617,1 Schuetz, Ch.P., 619,50 Sazhin, P.S., 616,8 Schuller, F.P., 612,93 Scarpaci, J.-A., 619,82 Schultz-Coulon, H.-C., 616,31 Scarpaci, J.A., 613, 128 Schümann, J., 613, 20; 614, 27; 615, 39; 617, 141; 618,34 Schacher, J., 619,50 Schüttauf, A., 612, 173 Schael, S., 611, 66; 614,7 Schwanda, C., 614, 27; 618,34 Schäfer, C., 613, 118; 615, 19; 616, 145, 159; 619,71 Schwanenberger, C., 616,31 Schaile, O., 615, 175 Schwartz, A.J., 618,34 Schakel, A.M.J., 611, 182 Schwartzman, A., 617,1 Cumulative author index to volumes 611–619 (2005) 389–427 419

Schweda, K., 612, 165, 181; 616,8 Shevchenko, A., 612, 165 Schweitzer, P., 612, 233 Shevchenko, O.Yu., 612, 154 Schwengner, R., 619,88 Shevchenko, S., 613, 118; 615, 19; 616, 145, 159; 619,71 Sciabà, A., 611, 66; 614,7 Sheviakov, I., 616,31 Sciacca, C., 613, 118; 615, 19; 616, 145, 159; 619,71 Shi, F., 614, 37; 619, 247 Sciascia, B., 619,61 Shi, X., 614, 37; 619, 247 Sciubba, A., 619,61 Shibata, T., 617, 141 Scotto Lavina, L., 613, 105; 614, 155 Shibuya, H., 613, 20, 105; 614, 27, 155; 615, 39; 617, 141, 198; Scuri, F., 619,61 618,34 Sedgbeer, J.K., 611, 66; 614,7 Shimada, K., 615, 186 Sedlák, K., 616,31 Shimanskii, S.S., 612, 181 Sefkow, F., 616,31 Shimanskiy, S.S., 616,8 Seger, J., 612, 181; 616,8 Shimizu, Y., 615, 193 Segoni, I., 618,23 Shimoda, T., 611, 239 Sekiguchi, K., 615, 193 Shimoura, S., 614, 174 Sekiguchi, M., 619, 255 Shindler, A., 612, 313; 619, 184 Selvaggi, G., 611, 66; 614,7 Shiozawa, M., 619, 255 Semenov, S., 614, 27; 617, 198 Shiraishi, K.K., 619, 255 Sen, S., 618, 237 Shishkin, A.A., 612, 154 Senyo, K., 613, 20; 615, 39; 617, 141; 618,34 Shivarov, N., 613, 118; 615, 19; 616, 145, 159; 619,71 Shivpuri, R.K., 617,1 Seres, Z., 612, 173 Shoutko, V., 613, 118; 615, 19; 616, 145, 159; 619,71 Serin, L., 611, 66; 614,7 Shpakov, D., 617,1 Servoli, L., 613, 118; 615, 19; 616, 145, 159; 619,71 Shrivastava, A., 613, 128 Setare, M.R., 612, 100 Shtarkov, L.N., 616,31 Seth, K.K., 612,1 Shu, F.-W., 614, 195; 619, 340 Settles, R., 611, 66; 614,7 Shumilov, E., 613, 118; 615, 19; 616, 145, 159; 619,71 Seuster, R., 614, 27; 618,34 Shupe, M., 617,1 Sevior, M.E., 614, 27; 617, 141; 618,34 Shvorob, A., 613, 118; 615, 19; 616, 145, 159; 619,71 Seweryniak, D., 618,51 Shwartz, B., 613, 20; 615, 39; 617, 141 Seyboth, P., 612, 181; 616,8 Shwartz, B.A., 613,29 Sﬁligoi, I., 619,61 Si, Z.G., 615,68 Sguazzoni, G., 611, 66; 614,7 Sibidanov, A.L., 613,29 Shabalina, E., 617,1 Sichtermann, E., 616,8 Shahaliev, E., 612, 181; 616,8 Sida, J.-L., 619,82 Shajesh, K.V., 613,97 Sidorov, A., 619,50 Shamanov, V., 613, 105; 614, 155 Sidorov, V., 613, 20; 617, 141 Shamov, A.G., 613,29 Sidorov, V.A., 613,29 Shan, L.Y., 614, 37; 619, 247 Sidwell, R.A., 617,1 Shang, L., 614, 37; 619, 247 Siebert, H.-W., 612, 154 Shao, M., 612, 181; 616,8 Siegel, E.R., 612, 122 Shao, W., 612, 181; 616,8 Sikora, B., 612, 173 Sharatchandra, H.S., 611, 289 Silva-Marcos, J.I., 614, 187 Sharkey, E., 619, 255 Silvestris, L., 611, 66; 614,7 Sharma, M., 612, 181; 616,8 Sim, K.S., 612, 173 Shatunov, Yu.M., 613,29 Simak, V., 617,1 Shcherbakov, A., 612, 283 Simion, V., 612, 173 Sheaff, M., 618,23 Simon, F., 612, 181; 616,8 Sheldon, P.D., 618,23 Simonov, Yu.A., 619, 293 Shen, D.L., 614, 37; 619, 247 Simopoulou, E., 611, 66; 614,7 Shen, W.Q., 616,8 Simpson, K.M., 619, 271 Shen, X.Y., 614, 37; 619, 247 Sin, S.-J., 614, 113 Shen, Y.-G., 612, 61; 614, 195; 619, 340 Singaraju, R.N., 612, 181; 616,8 Shende, S.V., 617,18 Singh, J.B., 614, 27; 615, 39; 617, 141; 618,34 Sheng, H.Y., 614, 37; 619, 247 Sinha, B., 611,27 Shestermanov, K.E., 612, 181; 616,8 Sinha, L., 612, 154 420 Cumulative author index to volumes 611–619 (2005) 389–427

Sinha, M., 618, 115 Sood, G., 612, 181; 616,8 Sinnis, G., 619, 271 Sorensen, P., 612, 181; 616,8 Sioli, M., 615,14 Sorín, V., 617,1 Siopsis, G., 613, 189 Sorrentino, M., 613, 105 Sirignano, C., 613, 105; 614, 155 Sorrentino, S., 613, 105; 614, 155 Sirois, Y., 616,31 Sosebee, M., 617,1 Sirotenko, V., 617,1 Sotnikova, N., 617,1 Sirri, G., 615,14 Souga, C., 613, 118; 615, 19; 616, 145, 159; 619,71 Sissakian, A.N., 612, 154 Soustruznik, K., 617,1 Sitnik, I., 619, 281 South, D., 616,31 Siudak, R., 619, 281 Souza, M., 617,1 Siwek-Wilczynska,´ K., 612, 173 Sowinski, J., 612, 181; 616,8 Skachkova, A., 612, 154 Sozzi, F., 612, 154 Skaza, F., 619,82 Sozzi, M., 615,31 Skibinski, R., 617,18 Spada, F.R., 613, 105; 614, 155 Skoro, G., 612, 181; 616,8 Spadaro, T., 619,61 Skrinsky, A.N., 613,29 Spagnolo, P., 611, 66; 614,7 Slattery, P., 617,1 Spaskov, V., 616,31 Slifer, K., 613, 148 Specka, A., 616,31 Sloan, T., 616,31 Speltz, J., 616,8 Slunecka, M., 612, 154 Spillantini, P., 613, 118; 615, 19; 616, 145, 159; 619,71 Smiechowicz, M., 619, 281 Spinka, H.M., 612, 181; 616,8 Smirnov, G.I., 612, 154 Spitzer, H., 616,31 Spooner, N.J.C., 616,17 Smirnov, N., 612, 181; 616,8 Springer, R.W., 619, 271 Smirnov, P., 616,31 Spurio, M., 615,14 Smith, J.D., 619, 271 Srivastava, B., 612, 181; 616,8 Smith, J.F., 618,51 Srivastava, S.K., 619,1 Smith, N.J.T., 616,17 Srnka, A., 612, 154 Smith, P.F., 616,17 Stadnik, A., 616,8 Smith, R.P., 617,1 Stamen, R., 613, 20; 614, 27; 615, 39; 616, 31; 617, 141, 198; Smizanska, M., 611, 66; 614,7 618,34 Smolik, J., 619,50 Stanic,ˇ S., 613, 20; 614, 27; 615, 39; 617, 141, 198; 618,34 Smolyankin, V., 612, 173 Stanislaus, T.D.S., 612, 181; 616,8 Smy, M., 619, 255 Stanton, N.R., 617,1 Snellings, R., 612, 181; 616,8 STAR Collaboration, 612, 181; 616,8 Snopkov, I.G., 613,29 Staric,ˇ M., 613, 20; 614, 27; 615, 39; 617, 141, 198; 618,34 Snow, G.R., 617,1 Steffen, F.D., 617,99 Snow, J., 617,1 Steffens, F.M., 612,5 Snow, R., 619, 271 Stein, H.J., 619, 281 Snyder, S., 617,1 Steinbrück, G., 617,1 Sobel, H.W., 619, 255 Stella, B., 616,31 Soffer, J., 612, 258 Stenson, K., 618,23 Sokolsky, P., 619, 271 Stenzel, H., 611, 66; 614,7 Soldner, T., 619, 263 Stepantsov, S., 619,82 Solodov, E.P., 613,29 Sterman, G., 613,45 Solomon, J., 617,1 Stern, M., 616,25 Soloviev, Y., 616,31 Steuer, M., 613, 118; 615, 19; 616, 145, 159; 619,71 Sommer, R., 612, 313 Stichel, P.C., 615,87 Somov, A., 613, 20; 614, 27; 615, 39; 617, 141, 198; 618,34 Stickland, D.P., 613, 118; 615, 19; 616, 145, 159; 619,71 Son, D., 613, 118; 615, 19; 616, 145, 159; 619,71 Stiewe, J., 616,31 Sona, P., 619, 240 Stinzing, F., 612, 154 Song, C., 619, 271 Stock, R., 612, 181; 616,8 Song, J.S., 613, 105; 614, 155 Stockmeier, M.R., 612, 173 Song, Y., 617,1 Stoicea, G., 612, 173 Soni, N., 613, 20; 614, 27; 615, 39; 617, 141, 198; 618,34 Stoker, D., 617,1 Cumulative author index to volumes 611–619 (2005) 389–427 421

Stokes, B.T., 619, 271 Swain, J.D., 613, 118; 615, 19; 616, 145, 159; 619,71 Stolarski, M., 612, 154 Swiderski,´ Ł., 615,55 Stolin, V., 617,1 Symons, T.J.M., 612, 181; 616,8 Stolpovsky, A., 612, 181; 616,8 Syritsyn, S.N., 613,52 Stone, A., 617,1 Szanto de Toledo, A., 612, 181; 616,8 Stone, J.L., 619, 255 Szarwas, P., 612, 181; 616,8 Stoyanov, B., 613, 118; 615, 19; 616, 145, 159; 619,71 Szillasi, Z., 613, 118; 615, 19; 616, 145, 159; 619,71 Stoyanova, D.A., 617,1 Szleper, M., 615,31 Straessner, A., 613, 118; 615, 19; 616, 145, 159; 619,71 Sznajder, A., 617,1 Strang, M.A., 617,1 Szydłowski, M., 619, 219 Strauch, I., 616,31 Straumann, U., 616,31 Tadsen, A., 615, 153 Strauss, M., 617,1 Tai, A., 612, 181; 616,8 Strikhanov, M., 612, 181; 616,8 Tain, J.L., 619,88 Strikman, M., 616, 59; 619,95 Tajima, O., 613, 20; 614, 27; 615, 39; 617, 141; 618,34 Stringfellow, B., 612, 181; 616,8 Takabayashi, N., 612, 154 Strolin, P., 613, 105; 614, 155 Takahashi, F., 618,1;619, 233 Strong, J.A., 611, 66; 614,7 Takahashi, J., 612, 181; 616,8 Strovink, M., 617,1 Takasaki, F., 613, 20; 614, 27; 615, 39; 617, 141; 618,34 Struck, C., 612, 181 Takasugi, E., 611,27 Stuchbery, A.E., 611,81 Takenaga, K., 615, 247 Takeshita, E., 614, 174 Stutte, L., 617,1 Takeuchi, S., 614, 174 Stutz, A., 615, 153 Takeuchi, Y., 619, 255 Su, F., 618,97 Takeutchi, F., 619,50 Su, R.-K., 611,21 Takook, M.V., 613,74 Suaide, A.A.P., 612, 181; 616,8 Talby, M., 617,1 Suda, K., 615, 193 Tamai, K., 617, 141, 198; 618,34 Sudhakar, K., 613, 118; 615, 19; 616, 145, 159; 619,71 Tamhankar, S., 612, 21; 617,49 Suga, Y., 619, 255 Tamii, A., 615, 193 Sugarbaker, E., 612, 181; 616,8 Tamura, N., 613, 20; 614, 27; 615, 39; 617, 141, 198; 618, 34; Sugiyama, A., 614,27 619, 255 Sugonyaev, V.P., 612, 154 Tanaka, M., 613, 20; 614, 27; 615, 39; 617, 141, 198; 618, 34; Suire, C., 612, 181; 616,8 619, 255 Sulak, L.R., 619, 255 Tang, A.H., 612, 181; 616,8 Sulc, M., 612, 154 Tang, X., 614, 37; 619, 247 Sulej, R., 612, 154 Tang, X.W., 613, 118; 615, 19; 616, 145, 159; 619,71 Sultanov, G., 613, 118; 615, 19; 616, 145, 159; 619,71 Tanihata, I., 614, 174 Šumbera, M., 612, 181 Tao, N., 614, 37; 619, 247 Sumbera, M., 616,8 Tarasov, A., 619,50 Sumisawa, K., 613, 20; 614, 27; 615, 39; 617, 141 Tarjan, P., 613, 118; 615, 19; 616, 145, 159; 619,71 Sumiyoshi, T., 613, 20; 614, 27; 615, 39; 617, 141, 198; 618,34 Tarnowsky, T., 616,8 Sumner, T.J., 616,17 Tatischeff, V., 615, 167 Sun, H.S., 614, 37; 619, 247 Tatishvili, G., 615,31 Sun, L.Z., 613, 118; 615, 19; 616, 145, 159; 619,71 Taureg, H., 615,31 Sun, S.S., 614, 37; 619, 247 Taurok, A., 615,31 Sun, Y., 617, 150 Tauscher, L., 613, 118; 615, 19; 616, 145, 159; 619, 50, 71 Sun, Y.Z., 614, 37; 619, 247 Tavartkiladze, Z., 613,83 Sun, Z.J., 614, 37; 619, 247 Taylor, G., 611, 66; 614,7 Surrow, B., 612, 181; 616,8 Taylor, L., 613, 118; 615, 19; 616, 145, 159; 619,71 Sushkov, S., 613, 118; 615, 19; 616, 145, 159; 619,71 Taylor, W., 617,1 Suter, H., 613, 118; 615, 19; 616, 145, 159; 619,71 Tchalishev, V.V., 612, 154 Suzuki, A., 619, 255 Tchoulakov, V., 616,31 Suzuki, S., 614, 27; 618,34 Teixeira-Dias, P., 611, 66; 614,7 Suzuki, S.Y., 613, 20; 614, 27; 615, 39; 618,34 Tellili, B., 613, 118; 615, 19; 616, 145, 159; 619,71 Suzuki, Y., 619, 255 Tempesta, P., 611, 66; 614,7 422 Cumulative author index to volumes 611–619 (2005) 389–427

Tenchini, R., 611, 66; 614,7 Totsuka, Y., 619, 255 Tengblad, O., 618,43 Tovey, D.R., 616,17 Tentindo-Repond, S., 617,1 Trabelsi, K., 618,34 Teramoto, Y., 613, 20; 614, 27; 615, 39; 617, 141, 198; 618,34 Trainor, T.A., 612, 181; 616,8 Tessarotto, F., 612, 154 Traynor, D., 616,31 Testa, M., 619,61 Trentalange, S., 612, 181; 616,8 Teubert, F., 611, 66; 614,7 Tretjakov, P.V., 619, 193 Teufel, A., 612, 154 Tretyak, V.I., 612, 154 Teyssier, D., 613, 118; 615, 19; 616, 145, 159; 619,71 Triantafyllopoulos, D.N., 615, 221 Tezuka, I., 613, 105; 614, 155 Tribble, R.E., 612, 181; 616,8 Thacker, H.B., 612, 21; 617,49 Tricomi, A., 611, 66; 614,7 Thein, D., 612, 181; 616,8 Trippe, T.G., 617,1 Thers, D., 612, 154 Trocmé, B., 611, 66; 614,7 Thirolf, P.G., 615, 175 Troncoso, R., 615, 277 Thomas, J.H., 612, 181; 616,8 Trousov, S., 612, 154 Thomas, J.R., 619, 271 Truöl, P., 616,31 Thomas, S.B., 619, 271 Trusov, S., 619,50 Thompson, A.S., 611, 66; 614,7 Trzaska, W.H., 615,55 Thompson, G., 616,31 Tsai, O., 612, 181 Thompson, J.A., 613,29 Tsai, O.D., 616,8 Thompson, J.C., 611, 66; 614,7 Tsenov, R., 613, 105; 614, 155; 619, 281 Thompson, L.F., 611, 66; 614,7 Tsipolitis, G., 616,31 Thompson, P.D., 616,31 Tsuboyama, T., 613, 20; 615, 39; 617, 198 Thomson, G.B., 619, 271 Tsujikawa, S., 619, 193 Tian, X.C., 613, 20; 614, 27; 615, 39; 617, 141; 618,34 Tsukamoto, T., 613, 20; 614, 27; 615, 39; 617, 141, 198; 618,34 Tian, Y.R., 614, 37; 619, 247 Tsukerman, I., 613, 105; 614, 155 Tiburzi, B.C., 617,40 Tsukui, M., 615, 186 Tilquin, A., 611, 66; 614,7 Tsurin, I., 616,31 Timmermans, C., 613, 118; 615, 19; 616, 145, 159; 619,71 Tsushima, K., 612,5 Timoshenko, S., 612, 181; 616,8 Tuchming, B., 611, 66; 614,7 Ting, S.C.C., 613, 118; 615, 19; 616, 145, 159; 619,71 Tully, C., 613, 118; 615, 19; 616, 145, 159; 619,71 Ting, S.M., 613, 118; 615, 19; 616, 145, 159; 619,71 Tung, K.L., 613, 118; 615, 19; 616, 145, 159; 619,71 Tioukov, V., 613, 105; 614, 155 Tupa, D., 619, 271 Tittel, K., 611, 66; 614,7 Tupper, G.B., 612, 293 Tkatchev, L.G., 612, 154 Turcot, A.S., 617,1 Tobar, M.J., 619,50 Turlay, R., 615,31 Tobe, K., 615, 120 Tödtli, B., 616,93 Turnau, J., 616,31 Toeda, T., 612, 154 Turyshev, S.G., 613,11 Togano, Y., 614, 174 Tuts, P.M., 617,1 Togo, V., 615,14 Tyminski, Z., 612, 173 Tokarev, M., 612, 181; 616,8 Tzamariudaki, E., 616,31 Toki, H., 611,27 Tziaferi, E., 616,17 Tolla, D.D., 619, 359 Tolun, P., 613, 105; 614, 155 Uchigashima, N., 615, 193 Tomalin, I.R., 611, 66; 614,7 Ueda, S., 619, 255 Tomasz, F., 616,31 Uehara, S., 613, 20; 614, 27; 615, 39; 617, 141, 198; 618,34 Tomlin, B.E., 611,81 Ueno, H., 615, 186 Tong, G.L., 614, 37; 619, 247 Ueno, K., 614, 27; 617, 141, 198; 618,34 Tonjes, M.B., 612, 181 Uggerhøj, E., 619, 240 Tonwar, S.C., 613, 118; 615, 19; 616, 145, 159; 619,71 Uggerhøj, U.I., 619, 240 Toporensky, A., 619, 193 Uglov, T., 613, 20; 614, 27; 615, 39; 617, 141; 618,34 Törnqvist, N.A., 619, 145 Uiterwijk, J.W.E., 613, 105; 614, 155 Tortora, L., 619,61 UK Dark Matter Collaboration, 616,17 Toshito, T., 613, 105; 614, 155 Ulbricht, J., 613, 118; 615, 19; 616, 145, 159; 619,71 Tóth, J., 613, 118; 615, 19; 616, 145, 159; 619,71 Ulery, J., 616,8 Cumulative author index to volumes 611–619 (2005) 389–427 423

Ulicny, M., 619, 281 Varvell, K.E., 614, 27; 617, 141, 198; 618,34 Ullrich, T., 612, 181; 616,8 Vary, J.P., 617,92 Ulrych, S., 612, 89; 618, 233 Vasilevski, I., 612, 181 Unal, G., 615,31 Vasilevski, I.M., 616,8 Underwood, D.G., 612, 181; 616,8 Vasiliev, A.N., 612, 181; 616,8 Uno, S., 613, 20; 614, 27; 615, 39; 617, 141, 198; 618,34 Vasquez, R., 613, 118; 615, 19; 616, 145, 159; 619,71 Ur, C.A., 619,88 Vayaki, A., 611, 66; 614,7 Uraev, A., 616,31 Vazdik, Y., 616,31 Urbach, C., 619, 184 Vázquez, F., 618,23 Urban, J., 619, 281 Vázquez, P., 619,50 Urban, M., 616,31 Vázquez Doce, O., 619,50 Uribe, C., 618,23 Veelken, C., 616,31 Urkinbaev, A., 616,8 Veillet, J.-J., 611, 66; 614,7 Urquijo, P., 617, 141 Velasco, M., 615,31 Ushida, N., 613, 105; 614, 155 Velázquez, V., 613, 134 Ushiroda, Y., 617, 198; 618,34 Venanzoni, G., 619,61 Usik, A., 616,31 Veneziano, S., 619,61 Utkin, D., 616,31 Ventura, A., 619,61 Utkin, V., 619,50 Venturi, A., 611, 66; 614,7 Uzawa, K., 619, 333 Verdini, P.G., 611, 66; 614,7 Vernet, R., 616,8 Versaci, R., 619,61 Vaandering, E.W., 618,23 Vest, A., 616,31 Vagins, M.R., 619, 255 Veszpremi, V., 613, 118; 615, 19; 616, 145, 159; 619,71 Valassi, A., 611, 66; 614,7 Vesztergombi, G., 613, 118; 615, 19; 616, 145, 159; 619,71 Valente, E., 613, 118; 615, 19; 616, 145, 159; 619,71 Vetlitsky, I., 613, 118; 615, 19; 616, 145, 159; 619,71 Valente, P., 619,61 Videau, H., 611, 66; 614,7 Valeriani, B., 619,61 Videau, I., 611, 66; 614,7 Valishev, A.A., 613,29 Viertel, G., 613, 118; 615, 19; 616, 145, 159; 619,71 Valkár, S., 616,31 Vigdor, S.E., 612, 181; 616,8 Valkárová, A., 616,31 Vigezzi, E., 615, 160 Vallage, B., 611, 66; 614,7;615,31 Vilain, P., 613, 105; 614, 155 Vallée, C., 616,31 Vilasi, G., 614, 131 Van Buren, G., 612, 181; 616,8 Vilja, I., 619, 155 Van Dantzig, R., 613, 105; 614, 155 Villa, S., 613, 20, 118; 615, 19, 39; 616, 145, 159; 617, 141, 198; Van den Berg, A.M., 612, 165; 615, 167; 617,18 618, 34; 619,71 Van der Aa, O., 611, 66; 614,7 Villar, V., 616,25 VanderMolen, A.M., 612, 181 Villegas, M., 611, 66; 614,7 Vander Molen, A.M., 616,8 Villeneuve-Seguier, F., 617,1 Van de Vyver, B., 613, 105; 614, 155 Vinokurova, S., 616,31 Van de Walle, R.T., 613, 118; 615, 19; 616, 145, 159; 619,71 Viollier, R.D., 612, 293 Van Garderen, E.D., 617,18 Virius, M., 612, 154 Vangioni, E., 619,30 Vitulo, P., 618,23 Vaniev, V., 617,1 Vivargent, M., 613, 118; 615, 19; 616, 145, 159; 619,71 Van Kooten, R., 617,1 Viyogi, Y.P., 612, 181; 616,8 Vankova, G., 619, 281 Vlachos, S., 613, 118; 615, 19; 616, 145, 159; 619, 50, 71 Van Leeuwen, M., 616,8 Vlassov, N.V., 612, 154 Van Mechelen, P., 616,31 Vodopianov, I., 613, 118; 615, 19; 616, 145, 159; 619,71 Van Remortel, N., 616,31 Vogel, C., 619, 263 Varanda, M., 612, 154 Vogel, H., 613, 118; 615, 19; 616, 145, 159; 619,71 Varela, O., 615, 127 Vogt, H., 613, 118; 615, 19; 616, 145, 159; 619,71 Varelas, N., 617,1 Vokal, S., 616,8 Vargas Trevino, A., 616,31 Volchinski, V., 616,31 Varma, R., 612, 181; 616,8 Volk, J., 617,11 Varner, G., 613, 20; 614, 27; 615, 39; 617, 141, 198; 618,34 Volkov, A.A., 617,1 Varner, G.S., 614, 37; 619, 247 Voloshin, S.A., 612, 181; 616,8 424 Cumulative author index to volumes 611–619 (2005) 389–427

Von Feilitzsch, F., 616, 174 Wang, Z.Y., 614, 37; 619, 247 Von Harrach, D., 612, 154 Wang, Zheng, 619, 247 Von Hodenberg, M., 612, 154 Wanke, R., 615,31 Von Neumann-Cosel, P., 612, 165 Warchol, J., 617,1 Von Rossen, P., 619, 281 Ward, H., 612, 181; 616,8 Von Wimmersperg-Toeller, J.H., 611, 66; 614,7 Ward, J.J., 611, 66; 614,7 Vorobiev, A.P., 617,1 Ward, R.S., 619, 177 Vorobiev, I., 613, 118; 615, 19; 616, 145, 159; 619,71 Wasserbaech, S., 611, 66; 614,7 Vorobyov, A.A., 613, 118; 615, 19; 616, 145, 159; 619,71 Watanabe, H., 615, 186 Vuilleumier, J.-L., 615, 153 Watanabe, M., 613, 20; 614, 27; 615, 39; 617, 141; 618,34 Vznuzdaev, M., 612, 181; 616,8 Watanabe, Y., 614, 27; 615, 39; 618,34 Watson, J.W., 612, 181; 616,8 Wachsmuth, H., 611, 66; 614,7 Watts, G., 617,1 Wacker, K., 616,31 Wayne, M., 617,1 Wadhwa, M., 613, 118; 615, 19; 616, 145, 159; 619,71 Webb, J.C., 612, 181; 616,8 Waggoner, W., 612, 181 Webb, R., 612, 154 Waggoner, W.T., 616,8 Weber, G., 616,31 Wagner, J., 616,31 Weber, M., 613, 118; 615, 19; 616, 145, 159; 619,71 Wagner, M., 612, 154 Weber, R., 616,31 Wagner, P., 612, 173 Webster, M., 618,23 Wahl, H., 615,31 Weerts, H., 617,1 Wahl, H.D., 617,1 Wegener, D., 616,31 Wakasa, T., 615, 193 Wei, C.L., 614, 37; 619, 247 Walker, A., 615,31 Wei, D.H., 614, 37; 619, 247 Walker, J.W., 611, 156 Weise, E., 612, 154 Wallenius, M., 616, 174 Weitzel, Q., 612, 154 Walter, C.W., 619, 255 Wells, R., 612, 181; 616,8 Wambach, J., 612, 165 Werner, C., 616,31 Wan, S.-L., 615,79 Werner, N., 616,31 Wang, B., 611,21 Wessels, M., 616,31 Wang, C.C., 613, 20; 614, 27; 615, 39; 617, 141, 198; 618,34 Wessling, B., 616,31 Wang, C.H., 613, 20; 614, 27; 615, 39; 617, 141, 198; 618,34 Wessling, M.E., 618, 269 Wang, D.Y., 614, 37; 619, 247 Westerhoff, S., 619, 271 Wang, F., 612, 181; 616,8 Westfall, G.D., 612, 181; 616,8 Wang, G., 612, 181, 181; 616,8,8 Wetzler, A., 616,8 Wang, J.Z., 614, 37; 619, 247 Wetzorke, I., 619, 184 Wang, K., 614, 37; 619, 247 Wheaton, S., 615,50 Wang, L., 614, 37; 619, 247 Whisnant, K., 617,78 Wang, L.S., 614, 37; 619, 247 White, A., 617,1 Wang, M., 614, 37; 618, 23; 619, 247 White, C.G., 617,11 Wang, M.-Z., 614, 27; 617, 141; 618,34 White, R., 611, 66; 614,7 Wang, P., 614, 37; 619, 247 White, S.L., 617,11 Wang, P.L., 614, 37; 619, 247 White, T.O., 615,31 Wang, Q., 613, 39, 118; 615, 19; 616, 145, 159; 619,71 Whiteson, D., 617,1 Wang, S.Z., 614, 37; 619, 247 Whitten Jr., C., 612, 181; 616,8 Wang, T., 611, 66; 614,7 Widhalm, L., 615,31 Wang, W., 619, 255 Wiedenhöver, I., 618,51 Wang, W.F., 614, 37; 619, 247 Wiedenmann, W., 611, 66; 614,7 Wang, X.L., 612, 181; 613, 118; 615, 19; 616, 8, 145, 159; 619,71 Wieders, L.H., 612, 223 Wang, Y., 612, 181; 616,8,8 Wiedner, U., 612, 154 Wang, Y.F., 614, 37; 619, 247 Wieland, O., 615, 160 Wang, Z., 614, 37, 37, 37; 619, 247, 247 Wieman, H., 612, 181; 616,8 Wang, Z.-G., 615,79 Wiencke, L.R., 619, 271 Wang, Z.-M., 617,1 Wiesmann, M., 612, 154 Wang, Z.M., 612, 181; 613, 118; 615, 19; 616, 8, 145, 159; 619,71 Wijngaarden, D.A., 617,1 Cumulative author index to volumes 611–619 (2005) 389–427 425

Wilhelmsen, K., 618,43 Wu, X., 611, 66; 614,7 Wilkes, R.J., 619, 255 Wu, X.-G., 611, 260 Wilkin, C., 619, 281 Wu, Y.M., 614, 37; 619, 247 Willenbrock, S., 616, 215 Wu, Z.C., 612, 115; 613,1 Williams, A.G., 616, 196 Wünsch, E., 616,31 Willis, A., 615, 167 Wunsch, M., 611, 66; 614,7 Willis, S., 617,1 Wynhoff, S., 613, 118; 615, 19; 616, 145, 159; 619,71 Willson, R., 612, 181 Wilquet, G., 613, 105; 614, 155 Xella, S., 616,31 Wilschut, H.W., 615, 167 Xia, L., 613, 118; 615, 19; 616, 145, 159; 619,71 Wilson, A.N., 611,81 Xia, X.M., 614, 37; 619, 247 Wilson, J., 615, 160 Xiao, Z.-G., 612, 173 Wilson, J.R., 618,23 Xie, Q.L., 617, 141, 198 Wiltshire, D.L., 619, 201 Xie, X.X., 614, 37; 619, 247 Wimpenny, S.J., 617,1 Xie, Y., 611, 66; 614,7 Windmolders, R., 612, 154 Xin, B., 614, 37; 619, 247 Wingerter-Seez, I., 615,31 Xing, Y.Z., 617,24 Winhart, A., 615,31 Xing, Z.-Z., 618, 131, 141 Winter, G.-G., 616,31 χLF Collaboration, 619, 184 Winter, K., 613, 105; 614, 155 Xu, G., 619,61 Winter, W., 613,67 Xu, G.F., 614, 37; 619, 247 Wirth, H.-F., 615, 175 Xu, H., 614, 37; 619, 247 Wirth, S., 612, 154 Xu, N., 612, 181; 616,8 Wise, M.B., 611, 53; 613,5 Xu, Q., 617,1 Wislicki,´ W., 612, 154 Xu, R., 611, 66; 614,7 Wislicki, W., 615,31 Xu, Y., 614, 37; 619, 247 Wisniewski,´ K., 612, 173 Xu, Z., 612, 181; 616,8 Wiss, J., 618,23 Xu, Z.Z., 612, 181; 613, 118; 615, 19; 616, 8, 145, 159; 619,71 Wissing, Ch., 616,31 Xue, S., 611, 66; 614,7 Wissink, S.W., 612, 181; 616,8 Xue, S.T., 614, 37; 619, 247 Witała, H., 617,18 Witecki, M., 615,55 Yabsley, B.D., 614, 27; 615, 39; 617, 141, 198 Witt, R., 612, 181; 616,8 Yager, P.M., 618,23 Wittgen, M., 615,31 Yagi, M., 611, 239 Woehrling, E.-E., 616,31 Yako, K., 615, 193 Wohlfarth, D., 612, 173 Yamada, R., 617,1 Wohlfarth, M.N.R., 612,93 Yamada, S., 619, 255 Wojcik, M., 616, 174 Yamaguchi, A., 613, 20; 614, 27; 615, 39; 617, 141, 198; 618,34 Wolf, G., 611, 66; 614,7 Yamaguchi, Y., 614, 174 Wolf, R., 616,31 Yamamoto, E., 612, 181; 616,8 Wolski, R., 619,82 Yamamoto, H., 618,34 Womersley, J., 617,1 Yamamoto, J., 613, 118; 615, 19; 616, 145, 159; 619,71 Wood, D.R., 617,1 Yamamoto, S., 619, 255 Wood, J., 612, 181; 616,8 Yamanaka, T., 618,34 Wörtche, H.J., 612, 165; 617,18 Yamashita, T., 615, 247 Worthy, L.A., 611, 199 Yamashita, Y., 613, 20; 614, 27; 615, 39; 617, 141, 198; 618,34 Wosiek, J., 619, 171 Yamauchi, M., 613, 20; 614, 27; 615, 39; 617, 141; 618,34 Wotton, S.A., 615,31 Yamazaki, T., 613, 140 Wronka, S., 615,31 Yan, M.L., 614, 37; 619, 247 Wu, C., 614, 174 Yan, W., 616,31 Wu, C.Y., 618,51 Yanagida, T., 615, 120 Wu, J., 611, 66; 612, 181; 614,7;616,8 Yanagisawa, C., 619, 255 Wu, N., 614, 37; 619, 247 Yanagisawa, Y., 614, 174 Wu, P., 618, 209 Yang, B.Z., 613, 118; 615, 19; 616, 145, 159; 619,71 Wu, S.L., 611, 60, 66; 614,7 Yang, C.G., 613, 118; 615, 19; 616, 145, 159; 619,71 426 Cumulative author index to volumes 611–619 (2005) 389–427

Yang, F., 614, 37; 619, 247 Zachariadou, K., 611, 66; 614,7 Yang, H., 613, 20; 614, 27; 617, 141 Zaitsev, A.S., 613,29 Yang, H.J., 613, 118; 615, 19; 616, 145, 159; 619,71 Zakharov, V.I., 613,52 Yang, H.X., 614, 37; 619, 247 Zálešák, J., 616,31 Yang, J., 614, 37; 619, 247 Zalipska, J., 619, 255 Yang, J.-C., 613, 148 Zalite, An., 613, 118; 615, 19; 616, 145, 159; 619,71 Yang, J.-J., 612, 258 Zalite, Yu., 613, 118; 615, 19; 616, 145, 159; 619,71 Yang, M., 613, 118; 615, 19; 616, 145, 159; 619,71 Zallo, A., 618,23 Yang, M.-Z., 619, 105 Zanabria, M., 617,1 Yang, S.D., 614, 37; 619, 247 Zanelli, J., 615, 277 Yang, W.-M., 615,79 Zanetti, A.M., 612, 154 Yang, Y., 618,97 Zanevski, Y.V., 612, 181 Yang, Y.X., 614, 37; 619, 247 Zanevsky, Y.V., 616,8 Yano, H., 611, 239 Zang, S.L., 614, 37; 619, 247 Yasuda, T., 617,1 Zanotti, J.M., 616, 196 Yatsunenko, Y.A., 617,1 Zanotti, L., 616, 174 Yazkov, V., 619,50 Zaremba, K., 612, 154 Ye, M., 614, 37; 619, 247 Zech, A., 619, 271 Ye, M.H., 614, 37; 619, 247 Zeitnitz, C., 611, 66; 614,7 Ye, Y.X., 614, 37; 619, 247 Zeng, Y., 614, 37, 37; 619, 247, 247 Yeganov, V., 616,31 Zerguerras, T., 613, 128 Yeh, S.C., 613, 118; 615, 19; 616, 145, 159; 619,71 Zhabitsky, M., 619,50 Yepes, P., 612, 181; 616,8 Zhalov, M., 616,59 Yi, L.H., 614, 37; 619, 247 Zhang, B.X., 614, 37; 619, 247 Yi, Z.Y., 614, 37; 619, 247 Zhang, B.Y., 614, 37; 619, 247 Zhang, C.C., 614, 37; 617, 198; 619, 247 Yim, K.K., 615, 134 Zhang, D.H., 614, 37; 619, 247 Yin, Q.-J., 613,91 Zhang, H., 612, 181; 616,8;618, 131 Ying, J., 614, 27; 615, 39; 617, 141, 198; 618,34 Zhang, H.Y., 614, 37; 619, 247 Yip, K., 617,1 Zhang, J., 611, 66; 614, 7, 27, 37; 617, 198; 618, 14, 34; 619, 247 Ynduráin, F.J., 612, 245 Zhang, J.-Z., 613,91 Yokoyama, H., 619, 255 Zhang, J.B., 612, 21; 617,49 Yoo, J., 619, 255 Zhang, J.W., 614, 37; 619, 247 Yoon, C.S., 613, 105; 614, 155 Zhang, J.Y., 614, 37; 619, 247 Yoshida, A., 614, 174 Zhang, L., 611, 66; 614,7 Yoshida, K., 611, 269; 614, 174; 619, 333 Zhang, L.M., 613, 20; 614, 27; 615, 39; 617, 141, 198; 618,34 Yoshida, M., 619, 255 Zhang, L.S., 614, 37; 619, 247 Yoshimi, A., 615, 186 Zhang, Q.J., 614, 37; 619, 247 Yoshimura, Y., 619,50 Zhang, R.-Y., 618, 209 Yu, C.S., 614, 37; 619, 247 Zhang, S.N., 612, 127 Yu, G.W., 614, 37; 619, 247 Zhang, S.Q., 614, 37; 619, 247 Yu, H.-b., 615, 231 Zhang, W.M., 612, 181; 616,8 Yu, J., 617,1 Zhang, X., 611,1;617,1 Yuan, C.Z., 614, 37; 619, 247 Zhang, X.M., 614, 37; 619, 247 Yuan, J.M., 614, 37; 619, 247 Zhang, X.Y., 614, 37; 619, 247 Yuan, Y., 613, 20; 614, 37; 619, 247 Zhang, Y., 614, 37; 618, 23; 619, 247 Yudin, Yu.V., 613,29 Zhang, Y.J., 614, 37; 619, 247 Yue, Q., 614, 37; 619, 247 Zhang, Y.Y., 614, 37; 619, 247 Yüksel, H., 613,61 Zhang, Z., 612, 207; 616, 31; 617, 157 Yurevich, V.I., 612, 181; 616,8 Zhang, Z.P., 612, 181; 613, 20, 118; 614, 27, 37; 615, 19, 39; Yusa, Y., 614, 27; 615, 39; 617, 198; 618,34 616, 8, 145, 159; 617, 141, 198; 618, 34; 619, 71, 247 Yushmanov, I., 612, 173 Zhang, Z.Q., 614, 37; 619, 247 Yuting, B., 612, 181 Zhao, D.X., 614, 37; 619, 247 Zhao, J., 612, 154; 613, 118; 615, 19; 616, 145, 159; 619,71 Žácek,ˇ J., 616,31 Zhao, J.B., 614, 37; 619, 247 Zacek, V., 615, 153 Zhao, J.W., 614, 37; 619, 247 Cumulative author index to volumes 611–619 (2005) 389–427 427

Zhao, M.G., 614, 37; 619, 247 Zhuang, H.L., 613, 118; 615, 19; 616, 145, 159; 619,71 Zhao, P.P., 614, 37; 619, 247 Zhuang, P., 615,93 Zhao, W., 611, 66; 614,7 Zichichi, A., 613, 118; 615, 19; 616, 145, 159; 619,71 Zhao, W.-Q., 612, 207; 617, 157 Ziegler, R., 612, 154 Zhao, W.R., 614, 37; 619, 247 Ziegler, T., 611, 66; 614,7 Zhao, X.J., 614, 37; 619, 247 Zielinski, M., 617,1 Zhao, Y.B., 614, 37; 619, 247 Zieminska, D., 617,1 Zhao, Y.L., 617,33 Zieminski, A., 617,1 Zhao, Z., 618,14 Zimmermann, B., 613, 118; 615, 19; 616, 145, 159; 619,71 Zhao, Z.G., 614, 37; 619, 247 Zinchenko, A., 615,31 Zhaomin, Z.P., 612, 181 Zinner, N., 616,48 Zhelezov, A., 616,31 Ziolkowski, M., 615,31 Zheng, H.Q., 614, 37; 619, 247 Zito, G., 611, 66; 614,7 Zheng, J.P., 614, 37; 619, 247 Zizong, Z.P., 612, 181 Zheng, L.S., 614, 37; 619, 247 Zobernig, G., 611, 66; 614,7 Zheng, Z.P., 614, 37; 619, 247 Zohrabyan, H., 616,31 Zhilich, V., 613, 20; 614, 27; 615, 39; 617, 141, 198; 618,34 Zöller, M., 613, 118; 615, 19; 616, 145, 159; 619,71 Zhilin, A., 612, 173 Zołnierczuk,˙ P.A., 612, 181 Zhokin, A., 616,31 Zomer, F., 616,31 Zhong, X.C., 614, 37; 619, 247 Žontar, D., 613, 20; 614, 27; 615, 39; 617, 141, 198; 618,34 Zhou, B., 617,1 Zou, B.S., 611, 123; 614, 37; 619, 247 Zhou, B.Q., 614, 37; 619, 247 Zoulkarneev, R., 612, 181; 616,8 Zhou, G.M., 614, 37; 619, 247 Zoulkarneeva, J., 612, 181 Zhou, H.Q., 611, 123 Zoulkarneeva, Y., 616,8 Zhou, L., 614, 37; 619, 247 Zrelov, P., 619,50 Zhou, M.-Z., 611, 260 Zubarev, A.N., 612, 181; 616,8 Zhou, N.F., 614, 37; 619, 247 Zubkov, M.A., 616, 221 Zhou, Z., 617,1 Zucchelli, P., 613, 105; 614, 155 Zhu, G., 619, 313 Zucchiati, A., 615, 160 Zhu, G.Y., 613, 118; 615, 19; 616, 145, 159; 619,71 Zhu, K.J., 614, 37; 619, 247 Zuker, A.P., 613, 134 Zhu, Q.M., 614, 37; 619, 247 Zuo, W., 617,24 Zhu, R.Y., 613, 118; 615, 19; 616, 145, 159; 619,71 Zutshi, V., 617,1 Zhu, S., 618,51 Zverev, E.G., 617,1 Zhu, Y., 614, 37; 619, 247 Zverev, S.G., 613,29 Zhu, Y.C., 614, 37; 619, 247 Zvyagin, A., 612, 154 Zhu, Y.S., 614, 37; 619, 247 Zyla, P., 617,11 Zhu, Z.A., 614, 37; 619, 247 Zylberstejn, A., 617,1 Zhuang, B.A., 614, 37; 619, 247 Zylicz,˙ J., 619,88