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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 628, 2005

Amsterdam – Boston – Jena – London – New York – Oxford Paris – Philadelphia – San Diego – St. Louis Physics Letters B 628 (2005) 1–10 www.elsevier.com/locate/physletb

Stringy dark energy model with cold dark matter

I.Ya. Aref’eva a, A.S. Koshelev a,S.Yu.Vernovb

a Steklov Mathematical Institute, Russian Academy of Sciences, Russia b Skobeltsyn Institute of Nuclear Physics, Moscow State University, Russia Received 15 July 2005; accepted 7 September 2005 Available online 27 September 2005 Editor: N. Glover

Abstract Cosmological consequences of adding the cold dark matter (CDM) to the exactly solvable stringy dark energy (DE) model are investigated. The model is motivated by the consideration of our Universe as a slowly decaying D3-brane. The decay of this D-brane is described in the string field theory framework. Stability conditions of the exact solution with respect to small fluctuations of the initial value of the CDM energy density are found. Solutions with large initial value of the CDM energy density attracted by the exact solution without CDM are constructed numerically. In contrast to the CDM model the Hubble parameter in the model is not a monotonic function of time. For specific initial data the DE state parameter wDE is also not monotonic function of time. For these cases there are two separate regions of time where wDE being less than −1 is close to −1.  2005 Elsevier B.V. All rights reserved.

1. Introduction dominated by smoothly distributed slowly varying dark energy (DE) component (see [7] for reviews), 1 Nowadays strings and D-branes found cosmologi- for which the state parameter wDE is negative. Con- cal applications related with the cosmological acceler- temporary experiments give strong support that cur- − = ation [1–3]. The combined analysis of the type Ia su- rently the state parameter wDE is close to 1, wDE − ± pernovae, galaxy clusters measurements and WMAP 1 0.1 [6,8–11]. data provides an evidence for the accelerated cos- From the theoretical point of view the speciﬁed mic expansion [4–6]. The cosmological acceleration domain of w covers three essentially different cases: − =− − strongly indicates that the present day Universe is w> 1,w 1 and w< 1(see[12], and references therein). The most exciting possibility would be the case w<−1 corresponding to the so-called phan- E-mail addresses: [email protected] (I.Ya. Aref’eva), [email protected] (A.S. Koshelev), [email protected] 1 (S.Yu. Vernov). Here wDE is usual notation for the pressure to energy ratio.

0370-2693/$ – see front matter  2005 Elsevier B.V. All rights reserved. doi:10.1016/j.physletb.2005.09.017 2 I.Ya. Aref’eva et al. / Physics Letters B 628 (2005) 1–10 tom dominated Universe. In phenomenological mod- 2. Exactly solvable phantom model els describing this case the weak energy conditions >0,+ p>0 are violated and there are problems We start by recalling the main facts related to the with stability at classical and quantum levels [13]. model considered in [12]. This is a model of Ein- Thus, a phantom becomes a great challenge for the stein gravity interacting with a single phantom scalar theory while its presence according to the supernovae field in the spatially flat Friedmann Universe. Since data is not excluded. the phantom field comes from the string field theory A possible way to evade the stability problem for the string mass Ms and a dimensionless open string a phantom model is to yield the phantom as an effec- coupling constant go emerges. The action is tive model of a more fundamental theory which has √ M2 no such problems at all. It has been shown in [3] that S = d4x −g P R 2M2 such a model does appear in the string theory frame- s 1 1 work. This DE model assumes that our Universe is + + gµν∂ φ∂ φ − V(φ) , 2 µ ν (1) a slowly decaying D3-brane which dynamics is de- go 2 scribed by the tachyonic mode of the string field theory where MP is the reduced Planck mass, gµν is a spa- (SFT). The notable feature of the SFT description of tially flat Friedmann metric the tachyon dynamics is a non-local polynomial in- 2 =− 2 + 2 2 + 2 + 2 teraction [14–18]. It turns out the string tachyon be- ds dt a (t) dx1 dx2 dx3 havior is effectively described by a scalar field with and coordinates (t, xi) and field φ are dimensionless. a negative kinetic term (phantom) however due to Hereafter we use the dimensionless parameter mp for the string theory origin the model is stable at large short: times. g2M2 In [12] we have found an exactly solvable stringy m2 = o P . p 2 (2) DE model in the Friedmann Universe. This model is Ms a modified version of the effective SFT model [3] and If the scalar field depends only on time, i.e., φ = φ(t), is inspired by super-SFT calculations [17]. First level then independent equations of motion are calculations in the SFT give fourth order polynomial 2 1 1 ˙2 interaction. Higher levels increase a power of the inter- 3H = DE,DE =− φ + V(φ), m2 2 action. Exactly solvable model has a particular six or- p 1 1 der polynomial interaction potential. However, small H 2 + H˙ =− p ,p =− φ˙2 − V(φ). 3 2 2 DE DE fluctuations of coefficients in that potential do not mp 2 change the solution qualitatively and one can say that (3) the model [12] represents the behavior of non-BPS D3 Here dot denotes the time derivative, H ≡˙a(t)/a(t), brane in the Friedmann Universe rather well. It is in- DE and pDE are energy and pressure densities of the teresting to investigate the dynamics of the model in DE respectively. One can recast the system (3) to the the presence of the dark matter. This is a subject of the following form present Letter. 1 It turns out from the observational data that DE H˙ = φ˙2, 2m2 forms about 73% and the dark matter forms about 23% p of our Universe. Thus because of a significance of the 1 1 H 2 = − φ˙2 + V(φ) . 3 2 (4) dark matter component in the Universe in the present mp 2 Letter we investigate an interaction of the phantom Besides of this there is an equation of motion for the matter considered in [12] with the CDM. It seems im- field φ which is in fact a consequence of system (3). possible to find exact solutions in the presence of the Following the superpotential method [20] (see also CDM, except the case when the DE state parameter is [21]) we assume that H(t)is a function (named as su- a constant [19], so we use numeric methods to analyze perpotential) of φ(t): the behavior of the phantom field and cosmological parameters in our model. H(t)= W φ(t) . (5) I.Ya. Aref’eva et al. / Physics Letters B 628 (2005) 1–10 3

Fig. 1. The time evolution of the Hubble parameter H(t)(left), the deceleration parameter q(t) (middle) and the state parameter wDE(t) (right) 2 = in the exactly solvable model for mp 0.2.

2 2 This still does not give a systematic way to find goes asymptotically to ωA /(3mp) when t goes to general solutions to the system (4) but allows one to infinity. Once H(t) is known one readily obtains the construct W(φ) and V(φ)for a known function φ(t). scale factor We take for φ(t) A2 2 2 2 A (cosh(ωt) − 1) a(t) = a (ωt) 3mp , φ(t)= A tanh(ωt). (6) 0 cosh exp 2 2 12mp cosh(ωt) This function is known to describe effectively the late (10) time behavior of the tachyon in the 4-dimensional flat where a0 is an arbitrary constant, and the deceleration case [22,23]. The function φ(t) satisfies the following parameter equation aa¨ 1 q(t) =− φ˙ = ω A − φ2 . a˙2 A =−1 Hence, we obtain 18m2 (cosh(ωt))2 − p ω 1 . W = Aφ − φ3 , A2((cosh(ωt))2 − 1)(2(cosh(ω t))2 + 1)2 2 (7) 2mp 3A (11) and corresponding potential It follows from formula (11) that the Universe in this scenario is accelerating. ω2 ω2φ2 V(φ)= A2 − φ2 2 + A2 − φ2 2. The expression for the state parameter is the fol- 2 2 2 3 2A 12A mp lowing (8) p (φ) (A2 − φ2)2 We have omitted an integration constant in (7) to yield w (φ) = DE =− − m2 . DE 1 12 p 2 2 2 2 an even potential (8). It is typical that to keep the form DE(φ) φ (3A − φ ) of solutions to the scalar field equation in the presence (12) of Friedmann metric one has to modify the potential Point φ = A corresponds to an infinite future and adding a term proportional to the inverse of the re- therefore wDE →−1ast →∞. 2 duced Planck mass MP [12,24]. Plots for the Hubble, deceleration and state pa- The described solution leads to a number of cosmo- rameters are drawn in Fig. 1. (Hereafter we assume logical consequences. The Hubble parameter A = ω = 1 for all plots.) Thus, we conclude by noting that in our model the ωA2 1 H = (ωt) − (ωt) 2 phantom field provides the DE dominated accelerating 2 tanh 1 tanh (9) 2mp 3 Universe. 4 I.Ya. Aref’eva et al. / Physics Letters B 628 (2005) 1–10

3. Interaction with cold dark matter Either one can perform calculations using nonau- tonomous system of equations [25] which can be ob- 3.1. The model tained from Eqs. (15) and (18) in the following form

dφ 1 Now we are going to couple in a minimal way a = ψ, (21) pressureless matter of energy density M (the CDM) dn H(φ,ψ,n) to our model such that the Friedmann equations get an dψ =− + 1 dV(φ) extra term 3ψ , (22) dn H(φ,ψ,n) dφ 1 1 H 2 = − φ˙2 + V(φ)+ (a) , where 3 2 M (13) mp 2 1 1 1 H˙ = φ˙2 − (a) H(φ,ψ,n)= √ − ψ2 + V(φ)+ e−3n, 2 M (14) M,0 2mp 3mP 2 ˙ and the equation describing the evolution of scalar φ = ψ and n = ln(a/a0). field has previous form 3.2. Stability analysis for small fluctuations ¨ + ˙ − = φ 3Hφ Vφ 0. (15) From (13)–(15) we obtain the conservation of the In [12] we have analyzed the system (13)–(15) energy density for the CDM: without the CDM under condition A = ω = 1 and found that the exact solution φ = tanh(t) is stable with ˙ + 3H = 0, (16) M M respect to small fluctuations of the initial conditions if 2 that after integration gives and only if mp 1/2. −3 Let us consider the behavior of the solution of sys- −3 t H(τ)dτ a tem (19)–(20) in the neighborhood of the exact solu- M = M,0e = M,0 , (17) a0 tion where constants and a are initial values of M,0 0 M = and a correspondingly. From (17) we obtain Eq. (13) φ0(t) tanh(t), ˙ 2 in the following form: φ0(t) ≡ ψ0(t) = 1 − tanh(t) , ˙2 3 1 1 2 1 φ a0 2 3H = V(φ)− + . (18) H0(t) = tanh(t) 1 − tanh(t) . 2 M,0 2m2 3 mp 2 a p Following the lines of [12] we address to our analy- Substituting sis the questions of cosmological evolution and stability. H(t)= H0(t) + εH1(t), The straightforward way to study a stability of solu- φ(t)= φ (t) + εφ (t), tions to the system of equations (13)–(15) is to exclude 0 1 ˙ M from (13), (14) and obtain the following system: φ(t) ≡ ψ(t)= ψ0(t) + εψ1(t), (23) φ¨ + 3Hφ˙ − V = 0, (19) in (19) and (20) we obtain in the first order of ε the φ 1 1 following equations: H˙ + H 2 = φ˙2 + V(φ) . 2 3 2 (20) mp 2 ˙ φ1 = ψ1, ˙ Depending on the initial values of H , φ and φ, which 2 − 1 6mp 1 have been considered as independent, this system de- ψ˙ = 2 2m2 − 1 − φ 1 m2 p cosh(t)2 1 scribes our model either with or without the CDM. p In particular, the initial values: H = 0, φ = 0 and (2 cosh(t)2 + 1) tanh(t) 3 0 0 − ψ − H , φ˙ = Aω correspond to the exact solution (6). 2 2 1 2 1 0 2mp cosh(t) cosh(t) I.Ya. Aref’eva et al. / Physics Letters B 628 (2005) 1–10 5

= 2 = ˙ = Fig. 2. The time evolution of the scalar ﬁeld φ(t), the Hubble parameter H(t)and the state parameter wDE(t). M,0 1, mp 0.2andφ0 1.

= 2 = Fig. 3. The time evolution of the scalar ﬁeld φ(t), the Hubble parameter H(t) and the state parameter wDE(t). M,0 100, mp 0.2and ˙ φ0 = 1.

tanh(t) H˙ = + − m2 (t)2 φ ence of the CDM. To analyze the cosmological evolu- 1 2 4 1 2 4 p cosh 1 4mp cosh(t) tion it is instructive to plot phase curves for the scalar 1 (1 + 2 cosh(t)2) tanh(t) field as well as evolution of the state parameter wDE + ψ − H . 2 cosh(t)2 1 2 cosh(t)2 1 for the scalar matter. In addition we find numerically a (24) ratio of the energy densities for the CDM and the DE. System (24) has been solved with the help of the Experimental bounds for this ratio is known and es- computer algebra system Maple. The exact depen- timated to be near 1/3 so we can find the time point we live and a corresponding value of wDE in our ap- dence φ1(t), ψ1(t) and H(t)are too cumbersome to be 2 proach. presented here. The main result is that for mp 1/2 ˙ DuetoEq.(18) initial data φ0, φ0 and H0 do fix an functions φ1(t), ψ1(t) and H1(t) are bounded func- initial value of the CDM density. To have a given ini- tions and our exact solution is stable. ˙ 2 tial energy density of the CDM we take φ0 and φ0 and Note that numerical calculations show that if mp 1/2 then even for large initial values of the CDM en- find the corresponding value H0. In particular, to have = = ˙ = 2 = ergy density numerical solutions tend to the exact so- M,0 √1, φ0 0, φ0 1formp 0.2 we must take = ≈ lution as t tends to infinity. H0 5/3 1.29. For this initial values numeric solutions are presented graphically in Fig. 2.InFig. 3 we = 3.3. Numeric solutions. Time dependence present the√ same plots for M,0 100. Corresponding H0 = 10 5/3. Comparing Figs. 2 and 3 with Fig. 1 At this point we pass to numeric methods because one can see that our solutions with and without the it seems impossible to find exact solutions in the pres- CDM are different only in the beginning of the evolu- 6 I.Ya. Aref’eva et al. / Physics Letters B 628 (2005) 1–10

˙ Fig. 4. φ-dependence of the velocity φ (black line), the state parameter wDE (red line) and the CDM/DE ratio (blue line). Initial velocity of 2 = the scalar field is equal to 1 and mp 0.2. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this Letter.) tion where the CDM dominates (if exists). Note, that are interesting in solutions which approach the non- the behavior of the Hubble parameter in the presence perturbative vacuum during an infinite time. Thus, the of the CDM is not monotonic and the DE state para- point φ = 1 corresponds to an infinite future. The φ(t) meter may be not monotonic as well. dependence can be found numerically to pass from φ coordinate to the time. 3.4. Numeric solutions. φ-dependence In Figs. 4–6 we plot results of numeric solutions to Eqs. (21), (22) that allow us to find physical variables

It turns out that values of the wDE as well as ratio such as H , wDE, CDM/DE as function of the field φ. CDM/DE which are observational cosmological pa- These sets of plots differ in an initial velocity of the rameters [6,9,11] can be found easier using Eqs. (21), scalar field. Note that it follows from (18) that there (22) as functions of the e-folding number n. However, exists a maximal initial velocity ψ0m for our phantom it is more instructive to find a dependence on φ and field. ψ0m depends on values of φ0, M,0 and a0 and 2 = = not on n. does not depend on mp. In all plots a0 1 and φ0 0. Let us recall that from an analysis of our phan- All plots have three curves: black ones are phase tom model without the CDM we know that the scalar curves, red ones are w-s and blue ones are CDM/DE field interpolates between an unstable and a nonper- ratios. In Fig. 4 the initial velocity is equal to 1 (which 2 = turbative vacua during infinite time similar to the non- is the same as for the exact solution), mp 0.2 and BPS string tachyon [22,23]. In our notations nonper- M,0 is equal to 0.01, 1 and 100 from left to right. Here turbative vacuum corresponds to φ =+1. In the pure we see that the scalar field reaches +1. This indicates phantom model the evolution is described by φ(t) = a stability of the system with respect to fluctuations of 2 A tanh(ωt) function, where A and ω can be rescaled the initial CDM energy density for small mp.InFig. 5 to 1. This dependence is monotonic and this allows us the initial velocity is equal to 0.72. The first row there 2 = to find physical variables as functions of φ. corresponds to mp 0.2 and M,0 is equal to 0.01, 1 The situation is more complicated in the presence and 100 from left to right. The second row shows the 2 of the CDM. First, we do not know an exact time de- behavior of the system with mp equal to 0.6 and 1 and 2 pendence of the scalar field. Second, it is not evident M,0 equal to 0.01 and with mp equal to 1 and M,0 2 for arbitrary initial data and value of parameter mp equal to 1. One again sees from these plots that the 2 that the scalar field evolves monotonically. However, scalar field reaches 1 for small values of mp inawide in the particular cases presented in Figs. 2 and 3 our range of an initial CDM energy density. 2 solutions φ(t) are monotonic functions of time and This situation is broken for greater mp even for a moreover look like tanh(t) at large times. Below we small initial CDM energy density and the field does I.Ya. Aref’eva et al. / Physics Letters B 628 (2005) 1–10 7

˙ Fig. 5. φ-dependence of the velocity φ (black line), the state parameter wDE (red line) and the CDM/DE ratio (blue line). Initial velocity of the scalar ﬁeld is equal to 0.72. (For interpretation of the references to colour in this ﬁgure legend, the reader is referred to the web version of this Letter.)

not reach 1. In Fig. 6 M,0 is taken to be 1, the initial is no observational data indicated singular behavior of velocity is equal to its maximal possible value ψ0m and cosmological parameters and we do not consider cor- 2 mp is equal to 0.2, 0.6 and 1 from left to right. Here responding plots further. we again observe a stability for small mp and also find Hence, seeking for a situation where field φ ap- 2 out that for large mp the scalar field goes beyond the proaches 1 and there is no cosmological singularities point 1. Also for the maximal possible initial veloci- during this evolution we are left with the first row ties wDE and CDM/DE functions have a discontinu- in Fig. 4 and Fig. 5. In this plots phase curves show ity. One can understand this qualitatively because the that field φ indeed depends monotonically on time be- energy density for the DE has two terms with oppo- cause φ˙ is always positive during the evolution. Look- site signs. Indeed, the scalar field is a phantom and ing for specified plots we draw the reader’s attention its kinetic energy is negative while the potential term to the following interesting properties of our model. is positive. Thus at some point the energy density of First, CDM/DE ratio dependence is monotonic and the DE changes the sign and develops a discontinuity experimentally measured value 1/3 is close to the be- in wDE and CDM/DE ratio. Such a behavior is rather ginning of the evolution. For example, in Fig. 4 (left) undesirable from cosmological point of view, since the this point corresponds to wDE ≈−1.02 and φ ≈ 0.09. 8 I.Ya. Aref’eva et al. / Physics Letters B 628 (2005) 1–10

˙ Fig. 6. φ-dependence of the velocity φ (black line), the state parameter wDE (red line) and the CDM/DE ratio (blue line). Initial velocity of the scalar ﬁeld is equal to its maximal possible value. (For interpretation of the references to colour in this ﬁgure legend, the reader is referred to the web version of this Letter.)

Moreover, this is not a distinguished value and it fol- It is interesting to find an influence of the higher lows from the model that this ratio will decrease with open string mass levels as well as an influence of time. Second, for ρM,0 large enough wDE behaves the closed string excitations on the obtained picture. non-monotonically. Even in the flat space–time the dynamics of a D-brane change drastically when the closed string excitations are included [28,29]. We get a stable behavior and smooth cosmologi- 4. Discussion and conclusion cal parameters in the stringy inspired model only in 2 the case when the dimensionless parameter mp is less To summarize, let us also note that we get an ex- than 0.5. This restricts the parameters of the original istence of a region of the initial energy density of the theory [3] the model considered in this Letter comes from. Let us recall that m2 is related with the reduced CDM, for which wDE is not monotonic. Such a behav- p 2 ior is interesting and very surprising. We see that for Planck mass, the string mass parameter Ms and the 2 large initial energy densities of the CDM wDE grows open string coupling constant go: with time from minus infinity to approximately −1, g2M2 then goes down to a local minimum and after this m2 = o P . p 2 grows again asymptotically approaching −1. Note, Ms that it has been proved in [26] that under some (com- Therefore, to have an acceptable cosmological solu- pare with [27]) conditions the phantom matter can- tionwehavetoassumethat2g2M2

Acknowledgements [11] U. Seljak, et al., astro-ph/0407372. [12] I.Ya. Aref’eva, A.S. Koshelev, S.Yu. Vernov, astro- ph/0412619. This work is supported in part by RFBR grant [13] R.R. Caldwell, Phys. World 17 (5) (2004) 37; 05-01-00758, I.A. and A.K. are supported in part by R.R. Caldwell, Phys. Lett. B 545 (2002) 23, astro-ph/9908168; INTAS grant 03-51-6346 and by Russian Federation R.R. Caldwell, M. Kamionkowski, N.N. Weinberg, Phys. Rev. President’s grant NSh-2052.2003.1, S.V. is supported Lett. 91 (2003) 071301, astro-ph/0302506; in part by Russian Federation President’s Grant NSh- B. McInnes, JHEP 0208 (2002) 029, hep-th/0112066; S. Nesseris, L. Perivolaropoulos, Phys. Rev. D 70 (2004) 1685.2003.2 and by the grant of the scientiﬁc Program 123529, astro-ph/0410309; “Universities of Russia” 02.02.503. S.M. Carroll, M. Hoffman, M. Trodden, Phys. Rev. D 68 (2003) 023509, astro-ph/0301273; J.M. Cline, S. Jeon, G.D. Moore, Phys. Rev. D 70 (2004) 043543, hep-ph/0311312; References S.D.H. Hsu, A. Jenkins, M.B. Wise, Phys. Lett. B 597 (2004) 270, astro-ph/0406043; [1] A. Sen, JHEP 0204 (2002) 048, hep-th/0203211; V.K. Onemli, R.P. Woodard, Class. Quantum Grav. 19 (2002) A. Sen, JHEP 0207 (2002) 065, hep-th/0203265; 4607, gr-qc/0204065; A.V. Frolov, L. Kofman, A.A. Starobinsky, Phys. Lett. B 545 V.K. Onemli, R.P. Woodard, Phys. Rev. D 70 (2004) 107301, (2002) 8, hep-th/0204187; gr-qc/0406098; G.N. Felder, L. Kofman, A. Starobinsky, JHEP 0209 (2002) U. Alam, V. Sahni, A.A. Starobinsky, JCAP 0406 (2004) 008, 026, hep-th/0208019; astro-ph/0403687; G.W. Gibbons, Class. Quantum Grav. 20 (2003) S321, hep- T. Padmanabhan, Phys. Rev. D 66 (2002) 021301, hep- th/0301117; th/0204150; A. Sen, hep-th/0410103. T. Padmanabhan, T. Roy Choudhury, Phys. Rev. D 66 (2002) [2] C. Deffayet, G. Dvali, G. Gabadadze, Phys. Rev. D 65 (2002) 081301, hep-th/0205055; 044023, astro-ph/0105068; A. Melchiorri, L. Mersini, C.J. Odman, M. Trodden, Phys. Rev. R. Kallosh, A. Linde, Phys. Rev. D 67 (2003) 023510, hep- D 68 (2003) 043509, astro-ph/0211522; th/0208157; B. Feng, X. Wang, X. Zhang, astro-ph/0404224; Sh. Mukohyama, L. Randall, Phys. Rev. Lett. 92 (2004) B. Feng, M. Li, Y.-S. Piao, X. Zhang, astro-ph/0407432; 211302, hep-th/0306108; S. Nojiri, S. Odintsov, hep-th/0408170; V. Sahni, Y. Shtanov, JCAP 0311 (2003) 014, astro- W. Fang, H.Q. Lu, Z.G. Huang, K.F. Zhang, hep-th/0409080; ph/0202346; J.-G. Hao, X.-Z. Li, Phys. Rev. D 67 (2003) 107303, gr- Ph. Brax, C. van de Bruck, A.-C. Davis, Rep. Prog. Phys. 67 qc/0302100; (2004) 2183, hep-th/0404011; P. Singh, M. Sami, N. Dadhich, Phys. Rev. D 68 (2003) Th.N. Tomaras, hep-th/0404142; 023522, hep-th/0305110; E.J. Copeland, M.R. Garousi, M. Sami, Sh. Tsujikawa, hep- R.-G. Cai, A. Wang, hep-th/0411025; th/0411192; Z.-K. Guo, Y.-Z. Zhang, astro-ph/0411524; B. McInnes, Nucl. Phys. B 718 (2005) 55, hep-th/0502209; S.M. Carroll, A. De Felice, M. Trodden, astro-ph/0408081. G. Calcagni, Sh. Tsujikawa, M. Sami, hep-th/0505193. [14] E. Witten, Nucl. Phys. B 268 (1986) 253; [3] I.Ya. Aref’eva, astro-ph/0410443. E. Witten, Nucl. Phys. B 276 (1986) 291. [4] A.G. Riess, et al., Astron. J. 116 (1998) 1009, astro- [15] I.Ya. Aref’eva, P.B. Medvedev, A.P. Zubarev, Phys. Lett. B 240 ph/9805201. (1990) 356; [5] S.J. Perlmutter, et al., Astrophys. J. 517 (1999) 565, astro- C.R. Preitschopf, C.B. Thorn, S.A. Yost, Nucl. Phys. B 337 ph/9812133. (1990) 363; [6] D.N. Spergel, et al., Astrophys. J. Suppl. 148 (2003) 175, astro- I.Ya. Aref’eva, P.B. Medvedev, A.P. Zubarev, Nucl. Phys. ph/0302209. B 341 (1990) 464. [7] V. Sahni, astro-ph/0403324; [16] N. Berkovits, A. Sen, B. Zwiebach, Nucl. Phys. B 587 (2000) P. Frampton, astro-ph/0409166; 147, hep-th/0002211. T. Padmanabhan, Phys. Rep. 380 (2003) 235, hep-th/0212290. [17] I.Ya. Aref’eva, D.M. Belov, A.S. Koshelev, P.B. Medvedev, [8] J.L. Tonry, et al., Astrophys. J. 594 (2003) 1, astro-ph/ Nucl. Phys. B 638 (2002) 3, hep-th/0011117; 0305008. I.Ya. Aref’eva, D.M. Belov, A.S. Koshelev, P.B. Medvedev, [9] M. Tegmark, et al., Phys. Rev. D 69 (2004) 103501, astro- Nucl. Phys. B 638 (2002) 21, hep-th/0107197. ph/0310723; [18] K. Ohmori, hep-th/0102085; M. Tegmark, JCAP 0504 (2005) 001, astro-ph/0410281. I.Ya. Aref’eva, D.M. Belov, A.A. Giryavets, A.S. Koshelev, [10] A.G. Riess, et al., Astrophys. J. 607 (2004) 665, astro- P.B. Medvedev, hep-th/0111208; ph/0402512. W. Taylor, hep-th/0301094. 10 I.Ya. Aref’eva et al. / Physics Letters B 628 (2005) 1–10

[19] V. Sahni, A.A. Starobinsky, Int. J. Mod. Phys. D 9 (2000) 373, [23] Ya.I. Volovich, J. Phys. A 36 (2003) 8685, math-ph/0301028; astro-ph/9904398. V.S. Vladimirov, Ya.I. Volovich, Theor. Math. Phys. 138 [20] O. DeWolfe, D.Z. Freedman, S.S. Gubser, A. Karch, Phys. Rev. (2004) 297, math-ph/0306018. D 62 (2000) 046008, hep-th/9909134. [24] I.Ya. Aref’eva, L.V. Joukovskaya, hep-th/0504200. [21] T. Padmanabhan, Phys. Rev. D 66 (2002) 021301, hep- [25] C. Armendariz-Picon, V.Mukhanov, P.J. Steinhardt, Phys. Rev. th/0204150. D 63 (2001) 103510, astro-ph/0006373. [22] I.Ya. Aref’eva, L.V. Joukovskaya, A.S. Koshelev, hep- [26] A. Vikman, Phys. Rev. D 71 (2005) 023515, astro-ph/0407107. th/0301137; [27] A.A. Andrianov, F. Cannata, A.Y. Kamenshchik, gr-qc/ I.Ya. Aref’eva, Fortschr. Phys. 51 (2003) 652; 0505087. I.Ya. Aref’eva, L.V. Joukovskaya, Rolling Tachyon on non- [28] K. Ohmori, Phys. Rev. D 69 (2004) 026008. BPS brane, Lectures given at the II Summer School in Mod- [29] L. Joukovskaya, Ya. Volovich, math-ph/0308034. ern Mathematical Physics, Kopaonik, Serbia, 1–12 September 2002. Physics Letters B 628 (2005) 11–17 www.elsevier.com/locate/physletb

Role of the Brans–Dicke scalar in the holographic description of dark energy

Hungsoo Kim a,H.W.Leea, Y.S. Myung a,b

a Relativity Research Center, Inje University, Gimhae 621-749, South Korea b Institute of Theoretical Science, 5203 University of Oregon, Eugene, OR 97403, USA Received 5 July 2005; accepted 20 September 2005 Available online 28 September 2005 Editor: T. Yanagida

Abstract We study cosmological application of the holographic energy density in the Brans–Dicke theory. Considering the holographic energy density as a dynamical cosmological constant, it is more natural to study it in the Brans–Dicke theory than in general relativity. Solving the Friedmann and Brans–Dicke field equations numerically, we clarify the role of Brans–Dicke field during evolution of the universe. When the Hubble horizon is taken as the IR cutoff, the equation of state (wΛ) for the holographic 5 energy density is determined to be 3 when the Brans–Dicke parameter ω goes infinity. This means that the Brans–Dicke field plays a crucial role in determining the equation of state. For the particle horizon IR cutoff, the Brans–Dicke scalar mediates a transition from wΛ =−1/3(past)towΛ = 1/3 (future). If a dust matter is present, it determines future equation of state. In the case of future event horizon cutoff, the role of the Brans–Dicke scalar and dust matter are turned out to be trivial, whereas the holographic energy density plays an important role as a dark energy candidate with wΛ =−1.  2005 Elsevier B.V. All rights reserved.

1. Introduction standard cosmology is given by inﬂation and FRW universe [3]. Type Ia supernova observations [1] suggest that A typical candidate for the dark energy is the cos- our universe is in accelerating phase and the dark en- mological constant in general relativity. Recently Co- hen et al. [4] showed that in the effective theory of ergy contributes ΩDE 0.60–0.70 to the critical energy density of the present universe. Also cosmic mi- quantum ﬁeld theory, the UV cutoff Λ is related to crowave background observations [2] imply that the the IR cutoff LΛ, due to the limit set by forming a black hole. In other words, if ρΛ is the quantum zero- point energy density caused by the UV cutoff, the to- E-mail addresses: [email protected], tal energy density of the system with size LΛ should [email protected] (Y.S. Myung). not exceed the mass of the system-sized black hole:

3 LΛρΛ LΛ/G. Here the Newtonian constant G is parameter ω goes infinity. This implies that the Brans– = 2 related to the Planck mass by G 1/Mp. The largest Dicke framework is suitable for studying an evolution IR cutoff LΛ is chosen as the one saturating this in- of the holographic energy density. equality and the holographic energy density is then In this work, we introduce a dust matter to our con- = 2 2 2 given by ρΛ 3c Mp/8πLΛ with an appropriate fac- sideration and solve the equations numerically. Since tor 3c2/8π. Comparing with the cosmological con- the holographic energy density is dynamical, it is non- stant, we regard it as a dynamical cosmological con- trivial to solve the Friedmann and the Brans–Dicke stant. Taking LΛ as the size of the present universe field equation with three conservation laws. They can- (Hubble horizon RHH), the resulting energy density not be solved analytically. From this study we investi- is comparable to the present dark energy density [5]. gate the role of the holographic energy density, Brans– Even though this holographic approach leads to the Dicke scalar and dust matter for a given IR cutoff. data, this description is incomplete because it fails Especially, we wish to show why a combination of the to explain the equation of state for the dark energy- holographic energy density and future event horizon dominated universe [6]. In order to resolve this situa- could describe a dark energy-dominated era. tion, one introduces other candidates for the IR cutoff. One is the particle horizon RPH. This provides −2(1+1/c) ρΛ ∼ a , which means that the equation of 2. Brans–Dicke cosmology state is given by wΛ = 1/3forc = 1 [7]. However, it corresponds to a radiation-dominated universe and For cosmological purpose, we introduce the Brans– it is a decelerating phase. In order to find an accel- Dicke (BD) action with a matter erating phase, we need to introduce the future event = √ horizon RFH. In the case of LΛ RFH, one finds S = d4x −g ∼ −2(1−1/c) ρΛ a which could describe the dark energy with wΛ =−1forc = 1. This is close to the data [1] 1 ∇ Φ∇αΦ × ΦR − ω α + L , (1) and the related works appeared in Refs. [8–11]. 16π Φ M On the other hand, it is worthwhile to investigate the holographic energy density in the frame- where Φ is the BD scalar which plays the role of an work of the Brans–Dicke theory. The reasons are inverse of the Newtonian constant, ω is the parameter L as follows. Because the holographic energy density of BD theory, and M represents other matter which belongs to a dynamical cosmological constant, we takes a perfect fluid form. The field equations for met- need a dynamical frame to accommodate it instead of ric gµν and BD scalar Φ are general relativity. Further, taking LΛ = RHH, it fails ≡ − 1 = BD + 8π M to determine the equation of state wΛ in the gen- Gµν Rµν gµνR 8πTµν Tµν , eral relativity framework. In addition to these, the 2 Φ 8π Brans–Dicke scalar speeds up the expansion rate of ∇ ∇αΦ = T M α, α + α (2) a dust matter-dominated era (reduces deceleration), 2ω 3 while slows down the expansion rate of cosmolog- where the energy–momentum tensor for the BD scalar ical constant era (reduces acceleration) [12,13].The is defined by Brans–Dicke generalization was first studied by Gong 1 ω 1 [14]. Since the Brans–Dicke description of gravita- T BD = ∇ Φ∇ Φ − g (∇Φ)2 µν 8π Φ2 µ ν 2 µν tion is to replace the Newtonian constant G by a time varying scalar Φ(t), the holographic energy density is 1 α = 2 2 = + ∇µ∇νΦ − gµν∇α∇ Φ (3) given by ρΛ 3Φ/8πLΛ with c 1. Gong recov- Φ ered the same results as those in general relativity for a large ω. The present authors studied the same issue and the energy–momentum tensor for other matter by considering a Bianchi identity as a consistency con- takes the form dition [15]. The equation of state for Hubble IR cutoff M = + + = 5 Tµν pM gµν (ρM pM )UµUν. (4) is determined to be wΛ 3 when the Brans–Dicke H. Kim et al. / Physics Letters B 628 (2005) 11–17 13

Here ρM (pM ) denote the energy density (pressure) 3. Brans–Dicke framework with holographic of the matter and Uµ is a four velocity vector with energy density and dust matter α UαU = 1. Assuming that our universe is homogeneous and In this section, we investigate how the equation of isotropic, we work with the Friedmann–Robertson– state for the holographic energy density changes when Walker (FRW) spacetime an interaction between the BD field ρBD, holographic energy density ρΛ and dust matter ρm is included. In dr2 this case the Friedmann and BD field equations are ds2 =−dt2 + a2(t) 1 − kr2 Φ˙ ω Φ˙ 2 8π ρ H 2 + H − = t , + r2 dθ2 + sin2 θdφ2 . (5) Φ 6 Φ 3 Φ ˙ ¨ Φ 8π = Φ + 3H = (ρt − 3pt ). (11) We consider a spatially flat spacetime of k 0. In the Φ 2ω + 3 FRW spacetime, the field equations take the forms Here ρt = ρΛ + ρm and pt = pΛ + pm. The holographic energy density ρ and a dust matter ρ are Φ˙ ω Φ˙ 2 8π ρ Λ m H 2 + H − = M , chosen to be Φ 6 Φ 3 Φ 3 Φ − 8π ρ = ,ρ= ρ0 a 3 ¨ ˙ Λ 2 m m (12) Φ + 3HΦ = (ρM − 3pM ) (6) 8π L 2ω + 3 Λ with p = 0. In order to solve Eq. (11) with Eqs. (8) with the Hubble parameter H =˙a/a. Here we note m and (9), we define that the case of ω =−3/2 is not allowed when a matter with pM = ρM /3 comes into the BD theory. Φ x = ln a, ϕ = , Regarding the BD field as a perfect fluid, its energy Φ and pressure are given by [13] H R λ =− ,r= , (13) H R 1 Φ˙ 2 Φ˙ ρBD = ω − 6H , where means the derivative with respect to x and R ∈ 16πG0 Φ Φ { } RHH,RPH,RFH . From the definition of ϕ and λ,itis ˙ 2 ˙ ¨ granted that H and Φ are taken to be positive [16]. = 1 Φ + Φ + Φ pBD ω 4H 2 , (7) Then the Friedmann equation and BD field equation 16πG0 Φ Φ Φ become where G is the present Newtonian constant. Usually, 0 ω 8π ρ if one does not specify the parameter ω, one cannot de- H 2 1 + ϕ − ϕ2 = t , termine the BD equation of state exactly. The Bianchi 6 3 Φ 8π ρ − 3p identity leads to an energy transfer between BD field H 2 ϕ − λϕ + ϕ2 + 3ϕ = t t . (14) and other matter 2ω + 3 Φ ˙ Also the energy–momentum conservation law leads to 1 ρM Φ ρ˙BD + 3H(ρBD + pBD) = (8) the pressure G0 Φ Φ 1 and the matter evolves according to its conservation p =− (ϕ − 2r + 3)ρ (15) Λ 3 Λ law whose equation of state is given by ρ˙M + 3H(ρM + pM ) = 0. (9) 1 wΛ =− (ϕ − 2r + 3). (16) Their equations of states are given by 3 If other interaction between ρΛ and ρm is included, pBD pM then the equation of state w takes a different form wBD ≡ ,wM ≡ . (10) Λ ρBD ρM [17]. 14 H. Kim et al. / Physics Letters B 628 (2005) 11–17

Fig. 1. A plot of equation of state wΛ versus x = ln a for Hubble horizon. Here BD denotes the Brans–Dicke framework.

From now on we focus on the change of wΛ by is essential for determining wΛ and it goes well with choosing an IR cutoff LΛ. Firstly, we take Hubble LΛ = 1/H . horizon as the IR cutoff scale (LΛ = RHH = 1/H ). For particle horizon IR cutoff with LΛ = RPH ≡ We have λ = r and then eliminate λ to obtain a a da and future event horizon IR cutoff L = 0 Ha2 Λ ≡ ∞ da RFH a a 2 , r is given by ω(ω + 1)ϕ 6 1 Ha ϕ = ϕ − ϕ − , 6 ω ω + 1 ω r = 1 ± Ω 1 + ϕ − ϕ2 (18) 1 ωϕ ϕ + 3 Λ 6 r = − 1 (ω + 1)ϕ − 1 + . (17) 2 3 2 with ΩΛ ≡ ρΛ/ρt .Here‘+’ denotes particle horizon, − One can solve the above equation numerically. We while ‘ ’ represents future event horizon. Eliminating λ leads to two coupled equations have a plot for wΛ as is shown in Fig. 1.InFig. 1,an = 5 ω 2 upper dotted line represents a graph for wΛ 3 with- 1 + ϕ − ϕ →∞ ϕ =− 6 3 (ω + 1)ϕ − 1 out a dust matter in the limit of ω . This case was 2ω + 3 already found in [15]. Solid lines represent the change + (ϕ − 2r + 3)(ωϕ − 3)ΩΛ , of equation of state wΛ with a dust matter. When a dust = − + − matter is present, the BD theory allows two solutions ΩΛ (ϕ 2r 3)ΩΛ(1 ΩΛ). (19) = 5 = 1 in the far past: wΛ 3 for a large ω and wΛ 3 ,in- Here λ is related to r via dependent of ω. As the BD ﬁeld evolves, the equation ω 1 − ϕ of state for the holographic energy density converges λ = r + 3 ϕ . (20) + − ω 2 that of dust matter. The universe behaves as a dust 2(1 ϕ 6 ϕ ) matter-dominated phase in the far future, irrespective We solve the coupled equations numerically and of where it starts. If the BD scalar is turned off, one plot wΛ in Figs. 2 and. 3 for particle horizon and fu- cannot determine the equation of state for the holo- ture event horizon, respectively. = = 1 graphic energy density only [6]. In this sense, although For particle horizon IR cutoff LΛ RPH, wΛ 3 we do not obtain a dark energy era, the BD framework is found for the holographic energy density solely H. Kim et al. / Physics Letters B 628 (2005) 11–17 15

Fig. 2. A plot for wΛ as a function of x = ln a for particle horizon. Here GR (BD) denote the general relativity (Brans–Dicke) frameworks.

Fig. 3. A plot for wΛ as a function of x = ln a for future event horizon. Here GR (BD) denote the general relativity (Brans–Dicke) frameworks. when using general relativity. A thin line stands sents wΛ in the BD framework with a dust matter. In for wΛ in the general relativity framework together the general relativity framework, the equation of state =±1 with a dust matter. A medium line results from the of holographic energy density starts with wΛ 3 in DB framework without matter and a thick line repre- the far past and ends with wΛ = 0 like as a dust matter. 16 H. Kim et al. / Physics Letters B 628 (2005) 11–17

However, in the BD framework, equation of state of Table 1 =−1 Summary for future equation of state. Here three combinations for the holographic energy density starts with wΛ 3 at 1 holographic energy density (ρ ), Brans–Dicke scalar (ρ ), and the far past and then, transits to wΛ = . This implies Λ BD 3 dust matter (ρ ) are evaluated for IR cutoff (L ) as Hubble hori- that without matter, the holographic energy density be- m Λ zon (RHH), particle horizon (RPH) and future event horizon (RFH), comes a radiation. In the BD framework together with respectively a dust matter, the equation of state of holographic en- 1 Matter RHH RPH RFH ergy density starts with wΛ =− in the far past. The 3 ρ + ρ w = 5/3 w = 1/3 w =−1 BD field makes a transition to a radiation phase and Λ BD Λ Λ Λ ρΛ + ρm wΛ = 0 wΛ = 0 wΛ =−1 finally, wΛ transits to a dust matter in the far future. ρΛ + ρBD + ρm wΛ = 0 wΛ = 0 wΛ =−1 In this case, the dust matter determines a future equation of state for the holographic energy density. This means that a dust matter dominates in the holographic state for the holographic energy density (w ) is de- energy density with particle horizon. Finally we men- Λ termined to be 5 when the Brans–Dicke parameter ω tion that the BD scalar plays a role of the mediator 3 goes infinity. This means that the Brans–Dicke scalar between w =−1 and w = 1 . Λ 3 Λ 3 is crucial for determining the equation of state when For future event horizon cutoff L = R , one Λ FH comparing to the case of ρ . However, if a dust mat- finds w =−1 with c2 = 1forρ only in the gen- Λ Λ Λ ter is turned on, its future equation of state is deter- eral relativity framework. A thin line stands for a mined by w = 0, irrespective of the presence of the graph of w with a dust matter in general relativity. Λ Λ Brans–Dicke scalar. Actually, the equation of state for A medium/thick lines correspond to the BD theory ρ +ρ can be determined to be w = 0 by the Fried- framework without/with a dust matter. A general rel- Λ m Λ mann equation [6]. ativistic analysis was carried out by Li [7]. Equation For particle horizon IR cutoff, the Brans–Dicke of state of the holographic energy density starts with scalar mediates the transition from w =−1/3 (past) w =−1 in the far past and becomes a cosmological Λ Λ 3 to w = 1/3 (future). However, if a dust matter is constant with w =−1 in the far future. For this IR Λ Λ present, it determines future equation of state. Hence a cutoff, the holographic energy density serves as a dark dust matter plays an important role in the holographic energy and leads to an accelerating era. As is shown description with particle horizon. in Fig. 3, this feature persists even in the BD frame- In the case of future event horizon cutoff, the role work with or without a dust matter. This means that of the Brans–Dicke scalar and dust matter are trivial, the holographic energy density goes well with future whereas the holographic energy density plays an im- event horizon L = R . On the other hand, the role Λ FH portant role as a dark energy candidate with w =−1. of dust matter and BD scalar is trivial when comparing Λ Consequently, we find the major roles in the holo- with the holographic energy density. graphic description of an evolving universe: BD scalar in Hubble horizon; dust matter in particle horizon; holographic energy density in future event horizon. 4. Summary The BD scalar plays a role of the mediator, but it does not determine the future equation of state if a dust mat- We study cosmological application of the holo- ter or the holographic energy density is present. graphic energy density in the Brans–Dicke framework. Considering the holographic energy density as a dynamical cosmological constant, it is more natural to study it in the Brans–Dicke theory than in general Acknowledgements relativity. Solving the Friedmann and Brans–Dicke field equations numerically, we investigate the role of H. Kim and H. Lee were in part supported by Brans–Dicke field during evolution of the universe. KOSEF, Astrophysical Research Center for the Struc- We summarize future equation of state for the holo- ture and Evolution of the Cosmos. Y. Myung was graphic energy density in Table 1. When the Hub- supported by the Korea Research Foundation Grant ble horizon is taken as the IR cutoff, the equation of (KRF-2005-013-C00018). H. Kim et al. / Physics Letters B 628 (2005) 11–17 17

References B. Wang, E. Abdalla, R.-K. Su, hep-th/0404057; K. Enqvist, M.S. Sloth, Phys. Rev. Lett. 93 (2004) 221302, hep-th/0406019; [1] S.J. Perllmutter, et al., Astrophys. J. 517 (1999) 565, astro- S. Hsu, A. Zee, hep-th/0406142; ph/9812133; K. Ke, M. Li, Phys. Lett. B 606 (2005) 173, hep-th/0407056; A.G. Reiss, et al., Astron. J. 116 (1998) 1009, astro- P.F. Gonzalez-Diaz, hep-th/0411070; ph/9805201; S. Nobbenhuis, gr-qc/0411093. A.G. Reiss, et al., Astrophys. J. 607 (2004) 665, astro- [9] Y.S. Myung, Phys. Lett. B 610 (2005) 18, hep-th/0412224; ph/0402512. Y.S. Myung, Mod. Phys. Lett. A 27 (2005) 2035, hep-th/ [2] H.V. Peiris, et al., Astrophys. J. Suppl. 148 (2003) 213, astro- 0501023. ph/0302225; [10] A.J.M. Medved, hep-th/0501100. C.L. Benett, et al., Astrophys. J. Suppl. 148 (2003) 1, astro- [11] S. Nojiri, S.D. Odintsov, hep-th/0506212; ph/0302207; S. Nojiri, S.D. Odintsov, hep-th/0505215. D.N. Spergel, et al., Astrophys. J. Suppl. 148 (2003) 175, astro- [12] D. La, P.J. Steinhardt, Phys. Rev. Lett. 62 (1989) 376. ph/0302209. [13] H. Kim, Phys. Lett. B 606 (2005) 223, astro-ph/0408154. [3] A.H. Guth, Phys. Rev. D 23 (1981) 347; [14] Y. Gong, Phys. Rev. D 70 (2004) 064029, hep-th/0404030. A.D. Linde, Phys. Lett. B 108 (1982) 389; [15] H. Kim, H.W. Lee, Y.S. Myung, hep-th/0501118. A. Albrecht, P.J. Steinhardt, Phys. Rev. Lett. 48 (1982) 1220. [16] A.D. Rendall, Class. Quantum Grav. 21 (2004) 2445, gr-qc/ [4] A. Cohen, D. Kaplan, A. Nelson, Phys. Rev. Lett. 82 (1999) 0403070. 4971, hep-th/9803132. [17] R. Horvat, Phys. Rev. D 70 (2004) 087301, astro-ph/0404204; [5] P. Horava, D. Minic, Phys. Rev. Lett. 85 (2000) 1610, hep- P. Wang, X. Meng, Class. Quantum Grav. 22 (2005) 283, astro- th/0001145; ph/0408495; S. Thomas, Phys. Rev. Lett. 89 (2002) 081301. Y.S. Myung, hep-th/0502128; [6] S.D. Hsu, Phys. Lett. B 594 (2004) 13, hep-th/0403052. B. Guberina, R. Horvat, H. Stefancic, JCAP 0505 (2005) 001, [7] M. Li, Phys. Lett. B 603 (2004) 1, hep-th/0403127. astro-ph/0503495. [8] Q.C. Huang, Y. Gong, JCAP 0408 (2004) 006, astro-ph/ 0403590; Physics Letters B 628 (2005) 18–24 www.elsevier.com/locate/physletb

+ Conﬁrmation of the doubly charmed baryon Ξcc(3520) via its decay to pD+K−

SELEX Collaboration A. Ocherashvili l,1, M.A. Moinester l,J.Russc, J. Engelfried m, I. Torres m, U. Akgun p, G. Alkhazov k, J. Amaro-Reyes m, A.G. Atamantchouk k,, A.S. Ayan p, M.Y. Balatz h,, N.F. Bondar k, P.S. Cooper e, L.J. Dauwe q, G.V. Davidenko h,U.Derschi,2, A.G. Dolgolenko h, G.B. Dzyubenko h, R. Edelstein c, L. Emediato s, A.M.F. Endler d, I. Eschrich i,3,C.O.Escobars,4, A.V. Evdokimov h, I.S. Filimonov j,, F.G. Garcia s,e, M. Gaspero r, I. Giller l,V.L.Golovtsovk,P.Gouffons, E. Gülmez b, He Kangling g,M.Iorir,S.Y.Junc, M. Kaya p,5, J. Kilmer e,V.T.Kimk, L.M. Kochenda k, I. Konorov i,6, A.P. Kozhevnikov f,A.G.Krivshichk,H.Krügeri,7, M.A. Kubantsev h, V.P. Kubarovsky f, A.I. Kulyavtsev c,e, N.P. Kuropatkin k,e, V. F. Ku r s h et s ov f, A. Kushnirenko c,f,S.Kwane, J. Lach e,A.Lambertot, L.G. Landsberg f,,I.Larinh,E.M.Leikinj, Li Yunshan g, M. Luksys n, T. Lungov s, V. P. M al eev k,D.Maoc,8, Mao Chensheng g, Mao Zhenlin g,P.Mathewc,9, M. Mattson c, V. Matveev h, E. McCliment p, V.V. Molchanov f,A.Morelosm, K.D. Nelson p,10, A.V. Nemitkin j, P.V. Neoustroev k,C.Newsomp,A.P.Nilovh, S.B. Nurushev f, Y. Onel p, E. Ozel p, S. Ozkorucuklu p,11,A.Penzot,S.V.Petrenkof, P. Pogodin p,12, M. Procario c,13, V.A. Prutskoi h, E. Ramberg e, G.F. Rappazzo t, B.V. Razmyslovich k,14, V. I . R u d j,P.Schiavont, J. Simon i,15, A.I. Sitnikov h,D.Skowe, V.J. Smith o, M. Srivastava s,V.Steinerl,V.Stepanovk,14, L. Stutte e,M.Svoiskik,14, N.K. Terentyev k,c, G.P. Thomas a,L.N.Uvarovk, A.N. Vasiliev f, D.V. Vavilov f, E. Vázquez-Jáuregui m, V.S. Verebryusov h, V.A. Victorov f, V.E. Vishnyakov h, A.A. Vorobyov k,K.Vorwalteri,16,J.Youc,e, Zhao Wenheng g, Zheng Shuchen g, R. Zukanovich-Funchal s

a Ball State University, Muncie, IN 47306, USA b Bogazici University, Bebek, 80815 Istanbul, Turkey c Carnegie-Mellon University, Pittsburgh, PA 15213, USA d Centro Brasileiro de Pesquisas Físicas, Rio de Janeiro, Brazil e Fermi National Accelerator Laboratory, Batavia, IL 60510, USA

f Institute for High Energy Physics, Protvino, Russia g Institute of High Energy Physics, Beijing, PR China h Institute of Theoretical and Experimental Physics, Moscow, Russia i Max-Planck-Institut für Kernphysik, 69117 Heidelberg, Germany j Moscow State University, Moscow, Russia k Petersburg Nuclear Physics Institute, St. Petersburg, Russia l Tel Aviv University, 69978 Ramat Aviv, Israel m Universidad Autónoma de San Luis Potosí, San Luis Potosí, Mexico n Universidade Federal da Paraíba, Paraíba, Brazil o University of Bristol, Bristol BS8 1TL, United Kingdom p University of Iowa, Iowa City, IA 52242, USA q University of Michigan-Flint, Flint, MI 48502, USA r University of Rome “La Sapienza” and INFN, Rome, Italy s University of São Paulo, São Paulo, Brazil t University of Trieste and INFN, Trieste, Italy

Received 26 August 2005; accepted 21 September 2005

Available online 29 September 2005

Editor: W.-D. Schlatter

Abstract + + + − We observe a signal for the doubly charmed baryon Ξcc in the decay mode Ξcc → pD K to complement the previous + + − + reported decay Ξcc → Λc K π in data from SELEX, the charm hadroproduction experiment at Fermilab. In this new decay mode we observe an excess of 5.62 events over a combinatoric background estimated by event mixing to be 1.38 ± 0.13 events. The mixed background has Gaussian statistics, giving a signal significance of 4.8σ . The Poisson probability that a background − fluctuation can produce the apparent signal is less than 6.4 × 10 4. The observed mass of this state is 3518 ± 3MeV/c2, consistent with the published result. Averaging the two results gives a mass of 3518.7 ± 1.7MeV/c2. The observation of this new weak decay mode confirms the previous SELEX suggestion that this state is a double charm baryon. The relative branching ratio for these two modes is 0.36 ± 0.21.  2005 Elsevier B.V. All rights reserved.

E-mail address: jurgen@iﬁsica.uaslp.mx (J. Engelfried). 1 Present address: Sheba Medical Center, Tel-Hashomer, Israel. 2 Present address: Advanced Mask Technology Center, Dresden, Germany. 3 Present address: University of California at Irvine, Irvine, CA 92697, USA. 4 Present address: Instituto de Física da Universidade Estadual de Campinas, UNICAMP, SP, Brazil. 5 Present address: Kafkas University, Kars, Turkey. 6 Present address: Physik-Department, Technische Universität München, 85748 Garching, Germany. 7 Present address: The Boston Consulting Group, München, Germany. 8 Present address: Lucent Technologies, Naperville, IL, USA. 9 Present address: Baxter Healthcare, Round Lake IL, USA. 10 Present address: University of Alabama at Birmingham, Birmingham, AL 35294, USA. 11 Present address: Süleyman Demirel Universitesi, Isparta, Turkey. 12 Present address: Legal Department, Oracle Corporation, Redwood Shores, CA, USA. 13 Present address: DOE, Germantown, MD, USA. 14 Present address: Solidum, Ottawa, ON, Canada. 15 Present address: Siemens Medizintechnik, Erlangen, Germany. 16 Present address: Allianz Insurance Group IT, München, Germany. Deceased. 20 SELEX Collaboration / Physics Letters B 628 (2005) 18–24

PACS: 14.20.Lq; 14.40.Lb; 13.30.Eg

Keywords: Doubly charmed baryon

1. Introduction tion Radiation Detector identified each beam particle as meson or baryon with zero overlap. The three-stage In 2002, the SELEX Collaboration reported the magnetic spectrometer is shown elsewhere [3,4].The first observation of a candidate for a double charmed most important features are the high-precision, highly + → + − + baryon, decaying as Ξcc Λc K π [1]. The state redundant, vertex detector that provided an average had a mass of 3519 ± 2MeV/c2, and its observed proper time resolution of 20 fs for the charm decays, width was consistent with experimental resolution, a 10 m long Ring-Imaging Cerenkov (RICH) detec- less than 5 MeV/c2. The final state contained a charm- tor that separated π from K up to 165 GeV/c [5], and + ed hadron, a baryon, and negative strangeness (Λc a high-resolution tracking system that had momentum − and K ), consistent with the Cabibbo-allowed decay resolution of σp/p < 1% for a 150 GeV/c proton. + of a Ξcc configuration. In order to confirm the inter- The experiment selected charm candidate events pretation of this state as a double charm baryon, it is using an online secondary vertex algorithm. A scin- essential to observe the same state in some other way. tillator trigger demanded an inelastic collision with Other experiments with large charm baryon samples, at least four charged tracks in the interaction scin- e.g., the FOCUS and E-791 fixed target charm exper- tillators and at least two hits in the positive particle iments at Fermilab or the B-factories, have not con- hodoscope after the second analyzing magnet. Event firmed the double charm signal. This is consistent with selection in the online filter required full track recon- the SELEX results. The report in Ref. [1] emphasized struction for measured fast tracks (p 15 GeV/c). that this new state was produced by the baryon beams These tracks were extrapolated back into the vertex sil- (Σ−, proton) in SELEX, but not by the π − beam. It icon planes and linked to silicon hits. The beam track also noted that the apparent lifetime of the state was was measured in upstream silicon detectors. A three- + significantly shorter than that of the Λc , which was dimensional vertex fit was then performed. An event not expected in a model calculation based on heavy was written to tape if all the fast tracks in the event quark effective theory [2]. were inconsistent with having come from a single + Another way to confirm the Ξcc is to observe it in primary vertex. This filter passed 1/8 of all interac- a different decay mode that also involves a final state tion triggers and had about 50% efficiency for other- with baryon number and charm (not anti-charm). One wise accepted charm decays. The experiment recorded such mode, involving only stable charged particles, is data from 15.2 × 109 inelastic interactions and wrote + → + − × 9 the channel Ξcc pD K . Observing a mass peak 1 10 events to tape using both positive and neg- + → + − + 2 − near the Ξcc Λc K π peak at 3519 MeV/c re- ative beams. The sample was 65% Σ -induced, with ported in Ref. [1] in a channel combining a proton with the balance split roughly equally between π − and pro- a D+K− pair but not a D−K+ pair would confirm the tons. + existence of the Ξcc state. Here we report the first ob- The offline analysis selected single charm events + → + − servation of Ξcc pD K . with a topological identification procedure. Only charged tracks with reconstructed momenta were used. Tracks which traversed the RICH (p 22 GeV/c) 2. Experimental apparatus were identified as protons or kaons if those hypotheses were more likely than the pion hypothesis. All other The SELEX experiment used the Fermilab charged tracks were assumed to be pions. The primary ver- hyperon beam at 600 GeV to produce charm particles tex was refit offline using all found tracks. An event in a set of thin foil targets of Cu or diamond. The neg- was rejected if all tracks were consistent with one pri- ative beam composition was about 50% Σ−, 50% π −. mary vertex. For those events which were inconsistent The positive beam was 90% protons. A beam Transi- with a single primary vertex, secondary vertices were SELEX Collaboration / Physics Letters B 628 (2005) 18–24 21 formed geometrically and then tested against a set 2450 D− → K+π −π − decays in these samples. The of charge, RICH identification, and mass conditions D+ → K−π +π + events contribute to the signal chan- to identify candidates for the different single charm nel. The D− → K+π −π − events cannot come from states. Candidate events were written to a charm data the decay of a double charm baryon and will be used summary file. Subsequent analysis began by selecting as a topological background control sample. The yield particular single-charm species from that set of events. asymmetry stems from the d-quark contribution of the Σ− beam component that gives a sizable production asymmetry favoring leading D− production over D+ 3. Search strategy production, as we have reported for other charm systems [6]. In this study we began with the SELEX D± sam- The track-based search code is identical to that used + ple that has been used in lifetime and hadroproduction on the Λc sample in the original investigation [1].The studies [7]. The sample-defining cuts are described in premise is that a ccd state will make a secondary decay that reference. No new cuts on the D mesons were vertex between the primary production vertex in one introduced in this analysis. The D meson momen- of the thin foil targets and the observed D meson de- tum vector had to point back to the primary vertex cay vertex, which must lie outside material. We looked with χ2 < 12 (the double charm lifetime is known for intermediate vertices using all charge zero pairs to be much shorter than the D meson lifetime, so of tracks from the set of reconstructed tracks not as- the D meson pointback is not affected by having signed to the D-meson candidate. The additional posi- come from a secondary decay). The D meson decay tive track in this final state must be RICH-identified as point must have a vertex separation significance of a proton if it traverses the RICH. The negative track at least 10σ from the primary. Everywhere in these in the new vertex is assigned the kaon mass. (This analyses the vertex error used is the quadrature sum track typically missed the RICH acceptance by being of the errors on the primary and secondary vertices. too soft or too wide-angle, as confirmed by simula- The K was positively identified by the RICH detection.) We made background studies by (i) assigning tor. The pions were required to be RICH-identified if the negative track a pion mass, (ii) looking for proton they went into its acceptance. The D+ → K−π +π + plus positive track combinations with a D meson, and − − and D− → K+π −π − mass distributions are shown in (iii) looking at proton D K combinations (wrong- Fig. 1. There are 1450 D+ → K−π +π + decays and sign charm). We require a good 3-prong vertex fit with

+ − + + − + − − Fig. 1. D → K π π (left) and D → K π π (right) mass distributions with cuts used in this analysis. 22 SELEX Collaboration / Physics Letters B 628 (2005) 18–24 a separation signiﬁcance of at least 1.0σ from the primary vertex, the same requirement used in Ref. [1]. The primary position was recalculated from the beam track and secondary tracks assigned to neither the D nor the pK− vertices. Results presented in this Letter come from this analysis.

4. Search results and signiﬁcance

The signal search was based on a 10 MeV/c2 + 2 window centered on the Ξcc mass 3519 MeV/c from Ref. [1]. The expected mass resolution for the + → + − 2 decay Ξcc pD K is 4 MeV/c .Oursimula- tion correctly reproduces the observed widths of all our reported single charm mesons and baryons. The 10 MeV/c2 window should collect 80% of the events + in a Ξcc signal. The results are insensitive to changing bin boundaries by up to half a bin. The background, assumed to be flat, is evaluated outside a 20 MeV/c2 window, to avoid putting the remaining 20% of any signal into the background. The right-sign mass combinations in Fig. 2(a) show an excess of 5.4 events over a background of 1.6 events. The wrong-sign mass combinations (c¯ quark in the decay) for the pD−π + final state are also plotted in Fig. 2 (b), scaled by 0.6 for the D+/D− ratio. The wrong-sign background shows no evidence for a significant narrow structure near 3519 MeV/c2.The average wrong-sign occupancy is 0.4 events/bin, exactly the background seen in the right-sign channel. + + − This confirms the combinatoric character of the back- Fig. 2. (a) Ξcc → pD K mass distribution for right-sign mass ground population in the right-sign signal. We have combinations. Vertical dashed lines indicate the region of smallest fluctuation probability as described in the text. (b) Wrong-sign investigated all possible permutations of particle as- − + signments. The only significant structure observed is events with a pD K , scaled by 0.6 as described in the text. The + → + − horizontal line shows a maximum likelihood fit to the occupancy. in the channel Ξcc pD K , the place where a double charm baryon decay can occur. In order to assign a significance level to this peak, Fig. 2, we scale the mixed-event background down for we have combined statistical methods used in the orig- the multiple D+ usage. inal double charm paper [1] and the event-mixing The resulting background distribution predicts the method used in Ref. [8]. We set out to test the hypoth- observed distribution very well. The mean number of esis that the background events in Fig. 2 are random background events below, in, and above the 4-bin sig- combinatoric tracks associated with real D+ mesons. nal peak is 5.74± 0.26, 1.38± 0.13, and 10.60± 0.36. To mix events we took a D+ meson in the peak region This agrees well with the 8, 1.6, and 10 events that we (Fig. 1) and combined it with proton and K− tracks observe in the corresponding regions. A mass plot with extracted from other events. Each D+ was reused 25 the combinatoric background level is shown in Fig. 3. times. To compare to the combinatoric background in The background that we observe is completely consis- SELEX Collaboration / Physics Letters B 628 (2005) 18–24 23

+ + − + → + − + + → + − Fig. 3. Ξcc → pD K mass distribution from Fig. 2(a) with Fig. 4. Gaussian fits for Ξcc Λc K π and Ξcc pD K high-statistics measurement of random combinatoric background (shaded data) on same plot. computed from event-mixing. the data distribution around the signal peak. The fit tent in shape and normalization with random combi- mass is 3518 ± 3MeV/c2. This agrees beautifully + natoric tracks associated with real D mesons. with the measurement of 3519 ± 2MeV/c2 from We take the combinatoric background to give a the original double charm baryon report. We present proper measure of the expected background under these data as confirmation of the double charm state at 2 + → + − the 7-event signal region. The single-bin significance 3520 MeV/c in a new decay mode Ξcc pD K . of the signal, using√ the above background numbers, The weighted average mass is 3518.7 ± 1.7MeV/c2. is: S = (7 − 1.38)/ 1.38 + 0.132 = 4.8σ . The Pois- The mass distributions for the two channels are shown son probability of observing at least this much ex- in Fig. 4. cess, including the Gaussian uncertainty in the back- We have used the simulation to study the rela- × −4 + → ground, is 6.4 10 . Both of these statistical signif- tive acceptance for the two decay channels Ξcc + − + → + − + icance calculations use methods identical to those in pD K and Ξcc Λc K π in order to quote a Ref. [1]. This indicates a robust signal atop a combina- relative branching ratio. The overall acceptance, in- toric background whose shape and normalization are cluding the single charm selection and the proton ID + → + − very well understood. One can ignore the information requirements in the Ξcc pD K mode, is very from the combinatorial background study and estimate similar. SELEX measures the relative branching ratio + → + − + → + − + = ± the signal significance from the non-signal regions of Γ(Ξcc pD K )/Γ (Ξcc Λc K π ) 0.36 Fig. 2(a) alone. Using the same procedure as above, 0.21. The systematic error due to acceptances is well that significance estimate is 4.16σ with a Poisson ex- understood from single charm studies and is negligible cess probability of 0.0021. compared to the statistical error. In Ref. [1] we noted that all observed ccd events were produced by the baryon beams. None came from 5. Signal properties pions. In this sample, 1 event out of the 7 in the peak region seen in Fig. 2 is a pion beam event, and 1 of + → + − In order to estimate the mass of the Ξcc pD K the 19 sideband events comes from the pion beam state in light of the sparse statistics in Fig. 2,wefixed sample. This sample is consistent with the view that the width of the Gaussian to 4 MeV/c2 and fitted double charm baryons are produced dominantly by the 24 SELEX Collaboration / Physics Letters B 628 (2005) 18–24 baryon beams in SELEX. In another comparison, we nical support from the staffs of collaborating insti- + → + − + had noted that the Ξcc Λc K π decays had an tutions. This project was supported in part by Bun- exceptionally short reduced proper time distribution, desministerium für Bildung, Wissenschaft, Forschung + indicating a Ξcc decay lifetime 5–10 times shorter und Technologie, Consejo Nacional de Ciencia y Tec- + than the Λc lifetime. That feature is confirmed by the nología (CONACyT), Conselho Nacional de Desen- + → + − Ξcc pD K channel. As we noted in Ref. [1], our volvimento Científico e Tecnológico, Fondo de Apoyo lifetime resolution is excellent but we cannot exclude a la Investigación (UASLP), Fundação de Amparo à 0 lifetime (strong decay) for these events. The width of Pesquisa do Estado de São Paulo (FAPESP), the Is- this peak is completely consistent with simulation of a rael Science Foundation founded by the Israel Acad- zero-width state, unlikely for a strong decay of a mas- emy of Sciences and Humanities, Istituto Nazionale sive state. Also, we do not see an increase in the signal di Fisica Nucleare (INFN), the International Science when we reduce the vertex significance cut L/σ be- Foundation (ISF), the National Science Foundation low 1. If this were a strong decay, one would expect (Phy #9602178), NATO (grant CR6.941058-1360/94), as many events with L/σ of −1as+1, so the signal the Russian Academy of Science, the Russian Ministry should grow significantly. It does not. of Science and Technology, the Secretaría de Edu- + → + − + In Ref. [1] we noted that the Ξcc Λc K π cación Pública (Mexico) (grant number 2003-24-001- yield and acceptance implied that a large fraction of 026), the Turkish Scientific and Technological Re- + ˙ the Λc decays seen in SELEX came from double search Board (TÜBITAK), and the US Department of + → charm decays. That was a surprise. For the Ξcc Energy (DOE grant DE-FG02-91ER40664 and DOE pD+K− case that is not true. Only a few percent contract number DE-AC02-76CHO3000). of the SELEX D+ events are associated with double charm. References 6. Summary [1] M. Mattson, et al., Phys. Rev. Lett. 89 (2002) 112001. In summary, SELEX reports an independent con- [2] V.V.Kiselev, A.K. Likhoded, A.I. Onishchenko, Phys. Rev. D 60 + (1999) 014007; firmation of the double charm baryon Ξcc previously + → + − + V.V. Kiselev, A.K. Likhoded, A.I. Onishchenko, Eur. Phys. J. seen in the Ξcc Λc K π decay mode via the ob- C 16 (2000) 461; + → + − servation of its decay into the Ξcc pD K final B. Guberina, B. Melic, H. Stefancic, Eur. Phys. J. C 9 (1999) state. Using only very loose cuts gives the statistically- 213; significant signal shown in Fig. 2. A combinatoric B. Guberina, B. Melic, H. Stefancic, Eur. Phys. J. C 13 (2000) background from event mixing describes the non- 551. [3] M. Mattson, PhD thesis, Carnegie Mellon University, 2002. signal distribution well. The Gaussian significance us- [4] SELEX Collaboration, J.S. Russ, et al., in: A. Astbury, et al. ing this background estimate is 4.8σ . This decay mode (Eds.), Proceedings of the 29th International Conference on + confirms that the Ξcc lifetime is very short and that it High Energy Physics, vol. II, World Scientific, Singapore, 1998, is produced dominantly by baryon beams, as we re- p. 1259, hep-ex/9812031. ported in Ref. [1]. [5] J. Engelfried, et al., Nucl. Instrum. Methods A 431 (1999) 53. [6] F. Garcia, et al., Phys. Lett. B 528 (2002) 49; M. Kaya, et al., Phys. Lett. B 558 (2003) 34. [7] A. Kushnirenko, et al., Phys. Rev. Lett. 86 (2001) 5243, hep-ex/ Acknowledgements 0010014. [8] A.V. Evdokimov, et al., Phys. Rev. Lett. 93 (2004) 242001, hep- The authors are indebted to the staff of Fermi Na- ex/0406045. tional Accelerator Laboratory and for invaluable tech- Physics Letters B 628 (2005) 25–32 www.elsevier.com/locate/physletb

Hard photons from a chemically equilibrating quark–gluon plasma at ﬁnite baryon density

Z.J. He a,b,c, J.L. Long a,Y.G.Maa,b,X.M.Xud,B.Liue

a Shanghai Institute of Applied Physics, Chinese Academy of Sciences, PO Box 800-204, Shanghai 201800, China b CCAST (World Laboratory), PO Box 8730, Beijing 100080, China c Research Center of Nuclear Theory of National Laboratory of Heavy Ion Accelerator, Institute of Modern Physics, Chinese Academy of Sciences, Lanzhou 730000, China d Department of Physics, Shanghai University, Shanghai 200444, China e Institute of High Energy Physics, Chinese Academy of Sciences, Beijing 100039, China Received 10 September 2004; received in revised form 29 July 2005; accepted 14 September 2005 Available online 26 September 2005 Editor: J.-P. Blaizot

Abstract We study hard photon production in a chemically equilibrating quark–gluon plasma at ﬁnite baryon density, and ﬁnd that the effect of the system evolution on the photon production and large contribution of the bremsstrahlung and Compton qg → γq processes make the total photon yield as a strongly increasing function of the initial quark chemical potential.  2005 Elsevier B.V. All rights reserved.

PACS: 12.38.Mh; 25.75.-q; 24.85.+p

Keywords: Hard photon; Chemically equilibrating quark–gluon plasma; Finite baryon density

The thermal production of hard photons in ultrarel- tion of hard photons in a QGP at finite temperature [2]. ativistic heavy ion collisions has been proposed as a Assuming the formation of a QGP already at AGS and possible signature for the formation of a quark–gluon SPS energies, however, a finite quark chemical poten- plasma (QGP) [1]. Photons have large mean free paths tial µq has to be considered [3], and even at RHIC en- due to the small cross section for electromagnetic in- ergies the quark chemical potential µq may not be neg- teraction in the plasma, can provide a direct probe of ligible as indicated by RQMD simulation (µq ≈ 1 − the plasma. Previous authors have studied the produc- 2T) [4]. Recently, Hammon, Geiger, and co-workers [5,6] have indicated that the initial QGP system produced at the RHIC energies has finite baryon density, E-mail address: [email protected] (Z.J. He). Majumder and Gale [7] have discussed the dileptons

0370-2693/$ – see front matter  2005 Elsevier B.V. All rights reserved. doi:10.1016/j.physletb.2005.09.030 26 Z.J. He et al. / Physics Letters B 628 (2005) 25–32 from QGP produced at the RHIC energies at finite termediate region of the λg the deviation is quite baryon density, and Bass et al. [8] have pointed out significant. It shows that it is difficult to study the that the parton rescattering and fragmentation lead to a whole process of the chemical equilibration of the substantial increase in the net-baryon density at midra- system based on the previous approximation distrib- pidity over the density produced by initial primary ution functions of partons. In this work, we describe parton–parton scatterings. These show that one may the evolution and photon production of a chemically further study the effect of the quark chemical poten- equilibrating QGP system at finite baryon density on tial on the signature of the QGP formation. At given the basis of the Jüttner distribution function of par- energy density estimates based on lowest order pertur- tons. bation theory indicate a strong suppression of the pho- We describe the evolution of the system accord- ton production at non-vanishing quark chemical po- ing to [20,21]. For brevity, we do not give the de- tential µq as compared to the case µq = 0 [9]. Traxler, tailed steps of the derivation of the evolution equations Vija, and Thoma have computed the photon produc- of the system. Expanding densities of quarks (anti- tion rate of a QGP at finite quark chemical potential quarks) for a given temperature (also a given energy density) 2 gq(q)¯ λq(q)¯ p dp using the Braaten–Pisarski method [10]. In addition, nq(q)¯ = , (1) 2 + (p∓µq )/T authors of [11–14] have studied the photon production 2π λq(q)¯ e in a chemically equilibrating baryon-free QGP system. over quark chemical potential µq , we can get the Especially in [12,13] the resummation of the diagrams baryon density of the system nb,q and correspond- that lead to the Landau–Pomeranchuk–Migdal (LPM) ing energy density εqgp, where gq(q)¯ is the degen- effect is included. eracy factor of quarks (anti-quarks). We consider In this work, we study the hard photon produc- the reactions leading to chemical equilibrium: gg tion in a chemically equilibrating QGP system at fi- ggg, gg qq¯, gg ss¯ and qq¯ ss¯. Assuming nite baryon density to reveal the effect of the quark that elastic parton scatterings are sufficiently rapid chemical potential on the production. As pointed out to maintain local thermal equilibrium, the evolu- in Refs. [15–18], the distribution functions of par- tions of gluon, quark and s quark densities can be tons in a chemically equilibrating QGP system at given by the master equations, respectively. Combin- finite baryon density can be given by Jüttner distrib- ing these master equations together with equations (p∓µ )/T utions fq(q)¯ = λq(q)¯ /(e q + λq(q)¯ ) for quarks of energy–momentum and baryon number conservation of the system, for the longitudinal scaling ex- (anti-quarks) and f (p) = λ /(ep/T − λ ) for glu- g g g pansion of the system, one can get a set of coupled ons. When the parton fugacities λ are much less i relaxation equations describing evolution of the sys- than unity as may happen during the early evolu- tem on the basis of the thermodynamic relations of tion of the parton system, the quantum effect may the system, as mentioned above. One solve the set be neglected, the distributions have been approxi- − ∓ of evolution equations under given initial values ob- = (p µq )/T mated as Boltzmann form fq(q)¯ λq(q)¯ e tained from Hijing model or self-screened parton cas- [19]. However, this introduces an error of the order cade model calculation [22] to get the evolutions of of 40% when the distribution approaches chemical temperature T , quark chemical potential µq and fu- equilibrium as mentioned in Ref. [15]. It should be gacities λq for quarks, λg for gluons and λs for s stressed here that the most commonly used approx- quarks. imations are the factorized Fermi–Dirac distribution ∓ The earlier works [2,11,23] considered the hard = (p µq )/T + functions fq(q)¯ λq(q)¯ /(e 1) for quarks photon production due to annihilation qq¯ → gγ and (anti-quarks) and factorized Bose–Einstein distribu- QCD Compton (qg → qγ and qg¯ →¯qγ) scatter- p/T tion function fg(p) = λg/(e − 1) for gluons. As ing processes from a quark matter. Recently, au- can be also seen from the discussion in Ref. [15] thors of [24–26] have found that the near-collinear the calculated thermal screening mass under this bremsstrahlung and a new process called inelastic pair approximation coincides with that from the Jüttner annihilation (IPA), fully including the LPM effect, distribution only near λg = 1, however, in the in- give also significant contribution to photon produc- Z.J. He et al. / Physics Letters B 628 (2005) 25–32 27 tion. In this work, we shall also compute the photon and for Compton scatterings production rate from these two processes. We first consider photon production from quark annihilation dR ¯ → → E qq γg and Compton scatterings gq γq and d3p gq¯ → γ q¯. Their production rates may be calculated com − 2 by [2] ∞ kc 5 αα λ λ =− s q g ds dt dR 1 d3p d3p d3p 5 E = 1 2 3 f E f (E ) 36 π E 3 8 1 1 2 2 2 − + 2 d p 2(2π) 2E 2E 2E 2kc s kc 1 2 3 4 × 1 ± f3(E3) (2π) u2 + s2 dE dE × 1 2 µ µ µ µ 2 (E1∓µq )/T (E2)/T − × δ p + p − p − p |M| , (2) us e + λq e λg 1 2 3 λq where f(E) is the parton distribution function, × 1 − θ P(E ,E ) 2 (E +E ∓µ −E)/T 1 2 |M| the square of the matrix element for reaction e 1 2 q + λq processes summed over spins, colors and flavors, as × −1/2 seen in Ref. [10]. The plus sign is for the annihilation P(E1,E2) , (5) process and the minus for the two Compton processes. Following Refs. [2,10], above equation can be rewrite where α is the fine-structure constant, the running cou- = as pling constant αs 0.4, the plus sign is for the Comp- ton process gq¯ → γ q¯ and the minus for the Compton dR 1 E = dsdt |M|2 process gq → γq. 3 7 d p 16(2π) E Arnold, Moore, and their co-workers [24,25] have studied the hard photon production from near-collinear × dE dE f (E )f (E ) 1 2 1 1 2 2 bremsstrahlung and IPA processes in an equilibrated QGP at zero quark chemical potential. They have × 1 ± f3(E1 + E2 − E) θ P(E1,E2) pointed out that a correct treatment of the near- −1/2 × P(E1,E2) , (3) collinear bremsstrahlung and IPA which contribute to the leading-order emission rate requires a sum- where θ is the step function, P(E1,E2) =−(tE1 + 2 2 2 2 mation of all ladder graphs of the form shown in (s + t)E2) + 2Es((s + t)E2 − tE1) − s E + s t + st2.Thes, t and u are the Mandelstam variables. In- Fig. 3 of [25]. The total photon emission rate is given by an integral over all particle momenta of serting Jüttner distribution function of partons and the squared matrix elements |M|2 in (3), we obtain the the emission rate for that particle momentum. Fac- expressions of the photon production rates. We have toring out the same coefficient A(E) which appears for quark annihilation reaction in the leading logarithmic result (1.2) of [25],we obtain the contribution of bremsstrahlung and IPA dR processes to the leading-order emission rate, as shown E 3 d p ann by (2.1) of [25]. Similar to the treatment in the − 2 preceding part, we extend these expressions to the ∞ kc 5 αα λ2 chemically equilibrating QGP system at finite baryon = s q ds dt 36 π 5 E density to explore the effect of the quark chem- 2k2 −s+k2 ical potential on the photon production from the c c 2 + 2 bremsstrahlung and IPA processes. Determining di- × u t dE1 dE2 ˜ ≡ − + mensionless transverse momenta pT pT /mD and ut e(E1 µq )/T + λ e(E2 µq )/T + λ ˜ ≡ q q qT qT /mD, taking the dimensionless ratio of ther- 2 2 λg mal masses κ ≡ m∞/m , and noting that the ar- × 1 + θ P(E ,E ) D (E +E −E)/T 1 2 → e 1 2 + λg gument of the p integral is invariant under p −E − p, the equation of photon production rate is × −1/2 P(E1,E2) (4) rewritten as [25] 28 Z.J. He et al. / Physics Letters B 628 (2005) 25–32

∞ 2 2 dR A(E)T p + (p + E) Thus, we get the integral value (P,E)= 2Qextr.Fi- E = dp 3 3 2 2 nally, inserting Jüttner distributions in (6), from (6) we d p (2π) 2p(p + E) −E/2 gain the photon production rates of the bremsstrahlung n (p + E)[ − n (p)] and IPA processes, respectively. × f 1 f (p,E), (6) We integrate above photon production rates over nf (E) the space-time volume of the reaction according with to Bjorken’s model. The volume element d4x = 2 2 ˜ d xT dy τ dτ, where τ is the evolution time of the d pT ˜ (p,E)= Re 2p˜T .f(p˜T ,p,E) , (7) system and y the rapidity of the fluid element. We (2π)2 consider only central collisions so the integration where p and pT are the components of the momen- over transverse coordinates just yields a factor of tum p parallel or perpendicular to the photon direc- 2 = 2 d xT πRA, where RA is the nuclear radius. Eqs. (4), tion, respectively, E the energy of the photon, mD the (5), and also (6) can be related to the rapidity via ∞ 3 2 Debye mass, m the thermal mass of a hard quark, d p/E = dpx dpy dY = d pT dY. We finally obtain = 2 and A(E) 40π/9T ααsnf (E)/κ.In(6) the inte- the photon spectra gration from −E/2 to zero gives the contribution of dn dn the IPA process; the integration from zero to infinity = πR2 τdτ dy dY E , 2 T 3 4 gives the contribution of the bremsstrahlung process. d pT d pd x ˜ The function f(p˜T ,p,E) is the solution to the fol- (11) lowing integral equation derived in [24], where Y is the rapidity of photons. 197 197 ˜ In this work, we focus on discussing Au +Au 2p˜T = iδE(p˜T ,p,E)f(p˜T ,p,E) central collisions at the RHIC energies. Authors of 2 d q˜T [5] have calculated non-equilibrium initial conditions + C(q˜T ) (2π)2 from perturbative QCD within Glauber multiple scat- √ ˜ ˜ tering theory for s = 200 A GeV. Taking into ac- × f(p˜T ,p,E)− f(p˜T +˜qT ,p,E) , (8) count higher order contribution by a factor K = 2.5 ˜ ≡ 2 where δE(pT ,p,E) (T /mD)δE(pT ,p,E), and from comparison with experiment at the RHIC, they ˜ f(p˜T ,p,E)≡ (mD/T )f (pT ,p,E). have obtained the energy density and number densi- Authors of [25] have calculated (P,E) via solv- ties of gluons, quarks, anti-quarks as well as the initial ing the integral equation (8) by an adaptation of the temperature T0 = 0.552 GeV. From these densities we variational approach. In this work, following [25] we have also obtained initial temperature T0 = 0.566 GeV calculate (P,E). We first calculate the matrix ele- and initial quark chemical potential µq0 = 0.285 GeV ˜ ˜ r ˜ i based on thermodynamic relations of the chemically ments Sn, Cmn, Cmn, and δEmn for a particular set of parameters [κ,E,p]atagivenA value included equilibrating QGP system at finite baryon density, as = = in the trial function, then obtain the indicated Nr + Ni mentioned in the preceding part, at λg0 0.09, λq0 = component vector and symmetric matrix 0.02 and λs0 0.01. Obviously, these two initial temperatures are near the one from Hijing model calcu- ˜ ˜ r S C −δEmn S ≡ n ,M≡ mn . (9) lation, as shown in Ref. [15]. With the help of [15] − − ˜ i = = 0 δEnm Cmn we take initial values: τ0 0.70 fm, T0 0.57 GeV, λg0 = 0.09, λq0 = 0.02 and λs0 = 0.01. To further Subsequently, we get the Nr + Ni component vector M−1S, from which we obtain the variational coeffi- understand the effect of finite baryon density on the ˜2 photon production, we have solved the set of coupled cients (am,bm) and thus the resulting function χ(pT ) (see [25]). We further specify the value of A according relaxation equations for initial quark chemical poten- = to the criterion and method used in [25], and obtain the tials µq0 0.000, 0.285, 0.570 and 0.855 GeV, and extremal value for the specified A obtained the evolutions of the temperature T , quark chemical potential µq and fugacities λg, λq and λs 1 − Q = ST M 1S. (10) of the system. The calculated evolution paths of the extr 2 system in the phase diagram have been shown in Z.J. He et al. / Physics Letters B 628 (2005) 25–32 29

Fig. 1. The calculated evolution paths of the system in the phase Fig. 2. The calculated equilibration rates at the same initial conditions as given in Fig. 1. The gluon and quark equilibration rates diagram for initial values τ0 = 0.70, T0 = 0.57 GeV, λg0 = 0.09, are, in turn, denoted by the λg and λq . The solid, dotted, dashed λq0 = 0.02 and λs0 = 0.01, where the solid, dotted, dashed and dash-dotted lines are, in turn, the evolution paths for initial quark and dash-dotted lines denote, respectively, the equilibration rates for initial quark chemical potentials µq0 = 0.000, 0.285, 0.570 and chemical potentials µq0 = 0.000, 0.285, 0.570 and 0.855 GeV, and the dot-dot-dashed line is the phase boundary. The time interval be- 0.855 GeV. tween small circles is 0.6 fm (i.e., 30 × calculation-step 0.02 fm). The phase diagram is calculated at B1/4 = 0.25 GeV. work, both the quark chemical potential and the temperature of the system are functions of time, compared Fig. 1, where the solid, dotted, dashed, and dot-dashed with the baryon-free QGP it necessarily takes a long lines denote, in turn, the calculated paths for initial time for value (µq ,T) of the system to reach a certain quark chemical potentials µq0 = 0.000, 0.285, 0.570 point of the phase boundary to make the phase tran- and 0.855 GeV at initial values T0 = 0.57 GeV, τ0 = sition. Furthermore, in the calculation we have found 0.70 fm, λg0 = 0.09, λq0 = 0.02 and λs0 = 0.01. The that with increasing the initial quark chemical poten- dot-dot-dashed line is the phase boundary between the tial the production rate of gluons goes up, and thus quark phase and hadronic phase. The corresponding their equilibration rate goes down, leading to the little equilibration rates of gluons and quarks, λg and λq , energy consumption of the system, i.e., slow cool- are shown in Fig. 2. ing of the system. Since gluons are much more than The calculated spectra for the qq¯ → γg, gq → γq quarks in the system, with increasing the initial quark and gq¯ → γ q¯ processes are, in turn, shown in pan- chemical potential the cooling of the system further els (a), (b) and (c) of Fig. 3, for bremsstrahlung and slows down. These cause the quark phase lifetime to IPA processes in panels (a) and (b) of Fig. 4. With further increase, for example the calculated lifetimes the increase of the initial quark chemical potential the of the quark phase for initial quark chemical poten- anti-quark density goes down, thus the photon produc- tials µq0 = 0.000, 0.285, 0.570 and 855 GeV are, in tion for the qq¯ → γg process calculated from Eq. (4) turn, about 3.42, 3.58, 3.82 and 4.22 fm. These evo- will go down. One has well known that the baryon- lution features of the system can be clearly seen in free QGP converts into the hadronic matter only with Fig. 1. With the help of Eq. (11) one can note that decreasing the temperature along the temperature axis these factors obviously heighten the contribution of of the phase diagram, and the phase transition occurs the integration over the evolution of the system to the at a certain critical temperature Tc. However, in this photon production. Thus the calculated spectra for the 30 Z.J. He et al. / Physics Letters B 628 (2005) 25–32

Fig. 4. The calculated photon spectra for bremsstrahlung and IPA processes are, in turn, shown in (a) and (b) panel. The solid, dotted, dashed and dot-dashed lines are, respectively, the calculated spectra for initial quark chemical potentials µq0 = 0.000, 0.285, 0.570 and 0.855 GeV based on the evolution described in Fig. 1.

by Eq. (5) for the process gq¯ → γ q¯ will go down, and for the process gq → γq will go up. The calculations of Eq. (11) have shown that due to the effect Fig. 3. The calculated photon spectra for processes qq¯ → γg, of the evolution of the system, with increasing the gq → γq and gq¯ → γ q¯ are, in turn, shown in (a), (b) and (c) panel. initial quark chemical potential the increase of the The solid, dotted, dashed and dot-dashed lines are, respectively, the photon production for the process gq → γq becomes = calculated spectra for initial quark chemical potentials µq0 0.000, even faster, and the decrease of the photon production 0.285, 0.570 and 0.855 GeV based on the evolution described in ¯ → ¯ Fig. 1. for the process gq γ q obviously slows down, as shown in panels (b) and (c) of Fig. 3. Inserting Jüttner distributions in (6), we note that in (6) the part relat- qq¯ → γg from Eqs. (4) and (11) still go somewhat ing to the quark chemical potential is only the factor + − − − up with increasing the initial quark chemical poten- 1 / [e(p E µq )/T + 1]e(p µq )/T / [e(p µq )/T + 1]. tial, as seen in panel (a) of Fig. 3. One also knows For the bremsstrahlung process, the dependence of the that since with increase of the initial quark chemi- photon production on the quark chemical potential is + − cal potential the quark density goes up and anti-quark mainly given by the factor 1/[e(p E µq )/T + 1]× − density goes down, the photon production calculated 1/[e(p µq )/T + 1]. This factor is the product of Z.J. He et al. / Physics Letters B 628 (2005) 25–32 31

from Au197 + Au197 central collisions at the RHIC energies, we have first extended those expressions of the photon production for the Compton (gq → qγ and gq¯ →¯qγ), annihilation qq¯ → gγ, bremsstrahlung and IPA processes in [2,11,25] to the chemically equilibrating system at finite baryon density. Subsequently, based on the evolution of the chemically equilibrating QGP system at finite baryon density, we have computed the photon production of the system. We have found that the increase of the initial quark chemical potential will change the hydrodynamic behavior of the system to cause the quark phase lifetime to increase. This effect is to heighten the photon yield of the Compton (gq → qγ and gq¯ →¯qγ), annihilation qq¯ → gγ, bremsstrahlung and IPA processes. On the other hand, the increase of the initial quark chemical potential will directly heighten the photon yield of the Compton gq → qγ and bremsstrahlung processes. It is shown that the bremsstrahlung and Compton gq → Fig. 5. The total photon spectra. The solid, dotted, dashed and gγ are two main processes of the photon production. dot-dashed lines are, respectively, the calculated total photon spec- They not only compensate the photon suppression of tra for initial quark chemical potentials µq0 = 0.000, 0.285, 0.570 and 0.855 GeV. processes gq¯ →¯qγ, qq¯ → gγ, and IPA caused by the increase of the initial quark chemical potential, but also make the total photon yield as a strongly increas- the two quark distribution functions. With increasing function of the initial quark chemical potential. ing the quark chemical potential the quark density Since photons are produced in an evolving QGP sys- of the system goes up, thus photon production of the tem, obviously for comparison with the experiment it bremsstrahlung process goes rapidly up, too, as seen is necessary that one study the photon production in in panel (a) of Fig. 4. Compared with the process the evolving QGP system. In this work, our results has gq → γq, we have found that the contribution of the demonstrated the importance of the system evolution bremsstrahlung process is quite near the one of the for the photon production. gq → γq process at the lower pT region, and larger than the one of the gq → γq process at the higher pT (p−µ )/T region. For the IPA process, the factor e q be- Acknowledgements comes important. Due to the integration from −E/2 − to zero, the factor e(p µq )/T is indeed an anti-quark We are thankful to P. Arnold and G.D. Moore for distribution function. With increasing the quark chemical potential the anti-quark density goes down, thus their help. This work is supported in part by CAS knowledge Innovation Project No. KJCX2-N11, the the photon production of the IPA process goes down, National Natural Science Foundation of China No. as shown in panel (b) of Fig. 4. We find that the 10405031, No. 10328509 and No. 10135030, the Ma- bremsstrahlung and gq → γq processes make the to- jor State Basic Research Development Program in tal photon production as a strongly increasing function China under Contract No. G200077400. of the initial quark chemical potential. Overall we can see that the total photon production increases with the initial quark chemical potential, as shown in Fig. 5. In conclusion, in order to reveal the effect of finite References baryon density on the photon production in a chemically equilibrating QGP system which is produced [1] P.V. Ruuskanen, Nucl. Phys. A 544 (1992) 169c. 32 Z.J. He et al. / Physics Letters B 628 (2005) 25–32

[2] J. Kapusta, P. Lichard, D. Seibert, Phys. Rev. D 44 (1991) [16] P. Levai, B. Müller, X.N. Wang, Phys. Rev. C 51 (1995) 3326. 2774. [17] D. Dutta, K. Kumar, A.K. Mohanty, R.K. Choudhury, Phys. [3] S. Nagamiya, Nucl. Phys. A 544 (1992) 5c. Rev. C 60 (1999) 014905. [4] A. Dumitru, et al., Frankfurt preprint UFTP 319/1992, unpub- [18] D.M. Elliott, D.H. Rischke, Nucl. Phys. A 671 (2000) 583. lished. [19] B. Kämpfer, O.P. Pavlenko, A. Peshier, G. Soff, Phys. Rev. [5] N. Hammon, H. Stöker, W. Greiner, Phys. Rev. C 61 (1999) C 52 (1995) 2704. 014901. [20] Z.J. He, J.L. Long, Y.G. Ma, G.L. Ma, B. Liu, Phys. Rev. C 69 [6] K. Geiger, J.I. Kapusta, Phys. Rev. D 47 (1993) 4905. (2004) 034906. [7] A. Majumder, C. Gale, Phys. Rev. D 63 (2001) 114008. [21] Z.J. He, J.L. Long, W.Z. Jiang, Y.G. Ma, B. Liu, Phys. Rev. [8] S.A. Bass, B. Müller, D.K. Srivastava, Phys. Rev. Lett. 91 C 68 (2003) 042902. (2003) 052302. [22] K.J. Eskola, B. Müller, X.N. Wang, Phys. Lett. B 374 (1996) [9] A. Dumitru, D.H. Rischke, H. Stöcker, et al., Mod. Phys. Lett. 20. A 8 (1993) 1291. [23] R. Baier, H. Nakkagawa, A. Neigawa, K. Redlich, Z. Phys. [10] C.T. Traxler, H. Vija, H. Thoma, Phys. Lett. B 346 (1995) 329. C 53 (1992) 433. [11] C.T. Traxler, M.H. Thoma, Phys. Rev. C 53 (1996) 1348. [24] P. Arnold, G.D. Moore, L.G. Yaffe, J. High Energy Phys. 0111 [12] F. Gelis, H. Niemi, P.V. Ruuskanen, S.S. Räsänen, J. Phys. G: (2001) 057. Nucl. Part. Phys. 30 (2004) S1031, nucl-th/0403040. [25] P. Arnold, G.D. Moore, L.G. Yaffe, J. High Energy Phys. 0112 [13] F. Arleo, P. Aurenche, F. Bopp, et al., hep-ph/0311131, Sec- (2001) 09. tion 6.3. [26] M.G. Mustafa, M.H. Thoma, Phys. Rev. C 62 (2000) 014902. [14] M. Strickland, Phys. Lett. B 331 (1994) 245. [15] T.S. Biró, E.V. Doorn, B. Müller, et al., Phys. Rev. C 48 (1993) 1275. Physics Letters B 628 (2005) 33–39 www.elsevier.com/locate/physletb

Is there a crystalline state of nuclear matter?

U.T. Yakhshiev a,b, M.M. Musakhanov a,b,H.-Ch.Kima

a Department of Physics and Nuclear Physics & Radiation Technology Institute (NuRI), Pusan National University, 609-735 Busan, Republic of Korea b Theoretical Physics Department & Institute of Applied Physics, National University of Uzbekistan, Tashkent 174, Uzbekistan Received 10 October 2004; accepted 14 September 2005 Available online 26 September 2005 Editor: J.-P. Blaizot

Abstract A possibility of the crystalline state of nuclear matter is discussed in a medium-modified Skyrme model. The interaction energy per nucleon in nuclear matter is evaluated by taking into account the medium influence on single nucleon–skyrmion properties and the tensor part of the nucleon–nucleon potential, and by using a variational method of Hartree–Fock type including zero-point quantum fluctuations. It is shown that in this approach the ground state of nuclear matter has no crystalline structure due to quantum fluctuations as well as medium modifications of hadron properties.  2005 Elsevier B.V. All rights reserved.

PACS: 12.39.Dc; 12.39.Fe; 21.65.+f; 21.30.Fe

Keywords: Nuclear matter; Crystalline structure; Medium modiﬁcations; Skyrme model

1. After a pioneering work by Kutschera et al. [1] hypersphere [6–8]. Former studies are still under dis- related to the dense-matter properties in the Skyrme cussion [9] and have concentrated on investigating the model, Klebanov [2] discussed a possible formation classical field configuration, assuming face centered of the skyrmionic matter with the simple cubic crys- cubic (FCC) structure. Obviously, one should quantize talline structure due to tensor forces between the nu- the model to investigate the real nuclear matter, but it cleon of a lattice-point and the nearest six nucleons is not at all easy to do it as explained in Ref. [10]. of the unit cell. There has been a great amount of On the other hand, quantum fluctuations around work on energetically favorable crystalline structures a classical ground state can play an important role [3–5] and parallel studies of the Skyrme model on the in leading to nuclear matter which is different in its behavior from the classical skyrmion matter. A standard variational method of Hartree–Fock type used in E-mail addresses: [email protected] (U.T. Yakhshiev), [email protected] (M.M. Musakhanov), Ref. [11] provides a theoretical framework to make [email protected] (H.-Ch. Kim). it tractable to deal with nuclear matter with quan-

0370-2693/$ – see front matter  2005 Elsevier B.V. All rights reserved. doi:10.1016/j.physletb.2005.09.029 34 U.T. Yakhshiev et al. / Physics Letters B 628 (2005) 33–39 tum fluctuations. Taking into account only the tensor Table 1 part of the nucleon–nucleon (NN) interaction based Comparison of hadronic properties in free space with those in nuclear matter with density ρ. In the second column the corresponding on the one-pion-exchange potential (OPEP) and as- values in free space are listed, g stands for the correlation para- suming the FCC structure, Diakonov and Mirlin [11] −3 meter, ρ0 = 0.17 fm is the density of normal nuclear matter. showed that at densities which are several times higher ∗ ≡ ∗ − ∗ M∆N M∆ MN is the ∆–N mass splitting that of ordinary nuclear matter a condensed state of ρ/ρ0 00.5 1.0 nuclear matter may be still energetically favorable. Al- g – 0.33 0.6 1 0.33 0.6 1 ternatively, using time-dependent numerical calcula- ∗ g 12.49 9.48 9.76 10.08 6.83 7.75 8.66 tions on a quantized simple cubic lattice, Walhout [12] πNN∗ M [MeV] 868 743 756 770 635 675 715 evaluated the zero-point kinetic energy of nucleons ∗N mπ [MeV] 140 146 146 146 152 152 152 ∗ and found it to unbind the classical crystalline ground Λ [MeV] 528 484 489 494 448 462 477 ∗ state. Consequently, one can conclude that due to the M∆N [MeV] 243 211 214 218 186 196 206 increasing number of neighboring nucleons the FCC structure appears in a deeper bound state. However, above-mentioned studies did not con- the effective pion–nucleon scattering lengths [17], one sider possible medium modifications of single nucleon gets the density dependence of hadron properties as 1 and pion properties and the corresponding medium- listed in Table 1. In the present work, the asterisk modified NN interaction [13–16].Refs.[14–16] con- in notation indicates medium-modified quantities. The sidered the nucleon as a skyrmion and then studied the results are presented for three different values of the Lorentz–Lorenz parameter: the first and the second influence of baryonic matter on its properties phenomenologically via medium modifications of pion fields. ones, i.e., g = 0.33 and g = 0.6, correspond to the classical and phenomenological values [17], respec- The results of these studies were in a qualitative agreement with experimental indications, e.g., the swelling tively, and the third one is simply taken as g = 1. of the nucleon in nuclear medium and the decrease of One can see that hadronic properties as well as the its mass. It was also shown that due to the influence of pion–nucleon coupling constant are changed due to surrounding nuclear environment the tensor part of the the presence of nuclear medium. Consequently, the NN potential is decreased [14]. The changes in hadron nuclear-matter properties, being calculated by using static properties and form factors of meson–nucleon the NN potential between the nearest neighbors in nu- vertices also lead to the suppression of the one-boson- clear matter, are expected to be modified accordingly. exchange potential [15]. In this context, it is natural to In order to study a phase structure of nuclear matter, ask and answer the question: How do these changes we follow closely a variational procedure described in play a role in describing nuclear matter? Ref. [11]. We define the position vector of an FCC Since it is of great difficulty to quantize the skyr- lattice point n by r0n and the optimum orientation mionic matter itself [10], we follow the phenom- (the relative orientation matrix in internal space) of enological and variational method put forward by the nucleon at this lattice point by A0n. The general Ref. [11] in order to investigate how the medium mod- orientation matrix A is related to an arbitrary posi- ifications of the NN potential and of hadronic proper- tion vector in isospin space. The trial wave function ties change the ground-state energy of nuclear matter of the crystal must be chosen as an antisymmetrized product of single-lattice-point wave functions ψ(r − and reestimate the role of zero-point quantum fluctua- + tions around the minimum. r0n,A A0n) localized near the lattice sites and the corresponding optimum orientations. The overlap integrals of the non-nearest neighbors being neglected, the energy functional (interac- 2. In order to calculate the medium-modifications tion energy per nucleon) can be written as a following of single hadron properties, we start with the effective chiral Lagrangian for the medium-modified 1 We refer the reader to the works [14,15], where the medium skyrmion [14]. Using the phenomenological values modifications of hadronic properties have been investigated and in- = phen =− −1 = phen = −3 b0 b0 0.024mπ , c0 c0 0.15mπ of clusion of medium parameters (b0, c0 and g ) were discussed. U.T. Yakhshiev et al. / Physics Letters B 628 (2005) 33–39 35

Hartree–Fock form: Utilizing Eqs. (2)–(3) of the lattice-point trial wave function and the NN potential given in the form of ∗ = 3 ∗ − + E d rdAψ r r0,A A0 Eq. (4), one can arrive at the ﬁnal expression for the 2 2 energy functional (Eq. (1)): 1 ∂ T A + × − ∗ + ψ r − r0,A A0 2 M ∂r 2 I ∗ ∗ α ∗ 2 N 2 E = M 1 + α2 ∆N 1 12 + 3 3 ∗ − + 2 −1 2 d rd r dAdA ψ r r0,A A0 15η 11 2 3 2 + 1 + 3η + 1 + η + η ∗ n=1 4 4 4M ∗ + ∗ + N × ψ r − r0n,A A0n V r − r ,A A + ∗ + ∗ EH EF , + + × ψ r − r0,A A0 ψ r − r0n,A A0n 2 −2 ∗ = 3 + + 15η − − + − + EH 1 3η ψ r r0n,A A0n ψ r r0,A A0 , (1) π 2 4 ∗ ∞ where I denotes the moment of inertia of the skyr- 2 q ∗ mion in medium. This functional must be minimized × dq q2 exp − a2 V (q) 2 2 1 with respect to the single-lattice-point wave function 0 ψ for a given distance R =|r0n − r0| between the 2 ∗ 2 (1 + α) V (q) ∂ ∗ nearest neighbors. It is clear that in general the quan- + 2 − V (q) + 2 2 2 2 3 tities, M∗ ,I∗,V∗,areR-dependent, while the R is 9(1 α ) q ∂R N sin qR related directly to the nuclear-matter density. × , We use the single-lattice-point trial function as fol- qR 2 −2 2 2 lows: ∗ 3 15η 4(1 − α − α ) E =− 1 + 3η + + F 2 + 2 ψ r − r0,A A0 π 4 3(1 α ) ∞ 1/2 + + 3/2 + 2R2 q2 Dkk (A A0) αDkk (A A0) × 2 − − = ϕ(r − r0) √ , (2) dq q exp 1 + α2 2 2 2 0 2 3/4 2 −1/2 ϕ(r) = /π 1 + 3η + 15η /4 2 2 b 4bc 3c ∗ × + + V (q) × exp − 2r2/2 1 + η 2r2 , (3) 3 15 35 2 2 where measures the spatial extent of the wave func- 2bc c ∗ + b2 + + V (q) , (5) tion, α denotes the admixture of the ∆-resonance, 3 5 3 T Dab(A) is an SU(2) Wigner D function, and η stands = where the variables a, b, and c are deﬁned as for an additional variational parameter. Putting η 0, one can get the corresponding trial function similar to a ≡ 1 + 6 − q2/ 2 η/2 Ref. [11]. + 60 − 20q2/ 2 + q4/ 4 η2/16 , In general, NN potential can be presented in the following form: b ≡ a + R2 2 η/2 + 4 + R2 2 − 2q2/ 2 η2/16 , ∗ + c ≡ 4q2R2η2/16. (6) V R,A1 A2 = ∗ V1 (R) To compute the ground-state energy in nuclear mat- 1 ter, we keep only a tensor part of the potential in + + + ∗ ∗ Tr A1 A2σiA2 A1σj Eq. (4), putting V = V = 0. This tensor potential 2 1 3 ∗ ∗ arises from one-pion exchange, which is expressed in × (3Rˆ Rˆ − δ )V (R) + δ V (R) , (4) i j ij 2 ij 3 momentum space as follows: ˆ where R is the corresponding unit vector along the ∗ 2 2 ∗2 + ∗ 3gπNN q Λ line joining the centers of two nucleons and A A2 de- V (q) =− . (7) 1 2 ∗ 2 + ∗2 2 + ∗2 notes their relative orientation matrix in internal space. 2MN q mπ q Λ 36 U.T. Yakhshiev et al. / Physics Letters B 628 (2005) 33–39

The effective values of the potential parameters and their density dependence can be calculated by using the medium-modified Skyrme Lagrangian given in Ref. [14].2 We present two different sets of OPEP parameters in free space: OPEP I (gπNN = 14, MN = 940 MeV, mπ = 140 MeV, Λ = 500 MeV, M∆N = 300 MeV) used in Ref. [11] and OPEP II which coincides with the second (ρ = 0) column of Table 1.We emphasize that our aim is not to describe quantitatively the NN potential in free space but to study qualitative changes in interaction energy due to the medium modifications. These two parameter sets show almost the same dependence on R for the interaction energy per nucleon as illustrated in Fig. 1. The curve with asterisks which corresponds to the OPEP I set with η = 0is equivalent with that of Ref. [11]. The curve with stars η = corresponding to the OPEP II with 0drawsthere- ∗ sult of the present calculation. Though we are not able Fig. 1. The minimization of the interaction energy E (Eq. (5))by using two sets of the OPEP parameters. The OPEP I (η = 0) cor- to distinguish the results with the OPEP II from those responds to the results of Ref. [11]. Those with the OPEP II are with the OPEP I in shorter ranges, they start to get de- drawn for two different values of the additional variational parame- viated each other when R grows: The πNN coupling ter,whichisdefinedinEq.(3),i.e.η = 0andη = 0, respectively. constant in the OPEP I set is larger than that in the OPEP II, so that the interaction energy becomes more of nuclear matter [14]. We now treat the phenomenological effective S- and P -wave pion–nucleon scatter- attractive. Note that the solutions with the OPEP II (η = 0) are stable with respect to the change of the ing lengths b0, c0, and the correlation parameter g in lattice-point nucleon trial functions. The additional deriving the interaction energy per nucleon, respec- = parameter η plays an important role only at larger dis- tively. While g is fixed like g 1, we can examine tances, i.e., R 1.6 fm, where the energy is getting how the interaction energy per nucleon depends on the lower. parameters b0 and c0. Introducing the controlling vari- In order to see how the ground-state energy of nu- ables βS and βP , we can rewrite b0 and c0 as follows: clear matter is changed with the medium modifica- phen phen b0 = b βS,c0 = c βP . (8) tions of the potential parameters taken into account, 0 0 we begin with the OPEP II for both η = 0 and η = 0 They allow us to include the medium effects in a cases so that we can keep our investigation in a self- controlled manner: βS,P are varied in the range of 0 β 1. Putting β = β = 0, we restore them consistent manner. The density of the√ medium in the S,P S P FCC structure is defined as ρ = 2/R3 [11].The in free space. normal nuclear-matter density ρ = 0.17 fm−3 corre- Similarly, we introduce the overall kinetic factor sponds to a distance between the nearest neighbors: βkin 1 in the energy functional (5) to show the R ≈ 2.03 fm. The medium modifications of the single importance of zero-point kinetic and iso-rotational (∆ admixture) fluctuations: hadron properties are expressed via three phenomeno- 2 logical quantities such as b0, c0, g for a given density ∗ α ∗ E = M 1 + α2 ∆N 2 −1 + + + 15η + + 11 2 2 The medium-modified values of the OPEP parameters at some 1 3η 1 η η ∗ ∗ 4 4 densities are presented in Table 1,whereg and Λ are ex- πNN 2 πNN 3 ∗ ∗ tracted from the medium-modified form factor (see Ref. [15] × β + E + E . (9) for more details). ∗ kin H F 4MN U.T. Yakhshiev et al. / Physics Letters B 628 (2005) 33–39 37

Fig. 2. The effect of the medium modifications of hadron properties Fig. 3. The effect of the Lorentz–Lorenz parameter g on the inter- on the interaction energy per nucleon. The dotted curve represents action energy per nucleon. Dotted, solid, and dashed curves corre- the case, where the medium modifications of hadron properties are spond to the results for g = 1, g = 0.6 (phenomenological value), not taken into account (βS,P = 0), while the short-dashed one de- and g = 0.33 (classical value), respectively. The medium modifi- picts the case when we consider the empirical parameters b0 and cations of the OPEP parameters are fully taken into account, i.e. = c0 by 2%, i.e. βS,P 0.02. The long-dashed curve draws the result βS,P = 1. The main panel represents the case with η = 0. In the with βS,P = 0.1, whereas the solid one takes into account βS,P = 1, small panel, the results are presented for both η = 0andη = 0 cases i.e. the full effect of the medium modifications. In these curves, we with g = 0.6. always consider η = 0. In the small panel, two different curves are shown in the case of βS,P = 1, i.e. η = 0andη = 0, respectively. The effect of the Lorentz–Lorenz parameter on the 3. The effect of the medium parameters b0, c0 interaction energy per nucleon is drawn in Fig. 3.We on the interaction energy per nucleon (Eqs. (5), (9)) present here the results with quantum fluctuations and is depicted in Fig. 2. If the medium modifications medium modification effects fully taken into account are ignored, the quantum fluctuations are not enough (βkin = βS,P = 1). As shown in Fig. 3, the interaction to loose a close-packed FCC structure. Thus, nuclear energy per nucleon depends on the Lorentz–Lorenz matter can be formed as a FCC crystal when the den- parameter rather weakly. Moreover, when g decreases sity is several times higher than that of normal nuclear in such a way that it is taken to be the empirical matter [11], as shown in the dotted curve (βS,P = 0) (g = 0.6) or classic (g = 0.33), it is even more dif- in Fig. 2. With βS,P switched on, one can see that the ficult to find the crystalline structure of the ground corresponding modifications of the NN tensor poten- state. All curves in the main panel correspond to the tial essentially changes this picture of nuclear matter. full variational function, including parameter η.The It implies that including a small amount of the medium small panel of Fig. 3 demonstrates the role of the vari- modifications breaks the condensed (or solid) FCC ational parameter η, where g = 0.6 is chosen. structure, as presented in short and long dashed curves The present results imply that the medium-modifi- in Fig. 2. It is found that βS,P = 0.1 breaks already the cations dramatically changes the picture of nuclear FCC structure. Consequently, the full consideration of matter due to the following reasons: Firstly, the mod- the medium modifications, i.e., βS,P = 1 does not al- ifications of single-nucleon properties make the con- low nuclear matter to be found in the form of a FCC tribution of the quantum fluctuations increased, since crystal. This result is drawn in the solid curve in Fig. 2 the nucleon mass is decreased in the kinetic term (see and stable under the change of η, which is shown in Eq. (9)). Secondly, the tensor part of the NN potential the small panel of Fig. 2. is suppressed in nuclear matter. As a result, its negative 38 U.T. Yakhshiev et al. / Physics Letters B 628 (2005) 33–39

crystalline structure of the ground state of nuclear matter. We found that the present results are stable under the changes of numbers of variational parameters in the lattice-point nucleon trial functions. However, one has to note that there might be a possibility that a condensed state at the normal nuclear- matter density could exist within the framework of the medium-modified Skyrme model according to the following reasons: Firstly, because of the contribution of the medium modifications to the interaction energy, a desirable minimum of nuclear matter seems to appear around the normal nuclear-matter density if quantum fluctuations are suppressed. The dotted curve in Fig. 4 indicates this possibility. However, we want to emphasize that without the medium modifications one should not conclude the existence of the crystalline structure of nuclear matter. Secondly, since it is well known that the binding of nuclear matter re- Fig. 4. The effect of the parameter βkin on the interaction energy per nucleon (9). The medium modifications of hadron properties are sults from a strong cancellation between an attractive fully taken into account (βS,P = 1) and the Lorentz–Lorenz para- potential term and a repulsive kinetic one, the present meter is fixed to be g = 0.6. conclusion may be changed if one can include the central attractive NN potential, which is in the present work neglected. For example, it is asserted in Ref. [18] contribution to the total interaction energy is getting that the scalar–isoscalar degrees of freedom are impor- smaller. tant. One can include the contribution of the scalar– It is interesting to study whether the medium modi- isoscalar channel in the Skyrme model in a similar fications play a similar role, when the quantum fluctu- way as done for the pseudoscalar–isovector channel. ations are suppressed. In order to see the effect of the For example, Refs. [13,15,19] showed that the attrac- medium modifications of hadron properties, we artifi- tion of the central NN potential is modified in nuclear cially vary the parameter βkin to suppress the quantum matter in such a way that it is suppressed as the den- fluctuations. The results are drawn in Fig. 4. The sup- sity increases. It is clear that due to these modifica- pression of the quantum fluctuations leads obviously tions the whole attractive potential containing the cen- to lowering the interaction energy per nucleon. This tral and tensor parts will be lessened. However, if the effect is shown in the long-dashed and short-dashed scalar degrees of freedom are added in the medium- curves in Fig. 4. When the quantum fluctuations be- modified Skyrme model, the nucleon effective mass come weak enough (βkin ≈ 0.3), the ground-state en- is less changed [15]. Consequently, the kinetic part ergy of the system makes it possible for nuclear matter on the interaction energy should be suppressed (quan- to be in the solid or condensed FCC configuration, as tum fluctuations will be suppressed). Therefore, it will discussed already in Refs. [3–5,11]. This is shown by be of great importance to understand the interplay be- the dotted curve in Fig. 4. tween these terms in the interaction energy within the In the present work, we investigated the structure framework of the medium-modified Skyrme model, of nuclear matter in the framework of the medium- the scalar–isoscalar channel being considered. modified Skyrme model. We found that the medium modifications of the hadron properties and NN tensor potential are crucial to see whether the crystalline Acknowledgements structure of nuclear matter exist or not. It turned out that the contribution of the medium modifications to The present work is supported by the Korea Re- the interaction energy per nucleon breaks a possible search Foundation Grant (KRF-2003-070-C00015). U.T. Yakhshiev et al. / Physics Letters B 628 (2005) 33–39 39

The work of U.T.Y. was ﬁnancially supported by Pu- A.D. Jackson, C. Weiss, A. Wirzba, A. Lande, Nucl. Phys. san National University in program Post-Doc. 2004. A 494 (1989) 523. M.M.M. acknowledges the support of the Brain Pool [8] A.D. Jackson, N.S. Manton, A. Wirzba, Nucl. Phys. A 495 (1989) 499; program 2004 which makes it possible for him to visit A. Wirzba, H. Bang, Nucl. Phys. A 515 (1990) 571; Pusan National University. A.D. Jackson, C. Weiss, A. Wirzba, Nucl. Phys. A 529 (1991) 741; A. Wirzba, hep-ph/9211295. [9] H.-J. Lee, B.-Y. Park, D.-P. Min, M. Rho, V. Vento, Nucl. Phys. References A 723 (2003) 427; H.-J. Lee, B.-Y. Park, M. Rho, V. Vento, Nucl. Phys. A 726 (2003) 69. [1] M. Kutschera, C.J. Pethick, D.G. Ravenhall, Phys. Rev. [10] B.Y. Park, D.P. Min, M. Rho, V. Vento, Nucl. Phys. A 707 Lett. 53 (1984) 1041. (2002) 381. [2] I.R. Klebanov, Nucl. Phys. B 262 (1985) 133. [11] D.I. Diakonov, A.D. Mirlin, Sov. J. Nucl. Phys. 47 (1988) 421, [3] N.K. Glendenning, Phys. Rev. C 34 (1986) 1072. Yad. Fiz. 47 (1988) 662. [4] E. Wüst, G.E. Brown, A.D. Jackson, Nucl. Phys. A 468 (1987) [12] T.S. Walhout, Nucl. Phys. A 484 (1988) 397; 450; T.S. Walhout, Nucl. Phys. A 519 (1990) 816. A.S. Goldhaber, N.S. Manton, Phys. Lett. B 198 (1987) 231; [13] J.W. Durso, H.C. Kim, J. Wambach, Phys. Lett. B 298 (1993) M. Kugler, S. Shtrikman, Phys. Lett. B 208 (1988) 491; 267. M. Kugler, S. Shtrikman, Phys. Rev. D 40 (1989) 3421. [14] A. Rakhimov, M. Musakhanov, F. Khanna, U. Yakhshiev, [5] A.D. Jackson, J.J.M. Verbaarschot, Nucl. Phys. A 484 (1988) Phys. Rev. C 58 (1998) 1738. 419; [15] A. Rakhimov, F. Khanna, U. Yakhshiev, M. Musakhanov, L. Castillejo, P.S.J. Jones, A.D. Jackson, J.J.M. Verbaarschot, Nucl. Phys. A 643 (1999) 383. A. Jackson, Nucl. Phys. A 501 (1989) 801; [16] U.T. Yakhshiev, M.M. Musakhanov, A.M. Rakhimov, U.-G. H. Forkel, A.D. Jackson, M. Rho, C. Weiss, A. Wirzba, H. Meißner, A. Wirzba, Nucl. Phys. A 700 (2002) 403; Bang, Nucl. Phys. A 504 (1989) 818; U.T. Yakhshiev, U.-G. Meißner, A. Wirzba, Eur. Phys. J. A 16 W.K. Baskerville, Nucl. Phys. A 596 (1996) 611. (2003) 569. [6] N.S. Manton, P.J. Ruback, Phys. Lett. B 181 (1986) 137; [17] T. Ericson, W. Weise, Pions and Nuclei, Claredon, Oxford, N.S. Manton, Commun. Math. Phys. 111 (1987) 469. 1988. [7] A.D. Jackson, A. Wirzba, L. Castillejo, Phys. Lett. B 198 [18] B.D. Serot, J.D. Walecka, Adv. Nucl. Phys. 16 (1986) 1. (1987) 315; [19] U. Yakhshiev, U. Meißner, A. Wirzba, A. Rakhimov, M. A.D. Jackson, A. Wirzba, L. Castillejo, Nucl. Phys. A 486 Musakhanov, nucl-th/0409002. (1988) 634; Physics Letters B 628 (2005) 40–48 www.elsevier.com/locate/physletb

NLO forward–backward charge asymmetries in pp( _ ) → l−l+j production at large hadron colliders

F. del Aguila a, Ll. Ametller b, R. Pittau a,1 a Departamento de Física Teórica y del Cosmos and Centro Andaluz de Física de Partículas Elementales (CAFPE), Universidad de Granada, E-18071 Granada, Spain b Departament de Física i Enginyeria Nuclear, Universitat Politècnica de Catalunya, E-08034 Barcelona, Spain Received 8 August 2005; received in revised form 14 September 2005; accepted 20 September 2005 Available online 28 September 2005 Editor: N. Glover

Abstract

We consider the next-to-leading order corrections, O(αs), to forward–backward charge asymmetries for lepton-pair production in association with a large transverse momentum jet at large hadron colliders. We find that the leading order results are essentially confirmed. Although experimentally challenging and in practice with large backgrounds, these observables could 2 lept 2 provide a new determination of the weak mixing angle sin θeff (MZ) with a statistical precision for each lepton flavour of ∼ −3 × −3 b 10 (7 10 ) at LHC (Tevatron), and if b jets are identified, of the b quark Z asymmetry AFB with a statistical precision − − of ∼ 2 × 10 3 (4 × 10 2) at LHC (Tevatron).  2005 Elsevier B.V. All rights reserved.

PACS: 13.85.-t; 14.70.-e

Keywords: Hadron-induced high- and super-high-energy interactions; Gauge bosons

1. Introduction Large Hadron Collider (LHC) might give a chance to determine the electroweak parameters with high precision [1,2]. In practice, the experimental chal- The large cross sections for gauge boson pro- lenge is very demanding, but at any rate pursu- duction at the Fermilab Tevatron and the CERN ing such measurements will help to disentangle the strong physics contributing to the different processes. E-mail address: [email protected] (Ll. Ametller). In this Letter we calculate the forward–backward 1 On leave of absence from Dipartimento di Fisica Teorica, Torino charge asymmetries of lepton pairs in events with a ( _ ) ∗ and INFN Sezione di Torino, Italy. large transverse momentum jet pp → Z, γ + j →

− + 2 e e + j at next-to-leading order (NLO), O(αs) theoretical calculations not ambiguous, it seems very corrections. We make use of the Monte Carlo pro- hard to get rid of systematic uncertainties to the re- gram MCFM v4.1 [3], which includes the necessary quired level. In the following with a more exclusive processes at this order, and of ALPGEN [4]. The par- process we will need a more demanding experimental ticularly interesting case of a final b jet is discussed in performance. But we will stay on the very optimistic detail. We find that the leading order (LO) predictions side, emphasizing what we may learn if we were only [5] are essentially confirmed. limited by statistics. At LHC with an integrated lumi- ( _ ) Electron–positron pair production pp → Z,γ ∗ → nosity of 100 fb−1 the weak mixing angle precision e−e+ has a large cross section at large hadron col- would be hopefully further improved by a factor ∼ 3. liders, and as it is sensitive to the presence of vector This would be comparable to the current global fit pre- and axial-vector fermion couplings to neutral gauge cision, 0.00016, but for instance a factor ∼ 2 better bosons, in principle allows for their precise measure- than the effective weak mixing angle precision ob- ment. A prime example is the determination of the ef- tained from the bottom forward–backward asymmetry 2 lept at LEP and SLD, 0.00029 [12]. fective weak mixing angle sin θeff , that enters in their definition, the optimum observable being the forward– The associated production of a neutral gauge boson = ∗ backward charge asymmetry of the lepton pairs AFB V Z, γ and a jet j has also a large cross section, [6,7]. Indeed, the tree level Drell–Yan3 parton process especially at LHC, and can also allow for a precise ¯ → ∗ → − + 2 lept qq Z, γ e e gives an asymmetric polar angle determination of sin θeff . This NLO correction to V electron distribution relative to the initial quark, which production is a genuine new process when we require also depends on the lepton pair invariant mass Me−e+ the detection of the extra jet. In particular, gluons can At the Fermilab Tevatron Run I the Collider De- be also initial states, and the large gluon content of tector at Fermilab (CDF) reported an asymmetry AFB the proton at high energy tends to increase the Vj at the Z peak of 0.07 ± 0.02 [9], in agreement with production cross sections, although they stay almost the Standard Model (SM) prediction. A new measure- one order of magnitude smaller than the corresponding ment of neutral gauge boson production in pp¯ colli- V cross sections. In Table 2 of the section that col- sions at the√ upgraded Run II Fermilab Tevatron op- lects our numerical results we gather the different LO erated at s = 1.96 TeV has been recently presented and NLO contributions to V(→ e−e+)j production at by CDF, giving, at the Z peak, AFB = 0.07 ± 0.03, Tevatron, to be compared with the inclusive LO and the statistical error being large because only an inte- NLO V → e−e+ cross section for the same cuts, 127 grated luminosity of 72 pb−1 has been analysed [10], and 158 pb, respectively. At LHC we find for e−e+ less than a tenth of the luminosity collected so far.4 production 685 and 745 pb, respectively, to be com- A fit to these data where the quark and electron cou- pared to the e−e+j cross section, 53 and 57 pb, in plings to the Z boson are expressed as a function of Table 2. All the calculations throughout the Letter have 2 lept 2 lept = ± ± been performed with MCFM v4.1, and with ALPGEN sin θeff gives sin θeff 0.2238 0.0040 0.0030, where the errors stand for statistics and systematics, when necessary. They provide a good description of respectively. This is far away from the estimate of the these processes at hadron colliders. For instance, the expected statistical precision ∼ 0.0005 to be reached prediction at Tevatron for the ratio of the inclusive − ( _ ) at Run II with an integrated luminosity of 10 fb 1 [11]. cross section for pp¯ → Vb to pp¯ → Vj production Even if the experiment is well understood [10] and the is, according to the results in Table 2, 0.020 to NLO5 (0.0096 to LO [5]).6 This has to be compared to the

− + recent measurement of this ratio with the D0 detector 2 Throughout the Letter we will explicitly refer to the e e (+j) − + decay channel. The same analysis applies to l l + j production with l = µ. 3 A list of Drell–Yan cross section measurements at Tevatron Run I and II is given in Ref. [8]. 5 We neglect in this estimate the small fraction of events where b 4 ∈ ¯ In Run I the asymmetry found for the last bin, Me−e+ and b combine into the same jet. [300, 600] GeV, deviated from the SM prediction [9], what is not 6 In apparent agreement with the NLO prediction 0.018 ± 0.004 conﬁrmed by Run II [10]. by J.M. Campbell and Willenbrock quoted in Ref. [13]. 42 F. del Aguila et al. / Physics Letters B 628 (2005) 40–48

0.023 ± 0.005 [13], obtained with a similar, but not identical, set of cuts. As pointed out in Ref. [5] the forward–backward charge asymmetry of the lepton pairs can be measured in neutral gauge boson production with an accompany- Fig. 1. LO qq¯ contribution to the Drell–Yan process in Eq. (1). ing jet either relative to a direction fixed by the initial CS state AFB as in the inclusive neutral gauge boson production (Drell–Yan case), or relative to the final jet di- j rection AFB. The former is adapted to obtain the asym- ( _ ) metry from qq¯ events, and the latter from gq ones. 2 lept Both asymmetries give similar precision for sin θeff at LHC but not at Tevatron, where the precision for CS AFB is almost one order of magnitude higher. How- Fig. 2. LO gb contributions to the process in Eq. (2) for j = b. j ever, in principle AFB also allows for the measurement of flavour asymmetries. Thus, if we require the final jet to be a b quark, we can make a new measurement of at LO and define the different forward–backward b asymmetries we are interested in. Fig. 1 shows the AFB. This is especially interesting given its observed deviation at the Z pole from the SM prediction, 3σ LO diagram contributing to the Drell–Yan process in [12]. However, to approach a similar precision will Eq. (1). In the absence of gluonic radiation the trans- be a very demanding experimental challenge because verse momentum of the exchanged vector boson is we have not only to identify the heavy quark but to zero. Therefore, one must, for instance, expect that the − measure its charge. Being very optimistic the corre- direction of the initial quark state and the final e are sponding effective weak mixing angle precision to be correlated. When the initial quark line emits gluons, in principle expected at LHC, ∼ 10−3, is already lower such correlations, although still present, tend to dimin- − than the one reported by LEP and SLD, 2.9×10 4,but ish because of the transverse momentum pt acquired − + similar to the difference between the central values re- by the e e system. If one of those additional gluons b is hard enough and is emitted in the central region of sulting from AFB at the Z pole and the global fit to all data [12]. the detector, it gives rise to an extra jet resulting into In the following we study the LHC and the Teva- the process in Eq. (2). However, allowing for an ex- tron potentials in turn. First, we review the LO con- tra jet obliges to consider new subprocess initiated by ( _ ) tributions to pp → Vj production and introduce the a gluon and a(n) (anti)quark. We show in Fig. 2 the different asymmetries. Afterward we discuss the NLO relevant tree level diagrams in the simple case j = b. ∗ corrections, paying special attention to the case of a fi- In diagram (b) the decay products of the Z,γ system nal b jet. Finally, we present the numerical results and know very little about the direction of the final state b, draw our conclusions. because of the initial state gluon that also connects to the b fermionic line. Instead, in diagram (a), the pt of the b quark exchanged in the s-channel is zero, there- 2. LO processes and forward–backward charge fore one expects correlations between the final state 7 asymmetries leptons and the direction of the b-jet [5]. An optimal observable to quantify such correlations for the process of Eq. (2) is a forward–backward asymmetry: Let us thus compare the processes − ( _ ) ∗ − + F B pp → Z,γ → e e (1) AFB = , (3) F + B and 7 ( _ ) ∗ − + Analogous considerations also hold in the general case with pp → Z,γ + j → e e + j (2) j = b. F. del Aguila et al. / Physics Letters B 628 (2005) 40–48 43

|p | − + Table 1 extra sign factor z , p = pe + pe + pj , which pz The definitions of the various asymmetries at the pp and pp¯ collid- corresponds to assume that the largest rapidity par- ers ton is a (anti)quark if it is along the (anti)proton di- Collider Asymmetry Definition rection. This explains the factor in the third line of CS ¯ ppA¯ cos θ = cos θCS Table 1. Finally, because both in pp and pp collid- FB + − | e + e + j | ¯ CS = × pz pz pz ers are produced as many b as b, in order to obtain pp A cos θ cos θCS + − j FB pe +pe +p b + − z z z AFB one must use cos θj multiplied by a ( ) sign + − j j |pe +pe +p | for b (anti)quarks, − sign(Q ) with Q the b charge. ppA¯ cos θ = cos θ × z z z b b FB j e+ e− j pz +pz +pz In practice this means detecting the charge of the pro- j = pp AFB cos θ cos θj duced b jet. ¯ b = × − ppAFB cos θ cos θj ( sign(Qb)) Such asymmetries have been studied in detail, at b = × − LO, in Ref. [5]. What is important to realize is that, pp AFB cos θ cos θj ( sign(Qb)) in order to get reliable predictions, a priori small additional contributions must be carefully taken into ac- with count. Let us study, in particular, the effect of includ- 1 0 ing radiative O(αs), NLO, corrections. dσ dσ F = d cos θ, B = d cos θ. d cos θ d cos θ 0 −1 3. NLO contributions One can consider two possible angles: In the following we discuss all contributions nec- 1 = essary to compute the process in Eq. (2) at the NLO. cos θCS − + (pe + pe )2 The described structure is implemented in the MCFM − + + − 2(pe Ee − pe Ee ) code, that we used as it is in the case j = b.How- × z z , b − + ever, as already pointed out, the computation of AFB e− + e+ 2 + e + e 2 ¯ (p p ) (pT pT ) requires disentangling b from b final states. In MCFM − + this selection is not possible on an event by event basis, (pe − pe ) · pj cos θ = , because b and b¯ contributions are summed up. There- j e− + e+ · j (p p ) p fore, we modified the code to take this into account. where the four-momenta are measured in the labo- In addition, we included part of the remaining real µ ≡ ratory frame and pT (0,px,py, 0). The Collins– NLO contribution with the help of ALPGEN. We find Soper angle [14] θCS is, on average, the angle between convenient to list the different contributions in the fol- − e and the initial quark direction, while θj is the an- lowing, for the case j = b, because this will allow us gle between e− and the direction opposite to the jet in to discuss their relative size. Analogous considerations the e−e+ rest frame [5]. From the previous discussion apply to the general case. it should be clear that the former choice is adapted to The virtual contributions are drawn in Fig. 3,to- ( _ ) the qq¯ collisions and the latter to the gq ones. gether with the definition of all the graphical symbols Different asymmetries can be defined, according to and conventions used. In particular, we omit drawing ∗ the scheme given in Table 1. A comment is in order, explicitly the decay of the Z,γ system, but we al- ∗ − + with respect to the phases appearing in Table 1.Inpp ways understand Z,γ → e e , and the blob stands colliders the quark direction is fixed by the rapidity for the sum of all possible contributing Feynman di- of the jet plus the lepton pair. This implies defining agrams. For the case at hand, it means exchanging a − + |pe +pe +pj | virtual gluon in all possible ways in the two diagrams cos θ with an extra sign factor z z z ,asin CS e− + e+ + j pz pz pz of Fig. 2. the second line of Table 1. On the other hand, in pp¯ The real contributions are given in Fig. 4.The colliders there are produced as many quarks as anti- shorter lines on the right part of the drawings means j quarks and AFB vanishes unless some difference is that the corresponding outgoing partons are not seen made between them. Hence, cos θ is defined with an because they are too soft or too collinear to the ingo- 44 F. del Aguila et al. / Physics Letters B 628 (2005) 40–48

hard scattering to µ = MZ, the distribution function (0) fb (x) intrinsically sums up all contributions of the k k = order αs L [16], where L ln(µ/mb) is the large (0) collinear logarithm associated to the fact that fb (x) describes an exactly collinear gluon splitting g → bb¯. Fig. 3. NLO virtual gb contributions. Therefore, the leading order term 1 1 (0) = (0) (0) ˆ (0) dσ dx1 dx2 fg (x1)fb (x2) dσgb 0 0 + (g ↔ b), (7) coming from the diagrams in Fig. 2 contains all possi- k k ble αs L contributions ∞ (0) = k k dσ ckαs L . (8) k=1 Including the diagrams in Figs. 3, 4(a) and 4(b) takes ( _ ) ¯ (k+1) k Fig. 4. NLO real gb (a), q b (b), gg (c), qq (d) contributions. into account all corrections order αs L , because (0) the corresponding subprocesses multiply fb (x) in ing or outgoing b quark, therefore also leading to a Eq. (4). The corrections given in Fig. 4(c) corre- − + final state formed by an e e pair plus a jet contain- spond to the contributions O(αs) or O(αsL), depend- ¯ ing a b quark. The NLO fully differential cross section ing whether the final state b is or is not collinear to one dσ NLO is given by of the initial state gluons.8 They correct the leading order picture implicit in Eq. (7) of exactly collinearly NLO dσ produced b: now the initial state b can acquire a pt . 1 1 However, as discussed, all collinear contributions are = NLO NLO ˆ NLO dx1 dx2 fi (x1)fj (x2) dσij , already included in the LO process, in particular the = i,j 0 0 k 1 term in Eq. (8), therefore we are facing an appar- i, j = g,b,q,q,¯ (4) ent double counting. The key for understanding that this is not the case is noticing that other contributions NLO where fi (x) is the parton density, evoluted at the are present in Eq. (4), that correspond to the evolution NLO, relative to the ith initial state particle, carrying of the parton densities a fraction x of the proton (or antiproton) longitudinal (1) (0) + (0) (1) ˆ (0) momentum αs fg (x1)fb (x2) fg (x1)fb (x2) dσgb + (g ↔ b). (9) NLO = (0) + (1) fi (x) fi (x) αsfi (x), (5) Their effect is, among others, subtracting the c1αsL ˆ NLO and dσij are the differential cross sections corre- term from the LO contribution in Eq. (8), so that sponding to the subprocesses given in Figs. 3 and 4 adding the corrections of Fig. 4(c), does not imply computed at the one loop accuracy in QCD double counting. The structure described so far is implemented in MCFM, that we had to modify in order dσˆ NLO = dσˆ (0) + α dσˆ (1). (6) ij ij s ij to disentangle b and b¯ production, which is necessary, b The separation between virtual and real contributions in our case, for computing AFB. ˆ (1) in dσij is performed in MCFM with the help of a dipole formalism [15]. 8 Note that such collinear or almost collinear configurations con- It is worth discussing the relative size of all the tribute to the exclusive process we are studying when the final state terms appearing in Eq. (4). Setting the scale of the b¯ is lost in the forward or backward regions of the detector. F. del Aguila et al. / Physics Letters B 628 (2005) 40–48 45

O Finally, we computed the pure (αs) corrections in Table 2 √ − + − + = Fig. 4(d) with the help of ALPGEN. In the next section Estimates for the e e j√and e e b cross sections at LHC ( s = we discuss our numerical findings. 14 TeV) and Tevatron ( s 1.96 TeV) in pb. The jet transverse momenta are required to be larger than 50 (30) GeV at LHC (Teva- tron) and all pseudorapidities |η| smaller than 2.5. The pt of the leptons is larger than 20 GeV. The separations in the pseudorapidity- 4. Numerical results azimuthal angle plane satisfy R > 0.4andMe−e+ is within the ( _ ) range [75, 105] GeV. q means summing over q and q¯ contribu- We present our numerical results for e−e+j and tions − + e e b at LHC and Tevatron in turn. Our simulation Contributing process LHC Tevatron of the set up at LHC (Tevatron) is as follows LO NLO LO NLO ( _ ) → pe > 20 GeV,pj > 50 (30) GeV, gq Vj(j) 44.353.43.40 4.77 t t qq¯ → Vj(j) 8.43.74.61 2.76 e,j ( _ )( _ ) η < 2.5,Re,j > 0.4. q q → Vj(j) –3.7– 2.76 → For muon pairs the main difference would be the gg Vj(j) –3.7– 2.76 Total 52.757.18.01 7.53 pseudorapidity coverage [17,18].Weusethecteq- 6l1 (cteq6m) parton distributions at LO (NLO) [19]. gb → Vb(g) 1.81 1.81 0.038 0.049 ¯ The effect of smearing the lepton and jet energies has gg → Vb(b) –1.81 – 0.049 ( _ ) ( _ ) been studied at LO and found to be negligible [5], q b → Vb(q ) –1.81 – 0.049 therefore we do not include it here. On the other hand, qq¯ → Vb(b)¯ –0.06 – 0.025 the dominant background processes are expected to Total 1.81 1.87 0.038 0.074 be the same as for Drell–Yan production, namely, ( _ ) jets misidentified as e±, and pp → W +W −j → + − ¯ Table 3 e e νeνej. They are understood experimentally, at − + − + Estimates for the e e j and e e b cross sections and asymme- least at Tevatron, and can be considered under control [ ] tries defined in the text with Me−e+ in the range 75, 105 GeV. [10]. The LO and NLO production rates at LHC and The first row of each entry is the LO result, while the second one Tevatron are given in Table 2.InFigs. 5 (6) we show refers to the NLO. The integrated luminosity as well as the cuts can CS the corresponding charge asymmetries, AFB relative to be found in the text. The statistical precisions are also given, to be j b compared with the current effective weak mixing angle uncertain- the initial parton and A , A relative to the final jet. − FB FB ties at LEP and SLD from asymmetries only 1.6 × 10 4, and from O − The effect of the (αs) corrections is moderate for Ab at the Z pole 2.9 × 10 4 [12] CS j b FB A and A , but sizable for A , especially at Teva- lept FB FB FB LO σ (pb) A δA δ sin2 θ tron. This is basically due to the genuine new higher FB FB eff ¯ → ¯ NLO order process qq Vb(b) in Fig. 4(d), that tends to − − − LHC σ Vj = 53 ACS 8.7 × 10 3 4.4 × 10 4 1.3 × 10 3 wash out the asymmetry which is mainly associated to FB − − − 57 6.8 × 10 3 4.2 × 10 4 1.3 × 10 3 the (a) contribution in Fig. 2. This change is more pro- j − − − A 1.2 × 10 2 4.4 × 10 4 8.8 × 10 4 ¯ FB − − − nounced at Tevatron energies, where the qq content 1.1 × 10 2 4.2 × 10 4 1.1 × 10 3 − − − of the (anti)proton is larger (see Table 2). In all cases σ Vb = 1.8 Ab 7.5 × 10 2 2.3 × 10 3 8.7 × 10 4 j FB − − − 1.9 4.9 × 10 2 2.3 × 10 3 1.4 × 10 3 the asymmetries at NLO diminish, except for AFB at Tevatron where this asymmetry is smaller. Vj = CS × −2 × −3 × −3 Tevatron σ 8.0 AFB 6.4 10 3.5 10 1.4 10 − + ∼ − − − Near the Z pole, Me e MZ, the asymmetries 7.5 5.5 × 10 2 3.6 × 10 3 1.7 × 10 3 j − − − can be approximated by [6] A 9.9 × 10 3 3.5 × 10 3 8.1 × 10 3 FB − − − 1.1 × 10 2 3.6 × 10 3 7.2 × 10 3 lept = − 2 2 Vb = b × −2 × −2 × −2 A b a sin θeff MZ , (10) σ 0.04 A 5.5 10 5.1 10 2.5 10 FB − − − 0.07 2.7 × 10 2 3.7 × 10 2 4.7 × 10 2 translating then their measurement into a precise determination of sin2 θ lept(M2 ).InTable 3 we col- eff Z lept 2 − + lect the asymmetry estimates, their statistical pre- δ sin θeff of LHC and Tevatron for Me e in the cision, the cross sections and the precision reach range [75, 105] GeV, by assuming an integrated Lu- 46 F. del Aguila et al. / Physics Letters B 628 (2005) 40–48

Fig. 5. NLO (solid histogram) and LO (points) asymmetries at LHC. minosity L of 100 (10) fb−1 at LHC (Tevatron). In atic errors are also sizable. At any rate approaching Table 3 we assumed a b-tagging efﬁciency of 100%, the quoted precisions will be an experimental chal- no contamination ω and, in particular, no charge lenge. misidentiﬁcation. The statistical precisions δA and A second source of uncertainty, that is not ac- lept − 1 2 2 counted for in Table 3, is the dependence of the results δ sin θeff are proportional to , and the asymmetries A to 1 − 2ω. Therefore the contamination on the chosen set of parton densities. We investigated it 2 lept − −1 by recomputing asymmetries and statistical precisions multiplies δ sin θeff by (1 2ω) . Thus, if we only consider semileptonic b decays, implying ∼ 0.1 and using different parton distribution sets in the classes ∼ 2 lept ∼ cteq and mrst [22]. By doing so, variations of the ω 0, δA and δsin θeff increase by a factor 3. In practice we must try to maximize the quality fac- asymmetries of the order of 10% can be easily ob- 2 tor Q = (1 − 2ω) [20]. Although the exact value of served in the range 75

Fig. 6. NLO (solid histogram) and LO (points) asymmetries at Tevatron.

5. Conclusions part by MCYT under contract FPA2003-09298-C02- 01 and by Junta de Andalucía group FQM 101, and by In summary, the large Vj production cross section MIUR under contract 2004021808_009. at hadron colliders and the possibility of measuring the lepton asymmetries relative to the final jet allow for a precise determination of the effective electroweak References mixing angle. We have evaluated them to NLO, con- firming to a large extent the LO results. If there is [1] R. Brock, et al., in: U. Baur, R.K. Ellis, D. Zeppenfeld (Eds.), an efficient b-tagging and charge identification, these Proceedings of the Workshop on QCD and Weak Boson events with a b jet also allow for a new determination Physics in Run II, FERMILAB-PUB-00/297, 2000, p. 78. b [2] S. Haywood, et al., in: G. Altarelli, M.L. Mangano (Eds.), Pro- of AFB. The corresponding statistical precisions are collected in Table 3. As in Drell–Yan production [23], ceedings of the Workshop on Standard Model Physics (and these processes can be also sensitive to new physics more) at the LHC, CERN 2000-004, 2000, p. 117. [3] J.M. Campbell, R.K. Ellis, Phys. Rev. D 62 (2000) 114012. for large Me−e+ , especially to new gauge bosons. [4] M.L. Mangano, M. Moretti, F. Piccinini, R. Pittau, A. Polosa, JHEP 0307 (2003) 001; For other programs see W. Hollik, et al., Acta Phys. Pol. B 35 Acknowledgements (2004) 2533. [5] F. del Aguila, Ll. Ametller, P. Talavera, Phys. Rev. Lett. 89 (2002) 161802. We thank A. Bueno, J.M. Campbell and T. Rodrigo [6] J.L. Rosner, Phys. Lett. B 221 (1989) 85. for useful comments. This work was supported in [7] P. Fischer, U. Becker, J. Kirkby, Phys. Lett. B 356 (1995) 404. 48 F. del Aguila et al. / Physics Letters B 628 (2005) 40–48

[8] T. Affolder, et al., Phys. Rev. D 63 (2001) 011101(R); [18] G.L. Bayatian, et al., CMS: the TriDAS Project Technical B. Abbott, et al., Phys. Rev. Lett. 82 (2000) 4769; Design Report, CERN/LHCC 2000-38, http://cmsdoc.cern.ch/ D. Acosta, et al., Phys. Rev. Lett. 94 (2005) 091803; cms/TDR/TRIGGER-public/trigger.html. F. Déliot, hep-ex/0505023. [19] J. Botts, et al., Phys. Lett. B 304 (1993) 159; [9] T. Affolder, et al., Phys. Rev. Lett. 87 (2001) 131802; H.L. Lai, et al., Phys. Rev. D 51 (1995) 4763; See also F. Abe, et al., Phys. Rev. Lett. 77 (1996) 2616. H.L. Lai, et al., Phys. Rev. D 55 (1997) 1280; [10] D. Acosta, et al., Phys. Rev. D 71 (2005) 052002. H.L. Lai, et al., Eur. Phys. J. C 12 (2000) 375; [11] U. Baur, S. Keller, W.K. Sakumoto, Phys. Rev. D 57 (1998) J. Pumplin, et al., JHEP 0207 (2002) 012. 199; [20] BaBar Collaboration, B. Aubert, et al., Phys. Rev. D 66 (2002) U. Baur, O. Brein, W. Hollik, C. Schappacher, D. Wackeroth, 032003. Phys. Rev. D 65 (2002) 033007. [21] DELPHI Collaboration, P. Abreu, et al., Eur. Phys. J. C 9 [12] Latest results in http://lepewwg.web.cern.ch/LEPEWWG/. (1999) 367. [13] V.M. Abazov, et al., Phys. Rev. Lett. 94 (2005) 161801. [22] A.D. Martin, R.G. Roberts, W.J. Stirling, R.S. Thorne, Eur. [14] J.C. Collins, D.E. Soper, Phys. Rev. D 16 (1977) 2219. Phys. J. C 4 (1998) 463; [15] S. Catani, M.H. Seymour, Phys. Lett. B 378 (1996) 287; A.D. Martin, R.G. Roberts, W.J. Stirling, R.S. Thorne, Eur. S. Catani, M.H. Seymour, Nucl. Phys. B 485 (1997) 291; Phys. J. C 14 (2000) 133. D. Dicus, T. Stelzer, Z. Sullivan, S. Willenbrock, Phys. Rev. [23] F. del Aguila, M. Quirós, F. Zwirner, Nucl. Phys. B 287 (1987) D 59 (1999) 094016. 419; [16] J. Campbell, R.K. Ellis, F. Maltoni, S. Willenbrock, Phys. Rev. J.L. Rosner, Phys. Rev. D 54 (1996) 1078; D 67 (2003) 095002; T. Affolder, et al., Phys. Rev. Lett. 87 (2001) 131802. J. Campbell, R.K. Ellis, F. Maltoni, S. Willenbrock, Phys. Rev. D 69 (2004) 074021. [17] A. Airapetian, et al., ATLAS Detector and Physics Per- formance Technical Design Report, CERN-LHCC 99-14/15, http://atlasinfo.cern.ch/ATLAS/internal/tdr.html. Physics Letters B 628 (2005) 49–56 www.elsevier.com/locate/physletb

Problems in resumming interjet energy ﬂows with kt clustering

A. Banﬁ a,b, M. Dasgupta c

a Cavendish Laboratory, University of Cambridge, Madingley Road, Cambridge CB3 0HE, UK b DAMTP, Centre for Mathematical Sciences, Wiberforce Road, Cambridge CB3 0WA, UK c School of Physics and Astronomy, University of Manchester, Manchester M13 9PL, UK Received 16 August 2005; accepted 24 August 2005 Available online 19 September 2005 Editor: N. Glover

Abstract We consider the energy flow into gaps between hard jets. It was previously believed that the accuracy of resummed predictions for such observables can be improved by employing the kt clustering procedure to define the gap energy in terms of a sum of energies of soft jets (rather than individual hadrons) in the gap. This significantly reduces the sensitivity to correlated soft large-angle radiation (non-global leading logs), numerically calculable only in the large Nc limit. While this is the case, as we demonstrate here, the use of kt clustering spoils the straightforward single-gluon Sudakov exponentiation that multiplies the O 2 non-global resummation. We carry out an (αs ) calculation of the leading single-logarithmic terms and identify the piece that is omitted by straightforward exponentiation. We compare our results with the full O(α2) result from the program EVENT2 to + − s confirm our conclusions. For e e → 2jetsandDIS(1+ 1) jets one can numerically resum these additional contributions as we show, but for dijet photoproduction and hadron–hadron processes further studies are needed.  2005 Elsevier B.V. All rights reserved.

1. Introduction able us to test and further our knowledge of soft QCD dynamics. Moreover the hadronisation corrections are large, Energy flow into gaps between hard jets is a valu- making these spectra a useful testing ground for theo- able source of information on many aspects of QCD. retical ideas about power corrections within say a dis- Since this radiation is typically soft, a perturbative persive model for the QCD coupling [1]. Additionally calculation of the corresponding distribution contains at hadron colliders the activity away from jets has tra- large logarithms that need resummation. Comparisons ditionally been used to study the soft underlying event of resummed perturbative estimates with data then en- and to refine models thereof [2], which will be an important component of physics at the LHC. In the present Letter we analyse the first aspect E-mail address: abanfi@hep.phy.cam.ac.uk (A. Banfi). alone, that of perturbative resummation. This re-

0370-2693/$ – see front matter  2005 Elsevier B.V. All rights reserved. doi:10.1016/j.physletb.2005.08.125 50 A. Banfi, M. Dasgupta / Physics Letters B 628 (2005) 49Ð56 summation is challenging since the leading single- QΩ as the sum of energies of soft (mini) jets inside logarithms (there are no collinear enhancements due to the gap region significantly reduced the non-global the fact that we are away from the hard jets) are gener- component of the result. Defining the jets via a kt clus- ated both by direct angular-ordered emission into the tering procedure [10], which is also common practice gap by the hard parton (jet) system as well as multi- experimentally [11,12], had the effect of pulling soft ple energy-ordered correlated emission by a complex emissions out of the gap region by clustering with ensemble of soft emissions outside the gap [3,4],in harder emissions outside. As was shown in the sim- addition to the hard jets. This latter piece, which can- ple case of e+e− → 2jets[8], it is still possible for not yet be computed analytically and more worryingly emissions near the centre of the gap to escape cluster- beyond a large Nc approximation, is known as the ing. However since these emissions are well separated non-global component of the result. The non-global in rapidity from their nearest neighbours outside the term causes a much stronger suppression of the soft gap (to escape clustering) the magnitude of non-global energy flows than that obtained by vetoing direct emis- effects is reduced. This is because the bulk of the non- sion off hard partons, into the gap, the Sudakov or global piece arises in the region when the emitted soft bremsstrahlung component of the answer [4]. The fi- gluon does not have too large an opening angle rel- nal result, for the cross-section with gap energy less ative to those involved in the emitting ensemble [4]. than QΩ can be expressed as [3–5] Forcing a relatively large opening-angle/rapidity separation between the softest gluon and the harder emit- Σ(Q,Q ) = exp −R(Q/Q ) S Q/Q . (1) Ω Ω Ω ters, as is required to escape clustering, does reduce In the above exp[−R] is the Sudakov term ob- the size of the non-global effects [8]. tained by exponentiating the single gluon contribution We would like to point out, however, that using and accounting for gluon branching to reconstruct the kt clustering impacts the general form Eq. (1) and scale of the running coupling, while S is the non- does not leave the primary emission Sudakov piece global part of the answer obtained by running a large exp[−R] unchanged as has been assumed till now Nc Monte Carlo program that encodes soft evolution [8,9]. In fact we find the exact calculation of R be- of a system of dipoles to single-log (SL) accuracy, for comes non-trivial at higher orders since it depends emission into the gap Ω. Equivalently for such away- at nth order on the n gluon geometry and the use of from–jet energy flows one has to solve numerically a the clustering algorithm. The departure of R from its non-linear evolution equation [5] obtained in the large naive one-gluon form starts with two gluons and the Nc limit. resulting piece does not have the properties of non- Given that only large Nc approximations of the global logs, neither in the colour structure nor in the non-global component are calculable at present in con- dynamical properties. This conclusion is unfortunate junction with the fact that it dominates the full result at especially in the case of dijet photoproduction [9] or smaller QΩ values, it is important to reduce the sensi- hadron–hadron studies where the missing piece we tivity of the measurement to this effect. One method of compute at leading order here, will have a highly non- doing so is to study event-shape–energy-flow correla- trivial colour structure since it reflects the colour flow tions [6,7] where measuring an event shape V outside of the primary emission piece, computed, e.g., in [9, the gap at the same time as the energy flow QΩ inside 13]. This would impact accurate theoretical studies of it leads to non-global logarithms in the ratio V/QΩ . such observables even though the non-global part is Thus choosing V ∼ QΩ the magnitude of the non- reduced. global effects is reduced. However this procedure is The current Letter is organised as follows. In the more complex to implement in the case of several hard subsequent section we identify the problem with naive partons (e.g., jet production in hadron–hadron colli- one-gluon exponentiation to obtain the supposed Su- sions) and is quite restrictive experimentally, amount- dakov part of the answer, with kt clustering. We coming essentially to studying a different observable. pute the piece that will be missed by one-gluon expo- O 2 Another version of the measurement was suggested nentiation, at leading order (αs ). We then provide 2 in Refs. [8,9], to reduce the impact of non-global log- the full answer up-to order αs , for the leading loga- arithms. There it was shown that defining the energy rithms, and show via comparisons with EVENT2 [14] A. Banfi, M. Dasgupta / Physics Letters B 628 (2005) 49Ð56 51 that the extra piece we compute is needed to agree with ities. In the present case we shall examine instead the 2 2 fixed order estimates while the expansion to order αs independent emission CF part of the two gluon matrix of the Sudakov term, as currently computed in the lit- element along with the corresponding virtual correc- erature, is insufficient. We conclude by pointing out tions. We have for the independent emission of two that while analytical control over the Sudakov term real gluons by a dipole ab [15]: is lost, due to clustering, it is possible in the simpler + − 2 = 2 cases e e → two jets and DIS (1 + 1) jets, to nu- M (k1,k2) CF Wab(k1)Wab(k2) merically compute the additional piece at all orders, = 2 (ab) (ab) 4CF , (3) with existing programs [8], as we show. For higher jet (ak1)(bk1) (ak2)(bk2) topologies such as those studied in Ref. [9] it may only where W (k ) represents the emission of k off the be possible to numerically compute this term in the ab 1 1 hard dipole ab and similarly for k . large N limit, thereby reducing the accuracy of the 2 c Now we examine the region where the two real soft theoretical results compared to current expectations. gluons k1 and k2 are clustered by the jet algorithm. This happens when

2. Independent soft gluon emission 2 2 2 (η1 − η2) + (φ1 − φ2)

16 where we have exploited the freedom to set φ = 0 primary = 2 2 3 1 C2 CF L R , (12) and Q is the e+e− centre-of-mass energy. We have 3π also neglected the recoil of the hard partons a and b, with L = ln Q . Alternatively choosing a smaller gap QΩ against the soft emissions k1 and k2, which is valid for η R we get our aim of extracting the leading logarithms. Then the leftover real-virtual two-loop contribution primary = 16 2 2 1 3 − 2 − 2 3/2 C2 CF L R R (η) (α / π)2 π 3 reads (we compute the coefficient of s 2 ) η πR2 primary + − 2 − 2 C2 η R (η) 1 1 2 2 dx2 dx1 2QΩ = 2 − 2 −1 η 16CF Θ x2 − R tan . (13) x2 x1 Q R2 − (η)2 0 x2 2π dφ It is clear that although this piece has the same × dη dη colour structure as that for independent two gluon 1 2 2π k1∈/Ω k2∈Ω 0 emission, it cannot arise from expanding the single- gluon generated Sudakov equation (2). The expansion × Θ R2 − (η − η )2 − φ2 . (10) 2 1 of the naive Sudakov would give a term independent 2 O 2 The above equation requires some explanation. We of R and which goes as (η) at (αs ). have introduced the dimensionless scaled transverse In the following section we shall show that the momenta x1,2 = 2kt1,t2/Q and restricted the region expansion of the Sudakov equation (2) needs to be such that virtual emission k1 is integrated outside the supplemented with the results equations (12), (13) gap region while real emission k2 inside. We have as appropriate, as well as the correlated non-global 2 2 also inserted a step function that ensures that we are CF CAαs L piece computed in [8], in order to agree 2 2 integrating over the region of Eq. (4), where the cor- with the full αs L result, for Σ(Q, Q/QΩ ) generated responding double real emissions would be clustered by the program EVENT2. and k2 would be pulled out of the gap. The additional step function involving x2, that constrains the gap en- O 2 ergy, is the usual one that corresponds to computing 3. Full (αs ) result and comparison to EVENT2 the cross-section for events with energy in the gap greater than QΩ . This, by unitarity, is trivially related First we expand the Sudakov exponent equation (2) Q O 2 to that for events with gap energy less than Ω .From to (αs ). The result is the latter quantity the distribution is directly obtained Σ (t) = 1 −¯α L(4C η) by differentiation with respect to QΩ . We have de- Ω,P s F noted this term Cprimary as it is a second order in α 22 2 s +¯α2L2 8(η)2C2 − ηC C piece that has the colour structure and matrix element s F F A 3 for independent emission from the primary dipole ab. 4CF nf η However it is not derived by expanding the standard + , (14) 2 3 Sudakov result to order αs and is a companion to the ¯ = αs non-global correction term S2 of [8], but with different where we denoted αs 2π . 2 2 functional properties and colour structure. An additional CF CAαs L term is indeed the non- Performing the integration over φ in Eq. (10) we global term computed in Ref. [8]. We compute this get piece numerically for different values of the parameters η and R and add it to the result from Eq. (14) for R 16 comparison with the fixed order program EVENT2. primary = 2 2 2 − 2 C CF L dumin(u, η) R u . = = = 2 π For example, with R 1 and η 1 one gets S2 − 0 (11) 1.249CF CA where S2 is the first coefficient of the = + n Choosing for instance values of the gap size η non-global log contribution S 1 n=2 Snt , with R we get t defined as before. A. Banfi, M. Dasgupta / Physics Letters B 628 (2005) 49Ð56 53

2 2 Fig. 1. Comparison of the CF αs L term produced by EVENT2 and the analytical calculation (referred to as resummed since it is derived by primary = = = expanding the naive Sudakov resummation to NLO) with and without C2 . The ﬁgures are for R 1andη 1.0 (above) and η 0.5 (below). The agreement for the unclustered case is also shown for comparison. 54 A. Banﬁ, M. Dasgupta / Physics Letters B 628 (2005) 49Ð56

The comparison to EVENT2 for the distribution into primary and non-global components (and restor- −1 2 2 σ dσ/dL is shown in Fig. 1 for the CF αs L term, ing the proper colour factors where possible) needs to with L = ln QΩ /Q for R = 1 and η = 1, as well as be done with care, keeping in mind our findings. η = 0.5. If all leading (single) logarithms in the inte- The procedure to generate the most accurate theo- + − grated quantity Σ(Q,QΩ ) are correctly accounted for retical results for the e e → two jets and DIS (1+1) by the resummed result equation (2), we would expect jets is the following. We take the results as generated the difference between the EVENT2 results and the by the code used for Ref. [8], for a given gap geome- expansion to NLO of the resummation, to be a con- try. This is the full result in the large Nc approximation stant at small QΩ corresponding to large (negative) L. and we divide it by the result obtained using the same primary code for primary emissions alone (rather than dividing As we see this is only the case when C2 is included by adding it to the expansion of the Sudakov by the naive Sudakov result), which takes as the only equation (2). We have considered different values of source for emissions the original hard dipole, e.g., the + − R and η as mentioned, for example, in the caption outgoing qq¯ pair in e e annihilation. The result of for Fig. 1. The comparison for other colour channels this division is the non-global piece in the large Nc CF CA and CF TRnf shows agreement with EVENT2 limit. We can then make use of the fact that one can 2 easily compute the exact O(α2) non-global term with (see Fig. 2) which reflects the fact that only the CF s channel, corresponding to independent emission, is in- proper CF CA colour factor and parameterise the non- correctly described by Eq. (2). global Monte Carlo result, as a function of t,inaform that retains the correct colour structure for the lead- 2 2 ing αs ln Q/QΩ non-global term (see, e.g., Ref. [3]). 4. All orders and conclusions This is the non-global result, with the large Nc approx- O 3 3 imation starting only from (αs ln Q/QΩ ) terms. We conclude by pointing out that in the simple The overall result is obtained by multiplying the re- cases of e+e− → two jets and DIS (1 + 1) jets, the sultant parameterised form with the primary emission additional terms we describe here can be accounted result, as computed here with full colour factors. The for numerically, at all orders. This is done by using the large Nc approximation is then confined to the non- Monte Carlo program for large Nc dipole evolution de- global term and starting from the next-to-leading such veloped in [3] and implemented with the kt clustering piece (S3 in the notation of [8]). It is thus still an in [8]. By demanding multiple emissions from the pri- important finding that the non-global logarithms are mary dipole alone and restoring the colour factor for reduced considerably by kt clustering as demonstrated independent emission by changing CA/2 → CF one in Ref. [8], since this potentially reduces the impact obtains the result for primary emissions only, in the of unknown non-global terms beyond the large Nc ap- full theory. proximation. However the correct procedure for iden- Using this procedure we see that the primary emis- tifying the primary and non-global pieces, pointed out sion result, with kt clustering, differs from the Su- here, must be accounted for while comparing to ex- dakov result generated by single gluon exponentia- perimental data to enable accurate phenomenological tion, which is valid in the unclustered case. In Fig. 3, studies. we show the results we obtain for primary emission In the case of dijet photoproduction, studied, e.g., with clustering and the Sudakov (unclustered) result. in [9], and gaps between jets in hadron–hadron proces- The discrepancy grows with the single-log evolution ses, it is less straightforward to account for the missing 1 1 Q variable t = ln where λ = β αs(Q) ln independent emission terms we point out. They will 4πβ0 1−2λ 0 2QΩ and for t = 0.25 we note an increase of around 30% have a complex colour structure and existing large on inclusion of the terms we describe, that start with Nc numerical programs cannot be employed to gen- primary C2 computed analytically here. erate the full answer beyond the large Nc limit. This We wish to clarify that the full result in the large Nc would mean that the accuracy of the resummed result approximation, including the effect we point out here, is limited not just by the unknown beyond-large-Nc is readily obtained by the method described in Ref. [8] non-global logs but similarly in the primary emission and in fact computed there. However its separation terms which are not reduced by the use of cluster- A. Banfi, M. Dasgupta / Physics Letters B 628 (2005) 49Ð56 55

2 Fig. 2. Comparison of the CF CA (above) and CF nf αs L term (below) produced by EVENT2 and the expanded Sudakov result, supplemented with non-global logs for the CF CA term. The ﬁgures are for R = 1andη = 1.0 and as we expect the difference between the exact and resummed result expanded to NLO is a constant at large L. 56 A. Banﬁ, M. Dasgupta / Physics Letters B 628 (2005) 49Ð56

Fig. 3. The results for the primary emission resummation with and without kt clustering for R = 1, η = 1, using an adaptation of the program used for Ref. [8]. As can be seen, the clustering affects the primary emission term and the effect for t = 0.25 is an increment of over 30%. ing. In these cases further studies are therefore re- [3] M. Dasgupta, G.P. Salam, Phys. Lett. B 512 (2001) 323. quired to account correctly for the missing primary [4] M. Dasgupta, G.P. Salam, JHEP 0203 (2002) 17. emission terms before one can argue that use of the [5] A. Banfi, G. Marchesini, G.E. Smye, JHEP 0208 (2002) 006. [6] C.F. Berger, T. Kucs, G. Sterman, Phys. Rev. D 68 (2003) clustering method mitigates the uncertainty involved 014012. in the theoretical predictions, by reducing the non- [7] Yu.L. Dokshitzer, G. Marchesini, JHEP 0303 (2003) 040. global component significantly. This is currently work [8] R.B. Appleby, M.H. Seymour, JHEP 0212 (2002) 063. in progress [16]. [9] R.B. Appleby, M.H. Seymour, JHEP 0309 (2003) 056. [10] S. Catani, Yu.L. Dokshitzer, M.H. Seymour, B.R. Webber, Nucl. Phys. B 406 (1993) 187; S.D. Ellis, D.E. Soper, Phys. Rev. D 48 (1993) 3160; Acknowledgements J.M. Butterworth, J.P. Couchman, B.E. Cox, B.M. Waugh, Comput. Phys. Commun. 153 (2003) 85. We would like to thank Mike Seymour for useful [11] H1 Collaboration, C. Adloff, et al., Eur. Phys. J. C 24 (2002) discussions concerning this Letter and the work de- 517. [12] ZEUS Collaboration, M. Derrick, et al., Phys. Lett. B 369 scribed in Ref. [8] and Rob Appleby for supplying (1996) 55. us with the numerical code used in Ref. [8].Wealso [13] C.F. Berger, T. Kucs, G. Sterman, Phys. Rev. D 65 (2002) thank Gavin Salam for helpful comments. 094031. [14] S. Catani, M.H. Seymour, Phys. Lett. B 378 (1996) 287; S. Catani, M.H. Seymour, Nucl. Phys. B 485 (1997) 291. [15] Yu.L. Dokshitzer, G. Marchesini, G. Oriani, Nucl. Phys. B 387 References (1992) 675. [16] A. Banfi, G. Corcella, M. Dasgupta, in preparation. [1] Yu.L. Dokshitzer, G. Marchesini, B.R. Webber, Nucl. Phys. B 469 (1996) 93. [2] G. Marchesini, B.R. Webber, Phys. Rev. D 38 (1988) 3419; CDF Collaboration, Phys. Rev. D 70 (2004) 072002. Physics Letters B 628 (2005) 57–65 www.elsevier.com/locate/physletb

SCET sum rules for heavy-to-light form factors

Junegone Chay a, Chul Kim b, Adam K. Leibovich b

a Department of Physics, Korea University, Seoul 136-701, Republic of Korea b Department of Physics and Astronomy, University of Pittsburgh, PA 15260, USA Received 17 August 2005; accepted 13 September 2005 Available online 26 September 2005 Editor: H. Georgi

Abstract We consider a sum rule for heavy-to-light form factors in soft-collinear effective theory (SCET). Using the correlation function given by the time-ordered product of a heavy-to-light current and its hermitian conjugate, the heavy-to-light soft form factor ζP can be related to the leading-order B meson shape function. Using the scaling behavior of the heavy-to-light form factor in ΛQCD/mb, we put a constraint on the behavior of the B meson shape function near the endpoint. We employ the sum rule to estimate the size of ζP with the model for the shape function and ﬁnd that it ranges from 0.01 to 0.07.  2005 Elsevier B.V. All rights reserved.

PACS: 11.55.Hx; 13.25.Hw

Keywords: Soft-collinear effective theory; Sum rule; Heavy-to-light form factor

Information on heavy-to-light form factors is important to extract Standard Model parameters from experimental results. For instance, the B → π form factor is needed to extract Vub or the CKM angle γ from exclusive B decays [1]. Due to the nonperturbative nature of the form factor, techniques beyond perturbation theory need to be employed to try to evaluate these functions. Unquenched lattice results are starting to be available for the B → π form factor [2]. Due to the uncertainties in the lattice technique, the pion is restricted to have energies Eπ 1GeV, just in the region where the experimental uncertainty is largest. Another method for determining the form factor is light-cone sum rules [3].InRef.[4] (see also [5]), light-cone sum rules were investigated using soft-collinear effective theory (SCET) [6], where it is argued that using SCET allows for a more consistent scheme to calculate both factorizable and nonfactorizable contributions than the traditional light-cone sum rule approach. In this Letter, we investigate a novel sum rule using SCET, relating the B → π form factor at small q2 to the B meson shape function [7] which describes the motion of the b quark inside the B meson. The shape function

E-mail addresses: [email protected] (J. Chay), [email protected] (C. Kim), [email protected] (A.K. Leibovich).

Fig. 1. Feynman diagram for the correlation function Π in the timelike (spacelike) region. The double line is a heavy-quark field, the solid line is a hard-collinear quark with the residual momentum k(k ). The dashed line is a spectator quark in the B meson, which is an usoft field in SCETI. cannot be calculated, but it can be extracted from the data [8]. After relating the two nonperturbative functions using the sum rule, we use the scaling in ΛQCD/mb of the B → π form factor to put constraints on shape-function models near the endpoint region. By choosing a model for a shape function which satisfies the constraints, we can give model-dependent values for the B → π form factor at q2 = 0. Let us consider the correlation function in SCET [9], defined as I = 4 iq·z ¯ | † | ¯ Π(q) i d ze B T J0 (z)J0(0) B , (1) where the leading-order heavy-to-light current in SCET can be written as

i(p˜−mbv)·x ¯ † J0(x) = e ξnWY hv(x). (2) µ µ µ µ µ Here p˜ =¯n · pn /2 + p⊥ is the label momentum of the collinear quark ξn, which has momentum p =˜p + k , µ and we will set p⊥ = 0. We are interested in the correlation function under the same kinematic condition as the forward scattering amplitude of inclusive B decays in the endpoint region. We denote the residual momentum r as the remainder of the sum of large momenta of the heavy quark, the collinear quark, and the incoming momentum q, with µ µ µ r = mbv + q −˜p ∼ O(Λ), n · r = n · q + mb, n¯ · r = r⊥ = 0. (3) n · q is chosen to be negative, and Λ is a typical hadronic scale. Since n · r is of order Λ, the quark ﬁeld in the 2 intermediate state becomes hard-collinear with p ∼ O(mbΛ). If we take a cut of the correlation function Π(q), the ﬁnal state contains a pion or other collinear particles which include a hard-collinear quark and an ultrasoft (usoft) quark. The Feynman diagram for the correlation function at leading order is schematically shown in Fig. 1. The dispersion relation of the correlation function can be written as ∞ 2 1 Im Π[(p + q) ] Π (p + q )2 = d(p + q)2 B , B B 2 2 (4) π (pB + q) − (pB + q ) − i 2 → mπ 0 ¯ where pB is the momentum of the B meson. With Λ = mB − mb, we can write 2 2 ¯ 2 ¯ 2 (pB + q) = (mB v + q) = (mb + Λ)v + q =¯n · p(n · r + Λ) + O Λ , (5) + 2 =¯· · + ¯ + O 2 2 ∼ 2 and similarly (pB q ) n p(n r Λ) (Λ ). The pion mass is regarded as mπ Λ . In terms of the residual momentum n · r, the dispersion relation is ∞ 1 Im Π(n· r) Π(n· r ) = dn· r . (6) π n · r − n · r − i 2 ¯· − ¯ mπ /n p Λ J. Chay et al. / Physics Letters B 628 (2005) 57–65 59

2 ¯ · ¯ In the lower bound of the integral, mπ /n p can be neglected compared to Λ. Using the optical theorem, the correlation function in the timelike region can be written as = D iq·z ¯ | † | | | ¯ = S + C 2ImΠ(r) e B J0 (z) X X J0(0) B 2ImΠ 2ImΠ , (7) X X where the superscripts ‘S’ and ‘C’ mean the contributions from the single-pion state and from the continuum state, respectively. The imaginary part of the correlation function from the single-pion state is given by d3p 1 Π S = π ( π)4δ(p + q − p )B¯ |J †|π π |J |B¯ , 2Im 3 2 B π 0 n n 0 (8) (2π) 2Eπ where the matrix element between the B meson and the energetic pion πn is expressed in term of the heavy-to-light form factor ¯ πn|J0|B=¯n · pζP (µ). (9) Therefore, the correlation function for the single pion in the timelike region can be written as S = + 2 − 2 ¯ · 2 2 ∼ + 2 ¯ · 2 2 2ImΠ 2πδ (pB q) mπ (n p) ζP (µ) 2πδ (pB q) (n p) ζP (µ). (10) Using Eqs. (4), (5), and (10), the dispersion relation Eq. (6) can be rewritten as ∞ 2 n¯ · pζ 1 Im Π(w) Π(n· r ) = P + dw , (11) −Λ¯ − n · r − i π w − n · r − i ws = · = 2 ¯ · − ¯ where w n r and ws 4mπ /n p Λ is related to the onset of the continuum states, which we take as = 2 =¯· + ¯ s0 4mπ n p(ws Λ). At the parton level, the correlation function Π(n· r) can be obtained from SCET. From the matching between SCETI and SCETII, the time-ordered product of the heavy-to-light current and its hermitian conjugate can be expressed in terms of the jet function and the B meson shape function in SCETII. Starting from Eq. (1),the correlation function is dn¯ · z · − · ¯· n¯ · z n/ Π(n· r ) =−m dn· kei(n r n k)n z/2J (n · k)B¯ |hY¯ Y †h(0)|B¯ , (12) B 4π P v 2 2 v where the jet function JP with P =¯n · p is deﬁned as · † ¯ n/ dn k −in·kn¯·z/2 0|TW ξ (z)ξW(0)|0=i δ(n · z)δ(z⊥) e J (n · k). (13) n 2 2π P The jet function can be computed perturbatively and at tree level it is given by 1 J (0)(n · k) = . (14) P n · k + i The B meson shape function is deﬁned as n¯ · z · ¯· B¯ |hY¯ Y †h(0)|B¯ = dn· lein ln z/2B¯ |hY¯ δ(n · l − n · i∂)Y†h|B¯ v 2 v v v · ¯· 1 + v/ · ¯· = dn· lein ln z/2f(n· l)Tr = 2 dn· lein ln z/2f(n· l), (15) 2 60 J. Chay et al. / Physics Letters B 628 (2005) 57–65 where the residual momentum n·l should be less than Λ¯ since the shape function has support for n·l Λ¯. Because n · k = n · l + n · r from the phase-space integral in Eq. (12), the correlation function can be written as

Λ¯ Π(n· r ) =−mB dn· lf (n · l)JP (n · l + n · r + i). (16) −∞ At tree level, the correlation function becomes ∞ f(−w = n · l) Π(n· r ) = m dw , (17) B w − n · r − i −Λ¯ where we have changed the integral variable to w =−n · l to allow for an easier comparison with Eq. (11). From Eqs. (11) and (17), we obtain the tree-level relation ∞ ∞ − ¯ · 2 f( w) n pζP 1 Im Π(w) mB dw = + dw . (18) w − n · r − i −Λ¯ − n · r − i π w − n · r − i ¯ −Λ ws Taking a Borel transformation on each side of Eq. (18), we obtain ∞ ∞ − ¯ 1 − m dwf(−w)e w/wM =¯n · pζ 2 eΛ/wM + dwIm Π(w)e w/wM . (19) B P π ¯ −Λ ws From this relation, we can relate the heavy-to-light form factor to the B meson shape function as ∞ ∞ 1 − + ¯ 1 − + ¯ ζ 2 = m dwf(−w)e (w Λ)/wM − dwIm Π(w)e (w Λ)/wM P n¯ · p B π ¯ −Λ ws ∞ ∞ mB − + ¯ − + ¯ = dwf(−w)e (w Λ)/wM − dwf(−w)e (w Λ)/wM n¯ · p ¯ −Λ ws ws mB − + ¯ = dwf(−w)e (w Λ)/wM . (20) n¯ · p −Λ¯ By changing the variable w + Λ¯ → w, we ﬁnally obtain our main result

ws mB − ζ 2 = dwf(Λ¯ − w)e w/wM , (21) P n¯ · p 0 2 where the Borel parameter wM is given by M = wM n¯ · p ∼ mbΛ with wM ∼ ΛQCD. The other parameter ws is = + ¯ = 2 = ¯ · given by ws ws Λ with s0 4mπ wsn p. Note that ws is a measure of the distance to the continuum and 2 is roughly of order ΛQCD/mb. When we expand the integrand in a Taylor series near w = 0, Eq. (21) becomes

ws ¯ 2 ¯ ¯ m f(Λ) w 2f (Λ) f(Λ) ζ 2 = B dw f(Λ)¯ − w f (Λ)¯ + + f (Λ)¯ + + +··· P ¯ · 2 n p wM 2 wM wM 0 J. Chay et al. / Physics Letters B 628 (2005) 57–65 61 2 ¯ 3 ¯ ¯ m w f(Λ) w 2f (Λ) f(Λ) = B f(Λ)w¯ − s f (Λ)¯ + + s f (Λ)¯ + + +··· . (22) ¯ · s 2 n p 2 wM 6 wM wM Since f(Λ)¯ = 0, Eq. (22) simplifies to 2 3 ¯ 2 = mB −ws ¯ + ws ¯ + 2f (Λ) +··· ζP f (Λ) f (Λ) . (23) n¯ · p 2 6 wM 3/2 From various arguments [10,11] the soft form factor ζP scales as (ΛQCD/mb) , which puts a constraint on the behavior of the B meson shape function near Λ¯, though the functional form is unknown. Since the left-hand side 3 2 scales as (ΛQCD/mb) , we can constrain the scaling of the terms on the right-hand side. With ws ∼ Λ /mb,the shape function f(Λ)¯ and its derivatives of f(Λ)¯ should scale as f(Λ)¯ ∼ Λ, f (Λ)¯ ∼ 1/Λ, f (Λ)¯ ∼ 1/Λ3,..., based on Eq. (22). In general, the shape function has a width of order Λ¯ and is normalized to one, so it has a typical size of 1/Λ¯ ∼ 1/Λ. There is an additional factor of 1/Λ for each derivative of the shape function, from which we expect the naive scaling like f(Λ)¯ ∼ 1/Λ, f (Λ)¯ ∼ 1/Λ2. The constraint from the scaling behavior of the soft form factor does not allow this naive scaling. In fact, f(Λ)¯ is suppressed by Λ2 and f (Λ)¯ is suppressed by Λ compared to the naive scaling behavior due to this constraint. Practically, we put f(Λ)¯ = 0 since it is suppressed by Λ2 compared to the naive scaling behavior, and assume that the series in Eq. (23) converges rapidly such that we choose the terms with the first derivative. However, this scaling behavior, especially f (Λ)¯ ∼ 1/Λ puts a constraint on the endpoint behavior for the shape-function models. We can estimate the size of ζP from Eq. (21) given a model of the shape function. We adopt the model of Ref. [12], which is given as b b−1 b ω ¯− f(ω)= 1 − eb(ω/Λ 1), (24) Λ (b)¯ Λ¯ at leading order in αs . The constraint that the first derivative scales as 1/ΛQCD implies that b>2 for this model. The default choice of parameters in Ref. [12] is Λ¯ = 0.63 GeV and b = 2.93, consistent with this constraint. With = 2 = = this choice of the parameters, and using ws 4mπ /mb 0.016 GeV, wM 0.5 GeV, we obtain m ζ ≈ 0.01 B . (25) P n¯ · p

2 On the other hand, if we choose ws as 0.05 GeV, which can be considered as a typical scale ΛQCD/mb with ΛQCD ∼ 0.5 GeV, we get m ζ ≈ 0.045 B . (26) P n¯ · p

The behavior of ζP with respect to ws and b is illustrated in Fig. 2. The soft form factor ζP with the input parameter b = 2.93 is smaller than 0.1 in the region ws =[0, 0.1] GeV and the central value is ζP ∼ 0.05. However, when we set the input parameter b = 2.0, the soft form factor can be enhanced almost three times compared with the case of b = 2.93. This can be easily understood when we consider Eqs. (23) and (24). In the case with b = 2, ¯ 2 the shape function in Eq. (24) scales as f (Λ) ∼ 1/Λ , which makes ζP ∼ Λ overestimating the scaling behavior. The shape-function model in Eq. (24) with b ∼ 3 is reliable considering the scaling behavior of ζP near the tail of the shape function. We can estimate the size of the soft form factor using the shape-function model extracted from B → Xsγ [8]. The central ﬁtted values on the exponential model of the shape function in Eq. (24) are

¯ 2 Λ = 0.545 GeV,λ1 =−0.342 GeV , (27) 62 J. Chay et al. / Physics Letters B 628 (2005) 57–65

Fig. 2. Dependence of ζP on the input parameter ws and b. In the ﬁrst plot, the solid line denotes the soft form factor with b = 2.93 and the dashed line with b = 2.00. In the second plot, the solid line represents the variation on b with ws = 0.016 GeV and the dashed line with ws = 0.05 GeV.

Fig. 3. Tree-level estimate of ζP with the ﬁtted input parameters extracted from the B → Xs γ data. The solid line shows the soft form factor with ¯ 2 the central ﬁtted parameters (Λ = 0.545 GeV, λ1 =−0.342 GeV ). The two dashed lines represent the range of the soft form factor with 1σ ¯ 2 ¯ error, where the upper line is obtained with the input parameters (Λ = 0.781 GeV, λ1 =−1.13 GeV ), and the lower line with Λ = 0.485 GeV, 2 λ1 =−0.13 GeV .

¯ ¯ 2 ¯ where λ1 is deﬁned as Bv|hv(n · iDus) hv|Bv=−λ1/3 and, using the relation [7] Λ¯2 λ dωω2f(ω)= =− 1 , (28) b 3 we can obtain the value of the input parameter b = 2.605 under the choice of Eq. (27). Using the ﬁtted shape function from the data on B → Xsγ , the soft form factor at tree level with 1σ error is given by +0.0342 n¯ · p 0.0175− (ws = 0.016 GeV), ζ = 0.0167 (29) m P +0.0540 = B 0.0710−0.0580 (ws 0.05 GeV).

The behavior of ζP using this shape-function model is shown in Fig. 3. In the heavy-to-light soft form factor such 2 as ζP , it is reasonable to choose ws as 4mπ /mb since the invariant mass squared of the lowest continuum state 2 ∼ 2 → starts at p 4mπ . Therefore, with this choice of ws , we ﬁnd ζP (B π) at tree level can be small contrary to the prediction of the QCD factorization approach [13,14]. J. Chay et al. / Physics Letters B 628 (2005) 57–65 63

Fig. 4. Feynman diagrams for the lowest-order contributions to Π1(q) in SCETI.

The numerical analysis of the soft form factor ζP is based on the tree-level relation in Eq. (21). There can be a few sources of theoretical uncertainties. For example, there may be uncertainties in choosing the Borel parameters ws and wM . Since the forward scattering is computed in SCETI, the virtuality of the intermediate state, or the 2 pion state is of order M = wM n¯ · p ∼ mbΛ, and wM ∼ Λ.Andws is the scale from which the continuum states = 2 ¯ · ∼ 2 ≈ 2 ∼ 2 start. We take as ws 4mπ /n p or Λ /mb with Λ 0.5 GeV, assuming that mπ Λ , which is obviously an overestimation. This is in contrast to the choice of the Borel parameters in Ref. [4], in which they take ws ≈ wM ≈ 0.2 GeV. Their tree-level evaluation of the soft form factor corresponds to ζP ∼ 0.32, while our evaluation is about 0.01 (ws = 0.016 GeV) and 0.05 (ws = 0.05 GeV), which is smaller by an order of magnitude. We find that the variation of wM between 0.2 GeV and 0.6 GeV gives 1% (ws = 0.016 GeV), 10% (ws = 0.05 GeV). Therefore, we can argue that the large uncertainty arises from the choice of ws . Indeed, we obtain ζP ≈ 0.30 in our evaluation with ws = 0.2GeV,wM = 0.4 GeV, close to the result in Ref. [4]. Let us briefly comment on the higher-order corrections to the tree-level relation in Eq. (21).First,wehaveto include the radiative correction to Π(q), which include the radiative corrections of J0. In addition, we also have to include the spectator interactions. When we consider a hard-collinear gluon exchange with the spectator quark, the contributions can be separated into two parts, the soft form factor ζP and the hard form factor ζJ , based on the existence of the spin symmetry of the heavy-to-light current [11,14]. The hard spectator contributions for ζP satisfying the spin symmetry can be distinguished simply by the presence of the leading order current J0 in the time-ordered product [9]. In this case we need to include the higher order interaction terms of SCETI collinear Lagrangian in the correlation function. ⊥ The spin symmetry breaks down explicitly if subleading currents with iD/ c are included in the time-ordered product. Therefore, when the hard spectator contributions for ζJ are considered, we have to define a correlation function with the subleading current J1, which is given by ¯ ˜− · /n ⊥ J (x) = ei(p mbv) xξ¯ i/D WY†h (x). (30) 1 n 2 c v In this case, the intermediate state consists of hard-collinear particles and one of the correlation functions is given by 2 = 4 iq·z 4 4 ¯ | † L(1) L(1) | ¯ Π1 q i d ze d x d y B T J1 (z)J1(0)i ξq (x)i ξq (y) B , (31)

L(1) = ¯ P⊥ + ⊥ † where ξq ξn( / gA/n )WY qus is the ultrasoft-collinear Lagrangian. This correlation function contributes to the hard form factor. After integrating out the hard-collinear degrees of freedom, we have four-quark operators with hv and qus . As seen in Fig. 4, from the four possible contractions of the hard-collinear particles in the intermediate 64 J. Chay et al. / Physics Letters B 628 (2005) 57–65

2 Fig. 5. Examples on the nonlocal time-ordered products in SCETII. The internal loops consist of the collinear particles with offshellness Λ . The dotted vertices denote the four-particle operators obtained from the matching, respectively, by integrating out the hard-collinear gluons (a), (d) and quark ﬁelds (b), (c). state we obtain the operators in SCETII of the form ¯ + + n/ + n/ + O l ,l = hY¯ δ l − n · i∂ Y †h · (q¯ Y) δ l − n · i∂ Y †q , 1 1 2 a 2 1 b us b 2 2 us a ¯ + + ¯ n/ + † n/ αβ + † O l ,l = (hY ) σ⊥ δ l − n · i∂ Y h · (q¯ Y) σ δ l − n · i∂ Y q , 2 1 2 a 2 αβ 1 b us b 2 ⊥ 2 us a ¯ ¯ + + n//n + n/n/ + O l ,l = (hY¯ ) δ l − n · i∂ (Y q ) · (q¯ Y) δ l − n · i∂ Y †h , 3 1 2 a 4 1 us a us b 4 2 b ¯ ¯ + + ¯ n//n + n/n/ αβ + † O l ,l = (hY ) σ⊥ δ l − n · i∂ (Y q ) · (q¯ Y) σ δ l − n · i∂ Y h , (32) 4 1 2 a 4 αβ 1 us a us b 4 ⊥ 2 b αβ α β where σ⊥ = i[γ⊥,γ⊥ ]/2, and a, b are color indices. In addition, we also have to consider the possibilities that the hard-collinear loop diagrams in Fig. 4 can be matched onto the nonlocal time-ordered products such as Fig. 5 in SCETII. When calculating the hard-collinear loop corrections, we are confronted with nonvanishing infrared divergences which result from internal propagators with virtualities of order Λ2. In this case we expect the infrared divergences will be reproduced by the possible nonlocal time-ordered products in SCETII. Therefore, by matching onto SCETII, we obtain 4 = + + ¯ · · + + ¯ | + + | ¯ Π1 dl1 dl2 Ji n p,n r, l1 ,l2 B Oi l1 ,l2 B i=1 + 4 4 ⊗¯ | L L | ¯ +··· d xd y Ji Jk B T Oi (x), Ok (y) B , (33) i,k ⊗ + + L L where Ji,Ji are the jet functions and denotes convolutions of li ,lk which can appear in the operators Oi ,Ok . The matrix elements between the B meson in Eq. (33) are new nonperturbative parameters to be included by deﬁn- ing additional subleading shape functions of the B meson. Similarly, there are also hard spectator contributions to ζP from other time-ordered products with the subleading collinear Lagrangian, which introduce more nonperturbative parameters. However, we have no information on these subleading B meson shape functions, and hence we do not further consider higher-order corrections and other theoretical uncertainties such as the renormalization scale dependence in this Letter. We have derived the tree-level relation between the soft form factor for B → π and the leading B meson shape function using the sum rule approach. This is achieved by considering the forward scattering amplitude of the J. Chay et al. / Physics Letters B 628 (2005) 57–65 65 heavy-to-light current near the endpoint. We derived a constraint on the behavior of the B meson shape function 3/2 near the endpoint using the scaling behavior of the soft form factor, ζP ∼ (Λ/mb) . Since we work at tree level only, there may be a lot of uncertainties, and there may also be a model dependence on the form of the shape function. But the analysis suggests that the soft form factor can be about 0.01 to 0.07, considering various inputs from the models and using the experimental data on the B meson shape function. The numerical value of the soft form factor is approximately smaller by an order of magnitude than other sum rule predictions. Since we have not included higher-order corrections in αs , this evaluation may have a large theoretical uncertainty, but our analysis favors a small value of the soft form factor.

Acknowledgements

We thank Ed Thorndike for his help in providing the parameters for the experimentally extracted shape function. We also thank the Institute for Nuclear Theory at the University of Washington for its hospitality and the Department of Energy for partial support during the completion of this work. J. Chay was supported by Grant No. R01-2002-000-00291-0 from the Basic Research Program of the Korea Science and Engineering Foundation. C. Kim and A.K. Leibovich were supported by the National Science Foundation under Grant No. PHY-0244599. A.K. Leibovich was also supported in part by the Ralph E. Powe Junior Faculty Enhancement Award.

References

[1] C.W. Bauer, I.Z. Rothstein, I.W. Stewart, Phys. Rev. Lett. 94 (2005) 231802; M. Beneke, G. Buchalla, M. Neubert, C.T. Sachrajda, hep-ph/0411171; C.W. Bauer, D. Pirjol, I.Z. Rothstein, I.W. Stewart, hep-ph/0502094. [2] M. Okamoto, et al., Nucl. Phys. B (Proc. Suppl.) 140 (2005) 461; J. Shigemitsu, et al., hep-lat/0408019. [3] P. Ball, R. Zwicky, JHEP 0110 (2001) 019; P. Ball, R. Zwicky, Phys. Rev. D 71 (2005) 014015. [4] F. De Fazio, T. Feldmann, T. Hurth, hep-ph/0504088. [5] P. Ball, hep-ph/0308249. [6] C.W. Bauer, S. Fleming, M.E. Luke, Phys. Rev. D 63 (2001) 014006; C.W. Bauer, S. Fleming, D. Pirjol, I.W. Stewart, Phys. Rev. D 63 (2001) 114020; C.W. Bauer, D. Pirjol, I.W. Stewart, Phys. Rev. D 65 (2002) 054022. [7] M. Neubert, Phys. Rev. D 49 (1994) 3392; T. Mannel, M. Neubert, Phys. Rev. D 50 (1994) 2037. [8] A. Bornheim, et al., CLEO Collaboration, Phys. Rev. Lett. 88 (2002) 231803. [9] C.W. Bauer, D. Pirjol, I.W. Stewart, Phys. Rev. D 67 (2003) 071502. [10] V.L. Chernyak, I.R. Zhitnitsky, Nucl. Phys. B 345 (1990) 137. [11] M. Beneke, G. Buchalla, M. Neubert, C.T. Sachrajda, Nucl. Phys. B 591 (2000) 313. [12] S.W. Bosch, B.O. Lange, M. Neubert, G. Paz, Nucl. Phys. B 699 (2004) 335; S.W. Bosch, M. Neubert, G. Paz, JHEP 0411 (2004) 073. [13] M. Beneke, G. Buchalla, M. Neubert, C.T. Sachrajda, Phys. Rev. Lett. 83 (1999) 1914. [14] M. Beneke, T. Feldmann, Nucl. Phys. B 592 (2001) 3. Physics Letters B 628 (2005) 66–72 www.elsevier.com/locate/physletb

Renormalisation of one-link quark operators for overlap fermions with Lüscher–Weisz gauge action

QCDSF Collaboration R. Horsley a,H.Perltb,c, P.E.L. Rakow d,G.Schierholze,f, A. Schiller c

a School of Physics, University of Edinburgh, Edinburgh EH9 3JZ, UK b Institut für Theoretische Physik, Universität Regensburg, D-93040 Regensburg, Germany c Institut für Theoretische Physik, Universität Leipzig, D-04109 Leipzig, Germany d Theoretical Physics Division, Department of Mathematical Sciences, University of Liverpool, Liverpool L69 3BX, UK e John von Neumann-Institut für Computing NIC, Deutsches Elektronen-Synchrotron DESY, D-15738 Zeuthen, Germany f Deutsches Elektronen-Synchrotron DESY, D-22603 Hamburg, Germany Received 27 May 2005; received in revised form 23 June 2005; accepted 20 September 2005 Available online 28 September 2005 Editor: N. Glover

Abstract We compute lattice renormalisation constants of one-link quark operators (i.e., operators with one covariant derivative) for overlap fermions and Lüscher–Weisz gauge action in one-loop perturbation theory. Among others, such operators enter the calculation of moments of polarised and unpolarised hadron structure functions. Results are given for β = 8.45, β = 8.0and mass parameter ρ = 1.4, which are commonly used in numerical simulations. We apply mean ﬁeld (tadpole) improvement to our results.  2005 Elsevier B.V. All rights reserved.

1. Introduction Weisz, Iwasaki and DBW2 gauge actions. The results were given for a variety of ρ parameters. Furthermore, In a recent publication [1] we have computed lat- we showed how to apply mean ﬁeld (tadpole) im- tice renormalisation constants of local bilinear quark provement to overlap fermions. In this Letter we shall operators for overlap fermions and improved gauge extend our work to one-link bilinear quark operators. actions in one-loop perturbation theory. Among the Operators of this kind enter, for example, the calcula- actions we considered were the Symanzik, Lüscher– tion of moments of polarised and unpolarised hadron structure functions. The present calculations are much more involved than the previous ones, so that we shall E-mail address: [email protected] (G. Schierholz). restrict ourselves to the Lüscher–Weisz action, and to

0370-2693/$ – see front matter  2005 Elsevier B.V. All rights reserved. doi:10.1016/j.physletb.2005.09.036 QCDSF Collaboration / Physics Letters B 628 (2005) 66–72 67 parameters actually being used in numerical calcula- Table 1 tions. βc1 c3 r0/a The integral part of the overlap fermion action − − [2–4] 8.45 0.154846 0.0134070 5.29(7) 8.0 −0.169805 −0.0163414 3.69(4) am S = ψ¯ 1 − D + m ψ, (1) F 2 N The remaining parameters are [10]: m being the mass of the quark, is the Neuberger–Dirac operator c (1 + 0.4805α) c 0.03325α 1 =− , 3 =− , 2 2 ρ X ρ c0 20u0 c0 u0 DN = 1 + √ ,X= DW − , (2) a X†X a 1 c1 c3 = 1 + 8 + , (5) where DW is the Wilson–Dirac operator, and ρ is a c0 c0 c0 real parameter corresponding to a negative mass term. where At tree level 0 <ρ<2r, where r is the Wilson parameter. We take r = 1 and consider massless quarks. 1 1/4 log(u4) u = TrU ,α=− 0 . (6) Numerical simulations of overlap fermions are sig- 0 3 plaquette 3.06839 nificantly more costly than simulations of Wilson fermions. The cost of overlap fermions is largely deter- The final results cannot be expressed in analytic mined by the condition number of X†X. This number form (as a function of β and ρ) anymore. We there- is greatly reduced for improved gauge field actions [5]. fore have to make a choice. Here we consider two = For example, for the tadpole improved Lüscher–Weisz couplings, β 8.45 and 8.0, at which we run Monte action we found a reduction factor of 3 compared Carlo simulations at present [6,11]. The corresponding to the Wilson gauge field action [6]. The reason is that values of c1 and c3 are shown in Table 1 [12]. the Lüscher–Weisz action suppresses unphysical zero In Table 1 we also quote the corresponding force = modes, sometimes called dislocations [7]. A reduction parameters r0/a,asgivenin[12]. Assuming that r0 = of the number of small modes of X†X appears also to 0.5 fm, they translate into a lattice spacing of a = = = result in an improvement of the locality of the overlap 0.095 fm at β 8.45 and a 0.136 fm at β 8.0. = operator [5]. The mass parameter was chosen to be ρ 1.4. This appeared to be a fair compromise between optimising We consider the tadpole improved Lüscher–Weisz † action [8–10] the condition number of X X as well as the locality properties of D [13]. N 6 1 The Letter is organised as follows. In Section 2 we S = c Re Tr(1 − U ) G g2 0 3 plaquette give a brief outline of our calculations and present plaquette results for the renormalisation constants in one-loop 1 + c1 Re Tr(1 − Urectangle) perturbation theory. In Section 3 we tadpole improve 3 our results, and in Section 4 we give our conclusions. rectangle 1 + c Re Tr(1 − U ) , 3 3 parallelogram parallelogram 2. Outline of the calculation and one-loop results (3) where Uplaquette is the standard plaquette, Urectangle de- The Feynman rules specific for overlap fermions notes the loop of link matrices around the 1×2 rectan- [14,15] are collected in [1], while the gluon-operator gle, and Uparallelogram denotes the loop along the edges and the gluon–gluon-operator vertices (needed for the of the three-dimensional cube [1]. It is required that cockscomb and operator tadpole diagrams) are inde- c0 + 8c1 + 8c3 = 1 in the limit g → 0, in order to en- pendent of the fermion action and can be found in [16]. sure the correct continuum limit. We define We consider general covariant gauges, specified by the 6 gauge parameter ξ. The Landau gauge corresponds β = c0. (4) = g2 to ξ 1, while the Feynman gauge corresponds to 68 QCDSF Collaboration / Physics Letters B 628 (2005) 66–72

ξ = 0. In lattice momentum space the gluon propaga- Table 2 tor Dµν(k) is given by the set of linear equations βb Σ − ξ ˆ ˆ 8.45 17.429 G (k) − k k D (k) = δ , (7) 8.0 −17.054 µρ ξ − 1 µ ρ ρν µν ρ where MS,MOM MS,MOM where Zψ , ZO are calculable in con- = ˆ ˆ + ˆ2 − ˆ ˆ Gµν(k) kµkν kρδµν kµkρδρν dµρ (8) tinuum perturbation theory, and therefore are inde- ρ pendent of the particular choice of lattice gauge and fermion actions. and In [1] the wave function renormalisation constants = − − 2 ˆ2 − 2 ˆ2 + ˆ2 dµν (1 δµν) C0 C1a k C2a kµ kν , were found to be 2 akµ 2 ˆ = ˆ2 = ˆ2 MOM g CF kµ sin , k kµ. (9) Z (a, µ) = 1 − 2(1 − ξ)log(aµ) a 2 ψ 2 µ 16π + 4.79201ξ + bΣ (15) The coefficients {Ci} are related to the coefficients {ci} of the improved action by in the MOM scheme, and = + + C0 c0 8c1 8c3, g2C ZMS(a, µ) = 1 − F 2(1 − ξ)log(aµ) = = − ψ 2 C1 c3,C2 c1 c3. (10) 16π + + + The calculations are done analytically as far as this is 3.79201ξ bΣ 1 (16) possible using Mathematica. Part of the numerical re- in the MS scheme, with C = 4/3 and in Table 2. sults have been checked by an independent routine. F We consider the following one-link operators The bare lattice operators O(a) are, in general, divergent as a → 0. We define finite renormalised oper- i ←→ O = ψ(x)γ¯ D ψ(x), (17) ators by µν 2 µ ν ←→ S S 5 i ¯ O (µ) = ZO(a, µ)O(a), (11) O = ψ(x)γ γ D ψ(x), (18) µν 2 µ 5 ν where S denotes the renormalisation scheme. We have ←→ −→ ←− where D ν = Dν − Dν is the (symmetric) lattice co- assumed that the operators do not mix under renormal- variant derivative. While in our previous work [1], isation, which is the case for the operators considered which involved local bilinear quark operators, we only in this Letter. The renormalisation constants ZO are had to deal with the vertex diagram shown on the often determined in the MOM scheme first from the left-hand side of Fig. 1, we now obtain contributions gauge fixed quark propagator SN and the amputated from additional diagrams: the operator tadpole and the Green function ΛO of the operator O: cockscomb diagrams shown on the right-hand side of MOM = tree Fig. 1. Zψ (a, µ)SN p2=µ2 S , (12) The amputated Green function of the operator Oµν MOM ZO (a, µ) (Eq. (17)) turns out to be ΛO MOM Zψ (a, µ) p2=µ2 Λµν(a, p) = Λtree + other Dirac structures. (13) O 2 g CF 1 2 2 = = γµpν + + ξ log a p (Note that Zψ 1/Z2.) The renormalisation constants 16π 2 3 can be converted to the MS scheme, − 4.29201ξ + b γ p MS = MS,MOM MOM 1 µ ν Zψ (a, µ) Zψ Zψ (a, µ), MS MS,MOM MOM 4 2 2 1 ZO (a, µ) = Z ZO (a, µ), (14) + log a p + ξ + b2 γνpµ O 3 2 QCDSF Collaboration / Physics Letters B 628 (2005) 66–72 69 2 1 + − log a2p2 − ξ + b δ p/ (We have given one example operator in each repre- 3 2 3 µν sentation. A complete basis for each representation 4 p p can be found in [17].) The operators (20) and (21) + b δ γ p + − + ξ µ ν p/ , (19) 4 µν ν ν 3 p2 are widely used in numerical simulations [6,11,18, 19]. They correspond to the first moment of the par- where p is the external quark momentum, and the co- ton distribution. The operators (22) and (23) represent { } efficients bi are given in Table 3 for the tadpole im- higher twist contributions in the operator product ex- proved Lüscher–Weisz action and, for comparison, for pansion, and so are not used as much as operators in = = the plaquette action (with c1 c3 0) as well. The the first two representations. For completeness we give latter numbers are independent of β. The Green func- results for all four representations, so that the renor- 5 O5 tion Λµν(a, p) of the operator µν (Eq. (18)) is ob- malisation factors for all operators of the form (17) tained by multiplying the right-hand side of (19) by γ5 will be known. We denote the corresponding ampu- { 5} from the right. The coefficients bi turn out to be tated Green functions by Λ , Λ , Λ and Λ . { } v2a v2b v2c v2d identical to bi , as is expected for overlap fermions. From (19) we read off O O5 Thus, µν and µν have the same renormalisation 1 constants. In the following we may therefore restrict Λ = (γ p + γ p ) O v2a 1 4 4 1 ourselves to the operator µν . 2 It has been checked numerically that the gauge de- g2C 5 × 1 + F ξ + log a2p2 pendent part of (19) is universal (i.e., independent of 16π 2 3 the lattice gauge and fermion action), in accordance − + with the arguments presented in [1]. 3.79201ξ bv2a Under the hypercubic group H(4) the 16 operators of type (17) fall into the following four irreducible rep- g2C 4 p p + F − + ξ 1 4p,/ (24) resentations [17]: 16π 2 3 p2 3 (6) O ≡ 1 O + O 1 τ3 : v2a ( 14 41), (20) Λ = γ p − γ p 2 v2b 4 4 3 i i i=1 (3) 1 O ≡ O − O + O + O 2 τ1 : v2b 44 ( 11 22 33), (21) g C 5 3 × + F ξ + a2p2 1 2 log (1) O ≡ O + O + O + O 16π 3 τ1 : v2c 11 22 33 44, (22) (6) O ≡ O − O − 3.79201ξ + bv τ1 : v2d 14 41. (23) 2b 2 3 + g CF −4 + 2 − 1 2 p/ Table 3 ξ p4 pi , { } 16π 2 3 3 p2 The coefficients bi for the tadpole improved Lüscher–Weisz action i=1 at β = 8.45 and 8.0, as well as for plaquette action (25) 2 Action b1 b2 b3 b4 g CF 2 2 Λv =p/ 1 + (ξ − 1) log a p β = 8.45 −5.6115 −3.8336 2.7793 0.3446 2c 16π 2 β = 8.0 −5.2883 −3.7636 2.7310 0.3331 Plaquette −10.6882 −4.7977 3.4612 0.5267 − + 4.79201ξ bv2c , (26)

Fig. 1. The one-loop lattice Feynman diagrams contributing to the amputated Green function. From left to right: vertex, operator tadpole and cockscomb diagrams. 70 QCDSF Collaboration / Physics Letters B 628 (2005) 66–72 2 = − MS g CF 4 Λv2d (γ1p4 γ4p1) Z (a, µ) = 1 − − + b + b , (35) v2c 2 v2c Σ 2 16π 3 g CF 2 2 × 1 + (ξ − 1) log a p g2C 16π 2 ZMS(a, µ) = 1 − F [b + b ]. (36) v2d 2 v2d Σ 16π − + 4.79201ξ bv2d (27) 3. Tadpole improved results with A detailed discussion of mean field—or tadpole— b = b + b ,b= b + b + b , v2a 1 2 v2b 1 2 4 improvement for overlap fermions and extended gauge 4 actions has been given in [1]. Here we will briefly re- bv = b1 + b2 + 4b3 + b4 − ,bv = b1 − b2. 2c 3 2d call the basic idea, before presenting our results. (28) Tadpole improved renormalisation constants are defined by It is worth pointing out that with Wilson or clover fermions the Green functions Λ and Λ both show v2c v2d TI MF ZO 2 ZO = ZO , (37) perturbative mixing of O(g /a) with local operators. ZMF With overlap fermions these O(1/a) terms are com- O pert MF pletely absent, showing once again that overlap fermi- where ZO is the mean field approximation of ZO, ons behave much more like continuum fermions when while the right-hand factor is computed in perturbation mixing is a possibility. theory. For overlap fermions (with r = 1), and opera- Using (13) and (15), we obtain the renormalisation tors with nD covariant derivatives, we have constants in the MOM scheme: − ρu1 nD ZMF = 0 . (38) MOM O − − Zv ,v (a, µ) ρ 4(1 u0) 2a 2b 2 In our case n = 1. It is required that ρ>4(1 − u ), g CF 16 D 0 = 1 − log(aµ) + ξ + bv ,v + bΣ , which is fulfilled here (see Table 4). 16π 2 3 2a 2b To compute the right-hand factor in (37),wehave (29) to remove the tadpole contributions from the pertur- 2 MOM g CF bative expressions of ZO first. This is achieved if we Z (a, µ) = 1 − [bv ,v + bΣ ]. (30) v2c,v2d 16π 2 2c 2d re-express the perturbative series in terms of tadpole As already mentioned, the conversion factors improved coefficients: MS,MOM TI TI Zv ,v ,v ,v are universal [16]. They are given by c c c c 2a 2b 2c 2d 0 = u4 0 , i = u6 i ,i= 1, 3. (39) g2 0 g2 g2 0 g2 g2C 40 TI TI ZMS,MOM = 1 − F − ξ , (31) v2a,v2b 16π 2 9 This does not fix all parameters, but leaves us with some freedom of choice. The simplest choice is to de- g2C 4 ZMS,MOM = 1 − F − , (32) fine v2c 16π 2 3 g2 g2 = ,cTI = c ,cTI = u2c ,i= 1, 3. ZMS,MOM = 1. (33) TI 4 0 0 i 0 i v2d u0 (40) In the MS scheme we then find

MS Table 4 Zv ,v (a, µ) The coefﬁcient kTI and the average plaquette u4 at β = 8.45 and 8.0 2a 2b u 0 2 TI 4 g CF 16 40 βku u0 = 1 − log(aµ) + + bv ,v + bΣ , 16π 2 3 9 2a 2b 8.45 0.543338π2 0.65176 2 (34) 8.00.515069π 0.62107 QCDSF Collaboration / Physics Letters B 628 (2005) 66–72 71

With this choice Dividing (44) and (45) by (43) and inserting (38),we obtain mean field/tadpole improved renormalisation TI = + TI + TI C0 c0 8c1 8c3 , constants: CTI = u2C ,CTI = u2C . (41) ρ 1 0 1 2 0 2 ZTI = v2a,v2b ρ − 4(1 − u ) (Note that CTI = 1. However, CTI → 1 in the con- 0 0 0 2 g CF 16 tinuum limit.) This means that we have to replace × 1 − TI log(aµ) every g2 by g2 and every c and c by cTI and cTI, 16π 2 3 CTI TI 1 3 1 3 0 respectively, while keeping c0 unchanged. The effect + BTI , (46) of introducing tadpole improved coefficients (40) is v2a,v2b that the rescaled gluon propagator remains of the same 2 ρ g CF form as we change u0, thus ensuring fast convergence. ZTI = 1 − TI BTI , (47) MF v2c,v2d − − 2 v2c,v2d To compute ZO perturbatively, we need to know ρ 4(1 u0) 16π the perturbative expansion of u0 to one-loop order [10, where we have introduced the abbreviated notation 20]. We write TI = TI + 4 TI 2 B B ρ,C ku . (48) g CF ρ u = 1 − TI kTI. (42) 0 16π 2 u The coefficients B(ρ,CTI) are the analogue of B(ρ, TI TI TI In [1] we have computed ku for the Lüscher–Weisz C), with C0, C1 and C2 being replaced by C0 , C1 action with coefficients CTI, CTI and CTI. The num- TI 0 1 2 and C2 , respectively. In (46) and (47) only the gluon bers are given in Table 4 for our two values of β, propagator has been tadpole improved. 4 together with the ‘measured’ values of u0. Expanding To tadpole improved the fermion propagator as (38) then gives well, we must replace ρ by [1]

g2 C 4 ρ − 4(1 − u0) MF = + TI F TI ρTI = ZO 1 ku . (43) (49) pert 16π 2 ρ u0 Let us now rewrite the one-loop renormalisation in the right-hand perturbative factor of (37). This de- constants of Section 2 as ﬁnes ‘fully tadpole improved’ renormalisation constants Zv ,v ρ 2a 2b ZFTI = 2 v2a,v2b C g 16 ρ − 4(1 − u0) = − F (aµ) + B (ρ, C) , 1 2 log v2a,v2b 2 16π 3C0 g CF 16 × 1 − TI log(aµ) (44) 2 TI 16π 3C0 2 CF g Z = − B (ρ, C). FTI v2c,v2d 1 2 v2c,v2d (45) + B , (50) 16π v2a,v2b

Table 5 The constants B and ZMS at a = 1/µ for various levels of improvement

Operator βB ZMS BTI ZTI,MS BFTI ZFTI,MS v2a 8.45 −22.430 1.315 0.502 1.393 −0.077 1.411 v2b 8.45 −22.085 1.311 0.793 1.384 0.230 1.401 v2c 8.45 −18.079 1.254 2.985 1.318 1.829 1.353 v2d 8.45 −19.207 1.270 2.303 1.338 1.369 1.367 v2a 8.0 −22.036 1.310 0.603 1.390 −0.108 1.412 v2b 8.0 −21.703 1.305 0.892 1.381 0.199 1.402 v2c 8.0 −17.890 1.252 3.038 1.316 1.643 1.358 v2d 8.0 −18.954 1.266 2.371 1.336 1.239 1.371 72 QCDSF Collaboration / Physics Letters B 628 (2005) 66–72 2 ρ g CF ZFTI = 1 − TI BFTI (51) gie) and by the EU Integrated Infrastructure Initiative v2c,v2d 2 v2c,v2d ρ − 4(1 − u0) 16π Hadron Physics (I3HP) under contract RII3-CT-2004- with 506078. 4 BFTI = B ρTI,CTI + kTI. (52) ρTI u References In Table 5 we present our final results and compare tadpole improved and unimproved renormalisa- [1] R. Horsley, H. Perlt, P.E.L. Rakow, G. Schierholz, A. Schiller, tion constants. We see that the improved coefficients Nucl. Phys. B 693 (2004) 3; B are rather small in the case of the operators v2a and R. Horsley, H. Perlt, P.E.L. Rakow, G. Schierholz, A. Schiller, v2b, much smaller than for Wilson and clover fermi- Nucl. Phys. B 713 (2005) 601, Erratum. ons [21], which raises hope that the perturbative series [2] R. Narayanan, H. Neuberger, Nucl. Phys. B 443 (1995) 305; R. Narayanan, H. Neuberger, Phys. Lett. B 302 (1993) 62. converges rapidly. This furthermore means that the [3] H. Neuberger, Phys. Lett. B 417 (1998) 141; dominant contribution to the renormalisation constants H. Neuberger, Phys. Lett. B 427 (1998) 353. is given by the mean field factor (38). [4] F. Niedermayer, Nucl. Phys. B (Proc. Suppl.) 73 (1999) 105. [5] T. DeGrand, A. Hasenfratz, T.G. Kovács, Phys. Rev. D 67 (2003) 054501. [6] D. Galletly, M. Gürtler, R. Horsley, B. Joó, A.D. Kennedy, 4. Summary H. Perlt, B.J. Pendleton, P.E.L. Rakow, G. Schierholz, A. Schiller, T. Streuer, Nucl. Phys. B (Proc. Suppl.) 129 (2004) We have computed the renormalisation constants 453. of one-link quark operators for overlap fermions and [7] M. Göckeler, A.S. Kronfeld, M.L. Laursen, G. Schierholz, tadpole improved Lüscher–Weisz action for two val- U.-J. Wiese, Phys. Lett. B 233 (1989) 192. ues of the coupling, β = 8.45 and 8.0, being used in [8] M. Lüscher, P. Weisz, Commun. Math. Phys. 97 (1985) 59; M. Lüscher, P. Weisz, Commun. Math. Phys. 98 (1985) 433, current simulations. The calculations have been per- Erratum. formed in general covariant gauge, using the symbolic [9] M. Lüscher, P. Weisz, Phys. Lett. B 158 (1985) 250. language Mathematica. This gave us complete control [10] M.G. Alford, W. Dimm, G.P. Lepage, G. Hockney, P.B. over the Lorentz and spin structure, the cancellation of Mackenzie, Phys. Lett. B 361 (1995) 87. infrared divergences, as well as the cancellation of 1/a [11] M. Gürtler, R. Horsley, V. Linke, H. Perlt, P.E.L. Rakow, G. Schierholz, A. Schiller, T. Streuer, Nucl. Phys. B (Proc. singularities. However, the price is high. In intermedi- Suppl.) 140 (2005) 707. 5 ate steps we had to deal with O(10 ) terms due to the [12] C. Gattringer, R. Hoffmann, S. Schaefer, Phys. Rev. D 65 complexity of the gauge field action. (2002) 094503. To improve the convergence of the perturbative se- [13] D. Galletly, M. Gürtler, R. Horsley, H. Perlt, P.E.L. Rakow, ries and to get rid of lattice artefacts, we have applied G. Schierholz, A. Schiller, T. Streuer, in preparation. [14] Y. Kikukawa, A. Yamada, Phys. Lett. B 448 (1999) 265. tadpole improvement to our results. This was done in [15] M. Ishibashi, Y. Kikukawa, T. Noguchi, A. Yamada, Nucl. two stages. In the first stage we improved the gluon Phys. B 576 (2000) 501. propagator, while in the second stage we improved [16] M. Göckeler, R. Horsley, E.-M. Ilgenfritz, H. Perlt, P.E.L. both gluon and quark propagators. Rakow, G. Schierholz, A. Schiller, Nucl. Phys. B 472 (1996) Results at other β values, ρ parameters (also in- 309. [17] M. Göckeler, R. Horsley, E.-M. Ilgenfritz, H. Perlt, P. Rakow, cluding other gauge field actions with up to six links) G. Schierholz, A. Schiller, Phys. Rev. D 54 (1996) 5705. can be provided on request. [18] M. Göckeler, R. Horsley, E.-M. Ilgenfritz, H. Perlt, P. Rakow, G. Schierholz, A. Schiller, Phys. Rev. D 53 (1996) 2317. [19] M. Göckeler, R. Horsley, D. Pleiter, P.E.L. Rakow, G. Schier- Acknowledgements holz, hep-ph/0410187. [20] G.P. Lepage, P.B. Mackenzie, Phys. Rev. D 48 (1993) 2250. [21] S. Capitani, M. Göckeler, R. Horsley, H. Perlt, P.E.L. Rakow, This work is supported by DFG under contract FOR G. Schierholz, A. Schiller, Nucl. Phys. B 593 (2001) 183. 465 (Forschergruppe Gitter-Hadronen-Phänomenolo- Physics Letters B 628 (2005) 73–84 www.elsevier.com/locate/physletb

Asymptotics of Feynman diagrams and the Mellin–Barnes representation

Samuel Friot a,b, David Greynat a, Eduardo de Rafael a,c,d

a Centre de Physique Théorique1, CNRS-Luminy, Case 907, F-13288 Marseille Cedex 9, France b Laboratoire de Mathématiques d’Orsay Université Paris-Sud2, Bât. 425, F-91405 Orsay Cedex, France c Grup de Física Teòrica and IFAE, Universitat Autònoma de Barcelona, 08193 Barcelona, Spain d Institució Catalana de Recerca i Estudis Avançats (ICREA), Spain Received 9 May 2005; received in revised form 23 August 2005; accepted 24 August 2005 Available online 20 September 2005 Editor: G.F. Giudice

Abstract It is shown that the integral representation of Feynman diagrams in terms of the traditional Feynman parameters, when combined with properties of the Mellin–Barnes representation and the so-called converse mapping theorem, provide a very simple and efﬁcient way to obtain the analytic asymptotic behaviours in both the large and small ratios of mass scales.  2005 Elsevier B.V. All rights reserved.

1. Introduction

The problem of extracting the asymptotic behaviour of Feynman diagrams, when ratios of mass scales in a diagram become large or small is an old one in quantum ﬁeld theory. Since the early days of quantum electrodynamics (QED), one has been confronted with it in almost every practical calculation of phenomenological interest. Although the discovery of dimensional regularization, together with the conceptual development of effective quantum ﬁeld theory, have triggered several systematic technical approaches to this problem, the practical evaluation of even a few asymptotic terms remains still a rather painful task. The purpose of this Letter is to present a new approach to this problem, which appears to be astonishingly simple and provides a new physical insight.

E-mail address: [email protected] (E. de Rafael). 1 Unité Mixte de Recherche (UMR 6207) du CNRS et des Universités Aix Marseille 1, Aix Marseille 2 et sud Toulon-Var, afﬁliée à la FRUMAM. 2 Unité Mixte de Recherche (UMR 8628) du CNRS.

The starting point is a systematic use of the old Feynman identity 1 1 1 = dα , (1.1) ab [αa + (1 − α)b]2 0 1 1 which allows one to combine the products of all the propagators a , b ,...in a Feynman graph into a single denominator where the internal circulating momenta kl appear in a quadratic form. The kl-loop momenta integrals can then be done (if necessary in dimension D = 4 − ) by an appropriate shift of the loop momenta, which eliminates the terms of first degree in kl, and then using standard diagonalization methods (see, e.g., Ref. [1] for details). One can thus reduce a Feynman diagram (or set of diagrams) to the evaluation of a few dimensionless integrals over a fixed number F of Feynman parameters of the type 1 1 1 N(α ) F(ρ ) = dα dα ··· dα i , (1.2) j 1 2 F [ ]n+(p/q) j Dj (αi)ρj 0 0 0 where the ρj denote scalar products of the external momenta and squared masses normalized to a fixed mass scale in the diagram, so that ρ0 = 1, and n + (p/q) > 0 with n, p and q positive integers. The difficulties, usually, start at this point. In order to illustrate the basic problem, let us consider the simplest case where there is only one ρ-parameter Dj (αi)ρj = D0(αi) + D1(αi)ρ, (1.3) j and we want the behaviour of the integral F(ρ) for ρ 1. The root of the problem lies in the fact that the naive Taylor expansion (for n = 1, 2,...), − − D (α ) n (p/q) D (α ) 1 + 1 i ρ = 1 − n + (p/q) 1 i ρ + O ρ2 , (1.4) D0(αi) D0(αi) is useless because, practically always, the successive terms in these expansions lead to divergent integrals in the Feynman parameters αi . It is largely due to this fact that various techniques, other than the use of Feynman parameters, have been developed.3 The basic feature of the approach that we shall present here is that it provides a regularization of the naive Taylor expansion in the ρ-parameter and at the same time it treats exactly the -dependence of each coefficient in the ρ-expansion. The regulator is provided by viewing F(ρ) as an appropriate inverse Mellin transform associated to the simple analytic dependence that the Feynman parametric integrand provides on the ρ-parameter. As we shall see, the asymptotic behaviours of F(ρ), both, for ρ 1 and ρ 1, are then encoded in the so-called converse mapping theorem [4] which establishes a precise relation between the singularities in the Mellin s-plane and the asymptotic behaviour(s) one is looking for. The Letter is organized as follows. In the next section we explain the basic features of the new approach that we propose as well as the converse mapping theorem. Section 3 illustrates this technique with the simple example of the lowest order vacuum polarization contribution from a light lepton (e) or a heavy lepton (τ ) to the muon anomaly. Section 4 discusses the calculation of a typical master integral which often appears in quantum field theory calculations. These examples offer the possibility of comparing our approach with other methods (much more tedious) which have been used previously in the literature to obtain asymptotic expansions of Feynman integrals. Finally, Section 5 is dedicated to a three-loop calculation involving two ratios of mass scales, where an alternative type of a Mellin–Barnes representation is used.

3 For recent reviews, see, e.g., Refs. [2,3] where other references can also be found. S. Friot et al. / Physics Letters B 628 (2005) 73–84 75

2. The Feynman–Mellin–Barnes representation

Let us pursue the study of the generic integral

1 1 1 F = ··· N(αi) = + (ρ) dα1 dα2 dαF ν ,νn (p/q), (2.1) [D0(αi) + D1(αi)ρ] 0 0 0 with only one mass-ratio ρ; i.e., the integral which we were left with in the introduction.4 We propose to apply the Mellin–Barnes representation [5] c+i∞ 1 1 − (s)(ν − s) = ds(X) s , (2.2) (1 + X)ν 2πi (ν) c−i∞ with the integration path along the imaginary axis deﬁned in the fundamental strip: c = Re(s) ∈]0,ν[,tothe integrand in Eq. (2.1). This results in a new integral representation

1 1 1 c+i∞ s − F = 1 ··· N(αi) −s D0(αi) (s)(ν s) (ρ) dα1 dα2 dαF ν ds ρ . (2.3) 2πi [D0(αi)] D1(αi) (ν) 0 0 0 c−i∞ The crucial property of this representation is that it decouples the dependence on the ρ-parameter from the one on the Feynman parameters αi . The integrals over the Feynman parameters αi are now those of a massless theory, and they are regularized by the s-dependence of the integrand (much the same as dimensional regularization does with integrals over the loop momenta). In fact, as we shall see later in a few examples, these integrals are often rather simple and in many cases they result in products of -functions. Altogether, we obtain an integral representation c+i∞ 1 − F(ρ) = dsρ sM[F](s), (2.4) 2πi c−i∞ where 1 1 1 s − M[F] = ··· N(αi) D0(αi) (s)(ν s) (s) dα1 dα2 dαF ν (2.5) [D0(αi)] D1(αi) (ν) 0 0 0 is the Mellin transform of the Feynman integral at the start in Eq. (2.1). In full generality, the function M[F](s) is a meromorphic function in the complex s-plane and, in perturbation theory, its singularities (poles and/or multiple poles) lie in the Re(s)-axis; outside of the fundamental strip.5 The position of the poles on the r.h.s. of the fundamental strip, their multiplicity and their residues encode the asymptotic behaviour of F(ρ) for ρ 1; the poles on the l.h.s. and their residues, those of the asymptotic behaviour of F(ρ) for ρ 1. The precise form of the encoding [7] is spelled out by a theorem which in the mathematical literature goes under the name of the converse mapping theorem (see Ref. [4] for a proof of this theorem), and which we next discuss.

4 Although we shall later discuss a particular example with two mass ratios, we postpone the study of the general case with multiple mass scales to a forthcoming publication. 5 In some cases, as we shall see later, the singular structure of the function M[F](s) associated to a speciﬁc Feynman integral may restrict the width of the fundamental strip. The case where no fundamental strip exists which, so far, we have not encountered in the quantum ﬁeld theory examples that we have analyzed, has also been considered in the mathematical literature [6]. 76 S. Friot et al. / Physics Letters B 628 (2005) 73–84

2.1. The converse mapping theorem

The theorem in question relates the singularities of the function M[F](s) in the complex s-plane to the asymptotic behaviour of the Feynman graph F(ρ) as follows:

• Right-hand side singularities ⇒ expansion for ρ →∞ With ξ(ν)∈ R and k ∈ N, the function M[F](s) in the r.h.s. of the fundamental strip has a singular expansion 6 of the type (ordered in increasing values of ξ):

a M[F](s) ξ,k . (2.6) (s − ξ)k+1 ξ k The corresponding asymptotic behaviour of F(ρ) for ρ large (ordered in increasing powers of ξ) is then:

− k+1 ( 1) −ξ k F(ρ) ∼ aξ,kρ log ρ. (2.7) ρ→∞ k! ξ k • Left-hand side singularities ⇒ expansion for ρ → 0 With ξ(ν)∈ R and k ∈ N, the function M[F](s) in the l.h.s. of the fundamental strip has a singular expansion of the type (ordered in increasing values of ξ):

b M[F](s) ξ,k . (2.8) (s + ξ)k+1 ξ k The corresponding asymptotic behaviour of F(ρ) (ordered in increasing powers of ξ) is then:

− k ( 1) ξ k F(ρ) ∼ bξ,kρ log ρ. (2.9) ρ→0 k! ξ k

The basic steps which we propose are: (i) the old-fashioned Feynman parameterization, with explicit integration over the loop momenta, and only then to use a conveniently chosen Mellin–Barnes representation; (ii) the factorized Feynman parametric integrals, which are then s-regularized and ρ-independent, appear to be remarkably simple; (iii) the use of the converse mapping theorem which encodes the relation between the singular behaviour in the Mellin s-plane and the full asymptotic behaviours of the initial Feynman integral for small or large ratios of mass parameters.7 These asymptotic series can be obtained without the explicit knowledge of the exact analytic expressions. It is also possible, if necessary, to extract an arbitrary term in those series by looking at the residues of the appropriate pole (or multiple pole) at the right or at the left of the fundamental strip, without having to do the calculation of the corresponding lower order terms. The rest of this Letter is dedicated to a few interesting examples which illustrate the simplicity of this approach; but there are many other possible applications one can think of which we invite the reader to consider.

6 The singular expansion (or singular series) of a meromorphic function is a formal series collecting the singular elements at all poles of the function (a singular element being the truncated Laurent’s series (at O(1)) of the function at a given pole) and it is denoted by the symbol [4]. 7 To our knowledge, the use of the Mellin–Barnes representation in quantum ﬁeld theory was ﬁrst proposed in Refs. [8,9]. Most of the recent calculations in the literature which use this technique (see, e.g., Ref. [10] and Refs. [2,3]) introduce a Mellin–Barnes representation for each free propagator in a diagram. Calculations in an approach similar to the one we propose here can be found, e.g., in Refs. [11,12]. None of these references, however, uses the rigorous short-cut provided by the converse mapping theorem. S. Friot et al. / Physics Letters B 628 (2005) 73–84 77

3. Vacuum polarization contributions to the muon anomaly

Historically, this is the ﬁrst example where the problem of extracting an asymptotic behaviour in the ratio of two masses in quantum ﬁeld theory appeared. The relevant Feynman graphs are shown in Fig. 1, where the internal fermion l in the vacuum polarization loop can either be an electron (large ratio mµ/me)oraτ (small ratio 8 mµ/mτ ). Their contribution to the muon anomaly is known analytically [13]. Here we are only considering these contributions once more, as an illustration of the simplicity of our approach. For the purpose we are concerned with here, it is convenient to start with the Feynman parameterization [14,16] (which already takes into account the on-shell renormalization of the vacuum polarization subgraph):

1 1 2 v.p. α dx 1 aµ = (1 − x)(2 − x) dy y(1 − y) . (3.1) m2 π x + l 1−x 1 1 2 2 − 0 0 mµ x y(1 y) This is a particular case of an integral like the one in Eq. (2.1) corresponding to the case of two Feynman parameters m2 = = = l = with α1 x, α2 y, ρ 2 and ν 1. mµ The Mellin–Barnes representation in Eq. (2.2), when applied to the last factor in the integrand in Eq. (3.1),gives

+ ∞ c i − 1 1 m2 s x2 s = ds l y(1 − y) (s)(1 − s). (3.2) m2 2 − + l 1−x 1 2πi mµ 1 x 1 2 2 − − ∞ mµ x y(1 y) c i [ = (u)(v)] The integrals over the Feynman parameters reduce now to simple Beta functions B(u, v) (u+v) . Altogether, we have the simple representation

+ ∞ c i − 1 m2 s av.p. = ds l M av.p. (s), µ 2 µ (3.3) 2πi mµ c−i∞ with (1 − s)[(s)(1 − s)]2 M av.p. (s) = , (3.4) µ (2 + s)(1 + 2s)(3 + 2s) the function which encodes the asymptotic behaviours we are looking for. v.p. As illustrated in Fig. 2, the singularities of M[aµ ](s) in the r.h.s. of the fundamental strip: c = Re(s) ∈]0, 1[, appear as a pole at s = 1 and as double and single poles at s = n for n = 2, 3,.... The nearer singularities to the fundamental strip, from the right, ﬁx the leading terms in the asymptotic expansion when ml mµ and they can be obtained by simple inspection of Eq. (3.4). We cannot refrain, however, from giving the explicit form of the full

Fig. 1. Vacuum polarization contributions to the muon anomaly from a small internal mass (electron) and from a large internal mass (tau).

8 The history of vacuum polarization contributions can be traced back from the review articles in Refs. [14,15]. 78 S. Friot et al. / Physics Letters B 628 (2005) 73–84

v.p. Fig. 2. Singularities of the function M[aµ ](s) (see Eq. (3.4))inthecomplexs-plane. singular series, which can be obtained rather simply:

∞ 2 3 v.p. 1 −45 + 28n + 8n 1 (−n) M aµ (s) + . s − (n + 1) [(3 + n)(3 + 2n)(5 + 2n)]2 [s − (n + 1)]2 (3 + n)(3 + 2n)(5 + 2n) n=0 (3.5) According to the converse mapping theorem this singular series encodes the asymptotic expansion9: ∞ + α 2 m2 n 1 45 − 28n2 − 8n3 n m2 av.p. ∼ µ + τ , µ 2 2 ln 2 (3.6) mτ mµ π m [(3 + n)(3 + 2n)(5 + 2n)] (3 + n)(3 + 2n)(5 + 2n) m n=0 τ µ a result which allows for an immediate evaluation of as many terms as one wishes in the asymptotic expansion. v.p. The singularities of M[aµ ](s) at the l.h.s. of the fundamental strip: c = Re(s) ∈]0, 1[, appear as single poles at s =−1/2 and s =−3/2; a triple, double and single pole at s =−2; and double and single poles at s = 0, s =−1 and s =−n for n = 3, 4,.... Again, it is rather easy to obtain the corresponding singular series: 2 2 2 v.p. π 1 5π 1 1 7 1 44 π 1 M aµ (s) − + + + + 4 + 1 4 + 3 (s + 2)3 3 (s + 2)2 9 3 s + 2 s 2 s 2 ∞ 1 + n 1 −25 + 32n + 4n2 − 8n3 1 + + , (3.7) (2 − n)(1 − 2n)(3 − 2n) (s + n)2 (2 − n)2(1 − 2n)2(3 − 2n)2 s + n n=0 n=2 which according to the converse mapping theorem encodes the asymptotic expansion10 α 2 1 m2 25 π 2 m m2 m2 5π 2 m 3 av.p. ∼ ln µ − + e + e −2ln µ + 3 − e µ 2 2 2 me mµ π 6 me 36 4 mµ mµ me 4 mµ 2 2 2 2 2 + me 1 2 me − 7 me + 44 + π 2 ln 2 ln 2 mµ 2 mµ 3 mµ 9 3 ∞ m2 n 1 + n m2 −25 + 32n + 4n2 − 8n3 + e ln µ + . (3.8) m2 (2 − n)(1 − 2n)(3 − 2n) m2 (2 − n)2(1 − 2n)2(3 − 2n)2 n=3 µ e

9 The leading term of this expansion can be physically understood [17] as the convolution of the slope of the vacuum polarization at the origin with the contribution to the muon anomaly from a heavy-photon. 10 The leading log-term in this expansion is in fact ﬁxed by simple renormalization group arguments [19]. S. Friot et al. / Physics Letters B 628 (2005) 73–84 79

The results in Eqs. (3.6) and (3.8) agree with the asymptotic expansions given in Ref. [18], obtained from the exact analytic expression in [13]. The new feature which we have shown here, with this well-known example, is how a full asymptotic series can be obtained, directly and in a simple way, without knowledge of the exact analytic expression. The reader can judge the simplicity of our method, by comparing it to other ways that a few terms of the same asymptotic expansions have been obtained in the literature (see, e.g., Ref. [20], where the so-called method of regions [21] is used).

4. Calculation of a master integral

The next example we wish to discuss concerns the integral 1 1 J p2,m2,t = dDk dDk , (4.1) 2D 1 2 2 − [ − 2 − 2] − 2 (2π) (k1 t) (k1 k2) m (k2 p) in dimensional regularization D = 4 − . This is a typical master integral which appears in many two loop calculations as a result of a systematic use of recurrence relations obtained from integrations by parts in D-dimensions.11 Here we shall be concerned with the case where p2 = m2 and the dependence of the integral on the massless ratio = m2 ρ t for ρ 1. In fact, it is precisely the calculation of this integral which triggered our ideas on the approach that we are advocating here. A standard Feynman parameterization of the three propagators in Eq. (4.1) results, after integration over the loop momenta k1 and k2, in a two-dimensional Feynman integral which, a priori, looks rather complicated. We split it into two terms, so as to reduce it to integrals of the generic type in Eq. (2.1): i 2 () µ2 m2 m2 m2 J m2,m2,t = (4π) t F (1) + F (2) , (4.2) 16π 2 1 − t t t t with [Y = 1 − y(1 − y)]

1 1 − − −2+ m2 x 2 y1 (1 − xY) 2 F (1) = dx dy , (4.3) 2 − 2 − − t 1 + m (1 y) 1 x(1 y) 0 0 t y 1−xY and

1 1 − − −3+ m2 x 2 y (1 − xY) 2 (1 − y)2[1 − x(1 − y)] F (2) = dx dy . (4.4) 2 − 2 − − t 1 + m (1 y) 1 x(1 y) 0 0 t y 1−xY Following the methodology explained in Section 2, we apply the Mellin–Barnes representation in Eq. (2.2) to the denominator of the integrands in Eqs. (4.3) and (4.4):

+ ∞ c i − − 1 1 m2 s (1 − y)2 1 − x(1 − y) s (s)( − s) = ds , (4.5) 2 (1−y)2 1−x(1−y) − 1 + m 2πi t y 1 xY () t y 1−xY c−i∞

F (1) m2 F (2) m2 and next consider the two integrals ( t ) and ( t ) separately.

11 Examples of such calculations can be found, e.g., in Refs. [22,23] and references therein. 80 S. Friot et al. / Physics Letters B 628 (2005) 73–84

• F (1) m2 The integral ( t ) F (1) m2 The integral over the Feynman x-variable in ( t ) can be done in terms of a Gauss hypergeometric function which, for convergence purposes, keeping in mind the fact that we still have to integrate over the y-variable in the range [0, 1], we choose to express in the form [24]

1 − − − − −s −2+s+ −1 1 , 2 s dxx 2 1 − x(1 − y) (1 − xY) 2 = B 1, 1 − y 2 F 2 2 y . (4.6) 2 1 − 2 2 2 0 Expanding this hypergeometric function in powers of y, one can do the y-integration term by term. The resummation of the series leads then to a simple product of -functions [24]. The overall result is a simple F (1) m2 Mellin–Barnes representation for the ( t ) integral + ∞ c i − m2 1 m2 s F (1) = ds M F (1) (s), (4.7) t 2πi t c−i∞ with M[F (1)](s) deﬁned by the product of -functions: (1 − ) (s)( − s)(1 − 2s)(1 + s − )( − s) M F (1) (s) = 2 2 2 . − − − (4.8) () (2 s 2 )(1 s) − Notice that, because of the presence of the (2 s) factor, which appears as a result of the integration over the Feynman parameters, the fundamental strip in the Mellin–Barnes representation in Eq. (4.7) is now restricted = ∈] [ to c Re(s) 0, 2 . • F (2) m2 The integral ( t ) F (1) m2 A procedure entirely similar to the one just described for the function ( t ) leads to the corresponding representation

+ ∞ c i − m2 1 m2 s F (2) = ds M F (2) (s), (4.9) t 2iπ t c−i∞ with M[F (2)](s) deﬁned by the product of -functions: (1 − ) (s)( − s)(s − )(3 − 2s)(1 + − s) M F (2) (s) = 2 2 2 , − − − (4.10) () (3 s 2 )(2 s) = ∈] [ and the fundamental strip in the Mellin–Barnes representation in Eq. (4.9) restricted to c Re(s) 2 , .

M[F (1)] = ∈] [ The singularities of (s) in Eq. (4.8) in the l.h.s. of the fundamental strip c Re(s) 0, 2 , appear as a =− =− − + ∈ N double series of simple poles at s n and s n 1 2 , n , with residues given by the singular expansion ∞ (1 − ) (−1)n ( + n)(1 + 2n)(1 − n − )( + n) 1 M F (1) (s) 2 2 2 ! + − + + () = n (2 n 2 )(1 n) s n n 0 ∞ (−n − 1 + )(n + 1 + )(3 + 2n − ) 1 + (−1)n 2 2 . (4.11) (3 + n − )(2 + n − ) s + n + 1 − n=0 2 2 S. Friot et al. / Physics Letters B 628 (2005) 73–84 81

The converse mapping theorem associates to this singular series the asymptotic expansion ∞ 2 (1 − ) − n ( + n)(1 + 2n)(1 − n − )( + n) 2 n F (1) m ∼ 2 ( 1) 2 2 m t m2t () n! (2 + n − )(1 + n) t n=0 2 ∞ n+1− (−n − 1 + )(n + 1 + )(3 + 2n − ) m2 2 + (−1)n 2 2 . (4.12) (3 + n − )(2 + n − ) t n=0 2 M[F (2)] = ∈] [ The singularities of (s) in Eq. (4.10) in the l.h.s. of the fundamental strip c Re(s) 2 , , appear as =− =− + ∈ N a double series of simple poles at s n and s n 2 , n , with residues given by the singular expansion ∞ (1 − ) (−1)n ( + n)(−n − )(3 + 2n)(1 + + n) 1 M F (2) (s) 2 2 2 () n! (3 + n − )(2 + n) s + n n=0 2 ∞ (−n + )( + n)(3 + 2n − ) 1 + (−1)n 2 2 , (4.13) (3 + n − )(2 + n − ) s + n − n=0 2 2 with which, the converse mapping theorem associates the asymptotic behaviour ∞ 2 (1 − ) − n ( + n)(−n − )(3 + 2n)(1 + + n) 2 n F (2) m ∼ 2 ( 1) 2 2 m t m2t () n! (3 + n − )(2 + n) t n=0 2 ∞ n− (−n + )( + n)(3 + 2n − ) m2 2 + (−1)n 2 2 . (4.14) (3 + n − )(2 + n − ) t n=0 2 The results in Eqs. (4.12) and (4.14), when incorporated in Eq. (4.2) give the full asymptotic expansion of the 2 2 m2 → master integral J(m ,m ,t) for t 0, with coefﬁcients which have a full exact dependence in the -variable. Again, the relative simplicity of this calculation using our approach, should be compared to the complicated expressions obtained by other methods in the literature [25]. We have checked that our results coincide with those given in Ref. [23] for the terms of O(1/2) and O(1/), and in Ref. [26] for the term of O(0), when their corresponding expressions are expanded for m2 t.

5. Three loop calculation

In this last example, we shall consider the contribution to the muon anomaly from the vacuum polarization correction to the photon propagator induced, simultaneously, by an electron loop and a τ loop; i.e., the Feynman diagrams in Fig. 3. This contribution is given by the Feynman parametric integral: 1 α −x2 −x2 av.p.(Fig. 3) = dx(1 − x)2 −e2Π (e) m2 −e2Π (τ) m2 , (5.1) µ π R 1 − x µ R 1 − x µ 0 where 1 −x2 α x2 m2 (−ie)2Π (l) m2 =− dz 2z (1 − z ) ln 1 + µ z (1 − z ) , (5.2) R − µ l l l − 2 l l 1 x π 1 x ml 0 is a convenient one-Feynman parametric representation of the on-shell renormalized photon self-energy generated by a lepton l, at the one-loop level. 82 S. Friot et al. / Physics Letters B 628 (2005) 73–84

Fig. 3. Vacuum polarization contributions to the muon anomaly induced, simultaneously, by an electron-loop and a τ -loop.

We have here a slightly more complicated problem, with three Feynman parameters x, ze and zτ and two ρ-like 2 2 mµ mµ parameters: 2 1, and 2 1. In this case, it is better to apply another type of Mellin–Barnes representation me mτ to each log-factor in each photon self-energy as follows (see, e.g., Ref. [27]):

c+i∞ 2 2 −s m 1 m − π + µ X = ds µ [X ] s , ln 1 2 l 2 l (5.3) ml 2πi ml s sin(πs) c−i∞ where, here, c = Re(s) ∈]−1, 0[ defines the fundamental strip of the corresponding integration path along the imaginary axis. This is a very useful Mellin–Barnes representation which also factorizes the dependence on the mass-ratio in the log-factor and which, to our knowledge, has not been used before in quantum field theory calcu- v.p. lations. The resulting Feynman parametric integrals can then be done trivially. This way, one finds that aµ (Fig. 3) has a double Mellin–Barnes integral representation: + ∞ + ∞ cs i − ct i − α 3 1 m2 s 1 m2 t av.p. = ds µ dt µ M av.p. (s, t), µ 2 2 µ (5.4) π 2πi me 2πi mτ cs −i∞ ct −i∞ with (s)(1 − s) (t)(1 − t) [(2 − s)(2 − t)]2 (1 − 2s − 2t)(2 + s + t) M av.p. (s, t) = 8 . (5.5) µ s t (4 − 2s)(4 − 2t) (3 − s − t)

Here we are interested in the simultaneous singularities at the right of the fundamental strip cs = Re(s) ∈]−1, 0[ 2 2 = ∈]− [ (which encode the mµ me expansion) and at the left of the fundamental strip ct Re(t) 1, 0 (which encode 2 2 the mµ mτ expansion). One can then proceed as follows. First we evaluate the singular series in the t-variable, for ﬁxed s, and apply the converse mapping theorem to that series with the result + ∞ cs i − − + α 3 1 m2 s m2 n m2 s m2 n 2 av.p. ∼ ds µ A (s) µ + τ B (s) µ , µ 2 n 2 2 n 2 (5.6) mµmτ π 2πi m m m m e n1 τ e n0 τ cs −i∞

2 mτ −s B + + where the ( 2 ) factor in the -series has been induced by the singular expansion of the (2 s t) factor in me Eq. (5.5) which generates the shift: − − − + + − + m2 s m2 t m2 s m2 s n 2 m2 s m2 n 2 µ µ ⇒ µ µ = τ µ , 2 2 2 2 2 2 (5.7) me mτ me mτ me mτ and (−1)n+1 (2+n)2 (1−2s+2n)(2+s−n) A (s) (s)(1 − s)(2 − s)2 + − + n = 8 n (4 2n) (3 s n) . (5.8) n+1 2 Bn(s) s(4 − 2s) (−1) (−2−n−s)(3+n+s)(4+n+s) (5+2n) n! (s+n+2)(8+2n+2s)(5+n) S. Friot et al. / Physics Letters B 628 (2005) 73–84 83

The problem has now been reduced to one of the type already discussed in the previous sections. In fact, even in this more complex case, we have been able to ﬁnd the exact form of the full asymptotic expansion [28,29]. In practice, however, one is only interested in a few terms in that expansion. Applying once more the converse mapping theorem, one can immediately see that the singular behaviour: 2 1 1 1 A1(s) ∼ + , (5.9) s→0 135 s2 135 s 2 2 O mµ O mµ 2 controls the leading terms of ( 2 ). Terms of ( 2 ) have their source in the singular behaviours: mτ mτ 1 1 1 1 π 2 47 1 A2(s) ∼ − + − − , (5.10) s→0 210 s3 504 s2 630 30240 s 1 1 37 1 39379 1 B0(s) ∼ − + . (5.11) s→0 210 s3 22050 s2 2315250 s 2 O mµ 3 Terms of ( 2 ) have their source in the singular behaviours: mτ 4 1 1 1 4π 2 11 1 A3(s) ∼ − − − − , (5.12) s→0 945 s3 4725 s2 2835 94500 s 4 1 199 1 2735573 1 B1(s) ∼ − + , (5.13) s→0 945 s3 297675 s2 187535250 s and so on. The next singularity at the right of the fundamental strip of A1(s) is at s → 1: 2 1 A1(s) ∼ − , (5.14) s→1 15 s − 1 2 m2 O[ me µ ] − + which generates a term of ( 2 )( 2 ) . Because of the factor (1 2s 2n) in Eq. (5.8), the next-to-next mµ mτ singularity of A1(s) at the right of the fundamental strip is at s → 3/2: 4π 2 1 A1(s) ∼ , (5.15) s→3/2 45 − 3 s 2 2 m2 O[ me 3/2 µ ] which generates a term of ( 2 ) ( 2 ) , and so on. mµ mτ From these results one can reconstruct, in a straightforward way, the asymptotic behaviour: α 3 m2 2 m2 1 2 m2 4π 2 m3 av.p. ∼ µ ln µ − + e − e µ 2 2 2 2 me mµ mτ π m 135 m 135 15 m 45 m mµ τ e τ τ m2 2 2 m2m2 2 m2 2 + µ − 1 mτ τ µ − 37 mτ + 1 µ + π − 229213 2 ln 2 ln 4 ln 2 ln 2 mτ 420 mµ me 22050 me 504 me 630 12348000 m2 3 2 m2m2 2 m2 2 + µ − 2 mτ τ µ − 199 mτ − 1 µ + 4π − 1102961 2 ln 2 ln 4 ln 2 ln 2 mτ 945 mµ me 297675 me 4725 me 2835 75014100 m2 4 2 m2m2 2 m2 2 + µ − 1 mτ τ µ − 391 mτ − 19 µ + π 2 ln 2 ln 4 ln 2 ln 2 mτ 594 mµ me 2058210 me 31185 me 891 161030983 m2 5 m2 m2 m2 m2 m2 − + O µ τ τ µ + O e µ . 2 ln 2 ln 4 2 2 (5.16) 14263395300 mτ mµ me mτ mτ 84 S. Friot et al. / Physics Letters B 628 (2005) 73–84

The ﬁrst three lines are in agreement with the result obtained in Ref. [20] using the method of regions [21].The terms in the fourth line are new; we have incorporated them just to show how easily higher order terms can be obtained, if necessary. By contrast, the method of regions requires in this case the separate consideration of ﬁve different integration regions in the virtual loop-momenta with their appropriate Taylor expansions of propagators, plus the evaluation of a large number of integrals. That complexity is to be compared with the simplicity of the approach reported here.

Acknowledgements

We are grateful to Marc Knecht and Santi Peris for discussions and suggestions. This work has been supported in part by TMR, EC-Contract No. HPRN-CT-2002-00311 (EURIDICE).

References

[1] R.J. Eden, P.V. Landshoff, D.I. Olive, J.C. Polkinghorne, The Analytic S-Matrix, Cambridge Univ. Press, Cambridge, 1966. [2] V.A. Smirnov, Applied Asymptotic Expansions in Momenta and Masses, Springer Tracts in Modern Physics, Springer, Berlin, 2001. [3] V.A. Smirnov, Evaluating Feynman Integrals, Springer Tracts in Modern Physics, Springer, Berlin, 2004. [4] Ph. Flajolet, X. Gourdon, Ph. Dumas, Theor. Comput. Sci. 144 (1995) 3. [5] E.T. Whittaker, G.N. Watson, A Course of Modern Analysis, Cambridge Univ. Press, Cambridge, 1965. [6] N. Bleistein, R. Handelsman, Asymptotic Expansions of Integrals, Holt, Rinehart and Winston, New York, 1975. [7] G. Doetsch, Handbuch der Laplace Transformation, vols. 1–3, Birkhäuser, Basel, 1955. [8] J.D. Bjorken, T.T. Wu, Phys. Rev. 132 (1963) 2566. [9] T.L. Trueman, T. Yao, Phys. Rev. 132 (1963) 2741. [10] E.E. Boos, A.I. Davydychev, Theor. Math. Phys. 89 (1991) 1052. [11] Ch. Greub, T. Hurth, D. Wyler, Phys. Rev. D 54 (1996) 3350. [12] H.H. Asatryan, H.M. Asatrian, C. Greub, M. Walker, Phys. Rev. D 65 (2004) 074004. [13] H.H. Elend, Phys. Lett. 20 (1966) 682; H.H. Elend, Phys. Lett. 21 (1966) 720, Erratum. [14] B.E. Lautrup, A. Peterman, E. de Rafael, Phys. Rep. C 3 (1972) 193. [15] M. Passera, J. Phys. G 31 (2005) R75. [16] B.E. Lautrup, E. de Rafael, Nuovo Cimento 64A (1969) 322. [17] B.E. Lautrup, E. de Rafael, Phys. Rev. 174 (1968) 1835. [18] M.A. Samuel, G. Li, Phys. Rev. D 44 (1991) 3935; M.A. Samuel, G. Li, Phys. Rev. D 48 (1993) 1879, Erratum. [19] B.E. Lautrup, E. de Rafael, Nucl. Phys. B 70 (1974) 317. [20] A. Czarnecki, M. Skrzypek, Phys. Lett. B 449 (1999) 354. [21] M. Beneke, V.A. Smirnov, Nucl. Phys. B 522 (1998) 321. [22] D.J. Broadhurst, J. Fleischer, O.V. Tarasov, Z. Phys. C 60 (1993) 287. [23] M. Caffo, H. Czy˙z, S. Laporta, E. Remiddi, Nuovo Cimento A 111 (1998) 365. [24] P. Appell, J. Kampé de Fériet, Fonctions Hypergéométriques et Hypersphériques-Polynômes d’Hermite, Gauthier–Villars, Paris, 1926. [25] F. Jegerhehner, M.Yu. Kalmykov, Nucl. Phys. B 676 (2004) 365. [26] H. Czy˙z, A. Grzelinska,´ R. Zabawa, Phys. Lett. B 538 (2002) 52. [27] A. Erdélyi, W. Magnus, F. Oberhettinger, F.G. Tricomi, Tables of Integral Transforms, vol. 1, McGraw–Hill, New York, 1954. [28] S. Friot, PhD Thesis, 2005, in preparation. [29] D. Greynat, PhD Thesis, 2005, in preparation. Physics Letters B 628 (2005) 85–92 www.elsevier.com/locate/physletb

Naturally light right-handed neutrinos in a 3–3–1 model

Alex G. Dias a,C.A.deS.Piresb, P.S. Rodrigues da Silva b

a Instituto de Física, Universidade de São Paulo, Caixa Postal 66.318, 05315-970 São Paulo, SP, Brazil b Departamento de Física, Universidade Federal da Paraíba, Caixa Postal 5008, 58059-970 João Pessoa, PB, Brazil Received 19 August 2005; received in revised form 13 September 2005; accepted 15 September 2005 Available online 27 September 2005 Editor: G.F. Giudice

Abstract In this work we show that light right-handed neutrinos, with mass in the sub-eV scale, is a natural outcome in a 3–3–1 model. By considering effective dimension ﬁve operators, the model predicts three light right-handed neutrinos, weakly mixed with the left-handed ones. We show also that the model is able to explain the LSND experiment and still be in agreement with solar and atmospheric data for neutrino oscillation.  2005 Elsevier B.V. All rights reserved.

PACS: 14.60.St; 13.15.+g; 14.60.Pq; 12.60.Cn

Keywords: Neutrino mass and mix; Sterile neutrinos; 3–3–1 model

1. Introduction

With the exception of non-zero neutrino mass and mixing, all the other collected experimental data in particle physics are consistent with the predictions of the standard model of the electroweak and strong interactions [1]. Concerning neutrinos, the understanding of the smallness of their masses and the largeness of their mixing, dictated by the neutrino oscillations experiments [2,3], is a real puzzle in particle physics at the present. On the theoretical side, if only left-handed neutrinos exist, then the most economical way they can acquire small masses is through the effective dimension-ﬁve operator [4]

f L = ab Φ LC (Φ L ) + h.c., (1) MP Λ l am n bp lm np

E-mail address: cpires@ﬁsica.ufpb.br (C.A. de S. Pires).

+ 0 T T 0 where Φ = (φ ,φ ) and La = (νaL,eaL) . Thus when φ develops a non-zero vacuum expectation value (VEV), v, left-handed neutrinos automatically develop Majorana mass f v2 (M ) (ν )Cν , with (M ) = ab . (2) L ab aL bL L ab Λ On supposing that such effective operator is realized in some high energy GUT scale, then we naturally have light masses for the left-handed neutrinos. It could be that right-handed neutrinos also exist and they could be light too, but this feature is not naturally obtained in the standard model. For instance, to generate light right-handed neutrinos in the standard model, by effective dimension-ﬁve operators or by any other mechanism, an intricate combination of symmetries is required, usually accompanied by a considerable increasing in the particle content [5]. On the other hand, in the standard model, light right-handed neutrinos are interesting only if they can explain the LSND experiment [6]. This requires that the light right-handed neutrinos get weakly mixed with the left-handed ones. People usually refer to these light and weakly mixed right-handed neutrinos as sterile neutrinos. In this Letter we examine the problem of generating light right-handed neutrino masses in the framework of the 3–3–1 model where those neutrinos are already part of the spectrum. Interesting enough, we show that light right-handed neutrino masses are a natural outcome in this model. Our approach to the subject is through effective dimension-ﬁve operators. Basically, we construct all the effective operators allowed by the symmetries and particle content of the model and show that they yield light Majorana and Dirac mass terms for the neutrinos. Consequently, we will have three light active neutrinos and three light sterile ones,1 which makes this 3–3–1 model capable of easily explaining the LSND experiment.

2. The model

The model we consider is the 3–3–1 model with right-handed neutrinos [7,8]. It is one of the possible models allowedbytheSU(3)C ⊗ SU(3)L ⊗ U(1) gauge symmetry where the fermions are distributed in the following representation content. Leptons come in triplets and singlets ν aL 1 La = eaL ∼ 1, 3, − ,eaR ∼ (1, 1, −1), (3) c 3 (νaR) where a = 1, 2, 3 refers to the three generations. After the spontaneous breaking of the 3–3–1 symmetry to the T standard symmetry, the triplet above splits into the standard lepton doublet La = (νaL,eaL) plus the singlet C (νaR) . Thus this model recovers the standard model with right-handed neutrinos. It is not a trivial task to generate light masses to the right-handed neutrinos in any simple extension of the standard model. However, in the 3–3–1 model in question, right-handed neutrinos can naturally obtain small masses through effective dimension-ﬁve operators. This is due, in part, to the fact that, in the model, the right-handed neutrinos compose, with the left-handed neutrinos, the same triplet L. As we will see in the next section, it is this remarkable feature that turns feasible the raise of light right-handed neutrinos. In the quark sector, one generation comes in the triplet and the other two compose an anti-triplet with the following content, di = − ∼ ¯ ∼ 2 ∼ −1 ∼ −1 QiL ui (3, 3, 0), uiR 3, 1, ,diR 3, 1, ,diR 3, 1, , 3 3 3 di L

1 Although we refer to those neutrinos as sterile neutrinos, we call the attention to the fact that in the framework of the 3–3–1 model those neutrinos are not sterile because they interact with the active neutrinos and the charged leptons as showed in Eqs. (27) and (28). A.G. Dias et al. / Physics Letters B 628 (2005) 85–92 87 u3 = ∼ 1 ∼ 2 ∼ −1 ∼ 2 Q3L d3 3, 3, ,u3R 3, 1, ,d3R 3, 1, ,u3R 3, 1, , (4) 3 3 3 3 u3 L where i = 1, 2. The primed quarks are the exotic ones but with the usual electric charges. In the gauge sector, the model recovers the standard gauge bosons and disposes of ﬁve other gauge bosons called V ±, U 0, U 0† and Z [7,8]. Also, the model possesses three scalar triplets, two of them transforming as, η ∼ (1, 3, −1/3) and χ ∼ (1, 3, −1/3) and the other as, ρ ∼ (1, 3, 2/3), with the following vacuum structure [7] v 1 η 1 0 1 0 η0 = √ 0 , ρ0 = √ vρ , χ0 = √ 0 . (5) 2 2 2 0 0 vχ These scalars are sufﬁcient to engender spontaneous symmetry breaking and generate the correct masses for all massive particles. In order to have the minimal model, we assume the following discrete symmetry transformation over the full Lagrangian →− (χ,η,ρ,eaR,uaR,u3R,daR,diR) (χ,η,ρ,eaR,uaR,u3R,daR,diR), (6) where a = 1, 2, 3 and i = 1, 2. This discrete symmetry helps in avoiding unwanted Dirac mass term for the neutrinos [7] and implies a realistic minimal potential [9]. With this at hand, the model ends up with the following Yukawa interactions, LY = 1 ¯ ∗ + 2 ¯ + 3 ¯ ∗ + 4 ¯ + 5 ¯ ∗ λij QiLχ djR λ33Q3Lχu3R λiaQiLη daR λ3aQ3LηuaR λiaQiLρ uaR + 6 ¯ + ¯ + λ3aQ3LρdaR GabfaLρebR h.c., (7) which generate masses for all fermions, with the exception of neutrinos.

3. Neutrino masses

In this section we construct all possible effective dimension-five operators in the 3–3–1 model with right-handed neutrinos that lead to neutrino masses. The first one involves the triplets L and η. With these triplets we can form the following effective dimension-five operator f ∗ L = ab LCη η†L + h.c. (8) ML Λ a b 0 According to this operator, when η develops a VEV, vη, the left-handed neutrinos develop Majorana mass terms with the same form as in Eq. (2) but now with 2 fabv (M ) = η . (9) L ab Λ Due to the fact that right-handed neutrinos are not singlets in the model in question, a second effective dimension-five operator generating neutrino masses is possible. It is constructed with the scalar triplet χ and the lepton triplet L, h ∗ L = ab LCχ χ†L + h.c. (10) MR Λ a b 0 When χ develops a VEV, vχ , this effective operator provides Majorana masses for the right-handed neutrinos, 2 habv (M ) (ν )Cν , with (M ) = χ . (11) R ab aR bR R ab Λ 88 A.G. Dias et al. / Physics Letters B 628 (2005) 85–92

Remarkably, a third effective dimension-five operator generating neutrino mass is possible, but now involving the scalar triplets η and χ, g ∗ L = ab LCχ η†L + h.c., (12) MD Λ a b which, remarkably, leads to the following Dirac mass term for the neutrinos, gabv vη (M ) ν¯ ν , with (M ) = χ . (13) D ab aR bL D ab Λ Thus, we have Majorana and Dirac mass terms for the neutrinos both having the same origin, i.e., effective dimension-five operators. As in the standard case, on supposing that the three effective dimension-five operators above are realized in some high energy GUT scale, we have thus light Dirac and Majorana mass terms. In view of these neutrino mass terms, the usual manner of proceeding here is to arrange ML, MR and MD in the following 6 × 6 matrix, C ¯ M νL νL , νR C , (14) νR C = C C C in the basis (νL,νR ) (νeL,νµL,ντL,νeR,νµR,ντR), with M = ML MD T . (15) MD MR At this point, two comments are in order. First, as the VEV vχ is responsible for the breaking of the 3–3–1 symmetry to the standard symmetry, and that vη contributes to the spontaneous breaking of the standard symmetry, thus it is natural to expect that vχ >vη, which implies MR >MD >ML. This hierarchy among MR, MD and ML leads to a feeble mixing among the left- and right-handed neutrinos, characterizing the last as sterile neutrinos required to explain LSND experiment. Second, the model leads inevitably to three sterile neutrinos. C In order to check this, let us consider the case of one generation. In the basis (νeL νeR) we have the mas matrix 1 fv2 gv v η η χ . (16) 2 Λ gvηvχ hvχ By diagonalizing this matrix for vχ >vη, we obtain the eigenvalues 2 2 fh− g2 v v η ,hχ , (17) h Λ Λ and the correspondent eigenvectors − 2 − 2 fh g vη C C fh g vη N1 = νeL + (νR) ,N2 = (νR) − νeL. (18) gh vχ gh vχ

We see that the magnitude of the mixing is basically established by the VEVs vη and vχ through the ratio vη/vχ . 2 3 The typical values of such VEVs are vη ≈ 10 GeV and vχ ≈ 10 GeV. This leads to an active–sterile mixing of order of 10−1 which falls in the expected range of values required to explain LSND as discussed below. In order to explain LSND experiment, we need at least one sterile neutrino. In the 3–3–1 model with right- handed neutrinos we have necessarily three sterile neutrinos. The masses and mixing of the neutrinos is dictated by the matrix M in Eq. (15). As in the case of quarks and charged leptons, the masses and mixing angles of the active and sterile neutrinos is a question of an appropriate tunning of the couplings fab, gab and hab. Presently we have three kinds of experimental evidence for neutrino oscillation. One involves neutrino oscillation from atmosphere whose data are [2], × −3 2 2 × −3 2 2 90%CL 1.5 10 eV m23 3.4 10 eV , sin 2θ23 > 0.92, (19) A.G. Dias et al. / Physics Letters B 628 (2005) 85–92 89 while the other evidence involves solar neutrino oscillation. The data in this case are [3], × −5 2 2 × −5 2 2 90%CL 7.4 10 eV m12 8.5 10 eV , 0.33 tan θ12 0.50. (20)

The third evidence refers to the appearance of ν¯e in a beam of ν¯µ observed by the LSND experiment [6]. This experiment does not have the status of the solar and atmospheric ones, since it needs to be confirmed. The MiniBooNE experiment is in charge of this [10]. The analysis of the data from LSND depends on the number of sterile neutrinos we suppose. For the case of only one sterile neutrino we have, the so-called 3 + 1 scenario, where [11], 2 = 2 = = m41 0.92 eV ,Ue4 0.136 and Uµ4 0.205. (21) According to Ref. [12], for the case of two sterile neutrinos, called 3 + 2 scenario [13], we can have two possible schemes. In one case, the best fit leads to 2 = 2 = = m41 0.92 eV ,Ue4 0.121 and Uµ4 0.204, 2 = 2 = = m51 22 eV ,Ue5 0.036 and Uµ5 0.224, (22) in the other case, we have 2 = 2 = = m41 0.46 eV ,Ue4 0.090 and Uµ4 0.226, 2 = 2 = = m51 0.89 eV ,Ue5 0.125 and Uµ5 0.160. (23) We would like to provide a texture for M that solves neutrino oscillation, i.e., that recovers as close as possible the neutrino data showed above. But as the 3–3–1 model provides three sterile neutrinos and we dispose of analysis considering at most two sterile neutrinos, we have to make some assumptions. We will neglect CP violation; assume that the third sterile neutrino decouples from the others; take Ue3 = 0 and consider that the atmospheric angle is 2 exactly maximal. By an appropriate choice of the free parameters fab, gab and hab, and taking vη = 10 GeV, 3 14 vχ = 10 GeV [14] and Λ = 10 GeV, a possible texture for M that incorporates such assumptions is,   0.0465 0.0208 −0.0208 0.121 0.136 0.0  − −   0.0208 0.064 0.0166 0.0495 0.167 0.0   − − −  M =  0.0208 0.0166 0.064 0.0495 0.167 0.0   −  (eV). (24)  0.121 0.0495 0.0495 0.66 0.00.0  0.136 0.167 −0.167 0.00.851 0.0 0.00.00.00.00.01.7 This mass matrix is diagonalized by the following mixing matrix,   0.847 0.476 0.00.179 0.154 0.0  − −   0.344 0.581 0.71 0.0733 0.189 0.0   − −  (6) ≈  0.344 0.581 0.71 0.0733 0.189 0.0  U  −  , (25)  0.207 0.00.00.978 0.00.0  0.0 −0.309 0.00.00.951 0.0 0.00.00.00.00.01.0 which leads to the following neutrino masses, −5 −3 −2 m1 ≈−5.5 × 10 eV,m2 ≈ 9.3 × 10 eV,m3 ≈ 4.8 × 10 eV, −1 −1 m4 ≈ 6.9 × 10 eV,m5 ≈ 9.4 × 10 eV,m6 = 1.7eV. (26) The values for the neutrino masses in Eq. (26) and the pattern of U (6) above, Eq. (25), accommodate the solar and atmospheric oscillation data along with the LSND experiment altogether. For the sterile neutrinos, we did not recover exactly the best fit, which we believe is due only to the set of assumptions made above. 90 A.G. Dias et al. / Physics Letters B 628 (2005) 85–92

We also would like to call the attention to the fact that our sterile neutrinos present non-standard interactions. For example, they interact directly to the charged leptons through a new charged gauge boson according to √g ¯ C µ + + (eL) γ νRVµ h.c., (27) 2 and also couple to the active neutrinos through a new non-Hermitian neutral gauge boson according to, √g ¯ C µ 0† + (νL) γ νRUµ h.c. (28) 2 This turns the phenomenology of our sterile neutrinos much richer than usual. For example, although our sterile neutrinos couple directly to the active ones, see Eq. (28), they are still stable. The interactions in Eq. (28) allows the decay of the heavier sterile neutrinos in lighter neutrinos. For example, we can have the following channel ντR → ντLνeLνeR. In this case we have the following expression for the decay width m5 m4 Γ = G2 ντR W . (29) F 3 4 192π mU = = × 34 For mU 250 GeV and mντR 1.5 eV, we obtain a life-time of order of 3.4 10 s, leading, thus, to stable sterile neutrinos. Finally, we would like to remember that there is a conﬂict among cosmology and LSND result. For sterile neutrino disposes of an acceptable abundance, the active–sterile mixing and the sterile neutrino mass must lie in a range of values that get incompactible with LSND experiment [15]. On the other hand, depending on the temperature light sterile neutrinos termalize, they can play an important role in big bang nucleosynthesis (BBN) [16]. It is important to have in mind that such conﬂict turns more serious in scenarios with more than one sterile neutrino [17].

4. Conclusions

The main achievement of this work is to show that the right-handed neutrinos that appear in a version of the 3–3–1 model can be naturally light when dimension five effective operators are included. Such neutrinos can be identified as sterile ones, offering us the possibility of explaining the LSND data. We have checked that, although the model leads to a 3 + 3 scenario, the results from a 3 + 2 scenario can be easily recovered by making an appropriate choice of the Yukawa couplings in the mass matrix M. Remarkably, we have shown that besides LSND, this 3–3–1 model has all the necessary features to also explain solar and atmospheric neutrino oscillation data without adding any extra fields or intricate symmetries. This is an automatic property as far as the allowed dimension five operators here included can be embedded in some larger underlying theory, maybe a GUT or something else at higher energies than TeV scale. Moreover, it is possible that the new non-standard interactions involving neutrinos can reveal a very different perspective concerning the conflict between LSND results and neutrino cosmology. That is something to be further analyzed but it is out of the scope of this work. However, we should stress that all this characteristics of this 3–3–1 model are fairly appealing, considering the tiny amount of assumptions we had to rely on.

Acknowledgements

We thank O.G. Peres and V. Pleitez for useful discussion about sterile neutrinos phenomenology. This work was supported by Conselho Nacional de Pesquisa e Desenvolvimento—CNPq (C.A.S.P., P.S.R.S.), Fundação de Amparo à Pesquisa do Estado de São Paulo—FAPESP (A.G.D.) and by Fundação de Apoio à Pesquisa do Estado da Paraíba (FAPESQ) (C.A.S.P.). A.G. Dias et al. / Physics Letters B 628 (2005) 85–92 91

References

[1] S. Eidelman, et al., Phys. Lett. B 592 (2004) 1. [2] Super-Kamiokande Collaboration, Y. Fukuda, et al., Phys. Rev. Lett. 81 (1998) 1562; Y. Ashie, et al., Phys. Rev. Lett. 93 (2004) 101801. [3] SNO Collaboration, Q.R. Ahmed, et al., Phys. Rev. Lett. 87 (2001) 071301; K2K Collaboration, M.H. Alm, et al., Phys. Rev. Lett. 90 (2003) 041801; KamLAND Collaboration, T. Araki, et al., Phys. Rev. Lett. 94 (2005) 081801. [4] S. Weinberg, Phys. Rev. Lett. 43 (1979) 1566; F. Wilczek, A. Zee, Phys. Rev. Lett. 43 (1979) 1571. [5] R. Foot, R. Volkas, Phys. Rev. D 52 (1995) 6595; E. Ma, P. Roy, Phys. Rev. D 52 (1995) 4342; E.J. Chun, A. Joshipura, A. Smirnov, Phys. Rev. D 54 (1996) 4654; D. Suematsu, Phys. Lett. B 392 (1997) 413; P. Langacker, Phys. Rev. D 58 (1998) 093017; R.N. Mohapatra, Phys. Rev. D 64 (2001) 091301; A.P. Lorenzana, C.A. de S. Pires, Phys. Lett. B 522 (2001) 297; K.S. Babu, G. Seidl, Phys. Rev. D 70 (2004) 113014; M. Frank, I. Turan, M. Sher, Phys. Rev. D 71 (2005) 113001; J. Sayre, S. Wiesenfeldt, S. Willembrock, Phys. Rev. D 72 (2005) 015001; R.N. Mohapatra, S. Nasri, H.-B. Yu, hep-ph/0505021. [6] LSND Collaboration, C. Athanassopoulos, et al., Phys. Rev. Lett. 77 (1996) 3082; LSND Collaboration, A. Aguilar, et al., Phys. Rev. D 64 (2001) 112007. [7] J.C. Montero, F. Pisano, V. Pleitez, Phys. Rev. D 47 (1993) 2918; R. Foot, H.N. Long, T.A. Tran, Phys. Rev. D 50 (1994) R34. [8] For some recent works on 3–3–1 models, see: J.A. Rodriguez, M. Sher, Phys. Rev. D 70 (2004) 117702; M.A. Perez, G. Tavares-Velasco, J.J. Toscano, Phys. Rev. D 69 (2004) 115004; J.L. Garcia-Luna, G. Tavares-Velasco, J.J. Toscano, Phys. Rev. D 69 (2004) 093005; G. Tavares-Velasco, J.J. Toscano, Phys. Rev. D 70 (2004) 053006; C.A. de S. Pires, P.S. Rodrigues da Silva, Eur. Phys. J. C 36 (2004) 397; R.A. Diaz, R. Martinez, F. Ochoa, hep-ph/0411263; A.G. Dias, A. Doff, C.A. de S. Pires, P.S. Rodrigues da Silva, hep-ph/0503014; A.G. Dias, C.A. de S. Pires, V. Pleitez, P.S. Rodrigues da Silva, Phys. Lett. B 621 (2005) 151; F. Ochoa, R. Martinez, hep-ph/0505027; L.D. Ninh, H.N. Long, hep-ph/0507069; A.C. Cid, G. Tavares-Velasco, J.J. Toscano, hep-ph/0507135; J. Montano, G. Tavares-Velasco, J.J. Toscano, F. Ramirez-Zavaleta, hep-ph/0508166. [9] P.B. Pal, Phys. Rev. D 52 (1995) 1659; A.G. Dias, C.A. de S. Pires, P.S. Rodrigues da Silva, Phys. Rev. D 68 (2003) 115009. [10] MiniBooNE Collaboration, A.O. Bazarko, hep-ex/9906003; MiniBooNE Collaboration, H.L. Ray, hep-ex/0411022. [11] D. Caldwell, R. Mohapatra, Phys. Rev. D 46 (1993) 3259; J. Peltoniemi, J.W.F. Valle, Nucl. Phys. B 406 (1993) 409; J. Peltoniemi, D. Tommasini, J.W.F. Valle, Phys. Lett. B 298 (1993) 383; S. Bilenky, W. Grimus, C. Giunti, T. Schwetz, hep-ph/9904316; V. Barger, B. Kayser, J. Learned, T. Weiler, K. Whisnant, Phys. Lett. B 489 (2000) 345; O.L.G. Peres, A.Y. Smirnov, Nucl. Phys. B 599 (2001) 3; W. Grimus, T. Schwetz, Eur. Phys. J. C 20 (2001) 1; M. Maltoni, T. Schwetz, J.W.F. Valle, Phys. Lett. B 518 (2001) 252. [12] M. Sorel, J. Conrad, M. Shavitz, Phys. Rev. D 70 (2004) 073004. [13] K.S. Babu, G. Seidl, Phys. Lett. B 591 (2004) 127; K.L. McDonald, B.H.J. McKellar, A. Mastrano, Phys. Rev. D 70 (2004) 053012; W. Krolikowski, hep-ph/0506099. [14] To simplify the numerical manipulation, we decided to consider the order of magnitude of the VEVs that enter in the mass matrix M,but 2 + 2 = 2 we are aware of the constraint vη vρ (247 GeV) and that little-Higgs versions of the 3–3–1 model requires vχ around 5 TeV, see: G. Marandella, C. Schappacher, A. Strumia, Phys. Rev. D 72 (2005) 035014. [15] R. Barbieri, A. Dolgov, Phys. Lett. B 237 (1990) 440; 92 A.G. Dias et al. / Physics Letters B 628 (2005) 85–92

K. Enqvist, K. Kainulainen, J. Maalampi, Phys. Lett. B 249 (1990) 531; S. Dodelson, L.M. Widrow, Phys. Rev. Lett. 72 (1994) 17; M. Maltoni, T. Schwetz, M.A. Tortola, J.W.F. Valle, hep-ph/0305312; A. Pierce, H. Murayama, Phys. Lett. B 581 (2004) 218. [16] N. Okada, O. Yasuda, Int. J. Mod. Phys. A 12 (1997) 3669; S.M. Bilenky, C. Giunti, W. Grimus, T. Schwetz, Astropart. Phys. 11 (1999) 413; K.N. Abazajian, Astropart. Phys. 19 (2003) 303. [17] One possible solution for the LSND/cosmology conﬂict is the lowering of the reheating temperature see: G. Gelmini, S.P. Ruiz, S. Pascoli, Phys. Rev. Lett. 93 (2004) 081302; S. Hannestad, Phys. Rev. D 70 (2004) 043506. Physics Letters B 628 (2005) 93–105 www.elsevier.com/locate/physletb

Covariant extremisation of ﬂavour-symmetric Jarlskog invariants and the neutrino mixing matrix

P.F. Harrison a,W.G.Scottb

a Department of Physics, University of Warwick, Coventry CV4 7AL, UK b CCLRC Rutherford Appleton Laboratory, Chilton, Didcot, Oxon OX11 0QX, UK Received 15 August 2005; received in revised form 5 September 2005; accepted 8 September 2005 Available online 19 September 2005 Editor: N. Glover

Abstract We examine the possibility that the form of the lepton mixing matrix can be determined by extremising the Jarlskog ﬂavour invariants associated, e.g. with the commutator (C) of the lepton mass matrices. Introducing a strictly covariant approach, keeping masses ﬁxed and extremising the determinant (Tr C3/3) leads to maximal CP violation, while extremising the sum of the 2 × 2 principal minors (− Tr C2/2), leads to a non-trivial mixing with zero CP violation. Extremising, by way of example, a general linear combination of two CP-symmetric invariants together, we show that our procedures can lead to acceptable √ | | 2 2 − 2 mixings and to non-trivial predictions, e.g., Ue3 2/3 m12/m23(1 mµ/mτ ) 0.07.  2005 Elsevier B.V. All rights reserved.

1. Introduction

It is now almost twenty years since Jarlskog [1] first alerted us to the inherent U(3) invariance representing the freedom in any weak basis [2] to transform fermion mass matrices in the generation space, whilst keeping the charged-current weak-interaction diagonal and universal, i.e., whilst always staying in a weak basis. While all observables (e.g., masses and mixing angles) must of course be Jarlskog-invariant, we focus here on flavour- symmetric invariants, i.e., those invariants without flavour indices of any kind (i.e., involving no flavour projection operators in their definition). While it has long been clear [3] that any fundamental laws underlying the masses and mixings should be Jarlskog-covariant (as defined below), this principle may not always be easy to respect in practice. In this Letter

E-mail addresses: [email protected] (P.F. Harrison), [email protected] (W.G. Scott).

0370-2693/$ – see front matter  2005 Elsevier B.V. All rights reserved. doi:10.1016/j.physletb.2005.09.009 94 P.F. Harrison, W.G. Scott / Physics Letters B 628 (2005) 93–105 we will show that extremisation of ﬂavour-symmetric Jarlskog invariants [4,5] leads naturally to Jarlskog-covariant constraints, which (as we shall see) can even have viable phenomenology. The archetypal example of a ﬂavour-symmetric Jarlskog invariant is the famous determinant of the commutator [6], which for the charged-lepton (L) and neutrino (N) mass matrices (taken here to be Hermitian) may be written:

3 3 Det C = Tr C /3 = Tr i[L,N] /3 =−2 Det diag(∆l) Det diag(∆ν)J, (1) T = − − − T = − − − ∆l (mµ mτ ,mτ me,me mµ), ∆ν (m2 m3,m3 m1,m1 m2), (2) where C := −i[L,N], with me, mµ, mτ the charged-lepton masses and m1, m2, m3 the neutrino masses, and with the invariant measure of CP violation [6] given by = 2 = − 2 1/2 − 2 1/2 − 2 J c12s12c23s23c13s13sδ 1 s12 s12 1 s23 s23 1 s13 s13sδ. (3)

Extremising√ Eq. (3), with√ respect to√ the standard PDG [7] variables s12, s23, s13 and δ is known [8] to lead to s12 = 1/ 2, s23 = 1/ 2, s13 = 1/ 3 and δ = π/2, corresponding to maximal CP violation, i.e., to trimaximal mixing [9]. It should be remarked that the assumption that the mass matrices are Hermitian (see above) may in fact be realised in two distinct ways: one may simply imagine applying a suitable transformation to the right-handed fields (which are anyway inert to the charged-current weak-interaction) bringing L and N into Hermitian form, or one may instead everywhere reinterpret L and N as Hermitian squares of mass matrices L → LL†, N → NN†, operating between left-handed fields. In the latter case, masses need to be replaced by masses-squared throughout → 2 (ml/ν ml/ν, e.g., in Eqs. (1)–(2)). In this Letter, in order to circumvent this as yet unresolved ambiguity, we shall give results, where appropriate, for both the linear and quadratic cases. Of course we now know that lepton mixing is not trimaximal, it being actually much closer to the so-called tribimaximal form [10,11]. The question then arises, as to whether extremising some other invariant (or combination of invariants) aside from Tr C3, might perhaps yield a more realistic mixing prediction. In this Letter we seek to kindle interest in these kinds of questions, and we start out (Section 2) by rederiving the results for Tr C3 above, introducing and bringing to bear a strictly covariant (basis-independent) approach. We focus here on the leptonic case: in the context of extremisation, quasi-maximal mixing in the lepton sector looks a priori more comprehensible than anything involving the quarks (which are anyway subject to larger radiative corrections, e.g., dependent on the top mass). We proceed (Section 3) with a similar covariant extremisation of the only other independent invariant depending on C alone, i.e., Tr C2. The mixing prediction is now CP-conserving and, whilst not phenomenologically viable as it stands, is non-trivial, with even a suggestive resemblance to some previously proposed lepton-mixing ansatze. Finally, taking a particular linear combination of two CP-symmetric invariants as an illustrative example, we show (Section 4) how covariant extremisation of slightly more general combinations of invariants can readily lead to realistic mixing predictions. Our “perspective” section (Section 5) simply consolidates what we have learned, also briefly pointing forward to what one might hope to learn in future studies, e.g., in terms of incorporating CP violation, dropping mass constraints, etc.

2. Extremising the Jarlskog determinant, Tr C3/3

One might argue that the PDG variables are arbitrary, e.g., it matters (Eq. (3)) that we chose to extremise with respect to δ and not sδ = sin δ. We choose to extremise with respect to the Yukawa couplings themselves (i.e., with respect to the mass matrices) using some established results [13] on differentiation of matrix traces with respect to matrix variables (see Appendix A). In terms of matrix derivatives ∂L := ∂/∂L, etc., the extremisation conditions for, e.g., Tr C3 may then be written (see Appendix A) in a manifestly Jarlskog-covariant (i.e., basis-independent) P.F. Harrison, W.G. Scott / Physics Letters B 628 (2005) 93–105 95 way: 3 2 T ∂L Tr C /3 =−i N,C = 0, (4) 3 2 T ∂N Tr C /3 =+i L,C = 0, (5) in that both sides of the equation are evidently form-invariant under an arbitrary U(3) transformation L → U(3)LU(3)†, N → U(3)NU(3)†, i.e., we would obtain the same equations performing the extremisation in any basis. When the masses are to be held ﬁxed, the zeros on the RHS of Eqs. (4)–(5) must be replaced by arbitrary matrix polynomials in LT and N T respectively, representing the dependence on unknown Lagrange multipliers (see below), with the manifest covariance clearly maintained. The covariance property allows us to solve our equations in any basis, so we choose a basis convenient to us where the charged-lepton mass matrix is diagonal and where all imaginary parts of the off-diagonal elements of the neutrino mass matrix are equal (the “epsilon” basis [12]). The neutrino mass matrix may then be written

 eµτ eaz+ id y − id   N = µz − id b x + id , (6) τy+ id x − id c where the seven variables a, b, c, x, y, z and d determine the three neutrino masses and the four mixing parameters (the charged-lepton masses being directly the diagonal elements of L). In terms of these variables the Jarlskog determinant is given by 3 2 Det C = Tr C /3 = 2d(me − mµ)(mµ − mτ )(mτ − me) d − xy − yz − zx (7) so that CP violation vanishes in the case d = 0 and also in the case d2 = xy + yz + zx. The off-diagonal elements of −i[N,C2] must be set to zero for an extremum (Eq. (4)), leading to the following cyclically symmetric constraints, from the real (Re) parts Re −d(m − m )(m − m ) (y − z) − (b − c) (y + z) = 0, e µ τ e −d(m − m )(m − m ) (z − x) − (c − a) (z + x) = 0, µ τ e µ −d(mτ − me)(mµ − mτ ) (x − y) − (a − b) (x + y) = 0, (8) and from the imaginary (Im) parts Im −(m − m )(m − m ) (b − c) d2 − yz − y2 − z2 x = 0, e µ τ e −(m − m )(m − m ) (c − a) d2 − zx − z2 − x2 y = 0, µ τ e µ 2 2 2 −(mτ − me)(mµ − mτ ) (a − b) d − xy − x − y z = 0. (9) It should be remarked that the cyclic symmetry (e → µ → τ , a → b → c, x → y → z) will be seen to be a useful and important generic feature of our approach, resulting from starting with ﬂavour-symmetric invariants and working in the epsilon basis. A non-trivial solution to Eq. (8), i.e., that corresponding to the real parts above (putting aside the d = 0 possibility for one moment), may be written (x − y) = (a − b), a = σ + x, (y − z) = (b − c), i.e., b = σ + y, (z − x) = (c − a), c = σ + z, (10) 96 P.F. Harrison, W.G. Scott / Physics Letters B 628 (2005) 93–105 where σ is an undetermined overall constant offset. Eq. (10) is precisely the “S3 invariant constraint” [14] (or “magic square constraint” [12]) whereby the neutrino mass matrix is determined to be an “S3 group matrix” [14], i.e., a matrix having all row/column sums equal and hence (at least) one trimaximal eigenvector. Turning to Eq. (9) (corresponding to the imaginary parts) a non-trivial solution is

a = b and x = y, b = c and y = z, c = a and z = x (11) whereby the mass matrix is circulant [4] and the mixing takes the trimaximal form [9] (Eq. (10) being also satisfied). More trivially, coming to the d = 0 case (above), we have that twofold-maximal mixing, e.g., b = c, y = z = 0also satisfies both sets of equations. Of course, these solutions do not solve all the extremisation equations, Eqs. (4)–(5). It transpires however, that in the case that the masses are held fixed, all the remaining equations serve only to determine suitable Lagrange multipliers λLi , λNi (i = 0, 1, 2) 3 T 2 2 ∂L Tr C /3 =−i N,C = λL0 + λL1L + λL2L , (12) 3 T 2 2 ∂N Tr C /3 =+i L,C = λN0 + λN1N + λN2N (13) with the apparently excess constraints turning out to be redundant (and furthermore with this circumstance occurring for all the extremisations studied in this Letter). In the epsilon basis, L is diagonal and any polynomial in L is diagonal also, whereby Eqs. (8)–(9) (and their solutions, e.g., Eqs. (10)–(11)) remain valid, even with the Lagrange multipliers (Eq. (12)). In addition, the on- diagonal elements of +i[L,C2] vanish (as in Eq. (5)) due to L being diagonal, and in the case of non-zero Lagrange multipliers (Eq. (13)) the consistency requirement on the coefficients of the λNi takes the form 2 + 2 + 2 1 aa y z 2 + 2 + 2 = 1 bx b z 0. (14) 1 cx2 + y2 + c2 All our solutions above do indeed turn out to satisfy this determinant condition. (In fact any S3 group matrix, Eq. (10), automatically satisfies this condition, Eq. (14)). Covariance implies that the λLi/Ni themselves will be Jarlskog scalars, expressible in terms of, e.g., traces of powers of mass matrices

L1 := Tr L = me + mµ + mτ ,N1 := Tr N = m1 + m2 + m3, := 2 = 2 + 2 + 2 := 2 = 2 + 2 + 2 L2 Tr L me mµ mτ ,N2 Tr N m1 m2 m3, := 3 = 3 + 3 + 3 := 3 = 3 + 3 + 3 L3 Tr L me mµ mτ ,N3 Tr N m1 m2 m3 (15) or traces of commutators of mass matrices, etc. Note that the deﬁnitions (Eq. (15)) are our “constraint equations”, 2 2 T e.g., Tr L = L2 for ﬁxed L2, etc., which when differentiated, ∂L(Tr L − L2) = 2L , etc., lead to the polynomial forms on the RHS of Eqs. (12)–(13). The on-diagonal elements of Eq. (12) (which are purely real) are given by 2 2 2 2 2 2d(mµ + mτ − 2me)(mµ − mτ ) d − x − y − z = λL0 + λL1me + λL2m , e 2 2 2 2 2 2d(mτ + me − 2mµ)(mτ − me) d − x − y − z = λL0 + λL1mµ + λL2m , µ + − − 2 − 2 − 2 − 2 = + + 2 2d(me mµ 2mτ )(me mµ) d x y z λL0 λL1mτ λL2mτ . (16) P.F. Harrison, W.G. Scott / Physics Letters B 628 (2005) 93–105 97

2 Solving for the λLi gives the charged-lepton discriminant L∆ in the denominator of the solution (from the determinant-of-coefficients, Eq. (16), RHS), where L∆ is given by := 3 + 4 + − 2 2 − 2 − 3 − 6 L∆ L2/2 3L1L2/2 6L1L2L3 7L1L2/2 3L3 4L1L3/3 L1/6 = (me − mµ)(mµ − mτ )(mτ − me). (17) After some work to cast the numerators also into invariant form, we find that 5 3 2 2 −L /2 + 3L L2 − 7L1L /2 − 2L L3 + 3L2L3 λ = 1 1 2 1 C3, L0 2 Tr (18) 3L∆ 4 2 2 3L /2 − 7L L2 + 3L /2 + 6L1L3 λ = 1 1 2 C3, L1 2 Tr (19) L∆ 3 −9L3 + 9L1L2 − 2L λ = 1 C3. L2 2 Tr (20) 3L∆ Of course, these expressions can also be obtained in a fully covariant way simply by multiplying Eq. (12) by successive powers of L (I = L0, L = L1,L2) and taking traces (powers higher than L2 must always be reduced using the characteristic equation). Analogous L ↔ N expressions for the λNi solve all the remaining equations. 3 Recognising the coefficients of Tr C on the RHS above as ∂L/N (L∆N∆)/(L∆N∆), we see that we have, in 3 effect, extremised the Jarlskog invariant J = Tr C /(L∆N∆), recovering the known result [8], and validating our procedure. Finally, we note that the above analysis is essentially unchanged, taking L and N as the Hermitian squares of mass matrices (see Section 1) substituting masses-squared for masses throughout.

3. The sum of the 2 × 2 principal minors, − Tr C2/2

The other independent invariant function of C may be taken to be the sum of the 2 × 2 principal minors, := − νν expressible in terms of the K-matrix (Klν Kll [15])asfollows: =− 2 = [ ]2 = T Q11 Tr C /2 Tr L,N /2 ∆l diag(∆l)K diag(∆ν)∆ν. (21) The K-matrix is a key oscillation observable [15], carrying equivalent information to the moduli-squared of the := − + mixing elements [16].TheK-matrix comprises the real parts of the plaquette products [17]: lν Klν iJ (indices to be interpreted mod 3) and may be viewed as the CP-conserving analogue of the Jarlskog invariant J [6]. From Eq. (21), taking for simplicity initially the extreme hierarchical approximation me, mµ, m1, m2 → 0(i.e., the 2 × 2 mixing limit), we have Q → m2m2(K + K + K + K ) + O m2m2 +··· 11 τ 3 e1 e2 µ1 µ2 τ 2 = m2m2c2 c2 s2 + c2 s2 + O m2m2 +···. (22) τ 3 13 23 13 13 23 τ 2 √ Extremising this expression with respect to s23 and s13 gives s23 = s13 = 0ors23 = s13 = 1orc23c13 = 1/ 2 (where the latter looks at ﬁrst sight very promising phenomenologically, see below). Extremising rather with respect to θ23 and θ13 gives in addition s23 = 1ors13 = 1 as solutions, for arbitrary s13 and s23, respectively. Since mixing matrices which are just permutation matrices give no mixing as such, we have that 2 × 2 maximal mixing is the only non-zero mixing solution in the 2 × 2 case. Just as for Tr C3 above, however, we will extremise exactly here with respect to the mass matrices (see again Appendix A): 2 T 2 −∂L Tr C /2 =+ N,[L,N] = 0,λL0 + λL1L + λL2L , (23) 98 P.F. Harrison, W.G. Scott / Physics Letters B 628 (2005) 93–105 2 T 2 −∂N Tr C /2 =− L,[L,N] = 0,λN0 + λN1N + λN2N , (24) where the matrix polynomials replace the RHS zeros, in the case of mass constraints. In the epsilon basis the off-diagonal elements of Eq. (23) are, for the real parts: Re (m + m − 2m ) d2 − yz + (m − m )(b − c)x = 0, µ τ e µ τ (m + m − 2m ) d2 − zx + (m − m )(c − a)y = 0, τ e µ τ e 2 (me + mµ − 2mτ ) d − xy + (me − mµ)(a − b)z = 0 (25) and for the imaginary parts:

Im d(mµ + mτ − 2me)(y + z) + d(mµ − mτ )(b − c) = 0,

d(mτ + me − 2mµ)(z + x) + d(mτ − me)(c − a) = 0,

d(me + mµ − 2mτ )(x + y) + d(me − mµ)(a − b) = 0. (26) We observe that Eqs. (25)–(26) imply that either d = 0ord2 = xy + yz+ zx, and so from Eq. (7) we see immediately that there can be no CP violation in this case. It turns out that the d2 = xy + yz + zx solution is equivalent to the d = 0 solution, so that we need to consider only the d = 0 case (note that in the case d2 = xy + yz+ zx the epsilon basis is not unique, and the imaginary part can be rephased to zero). Setting d = 0 then, solves Eq. (26), corresponding to the imaginary parts. Clearly, setting any pair of x, y, z to zero and the corresponding pair of a, b, c equal, solves also Eq. (25) for the real parts, so that, just as for Tr C3 above, 2 × 2 maximal mixing in any sector gives an exact extremum of Tr C2, independent of the neutrino mass spectrum. A somewhat more interesting non-trivial solution to Eq. (25) (still with d = 0 from Eq. (26)) clearly reﬂects the inherent cyclic symmetry: (m − m )(m − m ) x =± (a − b)(c − a)E, E = e µ τ e , (me + mµ − 2mτ )(mτ + me − 2mµ) (m − m )(m − m ) y =± (b − c)(a − b)M, M = µ τ e µ , (mµ + mτ − 2me)(me + mµ − 2mτ ) (m − m )(m − m ) z =± (c − a)(b − c)T , T = τ e µ τ . (27) (mτ + me − 2mµ)(mµ + mτ − 2me) With the charged-lepton masses positive, we have E>0 >M,T, whereby either b

 ν1 ν2 ν3   ν1 ν2 ν3  e 0.33333 0.33333 0.33333 e 1/31/31/3 2     |Ulν| = µ  0.17079 0.16257 0.66663  µ  1/61/62/3 . (32) τ 0.49587 0.50409 0.00003 τ 1/21/20 While being closely a transpose/permutation of the tribimaximal form, and bearing a clear relation to some previously proposed ansatze (see in particular Acker et al. [18] and Karl et al. [19] as well as the Fritzsch–Xing ansatz [20]), this mixing (Eq. (32)) clearly cannot at this point be considered acceptable phenomenologically. We ﬁnd that the situation is not improved substituting masses-squared in place of masses throughout. Indeed 2 −9 the large discrepancy in |Uτ3| is considerably worsened, with |Uτ3| ∼ 3 × 10 (maintaining the mass hierarchy unchanged requires setting (b − a)/(a − c) ∼ 0.36). Having also explored less minimalist assumptions regarding the neutrino mass-spectrum, i.e., inverted mass hierarchy, large mass offsets, etc., we can report no phenomeno- 2 logically acceptable solutions to the problem of extremising Tr C .Form1 m2 m3, we see that all our exact solutions are consistent with the (approximate) constraints derived differentiating√ Eq. (22) with respect to PDG angles (although we learn that the initially promising Uτ3 = c23c13 = 1/ 2 condition in fact corresponds only to phenomenologically uninteresting 2 × 2 maximal mixing solutions).

4. Extremising more general mixing invariants

More general commutator invariants are readily constructed, involving higher powers of mass matrices [21].In this section we shall extremise a very simple composite function, comprising an arbitrary linear combination of two such invariants:

A := Q11 + qQ21 (33) 100 P.F. Harrison, W.G. Scott / Physics Letters B 628 (2005) 93–105 with q a scalar constant of suitable dimensionality (assumed Jarlskog-invariant) and 2 Q11 := Tr[L,N][L,N]/2,Q21 := Tr[L,N] L ,N /2, (34) 1 our Qmn notation being essentially self-explanatory at this point. Of course, Q11 has already been individually extremised in Section 3 (Eqs. (23)–(24)), so we only need in addition here the relevant derivatives of Q21 (see Appendix A): (∂Q /∂L)T = N, L2,N /2 + L, N,[L,N] /2, (38) 21 T 2 (∂Q21/∂N) =− L , [L,N] , (39) where again curly brackets signify the anticommutator. The off-diagonal terms derived from Q21, analogously to Eq. (25), are (Re parts) 2 2 2 2 2 2 m + m + mµmτ − mτ me − memµ − m d − yz + m − m (b − c)x, µ τ e µ τ 2 2 2 2 2 2 m + m + mτ me − memµ − mµmτ − m d − zx + m − m (c − a)y, τ e µ τ e 2 + 2 + − − − 2 2 − + 2 − 2 − me mµ memµ mµmτ mτ me mτ d xy me mµ (a b)z (40) and (Im parts) 2 2 2 2 2 d m + m + mµmτ − mτ me − memµ − m (y + z) + d m − m (b − c), µ τ e µ τ 2 2 2 2 2 d m + m + mτ me − memµ − mµmτ − m (z + x) + d m − m (c − a), τ e µ τ e 2 + 2 + − − − 2 + + 2 − 2 − d me mµ memµ mµmτ mτ me mτ (x y) d me mµ (a b) (41) and the combined constraint equations are obtained simply by adding these terms (weighted by q) into Eqs. (25)– (26). As in Section 3 we may again set d = 0, and the solutions to these combined equations are then readily written down from Eq. (27), by modifying the expressions for E, M and T in Eq. (27), making the (cyclically-symmetric) substitutions 2 2 (me − mµ) → (me − mµ) + q m − m , e µ 2 2 (mµ − mτ ) → (mµ − mτ ) + q m − m , µ τ − → − + 2 − 2 (mτ me) (mτ me) q mτ me , (42)

1 We are considering here ﬂavour-symmetric quadratic commutator invariants (vanishing in the case of zero mixing) which may be usefully arranged as a 3 × 3matrix:   Tr[L,N]2 Tr[L,N][L,N2] Tr[L,N2]2 1 Q =  Tr[L,N][L2,N] Tr[L,N][L2,N2] Tr[L,N2][L2,N2]  . (35) 2 Tr[L2,N]2 Tr[L2,N][L2,N2] Tr[L2,N2]2

As usual, powers of mass matrices higher than L2, N2 are not considered, by virtue of the characteristic equation (and Tr[L2,N][L,N2]≡ 2 2 Tr[L,N][L ,N ], so there are indeed only nine such invariants). Generalising Eq. (21),theQmn (m, n = 1, 2, 3) are the double moments of the K-matrix:

= T m−1 n−1 Qmn ∆l diag(∆l)(diag Σl) K(diag Σν ) diag(∆ν )∆ν , (36)

Σl = (mµ + mτ ,mτ + me,me + mµ), Σν = (m2 + m3,m3 + m1,m1 + m2). (37) 2 For known masses, the Q-matrix and the K-matrix are therefore equivalent, and both are equivalent to the |Ulν| -matrix (see [15,16]), with only four elements functionally independent in each case. P.F. Harrison, W.G. Scott / Physics Letters B 628 (2005) 93–105 101 2 2 (me + mµ − 2mτ ) → (me + mµ − 2mτ ) + q m + m + memµ − mτ L1 , e µ 2 2 (mµ + mτ − 2me) → (mµ + mτ − 2me) + q m + m + mµmτ − meL1 , µ τ + − → + − + 2 + 2 + − (mτ me 2mµ) (mτ me 2mµ) q mτ me mτ me mµL1 . (43) The determinant condition (14) remains valid since the equation E + M + T + 1 = 0 (see text just after Eq. (27)) itself remains valid under these substitutions. Fitting to the “mass” spectrum as in Section 3, we now have (replacing the unique mixing prediction Eq. (32))a trajectory of mixings, each mixing given by a different value of the parameter q. To locate the phenomenologically viable solution along this trajectory (supposing even that there is a viable solution), we will simply impose the “S3 constraint” (Eq. (10)), which although not forced on us here by the extremisation, is well known [12,14] to be consistent with the phenomenology. We shall see below how the S3-constraint (by its nature) can be imposed in a fully covariant way. In practice, we shall substitute the S3 constraint Eq. (10) into the solution Eq. (27) (as modified by the substitutions Eqs. (42)–(43)), thereby fixing all the parameters in the neutrino mass matrix, as functions of q, as follows: k3 a = x + σ, x = , 2 (mµ − mτ ) (1 + q(mµ + mτ )) k3 b = y + σ, y = , 2 (mτ − me) (1 + q(mτ + me)) k3 c = z + σ, z = , 2 (44) (me − mµ) (1 + q(me + mµ)) where the mixing clearly cannot depend on the scale factor k or the offset σ . Assuming a normal classic neutrino mass hierarchy as before, the operative parameter is now q, fixing both the 2 := 2 2 2 2 hierarchy ratio hν m12/m23 m2/m3 and the mixing simultaneously. We see immediately from Eq. (44) that, e.g., as q →−1/(mµ + mτ ) we have a pole where x →−∞while the hierarchy factor hν → 0, with the mixing approaching the exact tribimaximal form [10,11] in the limit. =− + + 2 + + Parametrising deviations from this pole in the form q (1 )/(mµ mτ ), we find: 4/3hl hν(1 hν/3 2 +··· 2 := 2 1/3 2/3 8hν/9 ).Forhν 0.03 (with hl mµ/mτ 0.06) we have (0.03) , and taking k 0.38 meV GeV and σ 25 meV, gives a neutrino mass spectrum (m1, m2, m3) (0, 8.7, 50) meV, and a mixing matrix

 ν1 ν2 ν3   ν1 ν2 ν3  e 0.662 0.333 0.005 e 2/31/30 2     |Ulν| = µ  0.219 0.333 0.448  µ  1/61/31/2 . (45) τ 0.120 0.333 0.547 τ 1/61/31/2 The mixing equation (45), being very close√ to the tribimaximal form, is certainly acceptable phenomenologically. 2 There is the non-trivial prediction |U√e3| 2/3hν(1 − hl) 0.07 (cf. Ref. [12]) and violations of µ–τ symmetry [22] also, e.g., θ23 π/4 −|Ue3|/ 2. Related poles in y and z, for which the mixing matrix would have rows correspondingly permuted with respect to Eq. (45), would seem not to be phenomenologically relevant. Repeating the analysis with masses-squared, in effect takes us closer to the pole (now ∼ 5 × 10−7) yielding a mixing correspondingly closer to the tribimaximal form

 ν1 ν2 ν3   ν1 ν2 ν3  e 0.6665 0.3333 0.0002 e 2/31/30 2     |Ulν| = µ  0.1768 0.3333 0.4898  µ  1/61/31/2 . (46) τ 0.1567 0.3333 0.5100 τ 1/61/31/2 102 P.F. Harrison, W.G. Scott / Physics Letters B 628 (2005) 93–105 √ | | | |∼ 2 ∼ The corresponding prediction for Ue3 (now Ue3 2/3hν 0.014) is clearly of less immediate phenomenological interest than that in the unsquared case above. Lagrange multipliers for Eq. (33) are found in the usual way. For the ∂L equations, solving the analogue of Eq. (28) (after the substitutions Eq. (42)), we ﬁnd

λ + λ q + λ q2 + λ q3 + λ q4 + λ q5 λ = N 2 L00 L01 L02 L03 L04 L05 , (47) L0 1 2 + + 2 2 2(q LP 6 2qLP 2LP 3 LP 2) λ + λ q + λ q2 + λ q3 + λ q4 + λ q5 λ = N 2 L10 L11 L12 L13 L14 L15 , (48) L1 1 2 + + 2 2 2(q LP 6 2qLP 2LP 3 LP 2) λ + λ q + λ q2 + λ q3 + λ q4 + λ q5 λ = N 2 L20 L21 L22 L23 L24 L25 , (49) L2 1 2 + + 2 2 2(q LP 6 2qLP 2LP 3 LP 2) where we have explicitly assumed the “S3 constrained” solution Eq. (44). Fully invariant and flavour-symmetric, the expressions for λL00, λL01,...,λL10, etc., as functions of the Li , i = 1,...,3, are given in Ref. [25]. Similarly, for the ∂N equations we find 2 + + 2 LP 2 2qLP 2LP 3 q LP 6 λN0 =−N2 , (50) 6N1(LP 2 + qLP 3) λN1 = 0, (51) 2 + + 2 LP 2 2qLP 2LP 3 q LP 6 λN2 = (52) 2N1(LP 2 + qLP 3) with no clear relation to the ∂L case, since our “action” (Eq. (33))isnotL ↔ N symmetric. The supplementary polynomials LP 2, LP 3, LP 6 are defined by := − 2 = 2 + 2 + 2 − − − LP 2 3L2 L1 /2 me mµ mτ memµ mµmτ mτ me, (53) L := (3L − L L )/2 P 3 3 2 1 = 3 + 3 + 3 − 2 + 2 − 2 + 2 − 2 + 2 me mµ mτ memµ mµme /2 mµmτ mτ mµ /2 mτ me memτ /2, (54) L := L2 − L2 /4 P 6 P 3 ∆ = 3 + 3 + 3 − 2 − 2 − 2 3 + 3 + 3 − 2 − 2 − 2 me mµ mτ memµ mµmτ mτ me me mµ mτ memµ mµmτ mτ me . (55) It should be emphasised that (up to the apparently spontaneous choice of pole) the extremisation conditions on the composite action Eq. (33), together with the particular Lagrange multipliers defined by Eqs. (47)–(52) and Ref. [25], constitute an entirely covariant and flavour-symmetric specification of the realistic, non-trivial mixing Eq. (45).

5. Perspective

Motivated by the notion (Section 1) that the fundamental laws underlying the fermion masses and mixings might come from a variational principle, applied to some as yet unspecified function of flavour-symmetric Jarlskog invariants, we have presented a manifestly covariant machinery for extremising such invariants as functions of the mass matrices. (Building on the analogy [4] with the field-strength tensor in terms of covariant derivatives, cf. Fµν =−i[Dµ, Dν], we see our extremisation equations, e.g., Eqs. (23)–(24) as analogous to the Yang–Mills µν 2 [23] equations, cf. [Dµ,F ]=0, which are themselves derivable from a quadratic Lagrangian L =−Tr F /2, whereby we see the Yukawa couplings as “dynamical variables”, somewhat analogous to the gauge fields.) P.F. Harrison, W.G. Scott / Physics Letters B 628 (2005) 93–105 103

Having tested-out our methods on the familiar case of Tr C3 (Section 2) we have been able to show (Section 3) that extremising Tr C2 does not lead to physically realistic lepton mixings. Focussing on quadratic “actions”, we see Eq. (33) (Section 4) as just an example of an (approximate) “effective” action, whose main particular merit is that we know how to solve its resulting extremisation equations analytically, and that it leads to at least one mixing (Eq. (45)) which is compatible with observation. Clearly the complete absence of CP violation is a significant deficiency of Eq. (33) (given the success of the standard explanation of CP violation in the quark sector) and incorporating CP violation is an obvious challenge for the future. Most of all, we would like to think that this Letter will stimulate the search for a natural and beautiful action function, determining not only the mixing but also the (relative) mass values m1 : m2 : m3 and me : mµ : mτ ,obvi- ating the need for those ugly Lagrange multipliers (and perhaps also shedding some light on the quark case at the same time). Regarding masses, it is clear that our procedures apply directly as they stand: one will only have to notice that for some “perfect” action, not only is the mixing correctly predicted, but the Lagrange multiplier functions, on the RHS of the extremisation equations, vanish numerically for the correct masses, effectively as in a free − 2 − 2 + 4 = extremisation. (Empirical relations like the Koide relation: 512L3L1 64L2 656L2L1 207L1 0 [24] could be relevant here, cf. Ref. [25].) In removing a large part of the arbitrariness inevitably associated with the flavour sector as we know it today, such an “action” (if it exists) would surely be welcomed as an economical adjunct to the present Standard Model of particle physics.

Acknowledgements

It is a pleasure to thank S. Pakvasa and T.J. Weiler for useful discussions and helpful comments. This work was supported by the UK Particle Physics and Astronomy Research Council (PPARC). One of us (P.F.H.) acknowledges the hospitality of the Centre for Fundamental Physics (CfFP) at CCLRC Rutherford Appleton Laboratory.

Appendix A

We are making use of a long-established result from matrix calculus [13], whereby, in terms of the matrix derivative ∂X = ∂/∂X, for some constant matrix A,wehave

T ∂X Tr XA = Tr(∂XX)A = A , (A.1) where the superscript T denotes the matrix transpose. Note that ∂XX is a matrix of matrices (each sub-matrix with a single unit entry), and that in particular ∂XX = I , where I is the identity matrix (while ∂X Tr X = Tr ∂XX = I ). We ﬁrst consider Tr C3 = Tr i[L,N]3 (where C := −i[L,N], see Sections 1, 2)

3 3 ∂L Tr C = Tr ∂LC (A.2) 2 = 3Tr(∂LC)C (A.3) 2 =−3i Tr (∂LL), N C (A.4) 2 =−3i Tr(∂LL) N,C (A.5) =−3i N,C2 T (A.6) making use of the cyclic property of the trace, Eqs. (A.2)–(A.3) and Eqs. (A.4)–(A.5), and the matrix-calculus theorem (Eq. (A.1)) for the ﬁnal step Eqs. (A.5)–(A.6). 104 P.F. Harrison, W.G. Scott / Physics Letters B 628 (2005) 93–105

The case of Tr C2 =−Tr[L,N]2 (see Section 3) follows entirely analogously:

∂ Tr C2 =−2i[N,C]T (A.7) L =−2 N,[L,N] T , (A.8) where in the last line we have removed the factor i using C =−i[L,N]. 2 Finally, we consider the case corresponding to Q21 = Tr[L,N][L ,N]/2 (Section 4) 2 2 ∂L Tr[L,N] L ,N = Tr (∂LL), N L ,N +[L,N] (∂LL)L, N + L(∂LL), N . (A.9) Taking each of the above three terms one at a time, we have 2 2 Tr (∂LL), N L ,N = Tr(∂LL) N, L ,N , (A.10) 2 2 2 Tr[L,N] (∂LL)L, N = Tr(∂LL) LNLN − L N − LN L + LNLN = Tr(∂LL) L N,[L,N] , (A.11) 2 2 2 Tr[L,N] L(∂LL), N = Tr(∂LL) NLNL− LN L − N L + NLNL = Tr(∂LL) N,[L,N] L (A.12) whereby adding the above equations: 2 2 ∂L Tr[L,N] L ,N = Tr(∂LL) N, L ,N + L, N,[L,N] = N, L2,N T + L, N,[L,N] T , (A.13) where the curly brackets denote the anticommutator (see main text Eq. (38)).

References

[1] C. Jarlskog, Phys. Rev. D 35 (1987) 1685; C. Jarlskog, Phys. Rev. D 36 (1987) 2128; C. Jarlskog, A. Kleppe, Nucl. Phys. B 286 (1987) 245; C.H. Albright, C. Jarlskog, B.-A. Lindholm, Phys. Lett. B 199 (1987) 553; C.H. Albright, C. Jarlskog, B.-A. Lindholm, Phys. Rev. D 38 (1988) 872; C. Jarlskog, Phys. Lett. B 615 (2005) 207, hep-ph/0503199. [2] G.C. Branco, M.N. Rebelo, Phys. Lett. B 173 (1986) 313; G.C. Branco, D. Emmanuel-Costa, R. Gonzalez Felipe, Phys. Lett. B 477 (2003) 147, hep-ph/9911418; G.C. Branco, Invited talk at Flavor Physics and CP-violation (FPCP 2003), Paris, France, 3–6 June 2003, hep-ph/0309215. [3] G.C. Branco, D.-D. Wu, Phys. Lett. 205 (1988) 353; L. Lavoura, Phys. Rev. D 41 (1990) 2275. [4] P.F. Harrison, W.G. Scott, Phys. Lett. B 333 (1994) 471, hep-ph/9406351. [5] E. Rodriguez-Jauregui, DESY-01-040, hep-ph/0104092. [6] C. Jarlskog, Z. Phys. C 29 (1985) 491; C. Jarlskog, Phys. Rev. Lett. 55 (1985) 1039; C. Jarlskog, Phys. Lett. B 609 (2005) 323, hep-ph/0412288. [7] S. Eidelman, et al., Phys. Lett. B 592 (2004) 1, http://pdg.lbl.gov/. [8] L. Dunietz, O.W. Greenberg, D.-D. Wu, Phys. Rev. Lett. 55 (1985) 2935. [9] P.F. Harrison, D.H. Perkins, W.G. Scott, Phys. Lett. B 349 (1995) 137, http://hepunx.rl.ac.uk/~scottw/papers.html; P.F. Harrison, D.H. Perkins, W.G. Scott, Phys. Lett. B 396 (1997) 186, hep-ph/9702243; C. Giunti, C.W. Kim, J.D. Kim, Phys. Lett. B 352 (1995) 357; N. Cabibbo, Phys. Lett. B 72 (1978) 333; L. Wolfenstein, Phys. Rev. D 18 (1978) 958. [10] P.F. Harrison, D.H. Perkins, W.G. Scott, Phys. Lett. B 530 (2002) 167, hep-ph/0202074; P.F. Harrison, D.H. Perkins, W.G. Scott, Phys. Lett. B 458 (1999) 79, hep-ph/9904297. P.F. Harrison, W.G. Scott / Physics Letters B 628 (2005) 93–105 105

[11] P.F. Harrison, W.G. Scott, Phys. Lett. B 535 (2002) 163, hep-ph/0203209; Z. Xing, Phys. Lett. B 533 (2002) 85, hep-ph/0204049; X. He, A. Zee, Phys. Lett. B 560 (2003) 87, hep-ph/0301092; E. Ma, Phys. Rev. Lett. 90 (2003) 221802, hep-ph/0303126; E. Ma, Phys. Lett. B 583 (2004) 157, hep-ph/0308282; E. Ma, Phys. Rev. D 70 (2004) 031901, hep-ph/0404199; E. Ma, New J. Phys. 6 (2004) 104, hep-ph/0405152; E. Ma, hep-ph/0506036; C.I. Low, R.R. Volkas, Phys. Rev. D 68 (2003) 033007, hep-ph/0305243; N. Li, B.-Q. Ma, Phys. Rev. D 71 (2005) 017302, hep-ph/0412126; G. Altarelli, F. Feruglio, Nucl. Phys. B 720 (2005) 64, hep-ph/0504165; G. Altarelli, in: Renc. Phys. Vallee d’Aoste, La Thuile, Italy, 2005, hep-ph/0508053; S.F. King, hep-ph/0506297; I. de M. Varzielas, G.G. Ross, hep-ph/0507176; F. Plentinger, W. Rodejohann, TUM-HEP-596/05, hep-ph/0507143. [12] P.F. Harrison, W.G. Scott, Phys. Lett. B 594 (2004) 324, hep-ph/0403278; P.F. Harrison, W.G. Scott, in: Proc. NOVE-II, Venice, 2003, hep-ph/0402006. [13] C.R. Rao, M.B. Rao, Matrix Algebra and its Applications to Statistics and Econometrics, World Scientiﬁc, Singapore, 1998; F.A. Graybill, Matrices with Applications in Statistics, Wadsworth, Belmont, CA, 1983. [14] P.F. Harrison, W.G. Scott, Phys. Lett. B 557 (2003) 76, hep-ph/0302025. [15] P.F. Harrison, W.G. Scott, Phys. Lett. B 535 (2002) 229, hep-ph/0203021; P.F. Harrison, W.G. Scott, T.J. Weiler, Phys. Lett. B 565 (2003) 159, hep-ph/0305175. [16] G.C. Branco, L. Lavoura, Phys. Lett. B 208 (1988) 123; C. Hamzaoui, Phys. Rev. Lett. 61 (1988) 35; G. Belanger, C. Hamzaoui, Y. Koide, Phys. Rev. D 45 (1992) 4186; G. Belanger, et al., Phys. Rev. D 48 (1993) 4275; D.J. Wagner, T.J. Weiler, Phys. Rev. D 59 (1999) 113007, hep-ph/9801327; T.J. Weiler, D.J. Wagner, in: Proceedings of the 2nd Latin American Symposium on High Energy Physics, San Juan, Puerto Rico, 1998, hep-ph/9806490. [17] J.D. Bjorken, I. Dunietz, Phys. Rev. D 36 (1987) 2109. [18] A. Acker, et al., Phys. Lett. B 298 (1993) 149. [19] G. Karl, J.J. Simpson, Guelph preprint, hep-ph/0205288. [20] H. Fritzsch, Z. Xing, Phys. Lett. B 372 (1996) 265, hep-ph/9509389. [21] G.C. Branco, M.N. Rebelo, New J. Phys. 7 (2005) 86, hep-ph/0411196; J. Bernabeu, G.C. Branco, M. Gronau, Phys. Lett. B 169 (1986) 243; G.C. Branco, L. Lavoura, M.N. Rebelo, Phys. Lett. B 180 (1986) 264. [22] C.S. Lam, Phys. Lett. B 507 (2001) 214, hep-ph/0104116; T. Kitabayashi, M. Yasue, Phys. Lett. B 524 (2002) 308, hep-ph/0110303; T. Kitabayashi, M. Yasue, Phys. Rev. D 67 (2003) 015006, hep-ph/0209294; P.F. Harrison, W.G. Scott, Phys. Lett. B 547 (2002) 219, hep-ph/0210197; W. Grimus, L. Lavoura, Phys. Lett. B 572 (2003) 189, hep-ph/0305046; W. Grimus, L. Lavoura, J. Phys. G 30 (2004) 73, hep-ph/0309050; W. Grimus, L. Lavoura, J. Phys. G 30 (2004) 1073, hep-ph/0311362; Y. Koide, Phys. Rev. D 69 (2004) 093001, hep-ph/0312207; R.N. Mohapatra, JHEP 0410 (2004) 027, hep-ph/0408187; R.N. Mohapatra, Phys. Lett. B 615 (2005) 231, hep-ph/0502026. [23] R.L. Mills, C.N. Yang, Phys. Rev. D 96 (1954) 191. [24] Y. Koide, Lett. Nuovo Cimento 34 (1982) 201; Y. Koide, Phys. Lett. B 120 (1983) 161; Y. Koide, Phys. Rev. D 28 (1983) 252. [25] P.F. Harrison, W.G. Scott, hep-ph/0508012. Physics Letters B 628 (2005) 106–112 www.elsevier.com/locate/physletb

Spontaneous Lorentz violation and nonpolynomial interactions

B. Altschul, V. Alan Kostelecký

Physics Department, Indiana University, Bloomington, IN 47405, USA Received 13 June 2005; received in revised form 9 September 2005; accepted 9 September 2005 Available online 23 September 2005 Editor: H. Georgi

Abstract Gauge-noninvariant vector field theories with superficially nonrenormalizable nonpolynomial interactions are studied. We show that nontrivial relevant and stable theories have spontaneous Lorentz violation, and we present a large class of asymptotically free theories. The Nambu–Goldstone modes of these theories can be identified with the photon, with potential experimental implications.  2005 Elsevier B.V. All rights reserved.

1. Introduction An interesting and challenging issue that has received little attention to date is the extent to which Lorentz violation might be generic or even ubiquitous Experiments show that nature is well described at in prospective fundamental theories. The present work presently accessible energies by two field theories: the initiates a study of this issue. For definiteness, we fo- Standard Model (SM) of particle physics, and Ein- cus attention on the elegant possibility that Lorentz stein’s General Relativity. These are expected to arise symmetry is spontaneously broken in the underlying as the low-energy limit of a fundamental theory of theory [6]. The basic idea is that interactions in the 19 quantum gravity at the Planck scale, MP 10 GeV. underlying theory induce nonzero vacuum expectation The discrepancy between MP and attainable energies values for one or more Lorentz tensors, which can makes experimental signals from this underlying the- be regarded as background quantities in the vacuum ory difficult to identify, but one promising class of throughout spacetime. observables involves violations of Lorentz symmetry The analysis in this work adopts the methods of arising from new physics at the Planck scale.1 Lagrangian-based quantum field theory, in which observable effects of Lorentz violation are described by an effective low-energy field theory [7–9], and it as- E-mail address: [email protected] (V.A. Kostelecký). sumes that the issue of obtaining the required hierar- 1 For a variety of recent reviews see, e.g., Refs. [1–5]. chy [7,10,4] for the associated coefficients for Lorentz

0370-2693/$ – see front matter  2005 Elsevier B.V. All rights reserved. doi:10.1016/j.physletb.2005.09.018 B. Altschul, V.A. Kostelecký / Physics Letters B 628 (2005) 106–112 107 violation can be addressed. In this language, the fun- through the use of the Polchinski formulation of the damental theory may have many types of fields and RG [28]. Since any Lorentz violation in nature is a interactions, including ones that are nonrenormaliz- weak effect, we restrict attention to the linearized form able at the level of the effective low-energy theory. of the RG transformation, in which only terms that A comprehensive study of the likelihood of Lorentz vi- are first-order in the interaction are retained. The lit- olation at this level appears infeasible at present. How- erature for the Lorentz-invariant scalar case contains ever, in Lorentz-invariant scalar field theories, certain some discussion about the persistence of the nonpoly- nonpolynomial and hence superficially nonrenormal- nomial interactions when the full nonlinear RG is con- izable interactions have been shown to be relevant in sidered [29]. However, a nonperturbative demonstra- the sense of the renormalization group (RG) by study- tion to the contrary would require showing that the ing the natural cutoff dependences of the coupling nonpolynomial potentials can be expanded as a sum of constants [11]. The Gaussian fixed point of the RG an infinite number of irrelevant RG modes, a challeng- flow is ultraviolet-stable along certain directions in the ing task. Moreover, there is evidence for persistence: parameter space of interactions, and these directions in the limit where the number of scalar-field compo- correspond to nontrivial asymptotically free theories. nents is large, it is known that the RG equations for the Here, we exploit this idea by generalizing the scalar nonpolynomial modes can be integrated into a region analysis to the case of vector fields and investigating far from the fixed point [30]. This suggests that the the occurrence of spontaneous Lorentz violation in the novel potentials exist outside the linearized regime for resulting theories. finite-component fields as well. Other generalizations The prototypical field theories for Lorentz viola- of the original results include Refs. [31,32] and a study tion with a vector field Bµ are the so-called bumblebee of the impact of Lorentz violation on asymptotically models [12].2 These involve a gauge-noninvariant po- free scalar and spinor field theories [33]. No evidence µ µ tential V(B Bµ) that has a minimum at nonzero B exists for relevant nonpolynomial theories involving inducing spontaneous Lorentz violation. We consider spinor fields, but a leading-order analysis shows that a generic model of this type with a conventional Lorentz violation is a prerequisite for their existence, a Maxwell-type kinetic term and an arbitrary nonpoly- conclusion compatible with the results for vector fields nomial potential, as might arise from a fundamental obtained below. theory, and we take RG relevance of the interactions and stability of the associated quantum field theory as practical criteria determining acceptable models for 2. Running the bumblebee our study. These assumptions make it possible to address the ubiquity of Lorentz violation in a definite Consider a theory for a vector-valued ‘bumblebee’ context. Surprisingly, we find that consistent stable potential field Bµ with Lagrange density relevant theories of this type must have spontaneous 1 µν µ Lorentz violation and that a large class of such theories L =− B Bµν + V B Bµ , (1) arises from superficially nonrenormalizable bumble- 4 bee models. Moreover, these theories naturally contain where Bµν = ∂µBν − ∂νBµ is the field strength. The Nambu–Goldstone (NG) modes [25] associated with potential V is assumed to be representable as a power spontaneous Lorentz violation that can be identified series in B2, and it violates gauge invariance. For the with the photon, a result with potential experimental simple case V = m2B2/2, the theory describes a free consequences. massive vector boson. Potentials such as V = λ(B2 − To perform the analysis, we study the flow in the b2)2/4 for constant b2 produce bumblebee models de- Wilson formulation of the RG [26], in which the the- scribing spontaneous Lorentz violation. Here, we con- ory is considered with a momentum cutoff. An analy- sider a theory with more general nonpolynomial V .In sis with a more general cutoff is also possible [27], what follows, we introduce a momentum cutoff Λ representing the only scale in the system, and we insert appropriate powers of Λ in V to render dimensionless 2 Recent literature includes Refs. [9,13–24]. all the couplings [11]. 108 B. Altschul, V.A. Kostelecký / Physics Letters B 628 (2005) 106–112

To study the RG flow for the theory (1), it is con- theories have nontrivial continuum limits. It is conve- venient to Wick rotate the variables and operate in nient to parametrize the coupling g as Euclidean space. For scalar fields, the Euclidean RG c equations are known [34]. The RG calculations for the g = , (6) κ − 2 Wick-rotated vector field parallel those for an SO(4) multiplet of four scalar fields, except for a minor mod- so the sign of c gives the sign of the slope of Vκ at = ification arising from the structure of the kinetic term. z 0. At tree level, the transversality of the bumblebee ki- Substantial additional complexities arise when the netic term ensures there are only three propagating theory in Euclidean space is reconverted to Minkowski 2 = 4 j 2 modes contained in the four-component field Bµ.This spacetime. For the scalar-field case, φ j=1(φ ) feature remains true in the RG analysis because no ki- is guaranteed to be positive, but the analogous quan- 2 µ tity −B =−B Bµ for vector fields can be either netic contributions to the two-point function for the µ µ fourth mode can arise. The relevant diagrams either positive (spacelike B ) or negative (timelike B ). have both external legs on the same vertex, yielding a This complicates the analysis of the stability of these tadpole and no kinetic contribution, or they have an in- theories. Furthermore, any nontrivial, stable, asymp- ternal line for the fourth mode, which vanishes. Since totically free theory of this type necessarily involves only three modes propagate rather than four, we must spontaneous Lorentz breaking. This follows because replace the zero-separation scalar propagator z can now be positive or negative. If Vκ either increases or decreases at z = 0, then there is a state 4 jk µ jk d p δ with nonzero B having lower energy than a state ∆ (0) = (2) F (2π)4 p2 with Bµ = 0, so Bµ develops a Lorentz-violating vac- |p|<Λ uum expectation value. If instead Vκ has a vanishing with the transverse propagator derivative at z = 0, then c = 0, the potential vanishes identically, and the theory is trivial. d4p (δµν − pµpν/p2) Dµν(0) =− F (2π)4 p2 |p|<Λ 3. Stability analysis 3 3Λ2 =− ∆µν(0) =− δµν. (3) 4 F 64π 2 To determine which κ correspond to stable theo- However, the extra factor of 3/4 relative to the scalar ries, we examine the asymptotic behavior of Vκ as case contributes only to the overall normalization of z →+∞and z →−∞. For large positive z, the as- the vector field. ymptotic formula The asymptotically free solutions of the linearized (β)zα−β ez RG equations are [11] M(α; β; z) ≈ (7) (α) 2 = 4 − ; ; − Vκ B gΛ M(κ 2 2 z) 1 . (4) holds, with all corrections being suppressed by powers − Here, M(α; β; z) is the confluent hypergeometric of z 1. Also useful is the Kummer formula (Kummer) function [35], − M(α; β;−z) = e zM(β − α; β; z), (8) α z α(α + 1) z2 M(α; β; z) = 1 + + +···, (5) which is an exact relation. β 1! β(β + 1) 2! Consider first the case of a spacelike expectation 2 2 2 µ 2 with z =−32π B /3Λ . The parameter κ in Eq. (4) value for B , so that the minimum of Vκ has −B > 0 describes the growth of the coupling constant g when and hence z>0. This parallels the case of positive the cutoff scale Λ is changed. If all modes with mo- vacuum expectation value for φ2 [11]. Suppose c<0. menta in the range Λ1 < |p| <Λ0 are integrated out The potential is then decreasing with z at z = 0, so of the theory and the fields rescaled accordingly, then if Vκ diverges to positive infinity for large z then at 2κ the renormalized g shifts to g(Λ0/Λ1) . Asymptot- least one stable minimum must exist for z>0. Eq. (7) ically free theories must have κ>0, and only these shows Vκ indeed diverges as z →+∞, and its sign B. Altschul, V.A. Kostelecký / Physics Letters B 628 (2005) 106–112 109 is determined by the sign of c/(κ − 2) (κ − 2) or, In summary, we find that theories having poten- equivalently, by the sign of − (κ − 1). This is pos- tials Vκ with 0 <κ<1 are stable, with minima lying itive when κ lies in one of the open intervals (0, 1), at spacelike values of Bµ. Theories with 1 κ<2 (−2, −1), (−4, −3), etc. Since a nontrivial theory and 2 <κ 4 are unstable, while the case κ = 2is must have κ>0, the only relevant range is 0 <κ<1. trivial. Stability is restored for κ>4, and the vacuum In contrast, if c>0, then there are no stable potentials value of Bµ becomes timelike. A timelike vacuum with local minima on the positive z-axis because either value for Bµ may in fact be favored because the po- Vκ →−∞as z →+∞or Vκ increases monotoni- tentials leading to this form of symmetry breaking are cally. more relevant. The potentials Vκ of interest that generate a space- The potentials for 0 <κ<1 are discussed in like expectation value for Bµ are therefore those with Ref. [11]. The hypergeometric functions M(κ−2; 2; z) c<0 and 0 <κ<1, corresponding to g>0inthe have minima with z<10 for nearly all κ, and the range |c/2| 4, the po- In the range 0 <κ<1, the hypergeometric func- tential may possess multiple local minima at negative | | |z| = ; ; tion grows faster with z than e M(2 2 z). Since values of z. However, the exponential damping factor g>0, it follows that Vκ is positive for all negative z. e−|z| in Eq. (9) ensures that the one with the small- These theories are therefore stable, with a spacelike est |z| is always the global minimum. As κ increases, expectation value for Bµ and spontaneous Lorentz vi- the wavelength of the oscillations in Vκ (z) decreases, olation. and the location of the minimum is pushed to smaller Next, consider the case of a timelike expectation values of |z|. This location may be calculated numer- µ − 2 value for B , for which the vacuum has B < 0 and ically, and it is roughly given by z ≈−6/(κ − 3). z<0. Any stable theories of this type must have κ min The value Vκ,min of Vκ at zmin is consistently close to 1, and also c>0 is required for stability as z →+∞. 2 Vκ, ≈−0.1gΛ (κ − 3). Moreover, as z →−∞the asymptotic behavior of the min hypergeometric function is determined by

2−κ 4. Features and implications c −| | c (2)|z| e z M − κ; ;|z| ≈ . − 4 2 − − κ 2 κ 2 (4 κ) We have shown that Bµ must develop a Lorentz- (10) violating vacuum expectation value in any nontrivial For 1 κ<2, this diverges to negative infinity for stable theory. There are many potential implications of large negative z, so the potentials in this range are un- this scenario. An immediate one concerns the interpre- stable. For κ = 2, the potential vanishes. For κ>2, we tation of the excitations about the vacuum. Denoting 2−κ µ= µ find Vκ →−c/(κ − 2) as z →−∞, since |z| → 0. the vacuum value as B b , we may parame- The theory is therefore stable if there exists a z<0for trize Bµ as which V −c/(κ − 2), which occurs if and only if κ Bµ = (1 + ρ)bµ + Aµ, (11) M(4 − κ; 2;|z|) has a root. This function cannot have µ a root unless κ>4 because otherwise all the terms in where b Aµ = 0. Defining Fµν = ∂µAν − ∂νAµ,the ; ;| | µ − 1 µν the sum (5) are positive. However, M(α 2 z ) does kinetic term for A is found to be 4 F Fµν . More- indeed have a root for α sufficiently large and nega- over, at lowest order only fluctuations in ρ cause tive. The absolute value of the smallest root decreases changes in the potential term in the energy, so there as α becomes more negative [36]. In fact, there is a is no mass term for Aµ. We therefore can identify Aµ root for any α<0, i.e., any κ>4: if −1 <α<0, with the photon field. then the asymptotic value of M(α; 2;|z|) is negative The notation in Eq. (11) is chosen to match that of and so it must possess a root. Ref. [19], which provides a general description of the 110 B. Altschul, V.A. Kostelecký / Physics Letters B 628 (2005) 106–112

2 µ 2 NG modes associated with spontaneous Lorentz viola- of B about its vacuum value b bµ = b . This curva- tion and presents the complete effective action for Aµ ture is always proportional to gΛ2. However, g must in various space–times. In this context, excitations be small for the linearized calculations to be valid, around the vacuum of the field Aµ are the NG modes while the natural physical cutoff scale Λ is expected to associated with spontaneous Lorentz breaking, while be very large, possibly of the order of MP . It is there- vacuum excitations of ρ are the NG modes for sponta- fore reasonable to expect that the particles associated neous diffeomorphism breaking. The masslessness of with the fluctuations of B2 are unobservable in a low- the photon follows directly from this interpretation as energy theory. A large value of κ implies both small b2 a consequence of the breaking of Lorentz invariance3 and a large mass for these particles, so weak Lorentz rather than the existence of gauge symmetry. The ef- violation is naturally associated with unobservability fective action also contains higher-order corrections of the massive mode. to conventional electrodynamics that could be sought For κ>4, the potential remains finite at infinite in experimental tests. The superficially nonrenormal- positive timelike values of B2. The energy density izable couplings are suppressed by powers of Λ and required to shift the expectation value of the field ar- vanish in the continuum limit. bitrarily far from its vacuum value therefore remains µ Since at leading order the potential A satisfies the finite. However, if Λ is of the order of MP , this energy µ = 4 orthogonality condition b Aµ 0, the equivalent con- density is proportional to gMP . This is suppressed by ventional electrodynamics must be defined in an axial only one power of g relative to the naive scale of the or generalized axial gauge. One check on the quantum cosmological constant. The energy density required to equivalence between electrodynamics and the theory generate these large field values exists only in the early (1) at leading order is provided by a comparison of Universe, when high-temperature corrections are ex- the corresponding transverse propagators. In fact, the pected to restore the broken Lorentz symmetry. Euclidean propagator for electrodynamics subject to Another interesting feature of the timelike case is µ 4 the gauge condition b Aµ = 0is the inverse variation with κ of the locations of the min- V z ≈− /(κ − ) µν ima of the κ potentials, min 6 3 . Suppose D (x − y) the underlying theory contains a sum of nonpolyno- F 4 d p − · − mial interactions Vκ with values of κ ranging to a max- =− e ip (x y) (2π)4 imum κmax and with coefficients for the different Vκ potentials controlled by the details of the fundamen- 1 b2 bµpν + bνpµ × δµν + pµpν − . tal physics. When the spontaneous Lorentz breaking p2 (b · p)2 (b · p) is studied in a lower-energy effective field theory with (12) a smaller value of the cutoff, then the effective poten- Within the subspace of propagating modes, this is tial is dominated by those Vκ with κ in the vicinity equivalent to the corresponding propagator for the the- of κmax because these potentials grow the most rapidly ory (1). as the cutoff decreases. The magnitude of the Lorentz-√ µ The formulation of the theory (1) involves three violating vector b is then proportional to 1/ κmax. propagating modes, and three modes also appear af- Since κmax represents the maximum in a potentially ter the spontaneous Lorentz breaking. Two are photon large collection of κ values, it can naturally be big. modes contained in Aµ. The third mode is massive, This could provide a partial explanation for the small with excitations that change the value of B2 and hence size of any Lorentz violation in nature. the potential. The curvature at the global minimum of Additional physical implications of our scenario the potential determines the mass of the fluctuations can be explored by extending the theory (1) to include couplings between Bµ and one or more other fields. For example, introducing a Dirac fermion field ψ of- 3 In the context of electrodynamics without physical Lorentz vi- fers various possibilities for interactions between Bµ olation, this interpretation has a long history. See, for example, Ref. [37]. and fermion bilinears. Note that gauge-noninvariant 4 For a discussion of the propagator in axial gauges see, for exam- couplings are acceptable here, unlike the usual case ple, Ref. [38]. of quantum electrodynamics (QED), because the ini- B. Altschul, V.A. Kostelecký / Physics Letters B 628 (2005) 106–112 111 tial theory (1) has no gauge invariance. Whatever the Acknowledgements nature of the other fields being introduced, Lorentz- violating terms for them appear following spontaneous This work was supported in part by DOE grant DE- Lorentz violation when Bµ is replaced with its vac- FG02-91ER40661 and NASA grant number NAG3- uum value bµ in the interactions. If the additional 2914. fields are identified with ones in the SM or in gravity, the resulting Lorentz violation is contained in the Standard-Model Extension (SME) [8,9]. For example, References ¯ µ a simple choice of interaction is La ∝ Bµψγ ψ, par- [1] V.A. Kostelecký (Ed.), CPT and Lorentz Symmetry III, World alleling the usual QED current coupling. When Bµ Scientific, Singapore, 2005. acquires a vacuum value, this interaction generates the [2] R. Bluhm, hep-ph/0506054. usual coupling of Aµ to the current along with a coef- [3] D. Mattingly, gr-qc/0502097. ficient for Lorentz violation of the aµ type in the min- [4] G. Amelino-Camelia, C. Lämmerzahl, A. Macias, H. Müller, imal Lorentz-violating QED extension. For a single gr-qc/0501053. [5] S. Sarkar, Mod. Phys. Lett. A 17 (2002) 1025. fermion, a constant coefficient aµ is unobservable, but [6] V.A. Kostelecký, S. Samuel, Phys. Rev. D 39 (1989) 683; when fermion flavor changes are present coefficients V.A. Kostelecký, R. Potting, Nucl. Phys. B 359 (1991) 545. of this type can produce observable effects [39,40]. [7] V.A. Kostelecký, R. Potting, Phys. Rev. D 51 (1995) 3923. More exotic couplings could also be countenanced, [8] D. Colladay, V.A. Kostelecký, Phys. Rev. D 55 (1997) 6760; ¯ µ such as an axial-vector coupling Lb ∝ Bµψγ5γ ψ D. Colladay, V.A. Kostelecký, Phys. Rev. D 58 (1998) 116002. [13]. When Lorentz symmetry is spontaneously bro- [9] V.A. Kostelecký, Phys. Rev. D 69 (2004) 105009. [10] J. Collins, A. Perez, D. Sudarsky, L. Urrutia, H. Vucetich, Phys. ken, this induces a coefficient for Lorentz violation of Rev. Lett. 93 (2004) 191301. the bµ type in the SME, along with a remnant inter- [11] K. Halpern, K. Huang, Phys. Rev. Lett. 74 (1995) 3526; action involving the fluctuations about bµ. Note that K. Halpern, K. Huang, Phys. Rev. D 53 (1996) 3252. all these couplings are known to be renormalizable at [12] V.A. Kostelecký, S. Samuel, Phys. Rev. Lett. 63 (1989) 224; one loop [41]. If the massless excitations are identified V.A. Kostelecký, S. Samuel, Phys. Rev. D 40 (1989) 1886. µ [13] V.A. Kostelecký, R. Lehnert, Phys. Rev. D 63 (2001) 065008. with the photon A as above, then a novel coupling to [14] T. Jacobson, D. Mattingly, Phys. Rev. D 64 (2001) 024028; the photon arises. C. Eling, T. Jacobson, Phys. Rev. D 69 (2004) 064005. Many other types of couplings for Bµ can also [15] P. Kraus, E.T. Tomboulis, Phys. Rev. D 66 (2002) 045015. be considered. Although beyond our present scope, it [16] J.W. Moffat, Int. J. Mod. Phys. D 12 (2003) 1279. would be of definite interest to explore features in- [17] B.M. Gripaios, JHEP 0410 (2004) 069. [18] J.L. Chkareuli, C.D. Froggatt, R.N. Mohapatra, H.B. Nielsen, troduced by derivative couplings within this frame- hep-th/0412225. work. These could provide insight about the struc- [19] R. Bluhm, V.A. Kostelecký, Phys. Rev. D 71 (2005) 065008. ture of higher-derivative terms in the effective action [20] S.M. Carroll, E.A. Lim, Phys. Rev. D 70 (2004) 123525; and hence about the causality and stability of theo- E.A. Lim, Phys. Rev. D 71 (2005) 063405. ries with Lorentz violation [13], and they could have a [21] M.L. Graesser, A. Jenkins, M.B. Wise, Phys. Lett. B 613 (2005) 5. bearing on predicted Lorentz-violating effects within [22] C. Heinicke, P. Baekler, F.W. Hehl, gr-qc/0504005. the photon sector [42]. It would also be of interest [23] O. Bertolami, J. Paramos, hep-th/0504215. to investigate gravitational couplings, including back- [24] J.W. Elliott, G.D. Moore, H. Stoica, hep-ph/0505211. ground spacetimes [9]. Incorporating gravity typically [25] Y. Nambu, Phys. Rev. Lett. 4 (1960) 380; makes RG calculations of the type adopted here im- J. Goldstone, Nuovo Cimento 19 (1961) 154; J. Goldstone, A. Salam, S. Weinberg, Phys. Rev. 127 (1962) practical, but comparatively simple cases such as con- 965. formally flat backgrounds may be tractable. In any [26] K.G. Wilson, K. Kogut, Phys. Rep. C 12 (1974) 75. event, the occurrence of spontaneous Lorentz viola- [27] V. Periwal, Mod. Phys. Lett. A 11 (1996) 2915. tion as a necessary feature in the above stable and [28] J. Polchinski, Nucl. Phys. B 231 (1984) 269. relevant theories with nonpolynomial potentials sug- [29] T.R. Morris, Nucl. Phys. B 495 (1997) 477; T.R. Morris, Phys. Rev. Lett. 77 (1996) 1658; gests that Lorentz violation might indeed be generic K. Halpern, K. Huang, Phys. Rev. Lett. 77 (1996) 1659. in a large class of underlying theories at the Planck [30] H. Gies, Phys. Rev. D 63 (2001) 065011. scale. [31] A. Bonanno, Phys. Rev. D 62 (2000) 027701. 112 B. Altschul, V.A. Kostelecký / Physics Letters B 628 (2005) 106–112

[32] V. Branchina, Phys. Rev. D 64 (2001) 043513. [40] V.A. Kostelecký, M. Mewes, Phys. Rev. D 69 (2004) 016005; [33] B. Altschul, Nucl. Phys. B 705 (2005) 593, hep-th/0407173. V.A. Kostelecký, M. Mewes, Phys. Rev. D 70 (2004) [34] F.J. Wegner, A. Houghton, Phys. Rev. A 8 (1973) 401; 031902(R); A. Hasenfratz, P. Hasenfratz, Nucl. Phys. B 270 (1986) 687. V.A. Kostelecký, M. Mewes, Phys. Rev. D 70 (2004) 076002. [35] M. Abramowitz, I.A. Stegun, Handbook of Mathematical [41] V.A. Kostelecký, C.D. Lane, A.G.M. Pickering, Phys. Rev. Functions, Dover, New York, 1977, pp. 504–505. D 65 (2002) 056006; [36] E. Jahnke, F. Emde, F. Lösch, Tables of Higher Functions, V.A. Kostelecký, A.G.M. Pickering, Phys. Rev. Lett. 91 (2003) McGraw–Hill, New York, 1960, p. 283. 031801. [37] J.D. Bjorken, Ann. Phys. 24 (1963) 174; [42] S.M. Carroll, G.B. Field, R. Jackiw, Phys. Rev. D 41 (1990) Y. Nambu, Prog. Theor. Phys. Suppl. Extra 190 (1968). 1231; [38] G. Leibbrandt, Rev. Mod. Phys. 59 (1987) 1067. V.A. Kostelecký, M. Mewes, Phys. Rev. Lett. 87 (2001) [39] V.A. Kostelecký, Phys. Rev. Lett. 80 (1998) 1818; 251304; V.A. Kostelecký, Phys. Rev. D 61 (2000) 016002; V.A. Kostelecký, M. Mewes, Phys. Rev. D 66 (2002) 056005. V.A. Kostelecký, Phys. Rev. D 64 (2001) 076001. Physics Letters B 628 (2005) 113–124 www.elsevier.com/locate/physletb

The pressure of the SU(N) lattice gauge theory at large N

Barak Bringoltz, Michael Teper

Rudolf Peierls Centre for Theoretical Physics, University of Oxford, 1 Keble Road, Oxford OX1 3NP, UK Received 1 July 2005; accepted 25 August 2005 Available online 22 September 2005 Editor: J.-P. Blaizot

Abstract We calculate bulk thermodynamic properties, such as the pressure, energy density, and entropy, in SU(4) and SU(8) lattice gauge theories, for the range of temperatures T 2.0Tc and T 1.6Tc, respectively. We find that the N = 4, 8 results are very close to each other, and to what one finds in SU(3), and are far from the asymptotic free-gas value. We conclude that any explanation of the high-T pressure (or entropy) deficit must be such as to survive the N →∞limit. We give some examples of this constraint in action and comment on what this implies for the relevance of gravity duals.  2005 Elsevier B.V. All rights reserved.

PACS: 12.38.Gc; 12.38.Mh; 25.75.Nq; 12.38.Gc; 11.15.Ha; 11.25.Tq; 11.10.Wx; 11.15.Pg

1. Introduction mental results have, however, challenged this ‘simple’ picture (for example, see [1] and references therein), The thermodynamic properties of quantum chro- and point to a picture of the ‘plasma’ as a very good modynamics (QCD), besides being of fundamental fluid in the accessible range of T above Tc. In fact, interest, are currently at the centre of intense exper- numerical lattice results had already demonstrated the imental research. One of the most interesting phe- inadequacy of the simple quark–gluon plasma pic- nomena has to do with the range of temperatures, T , ture some time ago. Such lattice calculations, both for the pure gauge case [2] and with different kinds of above the phase transition (or crossover) at T = Tc, where the theory deconfines and chiral symmetry is fermions [3], found a large deficit in the pressure and restored. Traditionally, the description of this transi- entropy as compared to the Stephan–Boltzmann pre- tion assumed that the hadronic phase gives way to dictions for a free gluon gas (for pure glue), which a plasma, whose physical degrees of freedom are remained at the level of more than 10% even at tem- ∼ weakly interacting quarks and gluons. Recent experi- peratures as high as T 4Tc. Further evidence that points in the same direction is the survival of hadronic states above Tc, as seen in recent lattice simulations E-mail address: [email protected] (B. Bringoltz). (for example, see [4] and references therein).

These lattice calculations, and more recent exper- priate. We then present our results for the pressure, imental observations, have attracted considerable at- entropy and related quantities. We discuss the impli- tention (see, e.g., [5] for a review). Approaches have cations of our findings in the concluding section. ranged from modeling the system in terms of non- interacting quasi-particles with the quantum numbers of quarks and gluons but with temperature-dependent 2. Lattice setup and methodology masses [6,7], to using higher-order perturbation theory (restricted by infrared divergences), sometimes in- The theory is defined on a discretised periodic cluding nonperturbative contributions on the dimen- 3 × Euclidean four-dimensional space–time with Ls Lt sionally reduced 3D Euclidean lattice [8], large re- sites. Here Ls,t is the lattice extent in the spatial and summations (e.g., [9] and references therein), or, more Euclidean time directions. The partition function recently, a description [10] in terms of a large num- E ber of loosely bound states that survive deconfinement Z(T,V ) = exp − s T and come in various representations of the gauge and s flavor groups, and where one can use, for example, the F fV lattice masses measured in [11]. = exp − = exp − (2.1) In this Letter we ask whether this pressure (and en- T T tropy) deficit is a dynamical feature not just of SU(3) defines the free energy F and the free energy den- but of all SU(N) gauge theories—and, in particular, sity, f , and can be expressed as a Euclidean path inte- whether it survives the N →∞ limit. In this limit gral the theory becomes considerably simpler, although not (yet) analytically soluble, and so what happens there Z(T,V ) = DU exp(−βSW). (2.2) should strongly constrain the possible dynamics underlying the phenomenon. For example, in that limit −1 3 Here T = (aLt ) is the temperature and V = (aLs) supersymmetric SU(N) gauge theories become dual to is the spatial volume. When we change β,soasto weakly coupled gravity models, and in that context we change the lattice spacing a(β), we change both T and recall the frequently mentioned prediction [12], that V ,ifL and L are kept fixed. In the large-N limit, N = s t the pressure in the strong-coupling limit of the 4 the ’t Hooft coupling λ = g2N is kept fixed, and so =∞ and N supersymmetric gauge theory is 3/4of we must scale β = 2N 2/λ ∝ N 2 in order to keep the its Stephan–Boltzmann value, which is similar to the lattice spacing fixed in that limit. We use the standard deficit, referred to above, that one finds in the non- Wilson action SW given by supersymmetric case. To address this question we calculate the pressure 1 SW = 1 − Re Tr UP . (2.3) for T 2Tc in SU(4), and SU(8) lattice gauge theo- N P ries and compare the results to similar SU(3) calculations available in the literature (which we supple- Here P is a lattice plaquette index, and UP is the ment where it is useful to do so). Recent calculations plaquette variable obtained by multiplying link vari- of various properties of SU(N) gauge theories [13] ables along the circumference of a fundamental pla- have demonstrated that SU(8) is in fact very close to quette. We perform Monte Carlo simulations, using SU(∞) for most purposes and have provided informa- the Kennedy–Pendelton heat bath algorithm for the tion on the location, βc, of the deconfining transition link updates, followed by five over-relaxations of all for various Lt and N [14,15]. Thus our calculations the SU(2) subgroups of SU(N). should provide us with an accurate picture of what happens to the pressure at N =∞. 2.1. The method used In the next section we summarise the lattice setup, the relevant thermodynamics, and provide numerical In lattice calculations of bulk thermodynamics, checks that our system is large and homogeneous one can choose to use either the “integral” method enough for our thermodynamic relations to be appro- (e.g., [2]) or the “differential” method (e.g., [16] or a B. Bringoltz, M. Teper / Physics Letters B 628 (2005) 113–124 115

= 3 new variant [17]) or one can attempt a direct evalu- where Np 6Lt Ls is the total number of plaquettes ation of the density of states (e.g., [18]). We choose and up ≡ Re Tr UP /N. So the pressure can be ob- to use the first of these methods since the numeri- tained by simply integrating the average plaquette over cal price involved in using larger values of N drives β. This pressure has been defined relative to that of the us to smaller Lt , which means that the lattice spac- unphysical ‘empty’ vacuum and will therefore be ul- ing is too coarse (about 0.15 fm) for the differential traviolet-divergent in the continuum limit. To remove method. We have performed preliminary checks for this divergence we need to define the pressure rela- the applicability of the Wang–Landau algorithm [19] tive to that of a more physical system. We shall follow for the evaluation of the density of states in the SU(8) convention and subtract from p(T ) its value at T = 0, gauge theory, but found it numerically too costly for calculated with the same value of the cut-off a(β). the present work. Thus our pressure will be defined with respect to its The properties we will concentrate on are the pres- T = 0 value. Doing so we obtain from Eqs. (2.9), (2.8) sure p, the energy density per unit volume , and the β entropy S, as a function of temperature. These are 4 − = − given by a p(T ) p(0) 6 dβ up T up 0 , (2.10) β ∂ T 0 p = T log Z(T,V ) = log Z(T,V ) =−f, (2.4) 4 ∂V V where up 0 is calculated on some L lattice which = T 2 ∂ is large enough for it to be effectively at T 0. We = log Z(T,V ), (2.5) replace p(T )−p(0) → p(T ), where from now on it is V ∂T understood that p(T ) is defined relative to its value at S − f + p −1 = = , (2.6) T = 0, and we use T = (aLt ) to rewrite Eq. (2.10) V T T as where the second equality in the first and last lines fol- β lows if the system is large and homogeneous, i.e., if V p(T ) = 6L4 dβ u −u . (2.11) is large enough. In addition, it is useful to consider the T 4 t p T p 0 quantity β0 3 ∂ p We remark that when our Ls Lt lattice is in the confin- ∆ ≡ − p = T 5 , 3 4 (2.7) ing phase, then up is essentially independent of Lt ∂T T 4 and takes the same value as on a Ls lattice (see below). which vanishes for an ideal gluon plasma. Again the This should become exact as N →∞but is accurate second equality requires a large enough V . To calcu- enough even for SU(3). Thus as long as we choose β0 late the pressure at temperature T in a volume V with in Eq. (2.11) such that a(β0)Lt > 1/Tc then the inte- lattice cut-off a(β), we express log Z in the integral gration constant, referred to earlier, will cancel. form Finally, we evaluate ∆ in Eq. (2.7) as follows: = T ∆ ∂ p p(T ) log Z(T,V ) = T (2.12) V T 4 ∂T T 4 β ∂β ∂ p 1 ∂ log Z = (2.13) = dβ . (2.8) ∂ log T ∂β T 4 a4(β)L3L ∂β s t ∂β β0 = 4 − 6Lt up(β) 0 up(β) T . (2.14) (There is in general an integration constant, but it will ∂ log a(β) disappear when we regularise the pressure later on in To evaluate ∂ log(a(β))/∂β we can use calculations of this section.) This integral form is useful because it is the string tension, σ , in lattice√ units. For example, in easy to see from Eqs. (2.2), (2.3) that [20] the calculated values of a σ are interpolated in β for various N and one can take the derivative of the in- ∂ log Z terpolated form to use in Eq. (2.14). One could equally =−SW=Npup, (2.9) ∂β well use the calculated mass gap or the deconfining 116 B. Bringoltz, M. Teper / Physics Letters B 628 (2005) 113–124 temperature. All these choices will give the same re- of β values corresponding to T/Tc ∈[0.89, 1.98] for 2 sult up to modest O(a ) differences. SU(4), and to T/Tc ∈[0.97, 1.57] for SU(8). Since we use Lt = 5, while the data for SU(3) in [2] is for 2.2. Average plaquette Lt = 4, 6, 8, we also performed simulations for SU(3) 3 on 20 × 5 lattices with T/Tc ∈[1, 2]. The results are We see from the above that what we need to do presented in Tables 1–3. is to calculate average plaquettes closely enough in β In addition to the finite T calculations we have per- so as to be able to perform the numerical integration formed ‘T = 0’ calculations on 204 lattices for SU(3), in β. And we need the average plaquettes not only and on 164 lattices for SU(4). These have the advan- 3 = on the Lt Ls lattice but also on a reference ‘T 0’ tage of being on the same spatial volumes as the corre- L4 lattice at each value of β. However, we mostly sponding finite T calculations, and we know from pre- need values for β βc, where a(βc)Lt = 1/Tc, since vious calculations [21,22] that, for the range of a(β) p(T ) − p(0) 0 once T

Table 1 Statistics and results of the Monte Carlo simulations for SU(4) βT>0 T = 0 −3 −3 sT (lattice sweeps) × 10 s0 (lattice sweeps) × 10 10.55 0.537478(84) 10 0.537487(81) 5 10.60 0.543862(58) 20 0.543797(25) 15 10.62 0.546212(64) 10 0.546068(33) 10 10.64 0.550279(70) 10 0.548208(16) 20 10.68 0.554213(32) 20 0.552177(16) 20 10.72 0.557649(30) 20 0.555861(14) 20 10.75 0.560051(27) 20 0.558462(13) 20 10.80 0.563923(32) 20 0.562587(16) 20 10.85 0.567592(24) 20 0.566453(17) 20 10.90 0.571107(17) 20 0.570118(16) 20 11.00 0.577707(17) 20 0.576981(11) 20 11.02 0.578985(18) 20 0.578279(11) 20 11.10 0.583911(20) 20 0.583352(12) 20 11.30 0.595398(13) 20 0.595039(10) 20

Table 2 Statistics and results of the Monte Carlo simulations for SU(8) T>0 T = 0 −3 −3 βsT Ls (lattice sweeps) × 10 βs0 (lattice sweeps) × 10 43.90 0.525330(80) 14 5 43.85 0.523819(37)>20 43.93 0.526873(79) 819.5440.528788(18)>20 44.00 0.531307(50) 10 > 20 44.35 0.538491(13)>20 44.10 0.534164(34) 12 7 44.85 0.549794(9)>20 44.20 0.536650(70) 14 5 45.70.565708(4)>20 44.30 0.539181(30) 820 44.45 0.542629(38) 830 44.60 0.545812(35) 820 44.80 0.549968(37) 830 45.00 0.553926(38) 820 45.50 0.562992(28) 12 10 B. Bringoltz, M. Teper / Physics Letters B 628 (2005) 113–124 117

Table 3 Statistics and results of the Monte Carlo simulations for SU(3) βT>0 T = 0 −3 −3 sT (lattice sweeps) × 10 s0 (lattice sweeps) × 10 5.800 0.568664(100) 10 0.567667(29) 11 5.805 0.569688(153) 20 0.568438(23) 11 5.810 0.570624(55) 10 0.569218(18) 11 5.815 0.571297(81) 10 0.569996(26) 11 5.820 0.572205(78) 10 0.570788(16) 11 5.900 0.583058(38) 10 0.581854(20) 11 6.150 0.609377(27) 10 0.608971(8) 11 6.200 0.613966(31) 10 0.613628(13) 11

= will see below. (The same is not true for the finite T Lt 6. This√ is consistent with the fact that the SU(3) 3 calculation on 8 × 5 lattices where it is 1/aT that sets value of Tc/ σ for Lt = 5, 6 are the same within one the scale for finite volume corrections.) We, therefore, sigma [15].√ This is true for SU(8) as well, where the take instead the SU(8) calculations on larger lattices value of Tc/ σ for a = 1/(5Tc), and a = 1/(8Tc),are in [22], and interpolate between the values of β used the same within one sigma [20], and we find no point there, to obtain average plaquettes at the values of β to perform similar√ comparisons there. For SU(4) the we require. To perform this interpolation we fit with value of Tc/ σ at a = 1/(5Tc), 1/(6Tc) is ∼ 5, and the ansatz ∼ 3.7 sigma away from the value at a = 1/(8Tc) [20], which may suggest that in this case T/T (β) at values 2 √ c π G2 4 u (β) =u P.T.(β) + a σ of β that correspond to T 8/(5Tc) will be smaller p 0 p 0 2 12 Nσ when fixing the physical scale with Tc rather than with 8 10 + c4g + c5g , (2.15) the string tension. Nevertheless, the shift between the ∼ P.T. two is at the level of 2%, and will not change the re- where up (β) is the lattice perturbative result to 0 sults presented here. In addition, to fix T/Tc(β) by fix- O 6 = 2 = (g ) from [23] and N 8. Our best fit has χ / dof ing T , requires a larger scale calculation of β (L ,L ) = = c c t s 0.93 with dof 2, and the best fit parameters are c4 that will include evaluation of finite volume correc- −6.92, c = 26.15, and a gluon condensate of G2 = 5 Nσ2 tions, similar to what was done for Lt = 5in[15]. 0.72. In view of the small shifts and the high calculational For the scaling of the lattice spacing with β, needed price, we shall ignore this potential ambiguity in this in Eq. (2.15) and Eq. (2.14) and in√ the temperature Letter. scale, we used the interpolation of a σ as a function of β,asgivenin[20].1 For the temperature scale we 2.3. Finite volume effects need in addition to locate the value of β that corresponds to T = T for the relevant value of L , and for c t = this we have used the values in [15,20]. In the case For N 4, 8, one is able to use lattice volumes much smaller than what one needs for SU(3) [2]. That of SU(3) we compared the resulting T/Tc(β) with this is so for the deconfinement transition, has been that of [2] where the physical scale was set by Tc.We find that the two functions lie on top of each other for explicitly demonstrated in [15,20], and is theoretically expected, much more generally, as N →∞.Themain remaining concern has to do with tunnelling between 1 This is excluding the first three β values in the case confined and deconfined phases near Tc. When V → of SU(4), which are outside the interpolation regime of ∞ tunnelling occurs only at β = βc (in a calculation of [20]. In that case we have performed a new interpola- sufficient statistics) and the system is in the appropri- tion fit to include√ these points as well. This gave the string tensions a σ = 0.3739(15), 0.3440(10), 0.3336(10) and ate pure phase for TTc. On a finite the derivatives −d log a/dβ = 1.83(7), 1.55(7), 1.48(5) for β = volume, where this is no longer true, one minimises 10.55, 10.60, 10.62. finite-V corrections by calculating the average plaque- 118 B. Bringoltz, M. Teper / Physics Letters B 628 (2005) 113–124

Table 4 Finite volume effects for plaquette average in the deconﬁned phase on a Lt = 5 lattice, for SU(8)

βLs = 8 Ls = 10 Ls = 12 Ls = 14 43.95 – 0.529788(100) 0.529944(65) – 44.00 0.531343(45) 0.531307(50) –– 44.10 0.534219(54) –0.534164(34) – 44.20 0.536714(33) –0.536689(54) 0.536650(70) 44.25 – – 0.537954(60) 0.537850(100) 44.30 0.539181(29) ––0.539220(100) 45.50 0.563093(41) –0.562992(28) –

ttes only in field configurations that are confining, for T = Tc one can reliably categorise field configurations TTc. This ensures that as confined or deconfined and hence calculate the av- the system is as close as possible to being ‘large and erage plaquette in just the deconfined phase if one so homogeneous’ as is required in the derivations of this wishes. For our supplementary SU(3) calculations we 2 section. Because the latent heat grows ∝ N [20] the use Ls = 20 which is much smaller in units of ξ.In region δT around Tc in which there is significant tun- practice this means that in this case we are unable to 2 nelling shrinks as δT ∝ 1/N for a given V . Hence, separate phases at T Tc. we can reduce V as N increases without increasing To explicitly confirm our control of finite volume the ambiguity of the calculation. For SU(3), where the effects, we have compared the SU(8) value of up(β) phase transition is only weakly first order, frequent as measured in the deconfined phase of the our 83 × 5 3 × tunneling occurs in the vicinity of Tc in the volume lattice with other Ls 5 results from other stud- we use, and it is not practical to attempt to separate ies [24]. As summarised in Table 4, the results are phases. This will smear the apparent variation of the consistent at the 2σ level. pressure across Tc in the case of SU(3). We now turn to a more detailed discussion of finite volume effects. If ξ is the longest correlation length, 2.3.2. The confined phase in lattice units, in a volume of length L, then finite As we remarked above (see below for explicit evi- volume effects will be negligible if ξ L. In addi- dence) we have upT up0 in the confined phase tion finite volume corrections will be suppressed as and so the contribution in Eq. (2.11) of the range of →∞ N . In our particular context, ξ is given by the in- β where the finite T system is confined is very small. verse mass of the lightest (non-vacuum) state that cou- Nonetheless, we include an integration over that range ples to the loop that winds around the temporal torus. for completeness and so we need to discuss possible In both the confined and deconfined phases, these finite V corrections for this case as well. masses decrease as T → Tc. Therefore, the largest 3 × = In the confined phase, on an Ls 5 lattice at T length scale is set by the masses at T = Tc.AsN − Tc ,thevalueofξ is about 9.5 lattice spacings for increases these masses increase towards their limits, SU(3), but drops to about 5 and 3.5 for SU(4) and 2 with 1/N corrections that are quite large [20]. SU(8), respectively [20]. This leaves our choice of Ls still reasonable for SU(4) but somewhat worse for 2.3.1. The deconfined phase SU(8).InTable 5 we provide a finite volume check for 3 × In the deconfined phase, on an Ls 5 lattice at the latter case that proves reassuring. = + T Tc ,thevalueofξ is about 12.5 lattice spac- Finally we return to our earlier comment that for the ings for SU(3), while it is about 5.2, and 2.4 lattice ‘T = 0’ L4 lattice calculations, a size L = 8inSU(8) spacings for SU(4), and SU(8), respectively [20].This would not be large enough. This is demonstrated, for suggests that our choice of Ls = 16 for SU(4) and our largest β-value, in Table 6, where we also present Ls = 8 for SU(8) should be adequate. In addition, it is the value of Lt × T/Tc(β) (in our Lt = 5 calcula- 4 known from calculations of Tc [14,15,20] that on such tions). In the confined Ls lattice, finite volume effects lattices the tunnelling is sufficiently rare that even at will be suppressed when the latter is much smaller B. Bringoltz, M. Teper / Physics Letters B 628 (2005) 113–124 119

Table 5 Finite volume effects for plaquette average in the conﬁned phase on a Lt = 5 lattice, for SU(8)

βLs = 8 Ls = 10 Ls = 12 Ls = 14 43.90 0.525750(87) –0.525613(54) 0.525425(90) 43.95 – 0.527240(34) 0.527275(48) 0.527280(50) 44.00––0.528867(33) 0.528810(50) 44.10––0.531880(45) 0.531900(60)

Table 6 4 Finite volume effects for plaquette average in the conﬁned phase on a L lattice, for SU(8). The last column is for Lt = 5

βLs = 8 Ls = 10 Ls = 16 Lt × T/Tc 44.00 0.528876(39) 0.528788(18) –5.05 45.70 0.566089(23) –0.565708(4) 8.20

than Ls . Clearly, for β = 45.70 and Ls = 8, this is not In presenting our results for the pressure, we shall the case. normalize to the lattice Stephan–Boltzmann result By contrast, for SU(4) the finite volume effects given by seems not to be large on the 164 lattice as we checked 2 for our largest value of β = 11.30. There the value 4 = 2 − π p/T Nc 1 RI(Lt ). (3.1) of the plaquette on a 204 lattice is 0.595014(4) [21], free-gas 45 ∼ which is consistent within 2.3σ with the value pre- Here RI includes the effects of discretization errors in sented in Table 1. This is in spite of the fact that for this the integral method [25,26]. For large values of Lt , coupling Lt × T/Tc = 10, and is not so much smaller and an infinite volume, it is given by than Ls = 16. 8 π 2 5 π 4 1 6 RI(Lt ) = 1 + + + O . 21 Lt 21 Lt Lt 3. Results (3.2) Since some values of Lt discussed in this Let- To obtain the pressure from the values of the av- ter are not very large, we shall use the full correc- erage plaquette presented in Tables 1–3 we need to tion, which includes higher orders in 1/Lt , instead perform the integration in Eq. (2.11), which we do of Eq. (3.2). This was calculated numerically for the by numerical trapezoids. We have already remarked infinite volume limit in [25] for Lt = 4, 6, 8, and we that the contribution to the pressure from the confined supplement this calculation, with the same numerical phase is negligible. In Table 7 we provide some accu- routines [26], for other values of Lt . A summary of rate evidence for this. We show the values of the aver- RI(Lt ) in the infinite volume limit is given in Table 8. 4 age plaquette on L lattices, corresponding to T 0, We find that the full correction for Lt = 5isa 3 × ∼ as well as the values on Ls 5 lattices at T Tc, 21% effect, which, without this normalisation, might with the latter obtained separately in the confined and obscure the physical effects that we are interested in. deconfined phases. (These volumes are large enough This is an appropriate normalisation because we ex- for there to be no tunnelling, or even attempted tun- pect Eq. (3.1) to provide the T →∞limit of p/T 4. nelling, within our available statistics.) We see that for The same applies to the internal energy density, since both SU(4) and SU(8) there is no visible difference → 3p as T →∞, and so when presenting our re- 4 between the plaquette at T = 0 and T = Tc in the con- sults for /T we normalise it with the expression in fined phase at, say, the 2σ level. Any difference (and Eq. (3.1). For similar reasons we shall use the same there obviously must be some difference) is clearly normalisation when presenting our results for the en- negligible when compared to the difference between tropy. For ∆/T 4 it is less clear what normalisation one the confined and deconfined phases at (and above) Tc. should use since ∆ = − 3p → 0asT →∞,butfor 120 B. Bringoltz, M. Teper / Physics Letters B 628 (2005) 113–124

Table 7 The plaquette average in the conﬁned phase, C,atT Tc compared to the T = 0 value and to the value in the deconﬁned phase, D,forSU(4) and SU(8)

βNLattice up Phase T 3 10.635 4 32 × 50.549563(33) D Tc 3 32 × 50.547689(11) C Tc 104 0.547640(27) C0 3 43.965 8 12 × 50.530352(23) D Tc 3 12 × 50.527725(27) C Tc 104 0.527648(24) C0

Table 8 The lattice discretisation errors correction factor RI(Lt ) in the inﬁnite volume limit

Lt = 2 Lt = 3 Lt = 4 Lt = 5 Lt = 6 Lt = 8 2.04526(4) 1.6913(2) 1.3778(1) 1.2129(6) 1.1323(1) 1.0659(1)

Fig. 1. The pressure, normalized to the lattice Stephan–Boltzmann pressure, including the full discretization errors. The symbol’s vertical sizes are representing the largest error bars (which are received for the highest temperature). The solid line is for SU(3) and Lt = 6 from [2]. ease of comparison we shall once again normalise us- We present our N = 4 and N = 8 results for ing Eq. (3.1). p/T 4 in Fig. 1. We also show there our calcula- To facilitate the comparison of our results with ear- tions of the SU(3) pressure for Lt = 5, as well as lier work on SU(3) [2], which was done for Lt = the Lt = 6 calculations from [2]. Although our errors 4, 6, 8, we have performed SU(3) simulations with on the SU(3) pressure are probably underestimated, Lt = 5. The spatial size is Ls = 20 which should be since the mesh in β is quite coarse, nonetheless one sufficiently large in the light of our above discussion can clearly infer that the pressure in the SU(4) and of finite volume effects (and we note that it satisfies an SU(8) cases is remarkably close to that in SU(3) and empirical rule that one needs Ls/Lt 4 [27]). hence that the well-known pressure deficit observed in B. Bringoltz, M. Teper / Physics Letters B 628 (2005) 113–124 121

4 4 = ∂p/T Fig. 2. Results for ∆(T )/T T ∂T , normalized by the same coefﬁcient as we normalize the pressure. The solid line is for SU(3) and Lt = 6 from [2].

Table 9 Plaquette average in the deconfined phase for lattice with fixed coupling, at different values of Lt , and with β that corresponds to roughly the deconfining temperature at Lt = 5: β = 5.800, 10.635, 44.00 for N = 3, 4, 8.ThedataforLt = 5 are obtained for L = 64, 32, 10 for N = 3, 4, 8 (for N = 3, δup is the difference between the plaquette as calculated within separate confined and deconfined sequences of field configurations) NL3 × 583 × 483 × 383 × 2104 −d log a/dβ

3 δup=0.00080(5) 0.570987(37) 0.573311(34) 0.578121(27) 0.567642(29) 2.075(17) 40.549563(33) 0.551604(33) 0.554047(27) 0.559163(24) 0.547640(27) 1.440(23) 80.531202(92) 0.533066(25) 0.535991(24) 0.541518(17) 0.528788(18) 0.384(20)

SU(3) is in fact a property of the large-N planar the- To see what is the behaviour of ∆/T 4 at even ory. higher temperatures, we use the plaquette averages on 4 In Fig. 2 we present our results for ∆/T as calcu- lattices with Lt = 2, 3, 4, 5, that have been calculated lated from Eq. (2.14). This quantity can be considered at ﬁxed couplings which correspond to T Tc for as a measure of the interaction and non-conformality Lt = 5 [20]. We present the results in Table 9.For of the theory, since it is identically zero both for the the evaluation of ∆ one needs d log a/dβ which we noninteracting Stephan–Boltzmann case, and for the present in the table as well. N = 4 supersymmetric SU(N) gauge theory. As re- In such calculations where one varies T by vary- marked above, we normalise with the expression in ing Lt , the lattice spacing varies as a = 1/Lt × 1/T Eq. (3.1). We also note that in this case there are no when expressed in units of the relevant temperature errors from a numerical integration, and this enables a scale, and so lattice spacing corrections will vary fair comparison with the SU(3) data of [2]. Compar- with T . ing the results for different N we see that, just as for The resulting values of ∆ in the case of SU(3) are the pressure, the results for all these gauge theories are plotted in Fig. 3 where they are compared to the re- very similar. sults obtained from calculations where one varies T 122 B. Bringoltz, M. Teper / Physics Letters B 628 (2005) 113–124

4 4 = ∂p/T = Fig. 3. Results for ∆(T )/T T ∂T , normalized to the free-gas result. The lines are for SU(3) and Lt 4, 6 from [2]. Triangles correspond to Lt = 5, and changing β, while circles correspond to changing Lt and keeping a ﬁxed β = 5.800.

4 4 = ∂p/T = = = = Fig. 4. Results for ∆(T )/T T ∂T for N 3, 4, 8, by fixing β βc(Lt 5), while changing Lt 2, 3, 4, 5. by varying β at fixed Lt . These calculations include pendence is very similar in all cases, and that the ours for Lt = 5 and those of [2] for Lt = 4, 6. remaining Lt dependence appears to be much the As we see from Fig. 3 our Lt = 5SU(3) results same for the different kinds of calculation. This gives do in fact lie between the Lt = 4, 6 results of [2] us confidence that performing calculations where we as one would expect. We observe that the T de- vary T by varying Lt at fixed β does not intro- B. Bringoltz, M. Teper / Physics Letters B 628 (2005) 113–124 123

Fig. 5. Results for energy density and entropy, normalized to the lattice Stephan–Boltzmann result, including the full discretization errors. The solid line is for SU(3) and Lt = 6 from [2]. duce any unanticipated and important systematic er- the quantity ∆ = − 3p (where we were able to ex- rors. plore temperatures up to T 2.5Tc). All this implies Having performed this check, we compare in Fig. 4 that the dynamics that drives the deconfined system our results for ∆ in the range Tc T 2.5Tc that far from its noninteracting gluon plasma limit, must corresponds to 5 Lt 2. This comparison confirms remain equally important in the N =∞planar the- what we observed in Fig. 2 over a smaller range of T : ory. This is encouraging since that limit is simpler to ∆ is very similar for all the values of N (except very approach analytically, in particular using gravity du- close to Tc), implying that this is also a property of the als. N =∞planar limit. Our results have been (mostly) obtained for lattice Finally, we present in Fig. 5 our results for the nor- spacings a = 1/(5T) and it would be useful to per- malized energy density = ∆ + 3p, and the entropy form a larger scale calculation that allows us to per- per unit volume s = ( + p)/T . The lines are the form an explicit continuum extrapolation. However, SU(3) result of [2] with Lt = 6. Again we see very past SU(3) calculations of the pressure, and calcula- little dependence on the gauge group, implying very tions in SU(N) of various physical quantities, strongly similar curves for N =∞. suggest that our choice of a already provides us with a reliable preview of what such a more complete calculation would produce. 4. Summary and discussion Our results imply that any explanation of the QCD pressure deficit must survive the large-N limit, and In this Letter we have analyzed numerically the so should not be driven by special features particu- bulk thermodynamics of SU(4) and SU(8) gauge the- lar to SU(3). This can provide a strong constraint on ories. We found that the pressure, when normalized to such explanations. For example, in approaches based the Stephan–Boltzmann lattice pressure, is practically on higher-order perturbation theory, it tells us that the the same as for SU(3), in the range Tc T 1.6Tc important contributions must be planar. In models fo- that we analyze. We found the same to be the case cussing on resonances and bound states, it must be for the internal energy and entropy, as well as for that the dominant states are coloured, since the con- 124 B. Bringoltz, M. Teper / Physics Letters B 628 (2005) 113–124 tribution of colour singlets will vanish as N →∞. [3] J. Engels, et al., Phys. Lett. B 396 (1997) 210, hep-lat/9612018. Models using ‘quasi-particles’ should place these in [4] P. Petreczky, Nucl. Phys. B (Proc. Suppl.) 140 (2005) 78, hep- colour representations that do not exclude their pres- lat/0409139. =∞ [5] F. Karsch, in: Lecture Notes in Physics, vol. 583, Springer, ence at N , and in fact give them T -dependent Berlin, 2002, p. 209, hep-lat/0106019. properties which depend weakly on N. Also, topologi- [6] P. Levai, U.W. Heinz, Phys. Rev. C 57 (1998) 1879, hep-ph/ cal fluctuations should play no role in this deficit since 9710463. the evidence is that there are no topological fluctua- [7] A. Peshier, B. Kampfer, O.P. Pavlenko, G. Soff, Phys. Rev. tions of any size in the deconfined phase at large N D 54 (1996) 2399. [8] Y. Schroder, hep-ph/0410130. [28,29]. [9] J.-P. Blaizot, E. Iancu, A. Rebhan, hep-ph/0303185. Finally, we emphasize that our conclusion that the [10] E.V. Shuryak, I. Zahed, Phys. Rev. D 70 (2004) 054507, hep- SU(3) pressure and entropy deficits are features of ph/0403127. the large-N gauge theory, means that these ‘observ- [11] P. Petreczky, F. Karsch, E. Laermann, S. Stickan, I. Wetzorke, able’ phenomena can, in principle, be addressed using Nucl. Phys. B (Proc. Suppl.) 106 (2002) 513, hep-lat/0110111. [12] S.S. Gubser, I.R. Klebanov, A.A. Tseytlin, Nucl. Phys. B 534 AdS/CFT gravity duals. Indeed, it is precisely where (1998) 202, hep-th/9805156. the deficit is large that the coupling must be strong [13] M. Teper, hep-th/0412005. and this is also precisely where, at large N, such [14] B. Lucini, M. Teper, U. Wenger, Phys. Lett. B 545 (2002) 197, dualities can be established. As has been frequently hep-lat/0206029. emphasized (see, for example, [16,17]) the deficit in [15] B. Lucini, M. Teper, U. Wenger, JHEP 0401 (2004) 061, hep- lat/0307017. the normalized entropy is not far from the value of [16] R.V. Gavai, S. Gupta, S. Mukherjee, Phys. Rev. D 71 (2005) s/sfree-gas = 3/4 given by the AdS/CFT prediction. In 074013, hep-lat/0412036. this Letter we have found that large-N gauge theories [17] R.V. Gavai, S. Gupta, S. Mukherjee, hep-lat/0506015. show the same behaviour, as we see in Fig. 5, where, [18] G. Bhanot, S. Black, P. Carter, R. Salvador, Phys. Lett. B 183 3 (1987) 331. for the entropy, the horizontal line snormalized/T = 3 = [19] F. Wang, D.P. Landau, Phys. Rev. E 64 (2001) 056101. would correspond to s/sfree-gas 3/4. Our results can [20] B. Lucini, M. Teper, U. Wenger, hep-lat/0502003. therefore serve as a bridge between the AdS/CFT ap- [21] B. Lucini, M. Teper, JHEP 0106 (2001) 050, hep-lat/0103027. proach to large-N and the observable world of QCD. [22] B. Lucini, M. Teper, U. Wenger, JHEP 0406 (2004) 012, hep- lat/0404008. [23] B. Alles, A. Feo, H. Panagopoulos, Phys. Lett. B 426 (1998) 361, hep-lat/9801003. Acknowledgements [24] B. Bringoltz, M. Teper, hep-lat/0508021. [25] J. Engels, F. Karsch, T. Scheideler, Nucl. Phys. B 564 (2000) We are thankful to Juergen Engels for useful dis- 303, hep-lat/9905002. cussions on the finite lattice spacing corrections of the [26] J. Engels, private communications. free gas pressure in the integral method, and in par- [27] J. Engels, J. Fingberg, F. Karsch, D. Miller, M. Weber, Phys. Lett. B 252 (1990) 625. ticular for giving us the numerical routines to calcu- [28] B. Lucini, M. Teper, U. Wenger, Nucl. Phys. B 715 (2005) 461, late them. Our lattice calculations were carried out on hep-lat/0401028. PPARC and EPSRC funded computers in Oxford The- [29] L. Del Debbio, H. Panagopoulos, E. Vicari, JHEP 0409 (2004) oretical Physics. B.B. acknowledges the support of a 028, hep-th/0407068. PPARC postdoctoral research fellowship.

References

[1] U.W. Heinz, nucl-th/0412094. [2] G. Boyd, et al., Nucl. Phys. B 469 (1996) 419, hep-lat/ 9602007. Physics Letters B 628 (2005) 125–130 www.elsevier.com/locate/physletb

Chiral and continuum extrapolation of partially-quenched lattice results

C.R. Allton a, W. Armour a, D.B. Leinweber b, A.W. Thomas c, R.D. Young c

a Department of Physics, University of Wales Swansea, Swansea SA2 8PP, Wales, UK b CSSM and Department of Physics, University of Adelaide, Adelaide, SA 5005, Australia c Jefferson Lab, 12000 Jefferson Ave., Newport News, VA 23606, USA Received 17 June 2005; received in revised form 10 September 2005; accepted 10 September 2005 Available online 27 September 2005 Editor: H. Georgi

Abstract The vector meson mass is extracted from a large sample of partially quenched, two-flavor lattice QCD simulations. For the first time, discretisation, finite-volume and partial quenching artefacts are treated in a unified framework which is consistent with the low-energy behaviour of QCD. This analysis incorporates the leading infrared behaviour dictated by chiral effective field theory. As the two-pion decay channel cannot be described by a low-energy expansion alone, a highly-constrained model for the decay channel of the rho-meson is introduced. The latter is essential for extrapolating lattice results from the quark-mass regime where the rho is observed to be a physical bound state.  2005 Elsevier B.V. All rights reserved.

Recent developments in lattice QCD have enabled of these approximations requires special attention in the first large-scale simulation of chiral, dynamical the extraction of physical observables from Monte fermions [1]. While this accomplishment is a sig- Carlo simulations. In this Letter we analyse a very nificant milestone in the progress towards an accu- large set of partially quenched data for the mass of rate description of physical QCD, the high demand the ρ meson. We show that a systematic analysis of on computing resources restricts practical calcula- this data enables us to remove the effects of par- tions to an unphysical domain of simulation para- tial quenching and to take both the continuum and meters. In particular, lattice QCD involves a discre- infinite volume limits. The resulting data lies on a tised space–time of finite spatial extent, with input single, well defined curve which extrapolates to a quark masses much larger than those in nature. Each value within ∼1% of the physical ρ mass. The contrast between the raw lattice data (note the scatter in Fig. 2) and the corrected data shown in Fig. 3 is E-mail address: [email protected] (R.D. Young). striking.

Spontaneous chiral symmetry breaking in QCD manding component of the calculation of standard dictates that, in the vicinity of the chiral limit, hadronic observables. By comparison, the matrix inversion re- observables exhibit nonanalytic dependence on the quired to obtain quark propagators is relatively effi- quark mass [2]. This feature places tight constraints cient. This enables the calculation of quark propaga- on the form of chiral extrapolations if they are to tors over a range of quark masses for a fixed gauge be consistent with the properties of low-energy QCD field ensemble. Such calculations are referred to as [3,4]. The most natural solution to this problem is to partially quenched QCD (pQQCD), where the valence use effective field theory (EFT) to describe the quark- quark masses no longer match those simulated in the mass dependence of hadron properties. Considering a sea. Although an unphysical approximation, the con- benchmark quantity, such as the nucleon mass, there nection to the physical theory in EFT has been demon- is substantial phenomenological information on the strated [28]. Most importantly, the partially quenched quark-mass expansion near the chiral limit [5].Inthe EFT does not require any new, unphysical parameters. context of lattice simulations, where quark masses are While EFT provides a systematic framework for the significantly far from the chiral limit, the expansion analysis of lattice results, the present analysis of the ρ is acutely sensitive to higher-order corrections. Fortu- meson requires one to go beyond the scope of EFT. nately, such issues can be alleviated by reformulating Near the chiral limit, the ρ decays to two energetic the EFT in the framework of finite-range regularisa- pions. The pions contributing to the imaginary part tion (FRR) [6]—with demonstrated success in the ef- of the ρ mass cannot be considered soft, and there- ficient extrapolation [7] of lattice calculations of the fore cannot be systematically incorporated into a low- nucleon mass [8]. energy counting scheme [29,30]. Because almost all Provided simulations are performed on a suitably the lattice simulation points in this analysis lie in the large box, finite-volume corrections will be exponen- region mπ >mρ/2, it is evident that in the extrapo- tially suppressed. Nevertheless, these leading correc- lation to the chiral regime will encounter a threshold tions can be described by the same low-energy effec- effect where the decay channel opens. To incorporate tive field theory used to understand the quark-mass this physical threshold, we model the ρ → ππ self- variation [9]. Finite-volume corrections are dominated energy diagram constrained to reproduce the observed by the suppression of the infrared component of chi- width at the physical pion mass. Including this con- ral loop diagrams—as observed in a recent study of tribution also provides a model of the finite volume volume dependence in lattice QCD [10] (using the corrections arising from the infrared component of the quark-mass dependence described in Ref. [11]). Cor- loop integral. In particular, we can also describe the rections of this type have previously been incorporated lattice results in the region mπ

Fig. 1. Left: Diagram providing the leading nonanalytic contribution to the chiral expansion of the ρ-meson mass (a) and its associated quark-flow (b). Middle: Two-pion contributions, (a), (c), to the ρ-meson self-energy and their associated quark flow, (b), (d), respectively. Right: The η contribution (a) and its associated quark flow diagrams in pQQCD. While diagram (c) appears in quenched QCD, in pQQCD (or full QCD) it is complemented by an infinite series of terms, the first two of which are depicted in diagrams (d) and (e). 128 C.R. Allton et al. / Physics Letters B 628 (2005) 125–130

−1 the ωρπ coupling, gωρπ = 16 GeV [33],byg2 = gωρπ fπ /2 = 0.74; k =|k|; finite-range regularisation is implemented with a dipole form, u(k) = (1 + − k2/Λ2) 2, and the ρππ coupling is preserved at the = −1 2 − 2 physical threshold, uππ(k) u(k)u ( µρ/4 µπ ). By demanding this physical threshold the infrared behaviour is no longer constrained and hence is not controlled by EFT. Therefore, such a model is essential to extrapolate lattice results from the regime mπ >mρ/2. The leading finite-volume corrections are trivially incorporated by replacing the continuum loop integrals in Eqs. (4)–(6) by a sum over discrete momenta Fig. 2. Partially-quenched vector meson masses plotted versus “de- (deg) = a (deg) = [12–14] generate” pion mass squared. Here MV Mijj and mπ ma . The simulation results are from CP-PACS [8]. ijj 2π 3 d3k → , (7) L k Table 1 The resulting global fit to the entire sample of lattice results (with where k = (2π/L)i for i ∈ Z3. This modification pro- Λ = 655 MeV) 2 duces an infrared suppression of the loop integrals, and α0 X2 α2 α4 χ /d.o.f. − − − is independent of the choice of ultra-violet regularisa- (GeV) (GeV fm 2) (GeV 1) (GeV 3) +4 − +3 +12 − +8 tion [16]. 0.832−4 1.40−4 0.494−11 0.061−9 39/76 The bracketed term in Eq. (2) describes the residual variation of the vector meson mass which is not contained in the one-loop Goldstone boson diagrams. enabling all of these points to be shown in physical The analytic variation of the quark-mass dependence units—as in Fig. 2. is characterised by the continuum parameters, αi .At The form provided by Eq. (2) allows a universal finite lattice spacing, all terms at order a and a2 must fit to all these points with just four free parameters be treated consistently with the symmetry breaking (plus regulator scale). This is therefore a highly con- patterns of the prescribed fermion action [24]. This can strained fit to the large sample of simulation results. potentially lead to more singular chiral behaviour in The best fit parameters are displayed in Table 1.The the effective field theory at finite lattice spacing [26]. χ2 indicates that Eq. (2) accurately describes this large In this study, the leading lattice spacing corrections to quantity of data. The regulator mass, Λ, has also been the terms analytic in the quark mass are investigated. optimised to produce a best fit to the data, namely The lattice simulation data considered in this analy- Λ = 655 ± 35 MeV, with the bound determined sta- sis come from a large sample of partially quenched tistically from the 1σ variation about the central fit. simulation results from the CP-PACS Collaboration Studying the variation of the fit over this domain in- [8]. These simulations were performed using mean- troduces a small additional uncertainty to the extrapo- field improved clover fermions at four different cou- lated result which is listed below in the error estimate. plings, β. For each value of the coupling, four different Variations of Eq. (2) have also been investigated. sea quark masses have been calculated, yielding a total Scaling corrections to the parameters α2 and α4 yield of 16 independent gauge field ensembles. On each en- coefficients which are consistent with zero. Linear cor- semble, the quark propagator has been evaluated for rections in the lattice spacing, taking the form of a five values of the valence quark mass. For the vec- term X1a, are observed to be small and hence do not tor mesons constructed of degenerate valence quarks improve on the fit with X1 = 0. Extending to higher there are a total of 80 “data” points. The lattice scale analytic order in the quark mass expansion, by a term = 6 is set via the QCD Sommer scale r0 0.49 fm [34], α6mπ , reduces the stability of the fit, indicating that C.R. Allton et al. / Physics Letters B 628 (2005) 125–130 129

values of the extrapolated result with the monopole and Gaussian forms differs by +3 and −6 MeV, respectively, from the dipole result. Each regularisation scheme produces a different model of the ρ → ππ vertex. This suggests that the model-dependence of this contribution is small, once constrained to produce the correct width at the physical point. This extrapolation of the lattice results also offers an estimate of the vector meson mass in the chiral 0 limit, Mρ . The central value of the preferred fit gives a value 775 MeV, with errors similar to those quoted above for the vector mass at the physical point. These Fig. 3. The same 80 lattice data points as in Fig. 2, after correction errors are from extrapolation of lattice results only, to restore the infinite-volume, continuum and quark-mass unitar- whereas for phenomenological purposes the physical ity limits. The central curve displays the best-fit from the global point provides a further constraint. We therefore report analysis. The dashed curves show the bounds on the FRR scale, the correlated difference between the physical and chi- 0.620 <Λ<0.690 GeV. ral limit value the data are consistent with α = 0—see Ref. [31] for − 0 = +4 6 Mρ Mρ 3.7(2)−4(8) MeV. (9) a complete account of these effects. The systematic error quoted below covers the range found with all of Interestingly, the chiral limit value of Mρ is very sim- these variations. ilar to that of the physical value. This feature is ob- The fit parameters shown in Table 1 allow one served in the reduction in slope of the extrapolation to shift the simulation results to the infinite-volume, curve in Fig. 3 as the chiral limit is approached. The continuum limit and to remove the effects of par- underlying physics giving rise to this reduced slope tial quenching—hence restoring unitarity in the quark is the presence of substantial spectral strength in the masses. Complete details of the procedure are out- low-energy two-pion channel below the rho-meson lined in Ref. [31]. The results are displayed in Fig. 3, mass [35]. This suggests a small sigma term for the ρ where we observe a remarkable result. The tremen- in comparison with the nucleon, where the curvature dous spread of data seen in Fig. 2 is dramatically reis enhanced near the chiral limit [7]. duced, with all 80 points now lying very accurately on This analysis demonstrates the ability to treat all a universal curve. lattice artifacts within a unified framework. Both scal- The curve through Fig. 3 displays the determined ing violations and finite-volume discrepancies can be variation of the ρ-meson mass with pion mass. This removed through the procedure outlined. The number curve also presents an extrapolation to the physical of simulation points can be increased dramatically by point, allowing extraction of the physical ρ-meson including partially quenched results. This in turn per- mass mits a highly constrained fit to produce an accurate extrapolation to the physical point. With minimal in- = +16 Mρ 778(4)−6 (8) MeV, (8) put, namely the ρππ and ωρπ coupling constants, the real part of the ρ-meson mass has been accurately where the first error is statistical, the second is system- determined in two-flavour QCD.1 The final result for atic and the third from the determination of Λ [31]. the pion mass variation, as described by the universal This result is in excellent agreement with the experi- curve in Fig. 3, sets a benchmark for the continuum, mentally observed mass. infinite-volume limit of the ρ mesonintwo-flavour The systematic uncertainty arises from the choice QCD. of fitting function, as outlined above, and also from the choice of finite-range regulator. In addition to the presented dipole form, the analysis has been repeated 1 This is two-flavour QCD with the qq¯ force normalised to the with monopole and Gaussian regulators. The central physical value at a length scale r0 = 0.49 fm. 130 C.R. Allton et al. / Physics Letters B 628 (2005) 125–130

Acknowledgements [12] D.B. Leinweber, A.W. Thomas, K. Tsushima, S.V. Wright, Phys. Rev. D 64 (2001) 094502. C.R.A. and W.A. would like to thank the CSSM [13] R.D. Young, D.B. Leinweber, A.W. Thomas, S.V. Wright, Phys. Rev. D 66 (2002) 094507. for their support and kind hospitality. W.A. would [14] R.D. Young, D.B. Leinweber, A.W. Thomas, Phys. Rev. D 71 like to thank PPARC for support. The authors would (2005) 014001. like to thank W. Melnitchouk, D. Richards, G. Shore [15] G. Colangelo, S. Durr, Eur. Phys. J. C 33 (2004) 543. and S. Wright for helpful comments. This work was [16] S.R. Beane, Phys. Rev. D 70 (2004) 034507. supported by the Australian Research Council and [17] S.R. Beane, M.J. Savage, Phys. Rev. D 70 (2004) 074029. [18] D.B. Leinweber, et al., Phys. Rev. Lett. 94 (2005) 212001. by DOE contract DE-AC05-84ER40150, under which [19] B. Borasoy, R. Lewis, Phys. Rev. D 71 (2005) 014033. SURA operates Jefferson Laboratory. [20] M. Luscher, S. Sint, R. Sommer, P. Weisz, U. Wolff, Nucl. Phys. B 491 (1997) 323. [21] R.G. Edwards, U.M. Heller, T.R. Klassen, Phys. Rev. Lett. 80 (1998) 3448. References [22] S.J. Dong, F.X. Lee, K.F. Liu, J.B. Zhang, Phys. Rev. Lett. 85 (2000) 5051. [1] Y. Aoki, et al., hep-lat/0411006. [23] J.M. Zanotti, B. Lasscock, D.B. Leinweber, A.G. Williams, [2] L.F. Li, H. Pagels, Phys. Rev. Lett. 26 (1971) 1204. Phys. Rev. D 71 (2005) 034510. [3] D.B. Leinweber, D.H. Lu, A.W. Thomas, Phys. Rev. D 60 [24] G. Rupak, N. Shoresh, Phys. Rev. D 66 (2002) 054503. (1999) 034014. [25] O. Bär, G. Rupak, N. Shoresh, Phys. Rev. D 70 (2004) 034508. [4] D.B. Leinweber, A.W. Thomas, K. Tsushima, S.V. Wright, [26] S. Aoki, Phys. Rev. D 68 (2003) 054508. Phys. Rev. D 61 (2000) 074502. [27] S.R. Beane, M.J. Savage, Phys. Rev. D 68 (2003) 114502. [5] B. Borasoy, U.G. Meissner, Ann. Phys. 254 (1997) 192. [28] S.R. Sharpe, N. Shoresh, Phys. Rev. D 62 (2000) 094503. [6] R.D. Young, D.B. Leinweber, A.W. Thomas, Prog. Part. Nucl. [29] J. Bijnens, P. Gosdzinsky, P. Talavera, Nucl. Phys. B 501 Phys. 50 (2003) 399. (1997) 495, hep-ph/9704212. [7] D.B. Leinweber, A.W. Thomas, R.D. Young, Phys. Rev. [30] P.C. Bruns, U.G. Meissner, Eur. Phys. J. C 40 (2005) 97, hep- Lett. 92 (2004) 242002. ph/0411223. [8] CP-PACS Collaboration, A. Ali Khan, et al., Phys. Rev. D 65 [31] W. Armour, et al., in preparation. (2002) 054505; [32] C.K. Chow, S.J. Rey, Nucl. Phys. B 528 (1998) 303. CP-PACS Collaboration, A. Ali Khan, et al., Phys. Rev. D 67 [33] M. Lublinsky, Phys. Rev. D 55 (1997) 249. (2003) 059901, Erratum. [34] R. Sommer, Nucl. Phys. B 411 (1994) 839; [9] J. Gasser, H. Leutwyler, Nucl. Phys. B 307 (1988) 763. R.G. Edwards, U.M. Heller, T.R. Klassen, Nucl. Phys. B 517 [10] QCDSF-UKQCD Collaboration, A. Ali Khan, et al., Nucl. (1998) 377. Phys. B 689 (2004) 175. [35] D.B. Leinweber, T.D. Cohen, Phys. Rev. D 49 (1994) 3512, [11] M. Procura, T.R. Hemmert, W. Weise, Phys. Rev. D 69 (2004) hep-ph/9307261. 034505. Physics Letters B 628 (2005) 131–140 www.elsevier.com/locate/physletb

Probing the CP-violating light neutral Higgs in the charged Higgs decay at the LHC

Dilip Kumar Ghosh a, R.M. Godbole b,D.P.Royc,d

a Institut für Theoretische Physik und Astrophysik, Universität Würzburg, D-97074, Würzburg, Germany b Centre for High Energy Physics, Indian Institute of Science, Bangalore 560 012, India c Department of Theoretical Physics, Tata Institute of Fundamental Research, Homi Bhabha Road, 400 005 Mumbai, India d Theory Division, Physics Department, CERN, CH-1211 Geneva 23, Switzerland Received 24 December 2004; received in revised form 25 August 2005; accepted 30 August 2005 Available online 26 September 2005 Editor: G.F. Giudice

Abstract The CP-violating MSSM allows existence of a light neutral Higgs boson (MH1 50 GeV) in the CPX scenario in the low tan β( 5) region, which could have escaped the LEP searches due to a strongly suppressed H1ZZ coupling. This parameter + space corresponds to a relatively light H (M +

1. Introduction existence of a light Higgs boson (Mh < 246 GeV at 95% C.L.) whereas direct searches rule out the case The search for Higgs bosons and study of their Mh < 114.4GeV[1,2]. Naturalness arguments along properties is one of the main goals of physics stud- with the indication of a light Higgs state suggest that ies at the Tevatron upgrade (run 2) and the upcoming Supersymmetry (SUSY) is a likely candidate for new Large Hadron Collider (LHC). The precision mea- physics Beyond the Standard Model (BSM). Even in surements with Electro-Weak (EW) data indicate the the SUSY case, a mass for the lightest neutral Higgs smaller than ∼ 90 GeV is ruled out [3] if the SUSY parameters as well as the SUSY breaking parameters E-mail address: [email protected] (R.M. Godbole). are real and CP is conserved. However, in presence

0370-2693/$ – see front matter  2005 Elsevier B.V. All rights reserved. doi:10.1016/j.physletb.2005.08.132 132 D.K. Ghosh et al. / Physics Letters B 628 (2005) 131–140 of CP-violation in the Higgs sector, the lower limit and hence the Higgs sector, can affect the Higgs de- can get diluted due to a reduction in the H1ZZ cou- cays as well as their production rates at the colliders pling [4]. substantially and has been a subject of many investi- CP-violation, initially observed only in the K0–K¯ 0 gations [18,20]. system, is one feature of the Standard Model (SM) It is interesting to note that in the same region of that still defies clear theoretical understanding. It is the parameter space where the coupling of the lightest in fact one of the necessary ingredients for generat- mass eigenstate H1 to a pair of Z-bosons: the H1ZZ + − ing the observed excess of baryons over antibaryons coupling, is suppressed the H W H1 coupling is en- in the Universe [5,6]. The amount of CP-violation hanced because these two sets of couplings satisfy a present in the quark sector described very satisfac- sum-rule. The strong suppression of the H1ZZ cou- torily in the CKM picture, is however, too small to pling also means that the H1 is dominated by the generate a baryon asymmetry of the observed level of pseudo-scalar component in this region and hence −10 NB /Nγ 6.1 × 10 [7]. New sources of CP viola- implies a light charged Higgs boson (MH + MSUSY). The mass scale coupling means an enhanced H H W coupling. 2 2 1 MSUSY is defined by (m˜ + m˜ )/2. After diagonaliz- t1 t2 This enhancement will play a significant role in our ing the 3 × 3 symmetric Higgs mass-squared matrix analysis. Equally important is the correlation between M2 ij by an orthogonal matrix O, the physical mass the mass of the charged Higgs MH ± and that of the eigenstates H1, H2 and H3 (in ascending order of pseudo-scalar state that exists in the MSSM. A sup- mass) are states of indefinite CP parity. In this case pressed H1VV coupling implies a light pseudo-scalar MH ± is more appropriate parameter for description of state, which in turn implies a light charged Higgs, with the MSSM Higgs sector in place of the MA used usu- MH + i=1 then discuss the phenomenology of the charged and 3 ↔ g − + the neutral Higgs search in the region of the low MH1 L ∓ ± = − + ,µ HH W gHi H W (Hi ∂ µ H )W , window that is still allowed by LEP [3,18,19] for the 2cW = i 1 (4) case of CP-violating MSSM. + − where, gHi VV, gHi Hj Z and gHi H W are Higgs gauge boson couplings normalized to the standard model value and can be written as, 3. Results and discussion = + gHi VV O1i cos β O2i sin β, (5) Recently the OPAL Collaboration [19] reported = − − ↔ their results for the Higgs boson searches in the gHi Hj Z O3i(cos βO2j sin βO1j ) (i j), (6) CP-violating MSSM Higgs sector using the parame- g + − = O cos β − O sin β + iO . (7) Hi H W 2i 1i 3i ters defined in the CPX scenario as mentioned above These couplings obey the following sum-rules: and found that for certain values of phases and MH + , the lower mass limit on the neutral Higgs is diluted, at 3 2 = times vanishing completely. This results in windows gH VV 1, (8) i in the tan β–M + plane which are still allowed by the i=1 H 134 D.K. Ghosh et al. / Physics Letters B 628 (2005) 131–140

LEP data. The LEP bounds are essentially evaded in and Mt = 175 GeV has been used by them to calculate this window as the lightest state is largely a pseudo- the couplings and the masses of the Higgs bosons. scalar with highly suppressed coupling to the ZZ As mentioned already, in the same region of the pa- pair. There exist two programs; CPSuperH [22] and rameter space where H1ZZ coupling is suppressed, + − FeynHiggs 2.0 [23] to calculate the masses and mix- the H W H1 coupling is enhanced because these ing in the Higgs sector in the CP-violating case. Due two sets of couplings satisfy a sum-rule as shown in to the different approximations made in the two cal- Eq. (9). Further, in the MSSM a light pseudo-scalar culations as well as differences in the inclusion of dif- implies a light charged Higgs, lighter than the top ferent higher order terms, at least in the CPX scenario, quark. Tables 1 and 2 show the behavior of the MH + , + → + the two programs give somewhat different results and MH1 and the BR(H H1W ), for values of tan β the experimentalists use the lower prediction of the corresponding to the above mentioned window in the two for the expected cross-sections to get the most tan β–MH1 plane, of Ref. [18]. It is to be noted here conservative constraints. The constraints also depend that indeed the H ± is light (lighter than the top) over sensitively on the mass of the top quark used in the cal- the entire range, making its production in t decay pos- ± culation [3]. The preliminary results from a combined sible. Further, the H decays dominantly into H1W , analysis of all the LEP results [3], provide exclusion with a branching ratio larger than 47% over the en- regions in the MH1 –tanβ plane for different values of tire range where the decay is kinematically allowed, the CP-violating phases, for the following values of which covers practically the entire parameter range of = ◦ ◦ the parameters: interest; viz. MH1 < 50(40) GeV for ΦCP 90 (60 ). It can be also seen from both the tables that the ± Arg At = Arg Ab = Arg Mg˜ = ΦCP, (14) BR(H → H1W ) is larger than 90% over most of the parameter space of interest. So not only that H + can M = 0.5TeV,M˜ = 1TeV, (15) SUSY g be produced abundantly in the t decay giving rise to a = = MB˜ MW˜ 0.2TeV, (16) possible production channel of H1 through the decay ± ± = ◦ ◦ ◦ ◦ H → H1W , but this decay mode will be the only ΦCP 0 , 30 , 60 , 90 . (17) ± decay channel to see this light (MH ±

Table 1 + + + Range of values for BR(H → H1W ) and BR(t → bH ) for different values of tan β corresponding to the LEP allowed window in the ◦ + CPX scenario, for the common phase ΦCP = 60 , along with the corresponding range for the H1 and H masses. The quantities in the bracket + + in each column give the values at the edge of the kinematic region where the decay H → H1W is allowed tan β 2 2.2 2.5 3.0 + + BR(H → H1W )(%)>90 (83.5) > 90 (80.32) > 90 (73.85) > 90 (63.95) + BR(t → bH )(%) 4.0–4.2 4.9–5.1 4.8–5.11 4.0–4.3 MH + (GeV) < 133.6 (135.1) < 122.7 (124.3) < 113.8 (115.9) < 106.6 (109.7) MH1 (GeV) < 50.97 (54.58) < 39.0 (43.75) < 27.97 (35.44) < 14.28 (29.21)

Table 2 ◦ Same as in Table 1 but for the value of common phase ΦCP = 90 tan β 3.64 4.65 + + BR(H → H1W )(%)>90 (87.45) > 90 (57.65) > 90 (50.95) > 90 (46.57) + BR(t → bH )(%) ∼ 0.7 0.7–1.1 0.9–1.3 1.0–1.3 MH + (GeV) < 148.5 (149.9) < 139 (145.8) < 130.1 (137.5) < 126.2 (134) MH1 (GeV) < 60.62 (63.56) < 49.51 (65.4) < 36.62 (57.01) < 29.78 (53.49)

In our parton level Monte Carlo analysis we em- and the two solutions coalesce. The hadronically de- ploy following strategies to identify final state jets and caying W is reconstructed from that pair of untagged leptons: jets, whose invariant mass is closest to MW .Onetopis then reconstructed from one of the reconstructed W ’s 1. |η| < 2.5 for all jets and leptons, where η denotes and one of the remaining jets chosen such that the in- pseudo-rapidity. variant mass mW jet is closest to Mt . Similarly the H1 2. pT of the hardest three jets to be higher than is then reconstructed from a pair from among the re- 30 GeV. maining jets, such that the invariant mass of the pair ± 3. pT of all the other jets, lepton, as well as the miss- is closest to MH1 . Then the H is reconstructed from ing pT to be larger than 20 GeV. this H1 and the remaining reconstructed W . In case of 4. A minimum separation of R = ( φ)2 + ( η)2 a quadratic ambiguity for the latter, the one giving in- = 0.4 between the lepton and jets as well as each variant mass closer to MH ± is chosen. Although the ± pair of jets. If R between two partons is less than massesoftheH1 and H may not be known, one 0.4 we merge them into a single jet. can select the right combinations on the basis of a 5. We impose Gaussian smearing on energies, with clustering algorithm. Finally, the second top is recon- √ ± E/E = 0.6/ E for jets. structed by combining this H with the remaining jet. 6. We demand three or more tagged b jets in the final The signal cross-sections shown in Figs. 1 and 2 are state assuming a b-tagging efficiency of 50%. obtained using mass window cuts of MW ± 15 GeV, ± ± ± ± 7. The missing pT is obtained by vector summation Mt 25 GeV, MH1 15 GeV and MH 25 GeV on ± of the transverse momenta of the lepton and the the reconstructed W , t, H1 and H masses. Only the jets after Gaussian smearing. MW and Mt mass window cuts are retained in Figs. 3 and 4, showing the distributions in the reconstructed + Below we outline the mass reconstruction strategy H1 and H masses. we employ. The leptonically decaying W in the above In Fig. 1 we show the variation of the cross-section + decay chain is reconstructed from the lepton momen- with MH (a) and MH1 (b) for the CP-violating phase = ◦ tum pl and the missing transverse momentum pT ΦCP 60 while the choice of other MSSM parame- within a quadratic ambiguity using the constraint that ters are defined through Eqs. (11)–(16).Wehaveused = the invariant mass of the ν pair mν = MW . In case the CPSuperH program [22] with Mt 175 GeV, to of complex solutions the imaginary part is discarded calculate the masses and the couplings of the Higgs 136 D.K. Ghosh et al. / Physics Letters B 628 (2005) 131–140

Fig. 1. Variation of the expected cross-section with M + (a) and MH (b) for four values of tan β = 2, 2.2, 2.5 and 3. The CP-violating phase ◦ H 1 ΦCP is 60 . See text for the values of the remaining MSSM parameters. The cross-sections are obtained after applying the mass window cuts as mentioned in the text. These numbers should be multiplied by ∼ 0.5 to get the signal cross-section as explained in the text.

Fig. 2. Variation of the cross-section with M + (a) and MH (b) for four values of tan β = 3.6, 4, 4.6 and 5. The CP-violating phase ΦCP ◦ H 1 is 90 . The other MSSM parameters are same as in Fig. 1. These numbers should be multiplied by ∼ 0.5 to get the signal cross-section as explained in the text. The same mass window cuts as mentioned in Fig. 1 have been used in this case. bosons in the CPX scenario. We have used the CTEQ sented in the ﬁgure contain neither the suppression 4L parametrization of the parton density distributions factor due to b-tagging efﬁciency nor the K-factor ¯ and the QCD scale chosen is 2Mt . The numbers pre- (1.3–1.4) due to the NLO corrections to the tt cross- D.K. Ghosh et al. / Physics Letters B 628 (2005) 131–140 137

(a) (b) ¯ ¯ ¯ Fig. 3. Clustering of the bb, bbW and bbbW invariant masses: (a) three-dimensional plot for the correlation between mbb¯ and mbbW¯ invariant = = ◦ mass distribution; (b) mbb¯, mbbW¯ and mbbWb¯ Mt invariant mass distributions for ΦCP 60 . Mt , MW mass window cuts have been = = applied as explained in the text. The other MSSM parameters are tan β 2, MH + 125.6 GeV and the corresponding light Higgs mass is = MH1 24.8GeV.

(a) (b) ¯ ¯ ¯ Fig. 4. Clustering of the bb, bbW and bbbW invariant masses. (a) Three-dimensional plot for the correlation between mbb¯ and mbbW¯ invariant = = ◦ mass distribution. (b) mbb¯ , mbbW¯ and mbbWb¯ Mt invariant mass distributions for ΦCP 90 . Mt , MW mass window cuts have been applied = = as explained in the text. The other MSSM parameters are tan β 5, MH + 133 GeV, corresponding to a light neutral Higgs H1 with mass = MH1 51 GeV. 138 D.K. Ghosh et al. / Physics Letters B 628 (2005) 131–140 sections. Taking into account both, the numbers in the reduction is brought about by requiring that the invari- 1 figure should be multiplied by 5/16×1.3–1.4 ∼ 0.5to ant mass of the bbbW be within 25 GeV of Mt . This get the signal cross-section at the LHC. It may also be makes it very clear that the detectability of the sig- stated that the expected cross-sections at the Tevatron nal is controlled primarily by the signal size. It is also are far too small for this process to be useful there. clear from Figs. 1 and 2 that indeed the signal size As can be seen from the figure the signal cross- is healthy over the regions of interest in the parameter section decreases with increase in tan β. This can be space. Thus using this process one can cover the region + + explained by the fact that H → H1W as well as of the parameter space in CP-violating MSSM, in the → + t bH branching ratio decreases with the increase tan β–MH1 plane which cannot be excluded by LEP-2 in tan β for a fixed H1 mass. In this scenario, the and where the Tevatron and the LHC have no reach largest signal cross-section (∼ 160 fb) can be ob- via the usual channels. Note further that this process tained for tan β = 2 and MH + = 135 GeV, which would be the only channel of discovery for the charged ± corresponds to MH = 54.3 GeV. The cross-section is Higgs boson H as well in this scenario, as the tradi- 1 ± ∼ 125 fb for MH + = 130 GeV corresponding to tional decay mode of H → ντ is suppressed by over = MH1 40 GeV. In principle there exists a physics an order of magnitude. background to the signal arising from the decay Fig. 4(a) shows the three-dimensional plot for the ± ± ¯ H → W bb, via the virtual tb channel, but over this correlation between m ¯ and m ¯ invariant mass dis- b◦b bbW particular range of MH ± and tan β the corresponding tribution for ΦCP = 90 , and somewhat higher values branching ratio is negligibly small [24]. of tan β and MH + ,tanβ = 5 and MH + = 133 GeV. In Fig. 2, we show variation of the signal cross- The light Higgs mass corresponding to this set of input section with MH + (a) and MH (b) for the CP-vio- parameter is 51 GeV. The Fig. 4(b) shows the same, ◦ 1 ¯ ¯ lating phase ΦCP = 90 keeping other MSSM para- in terms of cross-section distribution in bb, bbW and meters fixed as in Fig. 1. Apart from the choice of the bbbW¯ invariant masses for the signal. Both these fig- phase, the main difference from Fig. 1 is in the val- ures show similar clustering of the bb¯, bbW¯ invariant + ues of tan β. In this case we have somewhat larger masses at values corresponding to MH1 and MH ,re- values of tan β, namely 3.6, 4.0, 4.6 and 5.0, corre- spectively, as in Fig. 3. sponding to the light Higgs window of Ref. [18] for It should be mentioned here that the combinatorial ◦ ΦCP = 90 . The largest signal cross-section in this background has already been included in the inclu- case is ∼ 38 fb. Note that in both cases the signal sive bb¯ and bbW¯ invariant mass distributions plot- cross-section is 20 fb for MH1 15 GeV. ted in Figs. 3–4 whereas the three-dimensional plots In Fig. 3(a) we show the three-dimensional plot showing the correlation do not include this. Within for the correlation between m ¯ and m ¯ invari- the framework of the mass reconstruction strategy out- bb ◦ bbW ant mass distribution for ΦCP = 60 ,tanβ = 2 and lined before, after the reconstruction of t → bW, one MH + = 125.6 GeV. The light Higgs mass correspond- is left with three b jets and a W . The former corre- ing to this set of input parameter is 24.8 GeV. It is spond to three possible invariant bb¯ masses for each clear from Fig. 3 that there is simultaneous clustering MC point. It is seen from Figs. 3 and 4 that even af- in the mbb¯ distribution around MH1 and in the mbbW¯ ter inclusion of all the possible pairs at each point the distribution around MH ± . Fig. 3(b) shows the same, peak at the H1 mass is clearly visible. Now for further in terms of cross-section distribution in bb¯, bbW¯ and reconstruction one can choose the pair with invariant bbWb¯ invariant masses for the signal. The clustering mass closest to the peak and then calculate the bbW¯ feature can be used to distinguish the signal over the invariant mass by combining this pair with the remain- standard model background. As a matter of fact we ing W . In case of quadratic ambiguity for the W both estimated the background to the signal coming from the values for the Wbb¯ invariant mass are retained. the QCD production of ttb¯ b¯. Even though the starting LO cross-section for ttb¯ b¯ production is as high as ∼ 8.5 pb, once all the cuts (including the mass win- 1 Preliminary studies in ATLAS Collaboration presented at Les dow cuts) are applied we are left with a contribution Houches Workshop [25] also find that this background can be sup- to the signal type events of less than 0.5 fb. The major pressed to negligible levels by similar requirements. D.K. Ghosh et al. / Physics Letters B 628 (2005) 131–140 139

+ ± Again, we see a clear peak at the H mass. Finally that such a light H1 and light H , can be probed at the combining this with the remaining b gives the Wbbb LHC in tt¯ signal where one of the top quarks decays ¯ ± ± invariant mass which peaks at Mt . In case of quadratic into the bbbW channel, via t → bH , H → WH1 ¯ ambiguity for the W we have chosen the Wbb com- and H1 → bb. Our parton level Monte Carlo yields bination with invariant mass closer to the H + mass upto ∼ 1100–5000 events for a L = 30 fb−1 corre- ◦ peak. In the three-dimensional plot of Figs. 3–4 we sponding to the CP-violating phase ΦCP = 90 and show the pair of invariant masses corresponding to this 60◦, respectively. The events will show a very charac- combination of Wbb as well as the bb¯ invariant mass teristic correlation between the bb¯, bbW¯ and bbbW¯ closest to H1 mass. We have found that about 50% invariant mass peaks, indicating that the SM back- of the signal events will have more than one com- ground may be negligible. Further, in a considerable bination of the bb¯ and bbW¯ invariant masses in the part of this region, the branching ratio for the H ± → ± + ± window MH1 15 GeV and MH 25 GeV, respec- τν channel, that is normally used for the charged tively, when one includes all the combinations. Thus Higgs search, is reduced by over an order of magni- the combinatorial background is important but does tude. Thus, this tt¯ signal will be a probe of both a ± not seem to overwhelm the signal. light neutral H1 and a light charged Higgs H .Itis A comment about the Mt dependence of our re- imperative that this investigation is followed up with sults is in order. If the value of Mt used is increased a more exact simulation using event generator level from 175 to 178 GeV, typically the mass difference Monte Carlo and detector acceptance effects, which + − MH MH1 goes up by about 7–8 GeV and thus the is beyond our means. We hope that the encouraging curves in Figs. 1 and 2 will extend to MH1 values results from this parton level Monte Carlo study will higher by about 7–8 GeV. We, however, have used the induce the CMS and the ATLAS Collaborations to un- more conservative value of 175 GeV for Mt .Asthe dertake such investigations. window in the tan β–MH + window which we explore, has been obtained using Mt = 175 GeV in Ref. [18]. Since the size of the window where LEP has no reach Acknowledgements also gets bigger with an increased value of Mt [3,19], the above observation simply implies that the region We wish to thank the organizers of the Work- ± ¯ which the process t → bH → bH1W → bbbW can shop on High Energy Physics Phenomenology 8 probe will also be bigger in that case. (WHEPP8) in Mumbai, India (January 4–15, 2004) where this work was started and the Board for Re- search in Nuclear Sciences (BRNS) in India, for 4. Conclusions its support to organize the Workshop. The work of D.K.G. is supported by the Bundesministerium für Thus we have looked in the CPX scenario, in the Bildung und Forschung Germany, grant 05HT1RDA/6. CP-violating MSSM, at the region in the tan β–MH ± D.K.G. would also like to thank US DOE contract plane, where a light H1 signal might have been lost numbers DE-FG03-96ER40969 for financial support at LEP due to strong suppression of the H1ZZ cou- during the initial stages of this work. pling and where the Tevatron and the LHC will have no reach due to a simultaneous suppression of ¯ the H1tt coupling as well. Specifically, we concen- References trated in the MSSM parameter space 3.5 < tan β<5, MH 50 GeV and 2 < tan β<3, MH 40 GeV, [1] Particle Data Group Collaboration, S. Eidelman, et al., Phys. 1 1 ◦ for the common CP-violating phase ΦCP = 90 and Lett. B 592 (2004) 1, see also http://pdg.lbl.gov. ◦ 60 , respectively, which correspond to the light H1 [2] ALEPH, DELPHI, L3, OPAL, The LEP Higgs Working Group window of [18]. We find that a light charged Higgs for Higgs Boson Searches, Phys. Lett. B 565 (2003) 61, CERN- EP-2003-011. (MH ±

[4] J.F. Gunion, B. Grzadkowski, H.E. Haber, J. Kalinowski, Phys. [13] A. Pilaftsis, C.E. Wagner, Nucl. Phys. B 553 (1999) 3, hep-ph/ Rev. Lett. 79 (1997) 982, hep-ph/9704410. 9902371. [5] A.D. Sakharov, Pis’ma Zh. Eksp. Teor. Fiz. 5 (1967) 32; [14] D.A. Demir, Phys. Rev. D 60 (1999) 055006, hep-ph/9901389. A.D. Sakharov, JETP Lett. 6 (1967) 24. [15] S.Y. Choi, M. Drees, J.S. Lee, Phys. Lett. B 481 (2000) 57, [6] For a recent summary, see A.D. Dolgov, hep-ph/0211260. hep-ph/0002287. [7] C.L. Bennett, et al., Astrophys. J. Suppl. Ser. 148 (2003) 1, [16] M. Carena, J.R. Ellis, A. Pilaftsis, C.E. Wagner, Nucl. Phys. astro-ph/0302207. B 586 (2000) 92, hep-ph/0003180. [8] For a review, see e.g. M. Dine, A. Kusenko, Rev. Mod. Phys. 76 [17] G.L. Kane, L.-T. Wang, Phys. Lett. B 488 (2000) 383, hep- (2004) 1, hep-ph/0303065. ph/0003198. [9] P. Nath, Phys. Rev. Lett. 66 (1991) 2565; [18] M. Carena, J.R. Ellis, S. Mrenna, A. Pilaftsis, C.E. Wagner, Y. Kizukuri, N. Oshino, Phys. Rev. D 46 (1992) 3025; Nucl. Phys. B 659 (2003) 145, hep-ph/0211467. T. Ibrahim, P. Nath, Phys. Lett. B 418 (1998) 98, hep-ph/ [19] OPAL Collaboration, G. Abbiendi, et al., Eur. Phys. J. C 37 9707409; (2004) 49. T. Ibrahim, P. Nath, Phys. Rev. D 57 (1998) 478, hep-ph/ [20] A. Dedes, S. Moretti, Phys. Rev. Lett. 84 (2000) 22, hep- 9708456; ph/9908516; T. Ibrahim, P. Nath, Phys. Rev. D 58 (1998) 019901, Erratum; A. Dedes, S. Moretti, Nucl. Phys. B 576 (2000) 29, hep- T. Ibrahim, P. Nath, Phys. Rev. D 60 (1999) 079903; ph/990941; T. Ibrahim, P. Nath, Phys. Rev. D 60 (1999) 119901; S.Y. Choi, K. Hagiwara, J.S. Lee, Phys. Rev. D 64 (2001) M. Brhlik, G.J. Good, G.L. Kane, Phys. Rev. D 59 (1999) 032004, hep-ph/0103294; 115004, hep-ph/9810457; S.Y. Choi, K. Hagiwara, J.S. Lee, Phys. Lett. B 529 (2002) 212, A. Bartl, T. Gajdosik, W. Porod, P. Stockinger, H. Stremnitzer, hep-ph/0110138; Phys. Rev. D 60 (1999) 073003, hep-ph/9903402; A.G. Akeroyd, A. Arhrib, Phys. Rev. D 64 (2001) 095018, hep- D. Chang, W.-Y. Keung, A. Pilaftsis, Phys. Rev. Lett. 82 (1999) ph/0107040; 900, hep-ph/9811202; A. Arhrib, D.K. Ghosh, O.C.W. Kong, Phys. Lett. B 537 (2002) S. Pokorski, J. Rosiek, C.A. Savoy, Nucl. Phys. B 570 (2000) 217, hep-ph/0112039; 81, hep-ph/9906206; S.Y. Choi, M. Drees, J.S. Lee, J. Song, Eur. Phys. J. C 25 E. Accomando, R. Arnowitt, B. Dutta, Phys. Rev. D 61 (2000) (2002) 307, hep-ph/0204200. 115003, hep-ph/9907446; [21] M. Drees, M. Guchait, D.P. Roy, Phys. Lett. B 471 (1999) 39, S. Abel, S. Khalil, O. Lebedev, Nucl. Phys. B 606 (2001) 151, hep-ph/9909266. hep-ph/0103320; [22] J.S. Lee, A. Pilaftsis, M. Carena, S.Y. Choi, M. Drees, J.R. El- U. Chattopadhyay, T. Ibrahim, D.P. Roy, Phys. Rev. D 64 lis, C.E.M. Wagner, Comput. Phys. Commun. 156 (2004) 283, (2001) 013004, hep-ph/0012337; hep-ph/0307377. D.A. Demir, M. Pospelov, A. Ritz, Phys. Rev. D 67 (2003) [23] S. Heinemeyer, Eur. Phys. J. C 22 (2001) 521, hep-ph/0108059. 015007, hep-ph/0208257. [24] E. Ma, D.P. Roy, J. Wudka, Phys. Rev. Lett. 80 (1998) 1162, [10] A. Pilaftsis, Nucl. Phys. B 644 (2002) 263, hep-ph/0207277. hep-ph/9710447. [11] T. Falk, K.A. Olive, M. Pospelov, R. Roiban, Nucl. Phys. B 560 [25] Talk presented by Markus Schumacher at the Workshop at TeV (1999) 3, hep-ph/9904393. colliders at Les Houches, 1–21 May, 2005, summary in the [12] A. Pilaftsis, Phys. Rev. D 58 (1998) 096010, hep-ph/9803297; talk by T. Lari: http://agenda.cern.ch/askArchive.php?base= A. Pilaftsis, Phys. Lett. B 435 (1998) 88, hep-ph/9805373. agenda&categ=a053250&id=a053250s10t1/moreinfo. Physics Letters B 628 (2005) 141–147 www.elsevier.com/locate/physletb

Probing universal extra dimension at the International Linear Collider

Gautam Bhattacharyya a, Paramita Dey a, Anirban Kundu b, Amitava Raychaudhuri b

a Saha Institute of Nuclear Physics, 1/AF Bidhan Nagar, Kolkata 700064, India b Department of Physics, University of Calcutta, 92 A.P.C. Road, Kolkata 700009, India Received 7 June 2005; received in revised form 29 August 2005; accepted 31 August 2005 Available online 15 September 2005 Editor: M. Cveticˇ

Abstract In the context of an universal extra-dimensional scenario, we consider production of the ﬁrst Kaluza–Klein electron positron + − pair in an e e collider as a case-study for the future International Linear Collider. The Kaluza–Klein electron decays into a nearly degenerate Kaluza–Klein photon and a standard electron, the former carrying away missing energy. The Kaluza–Klein electron and photon states are heavy with their masses around the inverse radius of compactiﬁcation, and their splitting is + − controlled by radiative corrections originating from bulk and brane-localised interactions. We look for the signal event e e + √ − large missing energy for s = 1 TeV and observe that with a few hundred fb 1 luminosity the signal will be readily detectable over the standard model background. We comment on how this signal may be distinguished from similar events from other new physics.  2005 Elsevier B.V. All rights reserved.

PACS: 12.60.-i; 14.60.Hi

Keywords: Universal extra dimension; International linear collider

1. Introduction consider such models with one extra dimension having inverse radius of compactification in the range R−1 = 250–450 GeV. We examine production of the first If extra-dimensional models in a few hundred GeV + − scale [1] are realised in nature, one cannot only un- Kaluza–Klein (KK) electron positron√ pair (E1 E1 )in a linear e+e− collider operating at s = 1 TeV. The dertake their precision studies at the proposed Inter- ± heavy modes E would decay into the standard (zero national Linear Collider (ILC) [2] but also can distin- ± 1 guish them from other new physics. In this Letter, we modes) e and the first KK photon (γ1), the latter carrying away missing energy. The splitting between ± E1 and γ1 comes from the bulk and brane-localised E-mail address: [email protected] (G. Bhattacharyya). radiative corrections. The cross section of the final

0370-2693/$ – see front matter  2005 Elsevier B.V. All rights reserved. doi:10.1016/j.physletb.2005.08.117 142 G. Bhattacharyya et al. / Physics Letters B 628 (2005) 141–147 state e+e− plus missing energy is quite large and the nario from g − 2 of the muon [4], flavour changing ¯ Standard Model (SM) background is√ tractable, so that neutral currents [5–7], Z → bb decay [8],theρ pa- even with a one year run of ILC at s = 1 TeV with rameter [3], other electroweak precision tests [9] and approximately 300 fb−1 enough statistics would ac- implications from hadron collider studies [10], all in- cumulate. Forward–backward asymmetry of the final dicate that R−1 a few hundred GeV. As a result, state electron mildly depends on the initial polarisa- even the second KK state having mass 2/R will be tions. Even though the mass spectrum of KK excita- beyond the pair-production reach of at least the first tions of different SM particles may resemble the su- phase of the planned linear collider. So, as mentioned persymmetric pattern, angular distribution of the final in the introduction, we consider the production of first electrons can be used to discriminate the intermedi- KK electron positron pair and their subsequent de- ate KK electrons from selectrons or other new physics cays into first KK photon plus the standard leptons; ± scalars. the degeneracy between E1 and γ1 being lifted by radiative corrections which we shall briefly touch upon below. 2. Simplest universal extra dimension

We consider the simplest realisation of the univer- 3. Radiative corrections and the spectrum sal extra dimension (UED) scenario in which there is only one extra dimension which is accessed by all Barring zero mode masses, the degeneracy (n/R) SM particles [3]. The extra dimension (y) is compact- at a given KK level is only a tree level result. Ra- ified on a circle of radius R along with a Z2 orb- diative corrections lift this degeneracy [11–14].For ifolding which renders all matter and gauge fields, intuitive understanding, we consider the kinetic term µ 5 viewed from a 4-dimensional (4d) perspective, depend of a scalar field as [11] Lkin = Z∂µφ∂ φ − Z5∂5φ∂ φ on y either as cos(ny/R) (even states) or sin(ny/R) (µ = 0, 1, 2, 3), where Z and Z5 are renormalisation (odd states), where n is the KK index. The tree level constants. Recall, tree level KK masses (Mn = n/R) mass of the nth state of a particular field is given by originate from the kinetic term in the y-direction. If 2 = 2 + 2 2 = Mn M0 n /R , where M0 is the zero mode mass Z Z5, there is no correction to those KK masses. of that field. Clearly, excepting the top quark, Higgs, But this equality is a consequence of Lorentz invari- W , and Z, the KK states of all other SM particles with ance. When a direction is compactified, Lorentz in- the same n are nearly mass degenerate at n/R.Now, variance is lost, so also is lost the equality between with all the fields propagating in the bulk, the mo- Z and Z5, leading to ∆Mn ∝ (Z − Z5). One actually mentum along the fifth direction, quantised as n/R, encounters two kinds of radiative corrections. remains a conserved quantity. A closer scrutiny however reveals that a remnant Z2 symmetry (different (a) Bulk corrections: These corrections are finite. from the previous Z2) in the effective 4d Lagrangian Moreover, they are nonzero only for bosons. They dictates that what actually remains conserved is the arise when the internal loop lines wind around the KK parity defined as (−1)n. As a result, level mix- compactified direction, sensing that compactifica- ings may occur which admit even states mix only with tion has actually occured, leading to the breaking even states, and odd with odd. Therefore, (i) the light- of Lorentz invariance. The correction to the KK est Kaluza–Klein particle (LKP) is stable, and (ii) a mass Mn works out to be independent of n and = 2 ∝ 4 2 single KK state (e.g., n 1 state) cannot be produced. goes like ∆Mn β/16π R , where β is a sym- These two criteria are reminiscent of supersymmetry bolic representation of the collective beta function with conserved R-parity where the lightest supersym- contributions of the gauge and matter KK fields metric particle (LSP) is stable, and superparticles can floating inside the loop. Since the beta function only be pair produced. If produced, the heavier KK contributions are different for particles in differ- modes can cascade decay to lighter ones, eventually ent representation, the KK degeneracy is lifted. to soft SM particles plus LKP carrying away missing One can understand the decoupling of the cor- energy. But low energy constraints on the UED sce- rection as inverse power of R by noting that the G. Bhattacharyya et al. / Physics Letters B 628 (2005) 141–147 143

Table 1 KK masses (n = 1) for different cases: excited electrons in SU(2) singlet and doublet representations, excited charged and neutral gauge bosons, respectively. All mass scales are in GeV −1 R ΛR M ˆ ME MW MZ Mγ E1 1 1 1 1 250 20 252.7 257.5 276.5 278.1 251.6 50 253.6 259.7 280.6 281.9 251.9 350 20 353.8 360.4 379.0 379.7 351.4 50 355.0 363.6 384.9 385.4 351.5 450 20 454.9 463.4 482.9 483.3 451.1 50 456.4 467.5 490.6 490.8 451.1

R →∞limit makes the fifth direction uncom- will be pair produced at the foreseeable collider en- pactified leading to exact Lorentz invariance. For ergy. As noted in Table 1, the orbifold corrections cre- the KK fermions this correction is zero. ate enough mass splitting between these states and γ1 (b) Orbifold corrections: Orbifolding additionally (dominantly B1) allowing the former to decay within breaks translational invariance in the fifth direc- the detector to e±+ missing energy which constitute E± Eˆ± tion. The corrections to the KK masses arising our signal. Below we denote 1 and 1 collectively from interactions localized at the fixed points are ± by E1 . not finite unlike the bulk corrections. These are Now we consider the pair production e+e− → logarithmically divergent. These boundary terms + − E1 E1 for different polarisations of the incident can be thought of as counterterms whose finite beams. The interaction proceeds through s- and t- parts are completely undetermined. A rather bold channel graphs. The s-channel processes are mediated but predictive hypothesis is to assume that these by γ and Z.Thet-channel processes proceed through corrections vanish at the cutoff scale Λ. Calcula- 5 5 γ1/Z1 gauge bosons and γ1 /Z1 scalars (fifth com- tion shows that the correction to Mn does depend ponents of 5d neutral gauge bosons). E1 decays into on Mn in this case, and a generic correction looks e and γ1. The splitting between E1 and γ1 masses is 2 2 2 like ∆Mn ∼ Mn(β/16π ) ln(Λ /µ ), where µ is sufficient for the decay to occur well within the de- the low energy scale where we compute these cor- tector with a 100% branching ratio (BR). It may be ˆ rections. The KK states are thus further split, this possible to observe even a displaced vertex (e.g., E1 time with an additional dependence on Λ. decays, for R−1 = 250 GeV). So in the final state we + − have e e + 2γ1 (≡ missing energy). The same final state can be obtained from e+e− → 3.1. Spectrum + − W1 W1 as well. Again, the interaction proceeds γ Z s N The mass spectra of the first excited electrons through and mediated -channel graphs, and 1L ± mediated t-channel graphs. Given the splittings (Ta- and the first excited W , Z and photon for different ± ± N ble 1), W1 can decay into ei and 1i ,aswellas choices of R and Λ are displayed in Table 1. While ± into E and νi , where i = 1 to 3 is the flavour in- thetreelevelKKmassisgivenby1/R, the radiative 1i ± dex. While N1i escapes undetected, E decays into corrections to them depend both on R and Λ (for the ± 1i exact expressions, see, e.g., [11]). ei and γ1. So, if we tag only electron flavours (plus + − → + − missing energy) in the final state, the e e W1 W1 cross section, which is in the same ball-park as the + − → + − 4. Production and decay modes of KK leptons e e E1 E1 cross section, should be multiplied by a BR of ∼ 1/9. Numerically, therefore, this channel The SU(2) doublet KK states appear with both left is not significant. Even more insignificant contribution L L = N E T ± and right chiralities as L,R, where ( n, n) ,so would come from (W 5) scalar (fifth component of ˆ 1 do the SU(2) singlets EL,R. All these states for n = 1 5d charged gauge bosons) pair production. 144 G. Bhattacharyya et al. / Physics Letters B 628 (2005) 141–147

5. SM background

The main background comes from γ ∗γ ∗ → e+e− events, where γ ∗s originate from the initial electron– positron pair while the latter go undetected down the beam pipe [15].Theγ ∗γ ∗ production cross section is ∼ 104 pb. About half of these events results in final state e+e− pair as visible particles. The background e+e− pairs are usually quite soft and coplanar with the beam axis [16]. An acoplanarity cut significantly removes this background. Such a cut, we have checked, does not appreciably reduce our signal. For example, excluding events which deviate from + − + − coplanarity within 40 mrad reduces only 7% of the Fig. 1. Cross section versus 1/R for the process e e → e e + signal cross section. In fact, current designs of LC en- missing energy. Plots are shown for unpolarised incident beams with ΛR = 2, 20 and 50, and for ‘optimum’ ILC polarisation (80% for visage very forward detectors to specifically capture − + = + − 1 e and 50% for e beams) for ΛR 20. The lower and upper the ‘would-be-lost’ e e pairs down the beam pipe. energy cuts on the final state leptons are set at 0.5 and 20 GeV, re- Numerically less significant backgrounds would spectively. The angular cuts with respect to the beam axis are set at + − + − + − ◦ come from e e → W W , eνW, e e Z, followed 15 . by the appropriate leptonic decays of the W and Z. mitting only those final state electrons which are away from beam pipe by more than 15◦. 6. Collider parameters

The study is√ performed in the context of the ILC 7. Cross sections [2], running at s = 1 TeV (upgraded option), and + − with a polarisation efficiency of 80% for e− and 50% Thecrosssectionfore e plus missing energy for e+ beams. We impose kinematic cuts on the lower final state has been plotted in Fig. 1. We have ne- and upper energies of the final state charged leptons as glected the events coming from excited W decay. No- 0.5 and 20 GeV, respectively. While the lower cut is a tice that varying the beam polarisations does create requirement for minimum energy resolution for iden- a detectable difference in the cross section, neverthe- tification, the (upper) hardness cut eliminates most of less, there is no special gain for any particular choice: − 3 the SM background. We also employ a rapidity cut ad- for left-polarised e beam, both B1 and W1 contribute, whereas for the right-polarised e− beam, only B1 contributes but with an enhanced coupling. The 1 To counter the two-photon background we may also advocate the following strategy. Instead of eliminating the background, we cross section enhances as we increase ΛR from 2 to + − calculate the number of e e events originating from two-photon 20; this is due to the change in θW1 (the weak angle + − production. For this we first count the number of µ µ plus miss- for n = 1 KK gauge bosons). Further increase of ΛR ing energy events. The number of such events coming from the does not change the cross section; a saturation point decay of KK muons, we have checked for 1/R ∼ 250–300 GeV, is reached. Additionally, the kinematic cuts tend to would be rather small, about a factor of 1/20 compared to the num- + − ber of e e plus missing energy events, due to strong s-channel reduce the cross section which is why the curve for + − suppression. So most of the observed µ µ events would have ΛR = 50 lies between the ones for ΛR = 2 and 20. ∗ ∗ sprung from γ γ . Thus the muon events serve as a normalisation + − to count the e e plus missing energy events originating from the two-photon background. Our signal events should be recognized as 8. Forward–backward (FB) asymmetries those which are in excess of that. Based on the estimates of two- photon events given by the Colorado group [15], we have checked − that for an integrated luminosity of 300 fb 1 the signal events would The FB asymmetries of the final state electrons, de- be about ten times larger than the square-root of background. fined as AFB = (σF − σB)/(σF + σB), are plotted in G. Bhattacharyya et al. / Physics Letters B 628 (2005) 141–147 145 √ scenario. Assume s m, where m is the mass of the heavy lepton/scalar, so that only the t-channel diagrams, with just a heavy gauge boson in case (a) and a heavy fermion in case (b) as propagators, are numerically dominant (we assume this only for the ease of analytic comparison). The heavy states are produced with sufficient boost, therefore the tagged leptons they decay into have roughly the same angular distributions as them. Take the mass of the t-channel propagator in either case to be about the same as the mass of the heavy lepton/scalar as m = 250 GeV. For these choices, the ratio of dσ/d cos θ (case (a)/case (b)) is observed to be (3.8 + 1.3 cos θ + 0.6 cos2 θ)/sin2 θ, Fig. 2. AFB versus 1/R for the same process. Plots are shown for un- clearly indicating that the two cases can be easily dis- = polarised incident beams with ΛR 2 and 20, and for ‘optimised’ tinguished from their angular distributions. Moreover, ILC polarisation for ΛR = 20. The cuts are as in Fig. 1. the UED cross section is found to be a factor of 4 to 5 larger than the scalar production cross section for Fig. 2 for different values of ΛR. The reason as to similar couplings and other parameters. For selectron why it falls with increasing 1/R is as follows. The production, indeed one must take the detailed neu- + − → + − first-stage process e e E1 E1 is forward-peaked, tralino structure and the exact couplings, but the basic and for smaller 1/R, i.e., lighter KK electrons, the fi- arguments that we advanced for distinguishing scalar- ± nal state e are boosted more along the direction of from the fermion-productions at the primary vertex us- ± the parent E .As1/R, or equivalently the KK mass, ing the toy model would still hold. increases the boost drops and the distribution tends to lose its original forward-peaked nature. Polarisation of the beams does not appear to have a marked advan- 10. Comparison with the CLIC Working Group tage. A point to note is that the electrons coming from study two-photon background will be FB symmetric. Our analysis is complementary to that in the CLIC multi-TeV linear collider study report [18]. While we 9. Discriminating UED from other new physics have electrons in the final state, the study in [18] involves muons. Clearly, the angular distribution in It is not our purpose in this brief note to discuss our case is dominated by t-channel diagrams, while at depth any specific version of new physics model the process studied in [18] proceeds only through s- and its possible discrimination from UED. Still, for channel graphs. Due to the inherently forward-peaked illustrative purposes, we recall that the spectrum of nature of the t-channel diagrams, we obtain a signif- KK excitations for a given level (here n = 1) may icantly larger FB asymmetry. Unlike in [18],wehave be reminiscent of a possible supersymmetry spectrum neither included the initial state radiation effect nor in- [17], where the KK parity is ‘like’ the R-parity. Even corporated detector simulation. in a situation when the LSP weighs above 250 GeV and conspires to be almost degenerate with the selectron, it is possible to discriminate a KK electron 11. LHC/ILC synergy decaying into the KK photon (LKP) from a selectron decaying into a neutralino LSP by studying the angu- Extensive studies have been carried out [19] ad- lar distribution pattern of the final state electron. We dressing the physics interplay between the LHC and demonstrate this with a simple toy example. Compare the ILC, in particular, how the results obtained at one the pair production of (a) generic heavy fermions and machine would influence the way analyses would be (b) generic heavy scalars in an e+e− collider in a toy carried out at the other. While LHC may serve as 146 G. Bhattacharyya et al. / Physics Letters B 628 (2005) 141–147 a discovery machine, precision measurements of the the Z2 peak can be discovered at the LHC through masses, decay widths, mixing angles, etc., of the dis- some hadronically quiet channels, γ2 will be hard to covered particles can be carried out at the ILC. To detect at the LHC because it immediately decays into illustrate this with an example, let us consider a selec- two jets which will be swamped by the QCD back- tron weighing around 200 GeV. The analysis2 shows ground [21]. Turning our attention now to the ILC, that while the uncertainty in its mass determination is given its proposed energy reach, these states will be around 5 GeV at the LHC, with inputs from the ILC too heavy to be pair produced, but as shown in [21], the uncertainty can be brought down to about 0.2 GeV single resonant productions of Z2 and γ2, despite sup- due to a significantly better edge analysis in the clean pression from KK number violating couplings, will ILC environment. Similar precisions may be expected have sizable cross sections. Precision measurements of for the masses of the KK electrons as well for a com- their peak positions and widths at the ILC will enable parable cross section. However, if R−1 is large and the one to extract R and Λ. cross section goes down by about a factor of 50–100, the sensitivity will also go down scaling inversely as the square root of the number of events. (Beam po- 12. Conclusion larisation will not be of much help, as can be seen from Fig. 1.) Another point that may play a signifi- We have shown that the ILC may have a significant cant role in determining the masses is the softness of role in not only detecting the presence of few-hundred- final-state electrons. We have applied adequate soft- GeV-size extra dimensions but also discriminating it ness cuts to remove very soft electrons. One needs a from other new physics options, like supersymme- detailed analysis to determine the exact accuracies at try. Even if the KK modes are first observed at the different benchmark points, but roughly the accuracy LHC, one needs the ILC for their proper identification of determining the KK masses at the ILC should be through precision measurements of the masses, cou- of the order of 1 GeV or even better. Even if the KK plings and spin correllations. The physics interplay of electrons are first observed at the LHC, their spin as- the LHC and the ILC will be quite important in this signments might not be possible. This particular issue, context. i.e., whether UED states can be distinguished from the supersymmetric states at the LHC, has been studied in [20]. Considering the decay chain in which a Acknowledgements KK quark (or, a squark) disintegrates into a quark, a lepton–antilepton pair and missing energy, and look- We thank H.C. Cheng for clarifying to us some ing at the spin correlations of the emitted quark with aspects of orbifold radiative corrections. We acknowl- one of the leptons, the authors of [20] conclude that edge very fruitful correspondences with M.E. Peskin for a quasi-degenerate (UED like) spectrum, spin as- on the two-photon background. We also thank J. Kali- signments of the discovered particles could hardly be nowski for presenting a preliminary version of this efficiently done: the best discriminator of UED from work in ICHEP 2004, Beijing [22]. Thanks are also supersymmetry in that case would be a significantly due to S. Dutta and J.P. Saha, who were involved in larger production cross sections for the UED particles the earlier stages of this work. Stimulating discussions than those of the supersymmetric ones. On the con- with the participants of the Study Group on Extra Di- trary, if the observed spectrum is hierarchical (e.g., mensions at LHC, held at HRI, Allahabad, are also a supersymmetric type), the prospects of observing acknowledged. G.B., P.D., and A.R. acknowledge hos- spin correlations would be better. However, the ILC pitality at Abdus Salam ICTP, Trieste, while G.B. also would provide a better environment for doing spin acknowledges hospitality at LPT, Orsay, and Theory studies. The clinching evidence of UED would of Division, CERN, at different stages of the work. G.B. course be the discovery of the n = 2 KK modes. While and A.R. were supported, in part, by the DST, India, project number SP/S2/K-10/2001. A.K. was supported by DST, Government of India, through the project 2 See, for example, Table 5.14 of Ref. [19]. SR/S2/HEP-15/2003. G. Bhattacharyya et al. / Physics Letters B 628 (2005) 141–147 147

References A. Muck, A. Pilaftsis, R. Rückl, Nucl. Phys. B 687 (2004) 55, hep-ph/0312186. [1] I. Antoniadis, Phys. Lett. B 246 (1990) 377. [11] H.C. Cheng, K.T. Matchev, M. Schmaltz, Phys. Rev. D 66 [2] D.J. Miller, hep-ph/0410306. (2002) 036005, hep-ph/0204342. [3] T. Appelquist, H.C. Cheng, B.A. Dobrescu, Phys. Rev. D 64 [12] M. Puchwein, Z. Kunszt, Ann. Phys. 311 (2004) 288, hep- (2001) 035002, hep-ph/0012100. th/0309069. [4] P. Nath, M. Yamaguchi, Phys. Rev. D 60 (1999) 116006, hep- [13] H. Georgi, A.K. Grant, G. Hailu, Phys. Lett. B 506 (2001) 207, ph/9903298. hep-ph/0012379. [5] D. Chakraverty, K. Huitu, A. Kundu, Phys. Lett. B 558 (2003) [14] G. von Gersdorff, N. Irges, M. Quiros, Nucl. Phys. B 635 173, hep-ph/0212047. (2002) 127, hep-th/0204223. [6] A.J. Buras, M. Spranger, A. Weiler, Nucl. Phys. B 660 (2003) [15] See the website: http://hep-www.colorado.edu/SUSY,inpar- 225, hep-ph/0212143; ticular, N. Danielson, COLO HEP 423; A.J. Buras, A. Poschenrieder, M. Spranger, A. Weiler, Nucl. See also M. Battaglia, D. Schulte, hep-ex/0011085; Phys. B 678 (2004) 455, hep-ph/0306158. H. Baer, T. Krupovnickas, X. Tata, JHEP 0406 (2004) 061, [7] K. Agashe, N.G. Deshpande, G.H. Wu, Phys. Lett. B 514 hep-ph/0405058. (2001) 309, hep-ph/0105084. [16] M.E. Peskin, private communications. [8] J.F. Oliver, J. Papavassiliou, A. Santamaria, Phys. Rev. D 67 [17] H.C. Cheng, K.T. Matchev, M. Schmaltz, Phys. Rev. D 66 (2003) 056002, hep-ph/0212391. (2002) 056006, hep-ph/0205314. [9] T.G. Rizzo, J.D. Wells, Phys. Rev. D 61 (2000) 016007, hep- [18] M. Battaglia, A. De Roeck, J. Ellis and D. Schulte (Eds.), ph/9906234; Physics at the CLIC multi-TeV linear collider, CERN Report A. Strumia, Phys. Lett. B 466 (1999) 107, hep-ph/9906266; No. CERN-2004-005; C.D. Carone, Phys. Rev. D 61 (2000) 015008, hep-ph/9907362. See also M. Battaglia, A. Datta, A. De Roeck, K. Kong, K.T. [10] T. Rizzo, Phys. Rev. D 64 (2001) 095010, hep-ph/0106336; Matchev, hep-ph/0502041. C. Macesanu, C.D. McMullen, S. Nandi, Phys. Rev. D 66 [19] G. Weiglein, et al., LHC/LC Study Group, hep-ph/0410364. (2002) 015009, hep-ph/0201300; [20] J.M. Smillie, B.R. Webber, hep-ph/0507170. C. Macesanu, C.D. McMullen, S. Nandi, Phys. Lett. B 546 [21] B. Bhattacherjee, A. Kundu, hep-ph/0508170. (2002) 253, hep-ph/0207269; [22] J. Kalinowski, hep-ph/0410137. H.-C. Cheng, Int. J. Mod. Phys. A 18 (2003) 2779, hep- ph/0206035; Physics Letters B 628 (2005) 148–156 www.elsevier.com/locate/physletb

On spectral ﬂow symmetry and Knizhnik–Zamolodchikov equation ✩

Gastón E. Giribet

Departamento de Física, Universidad de Buenos Aires, Buenos Aires, Argentina Instituto de Física, Universidad Nacional de La Plata, La Plata, Argentina Received 4 August 2005; received in revised form 14 September 2005; accepted 15 September 2005 Available online 27 September 2005 Editor: T. Yanagida

Abstract

It is well known that five-point function in Liouville field theory provides a representation of solutions of the SL(2, R)k Knizhnik–Zamolodchikov equation at the level of four-point function. Here, we make use of such representation to study some aspects of the spectral flow symmetry of sl(2)k affine algebra and its action on the observables of the WZNW theory. To illustrate the usefulness of this method we rederive the three-point function that violates the winding number in SL(2, R) in a very succinct way. In addition, we prove several identities holding between exact solutions of the Knizhnik–Zamolodchikov equation.  2005 Elsevier B.V. All rights reserved.

1. Introduction Spectral flow symmetry of sl(2)k affine algebra induces the identification between some states belonging to R D± discrete representations of SL(2, )k; namely, the identification between states of the of representations j and D∓ k/2−j (see Ref. [1]; see also Ref. [2] for previous studies on spectral flow symmetry). At the level of four-point correlation functions, such identification is realized by identities between different solutions of the Knizhnik– Zamolodchikov equation (KZ) (see Ref. [3]). In Ref. [4] it was pointed out that the description of the four-point functions of the SL(2, R)k CFT in terms of the Liouville five-point function leads one to interpret the action of the spectral flow as yielding a simple Liouville reflection transformation. Hence, this “Liouville description” of the SL(2, R)k correlators turns out to be a useful tool to work out the details of certain symmetries of KZ equation which, otherwise, would remain hidden within the original picture.

✩ Based on a talk given at the Workshop on Non-Perturbative Gauge Dynamics (SISSA, Trieste, 2005). E-mail address: [email protected] (G.E. Giribet).

Actually, the aim of this Letter is that of studying the manifestation of the spectral ﬂow symmetry as encoded in the symmetries of the four-point correlators by means of this method. Moreover, we extend the discussion to the case of other Z2 transformations.

1.1. The method

1.1.1. The five-point function in Liouville field theory Let us begin by considering the quantum Liouville field theory of central charge c = 1+6Q2, with Q = b+b−1. The Liouville field ϕ(z) transforms under holomorphic coordinate transformations z → z as follows: dz ϕ(z) → ϕ(z ) = ϕ(z) + Q log (1) dz and satisfies the asymptotic behavior ϕ(z) + 2Q log |z|∼O(1) for large |z|. This specifies the boundary conditions for the theory on the sphere (see [5] for an excellent review). Here, we want to consider correlation functions = − involving local primary operators Vα(z) with√ conformal dimension ∆α α(Q α). These operators can be re- 2ϕ(z) alized by the exponential form Vα(z) ∼ e , where the Liouville field that satisfy the free field correlator ϕ(z)ϕ(0)=−2log|z|. These obey −2α(Q−α) VQ−α(z) = Rb(α)Vα(z), Vα(z1)Vα(z2) =|z1 − z2| Rb(α), (2) where Rb(α) is the Liouville reflection coefficient, given by − − + 2 1+b 2−2αb 1 − 2 + −2 − −1 −2b2 (1 b ) −2 (2bα b )(1 b 2b α) Rb(α) =− πµb b , (3) (1 − b2) (2b−1α − b−2)(1 + b2 − 2bα) and where the Liouville cosmological constant µ can be chosen as µ = π −2 for convenience. A consequence of (2) is the following operator-valued relation ··· = ··· Vα1 (z1)Vα2 (z2) VαN (zN ) Rb(α1) VQ−α1 (z1)Vα2 (z2) VαN (zN ) , (4) and the same for any of the N vertex operators. For our purpose, we are interested in the five-point function (N = 5). This will be denoted as ALiouville (x, z) = V (0)V (z)V (x)V (1)V (∞) . (5) α1,α2,α5,α3,α4 α2 α1 α5 α3 α4 The conformal blocks of this correlators and the connection with the analogous quantities in WZNW theory was discussed in detail in [10]. Now, let us move to the WZNW theory which is the second ingredient for the method to be used.

1.1.2. The four-point functions in the SL(2, R)k WZNW model We consider the SL(2, R)k WZNW model. In particular, we concern about the four-point function. This observable is constructed by a sum over the conformal blocks of the theory. Conformal blocks are solutions of the Knizhnik–Zamolodchikov equation labeled by the internal quantum number j that is interpreted as the momentum of the interchanged states in the factorization procedure. Here, we are interested in correlators of the form AWZNW (x, z) = Φ (0, 0)Φ (x, z)Φ (1, 1)Φ (∞, ∞) (6) j1,j2,j3,j4 j2 j1 j3 j4 that involves the local primary ﬁelds Φj (x, z). These operators are associated to the analytic continuation of dif- + = C ferentiable functions on the H3 SL(2, )/SU(2). The complex variable x permits to order the representations of SL(2, R) as follows:

1 J a(z)Φ (x, z ) = DaΦ (x, z) +···, (7) j (z − z) x j 150 G.E. Giribet / Physics Letters B 628 (2005) 148–156 where a ∈{±, 3} correspond to the SL(2, R) generators J a, realized by − = 2 − + = 3 = − Dx x ∂x 2jx, Dx ∂x,Dx x∂x j. (8)

This representation is often employed in the applications to string theory in AdS3 space since it provides a clear interpretation of the AdS3/CFT2 correspondence (see [6] for a detailed description of functions Φj (x, z) within this context).

1.1.3. The Fateev–Zamolodchikov correspondence Once correlation functions in both Liouville and WZNW theories were introduced, we undertake the task of connecting them. Certainly, it is feasible to express the four-point functions AWZNW in terms of the ﬁve-point j1,j2,j3,j4 ALiouville =− 1 functions − 1 (including a particular ﬁfth state α5 2b ); namely, α1,α2, 2b ,α3,α4 AWZNW (x, z) = X (j ,j ,j ,j |x,z)F (j ,j ,j ,j )ALiouville (x, z), (9) j1,j2,j3,j4 k 1 2 3 4 k 1 2 3 4 − 1 α1,α2, 2b ,α3,α4 where cb represents a jµ-independent numerical factor whose explicit form can be found in the literature (see [4, 7–10]); besides, the quantum numbers of both correlators are related by −2 2α1 = b(j1 + j2 + j3 + j4 − 1), 2αi = b j1 − j2 − j3 − j4 + 2ji + b + 1 , (10) being i ∈{2, 3, 4}, and −1 −2 2α5 =−b ,b= k − 2. (11) The normalization factors are given by

2 2 −2α2/b −2α3/b −2α1/b −4(b j1j2−α1α2) −4(b j3j1−α3α1) Xk(j1,j2,j3,j4|x,z) =|x| |1 − x| |x − z| |z| |1 − z| , 2 1+j1−j2−j3−j4 4 2 (1 − b ) Υ (2j b − b) F (j ,j ,j ,j ) = c πb2b b µ , k 1 2 3 4 b 2 (12) (1 + b ) Υb(2αµ) µ=1 where the Υb function is deﬁned as follows: 2 τ 1 dτ − dτ sinh (Q − 2x) log Υ (x) = (Q − 2x)2e τ − 4 . b 4 τ τ bτ τ sinh 2 sinh 2b R> R> −1 −1 This function presents the zeros at x ∈−bZ0 − b Z0 and x ∈ bZ>0 + b Z>0.

1.1.4. Remarks Normalization factor (12) is the appropriate to connect the correlators of both models, leading to the correct structure constants when one of the momenta tends to zero. Besides, the scaling 2 1+j1−j2−j3−j4 2 (1 − b ) πb2b (1 + b2) corresponds to having set the Liouville cosmological constant to a speciﬁc value, namely, µ = π −2. Such factor is not symmetric under permutations of the symbol {j1,j2,j3,j4},asthemap(10) is not; this is to make the KPZ scaling of both correlators to match. =− 1 We can also understand the presence of the ﬁfth vertex at x with momentum α5 2b . This is certainly related to the existence of degenerate representations of SL(2, R)k (i.e., those representations containing null states in the modulo). Some of these representations are those having momentum j such that 1 − 2j = m ∈ Z>0. According to the conformal Ward identities, the correlators involving an operator Φ 1−m (x, z) are annihilated by the differential 2 G.E. Giribet / Physics Letters B 628 (2005) 148–156 151

m m − operator ∂x (and similarly for ∂x¯ ). This corresponds to the fact that, if 1 2j1 is a positive integer, then Φj1 (x, z) turns out to be a polynomial of degree m − 1. Besides, (10) implies that, when realizing the Liouville correlator (5) in terms of the Coulomb gas-like prescription, the amount of screening charges to be employed is exactly n =−2j = m − 1. Hence, a simple computation leads to obtain + m − ALiouville Dx Xk (1 m)/2,j2,j3,j4 x,z − 1 (x, z) α1,α2, 2b ,α3,α4 m −1 − 2 − 2 − − − =| | 4b j1j2 | − | 4b j1j3 m 2 | | 4α2b| − | 4α3b| − | 4α1b| − |2 = z 1 z ∂x d wr wr 1 wr z wr x wr 0, (13) r=1 − which is immediately obeyed due to the fact that the√ integrand√ is a polynomial of degree m 1inx (and the same ¯ 2α5ϕ(x) 2bϕ(wr ) ∼| − |2 +··· =− 1 for x); and this is a direct consequence of the OPE e e x wr which holds for α5 2b . The Liouville description of WZNW correlators was successfully employed in working out several details of the non-compact SL(2, C)k/SU(2) CFT. For instance, the crossing symmetry of the theory was proven in [9] and some aspects of the singularities in the WZNW observables were understood by means of this method (for instance, see [10] and [4]). Conversely, the act of thinking the four-point correlators (6) as a ﬁve-point function of other CFT (i.e., the Liouville CFT) permitted to understand the arising of certain poles at the middle of the moduli space; namely, at z = x. In terms of the function (5) these poles are understood as coming from the factorization limit when the operators Vα1 (z) and Vα5 (x) coincide.

1.2. Outline

We will study solutions of the Knizhnik–Zamolodchikov equation by means of their connection with correlators in Liouville theory. The Letter is organized as follows. In the next section we analyze the action of the spectral flow symmetry on the four-point correlation function in the WZNW model. We explicitly show how the identification D± D∓ R between states of discrete representations j and k/2−j of SL(2, )k turns out to correspond to the Liouville reflection of one particular vertex operator. This was pointed out in [4]. This simple observation leads us to rederive the formula for the three-point violating winding amplitude in AdS3. The key point in doing this is the normalization (12), which encodes the information of the WZNW structure constants. In Section 3 we apply the method of describing WZNW correlators in terms of their Liouville analogues to study other symmetries of KZ equation. Then, we are able to prove some identities between exact solutions in a rather simple way. This permits to visualize hidden symmetries of the KZ equation turning them “expectable”.

2. Spectral ﬂow symmetry and the KnizhnikÐZamolodchikov equation

2.1. The three-point function violating the winding number

First, let us consider the following 4-point correlation function in the SL(2, R)k WZNW model: AWZNW (x, z) = Φ (0, 0)Φ (x, z)Φ (1, 1)Φ (∞, ∞) . (14) k/2,j,j3,j4 j k/2 j3 j4

This particular quantity represents the three-string scattering amplitude in AdS3 space for the case of non-conserved total winding number. This was computed in [11], and the calculation employs the inclusion of the additional (auxiliary) vertex Φk/2(x, z) which is often called “spectral ﬂow operator” or “conjugate representation of the identity operator”. 152 G.E. Giribet / Physics Letters B 628 (2005) 148–156

By using the correspondence (9) and (10), we find the following expression: ∞ ∞ Φj (0, 0)Φk/2(x, z)Φj3 (1, 1)Φj4 ( , ) = Xk(k/2,j,j3,j4|x,z)Fk(k/2,j,j3,j4) 4 × ∞ Rb(αν) VQ−α2 (0)VQ−α1 (z)V− 1 (x)VQ−α3 (1)VQ−α4 ( ) , (15) 2b ν=1 −1 −2 −1 −2 −1 where 2α1 = b (j + j3 + j4 + b /2),2α2 = b (j − j3 − j4 + 3b /2 + 2),2α3 = b (−j + j3 − j4 + −2 + = −1 − − + + −2 + 3b /2 2) and 2α4 b ( j j3 j 4 3b /2 2). A crucial observation is that, according to (10),the = 4 = 5 −1 + condition j1 k/2 implies the constraint µ=1 αµ 2 b 3b. Hence, the corresponding Liouville five-point function can be realized by using a Coulomb-gas-like prescription with no insertion of screening charges; this is 4 − + − 1 = = due to the fact that the identity ν=1(Q αν) nb 2b Q is obeyed precisely for n 0. Then, we have the realization √ √ √ √ − − − − √1 ϕ(x) − ∞ e 2(Q α1)ϕ(z)e 2(Q α2)ϕ(0)e 2(Q α3)ϕ(1)e 2b e 2(Q α4)ϕ( ) − − − − −1 − −1 − −1 − =|z|4(Q α1)(α2 Q)|1 − z|4(Q α1)(α3 Q)|x − z|2b (Q α1)|x|2b (Q α2)|1 − x|2b (Q α3), and, from (10), we eventually find √ √ √ √ − √1 2(Q−α1)ϕ(z) 2(Q−α2)ϕ(0) 2(Q−α3)ϕ(1) ϕ(x) 2(Q−α4)ϕ(∞) Xk(k/2,j,j3,j4|x,z) e e e e 2b e − − + + − + − + − − − − =|x|2( j1 j2 j3 j4)|1 − x|2( j1 j2 j3 j4)|z|2j2 |1 − z|2j3 |x − z|2(k j1 j2 j3 j4), (16) where j1 = k/2 and j2 = j. Plugging (16) into (15) and rewriting the normalization factor Fk(k/2,j,j3,j4) by using −1 (bx)(b x) ±1− Υ (Q ∓ x) =±Υ (x) b2x(b b), (17) b b (±bx)(±b−1x) ±1 ±1 (b x) ±1∓2b±1x Υb b + x = Υb(x) b , (18) (1 − b±1x) we get the following expression: ∞ ∞ Φj (0, 0)Φk/2(x, z)Φj3 (1, 1)Φj4 ( , ) − − − 1 1 j j3 j4 1−2j 1 − − 1 + − = c π k 2 k 2 k + 1 2j−1 1 k−2 k−2 2(− k −j+j +j ) 2(− k +j−j +j ) 2j 2j 2( k −j−j −j ) ×|x| 2 3 4 |1 − x| 2 3 4 |z| 2 |1 − z| 3 |z − x| 2 3 4 G(1 − k/2 + j − j − j )G(j − k/2 + j − j )G(j − k/2 − j + j )G(k/2 − j − j − j ) × 3 4 3 4 3 4 3 4 , (19) G(−1)G(1 − k + 2j)G(1 − 2j3)G(1 − 2j4) where the k-dependent function G is defined as usual, namely, = −1 − −b2x2−(b2+1)x G(x) Υb ( bx)b , (20) and ck is certain j-independent factor which is completely determined by (17) and (20). Expression (19) is in exact agreement with the result obtained by Maldacena and Ooguri for the three-point violating winding correlator (cf. Eqs. (5.25), (5.33) and (E.14) of Ref. [11]). In fact, after Fourier transforming the expression above (with an 2k appropriate regularization realized by a factor limx→∞ |x| ) one gets the scattering amplitude of the three-string process that violates the winding number conservation in AdS3 space (see also [12]). G.E. Giribet / Physics Letters B 628 (2005) 148–156 153

It is remarkable that our deduction of (19) does not make use of the decoupling equation satisfied by the correlators that include a degenerate state with j1 = k/2. However, this information is implicit in the derivation above since it is codified in the factor Fk(k/2,j,j3,j4). Furthermore, the fact that the structure constants develop a delta singularity when one of the vertex involved in the OPE carries momentum j1 = k/2 is manifested in the condition n = 0 above. Recently, another relation between Liouville and WZNW theories was used to describe the string scattering amplitudes in AdS3 [13–15].In[14], the three-point function (19) was shown to be directly connected to the Liouville structure constant. It is worth mentioning that the connection of both theories employed here is a different one which, unlike the one in [13], is local in the five Liouville insertions on the sphere.

2.2. Spectral ﬂow and identities between exact solutions

First of all, let us notice that we can always consider a rescaling of the Liouville cosmological constant µ (and the WZNW coupling constant λ as well; see [16]) in such a way that the KPZ factor in (9) can be made to disappear. This corresponds to shifting the zero mode of the ﬁeld ϕ in an appropriate way. We adopt such convention here.

2.2.1. Identification between states of discrete representations ˜ = k − → ˜ Now, let us define Jµ 2 jµ. As it was pointed out in [4], the transformation jµ Jµ corresponds to reflecting the (four) quantum numbers in the Liouville correlation function, namely, αµ → Q−αµ. Then, according to (4), we find | 4 AWZNW = Xk(j1,j2,j3,j4 x,z) Fk(j1,j2,j3,j4) AWZNW j ,j ,j ,j (x, z) Rb(αν) ˜ ˜ ˜ ˜ (x, z) 1 2 3 4 ˜ ˜ ˜ ˜ | ˜ ˜ ˜ ˜ J1,J2,J3,J4 Xk(J1, J2, J3, J4 x,z) Fk(J1, J2, J3, J4) ν=1 Xk(j1,j2,j3,j4|x,z) + − + − + − − =| |2(j1 j2 k/2)| − |2(j1 j3 k/2)| |2(j4 j3 j2 j1) with ˜ ˜ ˜ ˜ z 1 z x Xk(J1, J2, J3, J4|x,z) − + − − − − − ×|1 − x|2(j4 j3 j2 j1)|x − z|2(k j1 j2 j3 j4). (21) − − − − Notice that a factor |x − z|2(k j1 j2 j3 j4) arises. This implies the existence of a singularity at the point x = z. The arising of this singularity in the solutions of the KZ equation was pointed out in Ref. [11]. Similar singularities appear in the solutions studied in [17], where an expansion in powers of (x − z) was considered. This was also discussed in [10] and [3], where the factors developing poles at x = z were studied in a similar context (cf. Eq. (42) of Ref. [10]). Moreover, in [4] logarithmic singularities at x = z for the configuration k = j1 + j2 + j3 + j4 were analyzed by using the same techniques. As it was remarked in [11], “[t]he presence of the singularity at z = x is very surprising from the point of view of the worldsheet theory since this is a point in the middle of the moduli spaces. [...] Theinterpretationofthis singularity is again associated with instantonic effects”. In Ref. [15], this “instantonic effects” in the worldsheet theory were studied in relation with Liouville theory as well, tough in a different framework. On the other hand, the normalization which connects both solutions takes the form − 4 4 1 + 1 2jµ Fk(j1,j2,j3,j4) =˜ k−2 Rb(αν) ck − , (22) F (J˜ , J˜ , J˜ , J˜ ) 2jµ 1 k 1 2 3 4 ν=1 µ=1 k−2 where c˜k is a k-dependent factor, determined by (17). This presents poles located at 2jµ = 1 + (m + 1)(k − 2) for any non-negative integer m. Besides, the zeros of this normalization factor arise at 2jµ = 1 − m(k − 2).

2.2.2. Formulae: Acting on two states Let us notice that making the change α1 → Q − α1, α2 → Q − α2, leaving α3 and α4 unchanged, is equivalent ˜ ˜ to transforming j1 → k/2 − j2 and j2 → k/2 − j1. In doing this, the factor Xk(j1,j2,j3,j4|x,z)/Xk(J2, J1,j3, 154 G.E. Giribet / Physics Letters B 628 (2005) 148–156

2(k−j −j −j −j ) j4|x,z) stands, and this develops a factor |x − z| 1 2 3 4 . Indeed, such factor appears every time the transformations of indices jµ correspond to reﬂecting the second Liouville vertex Vα1 (z). Then, we get

R (α )R (α )F (j ,j ,j ,j )X (j ,j ,j ,j |x,z) AWZNW (x, z) = b 1 b 2 k 1 2 3 4 k 1 2 3 4 AWZNW (x, z). (23) j1,j2,j3,j4 ˜ ˜ ˜ ˜ J˜ ,J˜ ,j ,j Fk(J2, J1,j3,j4)Xk(J2, J1,j3,j4|x,z) 2 1 3 4 Actually, this identity motivates the way of computing the violating winding correlator (19), since it precisely ˜ ˜ involves a transformation j1 = k/2 → J1 = 0,j2 = j → J2 = k/2 − j of its quantum numbers.

3. Hidden Z2 symmetry transformations in the four-point function

3.1. Liouville reﬂection and KZ equation

ˆ = Now, we can study a different (tough closely related) class of symmetry transformation. Let us define Jµ 1 4 − ={ } = = → ˆ 2 ν=1 jν jµ+2, with µ 1, 2, 3, 4 and jµ jµ if µ µ mod 4. The non-diagonal involution jµ Jµ is, according to (10), equivalent to doing α3 → Q − α3. This disentangles the symmetry transformation enabling us to understand it as a simple reflection (4). Consequently, we find the simple relation

| WZNW Xk(j1,j2,j3,j4 x,z) Fk(j1,j2,j3,j4) WZNW A (x, z) = Rb(α ) A (x, z) j1,j2,j3,j4 ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ 3 Jˆ ,Jˆ ,Jˆ ,Jˆ Xk(J1, J2, J3, J4|x,z) Fk(J1, J2, J3, J4) 1 2 3 4 Xk(j1,j2,j3,j4|x,z) 1 − 2− 1 − 2 1 − 2− 1 − 2 =| | k−2 (j1 j2) k−2 (j3 j4) | − | k−2 (j2 j4) k−2 (j1 j3) with ˆ ˆ ˆ ˆ z 1 z Xk(J1, J2, J3, J4|x,z) − + − ×|1 − x|2(j4 j3 j2 j1). (24)

In Ref. [3], Nichols and Sanjay proposed that both sides in Eq. (24) “can, presumably by uniqueness of the solution, beidentiﬁed[...];atleastuptosomeoverallscale”.Theidentity above proves such afﬁrmation presenting the precise overall scale, which is found to be

4 ˆ Fk(j1,j2,j3,j4) G(1 − 2Jµ) Rb(α3) =ˆck , (25) ˆ ˆ ˆ ˆ G(1 − 2jµ) Fk(J1, J2, J3, J4) µ=1

ˆ 4 − = − − − where ck is certain k-dependent factor. The factor above develops poles at ν=1 jν 2jµ+2 (1 m) n(k 2) 4 − = + + + − as well as at ν=1 jν 2jµ+2 (n 2) (m 1)(k 2), for any (m, n) pair of non-negative integers. Besides, analogous identities are obtained by considering αi → Q − αi with i ∈{2, 3, 4}, corresponding to

1 1 j → (j + j + j + j − 2j ), j → (j + j + j + j − 2j ), 1 2 1 2 3 4 i i 2 1 2 3 4 1 1 1 j → (j + j + j + j − 2j ), j → (j + j + j + j − 2j ), (26) j 2 1 2 3 4 k k 2 1 2 3 4 j for any even permutation of the symbol {i, j, k}. The conciseness of our deduction of (24) turns out to be surprising. In Ref. [8] Andreev stated the validity of such an identity and considered that the problem of “understand[ing] what underlies this mysterious relation” remained open. Regarding this, it was suggested that “[m]ay be there is a hidden symmetry in the theory”. Eq. (24) manifestly shows that such hidden symmetry actually corresponds to the Liouville reﬂection realized by (9). G.E. Giribet / Physics Letters B 628 (2005) 148–156 155

3.1.1. A working example The relation (24) also enables us to write down the explicit form of certain particular correlators. For instance, a simple computation leads to ∞ ∞ Φj2 (0, 0)Φj1 (x, z)Φj1+j2+j4 (1, 1)Φj4 ( , ) 4 4 − − j1j2 − − j1(1−j1−j2−j4) −4j1 = Fk(j1,j2,j1 + j2 + j4,j4)Rb(bj1 + Q/2)|z| k 2 |1 − z| k 2 |1 − x| , which is worked out in easy way since, as before, it is associated to a Liouville correlator with no insertion of screening charges. Besides, this correlator can be interpreted in terms of the operator product expansion of exponential operators in the SL(2, C)k/SU(2) WZNW model. Certainly, the asymptotic form of the differential functions + on H3 is governed by exponential functions which, for the conﬁguration above, correspond to the expectation value 2 j φ(0) 2 j φ(z) 2 (1−j )φ(1) 2 j φ(∞) e k−2 2 e k−2 1 e k−2 3 e k−2 4 . (27)

In a stringy theoretical context, this represents a four-string scattering process in AdS3 in which the third vertex (havingamassm ∼ j3 in the large k description) has a particular behavior with respect to the boundary variables (x3, x¯3) of the space on which the dual CFT is formulated (cf. Section 3 of [11] for a detailed discussion on the large k interpretation of this exponential functions).

3.2. A k-dependent Z2 symmetry transformation

Now, let us analyze a combination of the spectral ﬂow and the Z2 symmetry transformations considered above. → − R More precisely, let us perform the charge α1 Q α1. In terms of the SL(2, ) quantum numbers, this corre- → = 1 ˜ − ˜ sponds to doing jµ Jµ 2 ν=1 Jν Jµ. Hence, the following equality between correlators is found to hold: X (j ,j ,j ,j |x,z) F (j ,j ,j ,j ) AWZNW (x, z) = k 1 2 3 4 k 1 2 3 4 R (α ) AWZNW (x, z), (28) j1,j2,j3,j4 b 1 J1,J2,J3,J4 Xk(J1,J2,J3,J4|x,z) Fk(J1,J2,J3,J4) where | Xk(j1,j2,j3,j4 x,z) 4 (J J −j j ) 4 (J J −j j ) (k−j −j −j −j ) =|z| k−2 1 2 1 2 |1 − z| k−2 1 3 1 3 |z − x|2 1 2 3 4 . (29) Xk(J1,J2,J3,J4|x,z) 2(k−j −j −j −j ) Here, the factor |z − x| 1 2 3 4 stands again. In terms of the original indices jµ this transformation reads 1 1 j → (k + j − j − j − j ), j → (k − j + j − j − j ), 1 2 1 2 3 4 2 2 1 2 3 4 1 1 j → (k − j − j + j − j ), j → (k − j − j − j + j ). (30) 3 2 1 2 3 4 4 2 1 2 3 4 On the other hand, the relative normalization has the form 4 Fk(j1,j2,j3,j4) G(1 − 2Jµ) Rb(α1) = ck . (31) Fk(J1,J2,J3,J4) G(1 − 2jµ) µ=1 Involution (30) is a new symmetry of KZ equation, which results uncovered when is interpreted as a simple transformation (4). This enables us to write down the explicit expression of another special case; namely, ∞ ∞ Φj2 (0, 0)Φj1 (x, z)Φj1+k−j2−j4 (1, 1)Φj4 ( , ) 4 4 − − j1j2 − − j1(1−j1−k+j2+j4) −4j1 = Fk(j1,j2,j1 + k − j2 − j4,j4)Rb(bj1 + Q/2)|z| k 2 |1 − z| k 2 |z − x| , (32) which, again, can be thought of as the operator product expansion (27) of the operators in the SL(2, C)k WZNW − − − − model. This solution conﬁrms the appearance of the factor |x − z|2(k j1 j2 j3 j4) in the expression of several four-point functions [11]. 156 G.E. Giribet / Physics Letters B 628 (2005) 148–156

4. Discussion

Most of what we know about non-compact WZNW model is based on analogies with the Liouville ﬁeld theory. Moreover, the Fateev–Zamolodchikov dictionary (9) turned out to be one the principal tools in working out the formal aspects of this class of conformal models. Here, we have made use of the relation between correlators in both Liouville and WZNW theories to prove several identities between exact solutions of the Knizhnik–Zamolodchikov equation. Of course, such identities can, in principle, be proven by explicit manipulation of the KZ equation, at least up to a j-dependent overall factor [3]. However, the method of proving them by means of the connection with Liouville CFT turns out to be surprisingly concise and exploits the fact that the structure of the OPE of the models is encoded in the normalization factor (12). The derivation of the three-point function violating the winding number conservation we presented here turns out to be a good example of this conciseness.

Acknowledgements

This work was partially supported by Universidad de Buenos Aires. I am grateful to the people of Centro de Estudios Cientíﬁcos CECS, Valdivia, for the hospitality during my stay, where part of this work was done. I would also like to express my gratitude to Yu Nakayama for several discussions on related subjects. It is also a pleasure to thank Marco Matone and the organizers of the “Workshop on Non-Perturbative Gauge Dynamics”, at SISSA, Trieste, 2005.

References

[1] J.M. Maldacena, H. Ooguri, J. Math. Phys. 42 (2001) 2929, hep-th/0001053. [2] K. Bardakci, M.B. Halpern, Phys. Rev. D 3 (1971) 2493; A. Schwimmer, N. Seiberg, Phys. Lett. B 184 (1987) 191; J.K. Freericks, M.B. Halpern, Ann. Phys. 188 (1988) 258; J.K. Freericks, M.B. Halpern, Ann. Phys. 190 (1989) 212, Erratum; M.B. Halpern, E. Kiritsis, N.A. Obers, K. Clubok, Phys. Rep. 265 (1,2) (1996) 1. [3] A. Nichols, Sanjay, Nucl. Phys. B 597 (2001) 633, hep-th/0007007. [4] G. Giribet, C. Simeone, Int. J. Mod. Phys. A 20 (2005) 4821, hep-th/0402206. [5] Y. Nakayama, Int. J. Mod. Phys. A 19 (2004) 2771, hep-th/0402009. [6] D. Kutasov, N. Seiberg, JHEP 9904 (1999) 008, hep-th/9903219. [7] A. Zamolodchikov, V. Fateev, Sov. J. Nucl. Phys. 43 (4) (1986) 657. [8] O. Andreev, Phys. Lett. B 363 (1995) 166, hep-th/9504082. [9] J. Teschner, Phys. Lett. B 521 (2001) 127, hep-th/0108121. [10] B. Ponsot, Nucl. Phys. B 642 (2002) 114, hep-th/0204085. [11] J.M. Maldacena, H. Ooguri, Phys. Rev. D 65 (2002) 106006, hep-th/0111180. [12] G. Giribet, C. Núñez, JHEP 0106 (2001) 010, hep-th/0105200. [13] S. Ribault, J. Teschner, JHEP 0506 (2005) 014, hep-th/0502048. [14] S. Ribault, hep-th/0507114. [15] G. Giribet, Y. Nakayama, hep-th/0505203. [16] A. Giveon, D. Kutasov, Nucl. Phys. B 621 (2002) 303, hep-th/0106004. [17] P. Furlan, A. Ganchev, V. Petkova, Nucl. Phys. B 491 (1997) 635, hep-th/9608018. Physics Letters B 628 (2005) 157–164 www.elsevier.com/locate/physletb

On hidden broken nonlinear superconformal symmetry of conformal mechanics and nature of double nonlinear superconformal symmetry

Francisco Correa a, Mariano A. del Olmo b, Mikhail S. Plyushchay a

a Departamento de Física, Universidad de Santiago de Chile, Casilla 307, Santiago 2, Chile b Departamento de Física, Universidad de Valladolid, E-47011 Valladolid, Spain Received 5 September 2005; accepted 22 September 2005 Available online 29 September 2005 Editor: M. Cveticˇ

Abstract We show that for positive integer values l of the parameter in the conformal mechanics model the system possesses a hidden nonlinear superconformal symmetry, in which reﬂection plays a role of the grading operator. In addition to the even so(1, 2) ⊕ + + 1 u(1)-generators, the superalgebra includes 2l 1 odd integrals, which form the pair of spin-(l 2 ) representations of the bosonic subalgebra and anticommute for order 2l + 1 polynomials of the even generators. This hidden symmetry, however, is broken at the level of the states in such a way that the action of the odd generators violates the boundary condition at the origin. In the earlier observed double nonlinear superconformal symmetry, arising in the superconformal mechanics for certain values of the boson–fermion coupling constant, the higher order symmetry is of the same, broken nature.  2005 Elsevier B.V. All rights reserved.

1. Introduction vived recently in the context of the black hole physics, AdS/CFT correspondence and integrable Calogero– The conformal mechanics model of De Alfaro, Fu- Moser type systems [6–12]. This model provides an bini and Furlan [1,2] is the simplest nontrivial (0 + 1)- example of quantum mechanical system, in which the dimensional conformal ﬁeld theory. The interest to problem of a self-adjointness of the Hamiltonian arises it and to its supersymmetric extension [3–5] has re- in certain region of its parameter values [11,13–15]. The latter aspect is related to the problem of existence of bound quantum states and spontaneous breaking E-mail addresses: [email protected] (F. Correa), [email protected] (M.A. del Olmo), [email protected] of scale symmetry, that also ﬁnds applications in the (M.S. Plyushchay). physics of black holes [16].

Recently, it was observed that a certain change of 2. Hidden superconformal symmetry of a free the boson–fermion coupling constant in the supersym- particle on a line metric conformal mechanics model gives rise to a rad- ical change of symmetry: instead of the osp(2|2) su- Let us start with a free unit mass particle on a line, perconformal symmetry, the modified system is char- which is given by the Hamiltonian acterized by its nonlinear generalization [17,18].It 1 d2 was also found that when the boson–fermion coupling H =− . 2 (2.1) constant takes integer values, the system is described 2 dx simultaneously by the two nonlinear superconformal =− d A linear momentum p i dx is an integral of motion symmetries of the orders relatively shifted in odd num- satisfying the relation ber [17]. In particular, such nonlinear superconformal 2 symmetry arises in the original superconformal me- p = 2H. (2.2) | chanics model [3,4] in addition to the osp(2 2).How- As a consequence, every energy level with E> ever, the nature of this double superconformal symme- 0 is doubly degenerated: the√ two nonnormalizable try left to be mysterious. eigenstates√ ψE,+(x) = C+ cos 2Ex and ψE,−(x) = In the present Letter we show that when the para- C− sin 2Exbelong to it, while a nondegenerate state meter α of the purely bosonic conformal mechanics ψ0,+ = C0 = const corresponds to E = 0.Duetothe model (3.1) takes integer values, in addition to the energy levels structure, this elementary pure bosonic so(1, 2) conformal symmetry the system possesses a system possesses a hidden N = 1 supersymmetry in set of the integrals of motion which are odd differential exact, not spontaneously broken, phase. Indeed, take operators. Identifying the nonlocal reflection operator the reflection operator R, Rψ(x) = ψ(−x), satisfying as a grading operator, we find that these additional the relations {R,x}={R,p}=0, R2 = 1, where {., .} integrals extend the so(1, 2) to the nonlinear super- is an anticommutator. Then the operators Qa, a = 1, 2, conformal symmetry discussed in [17,18]. However, this hidden nonlinear superconformal symmetry of the 1 Q1 = √ p, Q2 = iRQ1, (2.3) conformal mechanics model is broken at the level of 2 the states. Similarly to the usual spontaneous super- can be identified as Hermitian supercharges, symmetry breaking mechanism, where a zero energy state loses the normalizability due to a violation of the {Qa,Qb}=2δabH, [Qa,H]=0. (2.4) boundary condition at infinity, here the odd genera- √1 Their linear combinations, Q± := (Q1 ± iQ2) = tors acting on the Hamiltonian eigenstates (being the 2 − d · 1 ± nonnormalizable, scattering states) produce its other i dx 2 (1 R), provide us with Hermitian conjugate 2 2 eigenstates, which violate the boundary condition at nilpotent supercharges, Q+ = Q− = 0, {Q+,Q−}= the origin. Then, we show that in the case of the dou- 2H , which mutually transform the even, ψE,+, and the ble superconformal symmetry observed in Ref. [17], odd, ψE,−, eigenstates with E>0, and annihilate the the symmetry of higher order has the same, broken na- ground state ψ0,+. In this construction the reflection ture. plays a role of a grading operator, which identifies the The Letter is organized as follows. In Section 2 we H as an even generator and the Q’s as odd generators show that a free particle on a line possesses a hid- of the N = 1 superalgebra (2.4). den osp(2|2) superconformal symmetry in the exact, The described hidden (“bosonized” [19–21]) N = 1 unbroken phase, in which reflection plays a role of a supersymmetry can be extended to the osp(2|2) su- grading operator. Section 3 is devoted to the discussion perconformal symmetry by supplying the set of the of a hidden broken nonlinear superconformal symme- integrals H and Qa with the dynamical odd, try of the conformal mechanics model. In Section 4 1 we analyse the double nonlinear superconformal sym- S1 = √ X, S2 = iRS1, 2 metry of superconformal mechanics. In Section 5 we = 1 { } = 1 2 discuss some problems to be interesting for further in- and even, D 4 X, p , K 2 X , integrals, where vestigation. X := x − tp.TheD and K are presented equivalently F. Correa et al. / Physics Letters B 628 (2005) 157–164 159 as Formally, for g = 0 the model is reduced to a free − 1 3 1 1 particle on a half-line. However, for 4 g<4 the D = {x,p}−tH, K = x2 − 2tD − t2H. operator (3.1) is not essentially self-adjoint (and there- 4 2 (2.5) fore cannot play the role of a Hamiltonian, for the details see Refs. [11,13–16]). On the other hand, Hamil- The dynamical integrals of motion satisfy the equation tonian (3.1) with g 3 is essentially self-adjoint, and of the form 4 in what follows we shall assume that the parameter α d ∂I takes the values corresponding to the latter case. I = − i[I,H]=0. (2.6) dt ∂t We shall show that analogously to the model of

The Qa, Sa, H , D, K and the operator the free particle on the line, for integer values of the parameter α = l, l = 1, 2,..., (or, equivalently, for 1 − = Σ =− R (2.7) α 2, 3,...) the conformal mechanics model is de- 2 scribed by the nonlinear superconformal symmetry satisfy the osp(2|2) superalgebra given by the nontriv- osp(2|2)2l+1. It generalizes the hidden superconfor- ial (anti)commutation relations mal symmetry of the free particle, and is produced by the even so(1, 2) ⊕ u(1) generators Hl, Dl, Kl [H,K]=−2iD, [D,H]=iH, + − and Σ, and by the set of odd operators Sn,m, Sn,m, [D,K]=−iK, (2.8) n = 2l + 1, m = 0, 1,...,n− 1. The odd generators + 1 constitute the pair of spin-(l 2 ) representations of the so(1, 2) ⊕ u(1), and anticommute for order 2l + 1 {Qa,Qb}=2δabH, {Sa,Sb}=2δabK, polynomials of the even generators. As we shall see, { }= − Sa,Qb 2δabD abΣ, (2.9) it is this hidden supersymmetry of the purely bosonic conformal mechanics model (3.1), (3.2) that explains [H,Sa]=−iQa, [K,Qa]=iSa, the origin and nature of the double superconformal i i symmetry in the superconformal mechanics at certain [D,Q ]= Q , [D,S ]=− S , (2.10) a 2 a a 2 a values of the boson–fermion coupling parameter [17]. On the other hand, it will be shown that unlike the case [Σ,Qa]=iabQb, [Σ,Sa]=iabSb, (2.11) of the model (2.1), the hidden superconformal nonlinear symmetry of the conformal mechanics model is of in which the even generators form the bosonic the broken nature. so(1, 2) ⊕ u(1) subalgebra, while the odd generators System (3.1) has two dynamical integrals of mo- form a pair of its spin- 1 representations. 2 tion, Dα and Kα, given by Eq. (2.5) with H changed for (3.1). Together with Hα, they form the so(1, 2) algebra (2.8). Deﬁne the operator 3. Hidden superconformal symmetry of conformal mechanics d γ † ∇ = + , ∇ =−∇− , (3.3) γ dx x γ γ Let us turn now to the conformal mechanics model where γ is a real parameter, and write down the rela- [1] described by the Hamiltonian tion 1 d2 g(α) H = − + , −2H = α 2 dx2 x2 α ∇ ∇ =∇ ∇ where g(α) := α(α + 1), (3.1) α+1 −(α+1) −α α =− 0

1 2 2l+1 γ = (n − 1), (3.6) (P + ) = (2H ) , (3.10) 2 l,2l 1 l i.e., when parameter γ takes integer or half-integer cf. the free particle relation (2.2) corresponding to values. When n takes an even value, n = 2l, l = l = 0. 1, 2,..., and in accordance with (3.6) γ is half-integer, In what follows, remembering the problem of self- (3.5) takes the form adjointness for operator (3.1), we shall assume that l P − 1 = (−1) ∇− − 1 ···∇− 1 ∇ 1 ···∇ − 1 . (3.7) α = l, l = 1, 2,.... (3.11) l 2 ,2l (l 2 ) 2 2 l 2 Making use of relation (3.4) starting from the center, Let us denote n = 2l + 1 and Sn,0 := Pl,2l+1.The we obtain the chain of equalities commutator of any two dynamical integrals is also a dynamical integral. Then we find that the subsequent ···∇− 3 ∇− 1 ∇ 1 ∇ 3 ··· 2 2 2 2 commutation of the integral Sn,0 with dynamical in- =···∇ ∇ ∇ ∇ ··· − 3 3 − 3 3 tegral Kl produces a set of the new n − 1 dynamical 2 2 2 2 integrals in accordance with relation =···∇5 ∇− 5 ∇ 5 ∇− 5 ···=···, 2 2 2 2 and find that operator (3.7) is reduced to the lth order [Kl,Sn,m]=−(n − m)Sn,m+1,m= 0,...,n− 1. of the operator H − 1 : (3.12) l 2 The n-times repeated commutator of Kl with Sn,0 pro- P = l l− 1 ,2l (2Hl− 1 ) . (3.8) duces finally zero and we obtain the finite chain of 2 2 = − Therefore, the Hermitian operator (3.7) is an integral dynamical integrals Sn,m, m 1,...,n 1 in addi- of motion for the system (3.1) with α = l − 1 , but it is tion to the integral Sn,0. These additional integrals, like 2 = + reduced to the lth order of the Hamiltonian itself. Sn,0, are also the order n 2l 1 differential opera- Consider now the case of the odd n = 2l + 1 and tors, and all together they explicitly can be presented integer γ = l. Then, we have the order (2l + 1) differ- in the form ential operator (3.5) of the form n Sn,m = (−i) (x + it∇−l)(x + it∇−l+1) ··· P := − 2l+1∇ ∇ ···∇ ···∇ ∇ l,2l+1 ( i) −l −l+1 0 l−1 l. (3.9) × (x + it∇−l+(m−1))∇−l+m ···∇l, Let us show that it is an integral of motion for sys- m = 0,...,n− 1. (3.13) tem (3.1) with α = l. First, this is so for l = 0 when To get this explicit form, we have used, in particular, Hamiltonian (3.1) is reduced to the formal free par- the equality ∇ x = x∇ + . The S satisfies the re- ticle Hamiltonian (2.1) (see the comment on self- γ γ 1 n,m † = − m adjointness above, which, however, is not important at lations Sn,m ( 1) Sn,m and the moment), and first order operator (3.9) is reduced ∂Sn,m = imS − . to the momentum operator. In a generic case, with tak- ∂t n,m 1 ing into account Eq. (3.4) we have Therefore, being the dynamical integral of motion, the [Pl,2l+1,Hl] Sn,m satisfies the commutation relation 1 =− [P + , ∇− ∇ ] [Hl,Sn,m]=−mSn,m−1. (3.14) 2 l,2l 1 l l 1 We have also the relation = ∇−l[Pl−1,2l−1, ∇l∇−l]∇l 2 n =−∇ [P ]∇ [D ,S ]=i − m S . (3.15) −l l−1,2l−1,Hl−1 l. l n,m 2 n,m F. Correa et al. / Physics Letters B 628 (2005) 157–164 161

According to (3.12), (3.14) and (3.15), the set of the functions satisfy the recursive differential relations n operators Sn,m forms the so(1, 2) spin- 2 representa- ∇−νJν(z) =−Jν+1(z), tion. All the Sn,m are the order n = 2l + 1 differential operators in x, and so, anticommute with the reflec- ∇νJν(z) = Jν−1(z), (3.21) tion operator R, while the so(1, 2) generators are even where ∇ν is the first order differential operator given operators commuting with R. Then, as in the free par- by Eq. (3.3) with x changed for z. Using repeatedly ticle case, one can extend the set of odd operators Sn,m the second relation, we get with the set of odd operators iRS , which satisfy the n,m P˜ = commutation relations with the so(1, 2) generators ex- ν,n+1Jν(z) Jν−(n+1)(z), ˜ actly of the same form as Sn,m. To distinguish these Pν,n+1 := ∇ν−n∇ν−n+1 ···∇ν. (3.22) two sets of odd operators, we, again, add the u(1) gen- = + 1 = P˜ When ν l and n 2l, the operator l+ 1 , l+ erator (2.7) to the set of even operators. All this set of 2 2 2 1 integrals forms the nonlinear superconformal algebra transforms the solution Jl+ 1 (z) into independent solu- 2 ∼ | + tion J− + 1 (z) of the same Eq. (3.20). Since Jν(z) osp(2 2)2l 1 given by nontrivial relations of the form (l 2 ) ν ∼ (2.8) and z for z 0, from the two solutions Jl+ 1 (z) and 2 ± ± 2 = + 1 2 Σ,S =±S , J−(l+ 1 )(z) of the same Eq. (3.20) with ν (l 2 ) n,m n,m 2 only the first one satisfies the boundary condition (3.2) ± n ± Dl,S = i − m S , (3.16) corresponding to the conformal mechanics model. Re- n,m 2 n,m membering relations (3.19), we find that the action of ˜ ± ± P 1 u(z) H ,S =∓mS , the operator l+ ,2l+1 on the function up to a l n,m n,m−1 2 ± ± numerical coefficient is reduced to the action of the Kl,S =∓(n − m)S + , (3.17) n,m n,m 1 supercharge Sn,0 on the corresponding function ψ(x). This means that when the integral Sn,0 acts on a physi- + − = m,m Sn,m,Sn,m P2l+1 (Hl,Kl,Dl,Σ) (3.18) cal state, which is an eigenstate of the Hamiltonian Hl + = 1 − − = − m 1 + satisfying boundary condition (3.2), it produces a state where Sn,m (1 R)Sn,m, Sn,m ( 1) (1 2 2 which formally still is an eigenstate of the same dif- = + † m,m = + R)Sn,m (Sn,m) , and P2l+1 is an order n 2l 1 ferential operator Hl but does not satisfy the boundary polynomial of its arguments, whose explicit form is condition at the origin. This picture is somewhat remi- not important for us here (see Ref. [18]). niscent to the spontaneously broken supersymmetry, in We have found the odd integrals of motion not which a zero energy state being nonnormalizable does taking into account the boundary condition (3.2).To not satisfy the boundary condition at infinity. On the clarify this aspect, we turn to the spectral problem for other hand, here, unlike the usual spontaneously bro- 3 Hamiltonian (3.1). Having in mind that for g 4 the ken supersymmetry, we have no pairing of the states system has no states with E 0, we introduce the no- even in a part of the spectrum. Because of this reason tations √ the hidden symmetry of the conformal mechanics can k = 2E, z = kx, be called a virtual superconformal symmetry. = + = + √ 1 Note also that taking in (3.22) n 2l 1, ν l 1, ψ(x)= zu(z), ν = α + , (3.19) l = 0, 1,..., and using the earlier observed relation 2 (3.8), we reproduce the well-known identity implying that E>0. Then the spectral equation (Hα − l E)ψ(x) = 0 is reduced to the Bessel equation J−l(z) = (−1) Jl(z). (3.23) d2u du z2 + z + z2 − ν2 u = 0. (3.20) dz2 dz 4. Double superconformal symmetry of ν α For noninteger values of (in this case is not half- superconformal mechanics integer, that includes (3.11)), the general solution of Eq. (3.20) is u(z) = AJν(z)+BJ−ν(z), where Jν(z) is The Hamiltonian of the system possessing super- Bessel function, and A, B are some constants. Bessel conformal symmetry of order n [17,18] can be pre- 162 F. Correa et al. / Physics Letters B 628 (2005) 157–164 sented in the form simultaneously be characterized by the two nonlinear superconformal symmetries of the orders shifted in H − 0 H = α n , (4.1) the odd number. More specifically, when α = n + p, n,α 0 H α p = 1, 2,..., the system (4.1) is characterized also, where the upper and lower Hamiltonian operators are in addition to the nonlinear superconformal symme- of the form (3.1), and the upper, ψ+, and lower, ψ−, try of the order n, by the superconformal symmetry components of the state Ψ T = (ψ+,ψ−), are sub- of the order n = n + 2p + 1. Let us show that this jected to the boundary condition (3.2). For the sake second supersymmetry really is of the broken, virtual + of definiteness we shall assume that the parameter α is nature. Indeed, the supercharge Sn+2p+1,0;n+p con- nonnegative. The set of odd generators of nonlinear su- structed in accordance with Eq. (4.2), commutes with perconformal symmetry associated with Hamiltonian Hamiltonian (4.1) and anticommutes with its conju- n+2p+1 (4.1) has the structure similar to (3.13), gate operator for the operator (Hn,n+p) . It can be presented in the form + S ; := (x + it∇α−n+1)(x + it∇α−n+2) ··· n,m α + = ∇ ···∇ + Sn+2p+1,0;n+p ( −p p)Sn,0;n+p × (x + it∇α−n+l)Pα,n−lσ+, (4.2) = 2p+1P + = 1 + P i p,2p+1Sn,0;n+p. (4.3) where σ+ 2 (σ1 iσ2), and operator α,n−l is defined by Eq. (3.5). These odd supercharges together The operator Pp,2p+1, as we have seen, commutes with conjugate operators anticommute for order n with the conformal mechanics model Hamiltonian Hp, polynomials in even generators Hn,α, Kn,α, Dn,α and but acting on an eigenstate of the latter which satisfies = 1 Σ 2 σ3 (for the details see Refs. [17,18]). In the case boundary condition (3.2), it produces a state violating n = 1 the system (4.1) corresponds to the supercon- (3.2). So, we conclude that the higher order symmetry formal mechanics model [3,4] possessing the osp(2|2) of the system with double superconformal symmetry = + 1 superconformal symmetry of the form (2.8)–(2.11). (like in the case 0 <α

[25] S.M. Klishevich, M.S. Plyushchay, Nucl. Phys. B 606 (2001) [26] H. Aoyama, M. Sato, T. Tanaka, M. Yamamoto, Phys. Lett. 583, hep-th/0012023; B 498 (2001) 117, quant-ph/0011009. S.M. Klishevich, M.S. Plyushchay, Nucl. Phys. B 628 (2002) [27] H. Falomir, P.A.G. Pisani, J. Phys. A 38 (2005) 4665, hep- 217, hep-th/0112158. th/0501083. Physics Letters B 628 (2005) 165–170 www.elsevier.com/locate/physletb

N = 4 supersymmetric quantum mechanics with magnetic monopole

Soon-Tae Hong a, Joohan Lee b, Tae Hoon Lee c, Phillial Oh d

a Department of Science Education, Ewha Womans University, Seoul 120-750, Republic of Korea b Department of Physics, University of Seoul, Seoul 130-743, Republic of Korea c Department of Physics, Soongsil University, Seoul 156-743, Republic of Korea d Department of Physics and Institute of Basic Science, Sungkyunkwan University, Suwon 440-746, Republic of Korea Received 20 July 2005; received in revised form 12 September 2005; accepted 13 September 2005 Available online 26 September 2005 Editor: M. Cveticˇ

Abstract We propose an N = 4 supersymmetric quantum mechanics of a charged particle on a sphere in the background of Dirac magnetic monopole and study the system in the CP(1) model approach. By using the Dirac quantization method, we explicitly calculate the symmetry algebra taking the operator ordering ambiguity into consideration. We find that it is given by the superalgebra su(1|2) × su(2)rot. We also show that the Hamiltonian can be written in terms of the Casimir invariant of su(2)rot algebra. Using this relation and analyzing the lower bound for angular momentum, we find the energy spectrum. We, then, examine the ground energy sector to find that the N = 4 supersymmetry is spontaneously broken to N = 2 for certain values of the monopole charge.  2005 Elsevier B.V. All rights reserved.

PACS: 11.30.Pb; 11.30.Qc; 14.80.Hv

Keywords: N = 4 supersymmetric quantum mechanics; Magnetic monopole; su(1|2) × su(2)rot algebra; Supersymmetry breaking

Since the pioneering work of Dirac the quantum matical properties. Its supersymmetric extension has mechanics of a charged particle in the background of been also proposed [3] and its various aspects has been a Dirac magnetic monopole [1,2] has attracted a great studied [4–7]. In the supersymmetric version, it was deal of attention due to its rich physical and mathe- found [4] that the system originally written in N = 1 formulation [3] possesses an additional (hidden) supersymmetry, making the system in fact N = 2 super- E-mail addresses: [email protected] (S.-T. Hong), [email protected] (J. Lee), [email protected] (T.H. Lee), symmetric. In Ref. [4] it was also pointed out that the [email protected] (P. Oh). hidden supercharge is related to the theory restricted

2 3 to S , the subspace of R representing a fixed distance δαz = ψα,δαz¯ = 0,δαψβ = 0, from the location of the monopole. On the other hand, ¯ ¯ ¯ ¯ δαψβ = 2i∇αβ z,¯ δαz = 0, δαz¯ = ψα, manifest N = 2 superspace formulation of the system ¯ = ∇ ¯ ¯ = is possible on S2 due to its Kähler structure, and the δαψβ 2i αβ z, δαψβ 0, (1) 2 system confined to S has been investigated in detail where recently [6,7]. In particular, in Ref. [6] the complete i ¯ ¯ energy spectrum with the corresponding wave func- ∇αβ z¯ = δαβ Dt z¯ − (ψβ · ψα − δαβ ψγ · ψγ )z,¯ (2) tions were found, and in Ref. [7] the issue of sponta- 2 neous symmetry breaking (whether or not the ground and its complex conjugate state is invariant under the supersymmetries) was dis- i ¯ ¯ cussed using the CP(1) model approach. In this con- ∇αβ z = δαβ Dt z + (ψα · ψβ − δαβ ψγ · ψγ )z. (3) 2 text, one of the interesting questions would be to ask The covariant derivative D is defined by whether the supersymmetry of the system could be ex- t tended further. Dt z = (∂t − ia)z, Dt z¯ = (∂t + ia)z,¯ (4) In this Letter, we propose an explicit model of N = 4 supersymmetric quantum mechanics of with a given by a charged particle on a sphere in the background i 1 a =− (z¯ ·˙z − z˙¯ · z) − ψ¯ · ψ . (5) of Dirac magnetic monopole. Following the previ- 2 2 α α ous work on N = 2 supersymmetric quantum me- Note a is invariant under the supersymmetric trans- chanics [7], we adopt the CP(1) model approach. formations of Eq. (1). One can check that the above This allows N = 4 formulation of the system in- supersymmetry transformations (1) preserve the su- cluding the monopole interaction in component for- persymmetric CP(1) constraints [8] malism. The dynamical variables consist of bosonic variables (z¯ ,z ) and fermionic variables (ψ¯ ,ψ ) ¯ i j αi βj z¯ · z − 1 = 0, z¯ · ψα = 0, ψα · z = 0. (6) (i, j = 1, 2; α, β = 1, 2) satisfying z¯ · z − 1 = 0, ¯ z¯ · ψα = 0, ψα · z = 0. Note that the fermionic vari- Our supersymmetric Lagrangian is then proposed ables are doubled in number compared to the N = 2 by case, as indicated by the internal indices, α, β.The i L = 2|D z|2 + (ψ¯ · D ψ − D ψ¯ · ψ ) bosonic variables are related to the space coordinate t 2 α t α t α α by the Hopf map x =¯zσz . 1 ¯ 2 ¯ 2 In quantizing the system there appears a parame- − (αβ ψα · ψβ ) + (ψα · ψα) − 2ga, (7) 4 ter associated with the choice of operator ordering in defining the basic commutation relations as in the which can be expressed in the following form: N = 2 case [7]. We study how physical quantities such i ˙ L = 2z˙ − (z¯ ·˙z)z2 + (ψ¯ · ψ˙ − ψ¯ · ψ ) as energy and angular momentum depend on this pa- 2 α α α α rameter. After quantization, we find that the symmetry i ˙ ¯ | × − (z¯ ·˙z − z¯ · z)ψα · ψα algebra of our system is given by su(1 2) su(2)rot. 2 The bosonic su(2) sector of su(1|2) is the internal rota- 1 ¯ 2 ¯ 2 α β ( ) − (αβ ψα · ψβ ) + (ψα · ψα) tions associated with , indices, whereas the su 2 rot 4 corresponds the angular momentum. Using the rela- + ig(z¯ ·˙z − z˙¯ · z − iψ¯ · ψ ). (8) tions of Hamiltonian with the supercharges and the α α Casimir invariant of the rotational algebra we find the In the above Lagrangian, we put the electric charge energy spectrum. We also investigate the ground state e =−1, and g is the magnetic monopole charge. Com- sector and find that the spontaneous supersymmetry pared to the previous N = 2 case [7], it has additional breaking occurs for some particular values of the mag- quartic fermionic interaction terms which are essential netic charge. for the existence of N = 4 supersymmetry. In order to proceed, we consider N = 4 supersym- Next, we perform the canonical quantization of the metric transformations of the following form system. We define the momenta p and p¯ conjugate, S.-T. Hong et al. / Physics Letters B 628 (2005) 165–170 167 respectively, to the fields z and z¯, with α + β = 1. The above brackets are supplemented by their Hermitian conjugates, and the remaining com- i ¯ p = 2Dt z¯ + (ψα · ψα + 2g)z,¯ mutators all vanish. They form a straightforward gen- 2 eralization of N = 2 case [7]. In the third line, the i ¯ p¯ = 2Dt z − (ψα · ψα + 2g)z. (9) operator ordering in the bracket is chosen by the con- 2 dition that the dynamical variables commute with the The classical Hamiltonian is second class constraint, C2, ordered as p ·z+¯z·¯p = 0. Note that this does not fix the operator ordering com- = ¯ − ¯ · Hc 2(Dt z)(Dt z) gψα ψα pletely and we still have undetermined α (or β)in 1 ¯ 2 ¯ 2 Eq. (15) as in the N = 2 case [7]. + (αβ ψα · ψβ ) − (ψα · ψα) . (10) 4 In order to obtain the symmetry algebra of the sys- It should be supplemented by the following six second tem, we compute the Noether charges associated with class constraints various global symmetries. The space rotations are generated by C1 =¯z · z − 1,C2 = p · z +¯z ·¯p, z1 − i waσ a z1 C =¯z · ψ ,C= ψ¯ · z, (11) → e 2 , (16) 3α α 4α α z2 z2 and one first class constraint, whose operator-ordered conserved charge is given by C =−i(z¯ ·¯p − p · z) − ψ¯ · ψ + 2g, (12) i i 1 0 α α K = zσ¯ p¯ − pσ z + γ zσ¯ z + ψ¯ σ ψ . (17) a 2 a 2 a a 2 α a α the Gauss law constraint corresponding to the local U(1) symmetry. Classically, two quartic terms in the Here we have added the third term associated with the fermion field in Eq. (10) cancel each other. We keep operator ordering ambiguity. We find that Ka’s gener- it here because they produce a quadratic term when ate the required rotation and satisfy the SU(2) algebra quantized. We quantize this theory following the Dirac [Ka,Kb]=iabcKc, (18) scheme. We start with the Poisson bracket relations provided the following conditions are satisfied {z ,p }={¯z , p¯ }=δ , {ψ¯ ,ψ }=−iδ δ , i j i j ij iα jβ ij αβ 1 3 (13) α = + γ, β = − γ. (19) 4 4 with the remaining brackets being zero. To incorporate The Noether charge associated with the phase sym- the second class constraints, we calculate the Dirac metry of the fermionic variables yields the conserved brackets using the definition given by charge ab {A,B}D ={A,B}−{A,Ca}Θ {Cb,B}, (14) ¯ NF = ψα · ψα. (20) ab ={ } where Θ is the inverse matrix of Θab Ca,Cb . The supercharges are given by From this result we obtain the following quantum ¯ ¯ commutation (and anti-commutation) relations upon Qα = p · ψα, Qβ = ψβ ·¯p. (21) replacing {A,B} →−i[A,B], D Note that the supercharges have no ordering ambigu- i i ity. The internal SU(2) rotations define isospin opera- [p ,z ]=−iδ + z¯ z , [p , z¯ ]= z¯ z¯ , a i j ij 2 i j i j 2 i j tors S by [ ]= i ¯ − ¯ 1 pi,pj (pizj pj zi), Sa = ψ¯ σ a ψ . (22) 2 2 α αβ β i ¯ ¯ [¯pi,pj ]= (z¯j p¯i − zipj ) − αψiαψjα + βψjαψiα, A straightforward calculation yields the following re- 2 lations ¯ [ψiα,ψjβ]=δαβ (δij −¯zizj ), 2 3 2 3 3 2 3 [ ¯ ]= ¯ ¯ S =− N + NF =− (NF − 1) + , (23) pi, ψjα iψiαzj , (15) 4 F 2 4 4 168 S.-T. Hong et al. / Physics Letters B 628 (2005) 165–170 where S2 = SaSa. Denoting the eigenvalue of S2 by with g˜ being treated as the monopole charge and our s(s + 1), we find the relations physical states are required to satisfy the above Gauss law constraint. s = 0 ↔ N = 0, F A complete algebraic structure of the symmetry 1 s = ↔ N = 1, generators can be computed to yield 2 F = ↔ = a b c a s 0 NF 2. (24) [S ,S ]=iabcS , [S ,NF ]=0, ¯ ¯ As the generator of local U(1) symmetry we choose [Qα,NF ]=Qα, [Qα,NF ]=−Qα, ≡− ¯ ·¯− · − ¯ · a 1 a a 1 a G0 i(z p p z) ψα ψα. (25) [S ,Q ]=− σ Q , [S , Q¯ ]= Q¯ σ , α 2 αβ β α 2 β βα Classically, this quantity being equal to minus the [Q ,Q ]=0, [Q¯ , Q¯ ]=0, twice of the monopole charge is the Gauss law con- α β α β straints, Eq. (12). To use this constraint to select the ¯ a a 1 [Qα, Qβ ]=2Hδαβ − 2g˜ S σ − NF δαβ . (28) physical Hilbert space, however, one should take care αβ 2 of the ordering problem. This amounts to adding an Also the Ka operators in Eq. (17) satisfy, appropriate constant before setting G0 equal to −2g. As our quantum mechanical Hamiltonian we choose [Ka,Kb]=iabcKc, [Ka,NF ]=0, the following expression ¯ [Ka,Qα]=0, [Ka, Qα]=0. (29) 1 i = ·¯− · ¯ ·¯ − ¯ ·¯− · ¯ · ˜ ≡ + ˜ ≡ − ˜ H (p p p zz p) (z p p z)ψα ψα Defining NF NF 2H/g and qα Qα/ 2g, 2 2 q¯ ≡ Q¯ / −2g(˜ g<˜ 0) we find the last equation of 1 ¯ β α − α + ψ · ψ Eq. (28) becomes 4 α α 1 [ ¯ ]=−1 ˜ + a a = (p ·¯p − p · zz¯ ·¯p) qα, qβ NF δαβ S σαβ . (30) 2 2 + 1 − + 1 ¯ · Other commutation relations remain the same because G0 2α ψα ψα a ˜ 2 2 [H,S ]=[H,NF ]=[H,Qα]=[H,Qβ ]=0 and we a ˜ ¯ | × 1 ¯ 2 1 ¯ find that Ka, S , NF , qα and qβ generate su(1 2) + (ψα · ψα) − ψα · ψα. (26) ˜ 2 2 su(2)rot symmetry [9]. (The case of g>0 can be a → 1 ¯ − aT → One can show that the above Hamiltonian commutes covered by redefining S 2 ψα( σαβ )ψβ , NF ¯ ¯ ¯ with the supercharges. It can be obtained by quantizing −ψα · ψα and interchanging Qα → Qα, Qα → Qα.) the classical Hamiltonian of Eq. (10) if G0 − 2α + 1/2 In order to examine the energy spectrum let us write is identified with −2g. This strongly suggests that the the Hamiltonian in terms of the supercharges, as aforementioned ordering constant is −2α + 1/2 and 1 ¯ 1 G0 − 2α + 1/2 should be interpreted as the monopole H = [Qα, Qα]+ − gN˜ F , (31) charge. However, for consistency of this interpreta- 4 2 + tion we must show that G0 − 2α + 1/2 is quantized where we have added the subscript ‘ ’ to emphasize according to the Dirac quantization condition of the that the bracket is the anti-commutator. This relation magnetic monopole charge. It will be shown shortly can be obtained by taking the trace of the last line (see Eq. (38) and discussions below) that this is in- of Eq. (28). [In this context, it is interesting to note deed the case. Still there is an ambiguity of adding an that in Ref. [10] a similar type of equation was used integer in the choice of the above mentioned ordering to demonstrate that shape invariance can best be un- constant. A different choice of this integer will corre- derstood as a BPS phenomenon.] Since the first term spond to a different theory (or a different interpretation is non-negative one finds that the energy is bounded of the theory). In this Letter we write the quantum me- from below by the second term in the above equation. chanical Gauss law constraint as Note that each sector with a definite fermion number 1 has a different bound. However, whether or not these G0 − 2α + =−2g,˜ (27) bounds can be saturated depends on the existence of 2 S.-T. Hong et al. / Physics Letters B 628 (2005) 165–170 169 the states in each sector that are invariant under the Comparing this with Eq. (36) leads to the minimum supersymmetry. In other words they may not be the op- value of the angular momentum quantum number timal bounds. In order to obtain the true energy bound k =|˜g − σ |. for each sector for a given parameter g˜, we proceed as min (38) follows. First, note that Since the angular momentum quantum number must be half-integer or integer and the spin σ is an integer, ·¯− · ¯ ·¯= − ¯ ¯ p p p zz p pi(δij zizj )pj g˜ must also be a half-integer or an integer. In particu- = piij z¯j zlklp¯k ≡ aa,¯ (32) lar, one finds that kmin is a half-integer if g˜ is a half- integer, and an integer if g˜ is an integer. This confirms, where we have defined the quantization of g˜,sog˜ can indeed be interpreted as a = ij piz¯j , a¯ = klzlp¯k. (33) the effective monopole charge. More generally, the angular momentum quantum number can be written as They satisfy the following commutation relation k = k + n(n= 0, 1, 2,...) (39) 3 min [a,a¯]=−G + 2α + − 2ψ¯ · ψ = 2(g˜ − Σ), 0 2 α α and the energy spectrum can be written as (34) 1 1 where Σ ≡ N − 1 can be regarded as the total spin E = k(k + 1) − g(˜ g˜ + 1). (40) F 2 2 along the radial direction, and Eq. (27) was used. We give the diagram for k versus g˜ in Fig. 1. Among Eigenvalues of the spin operator, Σ, consists of three the states satisfying Eq. (38), the ground states are rep- values, σ =+1, 0, −1, reflecting that we have two resented by the points marked with dots and circles. spin half degrees of freedom. Next, we need the bound The dots correspond to the supersymmetric ground for aa¯.Ifg˜ − σ 0, the bound is zero, and the state states for given values of g˜. The cases, g˜ =±1/2, are saturating this bound is obtained by imposing a¯ = 0. somewhat special. In these cases there is no N = 4 If g˜ − σ 0, aa¯ is bounded by 2(g˜ − σ) as one can supersymmetric states. In other words, there is no see from aa¯ =¯aa + 2(g˜ − σ) and aa¯ 0. In the lat- state killed by both Q and Q¯ .Forg˜ = 1/2, for ter case the bound is saturated by the states satisfying α α instance, the ground states, marked with a circle in a = 0. These two cases can be combined into Fig. 1, consist of two sectors, (k = 1/2, σ = 0) and aa¯ |˜g − σ |+(g˜ − σ). (35) (k = 1/2, σ = 1), each of which has a twofold angular momentum degeneracy. However, one can ar- This result together with Eqs. (26), (27) yields the gue in this case that the ground state still has N = 2 bound for the energy supersymmetry left over. Note that the supercharges 1 commute with the Hamiltonian and the angular mo- E = |˜g − σ |+(g˜ − σ) −˜g(σ + 1) min 2 mentum. Thus, applying the supercharges to any state 1 1 + (σ + 1)2 − (σ + 1) 2 2 1 1 = |˜g − σ | |˜g − σ |+1 − g(˜ g˜ + 1). (36) 2 2 − 1 ˜ By comparing this minimum energy with 2 gNF of Eq. (31) we conclude that the following types of supersymmetric ground states exist: σ =−1ifg˜ −1, σ = 0ifg˜ = 0, and σ = 1ifg˜ 1. One can obtain further information from the relation between the Hamiltonian and the angular momentum squared 1 H = K2 −˜g(g˜ + 1) . (37) 2 Fig. 1. Diagram for k versus g˜. 170 S.-T. Hong et al. / Physics Letters B 628 (2005) 165–170 will not change the energy and angular momentum dation grant (R01-2000-00015). P.O. was supported quantum numbers. On the other hand the commutation by Korea Research Foundation grant (R05-2004-000- ¯ ¯ relations [Σ,Qα]=−Qα and [Σ,Qα]=Qα tell us 10682-0). T.H.L. was supported by the Soongsil Uni- ¯ that Qα, Qα play the role of lowering and raising oper- versity Research Fund. We would like to thank the ators of the fermion number. Now, let |ψ be a ground ATCTP for the hospitality during our visit. Finally, state with σ = 0. Then Qα|ψ=0 because there is we would like to thank the referee for pointing out no σ =−1 state to be mapped in the ground energy Ref. [10] to us. ¯ sector. Similarly, Qα|ψ belongs to the σ = 1 sector of the lowest energy level. Simple counting together with the fact that the supercharges commute with Ka References ¯ suggests that a certain linear combination of Qα|ψ should vanish. To conclude, for g˜ =±1/2 the ground [1] P.A.M. Dirac, Proc. R. Soc. London A 133 (1931) 60. state is invariant under the half of the supersymmetry. [2] S.R. Coleman, The magnetic monopole fifty years later, in: Les Houches Summer School, 1981 p. 461; In summary, we have shown that the quantum me- P. Goddard, D.I. Olive, Rep. Prog. Phys. 41 (1978) 1357; chanics of a charged particle on a sphere in the back- R. Jackiw, Ann. Phys. 129 (1980) 183. ground of Dirac magnetic monopole allows N = 4 [3] E. D’Hoker, L. Vinet, Phys. Lett. B 137 (1984) 72; supersymmetric extension. Using the Dirac quantiza- E. D’Hoker, L. Vinet, Lett. Math. Phys. 8 (1984) 439. tion procedure, we explicitly calculated the symmetry [4] F. De Jonghe, A.J. Macfarlane, K. Peeters, J.W. van Holten, Phys. Lett. B 359 (1995) 114, hep-th/9507046. algebra and found that it is given by the superalge- [5] G.W. Gibbons, R.H. Rietdijk, J.W. van Holten, Nucl. Phys. bra su(1|2) × su(2)rot. We also investigate the spec- B 404 (1993) 42, hep-th/9303112; trum of the Hamiltonian which is given by the Casimir D. Spector, Phys. Lett. B 474 (2000) 331, hep-th/0001008; invariant of the su(2)rot symmetry. By analyzing the G. Papadopoulos, Class. Quantum Grav. 17 (2000) 3715, hep- ground energy sector we found that the supersymme- th/0002007; M.S. Plyushchay, Phys. Lett. B 485 (2000) 187, hep-th/ try is spontaneously broken for particular values of 0005122; g˜ =±1/2. There are a few unresolved aspects de- C. Leiva, M.S. Plyushchay, Phys. Lett. B 582 (2004) 135, hep- serving a further study: The first one is to look for th/0311150. a manifest superspace formulation of our system by [6] S. Kim, C. Lee, Ann. Phys. 296 (2002) 390, hep-th/0112120, using the N = 4 chiral superfield. The other is the con- and references therein. [7] S.T. Hong, J. Lee, T.H. Lee, P. Oh, Phys. Rev. D 72 (2005) struction of the complete wave functions by explicitly 015002, hep-th/0505018. realizing Eq. (15) as differential operators on an ap- [8] A. D’adda, P. Di Vecchia, M. Lüscher, Nucl. Phys. B 152 propriate function space. Some of these are currently (1979) 125, and references therein. under progress. [9] A. Pais, V. Rittenberg, J. Math. Phys. 16 (1975) 2062; P.G.O. Freund, I. Kaplansky, J. Math. Phys. 17 (1976) 228; M. Scheunert, W. Nahm, V. Rittenberg, J. Math. Phys. 18 (1977) 155; Acknowledgements M. Marcu, J. Math. Phys. 21 (1980) 1277. [10] M. Faux, D. Spector, J. Phys. A 37 (2004) 10397, quant-ph/ S.T.H. would like to acknowledge financial support 0401163. in part from the Korea Science and Engineering Foun- Physics Letters B 628 (2005) 171–175 www.elsevier.com/locate/physletb

On supersymmetry algebra based on a spinor-vector generator

Kazunari Shima, Motomu Tsuda

Laboratory of Physics, Saitama Institute of Technology, Okabe-machi, Saitama 369-0293, Japan Received 29 July 2005; received in revised form 5 September 2005; accepted 9 September 2005 Available online 26 September 2005 Editor: T. Yanagida

Abstract We study the unitary representation of supersymmetry (SUSY) algebra based on a spinor-vector generator for both massless and massive cases. A systematic linearization of nonlinear realization for the SUSY algebra is also discussed in the superspace formalism with a spinor-vector Grassmann coordinate.  2005 Elsevier B.V. All rights reserved.

Both linear (L) [1] and nonlinear (NL) [2] super- gested as corresponding supermultiplets to a spin-3/2 symmetry (SUSY) are realized based on a SUSY alge- NL SUSY action [7] through a linearization, although bra where spinor generators are introduced in addition those have not yet known at all. Also the linearization to Poincaré generators. The relation between the L and of the spin-3/2 NL SUSY is useful from the viewpoint the NL SUSY, i.e., the algebraic equivalence between towards constructing a SUSY composite unified the- various (renormalizable) spontaneously broken L su- ory based on SO(10) super-Poincaré (SP) group (the permultiplets and a NL SUSY action [2] in terms of superon–graviton model (SGM)) [8]. Indeed, it may a Nambu–Goldstone (NG) fermion has been investi- give new insight into an analogous mechanism with gated by many authors [3–6]. the super-Higgs one [9] for high spin fields which ap- An extension of the Volkov–Akulov (VA) model pear in SGM (up to spin-3 fields). [2] of NL SUSY based on a spinor-vector generator, In this Letter, we investigate the unitary represen- called the spin-3/2 SUSY, hitherto, and its NL realiza- tation of the spin-3/2 SUSY algebra in [7] towards tion in terms of a spin-3/2 NG fermion have been con- the linearization of the spin-3/2NLSUSY.Therole structed by Baaklini [7].Fromthespin-3/2NLSUSY of the spinor-vector generator as creation and anni- model, L realizations of the spin-3/2 SUSY are sug- hilation operators for helicity states is discussed explicitly. For both massless and massive representations for the spin-3/2 SUSY algebra, we show examples of E-mail addresses: [email protected] (K. Shima), [email protected] L supermultiplet-structure induced from those repre- (M. Tsuda). sentations. We also discuss on a systematic lineariza-

0370-2693/$ – see front matter  2005 Elsevier B.V. All rights reserved. doi:10.1016/j.physletb.2005.09.021 172 K. Shima, M. Tsuda / Physics Letters B 628 (2005) 171–175 tion of the spin-3/2 NL SUSY in the superspace for- invariant differential one-form is given by malism with a spinor-vector Grassmann coordinate. ωa = dxa + κ2abcdψ¯ γ γ dψ Let us begin with the brief review of the spin-3/2 b 5 c d a 2 acde ¯ b SUSY algebra and its NL realization in terms of the = δ + κ ψcγ5γd ∂bψe dx b spin-3/2 NG fermion. The spin-3/2 SUSY algebra in a a b = δ + t b dx [7], which satisfies all the Jacobi identities, is intro- b = a b duced based on a (Majorana) spinor-vector generator w b dx (7) a 1 Qα as and also a NL SUSY invariant action constructed from a b abcd Eq. (7) has the form Q ,Q = i (γ5γcC)αβ Pd , (1) α β 1 a b = S =− ω0 ∧ ω1 ∧ ω2 ∧ ω3 Qα,P 0, (2) 2κ2 a bc 1 bc β a ab c ac b 1 Q ,J = σ Q + i η Q − η Q , (3) =− d4x|w| α 2 α β α α 2 2κ a ab where P and J are translational and Lorentz gen- 1 4 a 1 a b a b =− d x 1 + t a + t at b − t bt a erators of the Poincaré group. 2κ2 2 The NL realization of the spin-3/2 SUSY which − 1 ef g d a b c reflects Eq. (1) is given by introducing the spin-3/2 abcd t et f t g 3! (Majorana) NG fermion ψa [7]. Indeed, supertrans- 1 ef g h a b c d lations of the ψa and the Minkowski coordinate xa − abcd t et f t gt h . (8) 4! parametrized by a global (Majorana) spinor-vector pa- 2 a rameter ζ a are defined as The second term in Eq. (8), i.e., −(1/2κ )t a,isthe ordinary Rarita–Schwinger kinetic term for ψa. 1 δψa = ζ a, Since (spontaneously broken) spin-3/2 L super- κ multiplets are suggested from a linearization of the a abcd ¯ δx = κ ψbγ5γcζd , (4) spin-3/2 NL SUSY, we investigate the structure of L supermultiplets induced from the spin-3/2SUSYal- where κ is a constant whose dimension is (mass)−2. gebra (1)–(3). For this purpose, we first focus on the Eq. (4) means a NL SUSY transformation of ψa at a relation (3) and discuss on the role of the spinor-vector fixed spacetime point, a generator Qα as creation and annihilation operators 1 for helicity states. When we choose the moving direc- δ ψa = ζ a − κbcdeψ¯ γ γ ζ ∂ ψa, (5) Q κ b 5 c d e tion of a massless particle as the 3-axis, Eq. (3) for the 12 which gives the closed off-shell commutator algebra, helicity operator J becomes 1 [ ]= a Q0 ,J12 =± Q0 , (9) δQ1,δQ2 δP Ξ , (6) α 2 α a + 1 + where δP (Ξ ) means a translation with a generator 12 = = Qα ,J Qα for α (1), (3), (10) Ξ a = 2abcdζ¯ γ γ ζ . 2 1b 5 c 2d + 3 + As parallel discussions in the VA model of the or- Q ,J12 = Q for α = (2), (4), (11) dinary (spin-1/2) NL SUSY [2],aspin-3/2NLSUSY α 2 α − 12 =−3 − = Qα ,J Qα for α (1), (3), (12) 1 = 2 Minkowski spacetime indices are denoted by a,b,... 0, 1, 1 2, 3 and four-component spinor indices are α,β,... = (1), (2), − 12 =− − = Qα ,J Qα for α (2), (4), (13) (3), (4). The Minkowski spacetime metric is 1 {γ a,γb}=ηab = 2 2 (+, −, −, −) and σ ab = i [γ a,γb]. We also use the spinor repre- 1 2 Q3 ,J12 =± Q3 , (14) − i α α sentation of the γ matrices, γ 0 = 0 I , γ i = 0 σ with σ i 2 I 0 σ i 0 ± 0 1 2 3 = 1 ± 2 being the Pauli matrices, γ5 = iγ γ γ γ and the charge conjuga- where Qα (1/2)(Qα iQα).Eqs.(9), (10), (13) =− 0 2 0 3 + tion matrix, C iγ γ . and (14) mean that the generators Qα, Qα, Qα (for K. Shima, M. Tsuda / Physics Letters B 628 (2005) 171–175 173

− 1 α = (1), (3)) and Q (for α = (2), (4)) raise or lower = √ 0 + 3 + − α a2 ξ1Q(3) ξ2Q(3) ξ3Q(4) , the helicity of states by 1/2 as the same discussions in the spin-1/2 SUSY algebra. On the other hand, † = √1 0 + 3 − + + a2 ξ1Q(2) ξ2Q(2) ξ3Q(1) , Eqs. (11) and (12) show that the operators Qα (for = − = α (2), (4)) and Qα (for α (1), (3)) raise or lower 1 − the helicity of states by 3/2 in contrast with the case a3 = √ Q , (3) of the spin-1/2SUSY. Next from Eq. (1) we study both massless and mas- † = √1 + a3 Q(2), (19) sive representations of the spin-3/2 SUSY algebra. 2 Let us first discuss on the massless case, P = 0. which are consistent with the Majorana condition of = a 2 By choosing a light-like reference frame, where Pa Q . In Eq. (19) (ξ1,ξ2,ξ3) and (ξ ,ξ ,ξ ) are arbi- 0 ± 3 α 1 2 3 (, 0, 0,),Eq.(1) for Qα, Qα , Qα becomes trary parameters which can be chosen as {a ,a†}=1 i i + − = Q0 ,Q = , Q0 ,Q = , for i 1, 2. (1) (3) (2) (4) Then Eqs. (16)–(18) become the following anti- 0 + 0 − Q ,Q =−, Q ,Q =−, commutation relations, (3) (1) (4) (2) 3 + =− 3 − =− Q(1),Q(3) , Q(2),Q(4) , † = − − a1,a1 2(ξ1 ξ2) ξ3 ξ3, 3 + = 3 − = Q(3),Q(1) , Q(4),Q(2) , † = − a2,a2 2 ξ1 ξ2 ξ3, + − = − + =− Q(2),Q(3) , Q(2),Q(3) , (15) { }= † † = ai,aj 0, ai ,aj 0, (20) and all other anticommutators vanish. Note that the 14 where i, j = 1, 2, while generators appear in Eq. (15) (i.e., the two generators, + − Q and Q , do not appear). In order to find an ex- † (4) (1) a3,a = 1, ample of the generators in the Fock space, we divide 3 { }= † † = Eq. (15) into the following three (irreducible) parts, a3,a3 0, a3,a3 0. (21) 0 + = 0 − =− Namely, Eqs. (20) and (21) are equivalent to Eq. (15) Q(1),Q(3) , Q(4),Q(2) , + − under the definition (19), although the physical mean- Q3 ,Q =−, Q3 ,Q = , (1) (3) (4) (2) ing and the mathematical structure of Eq. (19) are not − + =− known. Q(2),Q(3) , (16) If we choose the values of (ξ1,ξ2,ξ3) and (ξ1,ξ2, and ξ ) in Eq. (20) as {a ,a†}=1fori = 1, 2, the (a ,a†) 3 i i i i 0 − = 0 + =− mean the operators in the Fock space which raise or Q(2),Q(4) , Q(3),Q(1) , lower the helicity of states by 1/2. Also the (a ,a†) 3 − =− 3 + = 3 3 Q(2),Q(4) , Q(3),Q(1) (17) in Eq. (21) are the operators in the Fock space which for the generators which raise or lower the helicity of raise or lower the helicity of states by 3/2. Therefore, states by 1/2, while a massless irreducible representation for the spin-3/2 SUSY algebra induced from Eqs. (20) and (21) is + − Q ,Q = (18) (2) (3) 3 1 1 1 + , 2(+1), 1 + , 1(0), 2 − , 1(−1) for the generators which raise or lower the helicity of 2 2 2 states by 3/2. We further define creation and annihila- +[CPT conjugate]. (22) tion operators from appropriately rescaled generators as 2 In the two-component spinor formalism, the components, (Qa , 1 − (1) a = √ ξ Q0 + ξ Q3 + ξ Q , a a a 1 1 (1) 2 (1) 3 (2) Q(2)), correspond to an undotted spinor, while the (Q(3), Q(4)) are expressed as a dotted spinor. Also the Majorana condition of 1 + Qa means (Qa )† =−Qa and (Qa )† = Qa by the Hermitian a† = √ −ξ Q0 − ξ Q3 + ξ Q , α (1) (4) (2) (3) 1 1 (4) 2 (4) 3 (3) (complex) conjugation (for example, see [10]). 174 K. Shima, M. Tsuda / Physics Letters B 628 (2005) 171–175

1 + In Eq. (22) n(λ) means the number of states n for the a† = √ η Q3 − η Q , 2 m 1 (2) 2 (1) helicity λ. Let us second investigate the algebra (1) for the 2 − 2 = 2 a3 = Q , massive case, P m . By taking a rest frame mo- m (3) = ± mentum to be Pa (m, 0, 0, 0),Eq.(1) for Qα and 3 † = 2 + Qα becomes a3 Q(2), m 3 + =− 3 − =− Q(1),Q(3) m, Q(2),Q(4) m, 2 − a =− Q , 3 + = 3 − = 4 m (1) Q(3),Q(1) m, Q(4),Q(2) m, + − 1 − + 1 † = 2 + Q ,Q =− m, Q ,Q = m, a4 Q(4), (28) (1) (4) 2 (1) (4) 2 m + − = 1 − + =−1 which are consistent with the Majorana condition of Q(2),Q(3) m, Q(2),Q(3) m, (23) a 2 2 Qα.InEq.(28) (η1,η2) or (η1,η2) are arbitrary pa- { †}= = and all other anticommutators vanish. Note that in rameters which can be chosen as ai,ai 1fori 0 Eq. (23) Qα is completely decoupled from the alge- 1, 2. { + − } bra, while the anticommutator Q(1),Q(4) does not Then Eqs. (24)–(27) become the following anti- + − commutation relations: vanish and the two generators, Q(4) and Q(1), appear in contrast with Eq. (15). 1 a ,a† =− 2η + η η , As in the massless case, we divide Eq. (23) into four 1 1 1 2 2 2 (irreducible) parts as † =− − 1 + − a2,a 2η η η , Q3 ,Q =−m, Q3 ,Q = m, 2 1 2 2 2 (1) (3) (4) (2) {a ,a }=0, a†,a† = 0, (29) − + =−1 i j i j Q(2),Q(3) m, (24) 2 where i, j = 1, 2, while and † a3,a = 1, 3 − =− 3 + = 3 Q(2),Q(4) m, Q(3),Q(1) m, † =− a4,a4 1, + − 1 Q ,Q =− m (25) {a ,a }=0, a†,a† = 0, (30) (1) (4) 2 i j i j for the generators which raise or lower the helicity of where i, j = 3, 4. states by 1/2, while For the massive case, we choose the value of † (η1,η2) in Eq. (29) as {a1,a }=1, while the value + − 1 1 Q ,Q = m, (26) of (η ,η ) as {a ,a†} gives the negative norm, i.e., (2) (3) 2 1 2 2 2 { †}=− 3 † and a2,a2 1. Then the only (a1,a1) mean the operators in the Fock space which raise or lower the − + = 1 † Q(1),Q(4) m, (27) helicity of states by 1/2. Also the (a3,a3) in Eq. (30) 2 are the operators in the Fock space which raise or for the generators which raise or lower the helicity of { †} lower the helicity of states by 3/2, while the a4,a4 in states by 3/2. We also define creation and annihilation Eq. (30) gives the negative norm. Therefore, a (phys- operators from appropriately rescaled generators as ical) massive irreducible representation for the spin- † † 1 3 − 3/2 SUSY algebra induced from (a1,a ) and (a3,a ) a1 = √ η1Q + η2Q , 1 3 m (1) (2) 1 + † = √ − 3 + 3 Athough the values of (η ,η ) and (η ,η ) can be chosen as a1 η1Q(4) η2Q(3) , 1 2 1 2 m {a ,a†}=1fori = 1, 2, the excess of helicity-(±1) states in the i i 1 3 − resultant irreducible representation is not adequate for the massive a2 = √ η Q + η Q , m 1 (3) 2 (4) case. K. Shima, M. Tsuda / Physics Letters B 628 (2005) 171–175 175 uniquely has the following structure, in the case of the spin-1/2SUSY[3,5], and so the following conditions, 3 1 1 + , 1(+1), 1(0), 1 − ˜ 2 2 components of Φ = constant, (35) +[CPT conjugate], (31) may give SUSY invariant relations which connect a which represents massive spin-3/2, vector and scalar spin-3/2 L SUSY action, if it exists, with the spin-3/2 fields on shell. NL SUSY one (8). Explicit L realizations for the massless or the mas- We summarize our results as follows. We have stud- sive representations, e.g., Eq. (22) or Eq. (31),arenow ied the unitary representation of the spin-3/2SUSY algebra introduced in [7] for both massless and mas- under investigation. a Finally, we comment on the systematic lineariza- sive cases. The role of the spinor-vector generator Qα tion of the spin-3/2 NL SUSY in the superspace for- as creation and annihilation operators, which raise or malism by introducing a spinor-vector Grassmann co- lower the helicity of states by 1/2orby3/2, has ordinate θ a. Indeed, let us denote a L superfield on the been discussed explicitly in Eqs. (9)–(14). By defin- superspace coordinates (xa,θa) by Φ(xa,θa), and de- ing creation and annihilation operators from appropri- fine specific supertranslations as ately rescaled generators as in Eq. (19) or Eq. (28), the structure of the L supermultiplets induced from the a a abcd ¯ x = x − κ θbγ5γcψd , spin-3/2 SUSY algebra has been shown as Eq. (22) for the massless case or Eq. (31) for the massive θ a = θ a − κψa, (32) case. We have also shown the systematic linearization which are just the spin-3/2 SUSY version of the method by introducing the specific supertranslations specific supertranslations introduced in [3,5]. Then of Eq. (32). we can prove that the superfield on (x a,θ a), i.e., Φ(x a,θ a) = Φ(x˜ a,θa; ψa(x)), transforms homoge- neously; namely, according to superspace translations References of (xa,θa) generated by the global (Majorana) spinor- vector parameter ζ a,4 [1] J. Wess, B. Zumino, Phys. Lett. B 49 (1974) 52. [2] D.V. Volkov, V.P. Akulov, JETP Lett. 16 (1972) 438; a = a + abcd ¯ D.V. Volkov, V.P. Akulov, Phys. Lett. B 46 (1973) 109. x x θbγ5γcζd , [3] E.A. Ivanov, A.A. Kapustnikov, Relation between linear and θ a = θ a + ζ a (33) nonlinear realizations of supersymmetry, JINR Dubna Report No. E2-10765, 1977, unpublished; in addition to the spin-3/2 NL SUSY transformation E.A. Ivanov, A.A. Kapustnikov, J. Phys. A 11 (1978) 2375; (5),theΦ˜ transforms as E.A. Ivanov, A.A. Kapustnikov, J. Phys. G 8 (1982) 167. [4] M. Rocek,ˇ Phys. Rev. Lett. 41 (1978) 451. δ Φ˜ = ξ a∂ Φ,˜ (34) [5] T. Uematsu, C.K. Zachos, Nucl. Phys. B 201 (1982) 250. ζ a [6] K. Shima, Y. Tanii, M. Tsuda, Phys. Lett. B 525 (2002) 183; a abcd ¯ K. Shima, Y. Tanii, M. Tsuda, Phys. Lett. B 546 (2002) 162. where ξ = κ ζbγ5γcψd .Eq.(34) means that the ˜ [7] N.S. Baaklini, Phys. Lett. B 67 (1967) 335. components of Φ do not transform into each other as [8] K. Shima, Z. Phys. C 18 (1983) 25; K. Shima, Eur. Phys. J. C 7 (1999) 341; K. Shima, Phys. Lett. B 501 (2001) 237. 4 The specific supertranslations (32) correspond to the supertrans- [9] S. Deser, B. Zumino, Phys. Rev. Lett. 38 (1977) 1433. a − a lations (33) by replacing the ζ with κψ , while the NL SUSY [10] J. Wess, J. Baggar, Supersymmetry and Supergravity, Prince- a = transformation (5) (or Eq. (4)) is defined on a hypersurface θ ton Univ. Press, Princeton, NJ, 1992. κψa in Eq. (33). Physics Letters B 628 (2005) 176–182 www.elsevier.com/locate/physletb

Landau gauge Jacobian and BRST symmetry

M. Ghiotti, A.C. Kalloniatis, A.G. Williams

Centre for the Subatomic Structure of Matter, University of Adelaide, South Australia 5000, Australia Received 8 August 2005; received in revised form 7 September 2005; accepted 7 September 2005 Available online 21 September 2005 Editor: N. Glover

Abstract We propose a generalisation of the Faddeev–Popov trick for Yang–Mills fields in the Landau gauge. The gauge-fixing is achieved as a genuine change of variables. In particular, the Jacobian that appears is the modulus of the standard Faddeev– Popov determinant. We give a path integral representation of this in terms of auxiliary bosonic and Grassmann fields extended beyond the usual set for standard Landau gauge BRST. The gauge-fixing Lagrangian density appearing in this context is local and enjoys a new extended BRST and anti-BRST symmetry though the gauge-fixing Lagrangian density in this case is not BRST exact.  2005 Elsevier B.V. All rights reserved.

PACS: 11.15.Ha; 11.30.Ly; 11.30.Pb

Keywords: BRST; Gauge-ﬁxing; Ghosts; Determinant

1. Introduction

The elevation of Faddeev–Popov (FP) gauge-fixing of Yang–Mills theory beyond the realm of perturbation theory has been intensely pursued in recent years for many reasons. Nonperturbative gauge-fixed calculations on the lattice are being compared to analogous solutions of Schwinger–Dyson equations [1,2]. As well, the long-term goal of simulating the full Standard Model using lattice Monte Carlo requires the Ward–Takahashi identities associated with BRST symmetry [3] in order to control the lattice renormalisation. The main impediment to nonperturbative gauge-fixing is the famous Gribov ambiguity [4]: gauges such as Landau and Coulomb gauge do not yield unique representatives on gauge-orbits once large scale field fluctuations are permitted. To some extent one could live with such non-uniqueness if one could incorporate all Gribov copies in a computation. However, the no-go theorem of Neuberger [5] obstructs even this: (a naive generalisation of) BRST symmetry forces a complete cancellation of

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

0370-2693/$ – see front matter  2005 Elsevier B.V. All rights reserved. doi:10.1016/j.physletb.2005.09.015 M. Ghiotti et al. / Physics Letters B 628 (2005) 176–182 177 all Gribov copies in BRST invariant observables giving 0/0 for expectation values. In particular, Gribov regions contribute with alternating sign of the FP determinant. Here we shall propose an approach which takes seriously that gauge-fixing when seen as a change of variables involves a Jacobian being the absolute value of the Faddeev–Popov determinant. Usually the absolute value is dropped either because of an a priori restriction to perturbation theory or because of the identification of the determinant in terms of an invariant of a topological quantum field theory [6] such as the Euler character [7,8].In the latter case the Neuberger problem is encountered. The approach we describe in the following is not restricted to perturbation theory. Moreover, because it will be seen to involve a gauge-fixing Lagrangian density that is not BRST exact it falls outside the scope of the preconditions for the Neuberger problem. In the next section we shall derive the Jacobian associated with gauge- fixing in the presence of Gribov copies. We shall give a representation of the “insertion of the identity” in this case in terms of a functional integral over an enlarged set of scalar and ghost fields. The extended BRST symmetry of this new gauge-fixing Lagrangian density will be described though we will see that the final form of the gauge- fixing Lagrangian is not BRST exact.

2. Field theoretic representation for the Jacobian of FP gauge ﬁxing

In the following we shall formulate the problem in the continuum approach to gauge theory. Our aim is to generalise the standard formula from calculus for a change of variable: −1 ∂fi (n) det = dx1 ···dxn δ f(x) . (1) ∂xj f=0 Here one is changing from integration variables x to those satisfying the condition f( x) = 0 and where, for Eq. (1) to be valid, in the domain of integration of x there must be only one such solution. In the context of gauge-fixing of Yang–Mills theory the generalisation of Eq. (1) is [ g ] −1 δF A g det = Dgδ F A , (2) δg F =0 where Aµ represents the gauge field, g is an element of the SU(N) gauge group, Dg is the functional integration measure in the group and F gA = 0 (3) is the gauge-fixing condition. We shall be interested in Landau gauge F [A]=∂µAµ. As in the calculus formula, here Eq. (2) is only valid as long as Eq. (3) has a unique solution. This is known not to be the case for Landau g gauge. The FP operator nevertheless is MF [A]=(δF [ A]/δg)|F =0 and its determinant is ∆F [A]=det(MF ).For [ ]ab = ab[ ] ab[ ] a the Landau gauge MF A ∂µDµ A with Dµ A the covariant derivative with respect to Aµ in the adjoint representation. Now the standard FP trick is the insertion of unity in the measure of the generating functional of Yang–Mills theory realised via the identity (which follows from the above definitions): g g 1 = Dg∆F A δ F A . (4)

By analogy with standard calculus, in the presence of multiple solutions to the gauge-ﬁxing condition Eq. (4) must be replaced by g g NF [A]= Dgδ F A det MF A , (5) 178 M. Ghiotti et al. / Physics Letters B 628 (2005) 176–182

g where NF [A] is the number of different solutions for the gauge-fixing condition F [ A]=0 on the orbit characterised by A, where A is any configuration on the gauge orbit in question for which det MF = 0. It is known that Landau gauge has a fundamental modular region (FMR), namely a set of unique representatives of every gauge orbit which is, moreover, convex and bounded in every direction [9,10]. The following discussion can be found in more detail in [11]. Denoted Λ, the FMR is defined as the set of absolute minima of the functional 4 g 2 VA[g]= d x( A) with respect to gauge transformations g. The stationary points of VA[g] are those Aµ satisfying the Landau gauge condition. The boundary of the FMR, ∂Λ, is the set of degenerate absolute minima of VA[g]. Λ lies within the Gribov region Ω0 where the FP operator is positive definite. The Gribov region is comprised of all of the local minima of VA[g]. The boundary of Ω0, the Gribov horizon ∂Ω0, is where the FP operator MF (which corresponds to the second order variation of VA[g] with respect to infinitesimal g) acquires zero modes. When the degenerate absolute minima of ∂Λ coalesce, flat directions develop and MF develops zero modes. Such orbits cross the intersection of ∂Λ and ∂Ω0. The interior of the fundamental modular region is a smooth differen- tiable and everywhere convex manifold. Orbits crossing the boundary of the FMR on the other hand will cross that boundary again at least once corresponding to the degenerate absolute minima. Though, at present, there is no practical computational algorithm for constructing the FMR, it exists and we will make use of it for labelling orbits, i.e., Au are defined to be configurations in the FMR, Au ∈ Λ. Since every g orbit crosses the fundamental modular region once we are guaranteed to have NF 1. In turn, the Au fulfilling the constraint of Eq. (3) would be every other gauge copy of Au along its orbit. Eq. (5) is equal to the number of Gribov copies on a given orbit, NGC = NF − 1, except that copies lying on any of the Gribov horizons (∆F = 0) do not contribute to NF . The finiteness of NF in the presence of a regularisation leading to a finite number of degrees of freedom (such as a lattice formulation) can be argued as follows. Consider two neighboring Gribov copies corresponding to a single orbit. If they contribute to NF they cannot lie on the Gribov horizon. Therefore, they do not lie infinitesimally close to each other along a flat direction, namely they have a finite separation. This is true then for all copies on an orbit contributing to NF : all copies contributing to NF have a finite separation. But the g which create the copies of Au belong to SU(N) which has a finite group volume. Thus for each space–time point there is a finite number of such g. We conclude then for a regularised formulation that NF is finite. Consider then the computation of the expectation value of a gauge-invariant operator O[A] over an ensemble of gauge-field configurations Au which is this set of unique representatives of gauge orbits discussed above. Note that for a gauge-invariant observable, it makes no difference whether Au ∈ Λ or if the Au’s are any other unique representatives of the orbits. The expectation value on these configurations

− DA O[A ]e SYM [ ] = u u O A −S (6) DAu e YM is well-deﬁned. Since in any regularised formulation NF is a ﬁnite positive integer, we can legitimately use Eq. (5) to resolve the identity analogous to the FP trick and insert into the measure of integration for an operator expectation value. We thus have

1 g g −SYM[A] DAu [ ] Dgδ(F[ A])| det MF [ A]|O[A]e [ ] = NF Au O A − [ ] . (7) DA 1 Dgδ(F[gA])| det M [gA]|e SYM A u NF [Au] F

We can now pass NF [Au] under the group integration Dg and combine the latter with DAu to obtain the full g measure of all gauge ﬁelds D( Au) which we can write now as DA. NF is certainly gauge-invariant: it is a property g of the orbit itself. So NF [Au]=NF [ Au]=NF [A]. Thus we can write

1 g g −SYM[A] DA [ ] δ(F[ A])| det MF [ A]|O[A] e [ ] = NF A O A − [ ] . (8) DA 1 δ(F[gA])| det M [gA]|e SYM A NF [A] F M. Ghiotti et al. / Physics Letters B 628 (2005) 176–182 179

Perturbation theory can be recovered from this of course by observing that only A fields near the trivial orbit, containing A = 0 and for which SYM[A]=0, contribute significantly in the perturbative regime: the curvature of the orbits in this region is small so that the different orbits in the vicinity of A = 0 intersect the gauge-fixing hypersurface F = 0 the same number of times. Then the number of Gribov copies is the same for each orbit, NF is independent of Au and we can cancel NF out of the expectation value. In that case

− [ ] DAδ(F[A])| det M [A]|O[A]e SYM A [ ] = F O A −S [A] . (9) DAδ(F[A])| det MF [A]|e YM In turn, observing that fluctuations near the trivial orbit cannot change the sign of the determinant, the modulus can also be dropped and one recovers the usual starting point for a standard BRST invariant formulation of Landau gauge perturbation theory. Note that perturbation theory is built on the gauge-fixing surface in the neighbourhood of A = 0, which for a gauge-invariant quantity will be equivalent to averaging over the Gribov copies of A = 0 as in Eq. (9). For the nonperturbative regime, the orbit curvature increases significantly and in general there is no reason to expect that NF would be the same for each orbit. Moreover, the determinant can change sign. Let us focus on the partition function appearing in Eq. (8) − − Z = D 1[ ] [ ] [ ] SYM gauge-fixed ANF A det MF A δ F A e . (10)

The objective is to generalise the BRST formulation of Eq. (10) such that it is valid beyond perturbation theory taking into account the modulus of the determinant. We thus start with the following representation: det MF [A] = sgn det MF [A] det MF [A] . (11)

As mentioned, the factor det(MF [A]) in Eq. (11) is represented as a functional integral via the usual Lie algebra valued ghost and antighost fields in the adjoint representation of SU(N). Let us label these as ca, c¯a. It is usual also (see, for example, [12]) to introduce a Nakanishi–Lautrup auxiliary field ba. Thus the effective gauge-fixing Lagrangian density ξ L =−ba∂ Aa + baba +¯caMabcb (12) det µ µ 2 F yields [12] 4 a a a − d x Ldet lim Dc¯ Dc Db e = δ F [A] det MF [A] . (13) ξ→0

In order to write the factor sgn(det(MF [A])) in terms of a functional integral weighted by a local action, we consider the following Lagrangian density: 1 L = iBaMabϕb − id¯aMabdb + BaBa (14) sgn F F 2 with d¯a,da being new Lie algebra valued Grassmann ﬁelds and ϕa,Ba being new auxiliary commuting ﬁelds. Consider in Euclidean space the path integral 4 ¯a a a a − d x Lsgn Zsgn = Dd Dd Dϕ DB e . (15)

Completing the square in the Lagrangian density of Eq. (14),theB ﬁeld can be integrated out in the partition function leaving an effective Lagrangian density 1 L = ϕa (M )T abMbcϕc − id¯aMabdb, (16) sgn 2 F F F 180 M. Ghiotti et al. / Physics Letters B 628 (2005) 176–182

T where (MF ) denotes the transpose of the FP operator. Integrating all remaining fields now it is straightforward to see that the partition function Eq. (15) amounts to just det(MF ) Zsgn = = sgn det(MF ) . (17) T det((MF ) MF ) Thus the representation Eq. (15) can be used for the first factor of Eq. (11). The Lagrangian density of Eq. (14) therefore combines with the standard BRST structures of Eq. (12) coming from the determinant itself in Eq. (11) so that an equivalent representation for the partition function based on Eq. (10) is −1 − − − Z = D a D ¯a D a D ¯a D a D a D a [ ] SYM Sdet Ssgn gauge-fixed Aµ c c d d b ϕ NF A e (18) with Sdet and Ssgn the actions corresponding to the above Lagrangian densities Eqs. (12), (14).

3. A new extended BRST

The symmetries of the new Lagrangian density, Lsgn, are essentially a boson-fermion supersymmetry and can be seen from Eq. (14). In analogy to the standard BRST transformations typically denoted by s, we shall denote them by the Grassmann graded operator t tϕa = da,tda = 0,td¯a = Ba,tBa = 0, (19) such that

tLsgn = 0 (20) and trivially tLYM = 0. Eqs. (19) realise the infinitesimal form of shifts in the fields. The operation t is nilpotent: 2 t = 0. Using Eqs. (19) we can give the following form for the Lagrangian density Lsgn, 1 L = t d¯a iMabϕb + Ba . (21) sgn F 2 The question now is how to combine this with the standard BRST transformations 1 sAa = Dabcb,sca =− gf abccbcc,sc¯a = ba,sba = 0. (22) µ µ 2 The transformations due to t and s are completely decoupled except that the latter also act on the gauge field on which the FP operator MF depends. We propose the following unification of these symmetry operations. Consider an operation S block-diagonal in s and t: S = diag(s, t). The operator acts on the following multiplet fields: a a a a a Aµ a c ¯a c¯ a b A = , C = , C = ¯ , B = . (23) ϕa da da Ba We see that these fields transform under S completely analogously to the standard BRST operations: SAa = DabCb SCa = F abcCbCc SC¯a = Ba SBa = , i ij k j k , , 0, (24) where i, j, k = 1, 2 label the elements of the multiplets, and 1 Dab = diag Dab,δab F abc =− gf abc, F abc = 0forij k = 111. (25) µ 111 2 ij k Note that nilpotency is satisfied, S2 = 0. We shall refer to this type of operation as an extended BRST transformation which we distinguish from the BRST–anti-BRST or double BRST algebra of the Curci–Ferrari model M. Ghiotti et al. / Physics Letters B 628 (2005) 176–182 181

[13,14]. We can thus formulate the gauge-ﬁxing Lagrangian density for the Landau gauge as c¯aF a 0 L = Tr S . (26) gf ¯a ab b + 1 a 0 d (iMF ϕ 2 B ) This approach admits also an extended anti-BRST operation: SA¯ a = DabC¯b S¯C¯a = F abcC¯bC¯c SC¯ a =−Ba SB¯ a = , i ij k j k , , 0. (27) Writing S¯ = diag(s,¯ t)¯ we can extract the standard anti-BRST s¯-operations [14,15] 1 sA¯ a = Dabc¯b, s¯c¯a =− gf abcc¯bc¯c, sc¯ a =−ba,sba = 0, (28) µ µ 2 and those corresponding to t¯:

tϕ¯ a = d¯a, t¯d¯a = 0, td¯ a =−Ba, tB¯ a = 0. (29) Moreover, the ghosts and antighosts in this extended structure also fulﬁll the criteria for being Maurer–Cartan one-forms,

SC¯ + SC¯ = 0. (30) However, there is no extended BRST–anti-BRST (or double) symmetric form of the gauge-fixing Lagrangian density Eq. (26), unlike the two pieces of which it consists. Such a representation exists in the s-sector of Landau gauge: 1 L = ssA¯ a Aa . (31) gf,s 2 µ µ In the t-sector, the corresponding structure is 1 L = tt¯ ϕaMabϕb + d¯ada . (32) gf,t 2 F However, the complete Landau gauge-fixing Lagrangian density can only be expressed via a trace, namely, as 1 L = Tr SSW¯ (33) gf 2 with W = a a a ab b + ¯a a diag AµAµ,ϕ MF ϕ d d . (34) Nevertheless, this compact representation formulates the modulus of the determinant in Landau gauge fixing in terms of a local Lagrangian density and follows as closely as possible the standard BRST formulation without the modulus.

4. Discussion and conclusions

We have thus found a representation for Landau gauge-fixing corresponding to the FP trick being an actual change of variables with appropriate determinant. The resulting gauge-fixing Lagrangian density enjoys a larger extended BRST and anti-BRST symmetry. However, it cannot be represented rigorously as a BRST exact object, rather the sum of two such objects corresponding to different BRST operations. This means that some of the BRST machinery is not available to this formulation, such as the Kugo–Ojima criterion for selecting physical states. We discuss cursorily now the perturbative renormalisability of the present formulation of the theory. Note that the 182 M. Ghiotti et al. / Physics Letters B 628 (2005) 176–182 procedure leading to Eq. (26) does not introduce any new coupling constants; only the strong coupling constant g is present in MF [A] coupling the Yang–Mills field to both the new ghosts and scalars. The dimensions of the new fields are: − − [ϕ]=L0, [d]=[d¯]=L 1, [B]=L 2. (35) Most importantly in this context, the kinetic term for the new boson fields ϕa is quartic in derivatives: a 2 2 a Lkin = ϕ ∂ ϕ , (36) which is renormalisable, by power counting, since ϕa are dimensionless. Such a contribution is seemingly harm- less in the ultraviolet regime: for large momenta propagators will vanish like 1/p4. Moreover, it should play an important role in guaranteeing the decoupling of such contributions in perturbative diagrams. That such a decoupling should occur is clear from Eq. (11): in the perturbative regime fluctuations about Aµ = 0 will not feel the sgn(det MF [A]), so that the field theory constructed in this way must be equivalent to the perturbatively renormalisable Landau gauge fixed theory. For example, in the computation of the running coupling constant we expect that this property will lead to a complete decoupling of the t-degrees of freedom so that the known Landau gauge result emerges from just the gluon and standard ghost sectors. Naturally, the new degrees of freedom will be relevant in the infra-red regime, which will be the object of future study.

Acknowledgements

A.C.K. is supported by the Australian Research Council. We are indebted to discussions with Lorenz von Smekal, Mathai Varghese, Max Lohe and Martin Schaden.

References

[1] R. Alkofer, L. von Smekal, Phys. Rep. 353 (2001) 281, hep-ph/0007355. [2] P.O. Bowman, U.M. Heller, D.B. Leinweber, M.B. Parappilly, A.G. Williams, Phys. Rev. D 70 (2004) 034509, hep-lat/0402032. [3] C. Becchi, A. Rouet, R. Stora, Ann. Phys. 98 (1976) 287; I.V. Tyutin, LEBEDEV-75-39. [4] V.N. Gribov, Nucl. Phys. B 139 (1978) 1. [5] H. Neuberger, Phys. Lett. B 183 (1987) 337. [6] D. Birmingham, M. Blau, M. Rakowski, G. Thompson, Phys. Rep. 209 (1991) 129. [7] P. Hirschfeld, Nucl. Phys. B 157 (1979) 37. [8] L. Baulieu, M. Schaden, Int. J. Mod. Phys. A 13 (1998) 985, hep-th/9601039. [9] G. Dell’Antonio, D. Zwanziger, Commun. Math. Phys. 138 (1991) 291. [10] M.A. Semenov-Tyan-Shanskii, V.A. Franke, Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. Akad. Nauk SSSR 120 (1982) 159 (English translation: Plenum Press, New York, 1986). [11] P. van Baal, Nucl. Phys. B 369 (1992) 259. [12] For a thorough list of references, see: N. Nakanishi, I. Ojima, Covariant Operator Formalism Of Gauge Theories And Quantum Gravity, World Scientiﬁc Lecture Notes on Physics, vol. 27, World Scientiﬁc, Singapore, 1990, p. 1. [13] G. Curci, R. Ferrari, Nuovo Cimento A 35 (1976) 1; G. Curci, R. Ferrari, Nuovo Cimento A 47 (1978) 555, Erratum. [14] J. Thierry-Mieg, Nucl. Phys. B 261 (1985) 55. [15] L. Baulieu, J. Thierry-Mieg, Nucl. Phys. B 197 (1982) 477. Physics Letters B 628 (2005) 183–187 www.elsevier.com/locate/physletb

On integrating out heavy ﬁelds in SUSY theories

S.P. de Alwis

Physics Department, University of Colorado, Boulder, CO 80309, USA Received 7 July 2005; accepted 14 September 2005 Available online 26 September 2005 Editor: M. Cveticˇ

Abstract We examine the procedure for integrating out heavy fields in supersymmetric (both global and local) theories. We find that the usual conditions need to be modified in general and we discuss the restrictions under which they are valid. These issues are relevant for recent work in string compactification with fluxes.  2005 Elsevier B.V. All rights reserved.

PACS: 11.25.-w; 98.80.-k

In a theory containing both heavy (mass M) and In other words up to terms in the derivative expansion light (mass m M) fields, one may derive an effec- that maybe ignored at energy E M, Φ is a solution tive field theory [1] valid for energy scales E M, of the equation by doing the functional integral over the heavy fields ∂V = 0. (1) (as well as light field modes with frequencies greater ∂Φ than M). If one is just considering the classical ap- Consider now a globally supersymmetric theory of proximation then this just means solving the classi- chiral scalar fields with superspace action cal equation for the heavy field in terms of the light field and substituting back into the action. So if the S = d4xd4θ Φ¯ iΦi + d4xd2θW Φi + c.c. . potential for the heavy (Φ) and light (φ) fields is = 1 2 2 + ˜ V(Φ,φ) 2 M Φ V(Φ,φ)the equation of motion (2) for the heavy field gives, The superfield equations of motion are (with D2 = α 1 D Dαwhere Dα is the spinor covariant derivative) ˜ ˜ ˜ 1 ∂V 1 ∂V 1 ∂V 1 ¯ 2 ¯ i ∂W Φ = =− + O . − D Φ + = 0. (3) − M2 ∂Φ M2 ∂Φ M2 M2 ∂Φ 4 ∂Φi

1 We use the conventions of [4]. For the derivations below see E-mail address: [email protected] (S.P. de Alwis). also [3].

The component form of this is obtained by suc- effective potential for the light field—the only consis- cessive application of the spinorial covariant deriva- tent result of integrating out the heavy field is that all tive and setting θ = θ¯ = 0 (indicated below by the fields are sitting at the SUSY minimum (if it exists). symbol |). In particular the auxiliary field equation When there is more than one light field this is not nec- 0 − 1 2 (from D ) and the scalar field (from 4 D ) be- essarily the case but nevertheless the light field space come (after setting the fermions to zero and using is constrained. If we had kept the fermionic terms then D2D¯ 2Φ¯ = 16Φ¯ ) this constraint would be a relation between the bosonic components and squares of fermionic components of ∂W ¯ i + = the light fields. F i 0, (4) ∂Φ To see where this comes from let us write the su- ¯ i ∂W j perpotential for the heavy light theory as Φ + F = 0. (5) ∂Φi∂Φj j 1 W ΦH ,Φl = MΦH 2 + W˜ ΦH ,Φl . (9) Now suppose that we wish integrate out a heavy 2 field with say i = H to get an effective theory for the Operating on the (conjugate of the) equation of mo- = − 1 2 light fields with i l. For example one might have tion (3) with ( 4 D ), using (3) again and rearranging = 1 H 2 + 1 H + W 2 MΦ 2 λll Φ ΦlΦl WL(Φl). Then the we have analog of (1) (see for example [2]) is to require that ¯ 1 ∂W˜ D¯ 2 ∂W˜ ∂W ΦH = M + . = 0. (6) − M2 ∂ΦH 4 ∂Φ¯ H ∂ΦH Expanding the inverse Klein–Gordon operator as be- Taking components of this equation (and ignoring the fore we can rewrite this as fermionic terms) ¯ ∂W˜ D¯ 2 ∂W˜ ∂W MΦH + =− + O (···) . = 0, (7) H ¯ 2 ∂ΦH ∂Φ 4M ∂ΦH M 2 2 ¯ So to the lowest order in the space–time derivative ∂ W l ∂ W ∂W D2Φ + = 0, (8) ∂ΦH ∂Φl ∂ΦH ∂ΦH ∂Φ¯ H (momentum) expansion what we get for the equation j determining the heavy field in terms of the light is where in the second (obtained by acting with D2 and ¯ ∂W D¯ 2 ∂W˜ setting fermions to zero) we have used (4) for i = H . =− , (10) ∂ΦH 4M ¯ H The second condition is of course the requirement that ∂Φ = i ¯ i the potential V i F F be extremized with re- rather than (6). To get the latter one needs the addi- ˜¯ spect to ΦH , which is what one would impose in a tional assumption that the possible values of D¯ 2 ∂W ∂Φ¯ H non-supersymmetric theory (see (1)). However here are small compared to M. For instance in the above ex- we have the additional condition (7) which in conjunc- ample (in the paragraph after (5)) this means that we tion with (8) and the equitation of motion for the light need Φl M. This of course is what one would ex- field leads to pect. However as we pointed out earlier in this same 2 ∂ W ∂W approximation the light field space appears to be con- = 0. ∂ΦH ∂Φl ∂Φl strained. j Let us be even more specific and consider the model ˜ = 1 2 But (after solving for the heavy field using (6))this (9) with W 2 HL . (We have relabeled the heavy is a constraint on the light fields in theories where there field as H and the light field as L.) Then (10) becomes is a non-zero coupling between the heavy and light ∂W 1 D¯ 2 1 fields as in the above example. In particular if there is = MH + L2 =− L¯ 2 only one light field it imposes the condition ∂W |=0, ∂H 2 4M 2 ∂Φl meaning that the light scalar is also at the minimum 1 ¯ α˙ ¯ ¯ ¯ ¯ ¯ 2 ¯ =− D LD ˙ L + LD L . (11) of the potential. In other words one does not get an 4M α S.P. de Alwis / Physics Letters B 628 (2005) 183–187 185

Again we see that the strict imposition of ∂W/∂H = 0 If we had just imposed the usual condition ∂W/ leads to the constraint on the light field space that we ∂H = 0 we would not have got the |a|2/M2 term in found above. To see in what approximation this equa- the parenthesis. This means that this condition gives tion is valid let us solve it for H (giving H =−L2/M) the correct result for the potential only for small val- and then plug it back into the RHS of the last equality ues of the field |a|M. of (11) after using the light field equation. The bosonic To derive the analog of (10) in supergravity we | |2 term is then L2 L and is small when |L|M. use the formalism in Chapter 8 of [3] (though we M2 It is perhaps worthwhile looking at this example remain with the conventions of [4]). In this formal- in component form. The superspace Lagrangian in the ism the matter equations can be derived by replacing above example is the supervielbein determinant by the so-called chiral compensator field φ and treating the coupling of d4θ(HH¯ + LL)¯ the matter fields as in flat superspace. Thus the su- percovariant derivative Dα is the flat space one and 3 = + 2 1 2 + 2 + satisfies D 0. Also acting on chiral fields we have d θ MH HL c.c. . (12) 2 ¯ 2 = 2 = 2 (D D /16)Φ Φ. The action is (with Mp 1) In components one has, writing the scalar and F =− 4 4 ¯ −K/3 components of H = (A, F ) and of L = (a, f ) and ig- S 3 d xd θ φφe noring the fermion terms + d4xd2θφ3W + h.c. , (16) 1 L = A¯A +¯aa + FF¯ + ff¯ + Fa2 + F¯a¯2 2 where the superpotential is a holomorphic function of ¯ ¯ ¯ ¯ + (Aaf + Aa¯f)+ M(AF + AF). (13) the chiral scalar fields W = W(χi) and the Kähler potential is a real function K = K(χ,χ)¯ .Fromthis The heavy field equations are action one obtains the following equations of motion. −A¯ = MF + af, (14) 1 ¯ 2 ¯ k¯ + 1 k¯i − 2 ¯ α˙ ¯ j¯ ¯ ¯ l¯ a2 D χ K Kijl¯ Kij¯Kl¯ D χ Dα˙ χ F¯ =− − MA. (15) 4 4 3 2 ¯ α˙ ¯ 2 1 D φ ¯ k¯ K/3 φ ki¯ + Dα˙ χ¯ = e K DiW, Solving them we get after expanding in powers of 2 φ¯ φ¯ /M2, 1 − D¯ 2 φe¯ K/3 =−φ2W(χ). 1 1 4 A¯ = −Ma¯2 + 2af + O , 2M2 M2 M2 Using the identity above (16) we get ¯ a¯f 1 2 F¯ =− + O . k¯ D ki¯ 2 ¯ α˙ j¯ ¯ l¯ 2 2 χ¯ =− K K ¯ − K ¯K¯ D χ¯ D ˙ χ¯ M M M 16 ijl 3 ij l α 2 It is easily checked that these are precisely the − D ¯ − φ¯ 1 eK/3φ2DkW . (17) equations that would be obtained by looking at the 4 components of the superfield equation (10). Plugging Let us now consider the case with one heavy super- these equations into (13) we get the light field poten- field H which we will take to be canonically normal- tial, ized. So the Kähler potential becomes ¯ ¯ aa 1 3 ¯ 3 −V = ff 1 + − fa + f a¯ . K = HH¯ + Kl(L, L),¯ (18) M2 2M Eliminating the light auxiliary field we have where L stands for the light fields. Also the superpotential is taken to be 6 2 −1 |a| |a| 1 V = 1 + . W = MH2 + W(H,L),˜ 4M2 M2 2 186 S.P. de Alwis / Physics Letters B 628 (2005) 183–187 where M is a large mass parameter. so that the other O(M) terms which all have a factor of Then from (17) we have for the heavy field DαH also vanish (since the superpotential should not vanish at a generic point). So (20) is certainly a suffi- ¯ ¯ 2 K/3M2 2 −4φH = D e p φ DH W cient condition in the sense that it implies H = 0. ¯ However it is easy to see that the strict implementa- 2K/M2 2 2 HH ¯ + 4Me p φ φ¯ 1 + D ¯ W M2 H tion of the condition (20) leads to the conclusion that p the light fermion fields would have to be set to zero. HH¯ + 1 + Let us look at this in somewhat more general terms M2 than above. p α We assume that the Kähler potential is a sum of 2 2 Kl 2D φ × eK/3Mp φ2 Dαχl − 2 ¯ terms as in the string theory examples discussed in [5]. 3 M φ I p In particular if we call the heavy superfields H and ¯ α i HD H 2 the light superfields L assume that + + 2Dα eK/3Mp φ2 D H M2 α p K = Kh(H, H)¯ + Kl(L, L).¯ (21) ¯ φ 2 ¯ α˙ ¯ ¯ l¯ − D K¯D HD ˙ χ¯ I = i 2 l α Thus in the example in Section 3 of Ref. [5], H z 6Mp i = and L S,T . Then the generalization of (20) be- α K/3M2 2 ˜ + 2D e p φ DαDH W comes (note that capital letters I,J go over the heavy fields) K/3M2 2 2 ˜ + e p φ D DH W. (19) hIJ¯ = = + 2 K DJ W 0. (22) In the above DH W ∂H W KH W/Mp is the Kähler derivative of the superpotential with respect to Using the non-degeneracy of the metric on the the heavy field and we have restored the dependence heavy fields this becomes on the Planck mass. To integrate out a heavy field H + h = we have to set H¯ → p2H¯ → 0 and the condition for ∂I W KI W 0. (23) that is that the right-hand side of the above equation Taking the anti-chiral derivative of this equation and is set to zero. Note that this condition reduces to (10) using the chirality of the superpotential we get in the global limit Mp →∞and φ → 1. Here up to h ¯ ¯ J¯ WK D ˙ H = 0 (24) terms involving a factor of DαH we have IJ¯ α 2 ¯ ¯ ¯ h 2 K/3Mp 2 implying D ˙ H J = 0 since the metric is K is non- D e φ DH W α IJ¯ ¯ degenerate and W = 0 at generic points. 2K/M2 2 2 HH ¯ + Me p φ φ¯ + D ¯ W Now let us assume that there is a solution H = 4 1 2 H Mp H(L,L)¯ of Eq. (23). Using the chirality of H and L K/3M2 2 2 ˜ we get by differentiating this solution =−e p φ D DH W. I I I As in the case of the global SUSY discussion one D¯ = ∂H D¯ j + ∂H D¯ ¯ j¯ = ∂H D¯ ¯ j¯ = α˙ H α˙ L ¯ α˙ L ¯ α˙ L 0. may expect that with some restriction on the light field ∂Lj ∂L¯ j ∂L¯ j space (so that the right-hand side of the equation is (25) small compared to M) the relevant condition would be Similarly from the result (see (24)) that the chiral the natural generalization of (6) derivative of H also vanishes 1 I I I ∂H j ∂H j¯ ∂H j DH W = ∂H W + W∂H K = 0. (20) D = D + D ¯ = D = 2 αH j αL ¯ αL j αL 0. Mp ∂L ∂L¯ j ∂L Note that (by taking its spinor derivative) this con- (26) dition implies These two equations tell us that the chiral derivative of L should be zero—in other words the light ¯ ¯ WDα˙ H = 0, fermions should also be set to zero. This of course S.P. de Alwis / Physics Letters B 628 (2005) 183–187 187 means that the light field theory is not supersymmet- only for |a|2/M2 1 as in the global case discussed ric! However the problem is that the condition (20) or earlier. The point is that necessarily M Mp so its generalization is really too strong. It was obtained that supergravity corrections which in this model are | |2 2 by ignoring the DαH termsaswellastheO(1/M) O( a /Mp) should also be ignored for consistency of terms. Of course as was pointed out earlier, these terms the approximation. are zero if one imposes (20) so that one gets H = 0, However very often in string theoretic examples but the condition itself is not necessary. The actual (such as those with flux compactifications) there is a necessary condition which follows from (19) is a re- constant in the superpotential W = W0 + ··· where lation between the Kähler derivative terms and the the ellipses denote field dependent terms. In these fermionic superfield terms. This condition would then cases the supergravity corrections are indeed signif- = + 2 + express the bosonic component of the heavy superfield icant since DLW ∂LW KLW/MP ∂LW in terms of the light bosonic field as well as squares 2 KLW0/MP and the second term may even be of of light fermionic fields. However if we are interested O(1) even thought light field space is restricted to only in the scalar potential we do not need to keep |a|2 M2. So this does not necessarily force us to these terms. Thus the condition (20) can clearly be still the global limit since in many examples of interest used if one is just interested in computing the potential in string theory there would be a constant in the su- for the light chiral scalars. perpotential which is generically of the order of the Unlike the case of rigid supersymmetry (20) is not Planck/string scale. Thus the extra piece in the Käh- a holomorphic equation since the Kähler derivative in- ler derivative (as compared to the ordinary derivative) ¯ volves the real function K(Φ,Φ). This means that the of the superpotential has to be kept. So we may use solution for the heavy field will not in general be a the condition (22) with the understanding that it is to holomorphic function of the light fields and hence the be used only for the scalar components of the super- light field theory in general will have a superpotential fields for the purpose of calculating the light scalar that is just one (or a constant) and the whole effect of field potential, still remaining within the context of integrating out the heavy field will be accounted for supergravity. by changing the Kähler potential. In fact the original potential should be expressed in terms of the Kähler invariant function G = ln K + ln |W|2 before integrat- Acknowledgement ing out the heavy fields. In a companion paper [5] we show this explicitly in some examples that come from I am very grateful to Martin Rocek for several use- type IIB string theory compactifications with fluxes. ful suggestions. Now one might worry that the restriction on the range of the light field essentially forces us back to the global case. This would indeed be the case in an References example such as (12). Evaluating (20) in this case we have [1] S. Weinberg, Phys. Lett. B 91 (1980) 51. [2] K.A. Intriligator, N. Seiberg, Nucl. Phys. B (Proc. Suppl.) 45 1 2 ¯ 1 2 2 DH W = MH + L + H MH + HL = 0. (1996) 1, hep-th/9509066. 2 2 [3] S.J. Gates Jr., M.T. Grisaru, M. Rocek, W. Siegel, Superspace, Solving this for H we may compute the scalar po- or One Thousand and One Lessons in Supersymmetry, in: Fron- tential using the standard supergravity formula (ex- tiers in Physics, vol. 58, 1983, p. 1, hep-th/0108200. pressed in terms of G) but now the question is whether [4] J. Wess, J. Bagger, Supersymmetry and Supergravity, Princeton Univ. Press, Princeton, 1992. it is consistent to keep the supergravity corrections [5] S.P. de Alwis, hep-th/0506266. given that the integrating out formula above, is valid Physics Letters B 628 (2005) 189–196 www.elsevier.com/locate/physletb

The cosmological constant problem in codimension-two brane models

Jan-Markus Schwindt a, Christof Wetterich b

a Institut für Physik, Universität Mainz, Staudingerweg 7, 55128 Mainz, Germany b Institut für Theoretische Physik, Universität Heidelberg, Philosophenweg 16, 69120 Heidelberg, Germany Received 20 August 2005; accepted 18 September 2005 Available online 27 September 2005 Editor: L. Alvarez-Gaumé

Abstract We discuss the possibility of a dynamical solution to the cosmological constant problem in the context of six-dimensional Einstein–Maxwell theory. A deﬁnite answer requires an understanding of the full bulk cosmology in the early universe, in which the bulk has time-dependent size and shape. We comment on the special properties of codimension two as compared to higher codimensions.  2005 Elsevier B.V. All rights reserved.

The presence of extra dimensions provides a frame- of the general solution. The question remains why work in which the cosmological constant problem can a solution with small Λ4 should be dynamically se- be viewed from a different perspective. The question lected. Recently this problem was reconsidered in the is no longer why the full spacetime curvature is so context of codimension-two braneworlds with conical small, but rather why the four-dimensional (4D) cur- singularities [3–6], with a bulk stabilized by magnetic vature contains only such a small part of the total ﬂux [7]. higher-dimensional curvature. It has been known for We may view the solutions with static internal about twenty years [1,2] that in a certain subclass of space as candidates for asymptotic solutions for large six-dimensional solutions, namely, those with time- time t. For the approach to these asymptotic solu- independent size and shape of internal space, the 4D tions, however, internal space is not expected to be cosmological constant Λ4 is a free integration constant static. The evolution of the universe in this early pe- riod with time varying geometry will decide to which value of Λ4 the late universe will converge. Two sce- E-mail addresses: [email protected] narios are conceivable. The dynamical approach to (J.-M. Schwindt), [email protected] the ﬁnal value of the four-dimensional curvature (i.e., (C. Wetterich). of Λ4) may occur very early in cosmology, with a

fixed Λ4 since. This resembles the inflationary uni- Ref. [9]: the quintessence field φ of the effective 4D verse, where the final zero (or very tiny) value of the theory is given by the radius of the internal space and three-dimensional curvature is selected at a very early is therefore related to its curvature. On the other hand, stage. (For the Friedmann solutions the initial density the potential V(φ) and the time derivatives of φ in- is a free integration constant, and its value very close duce some curvature in the 4D world. The dynamics to the critical density is selected during the early infla- of φ is nothing else but an effective description of the tionary epoch.) interaction between the different parts (2D and 4D) of For the second alternative, the adjustment to the the total six-dimensional curvature. final value of Λ4 is still going on in present cosmol- The allowance for a more general geometry com- ogy. This will lead to a dynamical dark energy or plicates our subject immensely. The “football-shaped” quintessence [8]. In such a scenario the asymptotic cosmology was characterized by a few constants (the value of Λ4 is typically zero, and in present cosmology brane tension and the monopole number) and one sin- dark energy is expected to contribute of the same or- gle function φ(t). Within the more general geometry, der as matter. However, potential problems arise now there is an infinite number of degrees of freedom, cor- with time varying coupling constants. responding in the effective four-dimensional world to Particular exact dynamic solutions of the 6D Ein- an infinite set of scalar fields in the singlet representa- stein–Maxwell theory have been recently found for tion of the symmetry group. Below we derive the most both scenarios [9]. They are, however, special in the general ansatz for the six-dimensional metric with the sense that only the size, not the geometric shape of in- symmetries of three-dimensional rotations and trans- ternal space changes with time, and that there is no lations and internal U(1) isometry, and we have also warping. A dynamical solution of the cosmological computed the corresponding field equations. constant problem in early cosmology is not expected For an investigation of cosmological solutions we in such a restricted setting and actually not found. An aim to answer the question: is there a large class of investigation of this question requires a time-varying initial conditions for which the bulk comes to rest as- geometry and warping within the most general class ymptotically leaving a very small or zero 4D curva- of solutions consistent with the symmetries. Recent ture? To study this question, we adopt a bulk-based progress towards an understanding of this complicated point of view [10], in which the singularities (branes) dynamics was made by Vinet and Cline [4]. They con- are seen as properties of the bulk geometry (such as sidered different types of singularities, allowing for a the mass of a Schwarzschild black hole may be seen general equation of state on the brane. The authors as an integration constant of the vacuum geometry). find no self-tuning mechanism. However, the results This simplifies our task since only the field equations of Ref. [4] are by far not sufficient for a definite an- in the bulk need to be solved. The singularities will swer. The limitation of their work is the restriction to certainly play an important role in the development small perturbations around an essentially static bulk: of the bulk. It is of particular interest how bulk fields the “football-shaped” solution [5]. If the self-tuning of respond to the singularities, and if a part of the en- the 4D cosmological constant takes place in the very ergy of the fields may “fall” into them (like matter early universe, there is no reason to assume that the falls into a black hole). Due to the high complex- bulk was static, or even almost static, at that time. ity of the field equations we have not yet achieved A general analysis should take place in a bulk with to answer these questions in the present Letter. The the most general (non-perturbatively) time-dependent aim of the present Letter is therefore more modest, size and shape which is consistent with the symme- i.e., to clarify the problem for general bulk geometries. The curvature may then be dynamically shifted tries and develop a strategy for further investigations. between the 4D part, the 2D part and the warping, im- In the text we mainly refer to the Kaluza–Klein con- plying an effective time-dependent 4D cosmological text which implies that the two internal dimensions are constant. compactified on a scale not much different from the This can happen even if there are no time-dependent Planck scale. Nevertheless, most of our results remain sources at all, only the 6D cosmological constant valid in braneworld scenarios with large extra dimen- and a constant magnetic flux, as we demonstrated in sions. J.-M. Schwindt, C. Wetterich / Physics Letters B 628 (2005) 189–196 191

As our first general observation we note that codi- stant factor induces a change in the curvature which mension two is a very special case, for several reasons diverges as one approaches ρ = 0. on which we comment below. We already mentioned In fact, the deficit angle brane is a special type of the speciality of codimension one in an earlier pa- a Kasner singularity. Consider, for simplicity, a static per [10], in particular, the fact that the position of a vacuum singularity at ρ = 0, i.e., all metric compo- codimension-one brane cannot be detected by a “test nents are functions of ρ only. We may then normal- particle” in the bulk. In contrast, for codimension two ize gρρ to 1, and the most general metric (in 4 + D or higher the type and strength of the singularity can be dimensions) consistent with our symmetries (in par- inferred from the properties of the bulk geometry. Still ticular internal SO(D) isometry) is codimension two is special since there exists a type of 2 2 2 2 i 2 brane that is not possible for any higher codimension: ds =−c (ρ) dt + a (ρ) dx µ 2 α β 2 the deficit angle brane. Using coordinates x for the + b (ρ)g˜αβ (θ) dθ dθ + dρ . (2) four large dimensions and ρ and θ (with 0 θ<2π) as cylindric coordinates for the internal space, such a In the vicinity of the singularity, the vacuum Einstein brane, or conical singularity, can be described in the equations admit solutions of the form following way: the metric components g have well µν c ∼ ρp1 ,a∼ ρp2 ,b∼ ρp3 (3) defined, finite values at the position of such a singu- 2 larity (at ρ = 0, say), while gθθ is proportional to ρ with in the vicinity of the brane. This is the usual behav- + + − ior of cylindric coordinates with radial coordinate ρ. p1 3p2 (D 1)p3 The only effect of the singularity is a deficit angle ∆, = 2 + 2 + − 2 = p1 3p2 (D 1)p3 1. (4) which is expressed in the metric by the fact that The deficit angle brane corresponds to the very special solution with p1 = p2 = 0 and p3 = 1 which exists ∆ 2 3 gθθ = 1 − ρ + O ρ . (1) only for D = 2. In all other cases the gµν components 2π become irregular at ρ = 0 (either zero or infinite), and The infinite curvature at ρ = 0 is not “visible” from some components of the Riemann tensor diverge. outside, by which we mean that the curvature R (and In the presence of bulk matter, the singularities may in fact any invariant formed from the Riemann tensor) have an important influence on the cosmological evo- remains finite in the limit ρ → 0. The curvature and lution. Except for the deficit angle case, they may the corresponding brane tension are of the delta func- attract the matter and force it to fall into them, mak- tion type. The finiteness of the curvature implies that ing the singularities grow (as it is familiar for black = there are no attractive forces towards the brane, at least holes). As an example, consider p2 0 (i.e., con- none with a divergent behavior. stant a) where √ Such a type of singularity exists only in the co- 1 ± 1 + D(D − 2) p = , dimension-two case. Otherwise, all singularities are 1 D not of the delta function type, i.e., they are locally √ 1 1 + D(D − 2) “visible” from outside by Riemann tensor components p = ∓ . 3 − (5) that diverge as ρ → 0, and hence diverging forces as D D(D 1) one approaches them. The reason for the existence of For D = 3 one has a solution with a “black hole sin- =−1 = 2 deficit angle branes in codimension two is that one gularity” (p1 3 , p3 3 ) in internal space. One may “cut out” a part (i.e., the deficit angle) of the would expect the existence of solutions where mat- circle described by the θ coordinate at constant ρ ter falls into this singularity, thereby changing the without inducing any curvature on it (since it is a strength of the singularity or the associated brane one-dimensional object). For D>2, the sphere SD−1 tension (given by the “mass” of the black hole). Of described by coordinates θα at constant ρ does have course, the analogy of such a cosmological solution curvature, and “cutting out” some part of it does not with a black hole is only formal. In the effective four- work. Or, equivalently, multiplying the gθθ’s by a con- dimensional theory there is no local object since the 192 J.-M. Schwindt, C. Wetterich / Physics Letters B 628 (2005) 189–196 solution is actually a direct product of time-warped in- sistent with the symmetries, which should still be rep- ternal space and flat three-dimensional geometry. The resented by the new coordinates xi and θ . Trans- time singularity appears only for a particular point formations can never depend on θ, since this would in internal space. Integrating over internal space may lead to metric functions depending on θ ; for exam- lead to a perfectly regular time in the effective four- ple, t → t = t + δt(θ), θ → θ = θ would imply dimensional world. t = t − δt(θ), and so f(t)→ f (t,θ) for any func- We may take this discussion as a warning that re- tion f . Similarly, t, θ and ρ cannot depend on xi . sults for singularities with codimension two should Furthermore, for θ → θ , one has not be too easily generalized to higher codimension. 2 ∂θ It is well conceivable that the strength of singularities gθ θ = gθθ, (7) does not change with time for codimension two branes ∂θ whereas it generically does for higher codimension. and we impose ∂θ/∂θ = 1, since θ should be in the A second speciality of codimension two arises interval [0, 2π]. when one considers the most general metric consis- Transformations of xi cannot depend on t or ρ, tent with certain symmetries: we want to look for since this would lead to forbidden components via cosmological solutions with a general shape of the ∂t ∂xi two-dimensional internal space. (Static and de Sitter- gt i = gtt, (8) like solutions were described in Refs. [1,2,7].) The ∂t ∂t first step of a dynamical investigation is the selection and similarly for gρi. Obviously, the only effect of a of an appropriate ansatz for the metric. We will see transformation xi → xi (xj ) could be a rescaling of that the determination of the most general metric con- three-dimensional space, so we can forget about them sistent with the symmetries is nontrivial and actually in this context. We are left with the following possibil- extends beyond the metrics considered so far [4].We ities: require the following symmetries: three-dimensional i i translation and rotation invariance, acting on the coor- x → x ,θ→ θ + δ(t,ρ), i dinates x , and a U(1) symmetry, acting on the coor- t → t (t, ρ), ρ → ρ (t, ρ). (9) dinate θ ∈[0, 2π]. No metric function should depend on xi or θ, and no direction in the three-dimensional There are three off-diagonal metric components, gtρ, space should be preferred. (For simplicity, we will take gtθ and gρθ, and one might think that these can be removed by the three remaining coordinate transfor- this space to be flat, so that the metric components gij 2 mations. It turns out that this is in general not true. are a (t, ρ)δij .) We have to find the most general metric consistent with these symmetries. The reason for that is essentially the U(1) symmetry. (In fact, the metric can always be diagonalized, but Isotropy forbids metric components gti, gρi and then in general the new coordinate θ will not reflect gθi, since these would select preferred directions in the U(1) symmetry any more, and fields will depend three-space, e.g., by the three-vector (gt1,gt2,gt3). on θ .) To see this, consider the inverse of the metric. The other off-diagonal metric components gtρ, gtθ and The components gtθ and gρθ will be zero if and only if gρθ are allowed, as long as they are functions of t and tθ ρθ ρ only. Up to now we have identified the most general g and g are zero. The condition that this happens metric consistent with the symmetries as after a coordinate transformation of the type (9) is ∂t ∂θ ∂θ ds2 =−c2(t, ρ) dt2 gt θ = gtθ + gtt + gtρ 2 i i 2 2 ∂t ∂t ∂ρ + a (t, ρ) dx dx + b (t, ρ) dθ + ∂t ρθ + ∂θ ρt + ∂θ ρρ = + n2(t, ρ) dρ2 + 2w(t,ρ)dt dρ g g g 0, (10) ∂ρ ∂t ∂ρ + + 2u(t,ρ)dt dθ 2v(t,ρ)dρ dθ. (6) ∂ρ ∂θ ∂θ gρ θ = gtθ + gtt + gtρ The next step is to look how far this line element can ∂t ∂t ∂ρ be simplified by a coordinate transformation. There- ∂ρ ∂θ ∂θ + gρθ + gρt + gρρ = 0. (11) fore, one has to find the possible transformations con- ∂ρ ∂t ∂ρ J.-M. Schwindt, C. Wetterich / Physics Letters B 628 (2005) 189–196 193

A solution of these differential equations implies ei- gauge field AB in a monopole configuration (capital ther that the Jacobi determinant of the (ρ, t) transfor- indices run over all six dimensions). Six-dimensional mation vanishes, Einstein–Maxwell theory [7] according to the action ∂t ∂t √ 4 ∂t ∂ρ 6 M6 1 AB det = 0, (12) S = d x −g − R + λ + F F , (14) ∂ρ ∂ρ 2 6 4 AB ∂t ∂ρ which is not possible, or that the brackets vanish. But is a convenient toy model for higher-dimensional sce- the second possibility consists of two conditions for narios. Here M6 is the reduced Planck mass corre- the function θ , which can in general not be fulfilled sponding to six-dimensional gravity, λ6 is a cosmo- simultaneously. logical constant term and FAB is the field tensor of the One concludes that generally only one of the two gauge field. Including the gauge field into our consid- tθ ρθ components g and g can be set to zero (in contrast erations, we find that the three components At , Aρ and to [4]). A procedure to simplify the metric (6) could Aθ are allowed by the symmetries. One can choose to look as follows. Use the freedom for t and ρ to an- set either At or Aρ to zero by a gauge transformation. nihilate gtρ and for one further simplification, e.g., to This is similar to the choice between gtθ and gρθ de- arrange that gtt =−gii , i.e., to make time conformal scribed above. with respect to space. Then use the freedom for θ to Comparing this cosmological Einstein–Maxwell annihilate either gtθ or gρθ. The simplified line ele- system to the static case, one finds that the ordinary ment is then differential equations (containing only ρ-derivatives) are generalized to partial differential equations, con- 2 = 2 − 2 + i i + 2 2 ds a (t, ρ) dt dx dx b (t, ρ) dθ taining t- and ρ-derivatives. The three functions a, b + n2(t, ρ) dρ2 + 2u(t,ρ)dt dθ, (13) and Aθ , which are already present in the static case, are accompanied by three more functions: n, u or v, or similarly with 2v(t,ρ)dρ dθ instead of 2u(t, ρ) × and At or Aρ . dt dθ. In the effective four-dimensional picture u cor- We have computed the field equations for the six responds to the time component of an abelian gauge independent functions of t and ρ. As an example we field (hence some kind of electric potential), since gθµ give the (tt)-component of Einstein’s equations, in the integrated over internal space is the gauge field corre- gauge a = c and w = v = 0: sponding to the U(1) isometry. On the other hand, v corresponds to a scalar field. The fact that a degree of 1 a˙2 a˙b˙ a˙n˙ b˙n˙ Gt ≡− 3 + 3 + 3 + freedom can be shifted between a scalar field and the t a2(1 + q2) a2 ab an bn component of a gauge field is a familiar fact in particle 1 a2 an a physics. + 3 − 3 + 3 The presence of off-diagonal metric components n2 a2 an a gtθ, gρθ, which cannot be transformed away simulta- 1 a b b n + 3 − neously, is a special feature of codimension-two mod- n2(1 + q2) ab bn els. Consider D>2 internal dimensions, and D − 1of nuq2 b uq2 these dimensions, represented by coordinates θα,were − + + 2nu b 2u symmetric under, say, SO(D), then the gtθ and gρθ 2 2 components would be forbidden, because they would q a b b a u 3 2 − + + + 1 + q select preferred directions in the (D 1)-dimensional n2(1 + q2)2 ab b2 au 2 space, in conflict with the SO(D) symmetry. The dif- 3buq2 u2 ference is that a U(1) “rotation” is a translation rather − + 1 − q2 than a rotation. In this sense a codimension-two space- 2bu 4u2 time is more complicated than a higher-dimensional = t 8πG6Tt one. 4πG A˙2 A 2 A 2 A third special feature of codimension two is that ≡ 6 − θ − t − θ . (15) internal space can be compactified and stabilized by a 1 + q2 a2b2 a2n2 n2b2 194 J.-M. Schwindt, C. Wetterich / Physics Letters B 628 (2005) 189–196

Here dots and primes denote derivatives with respect use this freedom to make time conformal, i.e., −gtt = to t and ρ, respectively, and we use the abbreviation gii. We warn, however, that this may often not be the q2 ≡ u2/(a2b2). The equations for the other compo- convenient choice, because the corresponding time co- nents are of similar length and are not displayed here. ordinate may be different from the “physical” time. To A full numerical analysis of this system would involve explain this, remember that in usual four-dimensional as initial conditions twelve functions of ρ (four met- cosmology, time can be made conformal by a trans- ric and two gauge field components and their first time formation t → τ(t). In the six-dimensional model, we derivatives at some initial time t0) which are subject need instead transformations t,ρ → t (t, ρ), ρ (t, ρ) to three constraint equations, namely the (tt)-, (tρ)- in order to bring the metric into the required form. This and (tθ)-components of Einstein’s equations, which may mix the time and ρ coordinate to some extent. The contain no second time derivatives. The time evolu- effective four-dimensional Lagrangian is obtained by tion is determined by the (ii)-, (θθ)-, (ρρ)- and (θρ)- integrating out internal space in the form components of Einstein’s equations and two equations √ i i for the gauge field. Leff t,x ∼ dρ dθ gintL t,x ,ρ,θ , (17) Again we want to compare this to an Einstein– Maxwell system in higher codimensions. We already or similar, where gint is the determinant of the metric of the internal space. This effective Lagrangian obvi- showed that the metric components gtθα and gρθα are not consistent with a symmetry larger than U(1) acting ously depends on the choice of the (t, ρ) frame one on the θ coordinates. For the gauge field the situation uses. Nevertheless, all the effective Lagrangians de- is slightly different. For specific solutions (solitons) rived from different choices of t and ρ must describe the same physics because they are obtained from the a component Aθα may be allowed even if the internal symmetry is larger than U(1). An example is the same six-dimensional theory. Such an equivalence be- 2 monopole solution on S . Although Aρ = 0 and Aθ = tween different Lagrangians will in general not be seen 0, the θ-direction is not preferred physically. A coor- easily, because there is an infinite number of fields dinate transformation may be accompanied by a gauge mixed with each other when going from one frame to transformation, so that the transformed A-field lies in the other. the new θ-direction. An analogous procedure does not To illustrate this, we consider the example of five- work for the metric tensor, since the gauge transforma- dimensional Kaluza–Klein theory with action tions are the coordinate transformations themselves. √ =− 1 5 − But the components Aρ and At are not necessary in S d x gR. (18) 16πG5 the D>2 case: without the gtθ and gρθ metric components, the components Gt and Gρ of the Einstein The four large dimensions are again parametrized by θ θ µ tensor are identically zero. The corresponding compo- coordinates x , and the fifth coordinate y runs from 0 to 2πr. If one writes the metric in the form nents of the energy momentum tensor induced by the ˜ Maxwell field −1/3 gµνAµAνφAνφ gAB = φ , (19) 1 Aµφφ T (F ) = F F C − F F CDg , (16) AB AC B 4 CD AB integration over y leads to the following 4D action for = the zero modes: are then also zero, which implies Fρt 0 (as long (0) 1 as Aθα is non-trivial) and so Aρ and At are pure = 4 −˜ Seff d x g gauge. Compared to the codimension-two case, the 16πG 4 higher codimensions therefore involve two functions 1 × −R˜(0) + φ(0)F (0)F µν(0) less: only a, b, n and Aθ remain after appropriate sim- 4 µν plifications. 1 ∂ φ(0)∂µφ(0) We already mentioned that, after removing two of − µ . (20) (0) 2 the off-diagonal metric components, there remains still 6 (φ ) one degree of freedom for coordinate changes in order Here G4 = G5/2πr, Fµν = ∂µAν − ∂νAµ,thesu- to bring the metric into a pleasant form. We chose to perscript (0) denotes that in the Fourier expansion of J.-M. Schwindt, C. Wetterich / Physics Letters B 628 (2005) 189–196 195 components with respect to y only the zero modes are effective 4D world corresponding to such a solution. taken into account, and a tilde denotes a quantity con- By suitable (t, ρ) transformations one still finds lo- structed from g˜µν . cal charts with −gtt = gii, but these coordinates do Now we perform a small coordinate transforma- not represent the symmetry of the solution, since now tion, affecting only the time coordinate in the form functions depend on t and ρ: f(ρ)= f(ρ(t,ρ)). → = − y t t t sin r . The corresponding change of the Everything would look unnecessarily complicated in metric is (to order ) such a frame, which is therefore “unphysical”. Even for late cosmology, say the present epoch, i = i gty x ,t ,y gty x ,t(t ,y),y there is no exact timelike Killing vector, only an ap- i proximate one. This means that for some choice of + gtt(x ,t(t ,y),y) y cos , (21) coordinate frame variables vary only very slowly with r r i = i time. The identification of the “correct” time is more gyy x ,t ,y gyy x ,t(t ,y),y complicated in such a situation than in those with ex- i gty(x ,t(t ,y),y) y + 2 cos . (22) act Killing vector. For a wrong choice of (ρ, t)-frame r r variables may vary much too fast with time and the This transformation does not only mix the fields φ, Aµ geometry seems to get totally distorted. This is sim- and g˜µν in a complicated way. It also changes the line ilar to (but worse than) the unphysical gauge modes of y-integration defined by xµ = const. In the trans- appearing in some approaches of cosmological pertur- formation of the fields this is reflected by additional bation theory (e.g., in synchronous gauge). The iden- time derivatives, e.g., tification of the time coordinate relevant for the ef- ˙ fective four-dimensional physics is a serious task in A y φ y φ = φ 1 + 3 t cos + sin . (23) higher-dimensional cosmology. Conclusions based on r r φ r the “conformal gauge” (a = c in Eq. (6)) can easily be The 4D action for the zero modes of the new fields will misleading. In practice, this means that it may be ad- again be Eq. (20), but these zero modes are not only visable to work with a metric that is even more general combinations of the old zero modes. They contain ad- than the ansatz (13). mixtures of higher Fourier modes of the original fields In summary, we have computed the field equations y (due to the cos r term) and even of their time deriva- for the six-dimensional Einstein–Maxwell theory for tives. the most general ansatz of the metric and gauge fields For a given solution of the field equations, there consistent with the symmetries of three-dimensional are certainly choices of coordinates in which the four- rotations and translations and a U(1)-isometry. A cru- dimensional world looks simpler than in others. In cial issue for a possible dynamical solution of the some situations there may be a clear and unique pre- cosmological constant problem is the possibility that ferred frame which identifies a “physical” time coor- the brane tension changes with time. For a restricted dinate. The “physical” time is easily identified when ansatz it was found that this does not happen for this there is a timelike Killing vector. Returning to the six- system [3], but we would like to emphasize that a com- dimensional model, such a Killing vector is given for plete answer needs the most general ansatz for the met- the Kasner solutions mentioned before. In a situation ric. Therefore, a dynamical solution to the cosmologi- like the static football shaped solution (p1 = p2 = 0at cal constant problem in the context of six-dimensional the singularities), where time and three-space are not brane or Kaluza–Klein models is so far not ruled out, differently warped, an appropriate frame has necessar- not even in the case of infinitely thin deficit angle ily −gtt = gii. For Kasner solutions with p1 = p2, like branes. An answer to the question requires a much the aforementioned black hole type singularities, time more detailed understanding of the early universe dy- and three-space are warped differently, and the physi- namics with a time-dependent bulk geometry. As we cal time (with time axis parallel to the Killing vector) have shown, such an understanding is complicated by corresponds to a frame with −gtt = gii. In this frame the fact that the four-dimensional interpretation of the all metric components depend only on ρ (cf. Eq. (2)). six-dimensional dynamics is far from clear in the ab- In general, Lorentz invariance will be broken in the sence of a timelike Killing vector. 196 J.-M. Schwindt, C. Wetterich / Physics Letters B 628 (2005) 189–196

Furthermore, we have pointed out that codimension [2] C. Wetterich, Nucl. Phys. B 255 (1985) 480. two is a very special case for several reasons: [3] J.M. Cline, J. Descheneau, M. Giovannini, J. Vinet, JHEP 0306 (2003) 048. [4] J. Vinet, J.M. Cline, Phys. Rev. D 70 (2004) 083514. (i) There exist conical singularities (deﬁcit angle [5] S.M. Carroll, M.M. Guica, hep-th/0302067; branes) which do not induce any attractive forces I. Navarro, JHEP 0309 (2003) 004. in the bulk. [6] I. Navarro, Class. Quantum Grav. 20 (2003) 3603; (ii) The metric is relatively complicated even if one J. Garriga, M. Porrati, JHEP 0408 (2004) 028; requires that all quantities depend only on one in- A. Kehagias, Phys. Lett. B 600 (2004) 133; P. Bostok, R. Gregory, I. Navarro, J. Santiago, Phys. Rev. ternal coordinate ρ and on time. Lett. 92 (2004) 221601; (iii) A Maxwell ﬁeld can compactify and stabilize the H.M. Lee, Phys. Lett. B 587 (2004) 117; internal space. Y. Aghababaie, C.P. Burgess, S.L. Parameswaran, F. Quevedo, Nucl. Phys. B 680 (2004) 389; We have argued by analogy with the dynamical H.P. Nilles, A. Papazoglou, G. Tasinato, Nucl. Phys. B 677 (2004) 45; black hole geometries with infalling matter in the four- M.L. Graesser, J.E. Kile, P. Wang, Phys. Rev. D 70 (2004) dimensional world that in general the strength of a 024008; singularity can vary with time. Once such time vary- S. Randjbar-Daemi, V.A. Rubakov, JHEP 0410 (2004) 054; ing “brane tensions” can be described properly in a V.P. Nair, S. Randjbar-Daemi, JHEP 0503 (2005) 049; higher-dimensional world the issue of the cosmolog- I. Navarro, J. Santiago, JHEP 0502 (2005) 007; H.M. Lee, A. Papazoglou, Nucl. Phys. B 705 (2005) 152; ical constant or quintessence may show new, unex- P. Wang, X.H. Meng, Phys. Rev. D 71 (2005) 024023; pected facets. J. Vinet, J. Cline, Phys. Rev. D 71 (2005) 064011. [7] S. Randjbar-Daemi, A. Salam, J. Strathdee, Nucl. Phys. B 214 (1983) 491. References [8] C. Wetterich, Nucl. Phys. B 302 (1988) 668. [9] J. Schwindt, C. Wetterich, hep-th/0501049. [1] V.A. Rubakov, M.E. Shaposhnikov, Phys. Lett. B 125 (1983) [10] J. Schwindt, C. Wetterich, Phys. Lett. B 578 (2004) 409. 139. Physics Letters B 628 (2005) 197–205 www.elsevier.com/locate/physletb

Cosmological signiﬁcance of one-loop effective gravity

Domènec Espriu a, Tuomas Multamäki b, Elias C. Vagenas c

a DECM and CER for Astrophysics, Particle Physics and Cosmology, Universitat de Barcelona, Diagonal, 647, S-08028 Barcelona, Spain b NORDITA, Blegdamsjev 17, DK-2100 Copenhagen, Denmark c Nuclear and Particle Physics Section, Physics Department, University of Athens, GR-15771 Athens, Greece Received 18 August 2005; accepted 18 September 2005 Available online 30 September 2005 Editor: L. Alvarez-Gaumé

Abstract We study the one-loop effective action for gravity in a cosmological setup to determine possible cosmological effects of quantum corrections to Einstein theory. By considering the effect of the universal non-local terms in a toy model, we show that they can play an important role in the very early universe. We find that during inflation, the non-local terms are significant, leading to deviations from the standard inflationary expansion.  2005 Elsevier B.V. All rights reserved.

1. Introduction

Quantum corrections to Einstein action have received the attention of numerous authors for quite a long time. In the original works [1], the divergent structure of the theory was determined. It was seen that the theory required at the one-loop level O(R2) counter terms (albeit it turned out to be ﬁnite on shell if matter was absent). These divergences will actually be important for our discussion since, by unitarity arguments, they determine the non- local part of the one-loop effective action. Although it was soon found out [1] that higher order loops required more and more counter terms, making Einstein gravity non-renormalizable, not even on-shell, this does not mean that one-loop corrections are useless. In fact in a regime of small curvatures one-loop corrections will certainly dominate over two-loop corrections and so on. As a matter of fact this is not very different from the familiar expansion in chiral perturbation theory in powers 2 2 of (p /16πfπ ) (see e.g. [2]). Within this philosophy, the one-loop effective action for Einstein gravity has received quite a lot of attention recently [3]. For instance, the quantum corrections to Newton law have been considered by a number of authors [4],

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

0370-2693/$ – see front matter  2005 Elsevier B.V. All rights reserved. doi:10.1016/j.physletb.2005.09.033 198 D. Espriu et al. / Physics Letters B 628 (2005) 197–205 and after some controversy the correct result has been found. Quantum mechanics gives a correction to the classical result of O(1/r3) and positive; that is, at long distances gravity is more attractive that what Newton law predicts. Of course the correction vanishes anyhow at large values of r and it is accompanied by a very small coefficient, so for terrestrial or astronomical purposes this correction is most likely irrelevant and unlikely to be tested ever. However, the situation may be more promising from a cosmological view point. There is a cumulative effect of gravity and, given a fixed density of energy, the integration of this effect over large volumes could give an observable signal. This is, in short, the possibility that we would like to tentatively study in this work. We shall consider a flat Friedman–Robertson–Walker metric background and determine the effect of quantum corrections on the cosmological evolution of the scale factor. We shall concentrate here on the consequences of the non-local terms that necessarily appear in the effective action due to unitarity considerations. It is a well-known fact from pion physics that these terms dominate at large distances, but somewhat surprisingly they do not appear to have been considered before in the present context (with one exception to which we shall turn below). Somewhat to our surprise, the effect of these quantum corrections turns out to be relevant. In the present, exploratory, Letter we shall not consider the full one-loop effective action, but shall limit ourselves to a toy model where only the scalar curvature is included in the effective action and neglect the full Ricci or Riemann tensors. We shall also make for simplicity a number of additional approximations that we shall discuss in more detail below. The action of our toy model up to R2 terms is √ 2 2 2 2 S = dx −g κ R + αR ln ∇ /µ R + βR + SM , (1)

2 = = 2 where κ 1/(16πG) MPl/(16π) and SM includes the matter fields (and in particular the inflaton sector of the theory). Note that the constants α and β are dimensionless. The expansion parameter is curvature, R, which in the ˙ 2 approximation H = 0 is in a simple relation to the Hubble parameter, R ∼ H . Hence, when H/MPl 1e.g.at present times, local higher order terms can be neglected. The values of α and β are on a different theoretical footing. α is entirely determined from the analytical structure of quantum corrections induced by the lowest-dimensional term; in other words, its value is uniquely determined once one insists in the long distance description of gravity being provided by Einstein theory. β, on the other hand, is model dependent. Its value is fixed as a boundary condition upon integration of other degrees of freedom that do not have been included in (1). Furthermore, β is scale dependent so as to cancel the (local) ln µ2 dependence appearing normalizing the log in (1). We neglect higher orders in the expansion in derivatives of the metric. A power counting can be established here [5], in parallel to what is done in chiral perturbation theory. We shall therefore work with a precision where only up to four derivatives of the metric need to be included.

2. Determination of the equations of motion

We split the action (ignoring the matter part for now) into three parts and redeﬁne the constants for convenience √ √ √ 2 2 2 ˜ 2 2 ˜ S = κ dx −g R +˜α dx −g R ln ∇ /µ R + β dx −g R ≡ κ (S1 +˜αS2 + βS3), (2) where α˜ = κ−2α, β˜ = κ−2β. The dimensionful constant µ is actually a subtraction scale that is required for dimensional consistency. The coupling β˜ is µ dependent in such a way that the total action S is µ-independent. In conformal time, dt = adτ,wehave

a(τ) √ g = a2(τ)η , R =−6 , −g = a4(τ). (3) µν µν a3(τ) D. Espriu et al. / Physics Letters B 628 (2005) 197–205 199

Variation of the local action is straightforward and gives the following results

δS 1 =−12a (τ), (4) δa(τ) δS (a)2 aa (a)2a a(4) 3 = 72 −3 − 4 + 6 + . (5) δa(τ) a3 a3 a4 a2

The variation of S2 is more delicate and requires some discussion. The d’Alembertian in conformal space is related to the Minkowski space operator by [6]

− 1 ∇2 = a 32a + R. (6) 6 To the precision we are working in the curvature expansion we can neglect the R-term in the expansion of the d’Alembertian and commute the scale factor a with the ﬂat d’Alembertian; therefore we set − a 2 ∇2 = 2, (7) a0 ˜ where a0 = a(0). The rescaling (absorbable in β) ensures that at τ = 0 the d’Alembertian matches with the Minkowskian one. In fact, from now on we will set a0 = 1 for simplicity. We can now separate S2 into a local and a non-local piece √ = − − R R + R 2 2 R = I + II S2 dx g 2 ln(a) ln /µ S2 S2 . (8)

I Variating the ﬁrst term S2 is again straightforward δSI (a)2a aa (a)2 a(4) 2 =−72 2 6ln(a) − 5 + 4 1 − 2ln(a) + 3 1 − 2ln(a) + 2ln(a) . δa(τ) a4 a3 a3 a2 II We shall discuss the variation of S2 next.

2.1. Evaluation of the non-local contribution

The genuinely non-local piece in S2 is √ II = − R 2 2 R = − R − | 2 2 | R S2 dx g ln /µ dx g(x) (x) dy g(y) x ln /µ y (y). (9)

To evaluate this logarithmic term, we make use of the identity ln(x) ≈−1/ + x / , valid when is small. Using this, we see that we need to compute −1x|(2/µ2) |y, that has the integral representation − 1 − + 1 − 1 1x| 2/µ2 |y= 2π 2µ 2 dkk2 2 J k|x − y| ∼−8π 2µ 2 , (10) |x − y| 1 |x − y|4+2 where J1 is the Bessel function. As we are only interested in the time evolution of the scale factor we can integrate out the spatial dependence in Eq. (9) absorbing the constants into α˜ leaving − 1 SII = dτ −g(τ)R(τ) dτ −g(τ)µ 2 R(τ ). (11) 2 |τ − τ |1+2 The above integral is of course divergent; the divergence is local and is exactly cancelled by the −1/ in ln(x) ≈ −1/ + x / . By taking this term into account and taking → 0, (9) can be easily numerically computed for a given background. 200 D. Espriu et al. / Physics Letters B 628 (2005) 197–205

A ﬁnal word of caution has to do with the causal conditions to be imposed on our Green functions. We have been so far rather cavalier about that. When deriving evolution equations (as opposed to S-matrix elements) one has to be careful with the causal conditions. The usual Feynman rules lead to the so-called in–out effective action, relevant for S-matrix elements. When interested in causal evolution one has to consider the in–in effective action 2 [7]. This affects the causal deﬁnition of the Green function ∆(x,y) ≡x|(2/µ ) |y.If∆F ,∆D,∆+ and ∆− are the Feynman, Dyson, advanced and retarded Green functions, respectively, the combination that actually appears in the in–in effective action takes the matrix form (see e.g. [8] for a clear discussion on this formalism) ∆F −∆+ R+ (R+ R−) , (12) ∆− −∆D R− where the subscripts denote positive and negative frequency, respectively. This ensures that when taking a functional derivative w.r.t. a(t), only earlier times contribute to the evolution. This can be enforced by setting the limits in the time integrals appropriately.

2.2. Variation of the non-local contribution

Now we can proceed with the variation of the non-local part (for simplicity, we set = 0 here; although the expressions are all ill-deﬁned, we can replace back easily)

τ II δS δ 1 2 = 72 dτ a(τ )a (τ ) dτ a(τ )a (τ ) δa(τ) δa(τ) τ − τ 0 τ τ 1 1 = 72 2a dτ a(τ )a (τ ) + 2a ∂ dτ a(τ )a (τ ) τ − τ τ τ − τ 0 0 τ 1 + a∂2 dτ a(τ )a (τ ) . τ τ − τ 0 The derivatives can be easily performed by integration by parts

t τ 1 a0a 1 ∂ dτ a(τ )a (τ ) = 0 + dτ a (τ )a (τ ) + a(τ )a (τ ) , (13) τ τ − τ τ τ − τ 0 0 τ 1 ∂2 dτ a(τ )a (τ ) τ τ − τ 0 τ a0a a a + a0a 1 =− 0 + 0 0 0 + dτ a (τ )2 + 2a (τ )a (τ ) + a(τ )a(4)(τ ) . (14) τ 2 τ τ − τ 0 The poles at τ = 0 terms are an artifact arising from the fact that at τ = 0 we patch together Minkowski space and de Sitter space by starting inﬂation at that point. If this is done smoothly enough, the derivatives of the scale factor vanish at that point (this is evidenced by the fact that all these terms contain derivatives of a at τ = 0). Hence, we disregard these terms in the following. We have checked that modifying the matching has unobservable consequences in our results. D. Espriu et al. / Physics Letters B 628 (2005) 197–205 201

3. Effects on inﬂation

With the variated action, we can now look for solutions of the resulting equation. In general, the equation is a complicated integro-differential equation that is most suited for numerical studies. However, we expect that the new terms may have an effect during inflation when R is large and H˙ is small. In conformal time, the scale factor grows during inflation as 1 a (τ) = , (15) I 1 − Hτ where H is the inflationary Hubble rate as in a = a0 exp(H t). It is evident that in conformal time we must restrict the time interval considered to τ<1/H . Note that in order for this to be the solution of the varied gravitational action (without the higher curvature terms) one has to put in the appropriate matter terms into the action. Assuming a simple generic inflationary model, one can relate the Hubble rate to the inflaton potential by H 2 = 8πGV(φ)/3, where φ is the value of the inflaton during inflation (approximated with a constant). Here we consider H to be a constant and study how the new terms affect the inflationary expansion. In a more realistic treatment left for future work, one has to let the inflaton roll and solve the coupled system of equations of the inflaton equation along with the modified Einstein’s equation. We proceed by solving the varied gravitational action by a perturbative approximation, i.e., we consider the non-standard terms as a correction to the standard inflationary solution. This procedure is only valid as long as the correction is small compared to the unperturbed solutions, which sets the limits for the validity of our approach. For calculations it is useful to note the relation

= 2 aI (τ) HaI (τ), (16) which implies that

(n) = ! n n+1 aI n H aI (τ). (17) The 0th order equation, corresponding to including the inﬂaton sector in the action, is

−12a + 24H 2a3 = 0, (18) which Eq. (15) is a solution of.

3.1. Perturbing the 0th order solution

By using the 0th order solution (15) we see that the variation of S3 vanishes (this is not a general result, i.e., 2 n this only happens with the R term, other terms of the form R , n>2 are non-zero). The variation of S2 does not vanish and after a straightforward substitution we get for each part

δSI H 4 2 =−1152 =−1152H 4a3, (19) δa(τ) (1 − Hτ)3 I τ τ II δS 1 1 2 = 72 4H 2a3 2 dτ H 2a4(τ ) + 2Ha2 8 dτ H 3a5(τ ) δa(τ) I τ − τ I I τ − τ I 0 0 τ 1 + a 40 dτ H 4a6(τ ) . (20) I τ − τ I 0 202 D. Espriu et al. / Physics Letters B 628 (2005) 197–205

Hence, the 1st order equation of motion for a(τ) is τ τ τ 4 5 6 a (τ ) a (τ ) a (τ ) a − 2H 2a3 = 12α˜ −8H 4a3 + 4H 4a3 dτ I + 8H 4a2 dτ I + 20H 4a dτ I . I I τ − τ I τ − τ I τ − τ 0 0 0 (21) The r.h.s. of this equation (that depends on the unperturbed 0th order solution only) can be regarded as a driving external force for the evolution equation of the conformal factor. The integral terms can be explicitly computed (we now restore µ and ) τ n − a (τ ) 1 − µ 2 dτ I =− (τµ) 2 F(n,1, 1 − ,Hτ), (22) (τ − τ )1+2 0 where F is the hypergeometric function. In the limit → 0, Eq. (22) tends to τ n − a (τ ) 1 µ 2 dτ I =− F(n,1, 1,Hτ)+ ln(τµ)F (n, 1, 1,Hτ)+ ∂ F(n,1, 1 − ,Hτ) (τ − τ )1+2 →0 0

1 − − =− (1 − Hτ) n + (1 − Hτ) n ln(τµ) + ∂ F(n,1, 1 − ,Hτ) , (23) →0 indicating the need to regulate the integrals by the addition of an appropriate counter-term (this is the role of the −1/ term appearing in the logarithm representation). We are now ready to solve Eq. (21) numerically, using the 0th order solution (15) as an initial condition.

3.2. Numerical analysis

As we can see from the form of the inflationary expansion in conformal time, the scale factor is singular at 60 26 −26 τ = 1/H . Furthermore, if we wish that e.g. aI /a0 ∼ e ∼ 10 , we must require that 1−Hτ ∼ 10 , i.e., that τ is very close to the singular point. It is hence useful for numerical work to do a change of variables, es = 1/(1 − Hτ). s In these coordinates the inflationary expansion is simply aI (s) = e and the coordinate s ∈[0, ∞]. The equation of motion with the new dimensionless time coordinate is (after regularization) e2sa + e2sa − 2a3 = 12αH˜ 2 −8e3s + 4e3sG(4,s)+ 8e2sG(5,s)+ 20esG(6,s) , (24) ≡ ns µ − −s + − − −s | 2 where we have defined G(n, s) e ln( H (1 e )) ∂ F(n,1, 1 ,1 e ) →0 and divided both sides by H . In this form of the equation of motion, we note a number of interesting properties. First of all we see that the coupling constant appears in a combination αH˜ 2, indicating that the corrections to the standard evolution become extremely small at late times (recall that presently H ∼ 10−42 GeV and that α˜ is dimensionful and proportional to the inverse Planck mass). Secondly, the arbitrariness in the choice of the scale is exhibited by the presence of the µ/H term inside the logarithm. Finally, we note that the right-hand side is singular at s = 0 (τ = 0) but this is only a logarithmic divergence and can be avoided in numerical work by starting the calculation at some small non-zero value of s. For numerical work, we need to estimate the scale of inflation, V(φinf) ≡ V0, and hence the Hubble parameter 2 during inflation, H = 8πGV0/3. As an absolute lower limit, the energy density in the inflaton at the end of inflation must be enough to reheat the universe to a high enough temperature for nucleosynthesis to occur (T ∼ 1/4 16 1 MeV), but typical values can be much bigger than this, up to the CMB normalization limit V0 10 GeV, corresponding to H ∼ 1013 GeV. It should be stressed here that the model proposed in (1) is not a realistic one inasmuch as we only include R2 terms and neglect other possible curvature contributions. Our purpose here is to test in a simple setting whether D. Espriu et al. / Physics Letters B 628 (2005) 197–205 203

Fig. 1. The scale factor relative to the inflationary expansion for different values of α, µ and V0. these quantum contributions could be important at all. Therefore, albeit the actual values of α are known (see e.g. [9]), after integration of gravitons and the rest of massless particles in the Standard Model (massive particles are irrelevant for this discussion as they do not provide logs), we provide numerical results for several values of α around the value |α|∼10−4, which is the natural order of magnitude expected. A more accurate treatment would require the introduction of all the Riemann curvature tensor components. It would also require to establish which particles are exactly massless (or have a mass much smaller than the inverse horizon radius for that matter). There is also a built-in µ dependence in our results. As we have stressed, the results should be, in principle, µ-independent (this is, incidentally, a point that is often overlooked in this type of analysis). The dependence on µ in the non-local piece is exactly compensated by the (logarithmic) µ dependence of β. Of course we do not know the value of β as it contains contributions from all modes that have been integrated out and that are not explicitly included in the Lagrangian. We can, however, estimate the ‘natural’ scale for µ; i.e., the one that minimizes higher order corrections. Taking into account that massive particles are integrated out and that they do not generate logs, within a renormalization group approach it is natural to consider that the natural scale is that of the lightest particle that has been integrated out. This is similar to what is done in effective Lagrangians for the strong interactions such as chiral Lagrangians, where the optimal scale is somehow related to the scale of chiral symmetry breaking that separates ‘light’ degrees of freedom from ‘heavy’ ones. In the Standard Model we assume this lightest mass to be 1 meV, corresponding to the neutrino. To understand the effect of all of the previous choices, we have solved Eq. (24) numerically for different values of α, V0 and µ.TheyareshowninFigs. 1(a) and (b) as a ratio of the scale factor to the inflationary expansion, a/aI . The normalization scales are chosen to represent the minimum and maximum values. Note that here we only show results for positive α. For negative alpha, the curves simply turn in the opposite direction at the same value of s and therefore we choose not to shown them here. From the figures one can see how a higher inflationary scale leads to deviations from the inflationary expansion earlier than a lower scale. This is as expected since the source term in Eq. (24) is proportional to H and hence to V0. Similarly, a larger α has the same effect. The effect of changing the normalization scale µ has a mixed effect. If µ = MPl,theln(µ/H )-term in G(s, n) is positive for all considered values of V0. However, if µ = 1 meV, the 3 logarithmic term changes sign from negative to positive at V0 ∼ 2 × 10 GeV so that at large V0, the source term is negative. In the source term, the −8e3s term is subdominant compared to the other terms. The hypergeometric 204 D. Espriu et al. / Physics Letters B 628 (2005) 197–205

| R R R| |R 2 2 R R| Fig. 2. The scale factor relative to aI (solid line), log10( 2 I ln(a) I / ) (dotted line), log10( I ln( /µ ) I / ) (dashed line) for = −3 = 1/4 = 16 α 10 , µ 1 meV and V0 10 GeV.

function gives a negative contribution for all n values considered here. Hence, the sign of the logarithmic term is crucial, as is clear from the ﬁgures. The relative contributions to the action arising from the logarithmic term is shown for a particular case in Fig. 2, along with the corresponding numerical solution. From the ﬁgure one can see that the perturbative approximation we are making is appropriate as the action is still dominated by the Einstein term, R. Note that at the end of the corresponding calculation the relative scale factor has decreased to about 0.96. The new terms are also subdominant for the other choice of parameter values shown in Fig. 1.

4. Conclusions

In this Letter we have considered the effects of quantum corrections to gravity in a somewhat simplified setting. By considering the non-local terms that necessarily appear due to unitarity considerations in any effective Lagrangian involving massless particles, that typically dominate long-distance physics, we have found that during inflation the non-local effects are important and lead to deviations from the standard inflationary expansion. The effect is sizeable, as for typical inflationary parameter values the expansion rate is changed after only 10–20 e-foldings. The sign of the effect depends on the parameter values, and in particular on the sign of α and size of the normalization scale µ. Recalling that a change in µ is tantamount to a change in the coefficient of the local R2 term, pinning down the physical value for β (and its proper µ dependence) is very important. Taking into account that quantum corrections actually strengthen gravity at long distances, we believe that in the physically relevant situation, inflation would be slowed down or halted by the quantum corrections. This type of effects have not, to our knowledge been considered before, except for the studies presented in [10]. Although no detailed numerics are presented in this reference, the authors conclude that quantum effects slow inflation. Unfortunately the two approaches could hardly be more different and hence comparison is hopelessly difficult. It would be nice to make a clear contact between the two approaches. In doing the calculations, we have done a number approximations and simplifications in order to see whether the quantum effects can be important. As this has proven to be so in the toy model considered here, the effects of the non-local terms need to be studied more carefully in a more realistic model. We certainly believe that this issue deserves further studies. D. Espriu et al. / Physics Letters B 628 (2005) 197–205 205

Acknowledgements

The ﬁnancial support of EUROGRID and ENRAGE European Networks is gratefully acknowledged. D.E. would like to thank J. Garriga, A. Maroto, A. Roura and E. Verdaguer for discussions. The work of E.C.V. is supported by EPEAEK II in the framework of the grant PYTHAGORAS II—University Research Groups Support (co-ﬁnanced 75% by EU funds and 25% by National funds). E.C.V. would like to thank J. Russo and S. Odintsov for useful correspondences. The work of D.E. is supported by project FPA2004-04582.

References

[1] G. ’t Hooft, M. Veltman, Ann. Inst. Henri Poincaré, A 20 (1974) 69; S. Deser, P. van Nieuwenhuizen, Phys. Rev. D 10 (1974) 401; S. Deser, P. van Nieuwenhuizen, Phys. Rev. D 10 (1974) 411; S. Deser, H.-S. Tsao, P. van Nieuwenhuizen, Phys. Rev. D 10 (1974) 3337. [2] See e.g., S. Weinberg, Physica A 96 (1979) 327; J. Gasser, H. Leutwyler, Ann. Phys. 158 (1984) 142, and references therein. [3] C.P. Burgess, Living Rev. Relativ. 7 (2004) 5. [4] J.F. Donoghue, Phys. Rev. Lett. 72 (1994) 2996; J.F. Donoghue, Phys. Rev. D 50 (1994) 3874; H.W. Hamber, S. Liu, Phys. Lett. B 357 (1995) 51; A.A. Akkhundov, S. Bellucci, A. Shiekh, Phys. Lett. B 395 (1997) 16; I.B. Khriplovich, G.G. Kirilin, J. Exp. Theor. Phys. 95 (2002) 981; I.B. Khriplovich, G.G. Kirilin, J. Exp. Theor. Phys. 98 (2004) 1063; N.E.J. Bjerrum-Bohr, J.F. Donoghue, B. Holstein, Phys. Rev. D 67 (2003) 084033. [5] J.F. Donoghue, T. Torma, Phys. Rev. D 54 (1996) 1963; J.F. Donoghue, B. Holstein, Phys. Rev. Lett. 93 (2004) 201602. [6] N.D. Birrell, P.C.W. Davies, Quantum Fields in Curved Space, Cambridge Univ. Press, 1982. [7] R.D. Jordan, Phys. Rev. D 33 (1986) 444; E. Calzetta, B.L. Hu, Phys. Rev. D 35 (1987) 495. [8] A. Campos, E. Verdaguer, Phys. Rev. D 49 (1993) 1861. [9] A. Dobado, A. Maroto, Class. Quantum Grav. 16 (1999) 4057. [10] N.C. Tsamis, R.P. Woodard, Phys. Rev. D 54 (1996) 2621; J. Iliopoulos, T.N. Tomaras, N.C. Tsamis, R.P. Woodard, Nucl. Phys. B 534 (1998) 419. Physics Letters B 628 (2005) 206–210 www.elsevier.com/locate/physletb

Holographic dark energy and cosmic coincidence

Diego Pavón a, Winfried Zimdahl b

a Departamento de Física, Universidad Autónoma de Barcelona, 08193 Bellaterra (Barcelona), Spain b Institut für Theoretische Physik, Universität zu Köln, D-50937 Köln, Germany Received 5 May 2005; accepted 8 August 2005 Available online 30 September 2005 Editor: L. Alvarez-Gaumé

Abstract In this Letter we demonstrate that any interaction of pressureless dark matter with holographic dark energy, whose infrared cutoff is set by the Hubble scale, implies a constant ratio of the energy densities of both components thus solving the coincidence problem. The equation of state parameter is obtained as a function of the interaction strength. For a variable degree of saturation of the holographic bound the energy density ratio becomes time dependent which is compatible with a transition from decelerated to accelerated expansion.  2005 Elsevier B.V. All rights reserved.

Nowadays there is a wide consensus among cos- tials [2]. Most of the candidates, however, suffer from mologists that the Universe has entered a phase of ac- the coincidence problem, namely: Why are the matter celerated expansion [1]. The debate is now centered on and dark energy densities of precisely the same order when the acceleration did actually begin, whether it is today? [3]. to last forever or it is just a transient episode and, above Recently, a new dark energy candidate, based not all, which is the agent behind it. Whatever the agent, in any specific field but on the holographic princi- usually called dark energy, it must possess a negative ple, was proposed [4–9]. The latter, first formulated pressure high enough to violate the strong energy con- by ’t Hooft [10] and Susskind [11], has attracted much dition. A number of dark energy candidates have been attention as a possible short cut to quantum gravity put forward, ranging from an incredibly tiny cosmo- and found interesting applications in cosmology—see, logical constant to a variety of exotic fields (scalar, e.g., [12]—and black hole growth [13]. According to tachyon, k-essence, etc.) with suitably chosen poten- this principle, the number of degrees of freedom of physical systems scales with their bounding area rather than with their volume. In this context Cohen et al. E-mail addresses: [email protected] (D. Pavón), reasoned that the dark energy should obey the afore- [email protected] (W. Zimdahl). said principle and be constrained by the infrared (IR)

0370-2693/$ – see front matter  2005 Elsevier B.V. All rights reserved. doi:10.1016/j.physletb.2005.08.134 D. Pavón, Z. Zimdahl / Physics Letters B 628 (2005) 206–210 207 cutoff [14]. In line with this suggestion, Li has ar- This is exactly Li’s conclusion. What underlies this gued that the dark energy density should satisfy the reasoning is the assumption that ρM and ρX evolve in- 2 2 2 2 bound ρX 3Mpc /L , where c is a constant and dependently. However if one realizes that the ratio of 2 = −1 the energy densities Mp (8πG) [7]. He discusses three choices for the length scale L which is supposed to provide an IR 2 ρM 1 − c cutoff. The first choice is to identify L with the Hub- r ≡ = , (1) − ρ c2 ble radius, H 1. Applying arguments from Hsu [6], X Li demonstrates that this leads to a wrong equation of should approach a constant, finite value r = r0 for state, namely that for dust. The second option is the the coincidence problem to be solved, a different in- particle horizon radius. However, this does not work terpretation is possible, which no longer relies on an either since it is impossible to obtain an accelerated independent evolution of the components. Given the expansion on this basis. Only the third choice, the unknown nature of both dark matter and dark energy identification of L with the radius of the future event there is nothing in principle against their mutual in- horizon gives the desired result, namely a sufficiently teraction (however, in order not to conflict with “fifth negative equation of state to obtain an accelerated uni- force” experiments [16] we do not consider baryonic verse. matter) to the point that assuming no interaction at Here, we point out that Li’s conclusions rely on all is not less arbitrary than assuming a coupling. the assumption of an independent evolution of the en- In fact, this possibility is receiving growing atten- ergy densities of dark energy and matter which, in tion in the literature [17–19] and appears to be com- −3 patible not only with SNIa and CMB data [20] but particular, implies a scaling ρM ∝ a of the matter even favored over non-interacting cosmologies [21]. energy density ρM with the scale factor a(t).Anyin- teraction between both components will change, how- On the other hand, the coupling should not be seen ever, this dependence. The target of this Letter is to as an entirely phenomenological approach as differ- demonstrate that as soon as an interaction is taken into ent Lagrangians have been proposed in support of the account, the first choice, the identification of L with coupling—see [22] and references therein. H −1, can simultaneously drive accelerated expansion As a consequence of their mutual interaction nei- and solve the coincidence problem. We believe that ther component conserves separately, models of late acceleration that do not solve the coin- ρ˙M + 3HρM = Q, ρ˙X + 3H(1 + w)ρX =−Q, cidence problem cannot be deemed satisfactory (see, (2) however, [15]). though the total energy density, ρ = ρM + ρX, does. Let us reconsider the argument Li used to discard Here Q denotes the interaction term, and w the equa- the identification of the IR cutoff with Hubble’s radius. tion of state parameter of the dark energy. Without loss = −1 Setting L H in the above bound and working of generality we shall describe the interaction as a de- with the equality (i.e., assuming that the holographic cay process with Q = Γρ where Γ is an arbitrary = 2 2 2 X bound is saturated) it becomes ρX 3c MP H .Com- (generally variable) decay rate. Then we may write bining the last expression with Friedmann’s equation 2 2 = + ρ˙ + 3Hρ = Γρ (3) for a spatially flat universe, 3MP H ρX ρM ,re- M M X = − 2 2 2 sults in ρM 3(1 c )MP H . Now, the argument and 2 runs as follows: the energy density ρM varies as H , ρ˙X + 3H(1 + w)ρX =−ΓρX. (4) which coincides with the dependence of ρX on H . The energy density of cold matter is known to scale Consequently, the evolution of r is governed by as ρ ∝ a−3. This corresponds to an equation of state M 1 + r Γ pM ρM , i.e., dust. Consequently, this should be the r˙ = 3Hr w + . (5) equation of state for the dark energy as well. Thus, the r 3H dark energy behaves as pressureless matter. Obviously, In the non-interacting case (Γ = 0) and for a con- pressureless matter cannot generate accelerated expan- stant equation of state parameter w this ratio scales = −1 ∝ 3w = 2 2 2 sion, which seems to rule out the choice L H . as r a . If we now assume ρX 3c MP H ,this 208 D. Pavón, Z. Zimdahl / Physics Letters B 628 (2005) 206–210 definition implies ratio between the energy densities but requires it. In a sense, the holographic dark energy with L = H −1 w ρ˙ =− c2M2 H 3 + , together with the observational fact of an accelerated X 9 P 1 + (6) 1 r expansion almost calls for an interacting model. Note where we have employed Einstein’s equation H˙ = that the interaction is essential to simultaneously solve − 3 2[ + w ] the coincidence problem and have late acceleration. 2 H 1 1+r . Inserting (6) in the left-hand side of the balance equation (4) yields a relation between There is no non-interacting limit, since in the absence the equation of state parameter w and the interaction of interaction, i.e., Q = Γ = 0, there is no accelera- rate Γ , namely, tion. Obviously, a change of r demands a correspond- 1 Γ 0 w =− 1 + . (7) ing change of c2. Within the framework discussed so r 3H far, a dynamical evolution of the energy density ratio is Γ impossible. As a way out it has been suggested again The interaction parameter 3H together with the ratio r determine the equation of state. In the absence of to replace the Hubble scale by the future event hori- interaction, i.e., for Γ = 0, we have w = 0, i.e., Li’s zon [23]. Here we shall follow a different strategy to result is recovered as a special case. For the choice admit a dynamical energy density ratio. Motivated by 2 2 2 the relation (8) in the stationary case r = r = const, ρX = 3c M H an interaction is the only way to 0 P = 2 2 2 have an equation of state different from that for dust. we retain the expression ρX 3c MP H for the dark Any decay of the dark energy component (Γ>0) energy but allow the so far constant parameter c2 to into pressureless matter is necessarily accompanied vary, i.e., c2 = c2(t). Since the precise value of c2 by an equation of state w<0. The existence of an is unknown, some time dependence of this parameter interaction has another interesting consequence. Us- cannot be excluded. Then this definition of ρX implies ing the expression (7) for Γ in (5) provides us with · ˙ = = = w (c2) r 0, i.e., r r0 const. Therefore, if the dark en- ρ˙ =−9c2M2 H 3 1 + + ρ , (10) = 2 2 2 X P 1 + r c2 X ergy is given by ρX 3c MP H and if an interaction with a pressureless component is admitted, the ratio which generalizes Eq. (6). Using now the expression r = ρM /ρX is necessarily constant, irrespective of the (10) for ρ˙X on the left-hand side of the balance equa- specific structure of the interaction. Under this condition (4), leads to tionwehave[cf.(1)] · 1 (c2) r 1 + r Γ 2 = =−3H w + . (11) c . (8) 2 + 1 + r0 c 1 r r 3H At variance with [7,9], the fact that c2 is lower than A vanishing left-hand side, i.e., c2 = const, consis- unity does not prompt any conflict with thermodynam- tently reproduces (7). Comparing the right-hand sides ics. For the case of a constant interaction parameter of Eqs. (11) and (5) yields (c2)·/c2 =−˙r/(1 + r), Γ ≡ = 3H ν const, it follows that whose solution is −3m w ν 2 + = ρ,ρM ,ρX ∝ a m = 1 + = 1 − , c (1 r) 1. (12) 1 + r0 r (9) The constant has been chosen to have the correct be- n while the scale factor obeys a ∝ t with n = 2/(3m). havior (8) for the limit r = r0 = const. We conclude = 2 2 2 Consequently, the condition for accelerated expansion that if the dark energy is given by ρX 3c MP H and 2 is w/(1 + r0)<−1/3, i.e., ν>r0/3. c is allowed to be time dependent, this time depen- Accordingly, the expression for the holographic dence must necessarily preserve the quantity c2(1+r). dark energy with the identification L = H −1 fits well The time dependence of c2 thus fixes the dynamics into the interacting dark energy concept. The Hubble of r (and vice versa). Since r is expected to decrease radius is not only the most obvious but also the sim- in the course of cosmic expansion, r<˙ 0, this is ac- plest choice. It is not only compatible with a constant companied by an increase in c2, i.e., (c2)· > 0. D. Pavón, Z. Zimdahl / Physics Letters B 628 (2005) 206–210 209

Solving (11) for the equation of state parameter w References we find 1 Γ (c2)· [1] A.G. Riess, et al., Astron. J. 116 (1998) 1009; w =− + + . S. Perlmutter, et al., Astrophys. J. 517 (1999) 565; 1 2 (13) r 3H 3Hc S. Perlmutter, et al., in: V. Lebrun, S. Basa, A. Mazure (Eds.), For (c2)· = 0 one recovers expression (7). It is ob- Proceedings of the IVth Marseille Cosmology Conference 2 Where Cosmology and Particle Physics Meet, 2003, Frontier vious, that both a decreasing r and an increasing c Group, Paris, 2004; in (13) tend to make w more negative compared with A.G. Riess, et al., Astrophys. J. 607 (2004) 665. =− + 1 Γ 2 w (1 r ) 3H from (7). A variation of the c para- [2] S. Carroll, in: W.L. Freedman (Ed.), Measuring and Modeling meter can be responsible for a change in the equation the Universe, in: Carnegie Observatory, Astrophysics Series, of state parameter w. Such a change to (more) negative vol. 2, Cambridge Univ. Press, Cambridge, 2004; T. Padmanbhan, Phys. Rep. 380 (2003) 235; values is required for the transition from decelerated to J.A.S. Lima, Braz. J. Phys. 34 (2004) 194; accelerated expansion. For a specific dynamic model V. Sahni, astro-ph/0403324; assumptions about the interaction have to be intro- V. Sahni, in: P. Brax, et al. (Eds.), Proceedings of the IAP Con- duced. This may be done, e.g., along the lines of [18, ference on the Nature of Dark Energy, Frontier Group, Paris, 19]. However, as is well known, the holographic en- 2002. [3] P.J. Steinhardt, in: V.L. Fitch, D.R. Marlow (Eds.), Critical ergy must fulfill the dominant energy condition [24] Problems in Physics, Princeton Univ. Press, Princeton, NJ, whereby it is not compatible with a phantom equation 1997. of state (w<−1). This automatically sets a constraint [4] P. Horava,ˇ D. Minic, Phys. Rev. Lett. 85 (2000) 1610; on Γ and c2. P. Horava,ˇ D. Minic, Phys. Rev. Lett. 509 (2001) 138. It is noteworthy that in allowing c2 to vary, contrary [5] K. Enqvist, M.S. Sloth, Phys. Rev. Lett. 93 (2004) 221302. [6] S.D.H. Hsu, Phys. Lett. B 594 (2004) 13. to what one may think, the infrared cutoff does not [7] M. Li, Phys. Lett. B 603 (2004) 1. necessarily change. This may be seen as follows. The [8] Q.-G. Huang, Y. Gong, JCAP 08 (2004) 006. 2 2 2 holographic bound can be written as ρX 3c Mp/L [9] Q.-G. Huang, M. Li, JCAP 08 (2004) 013. with L = H −1. Now, Li and Huang [7–9]—as well as [10] G. ’t Hooft, gr-qc/9310026. [11] L. Susskind, J. Math. Phys. (N.Y.) 36 (1995) 6377. ourselves—assume that the holographic bound is sat- [12] B. Wang, E. Abdalla, T. Osada, Phys. Rev. Lett. 85 (2000) urated (i.e., the equality sign is assumed in the above 5507; expression). Since the saturation of the bound is not at M. Cataldo, N. Cruz, S. del Campo, S. Lepe, Phys. Lett. B 509 all compelling, and the “constant” c2(t) increases with (2001) 138; expansion (as r decreases) up to reaching the constant R. Horvat, Phys. Rev. D 70 (2004) 087301; + −1 = 2 2 2 G. Geshnizjani, D.J.H. Chung, N. Afshordi, Phys. Rev. D 72 value (1 r0) , the expression ρX 3c (t)MpH , (2005) 023517. in reality, does not imply a modification of the infrared [13] P.S. Custodio, J.E. Horvath, Gen. Relativ. Gravit. 35 (2003) − cutoff, which is still L = H 1. What happens is that, 1337. as c2(t) grows, the bound gets progressively saturated [14] A.G. Cohen, D.B. Kaplan, A.E. Nelson, Phys. Rev. Lett. 82 up to full saturation when, asymptotically, c2 becomes (1999) 4971. [15] B. McInnes, astro-ph/0210321; a constant. In other words, the infrared cutoff always R.J. Scherrer, Phys. Rev. D 71 (2005) 063519. −1 remains L = H , what changes is the degree of satu- [16] B. Ratra, P.J.E. Peebles, Phys. Rev. D 37 (1988) 3406; ration of the holographic bound. K. Hagiwara, et al., Phys. Rev. D 66 (2002) 010001. In this Letter we have shown that any interac- [17] L. Amendola, Phys. Rev. D 62 (2000) 043511; G. Mangano, G. Miele, V. Pettorino, Mod. Phys. Lett. A 18 tion of a dark energy component with density ρX = 2 2 2 2 = (2003) 831; 3c MP H (and c const) with a pressureless dark S. del Campo, R. Herrera, D. Pavón, Phys. Rev. D 70 (2004) matter component necessarily implies a constant ra- 043540; tio of the energy densities of both components. The G. Farrar, P.J.E. Peebles, Astrophys. J. 604 (2004) 1; equation of state parameter w is determined by the Z.-K. Guo, R.-G. Cai, Y.-Z. Zhang, JCAP 05 (2005) 002; interaction strength. A time evolution of the energy Z.-K. Guo, Y.-Z. Zhang, Phys. Rev. D 71 (2005) 023501; R.-G. Cai, A. Wang, JCAP 0503 (2005) 002; density ratio is uniquely related to a time variation of B. Gumjudpai, T. Naskar, M. Sami, S. Tsujikawa, JCAP 06 2 the c parameter. Under this condition a decreasing ra- (2005) 007; tio ρM /ρX sends w to lower values. R. Curbelo, T. Gonzáles, I. Quirós, astro-ph/0502141. 210 D. Pavón, Z. Zimdahl / Physics Letters B 628 (2005) 206–210

[18] W. Zimdahl, D. Pavón, L.P. Chimento, Phys. Lett. B 521 [21] M. Szydłowski, astro-ph/0502034. (2001) 133. [22] S. Tsujikawa, M. Sami, Phys. Lett. B 603 (2004) 113. [19] L.P. Chimento, A.S. Jakubi, D. Pavón, W. Zimdahl, Phys. Rev. [23] B. Wang, Y. Gong, E. Abdalla, Phys. Lett. B 624 (2005) 141. D 67 (2003) 083513. [24] D. Bak, S.-J. Rey, Class. Quantum Grav. 17 (2000) L83; [20] G. Olivares, F. Atrio, D. Pavón, Phys. Rev. D 71 (2005) E.E. Flanagan, D. Marolf, R.M. Wald, Phys. Rev. D 62 (2000) 063523. 084035. Physics Letters B 628 (2005) 211–214 www.elsevier.com/locate/physletb

Survival before annihilation in Ψ decays

P. Artoisenet, J.-M. Gérard, J. Weyers

Institut de Physique Théorique, Université catholique de Louvain, B-1348 Louvain-la-Neuve, Belgium Received 9 August 2005; received in revised form 19 September 2005; accepted 21 September 2005 Available online 28 September 2005 Editor: N. Glover

Abstract We extend the simple scenario for Ψ decays suggested a few years ago. The cc¯ pair in the Ψ does not annihilate directly into three gluons but rather survives before annihilating. An interesting prediction is that a large fraction of all Ψ decays could originate from the Ψ → ηc(3π) channel which we urge experimentalists to identify. Our model solves the problem of the apparent hadronic excess in Ψ decays as well as the ρπ puzzle since, in our view, the two-body decays of the Ψ are naturally of electromagnetic origin. Further tests of this picture are proposed, e.g., J/Ψ → b1η.  2005 Elsevier B.V. All rights reserved.

1. Introduction significantly annihilate into three gluons. In this Let- ter, we update and sharpen the arguments which led The wealth of recent data from BES and CLEO has to this somewhat unconventional point of view. In our scenario, all non-electromagnetic hadronic decays of led to a welcome revival of interest in charmonium physics. The data now provide an ideal testing ground the Ψ have a simple and general explanation: survival for theoretical expectations on the decay mechanisms amplitudes. By this, we mean transition amplitudes ¯ at work in the cc¯ system. from Ψ to lower-lying states which still contain a cc The conventional picture of a strong three-gluon pair. One original point of this Letter is the proposal → + annihilation of the Ψ runs into more and more dif- that the exclusive channel Ψ ηc (3π) could account for a significant fraction (possibly more than ficulties. The so-called ρπ puzzle and hadronic excess in Ψ decays pose indeed challenging problems [1]. 1%!) of all Ψ decays. We do urge our experimental A few years ago, two of us [2] proposed a sim- colleagues to actively search for this forgotten decay ple scheme for the decays of the J/Ψ and the Ψ . mode. In particular, it was suggested that the Ψ does not In Section 2 we briefly review and motivate our point of view on cc¯ annihilation into gluons. For the Ψ , survival precedes annihilation! More precisely, E-mail address: [email protected] (J.-M. Gérard). the cc¯ pair survives by spitting out two or three non-

0370-2693/$ – see front matter  2005 Elsevier B.V. All rights reserved. doi:10.1016/j.physletb.2005.09.041 212 P. Artoisenet et al. / Physics Letters B 628 (2005) 211–214 perturbative (i.e., with energy much less than 1 GeV) [4] survival decay gluons and the lower lying pair then annihilates into → 0 three or two perturbative (i.e., with energy 1GeV) Ψ π hc. (3) gluons, depending on the quantum numbers. These It is also of the type Eq. (1) where the (2NPg),theη0 2 + 3or3+ 2 annihilation scenarii get rid of the so- in this case, mixes with the π 0. The observed rate im- called hadronic excess problem in Ψ decays. Many plies that the effective Ψ hcη0 coupling is of the same experimental tests of these ideas are possible. order as the coupling Ψ J/Ψη0. At present [3],the To lowest order, survival amplitudes are not ex- survival radiative decays together with the three on- pected to hadronize in two-body channels. It follows shell channels (Eqs. (2) and (3)) account for more than that these decays of the Ψ should result from a di- 80% of all Ψ decays. rect electromagnetic annihilation of the cc¯ pair. Tests The success of the Gell-Mann, Sharp, Wagner off- and predictions of this assertion will be discussed in shell model [5] for the decay ω → 3π, namely ω → Section 3. In particular, as already emphasized in our π + “ρ”, has led us [2] to suggest that decay modes, earlier paper, the ρπ puzzle is then simply solved. still of the type Eq. (1), To conclude this Letter we comment very briefly on ++ +− Ψ → 2π 0 + “h ” 1 , (4a) some other issues in charmonium physics. c −+ +− Ψ → η 0 + “hc” 1 (4b) might also be the source of sizeable light hadron de- 2. Strong cc¯ annihilation cay modes of the Ψ . The observed [3] and large 5π −− hadronic decay of the Ψ could already correspond to It is well known that in the J/Ψ (1 ) decays the the pattern of Eq. (4a) where the “h ” is only slightly ¯ c cc pair mainly annihilates into three perturbative glu- off-shell. It would be nice if, for this decay, experimen- ons (3g) or into a photon. The least massive channel talists could identify a two-pion invariant mass with into which the 3g can materialize is ρπ which is in- the quantum numbers 0++. deed the strongest observed hadronic two-body decay There is of course another possibility for a two-step [3] of the J/Ψ. However, there is also a significant decay pattern cc¯ survival amplitude namely J/Ψ → ηcγ . Despite the cost of emitting a photon, this decay has the same Ψ → (3NPg) + (2g), (5) ρπ −+ ++ branching ratio as the channel. where the lower cc¯ configuration (0 or 0 ) anni- Ψ ( −−) For the 1 , the survival radiative decays hilates into 2g. The only on-shell channel for this type Ψ → γ + χ η ( ++, ++, ++ −+) c or c 0 1 2 or 0 are quite of decays is important. Similarly, the dominant strong decay channels have the structure Ψ → (3π)ηc (6a)

Ψ → (2NPg) + (3g). (1) to which one may again add the least off-shell amplitude The physical picture is as simple as can be: the ex- −− ++ Ψ → 3π 1 + “χ ” 0 . (6b) cited cc¯ pair in the Ψ does not annihilate directly but c0 rather in a two-step process. By spitting out two non- Eq. (6a) is an original ingredient of this Letter. It perturbative gluons (2NPg),itfirstsurvivesinalower is a genuine survival amplitude corresponding to the −− +− cc¯ configuration (1 or 1 ) which then eventually process where the cc¯ pair in the Ψ falls to a lower con- annihilates into 3g. The decays figuration (ηc) by radiating three non-perturbative gluons which hadronize in 3π. It could easily correspond Ψ → (2π)J/Ψ, (2a) to 1% or more of all Ψ decays. Unfortunately, an ef- Ψ → ηJ/Ψ (2b) fective calculation (i.e., at the hadronic level) of this decay amplitude requires some guesswork about cou- clearly follow the pattern of Eq. (1). Particularly im- plings which leaves room for considerable uncertainty. portant from our point of view is the recently observed Indeed, little is known about the (3NPg) hadronization P. Artoisenet et al. / Physics Letters B 628 (2005) 211–214 213 while (2NPg) hadronization is directly measured from the J/Ψ,theρ(770) almost saturates the two-pion in- Ψ → Ψππ which provides 50% of the Ψ full width. variant mass, while for the Ψ it is the ρ(2150) which It is not unreasonable to expect (3NPg) hadroniza- seems to dominate. Such a strong suppression of the tion to be of the same order within a factor of 2 or low-lying vector state contributions is not surprising 3. The extra pion in the final state should not cost in a high-energy electromagnetic process. These ob- more than a factor 10 to 20. Hence, a qualitative es- servations considerably strenghten the argument that timate for a Ψ → ηc3π branching ratio of the or- Ψ → VP is dominantly an electromagnetic process. der (1–2)% follows. It is interesting to point out that If such is the case, the so-called ρπ puzzle is solved. this estimate is quite compatible with the dominant Yet, the excess observed [7] for the K∗0K¯ 0 continuum [3] hadronic decay mode Ψ → 3(π +π −)π 0 since cross section and for the 3π events near the center of + − −2 Br(ηc → 2(π π )) 10 . We beg experimentalists the Dalitz plot has to be clarified. to search for a ηc peak in this multi-pion final state. The other available data collected by CLEO [7] In summary, Eqs. (1) and (5) are our explanation and BES [10] for the 1−−0−+ channels (i.e., ωπ, φπ, of the so-called hadronic excess in Ψ decays. If true, ρη, ωη, φη, ρη, ωη and φη) are compatible with there appears to be no need whatsoever for an impor- our suggestion that they are of electromagnetic origin. tant contribution of direct Ψ annihilation into three But in view of the present experimental uncertainties, gluons. Furthermore, the substitution of one photon a more precise statement about non-electromagnetic for one gluon in Eqs. (1) and (5) allows contributions at the level of a few 10−5 is premature. At the level of 10−3,the1+−0−+ channel b π → + + 1 Ψ (2NPg) 2g γ. (7) is physically much more interesting. For the mo- This 2+2+1 pattern corresponds to on-shell radiative ment [3], it is the largest light hadronic two-body decay of the Ψ but still of the same order as Ψ → decays such as + − 2(π π ) which is obviously of electromagnetic na- + − Ψ → π π ηcγ, (8a) ture. If the b1π channel comes from the hadroniza- → → tion of a photon, then both Ψ b1(1235)η and Ψ ηηcγ (8b) 0 Ψ → h1(1170)π should have branching ratios of the −3 which could be larger than the observed Ψ → ηcγ order of 10 . Indeed, for a virtual photon, SU(3) pre- mode. dicts

b1π : b1η : h1π = 1:2:3 (9) 3. Electromagnetic cc¯ annihilation if we assume sin θP =−1/3fortheη–η mixing and ideal mixing for h . From the recently measured [7] Whether the Ψ decays following the 2+3(Eq.(1)) 1 branching ratio Br(Ψ → b π) = (6.42 ± 1.47)10−4, or 3 + 2(Eq.(5)) pattern, it seems intuitively diffi- 1 Eq. (9) leads to cult to end up with a light hadronic two-body channel. This brings us to the suggestion that these channels − Br(Ψ → b η) = (1.3 ± 0.3)10 3, (10a) are of electromagnetic origin, namely they follow from 1 0 −3 the direct hadronization of a virtual photon. If such Br Ψ → h1π = (1.9 ± 0.4)10 . (10b) is the case, the e+e− → γ ∗ → hadrons continuum These processes are certainly welcome to saturate the [6] should be consistently subtracted for all two-body ∗ theoretically well-known Ψ → γ → light hadrons branching ratios. branching ratio (∼ 1.6%). Moreover, with the 12% In the SU(3) limit for hadrons, a well-known con- rule which holds true for a pure electromagnetic in- sequence of photon hadronization is that the ratio of teraction one obtains the surprising prediction that branching ratios into neutral and charged strange states is expected to be 4 (for d-coupling). The recent data 0 Br(J/Ψ → b1η) ≈ Br J/Ψ → h1π ≈ 1% (11) [7,8] on Ψ → K∗K¯ agree very well with this expectation. Another striking difference between J/Ψ and Ψ namely of the same order as the measured J/Ψ → ρπ decay modes is observed [7,9] in the 3π channel. For branching ratio. Theoretical uncertainties in Eq. (11) 214 P. Artoisenet et al. / Physics Letters B 628 (2005) 211–214 are related to a possible energy dependence of the elec- (2) to infer that two-body decays of Ψ into light tromagnetic form factor. This dependence has been hadrons are of electromagnetic origin. proven to be non-negligible for the 1−−0−+ decay modes [11]. These simultaneously solve the so-called hadronic excess and ρπ puzzle, respectively. Elegant as this may seem, experimental confirmation is still required. 4. Comments and conclusion The explicit identification of the decays Ψ → ηc(3π), ηc(2π)γ and a measurement of the branching ratios → 0 The main point of this Letter has been to reempha- for J/Ψ b1η, h1π would be important steps in size that there is no experimental necessity for a direct this direction. strong annihilation of the Ψ into 3g. Why is this annihilation process suppressed? We can only repeat the argument given earlier [2]: the putative (or theoretical) Acknowledgements 2S+1 cc¯ states (n LJ ) are one thing, the physical states are quite another! With strong annihilation of the 1−− ground state and its first “radial excitation”, mixing is This work was supported by the Belgian Federal expected. The QCD dynamics may be such that the Office for Scientific, Technical and Cultural Affairs physical states, presumably mixtures of the putative through the Interuniversity Attraction Pole P5/27. ones, are so built up that one of them strongly annihilates into three perturbative gluons while the other does not. Mixing of the 13S and 23S states via three 1 1 References perturbative gluons has little effect on the charmonium mass spectrum, but may be crucial for the decay pattern. If this explanation is correct, one may wonder [1] M. Suzuki, Phys. Rev. D 63 (2001) 054021; about the decay patterns of the 0−+ states below the Y.F. Gu, X.H. Li, Phys. Rev. D 63 (2001) 114019, and references therein. open charm threshold. For the ηc, we do expect sur- [2] J.-M. Gérard, J. Weyers, Phys. Lett. B 462 (1999) 324. vival to be significantly more important than a direct [3] Particle Data Group, S. Eidelman, et al., Phys. Lett. B 592 two-gluon annihilation. (2004) 1. Another comment concerns the 12% rule: we do [4] CLEO Collaboration, T. Skwarnicki, hep-ex/0505050. not see any reason for this rule to be valid. Contrary [5] M. Gell-Mann, D. Sharp, W.G. Wagner, Phys. Rev. Lett. 8 ¯ (1962) 261. to the electromagnetic annihilation of the cc into a [6] P. Wang, C.Z. Yuan, X.H. Mo, D.H. Zhang, Phys. Lett. B 593 photon which is a pointlike process, neither the J/Ψ (2004) 89. annihilation into 3g nor the 2 + 3or3+ 2 patterns for [7] CLEO Collaboration, N.E. Adam, et al., Phys. Rev. Lett. 94 the Ψ are of the same nature except, possibly, in the (2005) 012005. [8] BES Collaboration, M. Ablikim, et al., Phys. Lett. B 614 mc →∞limit, but then sizeable corrections are to be (2005) 37. expected. [9] BES Collaboration, M. Ablikim, et al., hep-ex/0408047. To conclude let us repeat that the main points of this [10] BES Collaboration, M. Ablikim, et al., Phys. Rev. D 70 (2004) short Letter have been: 112003; BES Collaboration, M. Ablikim, et al., Phys. Rev. D 70 (2004) (1) to revive and make more precise a very simple 112007. [11] J.-M. Gérard, G. Lopez Castro, Phys. Lett. B 425 (1998) 365. picture of all strong and radiative decay modes of the Ψ ; Physics Letters B 628 (2005) 215–222 www.elsevier.com/locate/physletb

Gluonic charmonium resonances at BaBar and Belle?

Frank E. Close a, Philip R. Page b

a Department of Physics—Theoretical Physics, University of Oxford, 1 Keble Road, Oxford OX1 3NP, UK b Theoretical Division, MS B283, Los Alamos National Laboratory, Los Alamos, NM 87545, USA Received 9 August 2005; accepted 7 September 2005 Available online 22 September 2005 Editor: N. Glover

Abstract We confront predictions for hybrid charmonium and other gluonic excitations in the charm region with recently observed structures in the mass range above 3 GeV. The Y(4260), if resonant, is found to agree with expectations for hybrid charmonium. The possibility that other gluonic excitations may be inﬂuencing the data in this region is discussed. Published by Elsevier B.V.

PACS: 12.39.Mk; 13.25.Gv; 14.40.Gx

Keywords: Y (4260); Charmonium; Hybrid; Glueball

1. Introduction with charmonia: Y(3940) seen in ψω [9], X(3940) ∗ ¯ seen in D D [10], χc2(3930) [11] and Y(4260) [12]. In a series of papers since 1995 we have deﬁned (The inclusion of charge-conjugated reactions is im- the properties of gluonic hybrid charmonium and de- plied throughout this Letter.) Furthermore there are veloped a strategy for producing and identifying such also three prominent enhancements X in e+e− → + states [1–8]. In this Letter we compare these predic- ψ X [10], which are consistent with being the ηc,ηc tions with recently observed structures in the mass and χ0. region above 3 GeV. We shall argue that the Y(4260), In this Letter we address the question of whether if resonant, has properties consistently in line with our any of these states may signal the excitation of gluonic historical predictions. degrees of freedom in the charmonium regime. Our starting point is that four recent experiments have reported the discovery of broad states consistent (i) The Y(3940) has been supposed to be hybrid charmonium [9]: we critically assess this claim. E-mail addresses: [email protected] (F.E. Close), (ii) By contrast, the Y(4260) [12] has mass, width, [email protected] (P.R. Page). production and decay properties, all in accord with

0370-2693/$ – see front matter Published by Elsevier B.V. doi:10.1016/j.physletb.2005.09.016 216 F.E. Close, P.R. Page / Physics Letters B 628 (2005) 215–222 those that we predicted historically for hybrid charmo- state discussed in Refs. [19,20]. The suppression of nium. this state among prominent C =+charmonium states (iii) The prominent enhancements X seen in [10] may thus be consistent with its molecular versus e+e− → ψ + X [10]: we suggest that these states be simple cc¯ nature. studied further as their production may be strongly af- A natural first guess is to associate the X/Y (3940) fected by C =+glueballs predicted to occur in this with radially excited 2P charmonium. In γγ produc- range, and there are prima facie inconsistencies with tion the radial charmonium χ2(3930) has now been simply associating them with known cc¯ states. reported [11], which gives a benchmark for comparing the other novel states. The nearness of X(3940) and We open with some brief remarks about points (iii) Y(3940) to the χc2(3930) suggests that these state(s) and then (i); the main thrust of this Letter will be to are consistent with radially excited P-wave charmonia discuss in detail the evidence related to the hybrid which are predicted in that mass region. In particu- charmonium hypothesis, point (ii). lar, decays to ψω and D∗D¯ with absence of DD¯ are In pQCD the amplitude for e+e− → ψ + cc¯ is the consistent with these two states being the same and + − 3 same order as e e → ψ + gg and has led to the sug- identified as being 2P( P1). This assignment has the gestion [13] that C =+glueballs could be produced advantage that the phase-space limited ψω mode is in at a significant level. Although the coupling of such S-wave, while another possibility, 0−+, does not share states to light flavours may give them large widths, and this feature. make a simple glueball description naive, it is nonethe- It is possible that the effect of nodes in the wave- less possible that their presence may enhance the pro- functions of radially excited states could cause an ac- PC ¯ 3 duction of cc¯ states with the same J . We note that cidental suppression of DD, for example in 2P( P0) in lattice QCD gluonic activity in the 0−+ channel is and confuse the identification of excited charmonium predicted ∼ 3.6GeV[14], which is potentially degen- states; this seems unlikely if the results of Ref. [21] are erate with the mass for the ηc(2S) [15]. Note there are a guide. However, the absence of a prominent χ1 state potentially different masses obtained for the state in in the data advises caution in identifying the prominent 3 electromagnetic (γγ) and B → Kηc decays [15]; thus X/Y (3940) as solely the radially excited 2P( P1) it is possible that different production mechanisms ex- state. If one interprets e+e− → ψ + X as a measure pose the presence of non-trivial mixing between the of the cross section for X(C =+), then the meson cc¯ and gluonic sectors here. Lattice QCD also predicts pair production thresholds with C =+are opening in activity in the gluonic waves 1++,1−+,2−+ and 3++ relative S-waves in this mass region and so some of in this mass region [14]. We advocate that until care- the structure may reflect the opening of such channels ful spin-parity analysis is done, one should be cautious rather than being simply resonant. Angular distribu- about identifying these enhancements naively with cc¯ tions of, for example, DD,D¯ ∗D¯ and D∗D¯ ∗ should states. be investigated to establish if there are specific res- Indeed, there are already some potential problems onances or alternatively threshold effects driving the with the specific fits to e+e− → ψ + X in Ref. [10], enhancement. which assigns the resonance bumps on the basis of Such information already exists qualitatively and the masses of states in the PDG [16].Asaresult can help to constrain interpretations. If the 3940 MeV they identify χ0 but no prominent χ1,2, though there enhancement consists of a single state, then the ob- may be room for these states in the small fluctu- servation of significant D∗D¯ in the decay of X(3940) ations around 3.5–3.6 GeV in Fig. 1 of Ref. [10]. suggests that this state is not simply a gluonic hy- The Born cross sections containing more than two brid charmonium [1]. However, the branching ratios ∗ ¯ ∗ ¯ charged tracks are approximately for X = ηc :25.6fb; into D D : ψω need to be established; if the D D is χ0 :6.4fb;ηc :16.5fb[17] and X(3940) :10.6fb[10]. small, then hybrid charmonium may be relevant. The There is no sign of the X(3872); this state now mass also is low compared to that predicted for hybrid appears to have C =+and be consistent with 1++ charmonia which are more generally expected to be at [18].ThisJ PC was first suggested in Ref. [19] and a ∼ 4.2GeV[3,22] unless, as we discuss later, there are dynamical picture of it as a quasi-molecular D∗0D¯ 0 significant J PC dependent mass shifts. F.E. Close, P.R. Page / Physics Letters B 628 (2005) 215–222 217

While the interpretation of these states may depend 2.2. The decay modes ψσ, ψf/a0(980) appear to rather critically on first establishing their J PC, the ap- dominate pearance of a further state, Y(4260) with J PC = 1−− has properties that appear uniformly to be consistent An S-wave phase space model of the three-body with those predicted earlier for hybrid charmonium. decay ψπ+π − [12] does not appear consistent with We now assess the experimental information about this the data (Fig. 3, Ref. [12]). On the other hand, two- state. body decay, which usually dominates three-body decay, would easily explain the data, which are consis- + − tent with π π peaks at the σ and f0(980)/a0(980) masses. These mesons are the only ones in the mass 2. Experimental information region 0.3–1.0 GeV displayed [12] with C =+,as required by C-parity conservation. The mode ψKK¯ The BaBar Collaboration recently observed a new should be searched for as the strong coupling of ± +2 ± ¯ ¯ structure at 4259 8−6 MeV with a width of 88 f/a0(980) to KK should manifest itself at the KK +6 threshold if these states are important in the de- 23−4 MeV and a significance greater than 8σ [12]. The structure is known to be produced in initial state cay. radiation from e+e− collisions and hence to have + − J PC = 1−−. It is seen decaying to ψπ+π − and 2.3. Γ(Y(4260) → e e ) is much smaller than all other 1−− charmonia + − + − Γ Y(4260) → e e B Y(4260) → ψπ π + − Noting that the cross-section σ(e e → Y → + → = 5.5 ± 1.0 0.8 eV. (1) X) into final state X is proportional to Γ(Y −0.7 e+e−)B(Y → X),

There are several consequences of the experimental Γ(Y → e+e−)B(Y → hadrons) work that are worth noting, and which align them- Γ(Y → e+e−)B(Y → ψπ+π −) selves most naturally with a hybrid charmonium in- σ(e+e− → Y → hadrons) terpretation. = , (2) σ(e+e− → Y → ψπ+π −) + − 2.1. The mass coincides with the D (2420)D¯ and using Eq. (1), σ(e e → Y → hadrons) 4% × 1 + − + − threshold 14.2nb[12,16], and σ(e e → Y → ψπ π ) ≈ 50 pb [12], it follows that ¯ + − The state can couple to D1(2420)D, and related 5.5 ± 1.3eV Γ Y(4260) → e e 62 ± 15 eV, thresholds to be discussed later, in S-wave. The ¯ (3) D1(2420)D thresholds are at 4287 MeV 0 ¯ 0 ± ∓ (D1(2420) D ) and 4296 MeV (D1(2420) D ) [15]. using that B(Y → hadrons) is very near to unity. (The At an S-wave threshold, re-scattering effects may drive lower bound on the width is obtained from Eq. (1).) the ψπ+π − signal. A resonance above 4.26 GeV, This e+e− width is at least a factor of 4 smaller ¯ −− which couples strongly to D1D, can through re- than that of the established 1 charmonium with ¯ scattering give an enhancement in ψππ at the D1D the smallest width, the ψ(3770) [16]. However, un- threshold (for example see Ref. [23]), in which case mixed radially excited D-wave (2D) cc¯ states can the true mass of Y(4260) could be O(100) MeV above have widths consistent with Eq. (3), as their widths 4.26 GeV. It is even possible for such a phenomenon to in potential models are typically 64 times lower than occur without any resonance. Therefore it is important 3S states [24]. The experimental width only just to establish that Y(4260) is resonant and not a thresh- overlaps with that in a four-quark interpretation of old effect. With these caveats, we shall now analyze Y(4260), which predicted that it should be 50–500 eV for the case where Y(4260) is resonant. [25]. 218 F.E. Close, P.R. Page / Physics Letters B 628 (2005) 215–222

2.4. Γ(Y(4260) → ψπ+π −) is much larger than all exotic J PC states for both light and heavy flavours, 1−− charmonia whose masses and spin dependent mass splittings are now being confirmed by lattice computations. The Using Eqs. (1) and (3) it is immediate that detailed production and decay signatures for hybrid + − states are still largely beyond the bounds of lattice B Y(4260) → ψπ π 8.8%, QCD, and for these we are still restricted to the model. + − Γ Y(4260) → J/ψπ π 7.7 ± 2.1MeV. (4) In due course we anticipate that these results will be tested by the lattice. In the meantime they are ar- → + − This is much larger than Γ(ψ(3770) ψπ π ) guably the nearest we have and it is on the basis which is in the 80–90 keV range [16,26].Itisalso of their implications that we proceed to examine the much larger than Γ(ψ(4040); ψ(4160); ψ(4415) → + − Y(4260). ψπ π ),asisnowshown.TheY(4260) is seen The eight low-lying hybrid charmonium states in the BaBar experiment and ψ(4040), ψ(4160) and (ccg¯ ) were predicted in the flux-tube model to oc- ψ(4415) not [12]. Using a ball-park estimate that the cur at 4.1–4.2GeV[3], and in UKQCD’s quenched error bars can mask the latter resonances if their cross- lattice QCD calculation with infinitely heavy quarks section is four times smaller than Y(4260) (Fig. 1, to be 4.04 ± 0.03 GeV (with unquenching estimated + − → → + − Ref. [12]), and noting that σ(e e X ψπ π ) to raise the mass by 0.15 GeV) [22]. The splittings of → into intermediate state X is proportional to Γ(X ccg¯ from the above spin-average was predicted model- + − B → + − e e ) (X ψπ π ),wehave dependently for long distance (Thomas precession) + − B ψ → ψπ π interactions in the flux-tube model [28], and for short distance (vector-one-gluon-exchange) interactions in → + − B → + − −− Γ(Y e e ) (Y ψπ π ) cavity QCD [2,7].Forthe1 state the long and + − , (5) 4Γ(ψ → e e ) short distance splittings are respectively 0 MeV and where ψ denotes any of ψ(4040), ψ(4160) or 60 MeV. ψ(4415). Together with Eq. (1) this can be used to These mass predictions are very much in accord calculate a bound on the branching ratio of each ψ. with the Y(4260) and somewhat removed from those Translating into widths [16] for X/Y (3940). A lattice inspired flux-tube model showed that the + − Γ ψ(4040); ψ(4160); ψ(4415) → ψπ π decays of hybrid mesons, at least with exotic J PC,are 100 ± 30, 140 ± 60, 130 ± 60 keV. (6) suppressed to pairs of ground state 1S conventional mesons [27,29]. This was extended to all J PC,for The simplest interpretation of this is that the ψππ light or heavy flavours in Ref. [1]. A similar selec- is not due to disconnected ψgg diagrams but instead tion rule was found in constituent gluon models [30] involves some strong affinity. This could be due to a and later in QCD sum rules [31], and their common four-quark interpretation [25] or due to intrinsic glu- quark model origin is now understood [32].Itwasfur- onic excitation in the initial state, as will be discussed ther shown that these selection rules for light flavoured below. hybrids are only approximate, but that they become very strong for cc¯ [1,2]. This implied that decays ¯ ¯ ∗ ¯ ∗ ∗ ¯ ∗ into DD, DsDs , D D and Ds Ds are suppressed ∗ ¯ ∗ ¯ ∗∗ ¯ 3. Y(4260) as hybrid charmonium whereas D D and Ds Ds are small, and D D,if above threshold, would dominate. (P-wave charmo- Lattice QCD inspired the flux-tube model of me- nia are denoted by D∗∗.) As ccg¯ is predicted around sons [27], which has been used to predict observ- the vicinity of D∗∗D¯ threshold, the opportunity for ables that, at the time, were beyond the bounds of anomalous branching ratios in these different classes lattice computation but which have subsequently been was proposed as a sharp signature [1,3]. (To the best largely confirmed and extended by these more funda- of our knowledge Ref. [1] was the first paper to pro- mental techniques. In particular and of relevance to pose such a distinctive signature for hybrid charmo- the present discussion, we cite the early prediction of nium.) F.E. Close, P.R. Page / Physics Letters B 628 (2005) 215–222 219

It has become increasingly clear recently that there 4. Implications of hybrid charmonium is an affinity for states that couple in S-wave to hadrons, to be attracted to the threshold for such chan- There are several of the theoretical expectations al- nels [33]. The hybrid candidate 1−− appearing at the ready given for ccg¯ that are born out by Y(4260): ¯ S-wave D1(2420)D is thus interesting. (1) Its mass is tantalizingly close to the prediction for More recently the signatures for hybrid charmo- the lightest hybrid charmonia; (2) The expectation that + − nia were expanded to note the critical region around the e e width should be smaller than for S-wave cc¯ D∗∗D¯ threshold as a divide between narrow states is consistent with Eq. (3); (3) The predicted affinity of ∗∗ ¯ with sizable branching ratio into cc¯ + light hadrons hybrids to D D could be related to the appearance of ∗∗ ¯ and those above where the anomalous branching ra- the state near the D D threshold. The formation of ∗∗ ¯ tios would be the characteristic feature [5,7,8].Here D D at rest may lead to significant rescattering into + − widths of order 10 MeV were anticipated around the ψπ π , which would feed the large signal (Eq. (4)). −− threshold. It was suggested to look in e+e− annihila- Quenched lattice QCD indicates that the ccg¯ 1 , −+ ++ +− tion in the region immediately above charm threshold (0, 1, 2) are less massive than 1 , (0, 1, 2) for state(s) showing such anomalous branching ratios [37]. The spin splitting for this lower set of hybrids −+ −+ −− [8]. The leptonic couplings to e+e−,µ+µ− and τ +τ − in quenched lattice NRQCD is 0 < 1 < 1 < −+ ¯ were expected to be suppressed [34] (smaller than ra- 2 [38], at least for bbg. This agrees with the order- dial S-wave cc¯ but larger than D-wave cc¯, but with ing found in the model-dependent calculations for qqg¯ ¯ some inhibition due to the fact that in hybrid vector [39] in the specific case of ccg¯ [2,7,28].Forbbg lat- mesons spins are coupled to the S = 0, whose cou- tice QCD predict substantial splittings ∼ 100 MeV or pling to the photon is disfavoured [8]). Even stronger greater [38], which become even larger in the model- suppression obtains for γγ couplings [35]. dependent calculations for ccg¯ [2,7,28]. Theory hence Small conventional charmonium mixing with ccg¯ strongly indicates that if Y(4260) is ccg¯ , and the split- or a glueball is expected. The latter is due to the tings are not due to mixing or coupled channel effects, −+ −+ penalty incurrent by the creation of a cc¯ pair, and then the J PC exotic 1 and non-exotic 0 ccg¯ are ∗∗ ¯ the former is due to the heaviness of the charm below D D threshold, making them narrow by virtue −+ quark which enable a Born–Oppenheimer approxima- of the selection rules. The 1 decay modes [5] and tion, separating conventional and hybrid charmonia by branching ratios [8] have extensively been discussed. ¯ virtue of their orthogonal gluonic wave functions [7]. The nearness of Y(4260) to the D1(2420)D thresh- −− ¯ However, for 1 hybrids, there is the possibility of old, and to the D1D threshold, with the broad D1 substantial mixing with the radially and orbitally ex- found at a mass of ∼ 2427 MeV and width ∼ 384 MeV cited cc¯ if mass degenerate: it was noted that hybrid [40], indicate that these states are formed at rest. Also, charmonia with 1−− can in principle mix [4,36] with these are the lowest open charm thresholds that can −− ¯ ∗ radially excited cc¯ states and a specific example was couple to 1 in S-wave (together with D0D , where discussed of what would occur if the hybrid mass is the D0 mass ∼ 2308 MeV and width ∼ 276 MeV ∼ 4.1GeV[4]. [40]). Flux-tube model predictions are that the D-wave −− + + ∗∗ The discovery of Y(4260) signals degrees of free- couplings of 1 ccg¯ to the 1 and 2 D are small dom beyond conventional cc¯. This is because the only [1,2,6]; and there is disagreement between various ver- such 1−− expected up to 4.4 GeV are 1S,2S,1D,3S, sions of the model on whether the S-wave couplings + 2D and 4S [24], and there are already established to the two 1 states are large. If these couplings are in candidates for these states. Thus even in the case of fact substantial, the nearness of Y(4260) to the thresh- mixing, the existence of Y(4260) hints that more than olds may not be coincidental, because coupled channel conventional cc¯ is needed. effects could shift the mass of the states nearer to a We now consider tests and implications of the idea threshold that it strongly couples to; and it would expe- that Y(4260) signals the onset of hybrid charmonium. rience a corresponding enhancement in its wave func- We describe these below, compare with the unmixed tion. The broadness of Y(4260) also implies that its ¯ ¯ ¯ ∗ hybrid charmonium hypothesis and propose further decay to D1(2420)D,D1D and D0(2308)D which tests. feed down to D∗Dπ¯ and DDπ¯ [41] would be allowed 220 F.E. Close, P.R. Page / Physics Letters B 628 (2005) 215–222 by phase space and should be searched for to ascertain thresholds then this could be investigated in ψKK¯ .If a significant coupling to D∗∗. Fig. 1 of Ref. [12] shows further structure beyond the ∗∗ ¯ Flux-tube model width predictions for other charm Y(4260) enhancement this may be due to the Ds Ds ∗ ¯ ¯ ¯ + − modes are 1–8 MeV for D D [6], with DD, DsDs , rescattering in ψf0(980) which yields a ψπ π sig- D∗D¯ ∗ and D∗D¯ ∗ even more suppressed. Thus a small nal. ¯ s¯ s DD and DsDs mode could single out the hybrid in- The nearness and S-wave coupling of Y(4260) to 0 0 terpretation. The hybrid decay pattern is very different specifically the D1(2420) D threshold, and also the ± ∓ from the css¯ c¯ four-quark interpretation for Y(4260) D1(2420) D threshold, together with the sizable ¯ which decays predominantly in DsDs [25]. Thus a width of the state, lead to the expectation that mix- search for the latter channel, or limit on its coupling, ing with both thresholds will be similar, and that the could be a significant discriminator for the nature of charmonium nature of Y(4260) should imply that it is the Y(4260). dominantly I = 0, as will be assumed in the remainder of this discussion. This can be established by search- ing for the isospin violating decays ψπ0 and π +π −. 5. Experimental searches and production If either the model dependent spin splittings are a guide, or if the states are attracted towards S-wave It is possible that Y(4260) is not a resonance, thresholds, then we would expect that the Y(4260) as ¯ ¯ −+ but reflects the opening of the D1(2420)D,D1D and vector hybrid ψg states will imply that the 0 ηcg and ¯ ∗ −+ D0(2308)D thresholds. The reason is that this is the exotic 1 will be at or below 4.3 GeV. The analyses lowest energy at which open charm thresholds can of Refs. [6,8] then imply the following: couple to e+e− (1−−) in S-wave. Thus there is the ∗ possibility that BaBar [12] is observing the process (i) Any decays into the disfavoured D D¯ channel + − ∗∗ ¯ + − + − −+ e e → D D → ψπ π , where ψπ π is pro- will be in the ratios 1 : ψg : ηcg = 1 : 2 : 4 apart from duced by rescattering. This could occur without a res- phase space effects. −+ ¯ ¯ onance, or with a resonance, as follows. The essential (ii) Γ(1 → D1D) > Γ (ψg → D1D). ¯ ingredients are (i) the presence of a non-resonant back- (iii) ηcg → D0D may be significant due to the ground (in this case ψππ); (ii) a resonance which broad width of the D0. Even if this is kinematically strongly couples to a channel (in this case D∗∗D¯ ); suppressed, significant re-scattering may result into (iii) rescattering between the latter channel and the ψω [20,42]. Hence the possibility that X/Y (3940) PC background. An example involving light quarks and an contains ηcg may be realized; establishing the J of earlier claimed signal for a hybrid meson (in that case the 3940 MeV structure(s) is thus important. −+ the 1 ) was discussed in Ref. [23]; a specific model (iv) ηcg → ηcππ may be anticipated [8] and of rescattering involving charmonium was applied to ηcg → ηcf0(980) may be a significant contributor. To ¯ the X(3872) in Ref. [20]. Hence the resonant nature of this end, a search for ηcg(3940) → ηcKK is also mer- Y(4260) should be confirmed. If a similar rescattering ited. The ψ{ω,φ} mode may be experimentally most ¯ ¯ ¯ ∗ ¯ ∗ effect occurs at the Ds1Ds , Ds1Ds , Ds0Ds and Ds2Ds tractable.

Table 1 Possible two-body hadronic decay modes of Y(4260), assuming that it has I = 0. Open charm modes may be suppressed by a selection rule discussed in the text. Hidden charm modes to low-lying charmonia are listed. For these modes, the charmonia tend to have the same C as that of the parent ccg¯ , since, barring non-perturbative effects, two gluons C =+are emitted in the lowest order process [5]. Electromagnetic modes like {ηc,ηc(2S),χc{0,1,2},hc,X(3872)}γ are expected to be small. Light hadron modes are restricted to hadrons up to the φ mass Open charm Hidden charm Light hadrons ¯ ¯ () DD; Ds Ds ηc{ω,φ,h1} η {ω,φ}; ρπ; a0(980)ρ ∗ ¯ ∗ ¯ { ()}{}{ } D D, Ds Ds J/ψ σ,f0(980), η σ,f0(980) ω,φ ∗ ¯ ∗ ∗ ¯ ∗ { }{∗} ¯ { ∗}¯ D D ; Ds Ds ψ(2S) σ,η K,K K; κ,K κ ¯ ¯ { } ∗ ¯ ∗; ¯ ¯ ¯ D1(2420)D; D1Dχc0ω; hc σ,η K K pp,pn, nn F.E. Close, P.R. Page / Physics Letters B 628 (2005) 215–222 221

It is singular that apart from the ψ(2S), no other brid charmonium assignment is favored [30,44].We states are visible in the BaBar data [12] until the thank S.J. Brodsky, J.J. Dudek and S.-L. Zhu for dis- Y(4260). Given that its e+e− coupling is small, this cussions. observation suggests that it is the special affinity of this state for the ψππ channel that gives its visibil- ity (Eqs. (3)–(4)). The possible decays of Y(4260) are References listed in Table 1. A further test for the ψg interpretation of Y(4260) would be that ψg →{σ,η}hc could be [1] F.E. Close, P.R. Page, Nucl. Phys. B 443 (1995) 233. significant. This would arise if the decay was driven [2] P.R. Page, PhD thesis, University of Oxford, 1995, unpub- by flux-tube deexcitation, with quark spin conserva- lished. [3] T. Barnes, F.E. Close, E.S. Swanson, Phys. Rev. D 52 (1995) tion. Such a mechanism is expected in the model [1], 5242. though its strength is currently unquantifiable; there [4] F.E. Close, P.R. Page, Phys. Lett. B 366 (1996) 323. are suggestions from lattice QCD that such deexcita- [5] F.E. Close, I. Dunietz, P.R. Page, S. Veseli, H. Yamamoto, tion modes may be significant for heavy flavours [43]. Phys. Rev. D 57 (1998) 5653. This particular mode could be detected by the isospin [6] P.R. Page, E.S. Swanson, A.P. Szczepaniak, Phys. Rev. D 59 (1999) 034016. violating mode of the hc → ψπ. + − [7] P.R. Page, in: Proceedings of Workshop on a Low-Energy A search for ψg → ψπ π at Belle and BaBar p¯ Storage Ring, Chicago, IL, 3–5 August 2000, p. 55, hep- in B → Kψg should be fruitful [5], even though the ph/0107016. + − small e e coupling of ψg suggests that its wave [8] F.E. Close, S. Godfrey, Phys. Lett. B 574 (2003) 210. function at the origin is tiny. Production in pp¯ an- [9] Belle Collaboration, S.K. Choi, et al., Phys. Rev. Lett. 94 ¯ → (2005) 182002. nihilation in the formation process pp ψg is also [10] Belle Collaboration, K. Abe, et al., hep-ex/0507019. feasible at future colliders. [11] Belle Collaboration, K. Abe, et al., hep-ex/0507033. If the Y(4260) is ψg, then the radiative transition [12] BaBar Collaboration, B. Aubert, et al., hep-ex/0506081. −+ ψg → 1 γ , though tiny, may reveal the exotic hybrid [13] S.J. Brodsky, A. Goldhaber, J. Lee, Phys. Rev. Lett. 91 (2003) −+ 112001; charmonium [4]. The decay 1 → χ{0,1,2}σ with → S. Dulat, et al., Phys. Lett. B 594 (2004) 118; σ (ππ)S should be an excellent search mode [5] F.E. Close, Q. Zhao, Phys. Rev. D 71 (2005) 094022. and is predicted to be large [43], although the ψ{ω,φ} [14] UKQCD Collaboration, G. Bali, et al., Phys. Lett. B 309 (1993) mode may be most tractable experimentally. However, 378; given the small e+e− width of the Y(4260), this may C. Morningstar, M.J. Peardon, Phys. Rev. D 60 (1999) 034509. [15] Particle Data Group, http://pdg.lbl.gov/2005/listings/mxxx require a dedicated search at BES or CLEOc.Anex- + − → + comb.html. citing possibility is that e e ψ X may reveal [16] Particle Data Group, S. Eidelman, et al., Phys. Lett. B 592 −+ 1 the 1 in the X around or above 4 GeV. (2004) 1. [17] P. Pakhlov, hep-ex/0412041; Belle Collaboration, K. Abe, et al., Phys. Rev. D 70 (2004) Note added in proof 071102. [18] CDF Collaboration, Measurement of the dipion mass spectrum → + − After this work was completed, an enhancement in X(3872) J/ψπ π decays, http://www-cdf.fnal.gov/ physics/new/bottom/050324.blessed.X/. consistent with Y(4260) was observed in B− → + − − [19] F.E. Close, P.R. Page, Phys. Lett. B 578 (2003) 119; J/ψπ π K [45]. Such a search was suggested ear- See also N.A. Tornqvist, hep-ph/0308277. lier in this Letter. [20] E. Swanson, Phys. Lett. B 588 (2004) 189. [21] T. Barnes, S. Godfrey, E. Swanson, hep-ph/0505002. [22] P. Lacock, C. Michael, P. Boyle, P. Rowland, Phys. Lett. B 401 Acknowledgements (1997) 308. [23] A. Donnachie, P.R. Page, Phys. Rev. D 58 (1998) 114012. [24] K. Heikkilä, N.A. Tornqvist, S. Ono, Phys. Rev. D 29 (1984) While this work was in preparation, a discussion of 110. the interpretations for Y(4260) suggested that the hy- [25] L. Maiani, F. Piccinini, A.D. Polosa, V. Riquer, hep-ph/ 0507062. [26] BES Collaboration, J.Z. Bai, et al., Phys. Lett. B 605 (2005) 1 We thank J.J. Dudek for this observation. 63. 222 F.E. Close, P.R. Page / Physics Letters B 628 (2005) 215–222

[27] N. Isgur, J. Paton, Phys. Rev. D 31 (1985) 2910. T. Manke, et al., Phys. Rev. D 57 (1998) 3829. [28] J. Merlin, J. Paton, Phys. Rev. D 35 (1987) 1668. [38] I.T. Drummond, et al., Phys. Lett. B 478 (2000) 151; [29] N. Isgur, R. Kokoski, J. Paton, Phys. Rev. Lett. 54 (1985) 869. T. Manke, et al., Nucl. Phys. B (Proc. Suppl.) 86 (2000) 397. [30] E. Kou, O. Pene, hep-ph/0507119. [39] T. Barnes, F.E. Close, Phys. Lett. B 116 (1982) 365; [31] S.-L. Zhu, Phys. Rev. D 60 (1999) 014008, Table VII. M. Chanowitz, S. Sharpe, Nucl. Phys. B 222 (1983) 211; [32] P.R. Page, Phys. Lett. B 402 (1997) 183. T. Barnes, F.E. Close, F. de Viron, Nucl. Phys. B 224 (1983) [33] F.E. Close, in: Proceedings of Hadron03, hep-ph/0311087; 241. F.E. Close, in: Proceedings of ICHEP04, hep-ph/0411396. [40] Belle Collaboration, K. Abe, et al., Phys. Rev. D 69 (2004) [34] S. Ono, et al., Z. Phys. C 26 (1984) 307; 112002. S. Ono, et al., Phys. Rev. D 34 (1986) 186. [41] F.E. Close, E.S. Swanson, hep-ph/0505206. [35] P.R. Page, Nucl. Phys. B 495 (1997) 268. [42] F.E. Close, E. Swanson, C. Thomas, in preparation. [36] S.B. Gerasimov, in: Proceedings of 11th International Confer- [43] UKQCD Collaboration, C. McNeile, C. Michael, P. Pennanen, ence on Problems of Quantum Field Theory, Dubna, Russia, Phys. Rev. D 65 (2002) 094505. 13–17 July 1998, hep-ph/9812509. [44] S.-L. Zhu, hep-ph/0507025. [37] J.J. Juge, J. Kuti, C.J. Morningstar, Nucl. Phys. B (Proc. [45] BaBar Collaboration, B. Aubert, et al., hep-ex/0507090. Suppl.) 83 (2000) 304; Physics Letters B 628 (2005) 223–227 www.elsevier.com/locate/physletb

Comment on evidence for new interference phenomena in the decay D+ → K−π +µ+ν

B. Ananthanarayan a,K.Shivarajb

a Centre for High Energy Physics, Indian Institute of Science, Bangalore 560 012, India b Department of Physics, Indian Institute of Science, Bangalore 560 012, India Received 18 August 2005; accepted 22 September 2005 Available online 29 September 2005 Editor: M. Cveticˇ

Abstract The experimental determination of low energy πK scattering phase shifts would assist in determining scattering lengths as well as low energy constants of chiral perturbation theory for which sum rules have been constructed. The FOCUS Collaboration has presented evidence for interference phenomena from their analysis of Dl4 decays based on decay amplitudes suitable for ∗ a cascade decay D → K → Kπ. We point out that if the well-known full ﬁve body kinematics are taken into account, πK scattering phases may be extracted. We also point out that other distributions considered in the context of Kl4 decays can be applied to charm meson decays to provide constraints on violation of |I|=1/2 rule and T-violation.  2005 Elsevier B.V. All rights reserved.

Keywords: Chiral perturbation theory; Semileptonic decay of charm mesons; πK scattering phase shifts

1. Chiral perturbation theory [1] as the low energy absence of pion targets. One important source of in- effective theory of the standard model is now in a re- formation comes from the rare kaon decay Kl4.Using markably mature phase. Several processes have been well-known techniques [3,4] one can extract the phase computed to two-loop accuracy and remarkable pre- difference for pion scattering of the iso-scalar S-wave 0 − 1 dictions exist for low energy processes. One of the and iso-triplet P-wave phase shifts δ0 δ1 from an important processes that has been studied is that of analysis of the angular distributions, where the ﬁnal ππ scattering, for a recent comprehensive review, see state or Watson theorem relates the phase of the decay Ref. [2]. It has been traditionally difﬁcult to study this form factors to the scattering phase shifts. Recently experimentally in the low-energy regime due to the the E865 Collaboration [5] at Brookhaven National Laboratory has carried out the analysis of data from a high statistics experiments which has brought about E-mail address: [email protected] (B. Ananthanarayan). a remarkable marriage between experiment and the-

0370-2693/$ – see front matter  2005 Elsevier B.V. All rights reserved. doi:10.1016/j.physletb.2005.09.047 224 B. Ananthanarayan, K. Shivaraj / Physics Letters B 628 (2005) 223–227 ory. There are preliminary measurements also from be exploited to determine the phase shifts of inter- NA48 for the semi-leptonic decays, Ref. [6]. Scatter- est. The data so obtained could be in conjunction with ing lengths will also be measured at high precision the recent accurate solutions to the Roy–Steiner equa- by the CERN experiment DIRAC from the lifetime tions [14]. of the pionium atom, and from the enormous statistics gathered by the NA48 Collaboration by employing the recent proposal of Cabibbo, see Refs. [7,8], 3. In this paragraph, we briefly recall the main of analyzing the cusp structure of the invariant mass features of the formalism of Ref. [13]. The process + of the dipion system produced in the reaction K → considered is π +π +π −. D(p1) → K(p2) + π(p3) + l(k) + ν(k ). (1) The authors give an explicit form for the 5-fold differ- 2. Chiral perturbation theory that involves the s- ential width quark degree of freedom is yet to be tested at a corresponding degree of precision. One sensitive labo- d5Γ ratory is the pion–kaon scattering amplitude [9].For 2 ∗ dq ds23 d cos θdχdcos θ recent studies on the comparison between the ampli- √ 2 | |2 2 tudes evaluated in chiral perturbation theory, and phe- GF Vcs q a2X = liHi, (2) nomenological determination, see Refs. [10,11]. It has ( π)6m3 96 2 1 i been pointed in these that it is desirable to have high = + precision phase shift determinations so that accurate where q k k , m1 is the mass of the√ D meson, 2 2 predictions for scattering lengths can be made. The s23 = (p2 + p3) , a2 = 4|p2| /s23, X = s23|p1|, θ is search for an experimental system where these phase the angle between the charged lepton and the D me- ∗ shifts can be measured, leads us naturally to an analog son in the dilepton center of mass frame, θ is the of the Kl4 decay in the charm-meson system, which angle between the K meson and the D meson in the 1 is the decay Dl4. It is clear that one might be able to dimeson center of mass frame, χ is the angle between extract information on the πK scattering amplitude as the lepton and meson decay planes, and the sum over well due to the final state or Watson theorem. What i runs over the symbols U, L, T , V , P , F , I , N, A, is required is an analog for the technique used in the with the Hi being the helicity structure functions, and case of Kl4 decay for the Dl4 decay. Note that in the the li given as follows for the case of massless charged Dl4 case, the dimeson pair in the final state is com- leptons: posed of unequal mass particles and that the iso-spin of 3 2 3 2 the system is different from that in the corresponding lU = 1 + cos θ ,lL = sin θ, ππ system. At leading order in the weak interaction, 8 4 one obtains only |I|=1/2 amplitudes and the sys- 3 2 lT = sin θ cos(2χ), tem yields information on the phase shift difference 4 δ1/2 − δ1/2. A comprehensive and self-contained ac- 3 3 0 1 l =− sin2 θ sin(2χ), l = cos θ, count of this is to be found in Ref. [13]. Note that for V 4 P 4 the moment, the analog of the pionium system for the 3 lF = √ sin(2θ)sin χ, πK atom is only in the planning stage, and determi- 2 2 nation of the πK scattering lengths from an analog of + → − + + 3 3 the proposal of Cabibbo from, say D K π π lI =− √ sin(2θ)cos χ, lN = √ sin θ sin χ, or K¯ 0π +π 0 would not be feasible due to limited sta- 2 2 2 tistics. As a result, it is imperative that the Dl4 decay 3 lA =−√ sin θ cos χ. 2

1 Indirect sources of information include pion production from We do not explicitly list all the Hi except for a few for scattering of kaons off nuclei, e.g., [12]. purposes of illustration (see below). B. Ananthanarayan, K. Shivaraj / Physics Letters B 628 (2005) 223–227 225

Writing the hadronic matrix element as result has been established. The decay amplitude has been adopted from Ref. [16] which considers the three p ,p |A + V |p 2 3 µ µ 1 body final state kinematics. In particular, the process 1 considered in [16] is the reaction of the type = f(p + p ) + g(p − p ) + rq 2 3 µ 2 3 µ µ m1 ∗ ∗ D(p ) → K p + l(k) + ν(k ) (3) ih 1 + qν(p + p )α(p − p )β , 2 µναβ 2 3 2 3 for which the hadronic part of the amplitude is written m1 down in terms of the matrix element where the form factors f , g, r, and h are in general functions of s , q2 and θ ∗ (r makes no contribution in ∗ ∗ + = ∗α 23 K p Aµ Vµ D(p1) 2 Tµα, (4) the case of massless charged leptons). The Hi can now be expressed in terms of the form factors. For instance, where √ = A + A + A X a2s23 Tµα F1 gµα F2 p1µp1α F3 qµp1α HF(A) = 2 2 2 V ρ ∗σ m 2q m + iF µαρσ p p , (5) 1 1 1 2 2 ∗ ∗ m − m and q = (p − p ) is the momentum transfer. Note × Im(Re) h Xf + gX 2 3 µ 1 µ s that F A contributes only in the case of decays with 23 3 2 2 massive charged leptons. The differential decay rates √ m − s23 − q ∗ ∗ + g a 1 cos θ sin θ , are expressed in terms of helicity amplitudes which 2 2 evaluate to Xa2s23 ∗ 2 ∗ HV =− Im h g sin θ . 1 4 = 2 − ∗2 − 2 A + 2 2 A m1 H0 M1 M q F1 2M1 p F2 , 2M∗ q2 It was shown first by Pais and Treiman that the = A ± V choice of variables made by Cabibbo and Maksymow- H± F1 M1pF , icz leads to the simple decomposition, Eq. (2) of the 5- where p is the momentum of the K∗ in the D rest sys- ∗ ∗ fold differential width and thus makes the determina- tem, M1 and M are the masses of the D and the K , tion of physical observables amenable. Furthermore, respectively. In Ref. [15],2 the expressions are pro- by parametrizing the functions f , g, h and identifying vided for the massless lepton case, for which case the their phases with πK phase shifts (a consequence of differential decay rate is written down as: Watson’s theorem), the partial wave expansion of f , ∗ g, and h read d4Γ(D→ K → Kπ) dq2 d cos θdχdcos θ ∗ 1/2 1/2 ∗ = ˜ iδ0 + ˜ iδ1 +··· ∗ f fse fpe cos θ , ∝ B K → Kπ 1/2 =˜ iδ1 +··· g gpe , 9 2 2 ∗ 2 2 × + | +| +| −| 1/2 1 cos θ sin θ H H = ˜ iδ1 +··· 32 h hpe . ∗ + 4sin2 θ cos2 θ |H |2 0 It may, therefore be seen from the above that an analy- 2 2 ∗ ∗ − 2sin θ cos 2χ sin θ Re H+H sis of the decay distribution would yield information − ∗ ∗ ∗ on the phase shifts of interest. − sin 2θ cos χ sin 2θ Re H+H + H−H 0 0 2 ∗ 2 2 + 2 cos θ sin θ |H+| −|H−| − ∗ ∗ − ∗ 4. The FOCUS Collaboration has recently pub- 2sinθ cos χ sin 2θ Re H+H0 H−H0 . lished “evidence for new interference phenomena in the decay D+ → K−π +µ+ν” [15]. By including an 2 In Ref. [16] a discussion is provided on the multipole behaviour S-wave in a straightforward manner into the decay am- A A ∗ that is expected of the functions F1 , F2 , FV ; the FOCUS Collab- plitude that is dominated by the P-wave K resonance, oration assumes all of them to have monopole behaviour in their and finding a superior fit to certain distributions, this analysis, Ref. [15]. 226 B. Ananthanarayan, K. Shivaraj / Physics Letters B 628 (2005) 223–227

We note here that in the above, (a) the result is ex- the procedure described at length for the case of Kl4 pressed in the notation of Ref. [16] along with the as- decays in Ref. [18]. sumptions stated therein on contributions proportional ∗ = to Im(HiHj ), i j, (b) and taking into account the remarks given in the last paragraph of Section 4 of 6. We recall here that in the context of Kl4 de- Ref. [13], and (c) and also that the FOCUS Collabo- cays, the original 5 body decay kinematics were dis- ration has taken lepton mass effects into account for cussedinRef.[3], where the authors discussed only the results presented in Ref. [15]. 1-dimensional distributions. In Ref. [4] 2-dimensional Note that in the treatment above, there will be no distributions were considered, and also analyzed in the contributions of the type i = F , N, V . The FOCUS context of limited statistics. (The latter was the ba- Collaboration in the analysis of its data, ﬁnds that a sis of the analysis of the events from the well-known simple analysis based on a 1−− does not ﬁt the data experiment, Ref. [18].) Subsequently Berends, Don- well. They make an ad hoc assumption and intro- nachie and Oades (BDO) [19], again considered 1- duce an amplitude with the properties of an S-wave dimensional distributions, but with limited statistics. A exp iδ. Introducing this generates interference terms They also discussed |I|=3/2, 5/2 transitions, and which would correspond to terms that appear as i = F , also looked at tests of T-invariance. Recently the NA48 N and a term of the i = L type. This assumption can- Collaboration [6] has observed some evidence for the not generate a term of the type i = V .3 In [15] the violation of the |I|=1/2 rule consistent with stan- ∗ narrow-width approximation for the K is replaced by dard model expectations in Kl4 decays using the tech- a Breit–Wigner and a full 5-fold distribution is written nique of BDO. down. BDO in the context of Kl4 decays consider the 2- fold distribution given by

2 5. It is our main comment here that the FOCUS d Γ ∗ Collaboration must account for the dynamics in its en- d cos θ dχ tirety by using the formalism of Ref. [13]. In this man- which could receive contributions from T-violating in- ner, they would also be able to determine the phase teractions assuming that higher wave contributions are shift difference which would allow us to pin down low absent. Here we point out that such a distribution for energy strong interactions observables to better pre- D-meson decays could receive additional contribu- cision. Note also that a complete description of the tions from T-violation in the decays. Also considered four body final state with lepton mass effects included in BDO are the distributions is presented in Ref. [13].4 By binning the data in the dΓ dΓ variable s23 and carrying out integrations in the vari- , ∗ ables θ ∗ and χ and fitting the resulting distribution dχ d cos θ to experimental data, it would be possible to deter- which could be used to fit the form factors in an analy- mine the phase shifts and the form factors themselves. sis independent of the Pais–Treiman type distributions. We note here that unlike in the Kl4 decay where the The work of BDO can be readily extended to D-meson dimeson system is composed of equal mass particles, decays to search for the violation of the |I|=1/2 in the present case a ratio of, e.g., HF /HA cannot rule if there is a sizable number events for other reac- 1/2 1/2 + ¯ 0 0 − tions including D → K + π + l + νl, but this need directly yield information on δ0 δ1 . Only a comprehensive fit to all the Hi can be used to extract this be pursued after a compelling analysis of presently quantity. In this regard, it would be useful to follow available data for the determination of phase shifts of interest. For a recent discussion on |I|=3/2 amplitudes, see Ref. [20]. 3 Note that this is consistent with HV of the previous paragraph vanishing in the S- and P-wave approximation. 4 In this regard, the FOCUS Collaboration has analyzed data with charged lepton mass effects with their modified formalism of 7. In summary, we point out that the FOCUS Col- Ref. [16] all along, and present the relevant expressions in Ref. [17]. laboration with its large sample of Dl4 decays can B. Ananthanarayan, K. Shivaraj / Physics Letters B 628 (2005) 223–227 227 carry out a determination of much sought after πK [4] A. Pais, S.B. Treiman, Phys. Rev. 168 (1968) 1858. phase shifts by adopting the methods of Pais and [5] BNL-E865 Collaboration, S. Pislak, et al., Phys. Rev. Lett. 87 Treiman, and those of Cabibbo and Maksymowicz, (2001) 221801, hep-ex/0106071; S. Pislak, et al., Phys. Rev. D 67 (2003) 072004, hep- and Berends, Donnachie and Oades, and go beyond es- ex/0301040. tablishing an interference phenomenon. This would be [6] NA48 Collaboration, J.R. Batley, et al., Phys. Lett. B 595 a valuable source of information for important low en- (2004) 75, hep-ex/0405010. ergy observables such as pion–kaon scattering lengths. [7] N. Cabibbo, Phys. Rev. Lett. 93 (2004) 121801, hep- ph/0405001. [8] N. Cabibbo, G. Isidori, JHEP 0503 (2005) 021, hep- ph/0502130. Acknowledgements [9] V. Bernard, N. Kaiser, U.G. Meissner, Nucl. Phys. B 357 (1991) 129. It is a pleasure to thank Prof. G. Colangelo for [10] B. Ananthanarayan, P. Buettiker, Eur. Phys. J. C 19 (2001) 517, suggesting this investigation and for invaluable discus- hep-ph/0012023. sions. B.A. thanks Prof. G. Dosch for a conversation, [11] B. Ananthanarayan, P. Buettiker, B. Moussallam, Eur. Phys. J. C 22 (2001) 133, hep-ph/0106230. Prof. G. Kramer for a useful clarification, and the hos- [12] D. Aston, et al., Nucl. Phys. B 296 (1988) 493. pitality of the Theory Group, Thomas Jefferson Na- [13] G. Kopp, G. Kramer, G.A. Schuler, W.F. Palmer, Z. Phys. C 48 tional Accelerator Facility, USA at the time this work (1990) 327. was initiated. This work is supported in part by the [14] P. Buettiker, S. Descotes-Genon, B. Moussallam, Eur. Phys. J. CSIR under scheme number 03(0994)/04/EMR-II. It C 33 (2004) 409, hep-ph/0310283. [15] FOCUS Collaboration, J.M. Link, et al., Phys. Lett. B 535 is a pleasure to thank Prof. D. Kim for comments on (2002) 43, hep-ex/0203031. the first version of the Letter. [16] J.G. Korner, G.A. Schuler, Z. Phys. C 46 (1990) 93. [17] FOCUS Collaboration, J.M. Link, et al., Phys. Lett. B 544 (2002) 89, hep-ex/0207049; References FOCUS Collaboration, J.M. Link, et al., Phys. Lett. B 607 (2005) 67, hep-ex/0410067. [18] L. Rosselet, et al., Phys. Rev. D 15 (1977) 574. [1] J. Gasser, H. Leutwyler, Ann. Phys. 158 (1984) 142; [19] F.A. Berends, A. Donnachie, G.C. Oades, Phys. Rev. 171 J. Gasser, H. Leutwyler, Nucl. Phys. B 250 (1985) 465. (1968) 1457. [2] G. Colangelo, J. Gasser, H. Leutwyler, Nucl. Phys. B 603 [20] L. Edera, M.R. Pennington, hep-ph/0506117. (2001) 125, hep-ph/0103088. [3] N. Cabibbo, A. Maksymowicz, Phys. Rev. 137 (1965) 438. Physics Letters B 628 (2005) 228–238 www.elsevier.com/locate/physletb

1/mQ corrections to B → ρlν decay and |Vub|

W.Y. Wang, Y.L. Wu, M. Zhong

Institute of Theoretical Physics, Academia Sinica, Beijing 100080, China Received 24 May 2005; accepted 18 September 2005 Available online 29 September 2005 Editor: T. Yanagida

Abstract

In the heavy quark effective ﬁeld theory of QCD, we analyze the order 1/mQ contributions to heavy to light vector decays. Light cone sum rule method is applied with including the effects of 1/mQ order corrections. We then extract |Vub| from B → ρlν decay up to order of 1/mQ corrections.  2005 Elsevier B.V. All rights reserved.

PACS: 11.55.Hx; 12.39.Hg; 13.20.Fc; 13.20.He

Keywords: B → ρlν;1/mQ correction; Heavy quark effective ﬁeld theory; Light cone sum rule

1. Introduction

Much effort has been devoted to discuss the heavy to light hadron semileptonic decays. In particular, B → π(ρ)lν decays attracted the most interest [1–9] because they can be used to determine the quark mixing matrix element |Vub|, a parameter of significance in particle physics. The heavy quark symmetry and relevant effective theory greatly simplify the study of hadrons each of which containing a single heavy quark and any number of light quarks, and provide relations between different processes. This symmetry is applied to study B(D)(s) → ∗ π(ρ,K,K )lν decays in Refs. [10–12], where the finite heavy quark mass (mQ) corrections are not considered. Ref. [13] extends the study on B → πlν decay up to the next to leading order of the heavy quark expansion. For a more complete knowledge of the magnitude of the finite mass corrections to heavy to light meson decays, and to the determination of |Vub|, one should also study the 1/mQ order corrections to semileptonic B decays to light vector mesons. In this short Letter we will apply the heavy quark effective field theory (HQEFT) developed in Refs. [14–18] to analyze the 1/mQ corrections to the B → ρlν decay. And the light cone sum rule method will be adopted to

E-mail address: [email protected] (W.Y. Wang).

0370-2693/$ – see front matter  2005 Elsevier B.V. All rights reserved. doi:10.1016/j.physletb.2005.09.038 W.Y. Wang et al. / Physics Letters B 628 (2005) 228–238 229 numerically estimate the nonperturbative functions, i.e., the heavy to light vector form factors with including 1/mQ order corrections. In Section 2 the 1/mQ order corrections are formulated in HQEFT framework. Section 3 devotes to evaluate wave functions using light cone sum rule method in HQEFT. And Section 4 is the numerical results and discussion.

2. B → ρlν decay in HQEFT

The transition matrix element responsible to the B → ρlν decay is generally parameterized by form factors as 2 ∗ µ 5 2 ∗µ A2(q ) ∗ µ ρ(p, ) uγ¯ 1 − γ b B(pB ) =−i(mB + mρ)A1 q + i · (p + q) (2p + q) mB + mρ 2 2 + A3(q ) ∗ · + µ + 2V(q ) µαβγ ∗ + i (p q) q α(p q)β pγ , (2.1) mB + mρ mB + mρ where q = pB − p is the momentum carried by the lepton pair. In the framework of HQEFT [14,15], the QCD quantum field Q for heavy quark is decomposed into particle field Q+ and antiparticle field Q−, so that the quark and antiquark fields are treated on the same footing in a symmetric way. The effective quark and antiquark fields in HQEFT are defined as ± · ± · ± = ivm/ Qv x ˆ = ivm/ Qv x Qv e Qv e P±Q , (2.2) ± = ± Rv P∓Q (2.3) 2 = ≡ ± ˆ ± with v being an arbitrary four-velocity satisfying v 1, and P± (1 v)// 2 being the projection operators. Qv ± defined above are actually the large components of the heavy quark and antiquark fields, respectively. Rv are the small components of the heavy quark and antiquark fields, respectively. The quantum field in QCD Lagrangian can ≡ + + − ≡ ˆ + + ˆ − + + + − be written as Q Q Q Qv Qv Rv Rv , which contains all large and small components of particle and antiparticle. The decomposition of Q is presented in detail in Refs. [14,15]. After the small components of particle and antiparticle fields being integrated out, QCD Lagrangian turns into L = L(++) + L(−−) + L(+−) + L(−+) Q,v Q,v Q,v Q,v Q,v (2.4) with L(±±) = ¯ ± D ± Q,v Qv i/ vQv , (2.5) −1 ±∓ 1 ± ←− · i/vv · D ∓ L( ) = ¯ D 2ivm/ Qv x − Q,v Qv (i /v)e 1 (iD/ ⊥)Qv 2mQ 2mQ ←− −1 1 ± ←− −i/vv · D − · ∓ = ¯ − − 2ivm/ Qv x D Qv ( iD/ ⊥) 1 e (i/ v)Qv , (2.6) 2mQ 2mQ where − 1 i/vv · D 1 iD/ v = i/vv · D + iD/ ⊥ 1 − iD/ ⊥, 2mQ 2mQ ←− − ←− ←− 1 ←− −i/vv · D 1 ←− iD/v =−i/vv · D + (−iD/ ⊥) 1 − (−iD/ ⊥), 2mQ 2mQ ←− ←− ←− iD/ ⊥ = iD/ − i/vv · D, −iD/ ⊥ =−iD/ + i/vv · D, (2.7) which is treated as HQEFT in the case that the longitudinal and transverse residual momenta, i.e., the operators iv · D and iD⊥ are at the same order of power counting in 1/mQ expansions. 230 W.Y. Wang et al. / Physics Letters B 628 (2005) 228–238

The heavy-light quark current qΓQ¯ with Γ being arbitrary Dirac matrices can be expanded in powers of 1/mQ as −im v·x 1 1 2 1 + qΓ¯ Q → e Q q¯ Γ + Γ (iD/ ⊥) + O Q . (2.8) · 2 v 2mQ i/vv D mQ Here the contributions from both heavy quark and antiquark ﬁelds have been considered. According to above expansions for effective Lagrangian and effective current, one can write the matrix element in Eq. (2.1) as the following form in powers of 1/mQ, ρ|¯uγ µ 1 − γ 5 b|B mB µ 5 + 1 µ 5 1 2 i αβ + = ρ|¯uγ 1 − γ Q |Mv+ ρ|¯uγ 1 − γ P+ D⊥ + σαβ F Q |Mv Λ¯ v 2m iv · D 2 v B Q + O 2 1/mQ , (2.9)

¯ αβ where ΛB = mB − mb, and F is the gluon field strength tensor. The effective heavy meson state |Mv satisfies the heavy quark spin–flavor symmetry. Its normalization is

| ¯ + µ +| = ¯ µ Mv Qv γ Qv Mv 2Λv (2.10) ¯ ≡ ¯ with the binding energy Λ limmQ→∞ ΛM being heavy flavor independent. It should be noted that the 1/mQ corrections in Eq. (2.9) include both contributions from the current expansion (2.8) and from the insertion of the effective Lagrangian (2.4).InEqs.(2.8) and (2.9) the operator 1/(iv · D) arises from the contraction of effective heavy quark and antiquark fields [15,18].Inthev · A = 0 gauge to be used in our calculation, this operator is tantamount to the heavy quark propagator. As can be seen in Eq. (2.9) that the 1/mQ order corrections to B → ρlν transition are only attributed to one kinematic operator and one chromomagnetic operator. The heavy quark symmetry enables us to parameterize the matrix elements in HQEFT as ∗ ¯ +| =− M ρ(p, ) uΓ Qv Mv Tr Ω(v,p)Γ v , (2.11) ∗ P+ + ρ(p, )uΓ¯ D2 Q |M =− Ω (v, p)Γ M , · ⊥ v v Tr 1 v (2.12) iv D ∗ P+ i αβ + αβ i ρ(p, ) uΓ¯ σ F Q |M =−Tr Ω (v, p)Γ P+ σ M , (2.13) iv · D 2 αβ v v 1 2 αβ v √ ¯ 5 where the pseudoscalar heavy meson spin wave function Mv =− ΛP+γ is independent of the heavy quark αβ flavor. Ω(v,p), Ω1(v, p) and Ω1 (v, p) can be decomposed into Lorentz scalar functions,

∗ ∗ ∗ ∗ Ω(v,p)= L (v · p)/ + L (v · p)(v · ) + L (v · p)/ + L (v · p)(v · ) p,/ˆ 1 2 3 4 ∗ ∗ ∗ ∗ Ω1(v, p) = δL1(v · p)/ + δL2(v · p)(v · ) + δL3(v · p)/ + δL4(v · p)(v · ) p,/ˆ αβ α β β α ∗ ∗ Ω (v, p) = pˆ γ −ˆp γ / (R1 + R2p)/ˆ + (v · )(R3 + R4p)/ˆ 1 ∗α β ∗β α ∗α β ∗β α + γ − γ [R5 + R6p/ˆ]+ pˆ − pˆ [R7 + R8p/ˆ]

αβ ∗ ∗ + iσ / (R9 + R10p)/ˆ + (v · )(R11 + R12p)/ˆ (2.14) ˆµ = µ · ≡ · = 2 + 2 − 2 with p p /(v p) and ξ v p (mB mρ q )/2mB . W.Y. Wang et al. / Physics Letters B 628 (2005) 228–238 231

Eqs. (2.9)–(2.14) yield ¯ ∗ µ 5 mB Λ µναβ ∗µ ρ(p, ) uγ¯ 1 − γ b B(pB ) =−2i iL νpˆαvβ + (L + L ) Λ¯ 3 1 3 B − − · ∗ ˆµ − · ∗ µ + O 2 (L3 L4)(v )p L2(v )v 1/mQ (2.15) with = + 1 = + 1 + Li Li δLi Li (δLi Ri), 2mQ 2mQ =− − + − + − ˆ2 R1 2R1 2R5 R7 3R9 (2R2 R8)p , =− − − − + ˆ2 R2 2R3 2R5 3R11 (2R4 R8)p , = − − − + − R3 2R1 2R2 2R6 R7 R8 3R10, =− − − − − R4 2R6 R7 2R3 2R4 3R12.

Comparison between Eqs. (2.1) and (2.15) gives relations between the form factors Ai (i = 1, 2, 3), V and the universal wave functions, ¯ 2 2 mB Λ A1 q = L (v · p) + L (v · p) +···, m + m Λ¯ 1 3 B ρ B m Λ¯ L (v · p) L (v · p) − L (v · p) A q2 = 2(m + m ) B 2 + 3 4 +···, 2 B ρ ¯ 2 · ΛB 2mB 2mB (v p) ¯ · · − · 2 mB Λ L2(v p) L3(v p) L4(v p) A3 q = 2(mB + mρ) − +···, Λ¯ 2m2 2m (v · p) B B B ¯ + 2 = mB Λ mB mρ · +··· V q ¯ L3(v p) , (2.16) ΛB mB (v · p) where the dots denote higher order 1/mQ contributions not to be taken into account in the following calculations.

3. Light cone sum rules in HQEFT

For the derivation of the 1/mQ order corrections to B → ρlν decay, we consider the two-point correlation function µ 4 −i(p −m v)·x ∗ µ 5 1 2 i αβ + F = i d xe B Q ρ(p, ) T u(¯ )γ − γ P+ D + σ F Q ( ), 0 1 · ⊥ αβ v 0 iv D 2 ¯ + 5 | Qv (x)iγ d(x) 0 (3.1)

¯ + 5 where Qv (x)iγ d(x) is the interpolating current for B meson. Inserting between the two currents in Eq. (3.1) a complete set of intermediate states with the B meson quantum number, one gets ¯ mB Λ 2iF µναβ ∗ ˆ + − · ∗ ˆµ − + ∗µ + · ∗ µ ¯ ¯ δL3 ν pαvβ i(δL3 δL4)(v )p i(δL1 δL3) iδL2(v )v mQΛB 2ΛB − ω ∞ ρ(ξ,s) + ds + subtraction (3.2) s − ω s0 232 W.Y. Wang et al. / Physics Letters B 628 (2005) 228–238 with ξ ≡ v · p and ω ≡ 2v · k, where k = pB − mQv is the residual momentum of the bottom quark. The second term in (3.2) represents the higher resonance contributions. F is the decay constant of B meson at the leading order of 1/mQ expansion, deﬁned by [17]

+ F 0|¯qΓ Q |B = Tr[Γ M ]. (3.3) v v 2 v In deep Euclidean region the correlator (3.1) can be calculated in effective ﬁeld theory. The result can be written also as an integral over a theoretic spectral density, ∞ ρ (ξ, s) ds th + subtraction. (3.4) s − ω 0 The standard treatment of sum rule is to assume the quark-hadron duality, and to equal the hadronic representation (3.2) and the theoretic one (3.4), which provides an equation: ¯ mB Λ 2iF µναβ ∗ ˆ + − · ∗ ˆµ − + ∗µ + · ∗ µ ¯ ¯ δL3 ν pαvβ i(δL3 δL4)(v )p i(δL1 δL3) iδL2(v )v mQΛB 2ΛB − ω ∞ ∞ ρ(ξ,s) ρ (ξ, s) + ds = ds th + subtraction. (3.5) s − ω s − ω s0 0 To ensure the reliability of sum rule estimates, one should enhance the importance of the ground state contribution, suppress higher order nonperturbative contributions and remove the subtraction. These can be achieved by performing the Borel transformation (−ω)n+1 d n Bˆ (ω) ≡ lim T −ω,n→∞ n! dω −ω/n=T to both sides of Eq. (3.5). With using the formulae 1 − 1 Bˆ (ω) = e s/T , Bˆ (ω)eλω = δ λ − , (3.6) T s − ω T T one gets ∗ ∗ ∗ ∗ − ¯ µναβ ˆ − + µ + − · ˆµ + · µ 2ΛB /T 2iF δL3 ν pαvβ i(δL1 δL3) i(δL3 δL4)(v )p iδL2(v )v e s0 − = dse s/T ρ(ξ,s), (3.7) 0 where the spectral density ρ(ξ,s) can be derived via double Borel transformations, = ˆ (−1/T) ˆ (ω) µ ρ(ξ,s) B1/s BT F (ξ, ω). (3.8) In calculating the three-point function (3.1), one may represent the nonperturbative contributions embeded in the hadronic matrix element in terms of light cone wave functions. Among them are the two-particle distribution functions and the three-particle ones. However, if we restrict our calculation to the lowest twist (twist 2) level, the three-particle functions do not contribute. As a result, the chromomagnetic operator in Eq. (3.1) could be neglected in the lowest twist approximation. The leading twist distribution functions are deﬁned by [2,4–6] 1 ∗ ¯ | =− ⊥ ∗ − ∗ iup·x ρ(p, ) u(0)σµνd(x) 0 ifρ (µpν ν pµ) due φ⊥(u), 0 W.Y. Wang et al. / Physics Letters B 628 (2005) 228–238 233

1 1 ∗ · ∗ · ∗ x iup·x ∗ x iup·x (v) ρ(p, ) u(¯ 0)γ d(x)|0=f m p due φ (u) + f m − p due g (u), µ ρ ρ µ p · x ρ ρ µ µ p · x ⊥ 0 0 1 ∗ 1 ∗ · ρ(p, )u(¯ 0)γ γ d(x)|0= f m νpαxβ dueiup xg(a)(u) (3.9) µ 5 4 ρ ρ µναβ ⊥ 0 (v,a) with φ⊥, , and g⊥ being functions with nonperturbative nature. + ¯ + Then the effective heavy quark ﬁelds Qv (x1)Qv (x2) can be contracted into a propagator of heavy quark, ∞ − − P+ 0 dt δ(x1 x2 vt). In the lowest twist approximation, only the kinematic operator contributes to the 1/mQ order corrections to B → ρlν decay. At v · A = 0 gauge we used, the correlation function (3.1) simpliﬁes as ∞ ∞ 4 4 −ik·x ∗ µ 5 i d x d y dl dt e ρ(p, ) u(¯ 0)γ 1 − γ P+δ(−y − vl)

0 0 × 2 − α β − α − α ∂(y) vαvβ ∂(y)∂(y) ∂(y)Aα(y) Aα(y)∂(y) i αβ α 5 + σ F (y) + A (y)A (y) P+δ(y − x − vt)γ d(x)|0 2 αβ α ∞ ∞ 2 2 − · ∂ ∂ → i d4x d4y dl dt e ik xδ(−y − vl) − vαvβ δ(y − x − vt) ∂y2 ∂yα∂yβ 0 0 ∗ 1 1 × ρ(p, )u(¯ 0) γ µγ 5 − γ µ − σ αβ iv g + v d(x)|0 (3.10) 2 β αµ 2 µναβ ν α ≡ with ∂(y) ∂/(∂yα). The ﬁnal formula in (3.10) includes only the terms related to the two-particle distribution functions (3.9). Now the transition matrix element can be evaluated through the distribution functions deﬁned in (3.9).Using Eqs. (3.6) and (3.9), the spectral function is found to be

− ρ(ξ,s)= Bˆ ( 1/T)Bˆ (ω)F µ 1/s T 1 µναβ ∗ 1 3 2 (a) ⊥ 2 1 2 = i pαvβ − fρm u g + f m u φ⊥ 2ξ ν 4ξ 2 ρ ⊥ ρ ρ ξ 1 3 + f m u2g(a) − f m2 ug(a) 4 ρ ρ ⊥ 4 ρ ρ ⊥ ∗µ 3 1 2 (v) ⊥ 2 2 (v) ⊥ 2 ⊥ 2 + f m u g − 2f m uφ⊥ − f m ξ u g − 2f ξ uφ⊥ + 4f ξ uφ⊥ ρ ρ ξ ⊥ ρ ρ ρ ρ ⊥ ρ ρ

∗ µ 1 3 2 (v) 1 3 2 2 3 (v) 2 3 + (v · )p − f m u g + f m u φ − f m ug + f m uφ ξ 2 ρ ρ ⊥ ξ 2 ρ ρ ξ 2 ρ ρ ⊥ ξ 2 ρ ρ 2 3 (v) 2 3 2 (v) 2 − f m ξG + f m ξΦ + f m u g − f m u φ ξ 3 ρ ρ ⊥ ξ 3 ρ ρ ρ ρ ⊥ ρ ρ

+ · ∗ µ ⊥ 2 − ⊥ 2 2 − ⊥ 2 + ⊥ 2 2 (v )v 2fρ mρuφ⊥ 2fρ mρ u φ⊥ 4fρ ξ uφ⊥ 2fρ ξ u φ⊥ , (3.11) → s u 2ξ 234 W.Y. Wang et al. / Physics Letters B 628 (2005) 228–238

(v) (v) where denotes derivative with respect to the variable u, while G⊥ (u) and Φ (u) are functions related to g⊥ and ∂ (v) = (v) ∂ = φ by ∂uG⊥ (u) g⊥ (u), ∂uΦ (u) φ (u). The detailed procedure in deriving Eq. (3.11) is similar to those in Refs. [10,11,13].

4. Numerical analysis

φ⊥ and φ are the lowest twist distributions in the fraction of total momentum carried by the quark in trans- 3/2 versely and longitudinally polarized mesons. They can be expanded in Gegenbauer polynomials Cn (x) whose coefﬁcients are renormalized multiplicatively. With the scale dependence explicitly, one has [4] = − + ⊥( ) 3/2 − φ⊥( )(u, µ) 6u(1 u) 1 an (µ)Cn (2u 1) , n=2,4,... ⊥( )− ⊥( ) (γn γ0 )/(2β0) ⊥( ) = ⊥( ) αs(µ) an (µ) an (µ0) , (4.1) αs(µ0) = − ⊥ where β0 11 (2/3)nf , and γn ,γn are the one-loop anomalous dimensions [19,20]. The nonperturbative ⊥ parameters an and an have been obtained in [4] with the values ⊥ = ± = ± a2 (1GeV) 0.2 0.1,a2(1GeV) 0.18 0.10 (4.2) ⊥( ) and an = 0forn = 2. (v) (a) The functions g⊥ and g⊥ describe transverse polarizations of quarks in the longitudinally polarized mesons. As in Ref. [6] they are parameterized as 3 3 3 g(v)(u, µ) = 1 + (2u − 1)2 + a (µ)(2u − 1)3 + a (µ) + 5ξ (µ) 3(2u − 1)2 − 1 ⊥ 4 2 1 7 2 3 9 15 + a (µ) + ξ (µ) 3ωV (µ) − ωA(µ) 3 − 30(2u − 1)2 + 35(2u − 1)4 , 112 2 64 3 3 3 1 5 3 9 g(a)(u, µ) = 6u(1 − u) 1 + a (µ)(2u − 1) + a (µ) + ξ (µ) 1 − ωA(µ) + ωV (µ) ⊥ 1 4 2 3 3 16 3 16 3 × 5(2u − 1)2 − 1 . (4.3)

All the nonperturbative parameters in Eqs. (4.3) have been estimated in Ref. [6]. The asymptotic form of these factors and the renormalization scale dependence are given by perturbative QCD [21,22].AsinRef.[11],the typical virtuality of the bottom quark ∼ 2 − 2 ≈ µb mB mb 2.4GeV, (4.4) is used for the energy scale for the current calculation. ⊥ ¯ ¯ The values of the hadron quantities fρ,fρ , ΛB , Λ and F have been extracted in the previous work (see, e.g., [4,17,23,24]). For consistency, here we use for them the same values as in Ref. [11], i.e.,

⊥ ± = ± = ± = ± fρ (195 7) MeV,fρ0 (216 5) MeV,fρ (160 10) MeV, ¯ ¯ 3/2 ΛB ≈ Λ = 0.53 GeV,F= (0.30 ± 0.06) GeV . (4.5) W.Y. Wang et al. / Physics Letters B 628 (2005) 228–238 235

Fig. 1. Variation of 1/mQ order wave functions δLi (i = 1, 2, 3, 4) with respect to the Borel parameter T at ξ = v · p = 2.5 GeV. The dashed, solid and dotted curves correspond to the thresholds s0 = 1.5, 2.1 and 2.7 GeV, respectively.

δLi as functions of ξ, T and s0 can be derived from Eqs. (3.7) and (3.11). Fig. 1 shows the variation of δLi as functions of the Borel parameter T at v · p = 2.5 GeV. The curves in each ﬁgure correspond to different values adopted for the threshold s0. The rule of LCSR method is to determine s0 from the stability of relevant curves in the reliable region of T , where both the higher nonperturbative corrections and the contributions from excited and continuum states should not be large. In the current case, we focus on the region around T = 1.5–2 GeV. As shown in Fig. 1, δLi are found to be stable with respect to the Borel parameter T .InRef.[11] the threshold s0 = 2.1 ± 0.6 GeV is adopted in evaluating the leading order wave functions Li . In calculating the decay width we will use for δLi the same threshold values as those for the leading order wave function Li , i.e., s0 ≈ 2.1GeV. With (2.16), the form factors A1, A2, A3 and V with including 1/mQ order corrections can be calculated. It is convenient to represent each of these form factors in terms of three parameters as

F(0) F q2 = , (4.6) − 2 2 + 2 2 2 1 aF q /mB bF (q /mB ) 2 2 2 2 2 where F(q ) can be any one of A1(q ), A2(q ), A3(q ) and V(q ). The parameters F(0), aF and bF presented in Table 1 are ﬁtted from the LCSR results at s0 = 2.1GeV.Fig. 2 shows the form factors as functions of the 236 W.Y. Wang et al. / Physics Letters B 628 (2005) 228–238

Table 1 Results of LCSR calculations up to leading (LO) and next leading order (NLO) in HQEFT. The leading order results are obtained in Ref. [10]

F(0)aF bF A1 0.26 0.37 −0.19 LO 0.27 0.32 −0.19 NLO

A2 0.26 1.11 0.26 LO 0.26 1.15 0.30 NLO

A3 −0.26 1.12 0.26 LO −0.26 1.11 0.25 NLO V 0.32 1.24 0.29 LO 0.31 1.26 0.32 NLO

Fig. 2. Form factors Ai (i = 1, 2, 3) and V obtained from light cone sum rules in HQEFT. The dashed curves are the leading order results in HQEFT [10,12], while the solid curves are the results with including 1/mQ order correction. momentum transfer, where the dashed curves are for the leading order results while the solid ones for the results with the 1/mQ order corrections included. W.Y. Wang et al. / Physics Letters B 628 (2005) 228–238 237

The differential decay width of B → ρlν with the lepton mass neglected is 2 | |2 dΓ GF Vub 1/2 2 2 2 2 = λ q H + H+ + H− (4.7) 2 3 3 0 dq 192π mB with the helicity amplitudes 1/2 2 λ 2 H± = (m + m )A q ∓ V q , B ρ 1 m + m B ρ 1 2 2 2 2 λ 2 H0 = m − m − q (mB + mρ)A1 q − A2 q (4.8) 2 B ρ + 2mρ q mB mρ and ≡ 2 + 2 − 2 2 − 2 2 λ mB mρ q 4mB mρ. (4.9) The total width of B → ρlν can be obtained by integrating (4.7) over the whole accessible region of q2. We get 2 −1 Γ(B→ ρlν) = (13.6 ± 4.0)|Vub| ps , (4.10) where the error results from the variation of the threshold energy s0 = 1.5–2.7GeV. The branching fraction of B0 → ρ−l+ν is measured to be Br(B0 → ρ−l+ν) = (2.6 ± 0.7) × 10−4 [25].This 0 and the world average of the B lifetime [25] τ 0 = 1.536 ± 0.014 ps yields B − + − − Γ B0 → ρ l ν = (1.69 ± 0.47) × 10 4 ps 1. (4.11)

|Vub| is then extracted from Eqs. (4.10) and (4.11).Itis −3 |Vub|=(3.53 ± 0.49 ± 0.52) × 10 , (4.12) where the ﬁrst and second errors correspond to the experimental and theoretical uncertainties, respectively. This value may be compared with the ones previously obtained [10,11,13,16,26]. From the exclusive semileptonic decays B → π(ρ)lν, we then have −3 |Vub|=(3.4 ± 0.5 ± 0.5) × 10 (B → πlν,LO), −3 |Vub|=(3.2 ± 0.5 ± 0.4) × 10 (B → πlν, to NLO), −3 |Vub|=(3.7 ± 0.6 ± 0.7) × 10 (B → ρlν, LO), −3 |Vub|=(3.5 ± 0.5 ± 0.5) × 10 (B → ρlν, to NLO), −3 |Vub|=(3.5 ± 0.6 ± 0.1) × 10 (B inclusive semileptonic decays).

Asasummary,wehavestudiedB → ρlν decay up to the 1/mQ order corrections in HQEFT. In HQEFT, 1/mQ order corrections from the effective current and from effective Lagrangian are given by the same operator forms, which simpliﬁes the structure of transition matrix elements. These 1/mQ order contributions have been calculated using light cone sum rules with considering the lowest twist distribution functions. Numerically, the 1/mQ order wave functions give only corrections lower than 10% to the transition form factors. Similar to the B → πlν case, the correction indicates a slightly smaller value of the CKM matrix element |Vub|. The discussion concerning 1/mQ order corrections to B → ρlν decay in this Letter is also applicable to other heavy to light vector meson decays.

Acknowledgements

This work was supported in part by the key projects of National Science Foundation of China (NSFC) and Chinese Academy of Sciences, and by the BEPC National Lab Opening Project. 238 W.Y. Wang et al. / Physics Letters B 628 (2005) 228–238

References

[1] V.M. Belyaev, A. Khodjamirian, R. Rückl, Z. Phys. C 60 (1993) 349. [2] A. Khodjamirian, R. Rückl, WUE-ITP-97-049, MPI-PhT/97-85, hep-ph/9801443. [3] A. Khodjamirian, R. Rückl, S. Weinzierl, C.W. Winhart, O. Yakovlev, Phys. Rev. D 62 (2000) 114002. [4] P. Ball, V.M. Braun, Phys. Rev. D 54 (1996) 2182. [5] P. Ball, V.M. Braun, Phys. Rev. D 55 (1997) 5561. [6] P. Ball, V.M. Braun, Phys. Rev. D 58 (1998) 094016. [7] M.A. Ivanov, Y.L. Kalinovsky, C.D. Roberts, Phys. Rev. D 60 (1999) 034018. [8] C.D. Roberts, S.M. Schmidt, Prog. Part. Nucl. Phys. 45 (2000) S1. [9] M.A. Ivanov, Y.L. Kalinovsky, C.D. Roberts, Preprint ANL-PHY-9594-TH-2000, hep-ph/0006189. [10] W.Y. Wang, Y.L. Wu, Phys. Lett. B 515 (2001) 57. [11] W.Y. Wang, Y.L. Wu, Phys. Lett. B 519 (2001) 219. [12] W.Y. Wang, Y.L. Wu, M. Zhong, Phys. Rev. D 67 (2003) 014024. [13] W.Y. Wang, Y.L. Wu, M. Zhong, J. Phys. G 29 (2003) 2743. [14] Y.L. Wu, Mod. Phys. Lett. A 8 (1993) 819. [15] W.Y. Wang, Y.L. Wu, Y.A. Yan, Int. J. Mod. Phys. A 15 (2000) 1817. [16] Y.A. Yan, Y.L. Wu, W.Y. Wang, Int. J. Mod. Phys. A 15 (2000) 2735. [17] W.Y. Wang, Y.L. Wu, Int. J. Mod. Phys. A 16 (2001) 377. [18] W.Y. Wang, Y.L. Wu, Int. J. Mod. Phys. A 16 (2001) 2505. [19] D.J. Gross, F. Wilczek, Phys. Rev. D 9 (1974) 980. [20] M.A. Shifman, M.I. Vysotsky, Nucl. Phys. B 186 (1981) 475. [21] V.M. Braun, I.B. Filyanov, Z. Phys. C 48 (1990) 239. [22] V.L. Chernyak, A.R. Zhitnitsky, Phys. Rep. 112 (1984) 173. [23] M.A. Benitez, et al., Phys. Lett. B 239 (1990) 1. [24] L. Montanet, et al., Phys. Rev. D 50 (1994) 1173. [25] S. Eidelman, et al., Phys. Lett. B 592 (2004) 1. [26] Y.L. Wu, Y.A. Yan, Int. J. Mod. Phys. A 16 (2001) 285. Physics Letters B 628 (2005) 239–249 www.elsevier.com/locate/physletb

QCD traveling waves at non-asymptotic energies

C. Marquet, R. Peschanski, G. Soyez 1

Service de physique théorique, CEA/Saclay, 91191 Gif-sur-Yvette cedex, France 2 Received 9 September 2005; received in revised form 19 September 2005; accepted 20 September 2005 Available online 30 September 2005 Editor: N. Glover

Abstract Using consistent truncations of the BFKL kernel, we derive analytical traveling-wave solutions of the Balitsky–Kovchegov saturation equation for both ﬁxed and running coupling. A universal parametrization of the “interior” of the wave front is obtained and compares well with numerical simulations of the original Balitsky–Kovchegov equation, even at non-asymptotic energies. Using this universal parametrization, we ﬁnd evidence for a traveling-wave pattern of the dipole amplitude determined from the gluon distribution extracted from deep inelastic scattering data.  2005 Elsevier B.V. All rights reserved.

1. Introduction observation. Indeed geometric scaling has a natural explanation [2] in terms of traveling-wave solutions In the past decade, there has been a large amount of of the Balitsky–Kovchegov (BK) non-linear equation work devoted to the description and understanding of [3,4]. quantum chromodynamics (QCD) in the high-energy/ The BK equation is an equation for the evolution density limit corresponding to saturation. On the the- in rapidity Y of the scattering amplitude N(r, b,Y)of oretical side, progress is made in obtaining non-linear a dipole (a colorless qq¯ pair) off a given target. r is QCD equations describing the evolution of scattering the transverse size of the dipole and b its impact para- amplitudes towards saturation. On the phenomenolog- meter. The rapidity Y is the logarithm of the squared ical side, the discovery of geometric scaling [1] in deep total center-of-mass energy of the collision. Assum- inelastic scattering (DIS) at HERA was a stimulating ing that the amplitude N is b-independent, the BK equation has been shown [2] to lie in the same university class as the Fisher–Kolmogorov–Petrovsky– E-mail addresses: [email protected] (C. Marquet), Piscounov (FKPP) equation [5], extensively studied in [email protected], [email protected] statistical physics. (R. Peschanski), [email protected] (G. Soyez). 1 On leave from the fundamental theoretical physics group of the The FKPP equation is an evolution equation for University of Liège. the function u(x, t) where x is a space variable and t 2 URA 2306, unité de recherche associée au CNRS. is the time. It admits [6] asymptotic solutions which

0370-2693/$ – see front matter  2005 Elsevier B.V. All rights reserved. doi:10.1016/j.physletb.2005.09.035 240 C. Marquet et al. / Physics Letters B 628 (2005) 239–249 are functions of the sole variable x − vt, namely phenomenology. Conclusions and outlook are given in traveling-wave solutions u(x − vt) where v is the Section 5. speed of the wave. It comes from a critical regime obtained by the competition of the exponential growth in the linear regime and the non-linear damping. This 2. The fixed-coupling case mechanism for the formation of traveling waves is more general [7] and applies to the BK equation for Let us consider the Fourier transform of the dipole which time is replaced by rapidity and the logarithm scattering amplitude defined as of the dipole size plays the role of the position. It ∞ has even been extended beyond the b-independent dr N (k, Y ) = J (kr)N(r, Y ). (1) case [8] to processes with non-zero momentum trans- r 0 fer. 0 However there are limitations for applying these so- The BK equation for N (k, Y ) then reads [4] lutions precisely because they are asymptotic, i.e., they N = − N − N 2 require very large values of Y to appear; also analyt- ∂Y χ( ∂L) , (2) ical expressions are only known for relatively small where L = log(k2/k2) and the rapidity Y is measured =|| 0 values of r r . This leads to a limitation of an- here in units of α¯ ≡ αs Nc where the strong coupling alytical predictions to the tail of the wave (N 1) π constant αs is kept fixed. which is essentially determined by the linear regime. For instance, the “interior” of the wave is not repro- χ(γ)= 2ψ(1) − ψ(γ)− ψ(1 − γ) (3) duced; it corresponds to the transition to saturation, a is the Balitsky–Fadin–Kuraev–Lipatov (BFKL) kernel intermediate regime between the tail (N 1) and the [10]. k is an arbitrary scale. ∼ 0 saturated region (N 1). That moderates the phenom- It has been proved [2] that Eq. (2) admits asymp- enological impact of those mathematically powerful totic traveling-wave solutions N (L − vY) where v is results. the speed of the wave. Starting with the initial condi- In order to circumvent these difficulties, it is neces- tion N (L, Y0) ∼ exp(−γiL), traveling-wave solutions sary to extend the study to the non-asymptotic regime are formed during the Y -evolution with the following in a region of larger N, which implies to use more in- value for the speed: formation on the non-linear term of the equation than the previous approaches. For this sake, a new approach • “pushed front”: 0 <γi <γc; v = χ(γi)/γi , to study the BK equation has been developed in [9].In • “pulled front”: γi γc; v = vc ≡ χ(γc)/γc, short, the guiding idea is to try and find exact solutions of approximate QCD equations rather than approxi- where γc is solution of the equation χ(γ)/γ = χ (γ ) mate solutions of exact QCD equations. and is called the critical anomalous dimension. Its The goal of this Letter is to derive these analytical value is γc = 0.6275 leading to the critical velocity solutions and to develop their applications in various vc = 4.883. Note that [2] only the large-L behavior cases including the important running-coupling case. of the initial condition matters to determine whether This allows to make a direct and model-independent one lies in a pushed- or pulled-front case. comparison for saturation between QCD predictions and phenomenology. 2.1. Exact traveling-wave solutions of truncated BK The plan of the Letter is as follows. In Section 2,we equations define consistently truncated BK equations for which we derive exact traveling-wave solutions in the fixed- In order to avoid mathematical difficulties related to coupling case (Section 2.1) and study the validity of the infinite-order differential equation (2), let us now the method by comparison with numerical simulations formally consider a set of truncated approximations of of the original BK equation (Section 2.2). In Section 3, the BK equation: we find solutions to the BK equation with running N = − N − N 2 coupling. In Section 4, we apply the results to DIS ∂Y χP,γ0 ( ∂L) , (4) C. Marquet et al. / Physics Letters B 628 (2005) 239–249 241

where each truncated kernel χP,γ0 isgivenbythe of traveling waves. Indeed, consider the ansatz expansion of the original kernel χ around γ0 (0 < N (L, Y ) = A U(s), (8) γ0 < 1) up to order P 2: 0 P with the scaling variable s given by χ(p)(γ ) χ (−∂ ) = 0 (−∂ − γ )p P,γ0 L p! L 0 λ λ p=0 s ≡ L − A + A Y, (9) c 0 c 1 P √ = − p p = ( 1) Ap∂L . (5) where λ A0/A2 and c is a parameter related to p=0 the speed of the wave. The equation for the traveling wave U then becomes the following ordinary differen- P and γ0 are then two parameters which define a given approximation of the original kernel. Each particular tial equation kernel χP,γ0 is polynomial in ∂L and its coefficients 1 U(1 − U)+ U + U Ap can be fully computed from the original kernel χ: c2 they are given by P p p (−1) λ Ap P −p + (p) = (i+p) p U 0. (10) i χ (γ0) i c A0 A = (− ) γ . p=3 p 1 ! ! 0 (6) = i p i 0 Let us define For the sake of simplicity, we do not write the sub- = + script P,γ0 of the coefficients Ap. We want to con- v(c) A1 c A0A2 (11) sider kernels χ that are somewhat close to χ.To P,γ0 such that s = λ(L − v(c)Y)/c and that v(c) is the be more specific, we shall limit our study to kernels speed of the traveling wave. For a given kernel χ , that are positive definite and feature A > 0, A < 0 √ P,γ0 0 1 the values of c>−A / A A for which Eq. (10) has and A > 0. We also require that, just as the original 1 0 2 2 a solution define the possible values of the speed v(c) BK equation (2),Eqs.(4) admit asymptotic traveling- of the traveling-wave solution (8). wave solutions. In other words, for a kernel χP,γ to be 0 It is crucial to notice that, in this approach, c is a considered, the equation χP,γ (γ )/γ = χ (γ ) > 0 0 P,γ0 free and thus adjustable parameter. It means that in P,γ0 should have a unique solution γc , which we shall general, there is a continuum of solutions with differ- call the critical anomalous dimension. We shall also ent speed to Eq. (4) or equivalently (7). For instance, P,γ0 denote the corresponding critical velocity vc . if one wants to describe asymptotic solutions, we are

For the kernels χP,γc , for which the expansion has led to choose the value of c such that P,γ been done around γ , one has of course γ c = γ c c c = P,γ0 ≡ P,γ0 P,γc v(c) vc χ (γc ) if γi γc, and v = v . Then to approximate the solutions of P,γ0 (12) c c = the BK equation (2) by solutions of (4), one consid- v(c) χP,γ0 (γi)/γi if 0 <γi <γc. ers kernels expanded around γ0 γc to insure that the Indeed, once the initial condition is fixed, the asymp- speed of the resulting traveling wave is close to vc.We P,γ0 totic speed is vc or χP,γ0 (γi)/γi depending on the also expect that a relevant truncation is obtained with initial condition, as confirmed by numerical simula- a small value of P . tions. We shall use the freedom on c to also examine Inserting (5) into (4), the truncated BK equation non-asymptotic properties. reads: Eq. (10) provides a 1/cp expansion in with the 1/c 2 A0N − N − ∂Y N − A1∂LN term missing. The key point of the method is that 1/c P is indeed a small parameter [9,11] allowing to find an p p iterative solution + (−1) Ap∂ N = 0. (7) L p=2 1 1 1 h(s) = h0(s) + h2(s) + hp(s) ≡ − U(s). The advantage of this equation w.r.t. the original BK c2 cp 2 p3 equation is that it can be exactly solved [9,11] in terms (13) 242 C. Marquet et al. / Physics Letters B 628 (2005) 239–249

Inserting it into (10) translates into a hierarchy of the corresponding traveling-wave equation: equations 1 U(1 − U)+ U + U = 0. (17) + 2 − = c2 h0 h0 1/4 0, = + + + + = One has χ2,γ0 (γi)/γi A0/γi A1 A2γi and h2 2h0h2 h0 0, √ 2,γ0 = + 3 vc A1 2 A0A2 and therefore, using (12) one h + 2h0h3 − λ A3h /A0 = 0, 3 0 should take c = λ/γi +γi/λ in the “pushed front” case + + 2 + + 4 = = h4 2h0h4 h2 h2 λ A4h0 /A0 0, and c 2 in the “pulled front” case to reproduce the correct critical speed. . . (14) In order to check the efficiency of the iterative solu- where we have written the equations up to O(1/c5). tion, we use the existence of an analytic solution√[11] c = / An iterative solution can easily be found with to Eq. (17), for the particular value of 5 6 2.04: initial conditions being appropriately chosen, i.e., ±∞ =±1 ±∞ = = √ −2 h0( ) 2 and hi=0( ) hi(0) 0. Note cs + U(s)= 1 + ( 2 − 1) exp √ . (18) that if h(s) is solution, also h(s s0) is solution; 6 this freedom corresponds to the possibility of arbitrar- This exact solution allows one to see that limiting ily redefining k0. The system is iteratively but fully solvable. One first solves the only non-linear equation the expansion after the second order is a very good = 1 s approximation: in Fig. 1 we have plotted the exact of the hierarchy, obtaining h0 2 tanh( 2 ).Usingthe property solution (18) along with the solution given by the expansion√(16) limited after first and second order with d c = 5/ 6. It is very hard to distinguish the exact solu- h (s) + 2h h (s) ds n 0 n tion from the solution at second order. 1 d 2 = cosh (s/2)hn(s) , (15) cosh2(s/2) ds 2.2. Comparison with numerical solutions of the BK equation all the other linear equations reduce to simple integrations of functions defined recursively from the hi- In terms of physical variables, the solution (16) of erarchy. The solution of the first three terms of the Eq. (4) reads expansion is k2 λ/c s + s 2 A0 A0 Q2(Y ) 1 1 e (1 e ) N (k, Y ) = − s U(s)= − log 2 2 + s 2 s 2 s + k λ/c c2 + k λ/c 2 1 e c (1 + e ) 4e 1 2 1 2 Qs (Y ) Qs (Y ) 3 s − s 2 λ A3 e (1 e ) + k λ/c 2 − + s 1 2 3 s 2 3 s Q (Y ) 1 c A0 (1 + e ) (1 + e ) × log s + O , (19) k2 λ/c c3 1 4 2 + O . Qs (Y ) 4 (16) c 2 = 2 [¯ ] where Qs (Y ) k0 exp αv(c)Y plays the role of the The first two terms of the expansion (16) (order 1/c0 saturation scale and where we have restored the α¯ and 1/c2)areuniversal: they do not depend neither dependence of Y . In the following we fix √α¯ = 0.2. The A coefficients are given by (6), λ = A /A on P nor on γ0. Indeed Eq. (4) with any kernel χP,γ0 p 0 2 admits solutions whose first two terms (in the 1/c ex- and c is√ chosen so that the traveling speed v(c) = pansion) are those of (16). The solutions for different A1 + c A0A2 takes the value of the critical speed P,γ0 kernels only differ through the definition (9) of the vc in the “pulled front” case and χP,γ0 (γi)/γi in scaling variable s which depends on P and γ0.Inthis the “pushed front” case, see (12). In the following, we sense, we obtain a universal parametric solution. shall only use the first two terms given in (19) which Let us concentrate on the case P = 2. Eq. (4) re- provide a good approximation of the full traveling- duces to the known FKPP equation and one obtains wave solution. C. Marquet et al. / Physics Letters B 628 (2005) 239–249 243

√ Fig. 1. The function U(s) solution of the equation U(1 − U)+ U + U /c2 = 0forc = 5/ 6 with U(−∞) = 1andU(0) = 1/2. Full line: exact solution (18). Dashed line: solution (16) containing only the ﬁrst order. Dotted line (hardly distinguishable from the full line): solution (16) containing only the ﬁrst and second order.

Let us compare numerical solutions of the BK For our purpose, it is useful to investigate the non- equation with the solution (19) for P = 2. For the nu- asymptotic regime since it is of interest for phenom- merical simulations, the initial condition we chose3 enological applications. In Fig. 3, we compare the −γ L is N (L, Y0) = e i /(2γi) if L 0 and N (L, Y0) = numerical solutions of the BK equation with the so- (1 − γiL)/(2γi) if L<0. As already mentioned, only lution (19), in the pulled front case, where we adjusted the large-L behavior of the initial condition is im- the parameter c to describe the speed of the wave v at portant to determine whether one lies in the pushed- moderate values of the rapidity Y up to 20. Again the or pulled-front case. For the analytical solution (19), agreement in the interior region is satisfactory and it the value of c that gives the critical speed is c = shows that the solution we obtained can be used also λ/γi + γi/λ in the pushed-front case and c = 2in in the physical range of non-asymptotic rapidities. the pulled-front case for which we recall the value Let us finally point out the role of the kernel trunca- γc = 0.6275. tion. First, the truncation up to a finite P is mathemat- The comparison is shown in Fig. 2: the “pulled ically necessary to apply our method. Indeed, when front” case is represented on the plot (a) for γi = 1 considering the full analytic expansion to define the and with the kernel expanded around γ0 = γc, the crit- coefficients Ai (i.e., P →∞), they are infinite due ical value. The “pushed front” case is represented on to the γ = 0 singularity of the BFKL kernel. This the plot (b) for γ0 = γi = 0.5. One sees that as soon as explains why the solutions (19) describe the interior the asymptotic traveling-wave regime is reached, there region of the BK front but not the whole k range. is a good description of the “interior” of the wave, For instance, one does not describe the saturated re- as defined in Ref. [7]. In both cases, one sees that gion [12]:formula(19) describes fronts which sat- Eq. (19) does not describe neither the saturated region urate at A0 while the BK front behaves as log 1/k (small k) nor the “leading-edge” (large k). Indeed, the when k is very small. A second remark is that we exact scaling properties of traveling waves are mathe- observed, e.g., by comparison of the results of the matically expected only in the interior region [7]. P = 2 and P = 4 truncations, that a good description of the interior region of the wave is already obtained for P = 2, i.e., the diffusive approximation of 3 This form ensures correct infrared and ultraviolet behaviors with the BFKL kernel. In this Letter, we shall thus stick to an appropriate matching. The details of this matching are not rele- = vant for the asymptotic properties. P 2. 244 C. Marquet et al. / Physics Letters B 628 (2005) 239–249

(a)

(b)

Fig. 2. The traveling wave N (L, Y ) (ﬁxed coupling) at large Y as a function of L. The different values of Y are 0, 20, 40,...,100. The full lines are numerical solutions of the BK equation and the dashed lines are the scaling solutions (19) for P = 2. (a) “Pulled-front” case (γ0 = γc, γi = 1): the asymptotic traveling-wave regime is reached after some evolution with the critical speed vc = 4.883. (b) “Pushed front” case (γ0 = γi = 0.5 <γc): the asymptotic regime is driven by the initial condition and the critical speed is χ2,1/2(1/2)/(1/2) = 5.55. 11N − 2N k2 3. The running-coupling case b = c f ,L= , log 2 (20) 12Nc ΛQCD Let us now perform the analysis in the running- and ΛQCD = 200 MeV. We thus write the following coupling case with form of the BK equation with running-coupling con- N 1 stant and leading-order BFKL kernel (3): c α k2 = , s 2 π bL bL∂Y N = χ(−∂L)N − N . (21) C. Marquet et al. / Physics Letters B 628 (2005) 239–249 245

Fig. 3. The traveling wave N (L, Y ) (ﬁxed coupling) at moderate Y as a function of L. The different values of Y are 0, 2, 4,...,20. The full lines are numerical solutions of the BK equation and the dashed lines are the scaling solutions (19) for P = 2, γ0 = γc,andγi = 1. The speed of the wave has been adjusted to the value v(c) = 4.2.

It has been proved [2] that this equation admits as- where c˜ is a free parameter. Putting this into (22) gives √ymptotic traveling-wave solutions of the form N (L − (2v/b)Y ) where as before v = vc in the “pulled P p A0 Ap (p) 1 front” case and v = χ(γi)/γi in the “pushed front” U(1 − U)+ U + U + O = 0. A1 A0 c˜ case. p=2 Let us consider the same truncated kernels than in (25) the previous section. That transforms the equation into The leading-order terms in 1/c˜ are of the same generic form as Eq. (10) and thus one can apply the previ- N − N 2 − N − N A0 bL∂Y A1∂L ous method. In this running-coupling case, the O(1/c)˜ P terms that we shall neglect contain scaling-violation + − p pN = ( 1) Ap∂L 0 (22) corrections decreasing as 1/L. Using the method de- p=2 scribed in the previous section, one obtains the follow- and we consider again the following form ing solution

N = ˜ 1 (L, Y ) A0U(s). (23) U(s)˜ = 1 + es˜ Our strategy is to reduce the problem to the one we A 2 A es˜ (1 + es˜)2 solved in the previous section through an appropri- − 0 2 log s˜ 2 s˜ ate change of variables. We require that the univer- A1 A0 (1 + e ) 4e sal terms in the corresponding equation for U are A3 the same as before: U(1 − U)+ U . Postulating s˜ = + O 0 , (26) A3 Lφ(Y/L2) determines a solution for φ and leads to the 1 scaling variable where the expansion parameter is A0/A1. As before, we shall only consider two first terms of the expansion. A 1 Y s˜ ≡ L − 0 − b − 2A , (24) In order to describe solution to the BK equation (21), ˜ 1 2 A1 c L one has now to select the speed of the wave by fixing 246 C. Marquet et al. / Physics Letters B 628 (2005) 239–249 the parameter c˜. Writing at large Y Fig. 4(b) which are both significantly smaller than the critical value v 4.88. It confirms that the estab- A A c s˜ − 0 L + 1 − A Y lishment of the asymptotic regime is much slower in ˜ 2 1 (27) A1 A0c the running-coupling case than in the fixed-coupling √and matching with the asymptotic form N (L − case [2]. Indeed, the theoretical sub-asymptotic cor- (2v/b)Y ), one gets rections to the critical speed have been computed [2] and predict such a behavior. There is a good descrip- bA3 v(c)˜ =− 1 . (28) tion of the interior of the wave for the different ranges 2 ˜2 A0c of rapidity, provided we adjust the speed of the wave. As expected, the parametrization does not describe the Adjusting c˜ to obtain a given speed, we check that saturated and the leading-edge regions. Note that the 1/c˜ is indeed small. Defining Q˜ 2(Y ) = Λ2 × s QCD scaling region at low values of Y is less extended in the ˜ exp( (2v(c)/b)Y ) that plays the role of the saturation running-coupling case (Fig. 4(b)) than for fixed cou- scale, the scaling variable reads in terms of physical pling (Fig. 3), this could be due to the scaling-violation variables terms that we had to neglect in (25). However it is clear A k2 that an approximative traveling-wave pattern already s˜ =− 0 log 2 emerges at moderate rapidities. A1 Λ QCD 2 1 k2 A c˜2 Q2(Y ) − b log2 + 0 log2 s . ˜ 2 2 2 c ΛQCD A1 ΛQCD 4. Application to deep inelastic scattering (29) Formula (29) along with We shall now draw a link between traveling waves

2 s˜ s˜ 2 and phenomenology. The observation of geometric A A A2 e (1 + e ) N (k, Y ) = 0 − 0 log scaling asks the question whether it can be viewed as + s˜ 2 + s˜ 2 s˜ 1 e A1 (1 e ) 4e a consequence of QCD traveling waves. The known (30) problem is that the range of rapidity for which asymp- is the traveling-wave solution of the truncated BK totic predictions of QCD non-linear evolution equa- equation with running coupling (21) at O(1/c)˜ . tions are available remains far from the moderate ra- Let us compare numerical solutions of the full BK pidity range accessible to experiments. Hence the phe- equation (21) with that parametrization. For the nu- nomenological interest of our method is to open a way merical simulations, the initial condition is the same as of confronting theory and experiment in some physical in the fixed-coupling case. The problem of the Landau kinematic range. pole of the running coupling is dealt with by introduc- The formula relating the dipole scattering ampli- 2 2 → + 2 2 ing a regulator: log(k /ΛQCD) log(κ k /ΛQCD) tude in coordinate space N(r, b,Y)to the unintegrated with κ = 3. As well as the details of the initial condi- gluon distribution of the target f(Y,k2) is (see, e.g., tion, this does not modify the traveling-wave pattern. [13]) The comparison is represented in Fig. 4 where the “pulled front” case is considered. The values of the 2 coefficients A0 = 9.55, A1 =−25.6 and A2 = 24.3 d bN(r, b,Y) = = have been fixed by considering P 2 and γ0 γc. We use the free parameter c˜ to adjust the speed of the 2 = 4π αs dk 2 − | | = f Y,k 1 J0 k r . (31) wave. On Fig. 4(a), high values of the rapidity Y Nc k 0, 10, 20,...,100 are considered while on Fig. 4(b), moderate values Y = 0, 2, 4,...,20 are represented. In the b-independent approximation, one writes 2 = 2 In both cases, in order to describe the numerical solu- d bN(r, b,Y) πRpN(r,Y) where Rp is the tions, one has to adjust the actual speed of the wave radius of the proton target. Through formulae (1) v(c)˜ to the value of 3.1forFig. 4(a) and 2.3for and (31) a link is established between N (k, Y ) and C. Marquet et al. / Physics Letters B 628 (2005) 239–249 247

(a)

(b)

Fig. 4. The traveling wave N(L,Y) with running coupling as a function of L. The full lines are numerical solutions of the BK equation with running-coupling and the dashed lines are the parametrization (29) and (30). (a) High-energy regime Y = 0, 10, 20,...,100 and v(c)˜ = 3.1. (b) Moderate energy regime Y = 0, 2, 4,...,20 and v(c)˜ = 2.3. The initial condition is chosen in the pulled-front regime. f(Y,k2). One gets xg(x,k2) to obtain

∞ ∞ παs dt t 4παs dp 2 p N (k, Y ) = xg(x,t)log . (34) N (k, Y ) = f Y,p log . (32) 2 2 2 2 NcRp t ek NcRp p k 2 k k Finally, we use the relation Note that formula (33) is a well-known approximation which may only be valid at small values of x.Itis ∂ useful here as it makes the analysis simpler. For a more f Y,k2 = xg x,k2 , (33) detailed study which is beyond the scope of this work, ∂k2 one could consider more advanced prescriptions [14]. where Y = log(1/x) between the unintegrated gluon In order to compare (34) to our predictions, we use distribution f(Y,k2) and the gluon distribution the running-coupling case which is known to lead to 248 C. Marquet et al. / Physics Letters B 628 (2005) 239–249

Fig. 5. The dipole amplitude N(k,Y) as a function of L for different values of Y = 4, 5,...,12. The full lines are obtained from the MRST gluon distribution via (34) and the dashed lines represent the parametrization (29) and (30). Only one term is kept in (30). traveling waves with a speed compatible with phenom- A0 = 10, A1 =−25, and the speed 2.3 0.01 which is the region where the presence of next-leading or higher-order correc- παs traveling-wave patterns can be tested. Note that we tions to the QCD kernel. This deserves further study. | |∼ considered only the first term of (30) which depends (iv) Note that we obtain A0/A1 1 for the observed parameters which could invalidate the expan- on the parameters A0, A1, and v the speed of the wave. We discarded the second term of (30) since we found sion (26). We checked both numerically and from the analytical form that the second and higher order terms that the parameter A2 stays undetermined by the fit in the kinematical region we considered. In Fig. 5,we stay sufficiently small to be still neglected. show the result of the fit, the obtained parameters being A0 = 17.1, A1 =−15.8, and v = 1.76. These results call for comments. 5. Conclusion

(i) The dipole amplitude (34) obtained from the Let us summarize the main results of our study. MRST distribution (full lines on Fig. 5) displays a structure compatible with an evolution towards trav- • We found iterative traveling-wave solutions to eling waves, namely a steeper slope at small rapidities non-linear BK evolution equations obtained by finite which evolves toward a less steep and regular pattern truncation of the BFKL kernel for both fixed and at higher rapidities. running-coupling constant. (ii) Fig. 5 shows a compatibility with the paramet- • These solutions exhibit universality properties, ric form ((29), (30)) at high values of Y and moderate the first two dominant terms of the iteration have a values of k2. parametric form independent of the truncation of the (iii) On the theoretical ground, it is worth compar- kernel and of the fixed- or running-coupling cases. ing the fitted parameters to various predictions. For the • The scaling variable is found to be a combina- 2 truncated BK kernel with P = 2 and γ0 = γc, one finds tion of L ≡ log(k ) and rapidity Y which is given by C. Marquet et al. / Physics Letters B 628 (2005) 239–249 249 formula (9) (respectively (24)) for fixed (respectively [2] S. Munier, R. Peschanski, Phys. Rev. Lett. 91 (2003) 232001, running) coupling. hep-ph/0309177; • The obtained traveling-wave solutions match S. Munier, R. Peschanski, Phys. Rev. D 69 (2004) 034008, hep- ph/0310357; with the asymptotic solutions of the BK equation. S. Munier, R. Peschanski, Phys. Rev. D 70 (2004) 077503, hep- This was verified by analytical and numerical checks. ph/0401215. The remarkable new property is that they also match [3] I.I. Balitsky, Nucl. Phys. B 463 (1996) 99, hep-ph/9509348. with the non-asymptotic behavior of the interior of the [4] Y.V. Kovchegov, Phys. Rev. D 60 (1999) 034008, hep- wave, provided an adjustment of the speed of the wave ph/9901281; Y.V. Kovchegov, Phys. Rev. D 61 (2000) 074018, hep- which is a free parameter in the iterative approach. ph/9905214. • As an application of the method and its valid- [5] R.A. Fisher, Ann. Eugenics 7 (1937) 355; ity at non-asymptotic energies, we considered the di- A. Kolmogorov, I. Petrovsky, N. Piscounov, Moscow Univ. pole amplitude in momentum space N (k, Y ) obtained Bull. Math. A 1 (1937) 1. from the MRST parametrization of the gluon distri- [6] M. Bramson, in: Memoirs of the American Mathematical So- ciety, vol. 285, 1983. bution. We found evidence for an evolution pattern [7] U. Ebert, W. van Saarloos, Physica D 146 (2000) 1; compatible with the formation of traveling waves. The For a review, see W. van Saarloos, Phys. Rep. 386 (2003) 29. obtained dipole amplitude is well described by the uni- [8] C. Marquet, R. Peschanski, G. Soyez, Nucl. Phys. A 756 versal parametric form (30). The obtained parameters (2005) 399, hep-ph/0502020; seem to point towards traveling-wave solutions for a C. Marquet, G. Soyez, Nucl. Phys. A 760 (2005) 208, hep- ph/0504080. BFKL kernel modified by higher orders and/or non- [9] R. Peschanski, Phys. Lett. B 622 (2005) 178, hep-ph/0505237. perturbative corrections. [10] L.N. Lipatov, Sov. J. Nucl. Phys. 23 (1976) 338; E.A. Kuraev, L.N. Lipatov, V.S. Fadin, Sov. Phys. JETP 45 Our results help establishing a stronger link be- (1977) 199; tween the theory of saturation and the phenomenolog- I.I. Balitsky, L.N. Lipatov, Sov. J. Nucl. Phys. 28 (1978) 822. [11] J. David Logan, An Introduction to Nonlinear Partial Differen- ical observation of geometric scaling [1]. Moreover, tial Equations, Wiley, New York, 1994; simple analytical parametrizations of the solutions of See also P.L. Sachdev, Self-Similarity and Beyond, Exact Solu- the BK equation in momentum space could help ex- tions of Nonlinear Problems, Chapman and Hall, Boca Raton, tending the present phenomenology [17] to a broader 2000. The first two terms of the F-KPP solution appear in range of kinematics or observables. It would be in- P.L. Sachdev’s book. [12] For the deep saturation region, see, e.g., M. Kozlov, E. Levin, teresting to investigate where one could distinguish hep-ph/0504146. between a DGLAP structure and traveling waves. On [13] A. Bialas, H. Navelet, R. Peschanski, Nucl. Phys. B 593 (2001) a more theoretical level, it seems feasible to extend 438, hep-ph/0009248. the method to QCD evolution equations beyond the [14] M.A. Kimber, A.D. Martin, M.G. Ryskin, Eur. Phys. J. C 12 “mean field” BK equation (2). In particular the ex- (2000) 655, hep-ph/9911379; M.A. Kimber, A.D. Martin, M.G. Ryskin, Phys. Rev. D 63 tension to the stochastic versions of QCD evolution (2001) 114027, hep-ph/0101348; equations [18] would help our understanding of QCD M.A. Kimber, J. Kwiecinski, A.D. Martin, A.M. Stasto, Phys. in the high-energy limit. Rev. D 62 (2000) 094006, hep-ph/0006184. [15] D.N. Triantafyllopoulos, Nucl. Phys. B 648 (2003) 293, hep- ph/0209121. Acknowledgement [16] A.D. Martin, R.G. Roberts, W.J. Stirling, R.S. Thorne, Phys. Lett. B 531 (2002) 216, hep-ph/0201127. [17] K. Golec-Biernat, M. Wusthoff, Phys. Rev. D 59 (1999) G.S. is funded by the National Funds for Scientific 014017, hep-ph/9807513; Research (Belgium). E. Iancu, K. Itakura, S. Munier, Phys. Lett. B 590 (2004) 199, hep-ph/0310338. [18] E. Iancu, A.H. Mueller, S. Munier, Phys. Lett. B 606 (2005) References 342, hep-ph/0410018; E. Iancu, D.N. Triantafyllopoulos, Nucl. Phys. A 756 (2005) 419, hep-ph/0411405. [1] A.M. Stasto,´ K. Golec-Biernat, J. Kwiecinski, Phys. Rev. Lett. 86 (2001) 596, hep-ph/0007192. Physics Letters B 628 (2005) 250–261 www.elsevier.com/locate/physletb

Heavy Majorana neutrino production at e−γ colliders

Simon Bray, Jae Sik Lee, Apostolos Pilaftsis

Department of Physics and Astronomy, University of Manchester, Manchester M13 9PL, United Kingdom Received 15 August 2005; received in revised form 20 September 2005; accepted 20 September 2005 Available online 28 September 2005 Editor: N. Glover

Abstract − We study signatures of heavy Majorana neutrinos at e γ colliders. Since these particles violate lepton number, they can − − − + − + − − give rise to the reactions e γ → W W e and e γ → W µ µ ν. The Standard Model background contains extra light neutrinos that escape detection, and can be reduced to an unobservable level after imposing appropriate kinematical cuts. We − √ analyze the physics potential of an e γ collider, with centre of mass energies s = 0.5–1 TeV and a total integrated luminosity −1 of 100 fb , for detecting heavy Majorana neutrinos with masses mN = 100–400 GeV. Assuming that heavy neutrinos couple predominantly to only one lepton flavour at a time, we find that 10 expected background-free events can be established for a −3 −2 WeN-coupling |BeN | 4.6 × 10 or for a WµN-coupling |BµN | 9.0 × 10 . Instead, if no signal is observed, this will −3 imply that |BeN | 2.7 × 10 and |BµN | 0.05 at the 90% confidence level.  2005 Published by Elsevier B.V.

1. Introduction

Neutrino oscillation experiments [1–3] have established the fact that the observed light neutrinos are not strictly massless, as predicted in the Standard Model (SM), but have non-zero tiny masses. One well-motivated framework for giving small masses to neutrinos is to extend the SM by adding one right-handed neutrino per family. In such a scenario, the right-handed neutrinos are singlets under the SM gauge group, and so are allowed to have large Majorana masses in the Lagrangian. Apart from the three observable neutrinos ν1,2,3, the resulting spectrum will contain three additional heavy neutrinos N1,2,3, whose masses could range from the electroweak to the Grand Uniﬁed Theory (GUT) scale [4]. Direct searches at LEP put strong limits on the couplings of heavy Majorana neutrinos weighing less than about 100 GeV [5]. For larger heavy Majorana masses, the indirect limits are still signiﬁcant and usually arise from the non-observation of sizeable quantum effects on low-energy observables, as

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

0370-2693/$ – see front matter  2005 Published by Elsevier B.V. doi:10.1016/j.physletb.2005.09.037 S. Bray et al. / Physics Letters B 628 (2005) 250–261 251 well as from the absence of lepton flavour violating decays [6–14], e.g., µ → eγ , µ → eee and µ → e conversion in nuclei, etc. In this Letter we study the distinctive signatures of lepton number violation (LNV) at an e−γ collider, which are mediated by heavy Majorana neutrinos. In particular, we consider the LNV reactions e−γ → W −W −e+ and e−γ → W +µ−µ−ν. These processes are strictly forbidden in the SM, and the corresponding background will always involve additional light neutrinos. An e−γ collider provides a clean environment to look for such particles, since the initial state e−γ has a definite non-zero lepton number and hence any LNV signal can easily be detected. There are already realistic proposals [15] for the construction of an e−γ machine, which will run at 0.5–1 TeV centre of mass system (CMS) energies and a total integrated luminosity of 100 fb−1. Here, we will analyze the physics potential of such a machine to discover heavy Majorana neutrinos with masses between 100 and 400 GeV. Our study has been structured as follows. After briefly reviewing the SM with right-handed neutrinos in Section 2, we discuss in Section 3 the LNV signals due to heavy Majorana neutrinos and the associated backgrounds. In Section 4 we present numerical results for the cross sections of the reactions e−γ → W −W −e+ and e−γ → W +µ−µ−ν. In particular, taking into account the SM backgrounds, we place lower limits on the heavy neutrino couplings to the charged leptons and the W ± bosons, by requiring that an observable LNV signal is obtained. Finally, our conclusions are summarized in Section 5.

2. The Standard Model with right-handed neutrinos

We briefly review the relevant low-energy structure of the SM modified by the presence of right-handed neutrinos. The Lagrangian describing the neutrino masses and mixings reads: 1 0 c Lmass =− ¯ 0 ¯0 c 0 mD (νL) + ν νL (νR) T 0 h.c., (2.1) 2 mD MR νR 0 = T 0 = T where νL (νeL,νµL,ντL) and νR (νeR,νµR,ντR) collectively denote the left- and right-handed neutrino fields in the weak basis, and mD and MR are 3 × 3 complex matrices. The latter obeys the Majorana constraint = T MR MR and, without loss of generality, can be assumed to be diagonal and positive. The weak neutrino states are related to the Majorana mass eigenstates through: 0 νL = T νL U 0 c , (2.2) NL (νR) where U is a 6 × 6 unitary matrix. In typical seesaw scenarios [4], the Dirac mass terms mD are expected to be around the electroweak scale, e.g., mD ∼ Ml or mD ∼ Mu, where Ml (Mu) is the charged lepton (up-quark) mass matrix, whilst the Majorana mass MR being singlet under the SM gauge group may be very large, close to the GUT scale. Seesaw models ∼ 2 can explain the smallness of the observed light neutrino masses, generically predicted to be mD/MR, through the huge hierarchy between mD and MR. However, the couplings of the heavy neutrinos to SM particles will be generically highly suppressed ∼mD/MR, thus rendering the direct observation of heavy Majorana neutrinos impossible. Another and perhaps phenomenologically more appealing solution to the smallness of the light neutrino masses may arise due to approximate flavour symmetries that may govern the Dirac and Majorana mass matrices mD and MR [16–18]. In such models, the heavy-to-light Majorana couplings are also ∼mD/MR, but they can be completely =− −1 T unrelated to the light neutrino mass matrix mν , which is determined by the relation: mν mDMR mD. In such non-seesaw models, one may have mD ∼ Ml and MR ∼ 100 GeV, without being in contradiction with neutrino data. However, as we will see below, these models are constrained by electroweak precision data and other low- energy lepton-flavour/number violating observables. Our study of LNV signatures at an e−γ collider will focus on such non-seesaw realizations. 252 S. Bray et al. / Physics Letters B 628 (2005) 250–261

For our subsequent phenomenological discussion, we now exhibit the interaction Lagrangians of the heavy neutrinos to W ±, Z and Higgs (H ) bosons [17]: L =−√g − ¯ µ ν + W Wµ llγ PLBlj h.c., (2.3) 2 N j g ν L =− Z ( ν¯ N)¯ γ µ(i Im C − γ Re C ) , (2.4) Z 4c µ i ij 5 ij N w j g ¯ ν LH =− H ( ν¯ N )i (mi + mj ) Re Cij − iγ5(mi − mj ) Im Cij , (2.5) 4MW N j where mi,j (with i, j = 1, 2,...,6) denote the physical light and heavy neutrino masses, and 3 3 = L ∗ = ∗ Blj Vlk Ukj ,Cij UkiUkj , (2.6) k=1 k=1 with l = e,µ,τ.In(2.6), V L is a 3 × 3 unitary matrix relating the weak to mass eigenstates of the left-handed charged leptons. As was mentioned above, the mixing elements Blj can be constrained from LEP and low-energy electroweak data [6–14]. Following [19], we define 3 3 ∗ ∗ Ω ≡ δ − B B = B B . (2.7) ll ll lνi l νi lNi l Ni i=1 i=1 νe,µ,τ 2 The above definition is a generalization of the Langacker–London parameters (sL ) [6], with the identification: = νl 2 Ωll (sL ) . At the 90% confidence level (CL), the allowed values for the parameters Ωll are [12]:

Ωee 0.012,Ωµµ 0.0096,Ωττ 0.016. (2.8) These limits depend only weakly on the heavy neutrino masses. Lepton ﬂavour violating processes, e.g., µ → eγ [10], µ → eee, τ → eee [11,13,14], can be induced by heavy neutrino quantum effects. Such processes do depend on the heavy neutrino masses and Yukawa couplings [11]. For mN MW and mD MW , the limits derived after including the recent BaBar constraints [20] from B(τ → e,µγ ) 10−7 are

|Ωeµ| 0.0001, |Ωeτ | 0.02, |Ωµτ | 0.02. (2.9) Finally, we present the partial decay widths of a heavy Majorana neutrino N for its dominant decay channels [17]: 2 ± ∓ α |B | Γ N → l W = w lN m2 − M2 2 m2 + 2M2 , (2.10) 2 3 N W N W 16MW mN α |C |2 Γ(N→ ν Z) = w νi N m2 − M2 2 m2 + 2M2 , (2.11) i 2 3 N Z N Z 16MW mN α |C |2 Γ(N→ ν H)= w νi N m2 − M2 2, i 2 N H (2.12) 16M mN W = 2 O 4 4 | |2 = where αw g /(4π). Notice that up to negligible corrections (mD/MR), it can be shown that l BlN | |2 i Cνi N . Hence, for heavy neutrinos, with mN MH , decaying into all charged leptons and light neutrinos, one obtains the useful relation among the branching fractions: B(N → l+W −) = B(N → l−W +) = B(N → νZ) = B(N → νH) = 1/4. S. Bray et al. / Physics Letters B 628 (2005) 250–261 253

(a) (b) (c) (d)

− − − + Fig. 1. Feynman diagrams for the process e γ → W W l .

3. Lepton number violating signatures

Previous analyses of single heavy neutrino production have mainly been focused on e+e− linear colliders [19, 21,22], where the process e+e− → Nν → l±W ∓ν is considered. Such a production channel, however, has two problems not associated with an e−γ collider. Firstly, since the light neutrinos escape detection with their chirality undetermined, the possible Majorana nature of the heavy neutrinos has very little effect on the signal, which makes the reduction of the contributing SM background very hard. Secondly, an observable signal would require a heavy ± −2 − neutrino coupling to the electron and the W bosons of reasonable strength, i.e., |BeN | 10 , whereas at an e γ collider a signal can still be seen for |BµN | 0.1, even if BeN = 0. Another possible option which might have the same advantages as an e−γ collider is a γγ collider. Here, the heavy Majorana neutrinos can be produced and observed via γγ → W +Nl− → W +W +l−l− [23]. However, like the case of the e+e− collider, one still has to assess the impact of the contributing SM background. Depending on the relative strength of the heavy neutrino coupling to the electron BeN , we consider the two reactions: (i) e−γ → W −W −l+ and (ii) e−γ → W +l−l−ν provided that e−γ → W −W ∓l± is not observed (see also our discussion at the end of Section 3.3). Both processes are manifestations of LNV, and as such, they allow to eliminate major part of the SM background. The process (i) dominates for relatively large values of the mixing −2 factor BeN , i.e., BeN 10 , whereas the process (ii) becomes only relevant if BeN is unobservably small, e.g., −3 BeN 10 .

3.1. BeN = 0

− − − + −2 We first consider the process e γ → W W l , which becomes the dominant signal for BeN 10 .The Feynman diagrams pertinent to this process are shown in Fig. 1.1 We improve upon an earlier study of this reaction [25], by carefully considering the contributing SM background (see our discussion in Section 3.3). The reaction e−γ → W −W −l+ is dominated by the graphs (a) and (b) of Fig. 1, where the heavy Majorana neutrinos occur in the s-channel. Considering therefore the 2 → 2 sub-process e−γ → W −N, Fig. 2 gives the − − √ cross section σ(e γ → W N) as functions of s and mN . The differential cross sections related to Fig. 2,aswell as to Figs. 4 and 7, have been calculated using the FeynCalc package [26]. We have also checked that our results are in agreement with [25]. Approximate values for the cross section σ(e−γ → W −W −l+) can be obtained by multiplying the y-axis values from Fig. 2 with the branching fraction B(N → W −l+). However, as well as ignoring contributions from off-shell heavy neutrinos, this approximation makes the imposition of kinematical cuts on the three-momentum of the final charged lepton rather difficult. Therefore, in Section 4, we calculate the LNV signatures by considering the complete 2 → 3 process, as it is described by the full set of Feynman diagrams depicted in Fig. 1. To this end, we have extended CompHEP [27] to include heavy Majorana neutrino interactions.

1 All diagrams, including Figs. 3 and 5, are produced with Axodraw [24]. 254 S. Bray et al. / Physics Letters B 628 (2005) 250–261

− − √ Fig. 2. Numerical estimates of the cross section σ(e γ → W N) as functions of s (left frame) and mN (right frame), with BeN = 0.07.

(a) (b) (c) (d) (e)

− + − − Fig. 3. Feynman diagrams for the process e γ → W l l ν, with BeN = 0.

3.2. BeN = 0

If the coupling of the heavy neutrino N to the electron is either zero or very small, then the dominant process is e−γ → W +l−l−ν, where l = µ,τ. The Feynman diagrams for this process are shown in Fig. 3. Although the cross section of the new reaction is much smaller in this case, so is the background. Hence, the predicted LNV signal could still be within observable reach. This is a major beneﬁt compared to an e+e− collider, which requires sizeable heavy neutrino couplings to the electron for an observable signal.2 To the best of our knowledge, this is a novel possibility which has not been investigated before. The cross section for the LNV reaction e−γ → W +l−l−ν is dominated by the resonant s-channel exchange graphs (a)–(c) of the heavy Majorana neutrino N,asshowninFig. 3. Thus, we present in Fig. 4 the heavy neutrino − − √ production cross section σ(e γ → Nµ ν) as functions of s and mN . The 4-dimensional phase-space integration involved in the 2 → 3 process was performed numerically using Bases [28]. The cross section values obtained for this process are about 2 orders of magnitude smaller than those presented in Fig. 2, for the reaction e−γ → W −N. An approximate estimate of σ(e−γ → W +µ−µ−ν) may be obtained by σ(e−γ → Nµ−ν)B(N → W +µ−).As was mentioned in Section 3.1, this method has certain limitations when differential cross sections are considered. For this reason, the LNV signal in Section 4 is computed using the full set of the diagrams displayed in Fig. 3 and our extended version of CompHEP that includes heavy Majorana neutrino interactions.

− + ∗ + − 2 There may exist other processes as well, such as e e → Z → NN and e e → Nννν. The ﬁrst cross section is suppressed by the O 2 = O | |4 O 3 | |2 small Z-coupling to a pair of heavy neutrinos, i.e., it is (CNN) ( BlN ). The second reaction is sub-dominant (αw BlN ) due to − − − − the additional gauge coupling constants involved. Moreover, the cross section of the possible reaction e e → W W is suppressed of order 4 |BeN | , and severely constrained by the current 0νββ-decay data. S. Bray et al. / Physics Letters B 628 (2005) 250–261 255

− − √ Fig. 4. Numerical estimates of the cross section σ(e γ → Nµ ν) versus s (left frame) and mN (right frame), where BeN = 0, BµN = 0.1 − and an infrared angle cut 0.99 cos θe−µ− 0.99 is used.

− − − + Fig. 5. Typical dominant Feynman diagrams related to the SM reaction e γ → W W W νe.

3.3. The Standard Model background

The LNV reactions we have been considering are strictly forbidden in the SM. The contributing SM background that may mimic the signal will always involve additional light neutrinos. Specifically, the SM background arises ± − − − + from the resonant production and decay of three W bosons, i.e., e γ → W W W νe. Typical dominant graphs of this 2 → 4 scattering process are shown in Fig. 5. − − − + + + The background to the first LNV reaction e γ → W W l originates from the decay W → l νl. Corre- spondingly, the background to the second LNV process e−γ → W +l−l−ν results from the leptonic decays W − → − − − − − + l ν¯l for both of the W bosons in the final state. To compute the background process e γ → W W W νe,we first use CompHEP [27], which also generates the appropriate weighted events, and then use PYTHIA [29] inter- faced via the CPyth program [30]. The last step is necessary in order to properly describe the Lorentz-boosted W ±-boson decay products on which appropriate kinematical cuts were placed. These kinematical cuts will be presented in detail in Section 4. Although our method uses the branching fractions for the W decays [31] as an input, it proves by far more practical than generating events for the complete set of 2 → 7 background processes, where a vast number of off-resonant amplitudes give negligible contributions. 256 S. Bray et al. / Physics Letters B 628 (2005) 250–261

It is important to clarify at this point that observation of e−γ → W +l−l−ν does not constitute by itself a signature for LNV, even if l = e. If the heavy neutrino couples to the electron as well as to the muon or tau lepton, then the occurrence of such a reaction will only signify lepton ﬂavour violation. If one now assumes that e−γ → W −W −l+ is not observed, there are two possibilities that could result in an observation of e−γ → W +l−l−ν. Either the heavy neutrino is a Majorana particle that does not couple predominantly to the electron, or it is a Dirac particle that does. To distinguish between these two possibilities, one has to look for e−γ → W −W +l−, where both W ± bosons decay hadronically. This process cannot occur, unless the heavy neutrino has a relevant coupling to the electron. For a heavy Dirac neutrino, however, the latter will always be larger than e−γ → W +l−l−ν. Therefore, if the reactions e−γ → W −W ∓l± were not detected, observation of e−γ → W +l−l−ν would be a clear manifestation of a LNV signature, which is mediated by a heavy Majorana neutrino.

4. Numerical results

There are many theoretical parameters in the SM with right-handed neutrinos that could vary independently, such as heavy Majorana neutrino masses and couplings. In addition, the machine parameters of any future e−γ collider are still under discussion. Since it would be counter-productive to explore all possible situations that may occur, we will, instead, analyze two representative scenarios for each of the processes: (i) e−γ → W −W −e+ and (ii) e−γ → W +µ−µ−ν. | |2 ≈| |2 In our analysis, we take BlN Cνi N , i.e., we assume that only one ﬂavour will couple predominantly to the heavy Majorana neutrino N. The Higgs-boson mass value MH = 120 GeV is used throughout our estimates, but our results depend only very weakly on MH . Finally, we use the global kinematical cut,

−0.99 cos θe−l± 0.99, (4.1) − ± for the angle θe−l± between the e in the initial state and the ﬁnal-state charged leptons l . The global cut (4.1) was used to ensure that the produced charged leptons l± are detected.

4.1. e−γ → W −W −e+

We will consider the simplest case where BeN is sizeable, but BµN = BτN = 0. We analyze two scenarios:

Scenario 1. √ s = 500 GeV,mN = 200 GeV,BeN = 0.07, − + predicting ΓN = 0.04 GeV and B(N → W e ) ≈ 0.3;

Scenario 2. √ s = 1TeV,mN = 400 GeV,BeN = 0.07, − + predicting ΓN = 0.4 GeV and B(N → W e ) ≈ 0.25.

If the two W − bosons decay hadronically, the LNV signal will have no missing momentum, whilst the SM − − − + background e γ → W W e νeνe has two light neutrinos that will escape detection. Making use of this fact, one may apply the kinematical cut on the missing transverse momentum pT : max = pT pT 10 GeV. (4.2) This has no effect on the signal, but reduces the background signiﬁcantly. The value 10 GeV has been chosen simply as a reasonable limit of the detector resolution [15]. Fig. 6 shows the dependence of the background on S. Bray et al. / Physics Letters B 628 (2005) 250–261 257

− → − − + − − → max Fig. 6. The dependence of the background cross section σcut(e γ W W e νeνe)B(W W hadrons) on the pT cut, where max pT dσ σcut = dp . The dotted line shows the value of cut used in Table 1. 0 T dpT

Table 1 − − − + − − Cross sections (in fb) for the process e γ → W W e and its SM background. The branching fraction B(W W → hadrons) is included Process Scenario 1 Scenario 2 Signal (without cut) 22.923.3 Background (without cut) 0.11 1.54 Signal (with cut) 22.923.3 Background (with cut) 0.002 0.01

max max pT . The inaccuracies for low values of pT that are apparent in Fig. 6 should be regarded as numerical artifacts max due to the small fraction of events that have passed the pT selection criterion. In Table 1 we present comparative numerical values of cross sections for the signal and the background in Scenarios 1 and 2, with and without the pT cut. Our numerical estimates also include the branching fraction for the two W − bosons to decay into hadrons. Table 1 shows that the background can be drastically reduced to an almost unobservable level. For a given scenario, this enables one to place sensitivity limits to the mixing factor BeN . Since the LNV cross sections exhibited in Table 1 are roughly equal in Scenarios 1 and 2, the limits on |BeN | will be −1 −3 very similar. Assuming an integrated luminosity of 100 fb , one must have |BeN | 4.6 × 10 to obtain at least −3 10 background-free events. If no signal is observed, this will place the upper limit |BeN | 2.7 × 10 at 90% CL. − + − Observe that the sensitivity of an e γ collider to BeN is better than that of an e e linear collider [19] due to the far smaller SM background.

4.2. e−γ → W +µ−µ−ν

−3 As was mentioned in Section 3, this process becomes relevant, only when the mixing factor |BeN | 10 .We therefore consider cases where BµN is sizeable, but BeN = BτN = 0. As before, we analyze the following two scenarios: 258 S. Bray et al. / Physics Letters B 628 (2005) 250–261

− − Fig. 7. Differential cross section for the process e γ → Nµ ν (BeN = 0, BµN = 0.1). Our cuts veto the region to the right of the dotted line.

Scenario 3. √ s = 500 GeV,mN = 200 GeV,BµN = 0.1, + − yielding ΓN = 0.08 GeV and B(N → W µ ) ≈ 0.3;

Scenario 4. √ s = 1TeV,mN = 400 GeV,BµN = 0.1, + − yielding ΓN = 0.8 GeV and B(N → W µ ) ≈ 0.26.

Since the heavy Majorana neutrino will have a small decay width, one expects the invariant mass mµ−W + of one of the muons with the W + boson to be very close to the mass of the heavy neutrino. The other muon, which does not come from the decaying heavy neutrino, will generally have a preference to a small scattering angle from the direction of the incoming photon, due to the infrared property of the exchanged muon in the t-channel. This last feature is also shown in Fig. 7. In view of all the above reasons, the following cuts prove very effective:

mµ−W + = mN ± m, for one of the muons,

cos θe−µ− cos θmin =−0.5, for the other muon. (4.3) Here m = 10 GeV is again set by the detector resolution, which is much greater than the actual heavy neutrino decay width ΓN . Fig. 8 shows how the background would be affected by different choices of these cuts. In Table 2 we summarize our results for the signal and the background, before and after the kinematical cuts (4.3) have been implemented. In particular, it can be seen that the selected kinematical cuts were very effective to drastically reduce the background by 2 orders of magnitude, without harming much the signal. The limits that can be derived for Scenarios 3 and 4 are very similar. To observe 10 background-free events, one would need a −2 −1 WµN-coupling |BµN | 9.0 × 10 , assuming a total integrated luminosity of 100 fb . The absence of any LNV signal would put the upper limit |BµN | 0.05 at the 90% CL. It is important to remark that the sensitivity limits − + − that can be placed on |BµN | at an e γ collider will be much higher than the previously considered studies at e e and γγ colliders [19,23]. S. Bray et al. / Physics Letters B 628 (2005) 250–261 259

− + − − + Fig. 8. The dependence of the background cross section σ(e γ → W µ µ νeν¯µν¯µ)B(W → hadrons) on the kinematical cuts m (left panel) and cos θmin (right panel).

Table 2 − + − − + Cross sections (in fb) for e γ → W µ µ ν and its SM background, where B(W → hadrons) is included Process Scenario 3 Scenario 4 Signal (without cuts) 0.15 0.15 Background (without cuts) 0.02 0.25 Signal (with cuts) 0.13 0.12 Background (with cuts) 0.0002 0.004

5. Conclusions

We have analyzed the phenomenological consequences of electroweak-scale heavy Majorana neutrinos at − an e γ collider. We have found that heavy Majorana neutrinos, with masses mN = 100–400 GeV and cou- ∼ −2 ∼ −1 − √plings BeN 10 and BµN 10 , will become easily observable at an e γ collider, with CMS energies s = 0.5–1 TeV and a total integrated luminosity of 100 fb−1. Specifically, we have computed the cross sections for the LNV reactions e−γ → W −W −e+ and e−γ → W +µ−µ−ν. In our analysis, we have also considered the contributing SM background, which involves additional light neutrinos in the final state. After imposing realistic missing pT , invariant mass and angle cuts, we have been able to suppress the SM background by 2 orders of magnitude to an unobservable level, without harming much the signal. Observation of the reaction e−γ → W −W −e+ would be a clear signal for LNV and hence for unravelling the possible Majorana nature of heavy neutrinos. The process e−γ → W +µ−µ−ν is also a clear signal for LNV in the absence of observations of e−γ → W +W −µ−. The search for heavy Majorana neutrinos at an e−γ collider adds particular value to analogous searches at an e+e− linear collider. Although the latter may be built first, it would be more difficult to discern the Dirac or Majorana nature of possible heavy neutrinos. Another obstacle facing e+e− colliders is the large irreducible SM background which has similar kinematical characteristics as the signal. Instead, an e−γ collider provides a cleaner environment and has a unique potential to identify the possible Majorana nature of the heavy neutrinos, as well as probe lower coupling strengths thanks to its highly reducible background. The analysis presented in this Letter may contain some degree of model dependence. For example, models that include new heavy gauge bosons may add new relevant LNV interactions to the Lagrangian. As long as these 260 S. Bray et al. / Physics Letters B 628 (2005) 250–261 additional contributions are negligible, our results will be robust. In conclusion, an e−γ collider would provide valuable information about the properties of possible heavy neutrinos that could be relevant for low-scale resonant leptogenesis [32], and also shed light on the structure of the light neutrino masses and mixings.

Acknowledgements

We thank George Lafferty for pointing us out Ref. [20]. The work of S.B. has been funded by the PPARC studentship PPA/S/S/2003/03666. The work of J.S.L. has been supported in part by the PPARC research grant: PPA/G/O/2002/00471. Finally, A.P. gratefully acknowledges the partial support by the PPARC research grants: PPA/G/O/2002/00471 and PP/C504286/1.

References

[1] S. Fukuda, et al., Super-Kamiokande Collaboration, Phys. Lett. B 539 (2002) 179, hep-ex/0205075; S. Fukuda, et al., Phys. Rev. Lett. 86 (2001) 5656, hep-ex/0103033; S. Fukuda, et al., Phys. Rev. Lett. 86 (2001) 5651, hep-ex/0103032; Y. Fukuda, et al., Super-Kamiokande Collaboration, Phys. Rev. Lett. 82 (1999) 2644, hep-ex/9812014; S. Fukuda, et al., Phys. Rev. Lett. 82 (1999) 1810, hep-ex/9812009. [2] S.N. Ahmed, et al., SNO Collaboration, Phys. Rev. Lett. 92 (2004) 181301, nucl-ex/0309004; Q.R. Ahmad, et al., SNO Collaboration, Phys. Rev. Lett. 89 (2002) 011302, nucl-ex/0204009; Q.R. Ahmad, et al., SNO Collaboration, Phys. Rev. Lett. 89 (2002) 011301, nucl-ex/0204008; Q.R. Ahmad, et al., SNO Collaboration, Phys. Rev. Lett. 87 (2001) 071301, nucl-ex/0106015. [3] K. Eguchi, et al., KamLAND Collaboration, Phys. Rev. Lett. 90 (2003) 021802, hep-ex/0212021. [4] P. Minkowski, Phys. Lett. B 67 (1977) 421; M. Gell-Mann, P. Ramond, R. Slansky, in: P. van Nieuwenhuizen, D. Friedman (Eds.), Supergravity, North-Holland, Amsterdam, 1979, p. 315; T. Yanagida, in: O. Sawada, A. Sugamoto (Eds.), Proceedings of the Workshop on the Uniﬁed Theories and Baryon Number of the Universe, KEK, Tsukuba, 1979; R.N. Mohapatra, G. Senjanovic, Phys. Rev. Lett. 44 (1980) 912. [5] P. Achard, et al., L3 Collaboration, Phys. Lett. B 517 (2001) 67, hep-ex/0107014. [6] P. Langacker, D. London, Phys. Rev. D 38 (1988) 886. [7] J.G. Körner, A. Pilaftsis, K. Schilcher, Phys. Lett. B 300 (1993) 381, hep-ph/9301290; J. Bernabéu, J.G. Körner, A. Pilaftsis, K. Schilcher, Phys. Rev. Lett. 71 (1993) 2695, hep-ph/9307295. [8] C.P. Burgess, S. Godfrey, H. König, D. London, I. Maksymyk, Phys. Rev. D 49 (1994) 6115, hep-ph/9312291. [9] E. Nardi, E. Roulet, D. Tommasini, Phys. Lett. B 327 (1994) 319, hep-ph/9402224. [10] T.P. Cheng, L.F. Li, Phys. Rev. Lett. 45 (1980) 1908. [11] A. Ilakovac, A. Pilaftsis, Nucl. Phys. B 437 (1995) 491, hep-ph/9403398. [12] S. Bergmann, A. Kagan, Nucl. Phys. B 538 (1999) 368, hep-ph/9803305. [13] J.I. Illana, T. Riemann, Phys. Rev. D 63 (2001) 053004, hep-ph/0010193. [14] G. Cvetic, C. Dib, C.S. Kim, J.D. Kim, Phys. Rev. D 66 (2002) 034008; G. Cvetic, C. Dib, C.S. Kim, J.D. Kim, Phys. Rev. D 68 (2003) 059901, hep-ph/0202212. [15] B. Badelek, et al., ECFA/DESY Photon Collider Working Group, Int. J. Mod. Phys. A 19 (2004) 5097, hep-ex/0108012. [16] E. Witten, Nucl. Phys. B 268 (1986) 1642; R.N. Mohapatra, J.W.F. Valle, Phys. Rev. D 34 (1986) 1642. [17] A. Pilaftsis, Z. Phys. C 55 (1992) 275, hep-ph/9901206. [18] J. Gluza, Acta Phys. Pol. B 33 (2002) 1735, hep-ph/0201002; G. Altarelli, F. Feruglio, New J. Phys. 6 (2004) 106, hep-ph/0405048. [19] F. del Aguila, J.A. Aguilar-Saavedra, A. Martinez de la Ossa, D. Meloni, Phys. Lett. B 613 (2005) 170, hep-ph/0502189; F. del Aguila, J.A. Aguilar-Saavedra, JHEP 0505 (2005) 026, hep-ph/0503026. [20] B. Aubert, et al., BaBar Collaboration, Phys. Rev. Lett. 95 (2005) 041802, hep-ex/0502032; B. Aubert, et al., BaBar Collaboration, hep-ex/0508012. [21] W. Buchmüller, C. Greub, Nucl. Phys. B 363 (1991) 345; S. Bray et al. / Physics Letters B 628 (2005) 250–261 261

G. Cvetic, C.S. Kim, C.W. Kim, Phys. Rev. Lett. 82 (1999) 4761, hep-ph/9812525. [22] F.M.L. Almeida, Y.A. Coutinho, J.A. Martins Simoes, M.A.B. do Vale, Phys. Rev. D 63 (2001) 075005, hep-ph/0008201; J. Gluza, M. Zralek, Phys. Rev. D 55 (1997) 7030, hep-ph/9612227. [23] J. Peressutti, O.A. Sampayo, Phys. Rev. D 67 (2003) 017302, hep-ph/0211355. [24] J.A.M. Vermaseren, Comput. Phys. Commun. 83 (1994) 45. [25] J. Peressutti, O.A. Sampayo, J.I. Aranda, Phys. Rev. D 64 (2001) 073007, hep-ph/0105162. [26] R. Mertig, M. Bohm, A. Denner, Comput. Phys. Commun. 64 (1991) 345. [27] E. Boos, et al., CompHEP Collaboration, Nucl. Instrum. Methods A 534 (2004) 250, hep-ph/0403113; A. Pukhov, et al., hep-ph/9908288. [28] S. Kawabata, Comput. Phys. Commun. 88 (1995) 309. [29] T. Sjöstrand, P. Eden, C. Friberg, L. Lonnblad, G. Miu, S. Mrenna, E. Norrbin, Comput. Phys. Commun. 135 (2001) 238, hep-ph/0010017. [30] A.S. Belyaev, et al., hep-ph/0101232. [31] S. Eidelman, et al., Particle Data Group, Phys. Lett. B 592 (2004) 1. [32] For most recent studies, see: A. Pilaftsis, T.E.J. Underwood, hep-ph/0506107; G.C. Branco, R. Gonzalez Felipe, F.R. Joaquim, B.M. Nobre, hep-ph/0507092; S. Dar, Q. Shaﬁ, A. Sil, hep-ph/0508037; E.J. Chun, hep-ph/0508050. Physics Letters B 628 (2005) 262–274 www.elsevier.com/locate/physletb

Bulk Higgs with 4D gauge interactions

A. Kehagias a,K.Tamvakisb

a Physics Department, National Technical University, 15 780 Zografou, Athens, Greece b Physics Department, University of Ioannina, 45110 Ioannina, Greece Received 25 July 2005; accepted 23 September 2005 Available online 30 September 2005 Editor: L. Alvarez-Gaumé

Abstract We consider a model with an extra compact dimension in which the Higgs is a bulk field while all other Standard Model fields are confined on a brane. We find that four-dimensional gauge invariance can still be achieved by appropriate modification of the brane action. This changes accordingly the Higgs propagator so that, the Higgs, in all its interactions with Standard Model fields, behaves as an ordinary 4D field, although it has a bulk kinetic term and bulk self-interactions. In addition, it cannot propagate from the brane to the bulk and, thus, no charge can escape into the bulk but it remains confined on the brane. Moreover, the photon remains massless, while the dependence of the Higgs vacuum on the extra dimension induces a mixing between the graviphoton and the Z-boson. This results in a modification of the sensitive ρ-parameter.  2005 Elsevier B.V. All rights reserved.

1. Introduction

The quest for a unification of gravity with the rest of the fundamental interactions has led to the idea of extra spatial dimensions, first introduced in the Kaluza–Klein (KK) theory. Presently, the only consistent theoretical framework for the development of such a unified theory is String Theory or some form of it [1], which also requires extra spacetime dimensions. In this higher-dimensional context, gravity describes the geometry of a D = 4 + d spacetime possessing d extra spatial dimensions. Many aspects of particle physics have been considered in this general framework, giving the opportunity of a new and fresh look in old and challenging problems. In particular, the introduction of D-branes has led in a reformulation and reevaluation of the original hierarchy problem, by considering compact internal spaces of large radius R, possibly corresponding to a fundamental higher-dimensional Planck mass of O(TeV) [2], or even of infinite radius [3–5]. Many other issues have also been reexamined in this general setup, in which, as a general rule, all degrees of freedom of the Standard Model (SM) are assumed to

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

0370-2693/$ – see front matter  2005 Elsevier B.V. All rights reserved. doi:10.1016/j.physletb.2005.09.064 A. Kehagias, K. Tamvakis / Physics Letters B 628 (2005) 262–274 263 be confined on a 4-dimensional subspace (brane), while only gravity propagates in the full space (bulk). The construction of models in which additional fields besides the graviton field can propagate in the bulk has been proven to be quite challenging. It is common wisdom that, in a world with gauge field on a brane, charged fields may exist only on the brane as a result of gauge invariance. Exception to this is neutral particles, as for example in brane models with SM on the brane and a bulk right-handed neutrino [6–9]. The rule is that only fields which are neutral under the Standard Model gauge interactions can be introduced as bulk fields with brane-interactions. In the present Letter we reexamine the question of fields in the bulk. Considering the Standard Model Higgs, we will show that, Higgs fields propagating in the bulk but with localized gauge interactions may exist as well. We restrict ourselves in the case of one extra compact dimension. The Higgs is introduced as a bulk field while the gauge bosons and the rest of the Standard Model are strictly confined on the brane. We find that the constraint of four-dimensional gauge invariance modifies the Higgs propagator so that, the Higgs, in all its interactions with Standard Model behaves as an ordinary four-dimensional field and although it propagates within the bulk, it cannot propagate from the brane to the bulk. Despite the fact that no charge can escape into the bulk but it remains confined on the brane, the Higgs has a bulk kinetic term and bulk self-interaction. Moreover, a bulk Higgs field can have a vacuum expectation value dependent on the extra dimension. Such a VEV induces an interaction between the graviphoton and the Z-boson. This interaction amounts to a mixing of Z with the graviphoton, which has already acquired a mass due to the presence of the brane [10]. In Section 2, we introduce a model with a bulk 5D scalar field with 4D U(1) gauge interactions on a brane and we present a gauge invariant action. In Section 3, we discuss the graviphoton-gauge field mixing by a vacuum 5D scalar field configuration. In Section 4, we introduce the SM Higgs in the 5D bulk interacting with the rest of the SM fields, which are localized on the 4D brane, and we find that, although the photon remains massless, there exist a graviphoton-Z mixing and an associated modification of the ρ-parameter. In Section 5 we discuss quantum effects due to the 5D Higgs, and finally, in Section 6, we summarize our findings.

2. Bulk Higgs and 4D gauge invariance

Let us consider a 5D space M4 × S1 with the extra compact dimension x5 = y that takes values in the circle 0 y R. A brane is present at the location y = 0 of this space. The five-dimensional metric is taken to the flat metric η 0 G = µν . MN 01 Consider now a complex scalar field Φ(xµ,y)with the 5D canonical dimension 3/2. In addition to that, there is a four-dimensional U(1) gauge field Bµ(x) propagating on the brane. The action for the model is

S = S0 + SBr + Sint, (1) where 4 2 S0 =− d x dy |∂M Φ| is the free-action for the bulk scalar field 1 S =− d4xB (x)Bµν(x), Br 4 µν with Bµν(x) = ∂µBν(x) − ∂νBµ(x) is the action for the U(1) gauge field and 4 µ ∗ ∗ 2 Sint =−iag d xB (x) Φ (x, 0)∂µΦ(x,0) − ∂µΦ (x, 0)Φ(x, 0) − igBµ(x) Φ(x,0) , 264 A. Kehagias, K. Tamvakis / Physics Letters B 628 (2005) 262–274 is their interaction. In the interaction term g is the dimensionless four-dimensional gauge coupling while the parameter a has canonical dimension −1 and corresponds to a length. The action (1) describes a 5D scalar interacting with a 4D gauge field. The scalar field propagates in the bulk of the spacetime, while its interactions are localized on the brane at y = 0. Clearly, SBr is invariant under the standard U(1) transformation δBµ(x) = ∂µω(x). However, the bulk and the interaction Lagrangian are not gauge invariant as they stand. At this point let us define the transformation properties of the scalar field. An obvious guess is that the scalar field transforms only at the position of the brane, i.e., where it experiences the gauge interactions. Let us consider then the gauge transformations

δΦ(x,y) = iagω(x)δ(y)Φ(x, y), δBµ(x) = ∂µω(x). (2) Note that these transformations are singular on the brane, i.e., δΦ(x,0) = iagω(x)δ(0)Φ(x, 0). However, they can be regularized by taking a−1 = δ(0). Then, we can write1 δΦ(x,y) = igω(x)δ(y)Φ(x,y),ˆ (3) so that the gauge transformations on the brane are just

δΦ(x,0) = igω(x)Φ(x,0), δBµ(x) = ∂µω(x). Note that on the brane we can deﬁne a four-dimensional ﬁeld with the correct canonical dimension − φ(x)= a1/2Φ(x,0) = δ(0) 1/2Φ(x,0) in terms of which S is just int 4 µ ∗ ∗ 2 Sint =−ig d xB (x) φ (x)∂µφ(x)− ∂µφ (x)φ(x) − igBµ(x) φ(x) .

Under the gauge transformation (3), the action (1) transforms as δS = iag d4xω(x) Φ† (x, 0)Φ(x, 0) − Φ (x, 0)Φ†(x, 0) , and thus, it is not gauge invariant. However, gauge invariance can be maintained by adding appropriate terms. Indeed, let us consider the term 4 † † S1 =−a d x Φ (x, 0)Φ(x, 0) + Φ (x, 0)Φ (x, 0) =−a d4x dy δ(y) Φ† (x, y)Φ(x, y) + Φ (x, y)Φ†(x, y) .

It is easy to verify then that 4 † † δS1 =−iag d xω(x) Φ (x, 0)Φ(x, 0) − Φ (x, 0)Φ (x, 0) =−δS, and thus, the total action

Stot = S + S1 is gauge invariant. The above discussion may easily be generalized to the non-Abelian case.

1 δ(y − ) δ(0 − ) δ(y)ˆ = lim , δ(ˆ 0) = lim = 1. →0 δ( ) →0 δ( ) A. Kehagias, K. Tamvakis / Physics Letters B 628 (2005) 262–274 265

3. Bulk Higgs with brane gauge interactions

Let us consider now a Higgs ﬁeld H living in a 5D bulk. All its gauge interactions, as well as some of its 2 self-interactions, are localized at y = 0. Taking into account gravity with 5D Planck mass M5, we have the action √ √ S = 3 5 − R − 5 − MN † + (5) M5 d x G d x G G DM H DN H V (H ) 1 − d4x −G(i) T + V (4)(H ) + B Bµν , (4) 4 µν where the gauge ﬁeld strength is Bµν(x) = ∂µBν(x) − ∂νBµ(x) and the covariant derivative Dµ = ∂µ − igaδ(y)Bµ(x), DM ≡ (5) D5 = ∂5. Note the presence of both a bulk self-interaction potential V (5)(H ) as well as a potential on the brane V (4)(H ). The canonical dimensions of these terms are different, being [V (5)]=5 and [V (4)]=4. If we add to S the extra local term 4 † † S1 =−a d x H (x, 0)H (x, 0) + H (x, 0)H (x, 0) , (6) the total action S + S1 becomes invariant under the set of gauge transformations

δH(x,y) = igaω(x)δ(y)H (x, y), δBµ(x) = ∂µω(x). (7) Let us now consider the general KK-form of the 5D metric g + R2A A R2A gµν −A = µν µ ν µ MN = µ GMN 2 2 ,G −2 µ , (8) R Aµ R −Aµ R + A Aµ

1 where the radion field G55 has been friezed to its VEV R, the radius of the extra S dimension. With this form of the 5D metric, gµν describes the 4D graviton and Aµ the 4D graviphoton. In the absence of the brane, the graviphoton 1 Aµ is a massless boson corresponding to the translational invariance along S . However, in the presence of the brane, translational invariance is broken and the graviphoton becomes massive. This has been shown explicitly in [10] for a fat brane formed by a kink soliton. This is also the case for a brane with delta-function profile. To see this, let us turn off all fields except gravity. In this case, the 5D action in the presence of the brane is √ S = 3 5 − R − 4 − (i) M5 d x G T d x G , (9)

(i) where Gµν is the induced metric at the position of the brane (i) = + 2 Gµν gµν R AµAν. (10) Then, we have for the determinant of the induced metric − (i) = − ν + 2 ν = − Tr ln(1+R2A⊗A) det G det( gµν) det δµ R AµA det( g)e ∞ R2nA2n 2 2 = ln(1+R A ) 2 µ = det(−g)e n 1 n = det(−g)e = det(−g) 1 + R AµA . (11)

2 G(i) is the determinant of the induced metric on the brane. The parameter T is the brane tension and has canonical dimensions [T ]=4. 266 A. Kehagias, K. Tamvakis / Physics Letters B 628 (2005) 262–274

Moreover, with the KK form (8) of the metric, where gµν = gµν(x) and Aµ = Aµ(x), the 5D Ricci scalar R turns out to be R2 R = R¯ − F F µν. (12) 4 µν ¯ R is the 4D Ricci scalar of the 4D metric gµν and Fµν = ∂µAν − ∂νAµ is the field strength of the graviphoton field. Substituting, (11), (12) in (9), we get √ √ 1 √ S = 2πRM3 d4x −gR + 2πR3M3 d4x −g − F F µν − T d4x −g 1 + R2A Aµ 1/2, 5 5 4 µν µ (13) (4) (5) 2 where T ≡ T +V + RV + R|∂5H| includes the Higgs contribution to the vacuum energy. Linearizing the 3 ≡ 2 above expression, we arrive at the effective 4D action (2πRM5 MP ) √ √ √ 1 1 T S = M2 d4x −gR − T d4x −g + (RM )2 d4x −g − F F µν − A Aµ +··· P P µν 2 µ 4 2 MP (14) from where we read off the graviphoton mass T 1 T M2 = = . A 2 π 3 MP 2 RM5 Thus, we see that the brane breaks the U(1) symmetry of the background and gives a non-zero mass to the graviphoton. This mass depends on the brane tension as in the case where the brane is a kink of finite width, formed by a scalar field [10]. As we are only interested in the Higgs-gauge sector, we may ignore gravity and the brane tension. In this case, we may disregard the first two terms in (14). The complete action turns out to be √ √ − 2 4 − 1 µν + 1 2 µ − 5 − † µ + (5) (RMP ) d x g FµνF MAAµA d x g DµH D H V 4 2 √ √ 5 µ † † 5 2 µ † + R d x −gA DµH ∂5H + ∂5H DµH − d x −g 1 + R AµA ∂5H ∂5H √ 1 √ − d4x −gV (4) − d4x −g 1 + R2A Aκ B Bµν. (15) 4 κ µν It is obvious that due to the Higgs field, the graviphoton and the brane gauge field are mixed. We will discuss this mixing and its consequences below. For this we define the 4D Higgs field on the brane with the correct canonical dimensions − φ(x)≡ H(x,0)a1/2 = H(x,0)δ(0) 1/2 and the canonical graviphoton field

−1 Aµ(x) → (RMP ) Aµ(x). Now, let us note that there are terms quadratic to the gauge and graviphoton ﬁelds coming from the bulk scalar µ † covariant derivatives, as well as a coupling Aµ(D H)∂5H . We shall assume a y-dependent vacuum H(x,y).By setting H (0, 0) Im ≡ M H(0, 0) A. Kehagias, K. Tamvakis / Physics Letters B 628 (2005) 262–274 267 the vector mass-terms in (15) turns out to be √ − 4 − 2 µ| |2 + 1 2 µ + gM µ| |2 d x g g BµB φ MAAµA 2 AµB φ . (16) 2 MP The vacuum is determined from the Higgs ﬁeld equation 2 + 2 − 2 + 2 − | |2 = ∂5 µ0 a ∂5 ,δ(y) δ(y) µ4 λ4 H H(0,y) 0, where we have taken λ λ V (4) =−µ2|H |2 + 4 |H|4,V(5) =−µ2|H |2 + 5 |H |4 ≈ µ2|H |2 +···. 4 2 5 2 0 The above equation possesses solutions of the form3

H(0,y)= C1δ(y) + C2 sinh(µ0y). (17) By substituting, we obtain 2 + − 2 + 3 | |2 = ν0 aδ (0) δ(0)µ4 λ4δ (0) C1 0. [ 2]= [ ]=− Since µ4 1 and λ4 2, we may introduce the canonical 4D parameters ¯ 2 = 2 ¯ = 2 µ4 aµ4, λ4 a λ4. 2 ≡ 2 + Introducing the renormalized mass µ µ0 aδ (0), we get | |2 = a ¯ 2 − 2 C1 ¯ µ4 µ . λ4 The condition − v H(0, 0) = a 1/2 √ 2 gives

2 µ¯ 2 − µ2 = −1/2 √v v = 4 C1 a , ¯ . (18) 2 2 λ4 The graviphoton mixing parameter M introduced above is given by √ µ M = 2a 0 Im{C }. v 2

Taking for simplicity C2 to be purely imaginary, we may rewrite the vacuum solution as −1/2 v M H(0,y)= a √ aδ(y) + i sinh(µ0y) . (19) 2 µ0

The solution can be periodic in y by choosing a purely imaginary µ0. This is not a problem as the physical Higgs mass, as we will see, is shifted by aδ(0).

3 2 An alternative solution, with µ0 < 0, is

= + | | H(0,y) C1δ(y) C2 sin( µ0 y). 268 A. Kehagias, K. Tamvakis / Physics Letters B 628 (2005) 262–274

2 = 2 2 Denoting by MB g v the gauge boson mass in the absence of the graviphoton, we obtain for the mixed mass-matrix of vector bosons 2 2 2 M M tan ζ M(0) = B B V 2 2 , (20) MB tan ζMA where we have deﬁned 1 M tan ζ = . (21) g MP The mass eigenstates are the gauge ﬁelds (1) = − (2) = + Aµ cos ξBµ sin ξAµ,Aµ sin ξBµ cos ξAµ, (22) where 2tanζ tan 2ξ = (23) MA 2 − 1 MB (1,2) and the corresponding masses of Aµ are 1 2 = 2 + 2 + 2 − 2 2 + 4 2 M(1) MA MB MB MA 4MB tan ζ , (24) 2 1 M2 = M2 + M2 − M2 − M2 2 + 4M4 tan2 ζ . (25) (2) 2 A B B A B

4. The Standard Model with a bulk Higgs

Let us consider now the same 5D spacetime M4 × S1 with a 4D brane embedded in it. All Standard Model fields, except the Higgs SU(2)L doublet, are localized on the brane. The Higgs field experiences the full 5D bulk. Nevertheless, gauge interactions are strictly four-dimensional. The same is true for Yukawa interactions as well. The model is actually a minimal embedding of the Standard Model in extra dimensions with the Higgs doublet as the only field that lives in the 5D bulk. Ignoring, for the moment, gravity and the graviphoton, the action for this model may be written as √ 5 2 (5) SSM = d x −g −|DM H| − V (H ) − LSMδ(y) , (26) where H(x,y) is an SU(2)L isodoublet with hypercharge 1/2 H (+) H = , (27) H (0) and the covariant derivatives are given by i ˆ i ˆ Dµ ≡ ∂µ − g δ(y)Bµ − gδ(y)Wµ · τ, µ= 0,...,3. DM = 2 2 (28) D5 ≡ ∂5. The Higgs potential can be taken to be (5) =− 2 2 + 4 ≈ 2 2 +··· V µ5 H(x,y) λ5 H(x,y) µ0 H(x,y) . (29) (4) Localized Higgs self-interactions V of analogous form are present in LSM as well, namely λ V (4) =−µ2H(x,0)2 + 4 H(x,0)4. (30) 4 2 A. Kehagias, K. Tamvakis / Physics Letters B 628 (2005) 262–274 269

The theory is invariant under the set of 4D-gauge transformations of the SU(2)L × U(1)Y gauge group i δH(x,y) = δ(y)ˆ gω(x) · τ + g ω(x) H(x,y), 2

δWµ(x) = ∂µω(x) + gω(x) × Wµ(x), δBµ = ∂µω(x), (31) provided we add to the action the local term 4 † † S1 =−a d x H (x, 0)H (x, 0) + H (x, 0)H (x, 0) . (32)

The variation of this term cancels the variation i δS =− a d4x H † (x, 0)(gω · τ + g ω)H(x,0) − H †(x, 0)(gω · τ + g ω)H (x, 0) SM 2 so that the total action S = SSM + S1 is SU(2)L × U(1)Y -invariant. The graviphoton mixing to the neutral gauge fields proceeds as before. The relevant mixing terms are √ √ S =− 2 4 − 1 µν + 1 2 µ − 5 − † µ + m (RMP ) d x g FµνF MAAµA d x g DµH D H V 4 2 √ √ 5 2 µ † 5 µ † † − d x −g 1 + R AµA ∂5H ∂5H + R d x −gA DµH ∂5H + ∂5H DµH 1 √ 1 √ − d4x −g 1 + R2A Aκ B Bµν − d4x −g 1 + R2A Aκ W · W µν, (33) 4 κ µν 4 κ µν (5) (4) where Bµν, Wµν are the U(1) and SU(2) field strengths, respectively and V stands for V + δ(y)V . Let us now consider the Higgs vacuum solution (19) −1/2 M H(0,y)= a aδ(y) + i sinh(µ0y) φ (34) µ0 and introduce a four-dimensional Higgs field isodoublet φ(x) as − φ(x)≡ H(x,0)a1/2 = H(x,0)δ(0) 1/2. Then, the emerging mass terms for the graviphoton and the gauge vectors turns out to be √ 2 2 4 1 T µ g µ 2 g µ 2 gg µ † Smass =− d x −g AµA + Wµ · W |φ| + BµB |φ| + B Wµ · (φ τφ) 2 M2 4 4 2 P M 2 µ M µ † 2 2 2 µ + g |φ| AµB + g A Wµ · φ τφ + R M |φ| AµA . (35) MP MP

We have introduced a canonically normalized graviphoton ﬁeld through the rescaling Aµ → Aµ/(RMP ). Substi- tuting the VEV 1 0 φ =√ 2 v we obtain the mass terms √ S =− 4 − 1 2 µ + 2 + −µ + 1 2 µ − vMW M µ mass d x g MAAµA MW Wµ W MZZµZ AµZ , 2 2 cos θW MP where T g2v2 M2 M2 = + R2M2v2,M2 = ,M2 = W . A 2 W Z 2 (36) MP 4 cos θW 270 A. Kehagias, K. Tamvakis / Physics Letters B 628 (2005) 262–274

= g As usual, we have introduced tan θW g and µ = √1 µ ± µ = 3 − W± W iW ,Zµ cos θW Wµ sin θW Bµ. (37) 2 1 2 3 + The photon sin θW Wµ cos θW Bµ stays massless while the graviphoton is mixed with the neutral Z-boson. The neutral vector mass matrix is 2 2 2 M −RMvMZ M − tan ζM M2 = Z = Z Z (38) − 2 − 2 2 RMvMZ MA tan ζMZ MA and we have introduced the mixing angle v M tan ζ ≡ . (39) MZ MP The mass eigenstates are

Z1µ = cos ξZµ − sin ξAµ,Z2µ = sin ξZµ + cos ξAµ, (40) where 2tanζ tan 2ξ = . (41) M2 A − 2 1 MZ

The corresponding masses of Z1, Z2 are 1 M2 = M2 + M2 − M2 − M2 2 + 4tan2 ζM4 , Z1 A Z A Z Z 2 1 M2 = M2 + M2 + M2 − M2 2 + 4tan2 ζM4 . (42) Z2 2 A Z A Z Z For MA MZ, we get  M2 2 4  2 − 2 Z +··· 2 1 2 2 2 MZ 2 MZ MZ 1 tan ζ 2 , M ≈ M + M ∓ M 1 − + 2tan ζ ≈ MA Z1,Z2 A Z A  2 MA MA 2 +··· MA .

Identifying Z1µ with the neutral gauge boson produced at LEP, we may write M2 tan2 ζ M2 W ≈ 2 θ + W . 2 cos W 1 2 2 M cos θW M Z1 A As a result, the graviphoton–Z boson mixing, gives a departure for the Standard Model value (ρ = 1) of the ≡ 2 2 2 parameter ρ MW / cos θW MZ tan2 ζ M2 4 M 2 M 2 δρ = W = W 2 2 2 (43) cos θW MA g MP MA which is positive. Moreover, we may consider the neutral current Lagrangian of Zµ g 1 L = ψ¯ γ T (1 − γ ) − 2Q sin2 θ ψ Zµ. (44) NC cos θ 2 i µ 3i 5 i W i W i A. Kehagias, K. Tamvakis / Physics Letters B 628 (2005) 262–274 271

Changing to mass eigenstates, we get g 1 L = ψ¯ γ T (1 − γ ) − 2Q sin2 θ ψ cos ξZµ + sin ξZµ (45) NC cos θ 2 i µ 3i 5 i W i 1 2 W i µ which includes a coupling of the physical graviphoton Z2 to matter proportional to sin ξ.ForMA MZ,this coupling is of order M ξ ≈ (δρ)1/2 Z . MA

Then, constraints on the ratio M/MA may be obtained by considering the Z1-partial width to fermions Z1 → ¯ f f [12–14]. Indeed, with a shift of the ρ parameter, after taking account higher Higgs representations and mt effects, of order 10−3 [15], we get

M 15 −15 10 ⇒ MA 10 (g tan ζ)MP . (46) MA Thus, for g tan ζ ∼ O(1), we may have

MA > 10 TeV.

5. Quantum effects on the brane

Radiative processes of Standard Model particles will also, in general, involve virtual bulk fields that interact with them. In the present model the Higgs field has been introduced as a bulk field and we would expect that its KK excitations will contribute to loop processes on the brane. For example, the gauge couplings are expected to receive contributions from the Higgs that involve the bulk [11]. In the simplified U(1) model of Section 1, the lowest order Higgs contributions to the vacuum polarization are4 a d4q d4p g2 A˜ (q)A˜µ(−q) D(p; ω ,ω ) R2 ( π)2 µ ( π)2 n n 2 2 n n 2 4 4 2 a d q ˜µ ˜ν d p + g A (q)A (−q) (2qµ + pµ)(2qν + pν)D(p; ωn,ω ) R4 (2π)2 (2π)2 n n,n,n,n × D(q + p; ωn ,ωn ), where D(p; ωn,ωn ) is the Fourier transform of the Higgs propagator 1 1 1 4 ip·(x−x ) iωny iω y G(x − x ; y,y ) = d pe e e n D(p; ωn,ω ). (47) (2π)2 R R n n n Notice that only the propagator G(x − x; 0, 0) with end points on the brane appears in these graphs. The above sum can be written as 4 4 2 d q ˜ ˜µ d p ¯ g Aµ(q)A (−q) D(p) (2π)2 (2π)2 4 4 2 d q ˜µ ˜ν d p ¯ ¯ + g A (q)A (−q) (2qµ + pµ)(2qν + pν)D(p)D(q + p) (48) (2π)2 (2π)2

4 The frequencies ωn are 2πn/R. 272 A. Kehagias, K. Tamvakis / Physics Letters B 628 (2005) 262–274 in terms of 4 ¯ a d x −ip·(x−x) D(p) = D(p; ωn,ω ) = a e G(x − x ; 0, 0). (49) R2 n (2π)2 n,n The expression (48) is a standard four-dimensional expression for the vacuum polarization. The existence of the ﬁfth dimension is encoded in the Higgs propagator G(x − x; y,y). Note however that this is not the free 5D propagator, satisfying5 − 2 + µ + 2 G − ; = (4) − − µ0 ∂µ∂ ∂5 0(x x y,y ) δ (x x )δ(y y ).

In order to maintain four-dimensional gauge invariance, we have introduced the extra local piece S1 in the brane action. This has the effect to modify the Higgs propagator. The modified propagator satisfies the equation6 − 2 + µ + 2 − 2 + 2 G − ; = (4) − − µ0 ∂µ∂ ∂5 a ∂5 δ(y) δ(y)∂5 (x x y,y ) δ (x x )δ(y y ). (50) The solution of this equation, although lengthy,7 proceeds in a straightforward fashion and leads to 1 1 D(p)¯ =− (51) (2π)2 p2 + µ2 2 = 2 + with µ µ0 aδ (0). This is just a free four-dimensional propagator. It is in sharp contrast to the free propagator which should go as − a 1 1 ( π)3 R 2 + 2 + 2 2 n p µ0 ωn and contains the contributions of the infinite tower of KK states. The constraint of 4D-gauge invariance has removed all these contributions from processes on the brane. We should also note that for y,y = 0, we have 1 K(y)K(y ) G˜ (p; y,y ) =− K(y − y ) − 1 (2π)2 K(0)

5 2 The parameter µ0 plays the role of an effective mass. 6 Note that aδ(0) = 1. 7 As a shortcut, we may consider the Fourier transform ˜ 1 4 −ip·(x−x ) G1(p; y,y ) = d xe G(x − x ; y,y ) (2π)2 which satisﬁes the equation 2 2 2 − 2 2 ∂ ∂ ∂ ˜ δ(y y ) −µ − p + − a δ(y) + δ(y) G1(p; y,y ) = 0 ∂y2 ∂y2 ∂y2 (2π)2 and verify by substitution that its solution is ˜ 1 a K(y)K(y ) G1(p; y,y ) =− δ(y)δ(y ) + K(y − y ) − , (2π)2 p2 + µ2 K(0) where (−R/2 z R/2) +∞ 1 eiωnz 1 cosh(p(R/¯ 2 −|z|)) K(z)≡ = , R ¯2 + 2 2p¯ sinh(pR/¯ 2) n=−∞ p ωn ¯2 ≡ 2 + 2 2 = 2 + G˜ ¯ ¯ = G˜ p p µ0 and µ µ0 aδ (0). The Green’s function 1 is related to D(p) as D(p) a 1(p, 0, 0). A. Kehagias, K. Tamvakis / Physics Letters B 628 (2005) 262–274 273 and the Higgs can propagate within the bulk. However, in addition to processes on the brane, a Higgs ﬁeld cannot ˜ propagate from the brane to the bulk. This is clear from the propagator G1 for which

1 δ(y ) G˜ (p; 0,y ) =− . 1 (2π)2 p2 + µ2 + This is consistent with charge conservation, as charge cannot escape to the bulk through process like, i.e., Z1W → H +H 0.

6. Brief summary

The present Letter was motivated by the (im)possibility of propagation of fields (scalars or fermions) in the bulk if gauge fields are localized on a brane. Indeed, it seems that gauge invariance requires the presence of gauge fields (photons) in the bulk as well. Then, from the brane-point of view, gauge invariance would be violated and charge would escape to the bulk. However, one may try to define gauge invariance strictly as a local symmetry of four dimensions and keep gauge fields absolutely trapped on the brane. In such a formulation, gauge invariance turns out to modify the action of the charged bulk fields so that, although they can propagate freely within the bulk, they cannot propagate from the brane towards the bulk. Apart from their bulk self-interactions, in all other interactions occurring on the brane they behave as ordinary four-dimensional fields. From the point of view of the brane, such a theory would be indistinguishable from a theory in which all charged fields are confined on the brane. Nevertheless, in the particular example of the Standard Model with a bulk Higgs isodoublet field that we have considered, a bulk Higgs field opens the possibility of a vacuum expectation value that depends on the fifth dimension. Such a VEV induces new interaction terms of gravitational origin that can lead to observable results. The five dimensional metric 4 1 of the compactified M × S space of the model includes the graviphoton Aµ(x), a massive vector field that owes its mass to the breaking of translational invariance of the fifth dimension by the brane. The solution (19) for the µ † Higgs vacuum allows for the coupling Aµ(D H)∂5H . This coupling induces a mixing between the graviphoton and the massive neutral vector boson of the Standard Model. This coupling modifies slightly the mass eigenvalues and the sensitive mass ratio MW /MZ . An important issue of the model presented is the issue of singularities. It should be noted that the singular object δ(0) = a−1 does not appear in observable quantities. Similarly, aδ(0) gets absorbed in a mass redefinition. Although, we have not introduced a regularization scheme, we do not expect to encounter any further problem associated with singularities arising due to the zero thickness of the brane. In the framework of a regularization scheme the brane would effectively acquire a non-zero thickness. Nevertheless, we feel confident that the main results obtained within our treatment would not change. It should be noted that our treatment is analogous to the treatment of singularities in the Horava–Witten theory [1,16] where a number of cancellations among singular terms result in unambiguous physical expressions. Another issue is of course renormalizability of the model. In the presence of bulk Higgs interaction given by a potential V (5), the model is clearly non-renormalizable. How- ever, we may consider just a bulk mass term for the Higgs with no bulk interactions at all, while all interactions, including the self-interactions, to be strictly localized on the brane. This model should be renormalizable as in 4D is renormalizable in the usual sense and in the 5D bulk, it is a free field.

Acknowledgements

We would like to thank T. Gherghetta for correspondence. This work is co-funded by the European Social Fund (75%) and National Resources (25%)—(EPEAEK-B’)—PYTHAGORAS-II. 274 A. Kehagias, K. Tamvakis / Physics Letters B 628 (2005) 262–274

References

[1] P. Horava, E. Witten, Nucl. Phys. B 475 (1996) 94. [2] N. Arkani-Hamed, S. Dimopoulos, G. Dvali, Phys. Lett. B 429 (1998) 263; I. Antoniadis, N. Arkani-Hamed, S. Dimopoulos, G.R. Dvali, Phys. Lett. B 436 (1998) 257. [3] L. Randall, R. Sundrum, Phys. Rev. Lett. 83 (1999) 3370; L. Randall, R. Sundrum, Phys. Rev. Lett. 83 (1999) 4690. [4] A. Kehagias, Phys. Lett. B 469 (1999) 123, hep-th/9906204; A. Kehagias, hep-th/9911134; A. Kehagias, Phys. Lett. B 600 (2004) 133, hep-th/0406025. [5] A. Kehagias, K. Tamvakis, Phys. Lett. B 504 (2001) 38, hep-th/0010112; A. Kehagias, K. Tamvakis, Mod. Phys. Lett. A 17 (2002) 1767, hep-th/0011006; A. Kehagias, K. Tamvakis, Class. Quantum Grav. 19 (2002) L185, hep-th/0205009. [6] K.R. Dienes, E. Dudas, T. Gherghetta, Nucl. Phys. B 557 (1999) 25, hep-ph/9811428. [7] Y. Grossman, M. Neubert, Phys. Lett. B 474 (2000) 361, hep-ph/9912408. [8] N. Arkani-Hamed, S. Dimopoulos, G.R. Dvali, J. March-Russell, Phys. Rev. D 65 (2002) 024032, hep-ph/9811448. [9] T. Gherghetta, Phys. Rev. Lett. 92 (2004) 161601. [10] G. Dvali, I. Kogan, M.A. Shifman, Phys. Rev. D 62 (2000) 106001. [11] K. Dienes, E. Dudas, T. Gherghetta, Nucl. Phys. B 537 (1998) 55. [12] G. Altarelli, R. Casalbuoni, D. Dominici, F. Feruglio, R. Gatto, Mod. Phys. Lett. A 5 (1990) 495. [13] P. Langacker, M.X. Luo, Phys. Rev. D 45 (1992) 278. [14] K.S. Babu, C.F. Kolda, J. March-Russell, Phys. Rev. D 54 (1996) 4635, hep-ph/9603212. [15] Particle Data Group, S. Eidelman, et al., Phys. Lett. B 592 (2004) 1. [16] E. Mirabelli, M. Peskin, Phys. Rev. D 58 (1998) 065002. Physics Letters B 628 (2005) 275–280 www.elsevier.com/locate/physletb

Special symmetries of the charged Kerr–AdS black hole of D = 5 minimal gauged supergravity

Paul Davis, Hari K. Kunduri ∗, James Lucietti

DAMTP, Centre for Mathematical Sciences, University of Cambridge, Wilberforce Rd., Cambridge CB3 0WA, UK Received 16 September 2005; accepted 26 September 2005 Available online 30 September 2005 Editor: M. Cveticˇ

Abstract In this Letter we prove that the Hamilton–Jacobi equation in the background of the recently discovered charged Kerr–AdS black hole of D = 5 minimal gauged supergravity is separable, for arbitrary values of the two rotation parameters. This allows us to write down an irreducible Killing tensor for the spacetime. As a result, we also show that the Klein–Gordon equation in this background is separable. We also consider the Dirac equation in this background in the special case of equal rotation parameters and show it has separable solutions. Finally, we discuss the near-horizon geometry of the supersymmetric limit of the black hole.  2005 Elsevier B.V. All rights reserved.

It is a curious fact that the Kerr–Newman black hole possesses a hidden symmetry which renders geodesic motion integrable [1]. This is related to the existence of a second rank Killing tensor Kµν ; by definition such µ ν a tensor satisfies ∇(µKνρ) = 0. Given a Killing tensor one may construct the quantity K = Kµνx˙ x˙ which is conserved along geodesics xµ(τ). Carter was the first to systematically analyse the consequence of separability of solutions to Einstein’s equations, and indeed this is how the Kerr–(A)dS black hole and its charged counterpart were first discovered [2]. Higher-dimensional Kerr–(A)dS metrics have only been recently constructed [3]. The existence of a Killing tensor has been verified in five dimensions for arbitrary rotation parameters [4] and in all dimensions for the special cases of equal sets of rotation parameters [5,6]. This renders both the Hamilton–Jacobi (HJ) and Klein–Gordon (KG) equations separable. The charged counterparts of the Kerr–AdS black holes in D = 5 minimal gauged supergravity are far more difficult to construct. Progress was first made by tackling the special case where the rotation parameters are equal [7], and a reducible Killing tensor for this black hole was found in [8]. Further, the

* Corresponding author. E-mail addresses: [email protected] (P. Davis), [email protected] (H.K. Kunduri), [email protected] (J. Lucietti).

0370-2693/$ – see front matter  2005 Elsevier B.V. All rights reserved. doi:10.1016/j.physletb.2005.09.062 276 P. Davis et al. / Physics Letters B 628 (2005) 275–280 same Killing tensor was found for black holes with equal rotation parameters in the more general U(1)3 theory [9]. Only very recently has a charged Kerr–AdS black hole been found with arbitrary rotation parameters [10].The purpose of this Letter is to show that this black hole also has a Killing tensor rendering geodesic motion integrable and the KG equation separable. We also show that the Dirac equation admits separable solutions in the special case of equal rotation parameters. Finally, we discuss the near-horizon geometry of the supersymmetric limit of this black hole. In the case of equal rotation parameters we show that it has a symmetry algebra sl(2, R) × su(2) × u(1) as is the case for the BMPV black hole [11]. In [10] it was shown that D = 5 minimal gauged supergravity with the Lagrangian density 1 1 L = R + 12g2 ∗ 1 − F ∧∗F + √ F ∧ F ∧ A, (1) 2 3 3 where F = dA, admits a black hole solution parameterised by its mass, charge and two rotation parameters. Ex- plicitly, the metric is given by: ∆ [(1 + g2r2)ρ2 dt + 2qν] dt 2qνω f ∆ dt 2 ρ2 dr2 ρ2 dθ2 ds2 =− θ + + θ − ω + + 2 2 4 ΞaΞbρ ρ ρ ΞaΞb ∆r ∆θ r2 + a2 r2 + b2 + sin2 θdφ2 + cos2 θdψ2, (2) Ξ Ξ √ a b 3q ∆ dt A = θ − ω , 2 (3) ρ ΞaΞb where dφ dψ ν = b sin2 θdφ+ a cos2 θdψ, ω= a sin2 θ + b cos2 θ ,ρ2 = r2 + a2 cos2 θ + b2 sin2 θ, Ξa Ξb (r2 + a2)(r2 + b2)(1 + g2r2) + q2 + 2abq ∆ = 1 − a2g2 cos2 θ − b2g2 sin2 θ, ∆ = − 2m, θ r r2 2 2 2 2 2 2 2 2 f = 2mρ − q + 2abqg ρ ,Ξa = 1 − a g ,Ξb = 1 − b g . (4) The metric is written in Boyer–Lindquist type coordinates, although we should emphasise that it is in a non-rotating frame at asymptotic inﬁnity. For general rotation parameters a and b this metric has three commuting Killing vectors, namely ∂t , ∂φ and ∂ψ . Remarkably, one can check that the determinant of the metric is independent of the charge parameter q and is thus given by the same expression as in the uncharged case,

rρ2 sin θ cos θ − det g = . (5) ΞaΞb A tedious calculation allows one to write the inverse metric as: (a2 + b2)(2mr2 − q2) (r2 + a2)(r2 + b2)[r2(1 − g2(a2 + b2)) − a2b2g2] ρ2gtt =− − 2 2 r ∆r r ∆r 2ma2b2 2abqr2 a2 cos2 θΞ + b2 sin2 θΞ − − − a b , 2 2 r ∆r r ∆r ∆θ aq2 −[2ma + bq(1 + a2g2)](r2 + b2) ρ2gtφ = , 2 r ∆r bq2 −[2mb + aq(1 + b2g2)](r2 + a2) ρ2gtψ = , 2 r ∆r P. Davis et al. / Physics Letters B 628 (2005) 275–280 277

a2g2q2 Ξ Ξ ρ2gφφ = + a + a + g2r2 r2 + b2 b2 − a2 2 2 2 1 r ∆r sin θ r ∆r 2m 2abq − a2g2r2 + b2 − Ξ g2 r2 − a2 − b4g4 + , 2 2 b 1 r ∆r Ξbr ∆r b2g2q2 Ξ Ξ ρ2gψψ = + b + b + g2r2 r2 + a2 a2 − b2 2 2 2 1 r ∆r cos θ r ∆r 2m 2abq − b2g2r2 + a2 − Ξ g2 r2 − b2 − a4g4 + , 2 2 a 1 r ∆r Ξar ∆r abg2q2 − (1 + g2r2)(2mab + (a2 + b2)q) ρ2gφψ = , 2 r ∆r 2 θθ 2 rr ρ g = ∆θ ,ρg = ∆r . (6) An important fact, that we will use shortly, is that the component functions ρ2gµν are additively separable as functions of r and θ. The Hamiltonian describing the motion of free uncharged particles in the background metric gµν is simply = 1 µν H 2 g pµpν . The corresponding Hamilton–Jacobi equation is then ∂S 1 ∂S ∂S + gµν = 0, (7) ∂τ 2 ∂xµ ∂xν where S is Hamilton’s principal function and τ is the parameter along the worldline of the particle. Due to the presence of the isometries one may immediately separate out the dependence on t, φ, ψ leaving 1 S = M2τ − Et + L φ + L ψ + F(r,θ), (8) 2 1 2 2 where M , E and Li are constants. Remarkably, it turns out that S is completely separable so F(r,θ)= Sr (r) + 2 µν Sθ (θ). The proof of this simply relies on the non-trivial fact that ρ g is additively separable as a function of r and θ. This implies that the HJ equation is separable after multiplying it through by ρ2.Theθ-dependent part of the HJ equation is 2 2 2 2 dS L Ξa L Ξb E ∆ θ + 1 + 2 − a2Ξ 2 θ + b2Ξ 2 θ + M2 a2 2 θ + b2 2 θ = K, θ 2 2 a cos b sin cos sin (9) dθ sin θ cos θ ∆θ whilst the r-dependent part is dS 2 ∆ r + V(r; E,L ,M)=−K, (10) r dr i where K is the separation constant, and we have defined an “effective” potential V which is a complicated function of r; as we shall not use it directly, we shall not display it for the sake of brevity. From the θ equation one may µν µν 2 easily read off a Killing tensor for the spacetime using K = K pµpν and g pµpν =−M . This gives 1 Ξ Ξ Kµν =−gµν a2 2 θ + b2 2 θ − a2Ξ 2 θ + b2Ξ 2 θ δµδν + a δµδν + b δµδν cos sin a cos b sin t t 2 φ φ 2 ψ ψ ∆θ sin θ cos θ + µ ν ∆θ δθ δθ . (11) This is an irreducible Killing tensor. Note that this has a smooth limit as g → 0 and when q = 0 coincides with the Killing tensor found in [4], up to terms which are outer products of the Killing vectors. In contrast to [4], here it was unnecessary to add outer products of Killing vectors to the Killing tensor in order to obtain a smooth limit. This is presumably related to the fact that we are in a non-rotating frame at infinity, whereas the metric in [4] was in a rotating frame. It is a curious result that the Killing tensor does not depend explicitly on the charge, 278 P. Davis et al. / Physics Letters B 628 (2005) 275–280 although this does also occur for the four-dimensional Kerr–Newman solution. Moreover, as we will discuss shortly, there exist supersymmetric solutions with a = b. Such black holes thus possess an irreducible Killing tensor, as do the supersymmetric Kerr–Newman–AdS black holes in four dimensions [12]. We should note that from the Hamiltonian point of view the functions H , K, pt , pφ, pψ are in involution thus establishing Liouville integrability. The general solution to geodesic motion can easily be deduced from the generating function S by differentiating 2 with respect to K, M , E, Li , respectively. As in the uncharged case, the additive separability of ρ2gµν allows for separable solutions to the Klein–Gordon equation which governs quantum field theory of massive, spinless particles on this background. Writing the KG equation as 1 µν 2 √ ∂µ − det gg ∂νΦ = M Φ, (12) − det g and taking the following standard ansatz Φ = e−iωteiαφeiβψ R(r)Θ(θ), renders the KG equation separable. The details of this are rather similar to the uncharged Kerr–(A)dS [4]. By making the change of variable z = sin2 θ,the θ equation can be rewritten as 2 d Θ + 1 + 1 + 1 dΘ dz2 z z − 1 z − d dz ω2(a2Ξ + z(b2Ξ − a2Ξ )) 1 α2Ξ β2Ξ M2 k + a b a − a + b + − Θ − 2 − − 2 − − 4z(1 z)∆z 4z(1 z)∆z z 1 z 4g z(1 z) 4z(1 z)∆z = 0, (13) 2 2 2 2 2 2 2 2 where d = Ξa/(g (b − a )), ∆z = Ξa + g z(a − b ) and k = k + M /g with k being the separation constant. This equation has four regular singular points and can easily be put in the form of Heun’s equation. The special case a = b simplifies this equation and the solutions are Jacobi polynomials. Having discussed the separability of the Klein–Gordon equation, the next thing to consider is the Dirac equation on this background. We find that the Dirac equation separates in the special case of equal rotation parameters, a = b, and can be written as

(Dr + Dθ )Ψ = 0, (14) where Dr and Dθ are linear differential operators depending only on r and θ respectively, once the following ansatz has been made: − Ψ = e iωteim1φ eim2ψ χ(r,θ ). (15) The angular coordinates (θ ,φ,ψ) are Euler angles following the notation of [8]. This then admits solutions which are separable in the sense that   R1(r)S+(θ )  R2(r)S−(θ )  χ(r,θ ) =   , (16) R3(r)S+(θ ) R4(r)S−(θ ) where the radial functions form a complicated, coupled system and the functions S± are eigenfunctions of the differential operators − 2 − 2 2 1 i(m1 cos θ m2) cot θ (m1 m2 cos θ ) ∂ + cot θ ∂θ − ∓ + + . (17) θ 2sin2 θ sin2 θ 4 sin2 θ In four dimensions the separability of the Dirac equation leads to the construction of an operator that commutes with the Dirac operator, and is intimately related to the existence of a Yano tensor for the spacetime [13]. Remark- ably, the four-dimensional Kerr–Newman Killing tensor admits a decomposition in terms of a Yano tensor, arising P. Davis et al. / Physics Letters B 628 (2005) 275–280 279 from the fact that there is a non-trivial supersymmetry on the worldline of a spinning particle [14]. It is not hard to show that the five-dimensional Schwarzschild’s Killing tensor K does not admit a Yano tensor, and hence this suggests that the full black hole we have been considering does not either. However, one actually should try to construct an operator that commutes with the Dirac operator. One expects this to exist due to the presence of an extra (separation) constant of the system. In four dimensions this is readily achieved, but seems to rely crucially on the existence of Weyl spinors, and we have been unable to find such an operator in the five-dimensional case. Finally, we will briefly discuss the near-horizon geometry of the supersymmetric limit of the black hole. As discussed in [10], the metric given in Eq. (2) can be rewritten as: ∆ ∆ r2 sin2 2θ dr2 dθ2 ds2 =− r θ dt2 + ρ2 + + B (dψ + ν dφ + ν dt)2 + B (dφ + ν dt)2, 2 ψ 1 2 φ 3 (18) 4(ΞaΞb) BφBψ ∆r ∆θ where 2 g gφψ gtψ gtφgψψ − gφψgtψ B = g ,B= g − φψ ,ν= ,ν= ,ν= . (19) ψ ψψ φ φφ 1 2 3 − 2 gψψ gψψ gψψ gφφgψψ gφψ In the supersymmetric limit, some simplification of the metric occurs due to the constraints imposed upon the parameters q and m, namely, m (a + b)(1 + ag)(1 + bg)(1 + ag + bg) q = ,m= . (20) 1 + ag + bg g 2 = −1 + + + = + + = With these restrictions in place, we find that at the horizon r0 g (a b abg), ν3 g 0, g gν1 ν2 0 and all the other functions in the metric are complicated functions of θ and the rotation parameters. To investigate the near-horizon geometry of this metric we first need to go to a frame which is corotating with the horizon. This ˜ ˜ ˜ ˜ is effected by the redefinitions t = t, φ = φ − gt, and ψ = ψ − gt. Then we set r − r0 = R and t = T/ and take the limit → 0. The near-horizon geometry is then dR2 dθ2 ds2 = ρ2(θ) −c R2 dT 2 + c + + B (θ) dψ˜ + ν (θ) dφ˜ + f(θ)RdT 2 NH 1 2 2 ψ 1 R ∆θ ˜ 2 + Bφ(θ)(dφ + c3RdT) , (21) where, in general, we denote F(θ)≡ F(r0,θ). The function f(θ)as well as the constants c1, c2, c3 are complicated and rather unenlightening. The resulting geometry is similar to the product of AdS2 with a squashed sphere, which appears√ to be a generic property of extremal, rotating (possibly charged) black holes [15]. A trivial time rescaling ˜ ˜ T = c2/c1T puts the TR part of the metric into a form conformal to AdS2 in Poincaré coordinates. Thus, in addition to the obvious isometries generated by ∂/∂T˜ , ∂/∂φ˜, and ∂/∂ψ˜ , (21) is also invariant under dilations ˜ ˜ T → αT , R → R/α. An obvious question is whether the near-horizon limit has all the symmetries of AdS2. Following [15], one might try to introduce global coordinates on the AdS2, in order to show the near-horizon limit has an (analogue) of the global time translation. This needs to be accompanied by a corresponding coordinate transformation for (ψ,˜ φ)˜ . We find that this method does not work in this case, due to the θ-dependence of the metric. Nevertheless, we can show that the near-horizon limit has all the symmetries of AdS2 in the special case a = b as follows. Let us write the near-horizon limit in terms of left-invariant forms on SU(2) as in [16].Itisoftheform dR2 ds2 =−(R dτ + jσ )2 + + L2 σ 2 + σ 2 + λ2σ 2, (22) 3 R2 1 2 3 where j, λ, L are constants related to the horizon radius and the cosmological constant. One should note that this metric is a deformation of the near-horizon limit of BMPV as found in [11]. One may easily check that in addition = ∂ =− ∂ + ∂ to the time translation k ∂τ and the dilation operator l τ ∂τ R ∂R, there is a third isometry analogous to the 280 P. Davis et al. / Physics Letters B 628 (2005) 275–280 one for pure AdS2: 2 j 2 4j m = 1 − ∂ + 2τ 2∂ − 4τR∂ + ∂ . (23) R2 λ2 τ τ R Rλ2 ψ One may check that these Killing vectors satisfy

[l,k]=k, [l,m]=−m, [k,m]=−4l. (24)

Furthermore, the gauge ﬁeld A is regular in the near-horizon limit and one can easily check that £kF = £lF = £mF = 0. Therefore, the algebra of the isometry group of the near-horizon limit which preserves the ﬁeld strength, in the a = b case, is sl(2, R) × su(2) × u(1). It would be most interesting to see whether the general case retains all the symmetries of AdS2. Further, an interesting problem is to determine the full superalgebra of the near-horizon limit, as was done for the BMPV case in [11]. The coordinates we are using are not really suitable on the horizon. One should really be using Gaussian null coordinates adapted to the Killing horizon, which would also allow direct comparison with the near-horizon geometries derived in [16]. One expects the “parameter” ∆ used therein to be non-constant for the metric at hand and thus would fall outside their analysis. While we have studied certain special symmetries of the general charged Kerr–AdS black holes, we doubt that the short list presented here is exhaustive. The existence of supersymmetric black hole solutions with spherical topology having non-equal angular momentum in two orthogonal planes seems unique to gauged supergravity. Given the natural link between supersymmetry and special geometric structures, it seems likely there are further non-trivial symmetries of these black holes.

Acknowledgements

P.D. would like to thank PPARC for ﬁnancial support. H.K.K. would like to thank St. John’s College for ﬁnancial support. We would like to thank Gary Gibbons and Harvey Reall for useful comments.

References

[1] B. Carter, Phys. Rev. 174 (1968) 1559. [2] B. Carter, Commun. Math. Phys. 10 (1968) 280. [3] G.W. Gibbons, H. Lu, D.N. Page, C.N. Pope, J. Geom. Phys. 53 (2005) 49, hep-th/0404008. [4] H.K. Kunduri, J. Lucietti, Phys. Rev. D 71 (2005) 104021, hep-th/0502124. [5] M. Vasudevan, K.A. Stevens, D.N. Page, Class. Quantum Grav. 22 (2005) 339, gr-qc/0405125. [6] M. Vasudevan, K.A. Stevens, gr-qc/0507096. [7] M. Cvetic,ˇ H. Lu, C.N. Pope, Phys. Lett. B 598 (2004) 273, hep-th/0406196. [8] H.K. Kunduri, J. Lucietti, Nucl. Phys. B 724 (2005) 343, hep-th/0504158. [9] M. Vasudevan, gr-qc/0507092. [10] Z.W. Chong, M. Cvetic,ˇ H. Lu, C.N. Pope, hep-th/0506029. [11] J.P. Gauntlett, R.C. Myers, P.K. Townsend, Class. Quantum Grav. 16 (1999) 1, hep-th/9810204. [12] V.A. Kostelecky, M.J. Perry, Phys. Lett. B 371 (1996) 191, hep-th/9512222. [13] B. Carter, R.G. Mclenaghan, Phys. Rev. D 19 (1979) 1093. [14] G.W. Gibbons, R.H. Rietdijk, J.W. van Holten, Nucl. Phys. B 404 (1993) 42, hep-th/9303112. [15] J.M. Bardeen, G.T. Horowitz, Phys. Rev. D 60 (1999) 104030, hep-th/9905099. [16] J.B. Gutowski, H.S. Reall, JHEP 0402 (2004) 006, hep-th/0401042. Physics Letters B 628 (2005) 281–290 www.elsevier.com/locate/physletb

Numerical study of the enlarged O(5) symmetry of the 3D antiferromagnetic RP2 spin model

L.A. Fernández a,b, V. Martín-Mayor a,b,D.Scirettic,b, A. Tarancón c,b, J.L. Velasco c,b

a Departamento de Física Teórica, Facultad de Ciencias Físicas, Universidad Complutense, 28040 Madrid, Spain b Instituto de Biocomputación y Física de Sistemas Complejos (BIFI), 5009 Zaragoza, Spain c Departamento de Física Teórica, Facultad de Ciencias, Universidad de Zaragoza, 50009 Zaragoza, Spain Received 23 July 2005; accepted 18 September 2005 Available online 30 September 2005 Editor: L. Alvarez-Gaumé

Abstract We investigate by means of Monte Carlo simulation and finite-size scaling analysis the critical properties of the three- dimensional O(5) non-linear σ model and of the antiferromagnetic RP2 model, both of them regularized on a lattice. High accuracy estimates are obtained for the critical exponents, universal dimensionless quantities and critical couplings. It is con- cluded that both models belong to the same universality class, provided that rather non-standard identifications are made for the momentum-space propagator of the RP2 model. We have also investigated the phase diagram of the RP2 model extended by a second-neighbor interaction. A rich phase diagram is found, where most of the phase transitions are of the first order.  2005 Elsevier B.V. All rights reserved.

PACS: 64.60.Fr; 05.10.Ln

Keywords: Universality; Spin models; Monte Carlo; Finite-size scaling

1. Introduction is the symmetry group of the high-temperature phase and H is the remaining symmetry group of the bro- Universality is sometimes expressed in a somehow ken phase (low temperature). As we shall discuss, the defectively simple way: some critical properties (the subtle point making the above statement not straight- G universal ones) of a system are given by space di- forward to use, is that needs not to be the symmetry mensionality and the local properties (i.e., near the group of the microscopic Hamiltonian, but that of the identity element) of the coset space G/H, where G coarse-grained fixed-point action. On the spirit of the above statement, some time ago [1] a seemingly complete classification was ob- E-mail address: laf@lattice.fis.ucm.es (L.A. Fernández). tained of the universality classes of three-dimensional

0370-2693/$ – see front matter  2005 Elsevier B.V. All rights reserved. doi:10.1016/j.physletb.2005.09.049 282 L.A. Fernández et al. / Physics Letters B 628 (2005) 281–290 systems where G = O(3). In this picture, a phase tran- the renormalization-group sense. To that end, those sition of a vector model, with O(3) global symmetry authors concentrated in producing extremely accurate and with an O(2) low-temperature phase symmetry, data on small lattices. in three dimensions must belong to the O(3)/O(2) Our purpose is to study in greater detail the critical scheme of symmetry breaking (classical Heisenberg properties of the three-dimensional O(5) non-linear 2 model). In addition, if H = O(1) = Z2 is the remain- σ model, and of the AFM RP model. We improve ing symmetry, the corresponding scheme should be over previous studies of both models, obtaining more O(4)/O(3) which is locally isomorphic to O(3)/O(1).1 accurate estimates for critical exponents, universal di- This classification has been challenged by the chiral mensionless quantities and non-universal critical cou- models [2]. However, the situation is still hotly de- plings. As symmetries play such a prominent role, we bated: some authors believe that the chiral transitions will also explore the possibilities of changing those are weakly first-order [3], while others claim [4] that of the low-temperature phase by adding a second- the chiral universality class exists, implying the rele- neighbors coupling to the Hamiltonian of the AFM vance of the global properties of G/H. RP2 model. In this Letter, we shall consider the three-dimensional antiferromagnetic (AFM) RP2 model [5–8], a model displaying a second-order phase transition 2. The models and escaping from the previously expressed paradigm. It is worth recalling [9,10] that one of the phase transi- We are considering a system of N-component nor- tions found in models for colossal magnetoresistance malized spins {vi} placed in a three-dimensional sim- oxides [11] belongs to the universality class of the ple cubic lattice of size L with periodic boundary con- AFM RP2 model. The microscopic Hamiltonian of ditions. The actions of our lattice systems are this model has a global O(3) symmetry group, while SO(N) =−β (v ·v ), the low-temperature phase has, at least, a remain- i j i,j ing O(2) symmetry [9]. We will show here that the RPN−1 2 model belongs to the universality class of the three- S =−β (vi ·vj ) , (1) dimensional O(5) non-linear σ model. Some ground i,j for this arises from a hand-waving argument, sug- where the sums are extended to all pairs of nearest gested to us by one of the referees of Ref. [9] (see neighbors. Our sign convention is fixed by the parti- below). tion function: The universality class of the three-dimensional −S O(5) non-linear σ model has received less attention Z = dvi e , (2) that O(N) models with N = 0, 1, 2, 3 and 4. In spite i of that, it has been recently argued that O(5) could dv being the rotationally invariant measure over the be relevant for the high-temperature superconduct- N-dimensional unit sphere. ing cuprates [12]. Nevertheless, perturbative field- To construct observables, in addition to the vector theoretic methods have been used to estimate the field vi , we consider the (traceless) tensorial field critical exponents [13–16]. From the numerical side, 1 only a rather unconvincing Monte Carlo simulation αβ = α β − αβ = τi vi vi δ ,α,β1,...,N. (3) [17] was available until very recently. Fortunately, N there has been a recent, much more careful study The interesting quantities related with the order pa- [18]. Yet, the scope of Ref. [18] was to determine rameters can be constructed in terms of the Fourier = whether an interaction explicitly degrading the O(5) transforms of the fields (fi vi,τi ) symmetry to an O(3) ⊕ O(2) group was relevant in 1 − · f(ˆ p) = e ip ri f . (4) L3 i i N−1 1 This statement assumes that the global properties of the coset For RP models, the local gauge invariance G/H are irrelevant, only the local properties matter. vi →−vi implies that the relevant observables are L.A. Fernández et al. / Physics Letters B 628 (2005) 281–290 283 constructed in terms of the tensor field. However, for We have computed β-derivatives of observables O(N) we have found very interesting to consider as through their connected correlation with the action. well quantities related with the tensor field. Furthermore, we have extrapolated mean-values from We construct the scalars (under global O(N) trans- the simulation coupling, to a neighboring value of β formations and spatial translations) using the standard reweighting techniques, that cover all the relevant part of the critical region [20]. = ˆ · ˆ∗ SV(p) v(p) v (p), The relationship between the O(5) model and the ∗ 2 ST(p) = tr τ(ˆ p)τˆ (p), (5) RP model arises from the Landau–Wilson–Fisher 2 which, in addition to the action, are the only quantities Hamiltonian for the RP system [9]. Indeed, at the measured during the simulation. Their mean values mean field level [6,9], the ferromagnetic quantities are yield the propagators: simple functions of the staggered ones. This suggests to construct the Landau–Wilson–Fisher Hamiltonian 3 GT,V(p) = L ST,V(p) . (6) from the staggered magnetization, which is a traceless, real, symmetric 3 × 3 matrix: In the thermodynamic limit and at the critical point, f = α,β x +y +z α,β the propagator is expected to have poles at p0 M = (−1) i i i τ . (12) s = s i (0, 0, 0) and, for the antiferromagnetic model, at p0 i (π,π,π)2: Note that Ms has 5 independent quantities. It is there- −η fore a simple matter to obtain a five-components real + ≈ Zξ G(p0 δp) − , (7) 2 = M2 (δp)2 + ξ 2 vector v such that v tr s . The less trivial part regards the fourth-order interaction terms. In principle, f s where ξ δp 1, and the exponents η and η corre- the O(3) symmetry of the microscopic Hamiltonian spond to independent wave function renormalization M4 [ M2]2 f would allow for a tr s term and a tr s one. Sur- at each pole. Note that close to the critical point ξ and (v2)2 s prisingly enough, both terms are proportional to . ξ are expected to remain proportional to each other Thus, assuming that sixth-order terms are irrelevant, (this will be explicitly checked numerically). the Landau–Wilson–Fisher Hamiltonian is expected to The (non-connected) susceptibilities are simply: have a O(5) symmetry group and both models belong = to the same universality class. This does not only im- χ G(p0). (8) plies that both models have the same critical exponents In a finite lattice an extremely useful definition of 2 but also that the L →∞limit of U O(5) and of U s,RP the correlation length can be obtained from the (dis- 4,V 4 (evaluated at their respective critical couplings) coin- crete) derivative of G(p).Usingδp = (2π/L,0, 0) cide. one obtains [19,20] G(p )/G(p + δp) − 1 1/2 ξ = 0 0 . (9) 4sin2(π/L) 3. Numerical methods We also compute the cumulants In the O(5) model we have studied lattice sizes S2 L = 6, 8, 12, 16, 24, 32, 48, 64, 96 and L = 128, at U4 = . (10) = S2 β 1.1812. We have combined a Wolff’s single cluster update with Metropolis. Our elementary Monte Finally, the energy per link is Carlo step (EMCS) consists of (10L+1) Wolff’s clus- E = S/ −3βL3 . (11) ter updates and then a full-lattice Metropolis sweep. We take measurements after every EMCS. Since the average size of clusters grows as L2−η ≈ L2, 80% 2 In the remaining part of this section, if a subindex V or T does of simulation time we are tracing clusters for all L, not explicitly appears, it will imply that the equation is valid both for vector or tensor quantities. The same convention will apply for while Metropolis accounts for 10% of time and mea- superscript (f) (ferromagnetic) and (s) (staggered). Staggered quan- surements for the remaining 10%. The total simulation tities are useful only for antiferromagnetic models. time has been the equivalent to 600 days of Pentium IV 284 L.A. Fernández et al. / Physics Letters B 628 (2005) 281–290 at 3.2 GHz. The number of EMCS ranges from 108 for where the dots stand for higher-order scaling cor- L = 6to1.4×106 at L = 128. The integrated autocor- rections, ν is the correlation-length critical exponent, relation times for the susceptibility and for the energy −ω is the (universal) first irrelevant critical exponent, are smaller than 1 EMCS for all the simulated lattice while AO is a non-universal amplitude. In a typical sizes. application, one fixes the ratio s = L2/L1 to 2, and Since we are interested in high accuracy estimates, consider pairs of lattices L and 2L. A linear extrapola- we have used double precision arithmetics. One also tion in L−ω is used to extract the infinite volume limit. needs to worry about the pseudo-random number One just needs to make sure that the minimum lattice generator. We have therefore implemented a Schwin- size included in the extrapolation is large enough to ger–Dyson test. It turned out that the 32-bits Parisi– safely neglect the higher-order corrections. Of course, Rapuano pseudo-random number generator [21] pro- any quantity scaling like ξ at the critical point, such duces biased results. Either the Parisi–Rapuano plus as LU4, may play the same role in Eq. (13). However, congruential generator [22] or the 64 bits Parisi– usually ξ yields smaller scaling corrections than U4. Rapuano generator cured this bias. The 64 bits Parisi– The extrapolation method based on Eq. (13) is Rapuano generator is faster and it has been our final feasible for the antiferromagnetic RP2 model. Un- choice. fortunately, for the O(5) model the amplitude AO For the antiferromagnetic RP2 model, no efficient is surprisingly small. In fact, resummation of the cluster method is available. We have simulated in lat- -expansion yields ω = 0.79(2) [13], while blind tice sizes from L = 8, 12, 16, 24, 32, 48 and L = 64 at use of Eq. (13) on our numerical data would predict β =−2.41. We used a multi-hit Metropolis sequential ω ≈ 2. We have then considered an additional cor- ˜ −σ algorithm. Making a new spin proposal completely in- rection term, AO L . The exponent σ is an effective dependent from the previous spin value, we achieve way of taking into account a variety of higher-order an acceptance of about 30%. We have used 2 hits what scaling corrections of similar magnitude (an L−2ω ensures a 50% acceptance. The observables have been contribution, subleading universal irrelevant critical measured every two Metropolis full-lattice sweep (our corrections, analytic corrections, effects of the non- EMCS). linearity of the scaling fields, etc. [20]). Its utility will The number of EMCS ranges from 108 for L = 8to be in that it allow us to give sensible error estimates for 7 × 108 for L = 64. In units of the integrated autocor- the infinite-volume extrapolations, instead of bluntly relation time τ (for the order parameter) we have more taking AO = 0. than 106τ for L = 64. The data up to L = 48 were The most precise way of extracting the critical ex- obtained in Pentium IV clusters (simulation time was ponent, ω, and the critical point βc is to consider the roughly equivalent to a 1000 days of a single proces- crossing point of dimensionless quantities such as ξ/L sor). For the largest lattice, data were obtained in the and U4. Indeed, comparing their values in lattices L1 Mare Nostrum computer of the Barcelona Supercom- and L2, one finds that they take a common value at puting Center (simulation time was roughly equivalent to 3000 days of a single processor). −ω 1 − (L2/L1) − − βL2,L1 = β + B L ω 1/ν +··· . We perform a finite-size scaling analysis, using the c c 1/ν 1 (L2/L1) − 1 quotients method [5,6,20]. In this approach, one com- (14) pares the mean value of an observable, O, in two systems of sizes L1 and L2,atthevalueofβ where the The non-universal amplitude B depends on the consid- correlation length in units of the lattice sizes coincides ered dimensionless quantity. Again, one usually take for both systems. If, for the infinite volume system, pairs of lattices L and 2L, and extrapolates to infinite −x O(β) ∝|β − βc| O , the basic equation of the quo- volume using Eq. (14), maybe performing a joint fit tient method is for the crossing points of several dimensionless quan- O(β,L ) tities. Again, for the O(5) model the amplitudes B are L1,L2 ≡ 2 QO exceedingly small, and we need to add to Eq. (14) an O(β,L1) ξ(L2,β) = L2 ξ(L1,β) L1 analogous higher-order term, where σ plays the role − = xO /ν + ω +··· ˜ (L2/L1) 1 AO L1 , (13) of ω, and with amplitude B. L.A. Fernández et al. / Physics Letters B 628 (2005) 281–290 285

4. Results for the O(5) model Table 1 L,2L Effective critical points, βc ,oftheO(5) model, from the cross- The first step is the location of the critical point ing points of the dimensionless quantities ξV/L and U4,V obtained and the scaling corrections exponent. In Table 1 we in lattices of sizes L and 2L. We also show the value of both quan- L,2L tities at their respective crossing point show the crossing points βc for the dimensionless L,2L L,2L Lβ ξV/L β U4,V quantities ξV/L and U4,V. To study the finite size cor- c,ξV/L c,U4,V rections, we need to fit them to 6 1.179331(10) 0.275961(19) 1.182619(19) 1.069593(17) 8 1.180656(8) 0.278757(18) 1.181896(12) 1.069544(16) L,2L ≈ + −ω−1/ν + ˜ −σ −1/ν βc βc BL BL . (15) 12 1.181202(4) 0.280487(17) 1.181492(7) 1.069673(14) ˜ 16 1.181313(4) 0.28105(2) 1.181410(6) 1.069705(18) Fixing B = 0 yields ω larger than 2 which is unac- 24 1.181353(3) 0.28137(2) 1.181371(5) 1.069746(17) ceptable given the field theory estimate ω = 0.79(2) 32 1.181365(3) 0.28146(3) 1.181371(4) 1.06976(2) [13]. Our interpretation is that B is too small to be ob- 48 1.181366(3) 0.28155(5) 1.181373(5) 1.06972(3) served even with our 6-digit accuracy. We have there- 64 1.181362(4) 0.28142(9) 1.181370(6) 1.06982(6) fore fixed ω = 0.79(4) and we have taken as fit parameters β , B, B˜ and σ . We have doubled the field c columns of Table 1. Although scaling-corrections are theory error in ω for safety. To further constrain β c tiny, they can be clearly observed. Using Eqs. (18), (and σ ), we have performed a joint fit of the crossing (19) we obtain points for both ξV/L and U4,V, with the same βc.For ∗ this model we always fit for L 8. The results are (in ξ χ2 2.9 V = 0.28145(13) = , all fits reported in this work, the full covariance matrix L dof 3 was used): 2 ∗ χ 2.8 = = U = 1.06978(5) = , (20) βc 1.1813654(19), σ 2.21(17), 4,V dof 3 χ2/dof = 7.1/8. (16) while the amplitudes of the leading scaling-corrections Notice that, for using Eq. (15), an estimate of ν is are needed. Fortunately, a rough estimate ν ≈ 0.78 (see = = below) is enough, given the uncertainty in ω and σ . CξV/L 0.002(3), CU4,V 0.0002(6). (21) As for the amplitudes of the leading correction term, To obtain the critical exponents, we consider the we find operators ∂β ξV and χV,T whose associated exponents = =− = + = = − BξV/L 0.004(6), BU4,V 0.001(3), (17) are x∂β ξV ν 1 and xχV,T γV,T ν(2 ηV,T). Taking the base 2 logarithm of the quotients, see while the amplitudes B˜ are of order one. We then see −σ Eq. (13), we obtain the effective size dependent expo- that the L term is crucial in order to obtain a sensi- nents shown in Table 2. In order to obtain their infinite →∞ ble error estimate in the L extrapolation. volume value, we use (13), including an explicit L−σ At this point we may obtain two universal quanti- term in the fit: ∗ ∗ →∞ ties, U4,V and ξV/L, namely, the L limit of U4,V L,2L − − and ξV/L evaluated exactly at the critical coupling. = xO /ν + ω + ˜ σ QO 2 AO L AO L (22) Again, due to the smallness of the leading scaling- corrections, we extrapolated to L →∞using the fol- (note that we have absorbed a constant factor 2xO /ν lowing functional forms: into the amplitudes A for scaling-corrections). We obtain L,2L ≈ ∗ + −ω + ˜ −σ U4,V βc ,L U4,V CU4,V L CU4,V L , (18) ν = 0.780(2), σ = 2.15(19), L,2L ∗ ξV(βc ,L) ξV −ω ˜ −σ =− 2 = ≈ + C L + C L . (19) A∂β ξV 0.04(7), χ /dof 8.4/8, L L ξV/L ξV/L ηV = 0.03405(3), σ = 2.27(19), Our numerical estimates for U (βL,2L,L) and 4,V c 2 L,2L Aχ = 0.0012(14), χ /dof = 8.5/8, ξV(βc ,L)/L are displayed in the third and fourth V 286 L.A. Fernández et al. / Physics Letters B 628 (2005) 281–290

− Fig. 1. Size-dependent estimators for the anomalous dimensions of the O(5) model as obtained from (L, 2L) pairs, versus L ω.Weplot estimates from the crossing points of U4,V and of ξV/L.Weusethevalueω = 0.79, from Ref. [13].

= Table 2 safely set A∂β ξV 0, the ﬁnal result would have L-dependent effective values of exponents ν, ηV and ηT for the been ν = 0.7813(4). L,2L O(5) model, calculated from QO Lν ηV ηT 6 0.7963(3) 0.04343(6) 1.3499(3) 5. Results for the antiferromagnetic RP2 model 8 0.7894(4) 0.03837(7) 1.34254(12) 12 0.7849(4) 0.03554(5) 1.33780(10) 16 0.7833(4) 0.03462(6) 1.33589(11) As we said before, qualitative arguments sug- 24 0.7819(7) 0.03430(10) 1.33455(19) gest that the antiferromagnetic RP2 model belongs 32 0.7802(14) 0.03396(16) 1.3332(2) to the O(5) universality class. Our aim is to make 48 0.7817(19) 0.0339(2) 1.3322(4) the most astringent possible test of this hypothe- 64 0.781(4) 0.0341(4) 1.3321(7) sis, thus we perform here an update of a previous study [6] of the RP2 critical quantities. We re- ηT = 1.3307(5), σ = 2.24(19), port here largely improved estimates for critical cou- 2 pling and exponents. Furthermore, we give estimates A =−0.0053(12), χ /dof = 9.6/8. (23) ∗ χT for the dimensionless quantity U s, , that can be di- ∗ 4 Less than a 5% of the total error is due to the error in rectly compared with the U4,V obtained for the O(5) ω = 0.79(4) for both ν and ηV.ForηT it is about a model. 30%. There are two points to be made about the ex- In this case, the extrapolation to the inﬁnite volume trapolation: limit is more standard (see, for instance, [24]) than for the O(5) model, because the amplitude of the leading • The O(5) model is not an improved action [23] scaling-corrections are much larger in most cases. To (in the sense of exactly vanishing leading scaling- estimate ω and βc, we consider pair of lattices L and

corrections), since AχT is clearly non-zero (this is 2L, performing a joint ﬁt to Eq. (14) of the crossing also illustrated in Fig. 1). Had we not considered points of all four dimensionless quantities, imposing a the tensorial operators, this would have been com- common value of βc and ω (see Fig. 2). To control for pletely missed. systematic errors due to higher-order corrections, we • It is somehow disappointing to compare the accu- follow the following procedure. We perform the ﬁt us- racy of the effective exponent ν in Table 2 with the ing data for L Lmin, seeking a value of Lmin where error in the extrapolation (23). Indeed, could we a reasonable value of χ2/dof is found. Furthermore, L.A. Fernández et al. / Physics Letters B 628 (2005) 281–290 287

− − Fig. 2. Crossing points of the different dimensionless quantities, for the (L, 2L) pairs, as a function of L ω 1/ν . Lines are ﬁts to Eq. (14), constrained to yield common values of βc and ω.Theω used in the X axis correspond to the optimal value. Note that the minimum pair plotted is (12, 24).

Table 3 Size-dependent estimators for the critical coupling and several universal quantities, as obtained from (L, 2L) pairs, in the RP2 model. The last row correspond to the inﬁnite volume extrapolations L,2L s s f Lβc,ξs/L U4 νηη 8 −2.39892(18) 1.06810(8) 0.7848(12) 0.0390(3) 1.4155(6) 12 −2.40487(13) 1.06805(11) 0.7871(16) 0.0364(4) 1.3902(7) 16 −2.40670(12) 1.06849(14) 0.782(2) 0.0357(6) 1.3776(11) 24 −2.40792(9) 1.06883(19) 0.779(3) 0.0351(7) 1.3655(13) 32 −2.40846(7) 1.06868(18) 0.783(3) 0.0337(7) 1.3572(14) ∞−2.40899(13) 1.0691(5) 0.780(4) 0.032(2) 1.328(4)

we require that the fit performed for L>Lmin yield antiferromagnetic model and it can be obtained us- compatible results. In that case, we report the central ing only the two-points correlation function. We have L,2L value from the L Lmin fit, but taking the enlarged er- checked that other choices for βc yield compatible rors from the L>Lmin fit. We found that Lmin = 12 is results, with slightly larger errors. Our extrapolations enough for the extrapolation of βc. We obtain: are shown together with the effective L-dependent estimates in Table 3. Error estimates in the extrapolation βc =−2.40899(13), include the effect of the uncertainty in ω. For expo- ω = 0.78(4), χ 2/dof = 8.5/10. (24) nent ν, scaling corrections are completely buried in the statistical errors. We extrapolated with a simple linear Once we have determined ω, we proceed to extrap- fit, using Lmin = 8. The situation is rather different for s,∗ s olate U , and the critical exponents, using the analog η . For that exponent, enlarging Lmin systematically 4 − of Eqs. (18), (19), (22) without the effective L σ term. increases the asymptotic estimate. On the other hand, Although one could consider all four types of crossing a fit quadratic in L−ω yields a linear term compatible L,2L points, βc , the resulting quotients would be highly with zero. The linear extrapolation with Lmin = 16 is correlated, making join fits scarcely useful. We con- identical to the quadratic extrapolation from Lmin = 8. centrate on the crossing point of ξ s/L, which seems This is the result indicated in Table 3.Asforηf,we the most natural quantity, as we are dealing with an have rather strong leading scaling corrections. Indeed, 288 L.A. Fernández et al. / Physics Letters B 628 (2005) 281–290

Fig. 3. Phase diagram of the extended antiferromagnetic RP2 model, Eq. (26). a fit linear in L−ω yields basically identical results for • PM: the usual (paramagnetic) disordered state, Lmin = 8 and Lmin = 12 (this is the result reported in where the O(3) symmetry of the action (26) is pre- Table 3). Furthermore, a fit quadratic in L−ω including served. all points, yielded ηf = 1.331(5). The extrapolation for • O(2): (say) even spins fluctuate almost parallel s,∗ U4 is equally simple. to (say) the Z axis, with random sense (local Z2 The extrapolation for other scale-invariant quanti- symmetry), while odd spins fluctuate in the per- ties, without an obvious correspondent in the O(5) pendicular plane (global O(2) symmetry). model, are: • O(1): two sublattices with ferromagnetic ordering in perpendicular directions, with random sense ∗ ∗ ξ s, ξ f, (the local Z2 symmetry, vi →−vi is always pre- = 0.5379(17), = 0.2236(15), L L served). • Skyrmion/flux: the spins are parallel to the diago- f,∗ = U4 1.3114(6). (25) nals of the unit cube, so that they point out from/to the center (i.e., the propagator show three peaks = at p0 (π, π, 0) and permutations). It is inter- 6. Next nearest neighbors coupling esting to note the vectorial version of this phase appear in models for colossal magnetoresistance oxides [25]. A rather subtle question regards the symmetry of the low-temperature RP2 antiferromagnetic phase The most relevant results can be summarized as fol- [6,9]. A way of investigating this problem is to study lows: the enlarged action • We have obtained critical exponents for several S =−β (v ·v )2 − β (v ·v )2, (26) points along the PM–O(2) critical lines, with sig- 1 i j 2 i j = i,j i,j nificantly less accuracy than for the β2 0 model. No variation was observed within errors. where an additional second-neighbors coupling is con- • The O(2)–O(1) critical line is repelled from the sidered. β2 = 0 axis by the second-neighbors coupling. The phase diagram for β1 < 0(Fig. 3) contains the We interpret this as a competition with the order- following regions (spins are classified as even or odd, from-disorder mechanism [6,9] behind the PM– according to the parity of xi + yi + zi ): O(2) transition. L.A. Fernández et al. / Physics Letters B 628 (2005) 281–290 289

Table 4 2 f 2 Summary of inﬁnite-volume estimates for the 3D antiferromagnetic RP ,O(5) and O(4) models. We call η to η for RP and to the ηT for O(N) models. FT stands for ﬁeld theory

Model βc νη ηU4 RP2 (this work) −2.40899(13) 0.780(4) 0.032(2) 1.328(4) 1.0691(5) O(5) (this work) +1.1813654(19) 0.780(2) 0.03405(3) 1.3307(5) 1.06978(5) O(5) (Ref. [18]) +1.18138(3) 0.779(3) 0.034(1) – 1.069(1) O(5) (FT [13])– 0.762(7) 0.034(4) –– O(4) +0.935858(8) [27] 0.749(2) [28] 0.0365(10) [28] 1.375(5) [27] –

• A naive analysis suggests that the O(2)–O(1) tran- sor representations improve significantly over previ- sition line should belong to the XY universality ous work (see Table 4). 2 class (consider first the limit β1 =−∞, then the For the RP model, the leading scaling corrections identity cos2 θ = (1 + cos 2θ)/2 for the less or- are sizeable. It is amusing that we are able to ob- dered face-centered cubic sublattice). However, tain numerically (for the first time, we believe) an we have found that this transition line is first- estimate for the (universal) scaling-correction expo- order, as revealed by the double-peaked histogram nent ω = 0.78(4), of accuracy comparable to the per- of the second neighbors energy. We are able to es- turbative field-theoretical estimate [13] ω = 0.79(2). timate a non-zero latent-heat up to β1 =−6. At As Table 4 shows, within the achieved accuracy, β1 =−4 the double-peak structure is still easy both models seem to belong to the same universal- to observe on small lattices. We presume that the ity class (for comparison, we also show results from whole line is first-order, although it could become the O(4) universality class). To conclude this, one very weak. needs to accept that in the RP2 model, the wave- • The skyrmion–PM transition lines turned out to be function renormalization for the propagator pole at = first-order at all the checked points. p0 (π,π,π) is as for the O(5) fundamental field, • = = Note that at β1 0, we have two decoupled fer- while at p0 (0, 0, 0) is as for the O(5) tensor romagnetic RP2 models on the face-centered cu- field. bic lattice (a model showing first-order transition, We have also obtained the phase-diagram of the well known in the liquids-crystal context [26]). RP2 model extended with a second nearest-neighbors We should remark that a precise location of the interaction. We have found a rich phase diagram. triple point O(2)–O(1)–PM is very difficult to achieve. Acknowledgements 7. Conclusions We are indebted with Juan Jesús Ruiz-Lorenzo for We have obtained high accuracy estimates of crit- discussions. We are grateful to the Barcelona Super- ical exponents and other universal quantities for the computing Center, for allowing us to use Mare Nos- three-dimensional O(5) and the antiferromagnetic trum in its installation phase. Isabel Campos was cru- RP2 models, by means of Monte Carlo simulation, cial in this respect. About a third of the total simula- finite-size scaling analysis and careful infinite-volume tion time was obtained from the cluster of the Insti- extrapolation. tuto de Biocomputación y Física de Sistemas Com- In the case of the O(5) model the coupling to the plejos, in Zaragoza, and other computing resources leading irrelevant operator is rather weak, but non- of the Departamento de Física Teórica of Univer- vanishing, and one needs to consider higher-order sidad de Zaragoza. We acknowledge partial finan- scaling corrections to obtain sensible error estimates. cial support from Ministerio de Educación y Ciencia In spite of that, our estimates for the critical coupling (Spain) though research contracts BFM2003-08532 and the anomalous dimensions for the vector and ten- and FIS2004-05073. 290 L.A. Fernández et al. / Physics Letters B 628 (2005) 281–290

References [13] P. Calabrese, A. Pelissetto, E. Vicari, Phys. Rev. B 67 (2003) 054505. [14] S.A. Antonenko, A.I. Sokolov, Phys. Rev. E 51 (1995) 1894. [1] P. Azaria, B. Delamotte, T. Jolicoeur, Phys. Rev. Lett. 64 [15] H. Kleinert, J. Neu, V. Schulte-Frohlinde, K.G. Chetyrkin, S.A. (1990) 3175; Larin, Phys. Lett. B 272 (1991) 39; P. Azaria, B. Delamotte, F. Delduc, T. Jolicoeur, Nucl. Phys. H. Kleinert, J. Neu, V. Schulte-Frohlinde, K.G. Chetyrkin, S.A. B 408 (1993) 485. Larin, Phys. Lett. B 319 (1993) 545, Erratum. [2] See, for example, H. Kawamura, Can. J. Phys. 79 (2001) 1447. [16] A. Butti, F. Parisen Toldin, Nucl. Phys. B 704 (2005) 527. [3] M. Tissier, B. Delamotte, D. Mouhanna, Phys. Rev. Lett. 84 [17] X. Hu, Phys. Rev. Lett. 87 (2001) 057004; (2000) 5208; X. Hu, Phys. Rev. Lett. 88 (2002) 059704. M. Itakura, J. Phys. Soc. Jpn. 72 (2003) 74. [18] M. Hasenbusch, A. Pelissetto, E. Vicari, cond-mat/0502327. [4] A. Pelissetto, P. Rossi, E. Vicari, Phys. Rev. B 65 (2002) [19] B. Cooper, B. Freedman, D. Preston, Nucl. Phys. B 210 (1982) 020403, and references therein. 210. [5] H.G. Ballesteros, L.A. Fernández, V. Martín-Mayor, A. Muñoz [20] D. Amit, V. Martin-Mayor, Field Theory, the Renormalization Sudupe, Phys. Lett. B 378 (1996) 207. Group and Critical Phenomena, third ed., World Scientiﬁc, Sin- [6] H.G. Ballesteros, L.A. Fernández, V. Martín-Mayor, A. Muñoz gapore, 2005. Sudupe, Nucl. Phys. B 483 (1997) 707. [21] G. Parisi, F. Rapuano, Phys. Lett. B 157 (1985) 301. [7] S. Romano, Int. J. Mod. Phys. B 8 (1994) 3389. [22] See, e.g., H.G. Ballesteros, V. Martín-Mayor, Phys. Rev. E 58 [8] G. Korhing, R.E. Shrock, Nucl. Phys. B 295 (1988) 36. (1998) 6787, and references therein. [9] J.L. Alonso, A. Cruz, L.A. Fernández, S. Jiménez, V. Martín- [23] H.G. Ballesteros, L.A. Fernández, V. Martín-Mayor, A. Muñoz Mayor, J.J. Ruiz-Lorenzo, A. Tarancón, Phys. Rev. B 71 (2005) Sudupe, Phys. Lett. B 441 (1998) 330; 014420. M. Hasenbusch, K. Pinn, S. Vinti, Phys. Rev. B 59 (1999) [10] J.M. Carmona, A. Cruz, L.A. Fernández, S. Jiménez, V. 11471. Martín-Mayor, A. Muñoz Sudupe, J. Pech, J.J. Ruiz-Lorenzo, [24] H.G. Ballesteros, L.A. Fernández, V. Martín-Mayor, A. Muñoz A. Tarancón, P. Tellez, Phys. Lett. B 560 (2003) 140. Sudupe, G. Parisi, J.J. Ruiz-Lorenzo, J. Phys. A: Math. Gen. 32 [11] See, e.g., J.M.D. Coey, M. Viret, S. von Molnar, Adv. Phys. 48 (1999) 1. (1999) 167; [25] See, e.g., J.L. Alonso, J.A. Capitán, L.A. Fernández, F. Guinea, E. Dagotto, T. Hotta, A. Moreo, Phys. Rep. 344 (2001) 1; V. Martín-Mayor, Phys. Rev. B 64 (2001) 54408. E. Dagotto, Nanoscale Phase Separation and Colossal [26] P.A. Lebwohl, G. Lasher, Phys. Rev. A 6 (1972) 426. Magneto-Resistence, Springer-Verlag, Berlin, 2002. [27] H.G. Ballesteros, L.A. Fernández, V. Martín-Mayor, A. Muñoz [12] S.-C. Zhang, Science 275 (1997) 1089; Sudupe, Phys. Lett. B 387 (1996) 125. S.-C. Zhang, J.-P. Hu, E. Arrigoni, W. Hanke, A. Auerbach, [28] M. Hasenbusch, J. Phys. A 34 (2001) 8221. Phys. Rev. B 60 (1999) 13070. Physics Letters B 628 (2005) 291 www.elsevier.com/locate/physletb Erratum Erratum to: “Relativistic quantum physics with hyperbolic numbers” [Phys. Lett. B 625 (2005) 313]

S. Ulrych

Wehrenbachhalde 35, CH-8053 Zürich, Switzerland Received 26 September 2005 Available online 30 September 2005

In the mentioned Letter [1] Table 1 is not correct with respect to the grade involution for the hypercomplex units i and I = e0e¯1e2e¯3 = ij . The correct form is given in this erratum. The new table is now consistent with † proposition 15.33 of Porteous [2] which requires that a¯ =ˆa . Note, that the element e0 = 1 is invariant under conjugation, reversion, and graduation

Table 1 Effect of conjugation, reversion, and graduation on the used hypercomplex units a aa¯ † aˆ ei −+ − σi ++ + I +− − i −− + j −+ −

References

[1] S. Ulrych, Phys. Lett. B 625 (2005) 313. [2] I. Porteous, Clifford Algebras and the Classical Groups, Cambridge Univ. Press, Cambridge, 1995.

DOI of original article: 10.1016/j.physletb.2005.08.072. E-mail address: [email protected] (S. Ulrych).

Cumulative author index to volumes 621–628

A2 Collaboration, 624, 173 Aliotta, M., 624, 181 Abazov, V.M., 622, 265; 626, 35, 45, 55 Alkhazov, G., 622, 265; 626, 35, 45, 55; 628,18 Abbott, B., 622, 265; 626, 35, 45, 55 Allaby, J., 622, 249; 623,26 Abdalla, E., 624, 141 Allton, C.R., 628, 125 Abe, K., 621, 28, 28, 41; 622, 218, 218; 624, 11, 11 Aloisio, A., 622, 249; 623,26 Abe, Y., 623, 126 Altieri, S., 624, 173 Ablikim, M., 622,6;625, 196 Alton, A., 622, 265; 626, 35, 45, 55 Abolins, M., 622, 265; 626, 35, 45, 55 Altschul, B., 628, 106 Achard, P., 622, 249; 623,26 Alverson, G., 622, 265; 626, 35, 45, 55 Acharya, B.S., 622, 265; 626, 35, 45, 55 Alves, G.A., 622, 265; 626, 35, 45, 55 Adachi, I., 621, 28, 41; 624,11 Alviggi, M.G., 622, 249; 623,26 Adam, C., 626, 235 Amarian, M., 622,14 Adamian, G.G., 621, 119 Amaro-Reyes, J., 628,18 Adams, M., 622, 265; 626, 35, 45, 55 Ambrosino, F., 626,15 Adams, T., 622, 265; 626, 35, 45, 55 Ametller, Ll., 628,40 Adler, J.-O., 626,65 Ananthanarayan, B., 628, 223 Adriani, O., 622, 249; 623,26 Anastasoaie, M., 622, 265; 626, 35, 45, 55 Agelou, M., 622, 265; 626, 35, 45, 55 Anchordoqui, L.A., 621,18 Agnello, M., 622,35 Andeen, T., 622, 265; 626, 35, 45, 55 Agostino, L., 621, 72; 622, 229, 239; 624, 22, 166 Anderhub, H., 622, 249; 623,26 Agram, J.-L., 622, 265; 626, 35, 45, 55 Anderson, S., 622, 265; 626, 35, 45, 55 Aguilar-Benitez, M., 622, 249; 623,26 Andersson, B.-E., 626,65 Aguilar-Saavedra, J.A., 625, 234 Andreev, V., 621,56 Ahn, S.H., 622, 265; 626, 35, 45, 55 Andreev, V.P., 622, 249; 623,26 Ahrens, J., 624, 173 Andrieu, B., 622, 265; 626, 35, 45, 55 Ahsan, M., 622, 265; 626, 35, 45, 55 Andrus, A., 622,14 Aihara, H., 621, 28, 41; 622, 218; 624,11 Anikin, I.V., 626,86 Airapetian, A., 622,14 Anjos, J.C., 621, 72; 622, 229, 239; 624, 22, 166 Akgun, U., 628,18 Annand, J.R.M., 624, 173; 626,65 Akkurt, I., 626,65 Ansari, A., 623,37 Akopov, N., 622,14 Anselmo, F., 622, 249; 623,26 Akopov, Z., 622,14 Anthonis, T., 621,56 Aktas, A., 621,56 Antinori, F., 623,17 Al-Khatib, A., 622,29 Anton, G., 624, 173 Alcaraz, J., 622, 249; 623,26 Antonelli, A., 626,15 Alemanni, G., 622, 249; 623,26 Antonelli, M., 626,15 Alexeev, G.D., 622, 265; 626, 35, 45, 55 Antonenko, N.V., 621, 119 Alimonti, G., 621, 72; 622, 229, 239; 624, 22, 166 Aoki, S., 626,24

0370-2693/2005 Published by Elsevier B.V. doi:10.1016/S0370-2693(05)01473-5 Cumulative author index to volumes 621–628 (2005) 292–321 293

Aoyama, S., 625, 127 Baksay, L., 622, 249; 623,26 Aplin, S., 621,56 Balatz, M.Y., 628,18 Aref’eva, I.Ya., 628,1 Baldew, S.V., 622, 249; 623,26 Areﬁev, A., 622, 249; 623,26 Baldin, B., 622, 265; 626, 35, 45, 55 Arena, V., 621, 72; 622, 229, 239; 624, 22, 166 Balin, D., 622,14 Arends, H.-J., 624, 173 Ball, P., 625, 225 Armour, W., 628, 125 Balm, P.W., 622, 265; 626, 35, 45, 55 Arnoud, Y., 622, 265; 626, 35, 45, 55 Ban, Y., 622,6;625, 196 Artamonov, A., 626,24 Banerjee, S., 622, 249, 265; 623, 26; 624, 11; 626, 35, 45, 55 Artamonov, A.V., 623, 192 Banerjee, Sw., 622, 249; 623,26 Artoisenet, P., 628, 211 Banﬁ, A., 628,49 Asakawa, E., 626, 111 Banu, A., 622,29 Asano, Y., 621, 28, 41; 622, 218; 624,11 Baran, V., 625,33 Aschenauer, E.C., 622,14 Baranov, P., 621,56 Askew, A., 622, 265; 626, 35, 45, 55 Barbera, R., 623,17 Åsman, B., 622, 265; 626, 35, 45, 55 Barberio, E., 621,28 Asmone, A., 621,56 Barberis, E., 622, 265; 626, 35, 45, 55 Assis Jesus, A.C.S., 622, 265; 626, 35, 45, 55 Barberis, S., 621, 72; 622, 229, 239; 624, 22, 166 Astvatsatourov, A., 621,56 Barbieri, R., 625, 189 Atamantchouk, A.G., 628,18 Barbuto, E., 626,24 Atchison, F., 625,19 Barczyk, A., 622, 249; 623,26 Atramentov, O., 622, 265; 626, 35, 45, 55 Bargassa, P., 622, 265; 626, 35, 45, 55 Auguste, M., 621, 233 Barger, V., 624, 233 Augustyniak, W., 622,14 Barillère, R., 622, 249; 623,26 Aulchenko, V., 621, 41; 622, 218 Baringer, P., 622, 265; 626, 35, 45, 55 Aulenbacher, K., 624, 173 Barnabé-Heider, M., 624, 186 Aushev, T., 621, 28, 41; 624,11 Barnes, C., 622, 265; 626, 35, 45, 55 Autermann, C., 622, 265; 626, 35, 45, 55 Barr, S.M., 622, 131 Avakian, R., 622,14 Barrelet, E., 621,56 Avetissian, A., 622,14 Barreto, J., 622, 265; 626, 35, 45, 55 Avetissian, E., 622,14 Bartalini, P., 622, 249; 623,26 Avila, C., 622, 265; 626, 35, 45, 55 Bartel, W., 621,56 Ayan, A.S., 628,18 Bartlett, J.F., 622, 265; 626, 35, 45, 55 Azemoon, T., 622, 249; 623,26 Basile, M., 622, 249; 623,26 Aziz, T., 622, 249; 623,26 Basrak, Z., 625,26 Bassalleck, B., 623, 192 Baba, H., 621,81 Bassler, U., 622, 265; 626, 35, 45, 55 Babaev, A., 621,56 Batalova, N., 622, 249; 623,26 Babu, K.S., 621, 160 Batchelder, J.C., 622,45 Bacchetta, A., 622,14 Battiston, R., 622, 249; 623,26 Bacci, C., 626,15 Battye, R.A., 626, 120 Backovic, S., 621,56 Baudrand, S., 621,56 Bacon, P.A., 623,17 Bauer, D., 622, 265; 626, 35, 45, 55 Badalà, A., 623,17 Bauer, F., 624, 250 Badaud, F., 622, 265; 626, 35, 45, 55 Baumann, T., 627,32 Baden, A., 622, 265; 626, 35, 45, 55 Baumgartner, S., 621,56 Baghdasaryan, A., 621,56 Bay, A., 621, 41; 622, 249; 623, 26; 624,11 Bagnaia, P., 622, 249; 623,26 Bean, A., 622, 265; 626, 35, 45, 55 Bahinipati, S., 621, 28, 41; 622, 218; 624,11 Beauceron, S., 622, 265; 626, 35, 45, 55 Bähr, J., 621,56 Becattini, F., 622, 249; 623,26 Bai, J.Z., 622,6;625, 196 Beck, R., 624, 173 Bailey, P., 622,14 Beck, T., 622,29 Bajo, A., 622, 249; 623,26 Becker, F., 622,29 Bakich, A.M., 621, 28, 41; 622, 218; 624,11 Becker, H.W., 624, 181 Baksay, G., 622, 249; 623,26 Becker, J., 621,56 294 Cumulative author index to volumes 621–628 (2005) 292–321

Becker, U., 622, 249; 623,26 Bianchi, N., 622,14 Beckingham, M., 621,56 Bianco, S., 621, 72; 622, 35, 229, 239; 624, 22, 166 Beckmann, M., 622,14 Biasini, M., 622, 249; 623,26 Bediaga, I., 621, 72; 622, 229, 239; 624, 22, 166 Bigi, I.I., 625,47 Bednarczyk, P., 622,29 Biglietti, M., 622, 249; 623,26 Bedny, I., 621, 41; 622, 218; 624,11 Biland, A., 622, 249; 623,26 Beer, G., 622,35 Binder, M., 622, 265; 626, 35, 45, 55 Begalli, M., 622, 265; 626, 35, 45, 55 Bingham, C.R., 622,45 Begel, M., 622, 265; 626, 35, 45, 55 Bini, C., 626,15 Behner, F., 622, 249; 623,26 Biscarat, C., 622, 265; 626, 35, 45, 55 Behnke, E., 624, 186 Bitenc, U., 621, 28, 41; 622, 218; 624,11 Behnke, O., 621,56 Bizjak, I., 621, 28, 41; 622, 218; 624,11 Behrendt, O., 621,56 Bizot, J.C., 621,56 Beisert, N., 622, 343 Bjerrum-Bohr, N.E.J., 621, 183 Bellavance, A., 622, 265; 626, 35, 45, 55 Black, K.M., 622, 265; 626, 35, 45, 55 Belle Collaboration, 621, 28, 41; 622, 218; 624,11 Blackler, I., 622, 265; 626, 35, 45, 55 Bellucci, L., 622, 249; 623,26 Blackmore, E.W., 623, 192 Belogianni, A., 623,17 Blaising, J.J., 622, 249; 623,26 Belostotski, S., 622,14 Blaizot, J.-P., 627,49 Belousov, A., 621,56 Blazey, G., 622, 265; 626, 35, 45, 55 Beltrame, P., 626,15 Blekman, F., 622, 265; 626, 35, 45, 55 Bencivenni, G., 626,15 Blessing, S., 622, 265; 626, 35, 45, 55 Bender, C.M., 625, 333 Bloch, D., 622, 265; 626, 35, 45, 55 Bentz, W., 621, 246 Bloise, C., 626,15 Benussi, L., 621, 72; 622, 35, 229, 239; 624, 22, 166 Blok, H.P., 622,14 Benzoni, G., 622,29 Bloodworth, I.J., 623,17 Berbeco, R., 622, 249; 623,26 Blumenschein, U., 622, 265; 626, 35, 45, 55 Berdugo, J., 622, 249; 623,26 Blyth, S., 621, 28, 41; 622, 218; 624,11 Berger, Ch., 621,56 Blyth, S.C., 622, 249; 623,26 Berger, N., 621,56 Bobbink, G.J., 622, 249; 623,26 Berges, P., 622, 249; 623,26 Boca, G., 621, 72; 622, 229, 239; 624, 22, 166 Beri, S.B., 622, 265; 626, 35, 45, 55 Bocci, V., 626,15 Bernard, V., 622, 141 Boehnlein, A., 622, 265; 626, 35, 45, 55 Bernardi, G., 622, 265; 626, 35, 45, 55 Boenig, M.-O., 621,56 Bernhard, R., 622, 265; 626, 35, 45, 55 Boeriu, O., 622, 265; 626, 35, 45, 55 Bertani, M., 621, 72; 622, 35, 229, 239; 624, 22, 166 Böhm, A., 622, 249; 623,26 Bertolucci, S., 626,15 Boland, M.J., 626,65 Bertram, I., 622, 265; 626, 35, 45, 55 Boldizsar, L., 622, 249; 623,26 Bertucci, B., 622, 249; 623,26 Bolton, T.A., 622, 265; 626, 35, 45, 55 Bertulani, C.A., 624, 203 Bombara, M., 623,17 Besançon, M., 622, 265; 626, 35, 45, 55 Bonatsos, D., 621, 102 BES Collaboration, 622,6;625, 196 Bondar, A., 621, 28, 41; 622, 218; 624,1,11 Betev, B.L., 622, 249; 623,26 Bondar, N.F., 628,18 Beuselinck, R., 622, 265; 626, 35, 45, 55 Bonnet, E., 623, 43; 627,18 Bezerra de Mello, E.R., 621, 318 Bonomi, G., 621, 72; 622, 229, 239; 624, 22, 166 Bezzubov, V.A., 622, 265; 626, 35, 45, 55 Boos, E., 622, 311 Bhalerao, R.S., 627,49 Boos, E.E., 622, 265 Bhang, H.C., 622,35 Borcherding, F., 622, 265; 626, 35, 45, 55 Bhat, P.C., 622, 265; 626, 35, 45, 55 Borderie, B., 623, 43; 627,18 Bhatnagar, V., 622, 265; 626, 35, 45, 55 Borghini, N., 627,49 Bhattacharya, P., 627,26 Borgia, B., 622, 249; 623,26 Bhattacharyya, G., 628, 141 Borissov, A., 622,14 Bhattacherjee, B., 627, 137 Borissov, G., 622, 265; 626, 35, 45, 55 Bhuyan, B., 623, 192 Borysenko, A., 622,14 Bian, J.G., 622,6;625, 196 Bos, K., 622, 265; 626, 35, 45, 55 Cumulative author index to volumes 621–628 (2005) 292–321 295

Boschini, M., 621, 72; 622, 229, 239; 624, 22, 166 Bunichev, V., 622, 265, 311 Bose, T., 622, 265; 626, 35, 45, 55 Bunyatyan, A., 621,56 Bossi, F., 626,15 Buontempo, S., 626,24 Botta, E., 622,35 Burdin, S., 622, 265; 626, 35, 45, 55 Bottai, S., 622, 249; 623,26 Bürger, A., 622,29 Böttcher, H., 622,14 Burger, J.D., 622, 249; 623,26 Boudry, V., 621,56 Burger, W.J., 622, 249; 623,26 Bougault, R., 627,18 Buric,´ M., 622, 183 Bourilkov, D., 622, 249; 623,26 Burke, S., 622, 265; 626, 35, 45, 55 Bourquin, M., 622, 249; 623,26 Burnett, T.H., 622, 265; 626, 35, 45, 55 Bouwhuis, M., 622,14 Busato, E., 622, 265; 626, 35, 45, 55 Bowring, D., 626,15 Buschhorn, G., 621,56 Boyarsky, A., 626, 184 Büsser, F.W., 621,56 Boyer, D., 621, 233 Busso, L., 622,35 Bozek, A., 621, 28, 41; 622, 218; 624,11 Buszello, C.P., 622, 265; 626, 35, 45, 55 Bozza, C., 626,24 Butler, J.M., 622, 265; 626, 35, 45, 55 Bozza, V., 625, 177 Butler, J.N., 621, 72; 622, 229, 239; 624, 22, 166 Braccini, S., 622, 249; 623,26 Bystritskaya, L., 621,56 Bracco, A., 622,29 Bracco, M.E., 624, 217 Cai, X., 622,6;625, 196 Bracinik, J., 621,56 Cai, X.D., 622, 249; 623,26 Bracko,ˇ M., 621, 28, 41; 622, 218; 624,11 Calderin, I.R., 622, 151; 625, 375 Bradtke, C., 624, 173 Caliandro, R., 623,17 Braghieri, A., 624, 173 Caloi, R., 626,15 Braguta, V.V., 625,41 Calvo, D., 622,35 Brambilla, S., 622,29 Camera, F., 622,29 Branchini, P., 626,15 Camerini, P., 622,35 Brandt, A., 622, 265; 626, 35, 45, 55 Cammin, J., 622, 265; 626, 35, 45, 55 Brandt, G., 621,56 Campana, P., 626,15 Branson, J.G., 622, 249; 623,26 Campbell, A.J., 621,56 Bray, S., 628, 250 Campbell, M., 623,17 Bregant, M., 622,35 Canfora, F., 625, 171, 277 Bressani, T., 622,35 Capell, M., 622, 249; 623,26 Bringel, P., 622,29 Capiluppi, M., 622,14 Bringoltz, B., 628, 113 Capitani, G.P., 622,14 Brisson, V., 621,56 Capon, G., 626,15 Brochu, F., 622, 249; 623,26 Caponero, M., 622,35 Brock, R., 622, 265; 626, 35, 45, 55 Caporale, F., 622,55 Brodzicka, J., 621, 28; 622, 218; 624,11 Capussela, T., 626,15 Brooijmans, G., 622, 265; 626, 35, 45, 55 Cara Romeo, G., 622, 249; 623,26 Bross, A., 622, 265; 626, 35, 45, 55 Cardoso, V., 621, 219 Browder, T.E., 621, 28; 624,11 Cardy, J., 622, 339 Brown, D.P., 621,56 Carena, W., 623,17 Brüll, A., 622,14 Carlino, G., 622, 249; 623,26 Bruncko, D., 621,56 Caron, S., 621, 56; 622, 265; 626, 35, 45, 55 Bruno, G.E., 623,17 Carone, C.D., 626, 195 Bruski, N., 626,24 Carpenter, M.P., 622, 151; 625, 203, 375 Bryman, D.A., 623, 192 Carrer, N., 623,17 Brys,´ T., 625,19 Carrillo, S., 621, 72; 622, 229, 239; 624, 22, 166 Bryzgalov, V., 622,14 Cartacci, A., 622, 249; 623,26 Buchanan, N.J., 622, 265; 626, 35, 45, 55 Cartas-Fuentevilla, R., 623, 165 Buchholz, D., 622, 265; 626, 35, 45, 55 Carvalho, W., 622, 265; 626, 35, 45, 55 Buehler, M., 622, 265; 626, 35, 45, 55 Casadio, R., 625,1 Buescher, V., 622, 265; 626, 35, 45, 55 Casalbuoni, R., 627,89 Bull, S.A., 623,17 Casaus, J., 622, 249; 623,26 296 Cumulative author index to volumes 621–628 (2005) 292–321

Casey, B.C.K., 622, 265; 626, 35, 45, 55 Chen, Y.B., 622,6;625, 196 Casimiro, E., 621, 72; 622, 229, 239; 624, 22, 166 Chensheng, M., 628,18 Cason, N.M., 622, 265; 626, 35, 45, 55 Cheon, B.G., 621, 28, 41; 622, 218; 624,11 Cassol-Brunner, F., 621,56 Cheu, E., 622, 265; 626, 35, 45, 55 Castilla-Valdez, H., 622, 265; 626, 35, 45, 55 Cheung, H.W.K., 621, 72; 622, 229, 239; 624, 22, 166 Castro, C., 626, 209 Chi, S., 626,15 Castromonte, C., 622, 229, 239; 624, 22, 166 Chi, S.P., 622,6;625, 196 Catanesi, M.G., 626,24 Chiang, I.-H., 623, 192 Cavaillou, A., 621, 233 Chiara, C.J., 625, 203 Cavallari, F., 622, 249; 623,26 Chiefari, G., 622, 249; 623, 26; 626,15 Cavallo, N., 622, 249; 623,26 Chikawa, M., 626,24 Cawlﬁeld, C., 621, 72; 622, 229, 239; 624, 22, 166 Chiodini, G., 621, 72; 622, 229, 239; 624, 22, 166 Cecchi, C., 622, 249; 623,26 Chistov, R., 621, 28, 41; 622, 218 Ceradini, F., 626,15 Chiu, T.-W., 624,31 Cerello, P., 622,35 Cho, D.K., 622, 265; 626, 35, 45, 55 Cerny, K., 621,56 Cho, K., 621, 72; 622, 229, 239; 624, 22, 166 Cerny, V., 621,56 Choi, S., 622, 265; 626, 35, 45, 55 Cerrada, M., 622, 249; 623,26 Choi, S.-K., 624,11 Cerutti, A., 621, 72; 622, 229, 239; 624, 22, 166 Choi, Y., 621, 28, 41; 622, 218; 624,11 Chakrabarti, S., 622, 265; 626, 35, 45, 55 Chong, Z.-W., 626, 215 Chakraborty, B., 625, 302 CHORUS Collaboration, 626,24 Chakraborty, D., 622, 265; 626, 35, 45, 55 Choudhary, B., 622, 265; 626, 35, 45, 55 Chamizo, M., 622, 249; 623,26 Chowdhury, P., 622, 151; 625, 375 Chan, K.M., 622, 265; 626, 35, 45, 55 Choy, P.T.W., 625, 203 Chandra, A., 622, 265; 626, 35, 45, 55 Christiansen, T., 622, 265; 626, 35, 45, 55 Chang, C.-H., 623, 218 Christidi, I.-A., 623, 192 Chang, H.-J., 624,31 Christofek, L., 622, 265; 626, 35, 45, 55 Chang, J.F., 622,6;625, 196 Chu, C.-S., 625, 145 Chang, M.-C., 621, 41; 622, 218; 624,11 Chu, Y.P., 622,6;625, 196 Chang, P., 621, 41; 624,11 Chun, E.J., 622, 112 Chang, Y.H., 622, 249; 623,26 Chung, Y.S., 621, 72; 622, 229, 239; 624, 22, 166 Chantler, H.J., 625, 203 Chuvikov, A., 621, 28, 41; 622, 218; 624,11 Chao, Y., 621, 28; 622, 218; 624,11 Ciambrone, P., 626,15 Chapin, D., 622, 265; 626, 35, 45, 55 Cifarelli, L., 622, 249; 623,26 Charles, F., 622, 265; 626, 35, 45, 55 Cindolo, F., 622, 249; 623,26 Chay, J., 628,57 Cinquini, L., 621, 72; 622, 229, 239; 624, 22, 166 Chbihi, A., 627,18 Ciullo, G., 622,14 Chekelian, V., 621,56 Civitarese, O., 626,80 Chemarin, M., 622, 249; 623,26 Claes, D., 622, 265; 626, 35, 45, 55 Chen, A., 621, 28, 41; 622, 218, 249; 623, 26; 624,11 Clare, I., 622, 249; 623,26 Chen, C.-H., 621, 253 Clare, R., 622, 249; 623,26 Chen, C.-M., 625,96 Clark, K., 624, 186 Chen, G., 622, 249; 623,26 Clarke, R.F., 623,17 Chen, G.M., 622, 249; 623,26 Clément, B., 622, 265; 626, 35, 45, 55 Chen, H.F., 622, 6, 249; 623, 26; 625, 196 Clément, C., 622, 265; 626, 35, 45, 55 Chen, H.S., 622, 6, 249; 623, 26; 625, 196 Clément, E., 622,29 Chen, H.X., 622,6;625, 196 Cloët, I.C., 621, 246 Chen, J., 622,6,6;625, 196, 196 Close, F.E., 628, 215 Chen, J.-W., 625, 165 Coadou, Y., 622, 265; 626, 35, 45, 55 Chen, J.C., 622,6;625, 196 Cocco, A.G., 626,24 Chen, K.-F., 621,28 Cognola, G., 624,70 Chen, M.L., 622,6;625, 196 Coignet, G., 622, 249; 623,26 Chen, S., 623, 192 Colangelo, P., 627,77 Chen, T., 622,14 Cole, S., 621, 28, 41; 624,11 Chen, W.T., 621, 28, 41; 622, 218; 624,11 Colino, N., 622, 249; 623,26 Cumulative author index to volumes 621–628 (2005) 292–321 297

Collar, J.I., 621, 233 Davidson, S., 626, 151 Colonna, M., 625,33 Davies, B., 622, 265; 626, 35, 45, 55 Conetti, S., 626,15 Davies, G., 622, 265; 626, 35, 45, 55 Conroy, J.M., 626, 195 Davis, G.A., 622, 265; 626, 35, 45, 55 Contalbrigo, M., 622,14 Davis, P., 628, 275 Contreras, J.G., 621,56 Dayras, R., 627,18 Cooke, M., 622, 265; 626, 35, 45, 55 De, K., 622, 265; 626, 35, 45, 55 Cooper, P.S., 623, 192; 628,18 Deacon, A.N., 622, 151; 625, 375 Cooper, W.E., 622, 265; 626, 35, 45, 55 De Alwis, S.P., 626, 223; 628, 183 Coppage, D., 622, 265; 626, 35, 45, 55 Dean, S., 622, 265; 626, 35, 45, 55 Corcoran, M., 622, 265; 626, 35, 45, 55 De Asmundis, R., 622, 249; 623,26 Correa, F., 628, 157 Debreczeni, J., 622, 249; 623,26 Costantini, S., 622, 249; 623,26 Deconinck, W., 622,14 Cotanch, S.R., 621, 269 Dedes, A., 627, 161 Cothenet, A., 622, 265; 626, 35, 45, 55 Degenhardt, J.D., 622, 265; 626, 35, 45, 55 Coughlan, J.A., 621,56 Déglon, P., 622, 249; 623,26 Cousinou, M.-C., 622, 265; 626, 35, 45, 55 Degrande, N., 624, 173 Cox, B., 622, 265; 626, 35, 45, 55 Degré, A., 622, 249; 623,26 Cox, B.E., 621,56 Dehmelt, K., 622, 249; 623,26 Cozzika, G., 621,56 Deiters, K., 622, 249; 623,26 Crawford, G.I., 626,65 De Jong, M., 626,24 Crépé-Renaudin, S., 622, 265; 626, 35, 45, 55 De Jong, P., 622, 249, 265; 623, 26; 626, 35, 45, 55 Cruz, J., 624, 181 De Jong, S.J., 622, 265; 626, 35, 45, 55 Cuautle, E., 621, 72; 622, 229, 239; 624, 22, 166 De la Cruz, B., 622, 249; 623,26 Cucciarelli, S., 622, 249; 623,26 De La Cruz-Burelo, E., 622, 265; 626, 35, 45, 55 Cui, X.Z., 622,6;625, 196 Del Águila, F., 627, 131; 628,40 Cumalat, J.P., 621, 72; 622, 229, 239; 624, 22, 166 Delbar, T., 626,24 Cutts, D., 622, 265; 626, 35, 45, 55 Delcourt, B., 621,56 Cvach, J., 621,56 De Lellis, G., 626,24 Cvetic,ˇ M., 626, 215 De Leo, R., 622,14 Delﬁno, A., 621, 109 D˛abrowski, M.P., 625, 184 Delgado, A., 627, 155 Dai, H.L., 622,6;625, 196 Déliot, F., 622, 265; 626, 35, 45, 55 Dai, Y.S., 622,6;625, 196 Dell’Agnello, S., 626,15 Dainese, A., 623,17 Della Volpe, D., 622, 249; 623,26 Dainton, J.B., 621,56 Delmeire, E., 622, 249; 623,26 Dalena, B., 622,35 Del Olmo, M.A., 628, 157 Dalpiaz, P.F., 622,14 De Lucia, E., 626,15 Dalseno, J., 621, 28, 41; 622, 218; 624,11 Demarteau, M., 622, 265; 626, 35, 45, 55 D’Ambrosio, N., 626,24 Demey, M., 622,14 Da Motta, H., 622, 265; 626, 35, 45, 55 Demina, R., 622, 265; 626, 35, 45, 55 D’Angelo, P., 621, 72; 622, 229, 239; 624, 22, 166 Demine, P., 622, 265; 626, 35, 45, 55 Danielewicz, P., 627,55 De Miranda, J.M., 621, 72; 622, 229, 239; 624, 22, 166 Danilov, M., 621, 28, 41; 624,11 Demirchyan, R., 621,56 Das, M., 622, 265; 626, 35, 45, 55 De Mori, F., 622,35 Dasgupta, M., 622, 23; 628,49 De Nardo, L., 622,14 Dash, M., 621, 28, 41; 622, 218; 624,11 Denes, P., 622, 249; 623,26 Da Silva, J.C., 624, 316 Deng, Z.Y., 622,6;625, 196 Dau, W.D., 621,56 Denig, A., 626,15 Daum, K., 621,56 Denisov, D., 622, 265; 626, 35, 45, 55 Daum, M., 625,19 Denisov, S.P., 622, 265; 626, 35, 45, 55 Dauwe, L.J., 628,18 DeNotaristefani, F., 622, 249; 623,26 Davenport III, T.F., 621, 72; 622, 229, 239; 624, 22, 166 De Oliveira Martins, C., 622, 265; 626, 35, 45, 55 Davidenko, G.V., 628,18 De Rafael, E., 628,73 Davids, C.N., 625, 203 D’Erasmo, G., 622,35 298 Cumulative author index to volumes 621–628 (2005) 292–321

Deriglazov, A.A., 626, 243 Dolgolenko, A.G., 628,18 Dermíšek, R., 622, 327 Dolgov, A.D., 621,1 De Roeck, A., 621,56 Dombrádi, Zs., 621,81 De Rosa, G., 626,24 Donà, R., 622,35 Dersch, U., 628,18 Dong, H., 622, 265; 626, 35, 45, 55 De S. Pires, C.A., 621, 151; 628,85 Dong, L.Y., 622,6;625, 196 Desai, B.R., 626, 167 Dong, Q.F., 622,6;625, 196 Desai, S., 622, 265; 626, 35, 45, 55 Donini, A., 621, 276 De Salvo, A., 622, 249; 623,26 Doornenbal, P., 622,29 De Sanctis, E., 622,14 Dore, U., 626,24 De Santis, A., 626,15 Doria, A., 622, 249; 623, 26; 626,15 Desch, K., 621,56 Dorn, H., 625, 117 De Simone, P., 626,15 Dorsner, I., 625,88 De Souza Dutra, A., 626, 249 Dos Reis, A.C., 621, 72; 622, 229, 239; 624, 22, 166 Detmold, W., 625, 165 Dotti, G., 627, 174 Dev, A., 624, 135 Doulas, S., 622, 265; 626, 35, 45, 55 Devitsin, E., 622,14 Dova, M.T., 622, 249; 623,26 Devlin, M., 625, 203 Dragic, J., 621, 28; 624,11 De Wolf, E.A., 621,56 Drees, M., 624,60 Dey, P., 628, 141 Dreschler, J., 622,14 De Zorzi, G., 626,15 Dreucci, M., 626,15 D’Hose, N., 624, 173 Drutskoy, A., 621, 28, 41; 622, 218 Di, Y.-Q., 624,39 Du, S.X., 622,6;625, 196 Diaconu, C., 621,56 Du, Z.Z., 622,6;625, 196 Dias, A.G., 621, 151; 628,85 Dubak, A., 621,56 Di Bari, D., 623,17 Duchesneau, D., 622, 249; 623,26 Di Capua, E., 626,24 Duda, M., 622, 249; 623,26 Di Capua, F., 626,24 Dudko, L.V., 622, 265; 626, 35, 45, 55 DiCorato, M., 621, 72; 622, 229, 239; 624, 22, 166 Duﬂot, L., 622, 265; 626, 35, 45, 55 Di Domenico, A., 626,15 Dugad, S.R., 622, 265; 626, 35, 45, 55 Di Donato, C., 626,15 Dumitru, A., 621, 89; 623, 200 Diefenthaler, M., 622,14 Dunbar, D.C., 621, 183 Diehl, H.T., 622, 265; 626, 35, 45, 55 Duperrin, A., 622, 265; 626, 35, 45, 55 Diehl, M., 622,69 Durand, D., 627,18 Diemoz, M., 622, 249; 623,26 Durell, J.L., 625, 203 Dierckxsens, M., 622, 249; 623,26 Düren, M., 622,14 Diesburg, M., 622, 265; 626, 35, 45, 55 Durin, B., 625, 291 Di Falco, S., 626,15 Dutta, B., 627, 145 Di Leva, A., 624, 181 Dutz, H., 624, 173 Di Liberto, S., 623,17 Dyer, J., 622, 265; 626, 35, 45, 55 Di Marco, M., 624, 186 Dyshkant, A., 622, 265; 626, 35, 45, 55 Di Micco, B., 626,15 Dzyubenko, G.B., 628,18 Di Nezza, P., 622,14 Dini, P., 621, 72; 622, 229, 239; 624, 22, 166 E949 Collaboration, 623, 192 Dionisi, C., 622, 249; 623,26 Eads, M., 622, 265; 626, 35, 45, 55 Di Santo, D., 622,35 Echenard, B., 622, 249; 623,26 Di Toro, M., 625,33 Eckerlin, G., 621,56 Dittmar, M., 622, 249; 623,26 Edelstein, R., 628,18 Divià, R., 623,17 Edera, L., 621, 72; 622, 229, 239; 623, 55; 624, 22, 166 Diwan, M.V., 623, 192 Edmunds, D., 622, 265; 626, 35, 45, 55 Djouadi, A., 622, 311; 624,60 Edwards, T., 622, 265; 626, 35, 45, 55 Doane, P., 624, 186 Efremenko, V., 621,56 DØ Collaboration, 622, 265; 626, 35, 45, 55 Egli, S., 621,56 Dodonov, V., 621,56 Ehrenfried, M., 622,14 Doidge, M., 622, 265; 626, 35, 45, 55 Eichler, R., 621,56 Cumulative author index to volumes 621–628 (2005) 292–321 299

Eidelman, S., 621, 28, 41; 622, 218; 624,11 Falkewicz, A., 621,56 Eisele, F., 621,56 Fang, F., 622, 218 Elalaoui-Moulay, A., 622,14 Fang, J., 622,6;625, 196 Elbakian, G., 622,14 Fang, S.S., 622,6;625, 196 Elekes, Z., 621,81 Fantoni, A., 622,14 El Hage, A., 622, 249; 623,26 Farakos, K., 621, 224 Elia, D., 622, 35; 623,17 Farchioni, F., 624, 324 Eline, A., 622, 249; 623,26 Farzan, Y., 621,22 Elizalde, E., 624,70 Faso, D., 622,35 Ellerbrock, M., 621,56 Fast, J., 622, 265; 626, 35, 45, 55 Ellinghaus, F., 622,14 Fatakia, S.N., 622, 265; 626, 35, 45, 55 Ellis, J., 624,47 Faulkner, P.J.W., 621,56 Ellison, J., 622, 265; 626, 35, 45, 55 Favara, A., 622, 249; 623,26 Ellwanger, U., 623,93 Favart, D., 626,24 El Mamouni, H., 622, 249; 623,26 Favart, L., 621,56 Elmsheuser, J., 622, 265; 626, 35, 45, 55 Fay, J., 622, 249; 623,26 Elschenbroich, U., 622,14 Fedin, O., 622, 249; 623,26 Elsen, E., 621,56 Fedotov, A., 621,56 Elvira, V.D., 622, 265; 626, 35, 45, 55 Feighery, W., 624, 186 Emediato, L., 628,18 Felawka, L., 622,14 Emmanuel-Costa, D., 623, 111 Felcini, M., 622, 249; 623,26 Enari, Y., 621, 41; 622, 218 Felici, G., 626,15 Endler, A.M.F., 628,18 Feliciello, A., 622,35 Engelfried, J., 628,18 Feligioni, L., 622, 265; 626, 35, 45, 55 Engh, D., 621, 72; 622, 229, 239; 624, 22, 166 Felizardo da Costa, M., 621, 233 Engler, A., 622, 249; 623,26 Felst, R., 621,56 Eno, S., 622, 265; 626, 35, 45, 55 Feoﬁlov, G.A., 623,17 Epifanov, D., 622, 218 Ferapontov, A.V., 622, 265; 626, 35, 45, 55 Eppling, F.J., 622, 249; 623,26 Ferbel, T., 622, 265; 626, 35, 45, 55 Erba, S., 621, 72; 622, 229, 239; 624, 22, 166 Ferencei, J., 621,56 Erdem, R., 621,11 Ferguson, T., 622, 249; 623,26 Erdmann, W., 621,56 Fernandes, A.C., 621, 233 Ermolaev, B.I., 622,93 Fernández, L.A., 628, 281 Ermolov, P., 622, 265; 626, 35, 45, 55 Fernández-Martínez, E., 621, 276 Eroshin, O.V., 622, 265; 626, 35, 45, 55 Ferrandes, R., 627,77 Eschrich, I., 628,18 Ferrari, A., 626,15 Escobar, C.O., 628,18 Ferrer, M.L., 626,15 Espriu, D., 628, 197 Fesefeldt, H., 622, 249; 623,26 Essenov, S., 621,56 Fiandrini, E., 622, 249; 623,26 Estrada, J., 622, 265; 626, 35, 45, 55 Fiedler, F., 622, 265; 626, 35, 45, 55 Eudes, Ph., 625,26 Field, J.H., 622, 249; 623,26 Evans, D., 623,17 Fierlinger, P., 625,19 Evans, H., 622, 265; 626, 35, 45, 55 Fileviez Pérez, P., 625,88 Evans, N., 622, 165 Filimonov, I.S., 628,18 Evdokimov, A., 622, 265; 626, 35, 45, 55 Filippi, A., 622,35 Evdokimov, A.V., 628,18 Filippini, V., 622,35 Evdokimov, V.N., 622, 265; 626, 35, 45, 55 Filthaut, F., 622, 249, 265; 623, 26; 626, 35, 45, 55 Extermann, P., 622, 249; 623,26 Finelli, F., 625,1 Fini, R., 622,35 Fabbri, F.L., 621, 72; 622, 35, 229, 239; 624, 22, 166 Fini, R.A., 623,17 Fabbri, R., 622,14 Finke, L., 621,56 Fadin, V.S., 621, 320 Finocchiaro, G., 626,15 Faessler, A., 622, 277 FINUDA Collaboration, 622,35 Falagan, M.A., 622, 249; 623,26 Fiore, M.E., 622,35 Falciano, S., 622, 249; 623,26 Fiore, R., 621, 320 300 Cumulative author index to volumes 621–628 (2005) 292–321

Fiorillo, G., 626,24 Gabyshev, N., 621, 28, 41; 622, 218; 624,11 Fisher, P.H., 622, 249; 623,26 Gadfort, T., 622, 265; 626, 35, 45, 55 Fisher, W., 622, 249, 265; 623, 26; 626, 35, 45, 55 Gaete, P., 625, 365 Fisk, H.E., 622, 265; 626, 35, 45, 55 Gaines, I., 621, 72; 622, 229, 239; 624, 22, 166 Fisk, I., 622, 249; 623,26 Galaktionov, Yu., 622, 249; 623,26 Fissum, K.G., 626,65 Galea, C.F., 622, 265; 626, 35, 45, 55 Fleck, I., 622, 265; 626, 35, 45, 55 Galichet, E., 627,18 Fleischer, M., 621,56 Gallas, E., 622, 265; 626, 35, 45, 55 Fleischmann, P., 621,56 Galyaev, E., 622, 265; 626, 35, 45, 55 Fleming, Y.H., 621,56 Gangopadhyay, S., 625, 302 Flucke, G., 621,56 Ganguli, S.N., 622, 249; 623,26 FOCUS Collaboration, 621, 72; 622, 229, 239; 624, 22, 166 Ganoti, P., 623,17 Fogli, G.L., 623,80 Gao, C.S., 622,6;625, 196 Fomenko, A., 621,56 Gao, Y.N., 622,6;625, 196 Fong, D., 622,45 Gapienko, G., 622,14 Fonseca, M., 624, 181 Gapienko, V., 622,14 Forconi, G., 622, 249; 623,26 Garbincius, P.H., 621, 72; 622, 229, 239; 624, 22, 166 Ford, C., 626, 139 Garcia, C., 622, 265; 626, 35, 45, 55 Foresti, I., 621,56 Garcia, F.G., 628,18 Formánek, J., 621,56 Garcia-Abia, P., 622, 249; 623,26 Forti, C., 626,15 Garcia-Bellido, A., 622, 265; 626, 35, 45, 55 Fortner, M., 622, 265; 626, 35, 45, 55 Gardner, J., 622, 265; 626, 35, 45, 55 Fossan, D.B., 625, 203 Gardner, R., 621, 72; 622, 229, 239; 624, 22, 166 Fox, H., 622, 265; 626, 35, 45, 55 Garibaldi, F., 622,14 Francia, D., 624,93 Garmash, A., 624,11 Frank, J.S., 623, 192 Garren, L.A., 621, 72; 622, 229, 239; 624, 22, 166 Frank, N.H., 627,32 Garrow, K., 622,14 Franke, G., 621,56 Garutti, E., 621,56 Frankland, J.D., 627,18 Gaspero, M., 628,18 Franzini, P., 626,15 Gataullin, M., 622, 249; 623,26 Fratina, S., 621, 28, 41; 622, 218; 624,11 Gatti, C., 626,15 Frederico, T., 621, 109 Gatto, R., 627,89 Freeman, S.J., 622, 151; 625, 203, 375 Gattringer, C., 621, 195 Frekers, D., 626,24 Gauzzi, P., 626,15 Freudenreich, K., 622, 249; 623,26 Gavrilov, G., 622,14 Freyhult, L., 622, 343 Gavrilov, V., 622, 265; 626, 35, 45, 55 Friot, S., 628,73 Gay, A., 622, 265; 626, 35, 45, 55 Frising, G., 621,56 Gay, P., 622, 265; 626, 35, 45, 55 Frisson, T., 621,56 Gayler, J., 621,56 Frullani, S., 622,14 GDH Collaboration, 624, 173 Fry, J.L., 626, 256 Ge, X.-H., 623, 141 Fu, C.D., 622,6;625, 196 Gehrmann, T., 622, 295 Fu, H.Y., 622,6;625, 196 Geissel, H., 622,29 Fu, S., 622, 265; 626, 35, 45, 55 Gelé, D., 622, 265; 626, 35, 45, 55 Fuess, S., 622, 265; 626, 35, 45, 55 Gelhaus, R., 622, 265; 626, 35, 45, 55 Fujioka, H., 622,35 Gelmini, G., 621,22 Fujiwara, T., 623, 192 Geltenbort, P., 625,19 Fulling, S.A., 624, 281 Genest, M.-H., 624, 186 Fülöp, Z., 624, 181 Geng, C.-Q., 621, 253 Fülöp, Zs., 621,81 Genser, K., 622, 265; 626, 35, 45, 55 Funel, A., 622,14 Gentile, S., 622, 249; 623,26 Furetta, C., 622, 249; 623,26 Gepner, D., 622, 136 Furman, A., 622, 207 Gérard, J.-M., 628, 211 Gerber, C.E., 622, 265; 626, 35, 45, 55 Gabathuler, E., 621,56 Gerhards, R., 621,56 Cumulative author index to volumes 621–628 (2005) 292–321 301

Gerl, J., 622,29 Gong, W.X., 622,6;625, 196 Gerlich, C., 621,56 Gong, Y., 624, 141 Gershon, T., 621, 28, 41; 622, 218; 624,1,11 Gong, Z.F., 622, 249; 623,26 Gershtein, Y., 622, 265; 626, 35, 45, 55 González Felipe, R., 623, 111 Gharibyan, V., 622,14 Gorbahn, M., 626, 151 Ghazaryan, S., 621,56 Gorbounov, S., 621,56 Ghidini, B., 623,17 Gorbunov, P., 626,24 Ghiotti, M., 628, 176 Görgen, A., 622,29 Ghosh, D.K., 628, 131 Gorini, E., 626,15 Ghosh, S., 623, 251 Gorišek, A., 621, 28, 41; 622, 218; 624,11 Ghosh, T.K., 627,26 Gornea, R., 624, 186 Giacosa, F., 622, 277 Górska, M., 622,29 Giagu, S., 622, 249; 623,26 Gottschalk, E., 621, 72; 622, 229, 239; 624, 22, 166 Gianini, G., 621, 72; 622, 229, 239; 624, 22, 166 Gouffon, P., 628,18 Gianotti, P., 622,35 Gounder, K., 622, 265; 626, 35, 45, 55 Gibelin, J., 621,81 Goussiou, A., 622, 265; 626, 35, 45, 55 Gillberg, D., 622, 265; 626, 35, 45, 55 Goyon, C., 621,56 Giller, I., 628,18 Grab, C., 621,56 Giller, S., 622, 192 Grabmayr, P., 624, 173 Ginter, T.N., 627,32 Grady, M., 626, 161 Ginther, G., 622, 265; 626, 35, 45, 55 Grannis, P.D., 622, 265; 626, 35, 45, 55 Ginzburgskaya, S., 621,56 Grawe, H., 622,29 Giovannella, S., 626,15 Graziani, E., 626,15 Giovannini, M., 622, 349 Grebeniouk, O., 622,14 Girard, T.A., 621, 233 Gr¸ebosz, J., 622,29 Giri, A.K., 621, 253 Greco, M., 622,93 Giribet, G.E., 628, 148 Greder, S., 622, 265; 626, 35, 45, 55 Gitman, D.M., 621, 295 Greenlee, H., 622, 265; 626, 35, 45, 55 Giudice, G.F., 627, 155 Greenshaw, T., 621,56 Giuliani, F., 621, 233 Greenwood, Z.D., 622, 265; 626, 35, 45, 55 Glazov, A., 621,56 Grégoire, G., 626,24 Gleiser, R.J., 627, 174 Gregoire, T., 624, 260 Glushkov, I., 621,56 Gregor, I.M., 622,14 Go, A., 621,41 Gregores, E.M., 622, 265; 626, 35, 45, 55 Göbel, C., 621, 72; 622, 229, 239; 624, 22, 166 Gregori, M., 621,56 Göckeler, M., 627, 113 Grella, G., 623, 17; 626,24 Godbole, R.M., 628, 131 Grenier, G., 622, 249; 623,26 Godłowski, W., 623,10 Greynat, D., 628,73 Goerlich, L., 621,56 Grigoriev, A., 622, 199 Goertz, S., 624, 173 Grimm, O., 622, 249; 623,26 Goettlich, M., 621,56 Grindhammer, G., 621,56 Gogitidze, N., 621,56 Grion, N., 622,35 Gogoladze, I., 622, 320 Gris, Ph., 622, 265; 626, 35, 45, 55 Gokhroo, G., 621, 28, 41; 622, 218; 624,11 Grivaz, J.-F., 622, 265; 626, 35, 45, 55 Golda, K.S., 627,26 Groer, L., 622, 265; 626, 35, 45, 55 Goldberg, H., 621,18 Gronau, M., 627,82 Goldberg, J., 626,24 Gruenewald, M.W., 622, 249; 623,26 Golling, T., 622, 265; 626, 35, 45, 55 Grünendahl, S., 622, 265; 626, 35, 45, 55 Gollub, N., 622, 265; 626, 35, 45, 55 Grünewald, M.W., 622, 265; 626, 35, 45, 55 Golob, B., 621, 28; 622, 218; 624,11 Grzywacz, R., 622,45 Golovtsov, V.L., 628,18 Gu, S.D., 622,6;625, 196 Golowich, E., 625,53 Guénette, R., 624, 186 Gómez, B., 622, 265; 626, 35, 45, 55 Guida, M., 622, 249; 623,26 Gonera, C., 622, 192 Guimarães, M.S., 625, 351 Gong, M.Y., 622,6;625, 196 Guinet, D., 627,18 302 Cumulative author index to volumes 621–628 (2005) 292–321

Güler, M., 626,24 Hastings, N.C., 621, 28; 624,11 Gülmez, E., 628,18 Hatsuda, T., 623, 208 Guo, Y.N., 622,6;625, 196 Hatzifotiadou, D., 622, 249; 623,26 Guo, Y.Q., 622,6;625, 196 Hauptman, J.M., 622, 265; 626, 35, 45, 55 Gupta, V.K., 622, 249; 623,26 Hauser, R., 622, 265; 626, 35, 45, 55 Gurtu, A., 622, 249; 623,26 Hayasaka, K., 621, 28, 41; 622, 218; 624,11 Gurzhiev, S.N., 622, 265; 626, 35, 45, 55 Hayashii, H., 621, 28, 41; 622, 218; 624,11 Gutay, L.J., 622, 249; 623,26 Hays, J., 622, 265; 626, 35, 45, 55 Gutierrez, G., 622, 265; 626, 35, 45, 55 Hazra, A.G., 625, 302 Gutierrez, P., 622, 265; 626, 35, 45, 55 Hazumi, M., 621, 28, 41; 622, 218; 624,11 Gutsche, Th., 622, 277 He, K.L., 622,6;625, 196 Gwilliam, C., 621,56 He, M., 622,6;625, 196 Gyürky, G., 624, 181 He, X., 622,6;625, 196 He, Z.J., 628,25 H1 Collaboration, 621,56 Hebbeker, T., 622, 249, 265; 623, 26; 626, 35, 45, 55 Haas, A., 622, 265; 626, 35, 45, 55 Hebecker, A., 624, 270 Haas, D., 622, 249; 623,26 Hedin, D., 622, 265; 626, 35, 45, 55 Haba, J., 621, 41; 622, 218; 624,11 Heenen, P.-H., 625, 203 Haddad, F., 625,26 Heid, E., 624, 173 Hadjidakis, C., 622,14 Heinmiller, J.M., 622, 265; 626, 35, 45, 55 Hadley, N.J., 622, 265; 626, 35, 45, 55 Heinson, A.P., 622, 265; 626, 35, 45, 55 Haﬁdi, K., 622,14 Heintz, U., 622, 265; 626, 35, 45, 55 Hägler, Ph., 627, 113 Heinzelmann, G., 621,56 Hagopian, S., 622, 265; 626, 35, 45, 55 Helbing, K., 624, 173 Haidt, D., 621,56 Hellström, M., 622,29 Hajduk, L., 621,56 Helstrup, H., 623,17 Hall, I., 622, 265; 626, 35, 45, 55 Hemmert, T.R., 622, 141 Hall, L.J., 625, 189 Henderson, R.C.W., 621,56 Hall, R.E., 622, 265; 626, 35, 45, 55 Heng, Y.K., 622,6;625, 196 Haller, J., 621,56 Henneck, R., 625,19 Halzen, F., 621,18 Henningson, M., 627, 203 Hamanaka, M., 625, 324 Henschel, H., 621,56 Hamilton, J.H., 622,45 Hensel, C., 622, 265; 626, 35, 45, 55 Hammond, G., 622,29 Henshaw, O., 621,56 Hammond, N.J., 622, 151; 625, 375 HERMES Collaboration, 622,14 Han, C., 622, 265; 626, 35, 45, 55 Hernandez, H., 621, 72; 622, 229, 239; 624, 22, 166 Han, L., 622, 265; 626, 35, 45, 55 Herrera, G., 621,56 Hanagaki, K., 622, 265; 626, 35, 45, 55 Hervé, A., 622, 249; 623,26 Handler, T., 621, 72; 622, 229, 239; 624, 22, 166 Hesketh, G., 622, 265; 626, 35, 45, 55 Hansen, K., 624, 173; 626,65 Hesselink, W.H.A., 622,14 Hansson, M., 621,56 Hetland, K.F., 623,17 Hao, G., 621, 139 Heule, S., 625,19 Hara, K., 624,11 Heydari-Fard, M., 626, 230 Hara, T., 621, 28; 626,24 Hildebrandt, M., 621,56 Harder, K., 622, 265; 626, 35, 45, 55 Hildreth, M.D., 622, 265; 626, 35, 45, 55 Harel, A., 622, 265; 626, 35, 45, 55 Hillenbrand, A., 622,14 Harikumar, E., 625, 156 Hiller, J.R., 624, 105 Harmsen, J., 624, 173 Hiller, K.H., 621,56 Harrington, R., 622, 265; 626, 35, 45, 55 Hinde, D.J., 622,23 Harrison, P.F., 628,93 Hinz, L., 621, 28; 622, 218; 624,11 Hartig, M., 622,14 Hirosky, R., 622, 265; 626, 35, 45, 55 Harty, P.D., 626,65 Hirschfelder, J., 622, 249; 623,26 Hasch, D., 622,14 Hisano, J., 624, 239 Hasegawa, S., 624, 173 Hitt, G.W., 627,32 Hasegawa, T., 624, 173 Hlavatý, L., 625, 285 Cumulative author index to volumes 621–628 (2005) 292–321 303

Hobbs, J.D., 622, 265; 626, 35, 45, 55 Iijima, T., 621, 28, 41; 622, 218; 624,11 Hoek, M., 622,14 Illingworth, R., 622, 265; 626, 35, 45, 55 Hoeneisen, B., 622, 265; 626, 35, 45, 55 Imoto, A., 621, 28, 41; 622, 218; 624,11 Hofer, H., 622, 249; 623,26 Inami, K., 621, 28, 41; 622, 218; 624,11 Hoffmann, D., 621,56 Incagli, M., 626,15 Hohlfeld, M., 622, 265; 626, 35, 45, 55 INDRA Collaboration, 627,18 Hohlmann, M., 622, 249; 623,26 Inzani, P., 621, 72; 622, 229, 239; 624, 22, 166 Hokuue, T., 621, 28, 41; 622, 218; 624,11 Iori, M., 628,18 Holler, Y., 622,14 Ippolito, N., 627,89 Holme, A.K., 623,17 Ireland, D.G., 626,65 Holvoet, H., 624, 173 Isaksson, L., 626,65 Holzner, G., 622, 249; 623,26 Ishida, M., 622, 286; 627, 105 Hommez, B., 622,14 Ishikawa, A., 621, 28, 41; 622, 218; 624,11 Hong, S.-T., 623, 135; 628, 165 Ishino, H., 624,11 Hong, S.J., 622, 265; 626, 35, 45, 55 Ismail, M., 621,56 Honma, M., 622,29 Ita, H., 621, 183 Hooper, R., 622, 265; 626, 35, 45, 55 Ito, A.S., 622, 265; 626, 35, 45, 55 Horikawa, N., 624, 173 Itoh, R., 621, 28, 41; 622, 218; 624,11 Horisberger, R., 621,56 Ivanilov, A., 622,14 Horsley, R., 627, 113; 628,66 Ivanov, E., 624, 304 Horváth, Á., 621,81 Ivanov, I.P., 622,55 Hosack, M., 621, 72; 622, 229, 239; 624, 22, 166 Ivanov, R.I., 623, 235 Hoshi, Y., 621, 28, 41; 622, 218; 624,11 Iwasa, N., 621,81 Hoshino, K., 626,24 Iwasaki, H., 621,81 Hou, S., 621, 28, 41; 622, 218; 624,11 Iwasaki, M., 621, 28, 41; 622, 218; 624,11 Hou, S.R., 622, 249; 623,26 Iwasaki, Y., 621, 28; 622, 218; 624,11 Hou, W.-S., 621, 28, 41; 622, 218; 624,11 Iwata, T., 624, 173 Houben, P., 622, 265; 626, 35, 45, 55 Izotov, A., 622,14 Hovhannisyan, A., 621,56 Hristova, I., 622,14 Jabeen, S., 622, 265; 626, 35, 45, 55 Hristova, I.R., 626,24 Jacholkowski, A., 623,17 Hsieh, T.-H., 624,31 Jackson, H.E., 622,14 Hu, H.M., 622,6;625, 196 Jacob, P., 627, 224 Hu, J., 623, 192 Jacquet, M., 621,56 Hu, T., 622,6;625, 196 Jaffe, D.E., 623, 192 Hu, Y., 622, 265; 626, 35, 45, 55 Jaffré, M., 622, 265; 626, 35, 45, 55 Huang, J., 622, 265; 626, 35, 45, 55 Jahan, A., 623, 179 Huang, X.P., 622,6;625, 196 Jahn, O., 624, 173 Huang, X.T., 622,6;625, 196 Jain, D., 624, 135 Hübel, H., 622,29 Jain, P., 621, 213 Huber, T., 622, 295 Jain, S., 622, 265; 626, 35, 45, 55 Hugonie, C., 623,93 Jain, V., 622, 265; 626, 35, 45, 55 Hwang, J.K., 622,45 Jakobs, K., 622, 265; 626, 35, 45, 55 Hynek, V., 622, 265; 626, 35, 45, 55 Janauschek, L., 621,56 Jansen, K., 624, 324, 334 Ianni, A., 627,38 Janssen, X., 621,56 Iarygin, G., 622,14 Janssens, R.V.F., 622, 151; 625, 203, 375 Iashvili, I., 622, 265; 626, 35, 45, 55 Jarlskog, C., 625,63 Ibadov, R., 627, 180 Jemanov, V., 621,56 Ibbotson, M., 621,56 Jenkins, A., 622, 265; 626, 35, 45, 55 Ichikawa, Y., 621,81 Jennewein, P., 624, 173 Ichinose, S., 625, 106 Jesik, R., 622, 265; 626, 35, 45, 55 Ideguchi, E., 621,81 Jesus, A.P., 624, 181 Idilbi, A., 625, 253 Jgoun, A., 622,14 Igi, K., 622, 286 Ji, X., 625, 253 304 Cumulative author index to volumes 621–628 (2005) 292–321

Ji, X.B., 622,6;625, 196 Karny, M., 622,45 Jiang, C.H., 622,6;625, 196 Kasper, J., 622, 265; 626, 35, 45, 55 Jiang, J., 624, 233 Kasper, P.H., 621, 72; 622, 229, 239; 624, 22, 166 Jiang, X.S., 622,6;625, 196 Kasprzak, M., 625,19 Jin, B.N., 622, 249; 623,26 Katayama, N., 621, 28; 622, 218; 624,11 Jin, D.P., 622,6;625, 196 Katzy, J., 621,56 Jin, S., 622,6;625, 196 Kau, D., 622, 265; 626, 35, 45, 55 Jin, Y., 622,6,6;625, 196, 196 Kaur, M., 622, 249; 623,26 Jindal, P., 622, 249; 623,26 Kaur, R., 622, 265; 626, 35, 45, 55 Johansson, E.P.G., 627, 203 Kavatsyuk, M., 622,29 Johns, K., 622, 265; 626, 35, 45, 55 Kavatsyuk, O., 622,29 Johns, W.E., 621, 72; 622, 229, 239; 624, 22, 166 Kawada, J., 626,24 Johnson, D.P., 621,56 Kawai, H., 621, 28, 41; 622, 218; 624,11 Johnson, M., 622, 265; 626, 35, 45, 55 Kawai, S., 621,81 Jonckheere, A., 622, 265; 626, 35, 45, 55 Kawamura, T., 626,24 Jones, G.T., 623,17 Kawasaki, M., 624, 162; 625,7 Jones, H.F., 625, 333 Kawasaki, T., 621, 28, 41; 622, 218; 624,11 Jones, L.W., 622, 249; 623,26 Kaya, M., 628,18 Jönsson, L., 621,56 Kayis-Topaksu, A., 626,24 Jonsson, P., 622, 265; 626, 35, 45, 55 Kehagias, A., 628, 262 Jorjadze, G., 625, 117 Kehoe, R., 622, 265; 626, 35, 45, 55 Josa-Mutuberría, I., 622, 249; 623,26 Keller, N., 621,56 Jovanovic, P., 623,17 Kelly, R., 627,71 Jun, S.Y., 628,18 Kelsall, N.S., 625, 203 Jung, H., 621,56 Kent, N., 624,11 Jusko, A., 623,17 Kenyon, I.R., 621,56 Juste, A., 622, 265; 626, 35, 45, 55 Keri, T., 622,14 Kermiche, S., 622, 265; 626, 35, 45, 55 Kabe, S., 623, 192 Kesisoglou, S., 622, 265; 626, 35, 45, 55 Kado, M.M., 626, 35, 45, 55 Kettell, S.H., 623, 192 Käfer, D., 622, 265; 626, 35, 45, 55 Kettner, K.U., 624, 181 Kageya, T., 624, 173 Khabibullin, M.M., 623, 192 Kahn, S., 622, 265; 626, 35, 45, 55 Khan, H.R., 621, 28, 41; 622, 218; 624,11 Kaiser, R., 622,14 Khanna, F.C., 624, 316 Kajfasz, E., 622, 265; 626, 35, 45, 55 Khanov, A., 622, 265; 626, 35, 45, 55 Kakizaki, M., 624, 239 Kharchilava, A., 622, 265; 626, 35, 45, 55 Kalinin, A.M., 622, 265; 626, 35, 45, 55 Kharzeev, D., 626, 147 Kalinin, S., 626,24 Kharzheev, Y.M., 622, 265; 626, 35, 45, 55 Kalk, J., 622, 265; 626, 35, 45, 55 Khotjantsev, A.N., 623, 192 Kalloniatis, A.C., 628, 176 Khovansky, V., 626,24 Kamermans, R., 623,17 Kichimi, H., 621, 28, 41; 622, 218; 624,11 Kaminski,´ R., 622, 207 Kiel, B., 624, 173 Kanagalingam, S., 624, 186 Kienzle-Focacci, M.N., 622, 249; 623,26 Kanemura, S., 626, 111 Kiesling, C., 621,56 Kang, J.H., 621, 28, 41; 622, 218; 624,11 Kihara, H., 621, 288 Kang, J.S., 621, 28, 41, 72; 622, 218, 229, 239; 624, 11, 22, 166 Kilmer, J., 628,18 Kang, K., 624, 125 Kim, C., 628,57 Kangling, H., 628,18 Kim, C.S., 621, 259; 623, 218 Kanno, S., 621,81 Kim, D.Y., 621, 72; 622, 229, 239; 624, 22, 166 Kanungo, R., 621,81 Kim, H., 622, 265; 626, 35, 45, 55; 628,11 Kapichine, M., 621,56 Kim, H.-Ch., 628,33 Kapusta, P., 621, 41; 622, 218; 624,11 Kim, H.J., 621, 28, 41; 622, 218; 624,11 Kar, S., 623, 244 Kim, H.O., 621,28 Karlsson, M., 621, 56; 626,65 Kim, J.K., 622, 249; 623,26 Karmanov, D., 622, 265; 626, 35, 45, 55 Kim, S.K., 621,28 Cumulative author index to volumes 621–628 (2005) 292–321 305

Kim, S.M., 621, 28, 41; 622, 218; 624,11 Korten, W., 622,29 Kim, T.J., 622, 265; 626, 35, 45, 55 Köse, U., 626,24 Kim, V.T., 628,18 Koshelev, A.S., 628,1 Kinney, E., 622,14 Kosinski,´ P., 622, 192 Kinoshita, K., 621, 28; 624,11 Kossert, K., 624, 173 Kinson, J.B., 623,17 Kostelecký, V.A., 628, 106 Kirch, K., 625,19 Kostka, P., 621,56 Kirkby, J., 622, 249; 623,26 Kotcher, J., 622, 265; 626, 35, 45, 55 Kisselev, A., 622,14 Kothari, B., 622, 265; 626, 35, 45, 55 Kitabayashi, T., 621, 133 Koubarovsky, A., 622, 265; 626, 35, 45, 55 Kitching, P., 623, 192 Koutouev, R., 621,56 Kittel, W., 622, 249; 623,26 Koutsenko, V., 622, 249; 623,26 Kleihaus, B., 623, 171; 627, 180 Kozelov, A.V., 622, 265; 626, 35, 45, 55 Klein, F., 624, 173 Kozhevnikov, A.P., 623, 192; 628,18 Klein, M., 621,56 Kozlov, V., 622,14 Kleinwort, C., 621,56 Kozminski, J., 622, 265; 626, 35, 45, 55 Klima, B., 622, 265; 626, 35, 45, 55 Kräber, M., 622, 249; 623,26 Klimentov, A., 622, 249; 623,26 Kraemer, R.W., 622, 249; 623,26 Klimkovich, T., 621,56 Králik, I., 623,17 KLOE Collaboration, 626,15 Kraml, S., 626, 175 Kluge, T., 621,56 Kraniotis, G.V., 625,96 Kluge, W., 626,15 Krasnoperov, A., 622,35 Klute, M., 626, 35, 45, 55 Krastev, K., 621,56 Krauss, B., 622,14 Kmiecik, M., 622,29 Krauss, C., 624, 186 Kneur, J.-L., 624,60 Kravcáková,ˇ A., 623,17 Knies, G., 621,56 Kretzschmar, J., 621,56 Knudson, K., 623,17 Kreymer, A.E., 621, 72; 622, 229, 239; 624, 22, 166 Knutsson, A., 621,56 Krimmer, J., 624, 173 Ko, B.R., 621, 72; 622, 229, 239; 624, 22, 166 Krivokhijine, V.G., 622,14 Ko, C.M., 624, 210 Krivshich, A.G., 628,18 Kobayashi, M., 623, 192 Krokovny, P., 621, 28, 41; 622, 218; 624,1,11 Kobayashi, T., 622,14 Krolas, W., 622,45 Kochenda, L.M., 628,18 Kropivnitskaya, A., 621,56 Kodama, K., 626,24 Krüger, A., 622, 249; 623,26 Kohli, J.M., 622, 265; 626, 35, 45, 55 Krüger, H., 628,18 Kohri, K., 625,7 Krüger, K., 621,56 Koike, T., 625, 203 Krusch, S., 626, 120 Kojouharov, I., 622,29 Kryemadhi, A., 621, 72; 622, 229, 239, 265; 624, 22, 166; 626, 45, Kolev, D., 626,24 55, 35 Koley, R., 623, 244 Krzywdzinski, S., 622, 265; 626, 35, 45, 55 Komatsu, M., 626,24 Kubantsev, M.A., 628,18 Komatsubara, T.K., 623, 192 Kubarovsky, V.P., 628,18 Konaka, A., 623, 192 Kubo, J., 622, 303 Kondo, Y., 621,81 Kückens, J., 621,56 Kondratiev, R., 624, 173 Kudenko, Yu.G., 623, 192 Kondratiev, V., 623,17 Kuijer, P., 623,17 König, A.C., 622, 249; 623,26 Kulasiri, R., 621, 28, 41 Konorov, I., 628,18 Kulik, Y., 622, 265; 626, 35, 45, 55 Konrath, J.-P., 622, 265; 626, 35, 45, 55 Kulikov, V., 626,15 Kopal, M., 622, 249, 265; 623, 26; 626, 35, 45, 55 Kulyavtsev, A.I., 628,18 Kopytin, M., 622,14 Kumar, A., 622, 265; 626, 35, 45, 55 Korablev, V.M., 622, 265; 626, 35, 45, 55 Kumar, S., 621, 28, 41; 622, 218; 624,11 Korbel, V., 621,56 Kundu, A., 622, 102; 627, 137; 628, 141 Korotkov, V., 622,14 Kunduri, H.K., 628, 275 Korpar, S., 621, 28; 622, 218 Kunin, A., 622, 249; 623,26 306 Cumulative author index to volumes 621–628 (2005) 292–321

Kunori, S., 622, 265; 626, 35, 45, 55 Lecoq, P., 622, 249; 623,26 Kunz, J., 623, 171; 627, 180 Le Coultre, P., 622, 249; 623,26 Kunze, K.E., 623,1 Leder, G., 621, 28, 41; 622, 218; 624,11 Kuo, C.C., 621, 28, 41; 622, 218; 624,11 Lee, F.X., 627,71 Kupco, A., 622, 265; 626, 35, 45, 55 Lee, H.W., 628,11 Kurca,ˇ T., 622, 265; 626, 35, 45, 55 Lee, J., 628, 165 Kuropatkin, N.P., 628,18 Lee, J.-Y., 624,31 Kurshetsov, V.F., 628,18 Lee, J.S., 628, 250 Kurz, N., 622,29 Lee, K.B., 621, 72; 622, 229, 239; 624, 22, 166 Kusenko, A., 621,22 Lee, S.E., 622, 218 Kushnirenko, A., 623, 192; 628,18 Lee, S.J., 621, 239 Kutschke, R., 621, 72; 622, 229, 239; 624, 22, 166 Lee, T.H., 628, 165 Kuzmin, A., 621, 41; 622, 218; 624,11 Lee, W.M., 622, 265; 626, 35, 45, 55 Kvita, J., 622, 265; 626, 35, 45, 55 Lee-Franzini, J., 626,15 Kwak, J.W., 621, 72; 622, 229, 239; 624, 22, 166 Leﬂat, A., 622, 265; 626, 35, 45, 55 Kwan, E., 627,32 Le Goff, J.M., 622, 249; 623,26 Kwan, S., 628,18 Lehner, F., 622, 265; 626, 35, 45, 55 Kwon, Y.-J., 621, 28, 41; 622, 218; 624,11 Leibovich, A.K., 628,57 Leikin, E.M., 628,18 L3 Collaboration, 622, 249; 623,26 Leinweber, D.B., 628, 125 Lacava, F., 626,15 Leißner, B., 621,56 Lach, J., 628,18 Leiste, R., 622, 249; 623,26 Ladron de Guevara, P., 622, 249; 623,26 Lemos, J.P.S., 621, 219 LaFosse, D.R., 625, 203 Lendermann, V., 621,56 Lagamba, L., 622,14 Le Neindre, N., 623, 43; 627,18 Lager, S., 622, 265; 626, 35, 45, 55 Lenis, D., 621, 102 Lahrichi, N., 622, 265; 626, 35, 45, 55 Lenisa, P., 622,14 Lai, Y.F., 622,6;625, 196 Lenti, V., 622, 35; 623,17 Laktineh, I., 622, 249; 623,26 Leone, D., 626,15 Lamberto, A., 628,18 Leonidopoulos, C., 622, 265; 626, 35, 45, 55 Landi, G., 622, 249; 623,26 Leroy, C., 624, 186 Landon, M.P.J., 621,56 Lesiak, T., 621, 28, 41; 622, 218; 624,11 Landsberg, G., 622, 265; 626, 35, 45, 55 Lesniak,´ L., 622, 207 Landsberg, L.G., 623, 192; 628,18 Lessard, L., 624, 186 Lanfranchi, G., 626,15 Leukel, R., 624, 173 Lang, M., 624, 173 Leveque, J., 622, 265; 626, 35, 45, 55 Langacker, P., 624, 233 Leveraro, F., 621, 72; 622, 229, 239; 624, 22, 166 Lange, J.S., 621,28 Levine, I., 624, 186 Lange, W., 621,56 Levonian, S., 621,56 Lannoy, B., 624, 173 Levtchenko, M., 622, 249; 623,26 Lanzalone, G., 627,18 Levtchenko, P., 622, 249; 623,26 Lapikás, L., 622,14 Lewis, B., 623, 192 Larin, I., 628,18 Lewis, P., 622, 265; 626, 35, 45, 55 Laštovicka,ˇ T., 621,56 Li, C., 622, 249; 623,26 Lauritsen, T., 622, 151; 625, 375 Li, F., 622,6;625, 196 Lautesse, P., 627,18 Li, G., 622,6;625, 196 Laycock, P., 621,56 Li, H.-n., 622,63 Laziev, A., 622,14 Li, H.H., 622,6;625, 196 Lazoﬂores, J., 622, 265; 626, 35, 45, 55 Li, J., 621, 28; 622, 6, 218, 265; 624, 11; 625, 196; 626, 35, 45, 55 Lebeau, M., 622, 249; 623,26 Li, J.C., 622,6;625, 196 Lebedev, A., 621, 56; 622, 249; 623,26 Li, K.K., 623, 192 Le Bihan, A.-C., 622, 265; 626, 35, 45, 55 Li, L., 621, 139 Lebrun, P., 622, 249, 265; 623, 26; 626, 35, 45, 55 Li, M., 626, 202 Lechtenfeld, O., 625, 145 Li, Q.J., 622,6;625, 196 Lecomte, P., 622, 249; 623,26 Li, Q.Z., 622, 265; 626, 35, 45, 55 Cumulative author index to volumes 621–628 (2005) 292–321 307

Li, R.Y., 622,6;625, 196 Lobodenko, A., 622, 265; 626, 35, 45, 55 Li, S.M., 622,6;625, 196 Lobodzinska, E., 621,56 Li, T., 622, 320; 624, 233 Lohmann, W., 622, 249; 623,26 Li, W.D., 622,6;625, 196 Loiseau, B., 622, 207 Li, W.G., 622,6;625, 196 Lokajicek, M., 622, 265; 626, 35, 45, 55 Li, X.L., 622,6;625, 196 Loktionova, N., 621,56 Li, X.Q., 622,6;625, 196 Long, J.L., 628,25 Li, Y.L., 622,6;625, 196 Longo, E., 622, 249; 623,26 Liang, Y.F., 622,6;625, 196 Lopes Pegna, D., 621, 72; 622, 229, 239; 624, 22, 166 Liao, H.B., 622,6;625, 196 Lopez, A.M., 621, 72; 622, 229, 239; 624, 22, 166 Liddick, S.N., 622,45 Lopez, O., 627,18 Liebing, P., 622,14 Lopez-Fernandez, R., 621,56 Lietava, R., 623,17 Lorenzon, W., 622,14 Liguori, G., 621, 72; 622, 229, 239; 624, 22, 166 Lounis, A., 622, 265; 626, 35, 45, 55 Likhoded, S., 622, 249; 623,26 Love, P., 622, 265; 626, 35, 45, 55 Lima, J.G.R., 622, 265; 626, 35, 45, 55 Loverre, P.F., 626,24 Limagne, C., 621, 233 Løvhøiden, G., 623,17 Limata, B., 624, 181 Lowe, D.A., 624, 275 Limosani, A., 621, 28; 624,11 Lozea, A., 624, 217 Lin, C.H., 622, 249; 623,26 Lozeva, R., 622,29 Lin, S.-W., 621, 28, 41; 622, 218; 624,11 Lu, F., 622,6;625, 196 Lin, W.T., 622, 249; 623,26 Lu, G.R., 622,6;625, 196 Lincoln, D., 622, 265; 626, 35, 45, 55 Lu, H., 622,14 Linde, F.L., 622, 249; 623,26 Lü, H., 626, 215 Linden-Levy, L.A., 622,14 Lu, H.J., 622,6;625, 196 Lindfeld, L., 621,56 Lu, J., 622,14 Link, J.M., 621, 72; 622, 229, 239; 624, 22, 166 Lu, J.G., 622,6;625, 196 Linn, S.L., 622, 265; 626, 35, 45, 55 Lu, S., 622,14 Linnemann, J., 622, 265; 626, 35, 45, 55 Lu, Y.S., 622, 249; 623,26 Lionti, R., 625,33 Lubatti, H.J., 622, 265; 626, 35, 45, 55 Lipaev, V.V., 622, 265; 626, 35, 45, 55 Lubimov, V., 621,56 Lipka, K., 621,56 Lucaci-Timoce, A.-I., 621,56 Lipkin, H.J., 621, 126 Lucherini, V., 622,35 Lipton, R., 622, 265; 626, 35, 45, 55 Luci, C., 622, 249; 623,26 Lisetskiy, A.F., 622,45 Lucietti, J., 628, 275 Lisi, E., 623,80 Ludovici, L., 626,24 Lisin, V., 624, 173 Lueders, H., 621,56 List, B., 621,56 Lueking, L., 622, 265; 626, 35, 45, 55 Lista, L., 622, 249; 623,26 Luiggi, E., 621, 72; 622, 229, 239; 624, 22, 166 Lister, C.J., 622, 151; 625, 375 Luis, H., 624, 181 Littenberg, L.S., 623, 192 Lüke, D., 621,56 Liu, B., 628,25 Lukierski, J., 624, 304 Liu, C.X., 622,6;625, 196 Luksys, M., 628,18 Liu, F., 622,6,6;625, 196, 196 Luminari, L., 622, 249; 623,26 Liu, H.H., 622,6;625, 196 LUNA Collaboration, 624, 181 Liu, H.M., 622,6;625, 196 Lundin, M., 626,65 Liu, J., 622,6;625, 196 Lungov, T., 628,18 Liu, J.B., 622,6;625, 196 Luo, C.L., 622,6;625, 196 Liu, J.P., 622,6;625, 196 Luo, L.X., 622,6;625, 196 Liu, P.-H., 624,31 Luo, M., 626,7 Liu, R.G., 622,6;625, 196 Luo, X.L., 622,6;625, 196 Liu, Z.A., 622, 6, 249; 623, 26; 625, 196 Lustermann, W., 622, 249; 623,26 Liu, Z.X., 622,6;625, 196 Lux, T., 621,56 Liventsev, D., 621, 28, 41; 622, 218; 624,11 Luzzi, M., 625,1 Lobo, L., 622, 265; 626, 35, 45, 55 Lynker, M., 622, 265; 626, 35, 45, 55 308 Cumulative author index to volumes 621–628 (2005) 292–321

Lyon, A.L., 622, 265; 626, 35, 45, 55 Mao, D., 628,18 Lytkin, L., 621,56 Mao, H.S., 622, 265; 626, 35, 45, 55 Lyubovitskij, V.E., 622, 277 Mao, Y., 622,14 Mao, Z.P., 622,6;625, 196 Ma, B.-Q., 622,14 Marage, P., 621,56 Ma, E., 625,76 Maravin, Y., 622, 265; 626, 35, 45, 55 Ma, F.C., 622,6;625, 196 Marcello, S., 622,35 Ma, H.L., 622,6;625, 196 Marianski, B., 622,14 Ma, J.M., 622,6;625, 196 Mariz, T., 625, 351 Ma, J.P., 625,67 Marlow, D., 624,11 Ma, L.L., 622,6;625, 196 Marotta, A., 626,24 Ma, Q.M., 622,6;625, 196 Marques, J.G., 621, 233 Ma, W.G., 622, 249; 623,26 Marquet, C., 628, 239 Ma, X.B., 622,6;625, 196 Marshall, R., 621,56 Ma, X.Y., 622,6;625, 196 Martelli, D., 621, 208 Ma, Y.G., 628,25 Martens, M., 622, 265; 626, 35, 45, 55 Macdonald, J.A., 623, 192 Martin, A.D., 627,97 Machado, A.A., 621, 72; 622, 229, 239; 624, 22, 166 Martin, J.-P., 624, 186 Maciel, A.K.A., 622, 265; 626, 35, 45, 55 Martin, J.P., 622, 249; 623,26 MacNaughton, J., 621, 28, 41; 624,11 Martín-Mayor, V., 628, 281 Madaras, R.J., 622, 265; 626, 35, 45, 55 Martini, M., 626,15 Madore, J., 622, 183 Martins, R.C., 621, 233 Magass, C., 622, 265; 626, 35, 45, 55 Martisikova, M., 621,56 Magerkurth, A., 622, 265; 626, 35, 45, 55 Martyn, H.-U., 621,56 Magnan, A.-M., 622, 265; 626, 35, 45, 55 Marukyan, H., 622,14 Magnin, J., 621, 72; 622, 229, 239; 624, 22, 166 Maruta, T., 622,35 Mahajan, N., 623, 119 Marzano, F., 622, 249; 623,26 Maiheu, B., 622,14 Masiero, A., 622, 112 Maître, D., 622, 295 Masip, M., 627, 131 Maj, A., 622,29 Maslanka,´ P., 622, 192 Majumder, G., 621, 28, 41; 624,11 Masoli, F., 622,14 Makankine, A., 621,56 Massafferri, A., 621, 72; 622, 229, 239; 624, 22, 166 Makhlioueva, I., 626,24 Massarotti, P., 626,15 Makins, N.C.R., 622,14 Matheus, R.D., 624, 217 Makovec, N., 622, 265; 626, 35, 45, 55 Mathew, P., 628,18 Mal, P.K., 622, 265; 626, 35, 45, 55 Mathieu, P., 627, 224 Malbouisson, A.P.C., 624, 316 Matsuda, T., 624, 173 Malbouisson, H.B., 622, 265; 626, 35, 45, 55 Matsumoto, T., 621, 28, 41; 622, 218; 624,11 Malbouisson, J.M.C., 624, 316 Mättig, P., 622, 265; 626, 35, 45, 55 Malden, N., 621,56 Mattingly, S.E.K., 622, 265; 626, 35, 45, 55 Maleev, V.P., 628,18 Mattson, M., 628,18 Malgeri, L., 622, 249; 623,26 Matveev, V., 628,18 Malik, S., 622, 265; 626, 35, 45, 55 Matyja, A., 621,41 Malinin, A., 622, 249; 623,26 Maxﬁeld, S.J., 621,56 Malinovski, E., 621,56 Mayes, V.E., 625,96 Malvezzi, S., 621, 72; 622, 229, 239; 624, 22, 166 Mayorov, A.A., 622, 265; 626, 35, 45, 55 Malyshev, V.L., 622, 265; 626, 35, 45, 55 Maziashvili, M., 627, 197 Maña, C., 622, 249; 623,26 Mazumdar, K., 622, 249; 623,26 Manashov, A., 622,69 Mazur, A.I., 621,96 Mandal, S., 622,29 Mazzocchi, C., 622,45 Mandl, F., 621, 28, 41; 622, 218; 624,11 Mazzoni, M.A., 623,17 Mangano, S., 621,56 McCarthy, R., 622, 265; 626, 35, 45, 55 Mans, J., 622, 249; 623,26 McCliment, E., 628,18 Mantica, P.F., 622,45 McCroskey, R., 622, 265; 626, 35, 45, 55 Manzari, V., 622, 35; 623,17 McGeorge, J.C., 624, 173; 626,65 Cumulative author index to volumes 621–628 (2005) 292–321 309

McNeil, R.R., 622, 249; 623,26 Milstead, D., 621,56 McNeile, C., 624, 334 Milton, K.A., 621, 309 Meddi, F., 623,17 Mimura, Y., 622, 320; 627, 145 Meder, D., 622, 265; 626, 35, 45, 55 Mineev, O.V., 623, 192 Meer, D., 621,56 Mirabelli, G., 622, 249; 623,26 Mehta, A., 621,56 Mirfakhrai, N., 622,35 Mei, J., 623, 227 Miscetti, S., 626,15 Mei, W., 626,15 Mishustin, I.N., 627,64 Meier, A., 624, 173 Mitaroff, W., 621, 28, 41; 622, 218; 624,11 Meier, K., 621,56 Mitchell, R., 621, 72; 622, 229, 239; 624, 22, 166 Meinhard, H., 626,24 Mitrevski, J., 622, 265; 626, 35, 45, 55 Meißner, U.-G., 622, 141 Miyabayashi, K., 621, 28; 624,11 Mekjian, A.Z., 621, 239 Miyachi, Y., 622,14 Mele, S., 622, 249; 623,26 Miyajima, M., 623, 192 Melnitchouk, A., 622, 265; 626, 35, 45, 55 Miyake, H., 621, 28, 41; 622, 218; 624,11 Menasce, D., 621, 72; 622, 229, 239; 624, 22, 166 Miyanishi, M., 626,24 Mendes, A., 622, 265; 626, 35, 45, 55 Miyata, H., 621, 28; 622, 218; 624,11 Mendez, H., 621, 72; 622, 229, 239; 624, 22, 166 Mizouchi, K., 623, 192 Menezes, R., 625, 351 Mizuk, R., 621, 28, 41; 622, 218 Menze, D., 624, 173 Mnich, J., 622, 249; 623,26 Meola, S., 626,15 Mo, X.H., 622,6;625, 196; 626,95 Merkin, M., 622, 265; 626, 35, 45, 55 Moch, S., 625, 245 Merlo, M.M., 621, 72; 622, 229, 239; 624, 22, 166 Mocko, M., 627,32 Mermod, P., 622, 249 Moffat, J.W., 627,9 Merola, L., 622, 249; 623,26 Mohamed, A., 621,56 Merritt, K.W., 622, 265; 626, 35, 45, 55 Mohanty, G.B., 622, 249; 623,26 Meschini, M., 622, 249; 623,26 Mohapatra, D., 621, 28; 624,11 Messina, M., 626,24 Mohapatra, R.N., 627, 124 Metzger, W.J., 622, 249; 623,26 Mohseni, M., 626, 230 Mexner, V., 622,14 Moinester, M.A., 628,18 Meyer, A., 622, 265; 626, 35, 45, 55 Molchanov, V.V., 628,18 Meyer, A.B., 621,56 Molina, J., 622, 265; 626, 35, 45, 55 Meyer, H., 621,56 Moloney, G.R., 621, 41; 622, 218; 624,11 Meyer, J., 621, 56; 622, 265; 626, 35, 45, 55 Mondal, N.K., 622, 265; 626, 35, 45, 55 Meyer, W., 624, 173 Montanino, D., 627,38 Meyners, N., 622,14 Montvay, I., 623, 73; 624, 324 Mezzadri, M., 621, 72; 622, 229, 239; 624, 22, 166 Moore, J.E., 621, 72; 622, 229, 239; 624, 22, 166 Michael, C., 624, 334 Moore, R.W., 622, 265; 626, 35, 45, 55 Michalon, A., 623,17 Morando, M., 623,17 Michaut, M., 622, 265; 626, 35, 45, 55 Moreau, F., 621,56 Michel, T., 624, 173 Morelos, A., 628,18 Michler, T., 622,14 Mori, T., 621, 28, 41 Miettinen, H., 622, 265; 626, 35, 45, 55 Morlat, T., 621, 233 Migliozzi, P., 626,24 Moroi, T., 625,7,79 Mihul, A., 622, 249; 623,26 Moroni, L., 621, 72; 622, 229, 239; 624, 22, 166 Mikami, Y., 621, 28; 622, 218; 624,11 Morozov, A., 621,56 Mikloukho, O., 622,14 Morra, O., 622,35 Mikocki, S., 621,56 Morris, J.V., 621,56 Milcent, H., 622, 249; 623,26 Morton, A.C., 622,45 Milcewicz-Mika, I., 621,56 Moshin, P.Yu., 621, 295 Mildenberger, J., 623, 192 Motobayashi, T., 621,81 Miley, H.S., 621, 233 Moulik, T., 622, 265; 626, 35, 45, 55 Miller, C.A., 622,14 Moulson, M., 626,15 Miller, G.J., 626,65 Mozer, M.U., 621,56 Million, B., 622,29 Muanza, G.S., 622, 249, 265; 623, 26; 626, 35, 45, 55 310 Cumulative author index to volumes 621–628 (2005) 292–321

Muccifora, V., 622,14 Naumann, J., 621, 56; 624, 173 Muciaccia, M.T., 626,24 Naumann, N.A., 622, 265; 626, 35, 45, 55 Mueller, W.F., 622,45 Naumann, Th., 621,56 Muijs, A.J.M., 622, 249; 623,26 Navarra, F.S., 624, 217 Mukhin, V.A., 623, 192 Navarro, I., 622,1 Mulders, M., 622, 265; 626, 35, 45, 55 Neal, H.A., 622, 265; 626, 35, 45, 55 Müller, K., 621,56 Neemann, U., 623, 171 Müller, S., 626,15 Negodaev, M., 622,14 Multamäki, T., 628, 197 Negret, J.P., 622, 265; 626, 35, 45, 55 Mundim, L., 622, 265; 626, 35, 45, 55 Nehring, M., 621, 72; 622, 229, 239; 624, 22, 166 Muralithar, S., 622,29 Nelson, K.D., 628,18 Muramatsu, N., 623, 192 Nelson, S., 622, 265; 626, 35, 45, 55 Murayama, A., 625, 106 Nemitkin, A.V., 628,18 Murín, P., 621,56 Neoustroev, P.V., 628,18 Murray, M., 622,14 Nessi-Tedaldi, F., 622, 249; 623,26 Murtas, F., 626,15 Neußer, A., 622,29 Musakhanov, M.M., 628,33 Neustroev, P., 622, 265; 626, 35, 45, 55 Musicar, B., 622, 249; 623,26 Newman, H., 622, 249; 623,26 Musielak, Z.E., 626, 256 Newman, P.R., 621,56 Musy, M., 622, 249; 623,26 Newsom, C., 628,18 Mutaf, Y.D., 622, 265; 626, 35, 45, 55 Nguyen, F., 626,15 Muthusi, C., 624, 186 Nie, J., 622,6;625, 196 Myung, Y.S., 624, 297; 626,1;628,11 Nie, Z.D., 622,6;625, 196 Niebuhr, C., 621,56 NA57 Collaboration, 623,17 Nielsen, M., 624, 217 Nagae, T., 622,35 Nikiforov, A., 621,56 Nagai, K., 624, 334 Nikitin, D., 621,56 Nagai, M., 624, 239 Nilov, A.P., 628,18 Nagaitsev, A., 622,14 Nilsson, B., 626,65 Nagamine, T., 621, 28; 622, 218; 624,11 Nisati, A., 622, 249; 623,26 Nagasaka, Y., 621, 28, 41; 622, 218 Nishida, S., 621, 28, 41; 622, 218; 624,11 Nagy, E., 622, 265; 626, 35, 45, 55 Nitoh, O., 621, 28, 41; 622, 218; 624,11 Nagy, S., 622, 249; 623,26 Niu, K., 626,24 Nakamura, A., 621, 171 Niwa, K., 626,24 Nakamura, M., 626,24 Noble, A., 624, 186 Nakano, E., 621, 28, 41; 622, 218; 624,11 Noeding, C., 622, 265; 626, 35, 45, 55 Nakano, T., 623, 192; 626,24 Nomachi, M., 623, 192 Nakao, M., 621, 28, 41; 622, 218; 624,11 Nomerotski, A., 622, 265; 626, 35, 45, 55 Nakazawa, H., 621, 41; 622, 218; 624,11 Nomura, T., 623, 192 Nandi, S., 622, 102, 320 Nonaka, N., 626,24 Nankov, K., 621,56 Norman, P.I., 623,17 Nanopoulos, D.V., 625,96 Notani, M., 621,81 Napolitano, M., 622, 249; 623, 26; 626,15 Noulty, R., 624, 186 Nappi, E., 622,14 Novaes, S.F., 622, 265; 626, 35, 45, 55 Nara, Y., 621,89 Novak, T., 622, 249; 623,26 Narain, M., 622, 265; 626, 35, 45, 55 Novosel, I., 625,26 Nardulli, G., 623, 65; 627,89 Nowacki, F., 622,29 Narison, S., 624, 223; 626, 101 Nowak, G., 621,56 Narita, K., 626,24 Nowak, H., 622, 249; 623,26 Naroska, B., 621,56 Nowak, W.-D., 622,14 Naryshkin, Y., 622,14 Nozaki, T., 621, 28; 624,11 Nasri, S., 627, 124 Nozicka, M., 621,56 Nastase, H., 624, 125 Numao, T., 623, 192 Natale, S., 622, 249; 623,26 Nunes, N.J., 623, 147 Natkaniec, Z., 621, 28, 41; 622, 218 Nunnemann, T., 622, 265; 626, 35, 45, 55 Cumulative author index to volumes 621–628 (2005) 292–321 311

Nurse, E., 622, 265; 626, 35, 45, 55 Pacetti, S., 622, 229, 239; 624, 22, 166 Nurushev, S.B., 628,18 Padilla, J.L., 627, 131 Nussinov, S., 627, 124 Padley, P., 622, 265; 626, 35, 45, 55 Paech, K., 623, 200 Obraztsov, V.F., 623, 192 Page, P.R., 628, 215 Ocherashvili, A., 628,18 Pahlavan, A., 627, 217 Oda, I., 623, 155 Pakhlov, P., 622, 218; 624,11 O’Dell, V., 622, 265; 626, 35, 45, 55 Pal, I., 622, 249; 623,26 Oeckl, R., 622, 172 Pal, S., 624, 210; 627, 26, 55 Oﬁerzynski, R., 622, 249; 623,26 Pal, S.S., 622, 136 Oganessyan, K., 622,14 Palazzo, A., 623,80 Oganezov, R., 621,56 Palichik, V., 621,56 Ogawa, S., 621, 28, 41; 622, 218; 624, 11; 626,24 Palka, H., 621, 28, 41; 622, 218; 624,11 Oguri, V., 622, 265; 626, 35, 45, 55 Pallotta, M., 622,35 Oh, P., 628, 165 Palmeri, A., 623,17 Oh, S., 621, 259 Palomares, C., 622, 249; 623,26 Ohlsson, T., 622, 159 Palomba, M., 622,35 Ohnishi, T., 621,81 Palutan, M., 626,15 Ohshima, T., 621, 28, 41; 622, 218; 624,11 Panman, J., 626,24 Ohsuga, H., 622,14 Panotopoulos, G., 623, 185 Okabe, T., 621, 28; 622, 218; 624,11 Pantaleo, A., 622,35 Okuno, S., 621, 28, 41; 622, 218; 624,11 Pantea, D., 621, 72; 622, 229, 239; 624, 22, 166 Okusawa, T., 626,24 Panzarasa, A., 622,35 Oldeman, R.G.C., 626,24 Panzeri, A., 624, 173 Olin, A., 622,35 Paolucci, P., 622, 249; 623,26 Olive, K.A., 624,47 Papadopoulou, T., 621,56 Oliveira, C., 621, 233 Papakonstantinou, P., 624, 195 Oliveira, N., 622, 265; 626, 35, 45, 55 Papinutto, M., 624, 334 Oliver, S.J., 625, 189 Pappalardo, G.S., 623,17 Olivier, B., 621,56 Paramatti, R., 622, 249; 623,26 Ollitrault, J.-Y., 627,49 Parashar, N., 622, 265; 626, 35, 45, 55 Olsen, S.L., 621, 28, 41; 622, 218; 624,11 Paris, A., 621, 72; 622, 229, 239; 624, 22, 166 Olsson, J.E., 621,56 Park, C.W., 621, 28; 622, 218; 624,11 Omata, K., 623, 192 Park, H., 621, 28, 41, 72; 622, 229, 239; 624, 22, 166 O’Neil, D.C., 622, 265; 626, 35, 45, 55 Park, S.K., 622, 265; 626, 35, 45, 55 Onel, Y., 628,18 Parlog, M., 627,18 Önengüt, G., 626,24 Parslow, N., 621, 28; 622, 218; 624,11 Onuki, Y., 621,28 Parsons, J., 622, 265; 626, 35, 45, 55 O’Reilly, B., 621, 72; 622, 229, 239; 624, 22, 166 Partridge, R., 622, 265; 626, 35, 45, 55 Organtini, G., 622, 249; 623,26 Parua, N., 622, 265; 626, 35, 45, 55 Osborne, A., 622,14 Pascaud, C., 621,56 Oshima, N., 622, 265; 626, 35, 45, 55 Pasipoularides, P., 621, 224 Osman, S., 621,56 Pasqualucci, E., 626,15 Ostrowicz, W., 621, 28, 41; 622, 218; 624,11 Passaleva, G., 622, 249; 623,26 Otero y Garzón, G.J., 622, 265; 626, 35, 45, 55 Passeri, A., 626,15 Otsuka, T., 622,29 Pastircák,ˇ B., 623,17 Outa, H., 622,35 Patalakha, D.I., 623, 192 Ozaki, H., 621, 28; 622, 218 Patel, G.D., 621,56 Ozawa, A., 621,81 Patera, V., 626,15 Ozel, E., 628,18 Pati, J.C., 621, 160 Ozerov, D., 621,56 Paticchio, V., 622,35 Ozkorucuklu, S., 628,18 Patricelli, S., 622, 249; 623,26 Patwa, A., 622, 265; 626, 35, 45, 55 Paar, N., 624, 195 Paul, E.S., 625, 203 Pace, E., 622,35 Paul, T., 622, 249; 623,26 312 Cumulative author index to volumes 621–628 (2005) 292–321

Pauluzzi, M., 622, 249; 623,26 Pitzl, D., 621,56 Paus, C., 622, 249; 623,26 Placakytˇ e,˙ R., 621,56 Pauss, F., 622, 249; 623,26 Platt, R.J., 623,17 Pavlopoulos, T.G., 625,13 Pleier, M.-A., 622, 265; 626, 35, 45, 55 Pavón, D., 628, 206 Pleiter, D., 627, 113 Pawloski, G., 622, 265; 626, 35, 45, 55 Pleitez, V., 621, 151 Pawlowski, J.M., 626, 139 Plentinger, F., 625, 264 Peak, L.S., 621, 41; 622, 218; 624,11 Plonka, C., 625,19 Pedace, M., 622, 249; 623,26 Plyaskin, V., 622, 249; 623,26 Pedrini, D., 621, 72; 622, 229, 239; 624, 22, 166 Plyushchay, M.S., 628, 157 Pedroni, P., 624, 173 Podesta-Lerma, P.L.M., 622, 265; 626, 35, 45, 55 Peez, M., 621,56 Podolyák, Z., 622,29 Peloso, M., 623, 147 Podstavkov, V.M., 622, 265; 626, 35, 45, 55 Peng, H.P., 622,6;625, 196 Pogodin, P., 628,18 Penionzhkevich, Yu.E., 621, 119 Pogorelov, Y., 622, 265; 626, 35, 45, 55 Pennington, M.R., 623,55 Pohl, M., 622, 249; 623,26 Pensotti, S., 622, 249; 623,26 Pojidaev, V., 622, 249; 623,26 Penzo, A., 628,18 Pol, M.-E., 622, 265; 626, 35, 45, 55 Pepe, I.M., 621, 72; 622, 229, 239; 624, 22, 166 Polarski, D., 627,1 Perea, P.M., 622, 265; 626, 35, 45, 55 Poluektov, A., 621, 41; 624,11 Perez, E., 621, 56; 622, 265; 626, 35, 45, 55 Polycarpo, E., 621, 72; 622, 229, 239; 624, 22, 166 Perez-Astudillo, D., 621,56 Pompili, F., 622,35 Perfetto, F., 626,15 Pompoš, A., 622, 265; 626, 35, 45, 55 Perieanu, A., 621,56 Ponomarev, V.Yu., 624, 195 Perlt, H., 628,66 Pontecorvo, L., 626,15 Perret-Gallix, D., 622, 249; 623,26 Pontoglio, C., 621, 72; 622, 229, 239; 624, 22, 166 Peschanski, R., 622, 178; 628, 239 Pope, B.G., 622, 265; 626, 35, 45, 55 Pestotnik, R., 621, 28, 41; 624,11 Pope, C.N., 626, 215 Peters, W., 627,32 Porod, W., 626, 175 Petrellis, D., 621, 102 Portheault, B., 621,56 Petrenko, S.V., 623, 192; 628,18 Pothier, J., 622, 249; 623,26 Pétroff, P., 622, 265; 626, 35, 45, 55 Potterveld, D.H., 622,14 Petrov, A.A., 625,53 Poutissou, R., 623, 192 Petrukhin, A., 621,56 Povh, B., 621,56 Petteni, M., 622, 265; 626, 35, 45, 55 Prado da Silva, W.L., 622, 265; 626, 35, 45, 55 Phaf, L., 626,45 Prelz, F., 621, 72; 622, 229, 239; 624, 22, 166 Pham, T.N., 623,65 Preobrajenski, I., 624, 173 Piano, S., 622,35 Prideaux, P., 621,56 Piantelli, S., 623, 43; 627,18 Primavera, M., 626,15 Piccolo, D., 622, 249; 623,26 Procario, M., 628,18 Pichlmaier, A., 625,19 Prodanov, E.M., 623, 235 Pickavance, J., 624, 334 Prokoﬁev, D., 622, 249; 623,26 Pickert, N., 622,14 Prosper, H.B., 622, 265; 626, 35, 45, 55 Piegaia, R., 622, 265; 626, 35, 45, 55 Protopopescu, S., 622, 265; 626, 35, 45, 55 Pierella, F., 622, 249; 623,26 Prutskoi, V.A., 628,18 Piilonen, L.E., 621, 28, 41; 622, 218; 624,11 Puibasset, J., 621, 233 Pilaftsis, A., 628, 250 Pinelli, T., 624, 173 QCDSF/UKQCD Collaboration, 627, 113 Pinsky, S., 624, 105 QCDSF Collaboration, 628,66 Pioline, B., 625, 291 Qi, N.D., 622,6;625, 196 Pioppi, M., 622, 249; 623,26 Qian, C.D., 622,6;625, 196 Pire, B., 622, 83; 626,86 Qian, J., 622, 265; 626, 35, 45, 55 Piroué, P.A., 622, 249; 623,26 Qiao, C.-F., 621, 139 Pistolesi, E., 622, 249; 623,26 Qin, H., 622,6;625, 196 Pittau, R., 628,40 Qiu, J.F., 622,6;625, 196 Cumulative author index to volumes 621–628 (2005) 292–321 313

Quadt, A., 622, 265; 626, 35, 45, 55 Reyes, M., 621, 72; 622, 229, 239; 624, 22, 166 Queiroz, H., 624, 316 Ribeiro, J.P., 624, 181 Quercigh, E., 623,17 Riccardi, C., 621, 72; 622, 229, 239; 624, 22, 166 Quinn, B., 622, 265; 626, 35, 45, 55 Riedl, C., 622,14 Quinones, J., 621, 72; 622, 229, 239; 624, 22, 166 Riemann, S., 622, 249; 623,26 Riggi, F., 623,17 Raby, S., 622, 327 Rigolin, S., 621, 276 Rador, T., 621, 176 Rijssenbeek, M., 622, 265; 626, 35, 45, 55 Radtke, E., 624, 173 Riles, K., 622, 249; 623,26 Raduta, Ad.R., 623,43 Rimmer, A., 621,56 Rahal-Callot, G., 622, 249; 623,26 Rimmer, S., 627, 161 Rahaman, M.A., 622, 249; 623,26 Ripp-Baudot, I., 622, 265; 626, 35, 45, 55 Rahimi, A., 621, 72; 622, 229, 239; 624, 22, 166 RISING Collaboration, 622,29 Raicevic, N., 621,56 Risler, C., 621,56 Raics, P., 622, 249; 623,26 Rith, K., 622,14 Raiola, F., 624, 181 Riva, V., 622, 339 Raithel, M., 622,14 Rivelles, V.O., 625, 156 Raja, N., 622, 249; 623,26 Rivers, R.J., 625, 333 Rajasekaran, G., 626, 167 Rivet, M.F., 623, 43; 627,18 Rakow, P.E.L., 627, 113; 628,66 Rizatdinova, F., 622, 265; 626, 35, 45, 55 Ralston, J.P., 621, 213 Rizvi, E., 621,56 Ramberg, E., 628,18 Robinson, S., 622, 265; 626, 35, 45, 55 Ramberg, E.J., 623, 192 Robmann, P., 621,56 Ramelli, R., 622, 249; 623,26 Rocha-Filho, T.M., 624, 316 Ramirez, J.E., 621, 72; 622, 229, 239; 624, 22, 166 Rodejohann, W., 625, 264 Ramos, A.R., 621, 233 Rodrigues, R.F., 622, 265; 626, 35, 45, 55 Rancoita, P.G., 622, 249; 623,26 Rodrigues da Silva, P.S., 621, 151; 628,85 Rani, K.J., 622, 265; 626, 35, 45, 55 Roe, B.P., 622, 249; 623,26 Ranieri, R., 622, 249; 623,26 Rohlof, Ch., 624, 173 Ranjan, K., 622, 265; 626, 35, 45, 55 Röhrich, D., 623,17 Ranquet, A., 627,1 Roland, B., 621,56 Rapidis, P.A., 622, 265; 626, 35, 45, 55 Rolfs, C., 624, 181 Rappazzo, G.F., 628,18 Romano, G., 623, 17; 626,24 Raspereza, A., 622, 249; 623,26 Romano, M., 624, 181 Rastogi, P., 621, 160 Romeo, A., 621, 309 Ratoff, P.N., 622, 265; 626, 35, 45, 55 Romero, L., 622, 249; 623,26 Rattazzi, R., 624, 260 Rong, G., 622,6;625, 196 Ratti, S.P., 621, 72; 622, 229, 239; 624, 22, 166 Ronga, F.J., 621, 28; 624,11 Raychaudhuri, A., 628, 141 Roosen, R., 621,56 Razis, P., 622, 249; 623,26 Root, N., 622, 218 Razmyslovich, B.V., 628,18 Rosa, G., 626,24 Redlinger, G., 623, 192 Rosato, E., 627,18 Reggiani, D., 622,14 Rosca, A., 622, 249; 623,26 Reichert, E., 624, 173 Rosemann, C., 622, 249; 623,26 Reicherz, G., 624, 173 Rosenbleck, C., 622, 249; 623,26 Reimer, P., 621,56 Rosenfeld, R., 624, 158 Reimer, P.E., 622,14 Rosiek, J., 627, 161 Reischl, A., 622,14 Rosier-Lees, S., 622, 249; 623,26 Reiter, P., 622, 29; 625, 203 Rosner, G., 622, 14; 624, 173 Ren, D., 622, 249; 623,26 Rossi, A., 622, 112 Ren, Z.Y., 622,6;625, 196 Rostomyan, A., 622,14 Reolon, A.R., 622,14 Rostomyan, T., 624, 173 Rescigno, M., 622, 249; 623,26 Rostovtsev, A., 621,56 Reshetnyak, A.A., 621, 295 Roth, S., 622, 249; 623,26 Reucroft, S., 622, 249, 265; 623, 26; 626, 35, 45, 55 Rotunno, A.M., 623,80 314 Cumulative author index to volumes 621–628 (2005) 292–321

Rouhani, S., 627, 217 Santangelo, P., 626,15 Rovere, M., 621, 72; 622, 229, 239; 624, 22, 166 Santoro, A., 622, 265; 626, 35, 45, 55 Roy, D.P., 628, 131 Santovetti, E., 626,15 Royon, C., 622, 265; 626, 35, 45, 55 Sanyal, A.K., 624,81 Rozanov, A., 626,24 Saracino, G., 626,15 Rubacek, L., 622,14 Sarangi, T.R., 624,11 Rubin, J., 622,14 Sarantites, D.G., 625, 203 Rubinov, P., 622, 265; 626, 35, 45, 55 Saremi, S., 622, 249; 623,26 Rubio, J.A., 622, 249; 623,26 Sarkar, S., 622, 249; 623,26 Ruchayskiy, O., 626, 184 Sarkar, U., 622, 118; 626, 167 Ruchti, R., 622, 265; 626, 35, 45, 55 Sasaki, K., 623, 208 Rud, V.I., 622, 265; 626, 35, 45, 55; 628,18 Sasaki, S., 623, 208 Ruggieri, M., 627,89 Satarov, L.M., 627,64 Ruggiero, G., 622, 249; 623,26 Sato, N., 621, 28, 41; 622, 218; 624,11 Rui, R., 622,35 Sato, O., 626,24 Ruijter, H., 626,65 Sato, T., 623, 192 Rurikova, Z., 621,56 Sato, Y., 626,24 Rusakov, S., 621,56 Satta, A., 626,24 Russ, J., 628,18 Sauer, M., 624, 173 Ryckbosch, D., 622, 14; 624, 173 Sauvan, E., 621,56 Rykaczewski, H., 622, 249; 623,26 Savage, G., 622, 265; 626, 35, 45, 55 Rykaczewski, K.P., 622,45 Savin, I., 622,14 Ryskin, M.G., 627,97 Savvidy, G., 625, 341 Sawyer, L., 622, 265; 626, 35, 45, 55 Šafarík,˘ K., 623,17 Scanlon, T., 622, 265; 626, 35, 45, 55 Sagawa, H., 621, 28, 41; 622, 218; 624,11 Schäfer, A., 622, 14, 69; 627, 113 Sagnotti, A., 624,93 Schäfer, C., 622, 249; 623,26 Saha, J.P., 622, 102 Schaile, D., 622, 265; 626, 35, 45, 55 Saito, N., 622,29 Schamberger, R.D., 622, 265; 626, 35, 45, 55 Saito, T., 621, 171 Schätzel, S., 621,56 Saito, T.R., 622,29 Schegelsky, V., 622, 249; 623,26 Saitta, B., 626,24 Schellman, H., 622, 265; 626, 35, 45, 55 Sajot, G., 622, 265; 626, 35, 45, 55 Schiavon, P., 628,18 Sakaguchi, M., 621, 288 Schieferdecker, P., 622, 265; 626, 35, 45, 55 Sakai, Y., 621, 28, 41; 622, 218; 624,11 Schierholz, G., 627, 113; 628,66 Sakharov, A., 622, 249; 623,26 Schietinger, T., 621, 28, 41; 622, 218; 624,11 Sakurai, H., 621,81 Schiller, A., 627, 32; 628,66 Sala, S., 621, 72; 622, 229, 239; 624, 22, 166 Schilling, F.-P., 621,56 Salicio, J., 622, 249; 623,26 Schillings, E., 623,17 Salomatin, Y., 622,14 Schmidt, S., 621,56 Salvaire, F., 621,56 Schmidt-Sommerfeld, M., 627, 161 Salwen, N., 624, 105 Schmitt, C., 622, 265; 626, 35, 45, 55 Sanchez, E., 622, 249; 623,26 Schmitt, S., 621,56 Sánchez-Guillén, J., 626, 235 Schmitz, C., 621,56 Sánchez-Hernández, A., 621, 72; 622, 229, 239, 265; 624, 22, 166; Schneider, O., 621, 28, 41; 622, 218; 624,11 626, 35, 45, 55 Schnell, G., 622,14 Sanda, A.I., 625,47 Schoch, B., 624, 173 Sandell, A., 626,65 Schoeffel, L., 621,56 Sanders, M.P., 622, 265; 626, 35, 45, 55 Scholtz, F.G., 625, 302 Šándor, L., 623,17 Scholz, E., 623,73 Sanjiev, I., 622,14 Scholz, E.E., 624, 324 Sankey, D.P.C., 621,56 Schöning, A., 621,56 Santacesaria, R., 626,24 Schönmeier, P., 622, 218 Santamaria, A., 626, 151 Schopper, H., 622, 249; 623,26 Santana, A.E., 624, 316 Schotanus, D.J., 622, 249; 623,26 Cumulative author index to volumes 621–628 (2005) 292–321 315

Schreiber, H.J., 622, 311 Shary, V., 622, 265; 626, 35, 45, 55 Schröder, B., 626,65 Shchukin, A.A., 622, 265; 626, 35, 45, 55 Schröder, V., 621,56 Sheaff, M., 621, 72; 622, 229, 239; 624, 22, 166 Schröder, Y., 622, 124 Shearer, C., 622,14 Schüler, K.P., 622,14 Sheldon, P.D., 621, 72; 622, 229, 239; 624, 22, 166 Schultz-Coulon, H.-C., 621,56 Shen, D.L., 622,6;625, 196 Schumacher, M., 624, 173 Shen, X.Y., 622,6;625, 196 Schümann, J., 622, 218 Shen, Y.-G., 623, 141 Schürmann, D., 624, 181 Sheng, H.Y., 622,6;625, 196 Schwanda, C., 621, 28; 624,11 Shephard, W.D., 622, 265; 626, 35, 45, 55 Schwanenberger, C., 621, 56; 622, 265; 626, 35, 45, 55 Shevchenko, S., 622, 249; 623,26 Schwartz, A.J., 624,11 Sheviakov, I., 621,56 Schwartzman, A., 622, 265; 626, 35, 45, 55 Shi, F., 622,6;625, 196 Schweizer, J., 625, 217 Shi, X., 622,6;625, 196 Schwienhorst, R., 622, 265; 626, 35, 45, 55 Shibata, T., 622, 218; 624,11 Schwindt, J.-M., 628, 189 Shibata, T.-A., 622,14 Sciacca, C., 622, 249; 623,26 Shibuya, H., 621, 28, 41; 622, 218; 624, 11; 626,24 Sciascia, B., 626,15 Shima, K., 628, 171 Sciretti, D., 628, 281 Shimizu, Y., 625,79 Sciubba, A., 626,15 Shimoura, S., 621,81 Scorzato, L., 624, 324 Shin, H., 627, 188 Scott, W.G., 628,93 Shindler, A., 624, 324, 334 Scotto Lavina, L., 626,24 Shinkawa, T., 623, 192 Scrucca, C.A., 624, 260 Shirokov, A.M., 621,96 Scuri, F., 626,15 Shivaraj, K., 628, 223 Sébille, F., 625,26 Shivarov, N., 622, 249; 623,26 Sedlák, K., 621,56 Shivpuri, R.K., 622, 265; 626, 35, 45, 55 Seele, J., 622,14 Shock, J.P., 622, 165 Sefkow, F., 621,56 Shoutko, V., 622, 249; 623,26 Segato, G., 623,17 Shpakov, D., 622, 265; 626, 35, 45, 55 Segoni, I., 621, 72; 622, 229, 239; 624, 22, 166 Shtarkov, L.N., 621,56 Seidl, G., 624, 250 Shuchen, Z., 628,18 Seidl, R., 622,14 Shumilov, E., 622, 249; 623,26 Seitz, B., 622, 14; 624, 173 Shutov, V., 622,14 Sekiguchi, T., 623, 192 Shvorob, A., 622, 249; 623,26 SELEX Collaboration, 628,18 Shwartz, B., 621, 28, 41; 622, 218 Sené, M., 623,17 Si, Z.G., 625,67 Sené, R., 623,17 Sidorov, V., 621,28 Sengupta, S., 622, 265; 626, 35, 45, 55 Sidwell, R.A., 622, 265; 626, 35, 45, 55 Senyo, K., 621, 41; 622, 218; 624,11 Signer, A., 626, 127 Sepangi, H.R., 626, 230 Simak, V., 622, 265; 626, 35, 45, 55 Servoli, L., 622, 249; 623,26 Simon, J., 628,18 Sethi, G., 624, 135 Simonetti, G., 622,35 Seuster, R., 621,28 Sims, D.A., 626,65 Severini, H., 622, 265; 626, 35, 45, 55 Singh, A.K., 622,29 Sevior, M.E., 621, 28, 41; 622, 218; 624,11 Singh, J.B., 621, 41; 622, 218; 624,11 Seweryniak, D., 622, 151; 625, 203, 375 Sinram, K., 622,14 Sfetsos, K., 625, 135 Sirignano, C., 626,24 Sﬁligoi, I., 626,15 Sirois, Y., 621,56 Shabalina, E., 622, 265; 626, 35, 45, 55 Sirotenko, V., 622, 265; 626, 35, 45, 55 Shamanov, V., 626,24 Sitnikov, A.I., 628,18 Shamim, M., 622, 265; 626, 35, 45, 55 Skow, D., 628,18 Shan, L.Y., 622,6;625, 196 Skubic, P., 622, 265; 626, 35, 45, 55 Shang, L., 622,6;625, 196 Slattery, P., 622, 265; 626, 35, 45, 55 Shaposhnikov, M., 626, 184 Sloan, T., 621,56 316 Cumulative author index to volumes 621–628 (2005) 292–321

Smirnov, A.Yu., 621,1 Steijger, J.J.M., 622,14 Smirnov, P., 621,56 Steiner, M., 622,45 Smith, J.F., 622, 151; 625, 203, 375 Steiner, V., 628,18 Smith, R.P., 622, 265; 626, 35, 45, 55 Steinhauser, M., 622, 124 Smith, V.J., 628,18 Stella, B., 621,56 Smolek, K., 622, 265; 626, 35, 45, 55 Stenson, K., 621, 72; 622, 229, 239; 624, 22, 166 Snoeys, W., 623,17 Stenzel, H., 622,14 Snow, G.R., 622, 265; 626, 35, 45, 55 Stepanov, V., 628,18 Snow, J., 622, 265; 626, 35, 45, 55 Steuer, M., 622, 249; 623,26 Snyder, S., 622, 265; 626, 35, 45, 55 Stevenson, K., 622, 265; 626, 35, 45, 55 So, H., 622,35 Stewart, J., 622,14 Solà, J., 624, 147 Stickland, D.P., 622, 249; 623,26 Solbrig, S., 621, 195 Stiewe, J., 621,56 Söldner-Rembold, S., 622, 265; 626, 35, 45, 55 Stinzing, F., 622,14 Soloviev, Y., 621,56 Stöcker, H., 627,64 Sommer, W., 622,14 Stöckinger, D., 626, 127 Somorjai, E., 624, 181 Stolin, V., 622, 265; 626, 35, 45, 55 Somov, A., 621, 28, 41; 622, 218; 624,11 Stolz, A., 622, 45; 627,32 Son, D., 622, 249; 623,26 Stone, A., 622, 265; 626, 35, 45, 55 Song, J.S., 626,24 Stoyanov, B., 622, 249; 623,26 Song, X., 622, 265; 626, 35, 45, 55 Stoyanova, D.A., 622, 265; 626, 35, 45, 55 Soni, N., 621, 41; 622, 218 Straessner, A., 622, 249; 623,26 Sonnenschein, L., 622, 265; 626, 35, 45, 55 Strand, R.C., 623, 192 Sopczak, A., 622, 265; 626, 35, 45, 55 Strandberg, J., 622, 265; 626, 35, 45, 55 Soramel, F., 623,17 Strang, M.A., 622, 265; 626, 35, 45, 55 Sorrentino, S., 626,24 Strauch, I., 621,56 Sosebee, M., 622, 265; 626, 35, 45, 55 Straumann, U., 621, 56; 625,19 Souga, C., 622, 249; 623,26 Strauss, M., 622, 265; 626, 35, 45, 55 Soustruznik, K., 622, 265; 626, 35, 45, 55 Strieder, F., 624, 181 South, D., 621,56 Strikman, M., 626,72 Souza, M., 622, 265; 626, 35, 45, 55 Ströhmer, R., 622, 265; 626, 35, 45, 55 Soyez, G., 628, 239 Strolin, P., 626,24 Spada, F.R., 626,24 Strom, D., 622, 265; 626, 35, 45, 55 Spadaro, T., 626,15 Strovink, M., 622, 265; 626, 35, 45, 55 Spanos, V.C., 624,47 Strumia, A., 625, 189 Sparks, J., 621, 208 Studenikin, A., 622, 199 Spaskov, V., 621,56 Stutte, L., 622, 265; 626, 35, 45, 55; 628,18 Specka, A., 621,56 Su, Q., 626,7 Speckner, T., 624, 173 Sudhakar, K., 622, 249; 623,26 Spillantini, P., 622, 249; 623,26 Sugimoto, S., 623, 192 Spurlock, B., 622, 265; 626, 35, 45, 55 Sugiyama, A., 621,41 Spyropoulou-Stassinaki, M., 623,17 Suhonen, J., 626,80 Srivastava, M., 628,18 Sultanov, G., 622, 249; 623,26 Stamen, R., 621, 28, 41; 622, 218; 624,11 Sumisawa, K., 621, 28; 622, 218; 624,11 Stancari, M., 622,14 Sumithrarachchi, C.S., 627,32 Stanic,ˇ S., 621, 28, 41; 622, 218; 624,11 Sumiyoshi, T., 621, 28, 41; 622, 218; 624,11 Stanton, N.R., 622, 265; 626, 35, 45, 55 Sumowidagdo, S., 622, 265; 626, 35, 45, 55 Staric,ˇ M., 621, 28, 41; 622, 218; 624,11 Sun, H.S., 622,6;625, 196 Stark, J., 622, 265; 626, 35, 45, 55 Sun, J.F., 622,6;625, 196 Staroba, P., 623,17 Sun, L.Z., 622, 249; 623,26 Starosta, K., 625, 203 Sun, S.S., 622,6;625, 196 Statera, M., 622,14 Sun, Y.Z., 622,6;625, 196 Steele, J., 622, 265; 626, 35, 45, 55 Sun, Z.J., 622,6;625, 196 Štefanciˇ c,´ H., 624, 147 Susaki, Y., 624, 115 Steffens, E., 622,14 Sushkov, S., 622, 249; 623,26 Cumulative author index to volumes 621–628 (2005) 292–321 317

Sutcliffe, P.M., 626, 120 Terziev, P.A., 621, 102 Suter, H., 622, 249; 623,26 Testa, M., 626,15 Suzuki, S., 624,11 Teyssier, D., 622, 249; 623,26 Suzuki, S.Y., 621, 28, 41 Tezuka, I., 626,24 Svoiski, M., 628,18 Thoennessen, M., 627,32 Swain, J.D., 622, 249; 623,26 Thomas, A., 624, 173 Swift, L.D., 626, 256 Thomas, A.W., 621, 246; 628, 125 Szillasi, Z., 622, 249; 623,26 Thomas, G.P., 628,18 Sznajder, A., 622, 265; 626, 35, 45, 55 Thompson, G., 621,56 Szydłowski, M., 623,10 Thompson, P.D., 621,56 Szymanowski, L., 622,83 Tian, X.C., 621, 28, 41; 622, 218; 624,11 Tian, Y.R., 622,6;625, 196 Tabor, S.L., 622, 151; 625, 375 Timmermans, C., 622, 249; 623,26 Tait, P., 622,14 Timóteo, V.S., 621, 109 Tajima, O., 621, 28, 41; 622, 218; 624,11 Ting, S.C.C., 622, 249; 623,26 Takabayashi, N., 624, 173 Ting, S.M., 622, 249; 623,26 Takasaki, F., 621, 28, 41; 622, 218; 624,11 Tioukov, V., 626,24 Takayama, Y., 624, 115 Tlapanco-Limón, J.F., 623, 165 Takeshita, E., 621,81 Togano, Y., 621,81 Takeuchi, S., 621,81 Tolun, P., 626,24 Takook, M.V., 627, 217 Tomassini, S., 622,35 Talby, M., 622, 265; 626, 35, 45, 55 Tomasz, F., 621,56 Tamagawa, Y., 623, 192 Tomio, L., 621, 109 Tamai, K., 621, 28, 41; 622, 218; 624,11 Tomoto, M., 622, 265; 626, 35, 45, 55 Tamain, B., 627,18 Tong, G.L., 622,6;625, 196 Tamas, G., 624, 173 Tonin, M., 623, 155 Tamburello, P., 622, 265; 626, 35, 45, 55 Tonwar, S.C., 622, 249; 623,26 Tamura, N., 621, 28, 41; 622, 218; 624,11 Toole, T., 622, 265; 626, 35, 45, 55 Tamvakis, K., 628, 262 Torborg, J., 622, 265; 626, 35, 45, 55 Tanaka, H., 622,14 Torres, I., 628,18 Tanaka, M., 621, 28; 622, 218; 624,11 Tortora, L., 626,15 Tang, X., 622,6;625, 196 Toshito, T., 626,24 Tang, X.W., 622, 249; 623,26 Tóth, J., 622, 249; 623,26 Tanihata, I., 621,81 Toublan, D., 621, 145; 623,48 Tao, N., 622,6;625, 196 Towers, S., 622, 265; 626, 35, 45, 55 Tarancón, A., 628, 281 Toyoda, A., 622,35 Tarjan, P., 622, 249; 623,26 Trabelsi, K., 624,11 Taroian, S., 622,14 Traynor, D., 621,56 Tauscher, L., 622, 249; 623,26 Trefzger, T., 622, 265; 626, 35, 45, 55 Tawﬁk, A., 623,48 Trincaz-Duvoid, S., 622, 265; 626, 35, 45, 55 Taylor, G.N., 621,41 Trittmann, U., 624, 105 Taylor, L., 622, 249; 623,26 Troyan, S.I., 622,93 Taylor, W., 622, 265; 626, 35, 45, 55 Truöl, P., 621,56 Tchoulakov, V., 621,56 Trzcinski, A., 622,14 Tchuiko, B., 622,14 Tsakov, I., 621,56 Telford, P., 622, 265; 626, 35, 45, 55 Tschirhart, R., 623, 192 Tellili, B., 622, 249; 623,26 Tsenov, R., 626,24 Temple, J., 622, 265; 626, 35, 45, 55 Tsipolitis, G., 621,56 Teper, M., 628, 113 Tsuboyama, T., 624,11 Teramoto, Y., 621, 28, 41; 622, 218; 624,11 Tsuda, M., 628, 171 Terentyev, N.K., 628,18 Tsukamoto, T., 621, 28, 41; 622, 218; 624,11 Tereshchenko, V., 622,35 Tsukerman, I., 626,24 Terkulov, A., 622,14 Tsunemi, T., 623, 192 Ternov, A., 622, 199 Tsurin, I., 621,56 Teryaev, O.V., 626,86 Tuchin, K., 626, 147 318 Cumulative author index to volumes 621–628 (2005) 292–321

Tuchming, B., 622, 265; 626, 35, 45, 55 Van Hoorebeke, L., 624, 173 Tully, C., 622, 249, 265; 623, 26; 626, 35, 45, 55 Van Kooten, R., 622, 265; 626, 35, 45, 55 Tung, K.L., 622, 249; 623,26 Van Leeuwen, W.M., 622, 265; 626, 35, 45, 55 Turcot, A.S., 622, 265; 626, 35, 45, 55 Van Mechelen, P., 621,56 Turnau, J., 621,56 Van Remortel, N., 621,56 Turrisi, R., 623,17 Varelas, N., 622, 265; 626, 35, 45, 55 Tuts, P.M., 622, 265; 626, 35, 45, 55 Vargas Trevino, A., 621,56 Tverskoy, M., 626,72 Varley, B.J., 622, 151; 625, 375 Tveter, T.S., 623,17 Varner, G., 621, 28, 41; 622, 218; 624,11 Tytgat, M., 622,14 Varnes, E.W., 622, 265; 626, 35, 45, 55 Tzamariudaki, E., 621,56 Vartapetian, A., 622, 265; 626, 35, 45, 55 Varvell, K.E., 621, 28; 624,11 Uehara, S., 621, 28, 41; 622, 218; 624,11 Vary, J.P., 621,96 Ueno, K., 621, 28, 41; 622, 218 Vascotto, A., 623,17 Uglov, T., 621, 28, 41; 622, 218; 624,11 Vasiliev, A.N., 628,18 Uiterwijk, J.W.E., 626,24 Vasilyev, I.A., 622, 265; 626, 35, 45, 55 Ukita, N., 624, 324 Vasquez, R., 622, 249; 623,26 Ulbricht, J., 622, 249; 623,26 Vasudevan, M., 624, 287 Ulrych, S., 625, 313; 628, 291 Vaupel, M., 622, 265; 626, 35, 45, 55 Uno, S., 621, 28, 41; 622, 218; 624,11 Vavilov, D.V., 623, 192; 628,18 Urbach, C., 624, 324, 334 Vazdik, Y., 621,56 Urbán, J., 623,17 Vázquez, F., 621, 72; 622, 229, 239; 624, 22, 166 Urban, M., 621,56 Vázquez-Jáuregui, E., 628,18 Uribe, C., 621, 72; 622, 229, 239; 624, 22, 166 Veelken, C., 621,56 Urquijo, P., 621, 28; 622, 218; 624,11 Velasco, J.L., 628, 281 Ushida, N., 626,24 Vempati, S.K., 622, 112 Ushiroda, Y., 622, 218; 624,11 Venanzoni, G., 626,15 Usik, A., 621,56 Veneziano, G., 625, 177 Utkin, D., 621,56 Veneziano, S., 626,15 Uvarov, L., 622, 265; 626, 35, 45, 55 Ventura, A., 626,15 Uvarov, L.N., 628,18 Venturi, G., 625,1 Uvarov, S., 622, 265; 626, 35, 45, 55 Verdier, P., 622, 265; 626, 35, 45, 55 Uzunyan, S., 622, 265; 626, 35, 45, 55 Verebryusov, V.S., 628,18 Vermaseren, J.A.M., 625, 245 Vaandering, E.W., 621, 72; 622, 229, 239; 624, 22, 166 Vernov, S.Yu., 628,1 Vachon, B., 622, 265; 626, 35, 45, 55 Versaci, R., 626,15 Vagenas, E.C., 628, 197 Vertogradov, L.S., 622, 265; 626, 35, 45, 55 Valente, E., 622, 249; 623,26 Verzocchi, M., 622, 265; 626, 35, 45, 55 Valente, P., 626,15 Vest, A., 621,56 Valeriani, B., 626,15 Veszpremi, V., 622, 249; 623,26 Valkár, S., 621,56 Vesztergombi, G., 622, 249; 623,26 Valkárová, A., 621,56 Vetlitsky, I., 622, 249; 623,26 Vallée, C., 621,56 Victorov, V.A., 628,18 Van Acoleyen, K., 622,1 Vient, E., 627,18 Van Dantzig, R., 626,24 Viertel, G., 622, 249; 623,26 Van den Berg, P.J., 622, 265; 626, 35, 45, 55 Vigilante, M., 627,18 Vandenbroucke, A., 622,14 Vik, T., 623,17 Van der Nat, P.B., 622,14 Vikhrov, V., 622,14 Van der Steenhoven, G., 622,14 Vilain, P., 626,24 Van de Ven, P., 623,17 Vilasi, G., 625, 171 Van de Vyver, B., 626,24 Villa, S., 621, 28, 41; 622, 218, 249; 623, 26; 624,11 Van de Vyver, R., 624, 173 Villalobos Baillie, O., 623,17 Vande Vyvre, P., 623,17 Villante, F.L., 627,38 Van de Walle, R.T., 622, 249; 623,26 Villeneuve-Seguier, F., 622, 265; 626, 35, 45, 55 Van Haarlem, Y., 622,14 Vincter, M.G., 622,14 Cumulative author index to volumes 621–628 (2005) 292–321 319

Vinogradov, L., 623,17 Wang, X.L., 622, 249; 623,26 Vinokurova, S., 621,56 Wang, Y.F., 622,6;625, 196 Viren, B., 623, 192 Wang, Z., 622,6,6,6;625, 196, 196, 196 Virgili, T., 623,17 Wang, Z.M., 622, 249; 623,26 Vishnyakov, V.E., 628,18 Wang, Z.Y., 622,6;625, 196 Vitulo, P., 621, 72; 622, 229, 239; 624, 22, 166 Warchol, J., 622, 265; 626, 35, 45, 55 Vivargent, M., 622, 249; 623,26 Watanabe, M., 621, 41; 622, 218; 624,11 Vlachos, S., 622, 249; 623,26 Waterson, T., 622, 165 Vlimant, J.-R., 622, 265; 626, 35, 45, 55 Watt, G., 627,97 Vodopianov, I., 622, 249; 623,26 Watts, D., 626,65 Vogel, C., 622,14 Watts, G., 622, 265; 626, 35, 45, 55 Vogel, H., 622, 249; 623,26 Wayne, M., 622, 265; 626, 35, 45, 55 Vogt, A., 625, 245 Waysand, G., 621, 233 Vogt, H., 622, 249; 623,26 Weber, G., 621,56 Volant, C., 627,18 Weber, M., 622, 249, 265; 623, 26; 626, 35, 45, 55 Volchinski, V., 621,56 Weber, R., 621,56 Volmer, J., 622,14 Weber, T.A., 621,96 Von Gersdorff, G., 624, 270 Webster, M., 621, 72; 622, 229, 239; 624, 22, 166 Von Harrach, D., 624, 173 Weerts, H., 622, 265; 626, 35, 45, 55 Von Toerne, E., 622, 265; 626, 35, 45, 55 Wegener, D., 621,56 Vorobiev, I., 622, 249; 623,26 Wei, C.L., 622,6;625, 196 Vorobyov, A.A., 622, 249; 623, 26; 628,18 Wei, D.H., 622,6;625, 196 Vorwalter, K., 628,18 Weick, H., 622,29 Votruba, M.F., 623,17 Weihofen, W., 624, 173 Vreeswijk, M., 622, 265; 626, 35, 45, 55 Weiler, T.J., 621,18 Vrláková, J., 623,17 Wendland, J., 622,14 Vu Anh, T., 622, 265; 626, 35, 45, 55 Wenger, U., 624, 324 Vujicic, B., 621,56 Wenheng, Z., 628,18 Wereszczynski,´ A., 621, 201 Wacker, K., 621,56 Wermelinger, C., 625,19 Wadhwa, M., 622, 249; 623,26 Wermes, N., 622, 265; 626, 35, 45, 55 Wadsworth, R., 625, 203 Werner, C., 621,56 Wagner, J., 621,56 Werner, N., 621,56 Wahl, H.D., 622, 265; 626, 35, 45, 55 Wessels, M., 621,56 Wakai, A., 624, 173 Wessling, B., 621,56 Walker, J.W., 625,96 Wetterich, C., 628, 189 Wambach, J., 624, 195 Wetzorke, I., 624, 324, 334 Wang, B., 624, 141 Weyers, J., 628, 211 Wang, C.C., 621, 28, 41; 622, 218; 624,11 Wheadon, R., 622,35 Wang, C.H., 621, 28, 41; 622, 218; 624,11 Wheldon, C., 622,29 Wang, D.Y., 622,6;625, 196 White, A., 622, 265; 626, 35, 45, 55 Wang, G.-L., 623, 218 White, V., 622, 265; 626, 35, 45, 55 Wang, J.Z., 622,6;625, 196 Whiteson, D., 626, 45, 55 Wang, K., 622,6;625, 196 Wichoski, U., 624, 186 Wang, L., 622, 6, 265; 624, 39; 625, 196; 626, 35, 45, 55 Wicke, D., 622, 265; 626, 35, 45, 55 Wang, L.S., 622,6;625, 196 Wieland, O., 622,29 Wang, M., 621, 72; 622, 6, 229, 239; 624, 22, 166; 625, 196 Wieleczko, J.P., 627,18 Wang, M.-Z., 621,28 Wigmore, C., 621,56 Wang, P., 622,6;625, 196; 626,95 Wijngaarden, D.A., 622, 265; 626, 35, 45, 55 Wang, P.L., 622,6;625, 196 Wilcox, W., 627,71 Wang, Q., 622, 249; 623,26 Williams, A.G., 628, 176 Wang, S., 622,14 Williams, R.A., 621, 269 Wang, S.Z., 622,6;625, 196 Wilquet, G., 626,24 Wang, W.F., 622,6;625, 196 Wilson, A.N., 625, 203 Wang, W.Y., 628, 228 Wilson, G.W., 622, 265; 626, 35, 45, 55 320 Cumulative author index to volumes 621–628 (2005) 292–321

Wilson, J.R., 621, 72; 622, 229, 239; 624, 22, 166 Yan, M., 622, 265; 626, 35, 45, 55 Wimpenny, S.J., 622, 265; 626, 35, 45, 55 Yan, M.L., 622,6;625, 196 Winger, J.A., 622,45 Yan, W., 621,56 Winkler, M., 622,29 Yanagida, T., 624, 162 Winter, G.-G., 621,56 Yanagisawa, Y., 621,81 Winter, K., 626,24 Yang, B.Z., 622, 249; 623,26 Wirschins, M., 627, 180 Yang, C.G., 622, 249; 623,26 Wiss, J., 621, 72; 622, 229, 239; 624, 22, 166 Yang, F., 622,6;625, 196 Wissing, Ch., 621,56 Yang, H., 621, 41; 622, 218; 624,11 Wissmann, F., 624, 173 Yang, H.J., 622, 249; 623,26 Wittlin, J., 622, 265; 626, 35, 45, 55 Yang, H.X., 622,6;625, 196 Wobisch, M., 622, 265; 626, 35, 45, 55 Yang, J., 622,6;625, 196 Wolf, R., 621,56 Yang, J.-F., 625, 357 Wollersheim, H.J., 622,29 Yang, M., 622, 249; 623,26 Womersley, J., 622, 265; 626, 35, 45, 55 Yang, S., 624,39 Wood, D.R., 622, 265; 626, 35, 45, 55 Yang, Y.X., 622,6;625, 196 Wotzasek, C., 625, 351, 365 Yasuda, T., 622, 265; 626, 35, 45, 55 Wu, C., 621,81 Yasuè, M., 621, 133 Wu, N., 622,6;625, 196 Yasui, Y., 621, 288 Wu, Y.L., 628, 228 Yatsunenko, Y.A., 622, 265; 626, 35, 45, 55 Wu, Y.M., 622,6;625, 196 Ye, M., 622,6;625, 196 Wünsch, E., 621,56 Ye, M.H., 622,6;625, 196 Wyatt, T.R., 622, 265; 626, 35, 45, 55 Ye, Y., 622,14 Wynhoff, S., 622, 249; 623,26 Ye, Y.X., 622,6;625, 196 Ye, Z., 622,14 Xella, S., 621,56 Yeganov, V., 621,56 Xia, L., 622, 249; 623,26 Yeh, S.C., 622, 249; 623,26 Xia, X.M., 622,6;625, 196 Yen, S., 622,14 Xie, Q.L., 621, 28, 41; 622, 218; 624,11 Yen, Y., 622, 265; 626, 35, 45, 55 Xie, X.X., 622,6;625, 196 Yershov, N.V., 623, 192 Xin, B., 622,6;625, 196 Yi, L.H., 622,6;625, 196 Xing, Z.-Z., 623, 227 Yi, Z.Y., 622,6;625, 196 χ F Collaboration, 624, 334 L Yigitoglu, I., 621, 102 Xu, F.R., 622, 151; 625, 375 Ying, J., 621, 28, 41; 622, 218; 624,11 Xu, G., 626,15 Yip, K., 622, 265; 626, 35, 45, 55 Xu, G.F., 622,6;625, 196 Yoo, H.D., 622, 265; 626, 35, 45, 55 Xu, H., 622,6;625, 196 Xu, Q., 622, 265; 626, 35, 45, 55 Yoon, C.S., 626,24 Xu, X.M., 628,25 Yoshida, A., 621,81 Xu, Z.Z., 622, 249; 623,26 Yoshida, K., 621, 81; 624, 115; 627, 188 Xuan, N., 622, 265; 626, 35, 45, 55 Yoshimura, Y., 623, 192 Xue, S.T., 622,6;625, 196 Yoshioka, T., 623, 192 Yotsuyanagi, A., 625,79 Yabsley, B.D., 621, 28, 41; 622, 218; 624,11 You, J., 628,18 Yacoob, S., 622, 265; 626, 35, 45, 55 Youn, S.W., 622, 265; 626, 35, 45, 55 Yager, P.M., 621, 72; 622, 229, 239; 624, 22, 166 Young, R.D., 628, 125 Yakhshiev, U.T., 628,33 Yu, C., 621, 259 Yamada, R., 622, 265; 626, 35, 45, 55 Yu, C.S., 622,6;625, 196 Yamada, Y., 623, 104 Yu, G.W., 622,6;625, 196 Yamaguchi, A., 621, 28, 41; 622, 218; 624,11 Yu, J., 622, 265; 626, 35, 45, 55 Yamaguchi, Y., 621,81 Yuan, C.Z., 622,6;625, 196; 626,95 Yamamoto, H., 621, 28; 624,11 Yuan, F., 625, 253 Yamamoto, J., 622, 249; 623,26 Yuan, J.M., 622,6;625, 196 Yamashita, Y., 621, 28, 41; 622, 218; 624,11 Yuan, Y., 622,6;625, 196 Yamauchi, M., 621, 28, 41; 622, 218; 624,11 Yue, C.-X., 624,39 Cumulative author index to volumes 621–628 (2005) 292–321 321

Yunshan, L., 628,18 Zhao, Y.B., 622,6;625, 196 Yurkewicz, A., 622, 265; 626, 35, 45, 55 Zhao, Z., 622, 265; 626, 35, 45, 55 Zhelezov, A., 621,56 Zabi, A., 622, 265; 626, 35, 45, 55 Zheng, H.Q., 622,6;625, 196 Žácek,ˇ J., 621,56 Zheng, J.P., 622,6;625, 196 Zacek, V., 624, 186 Zheng, L.S., 622,6;625, 196 Zálešák, J., 621,56 Zheng, Z.P., 622,6;625, 196 Zalite, An., 622, 249; 623,26 Zhenlin, M., 628,18 Zalite, Yu., 622, 249; 623,26 Zhilich, V., 621, 41; 622, 218 Zallo, A., 621, 72; 622, 229, 239; 624, 22, 166 Zhokin, A., 621,56 Zamick, L., 621, 239 Zhong, M., 628, 228 Zang, S.L., 622,6;625, 196 Zhong, X.C., 622,6;625, 196 Zanotti, J.M., 627, 113 Zhou, B., 622, 265; 626, 35, 45, 55 Zapadtka, F., 624, 173 Zhou, B.Q., 622,6;625, 196 Zatserklyaniy, A., 622, 265; 626, 35, 45, 55 Zhou, G.M., 622,6;625, 196 Závada, P., 623,17 Zhou, L., 622,6;625, 196; 627,71 Zaytsev, S.A., 621,96 Zhou, N.F., 622,6;625, 196 Zdrazil, M., 622, 265; 626, 35, 45, 55 Zhu, G.Y., 622, 249; 623,26 Zeitler, G., 624, 173 Zhu, J., 622, 265; 626, 35, 45, 55 Zeitnitz, C., 622, 265; 626, 35, 45, 55 Zhu, K.J., 622,6;625, 196 Zeng, Y., 622,6,6;625, 196, 196 Zhu, Q.M., 622,6;625, 196 Zenoni, A., 622,35 Zhu, R.Y., 622, 249; 623,26 Zerbini, S., 624,70 Zhu, S., 622, 151; 625, 375 Zhalov, M., 626,72 Zhu, S.-L., 625, 212 Zhang, B.X., 622,6;625, 196 Zhu, Y., 622,6;625, 196 Zhang, B.Y., 622,6;625, 196 Zhu, Y.C., 622,6;625, 196 Zhang, C.C., 622,6;625, 196 Zhu, Y.S., 622,6;625, 196 Zhang, D., 622, 265; 626, 35, 45, 55 Zhu, Z.A., 622,6;625, 196 Zhang, D.H., 622,6;625, 196 Zhuang, B.A., 622,6;625, 196 Zhang, H.Y., 622,6;625, 196 Zhuang, H.L., 622, 249; 623,26 Zhang, J., 621, 28; 622, 6, 218; 624, 11; 625, 196 Zhuang, X.A., 622,6;625, 196 Zhang, J.W., 622,6;625, 196 Zichichi, A., 622, 249; 623,26 Zhang, J.Y., 622,6;625, 196 Zielinski, M., 622, 265; 626, 35, 45, 55 Zhang, L.M., 621, 28, 41; 622, 218; 624,11 Zieminska, D., 622, 265; 626, 35, 45, 55 Zhang, Q.J., 622,6;625, 196 Zieminski, A., 622, 265; 626, 35, 45, 55 Zhang, S.Q., 622,6;625, 196 Zihlmann, B., 622,14 Zhang, X., 622, 265; 626, 35, 45, 55 Zimdahl, W., 628, 206 Zhang, X.M., 622,6;625, 196 Zimmermann, B., 622, 249; 623,26 Zhang, X.Y., 622,6;625, 196 Zimmermann, J., 621,56 Zhang, Y., 621, 72; 622, 6, 229, 239; 624, 22, 166; 625, 196 Zitoun, R., 622, 265; 626, 35, 45, 55 Zhang, Y.Y., 622,6;625, 196 Zoakos, D., 625, 135 Zhang, Z., 621,56 Zohrabyan, H., 621,56 Zhang, Z.P., 621, 28, 41; 622, 6, 218, 249; 623, 26; 624, 11; 625, 196 Zöller, M., 622, 249; 623,26 Zhang, Z.Q., 622,6;625, 196 Zomer, F., 621,56 Zhang, Z.Y., 624, 210 Žontar, D., 621, 28, 41; 622, 218; 624,11 Zhao, D.X., 622,6;625, 196 Zou, B.S., 622,6;625, 196 Zhao, J., 622, 249; 623,26 Zucchelli, P., 626,24 Zhao, J.B., 622,6;625, 196 Zukanovich-Funchal, R., 628,18 Zhao, J.W., 622,6;625, 196 Zupnik, B.M., 627, 208 Zhao, M.G., 622,6;625, 196 Zupranski, P., 622,14 Zhao, P.P., 622,6;625, 196 Zürcher, D., 621,41 Zhao, T., 622, 265; 626, 35, 45, 55 Zutshi, V., 622, 265; 626, 35, 45, 55 Zhao, W.R., 622,6;625, 196 Zverev, E.G., 622, 265; 626, 35, 45, 55 Zhao, X.J., 622,6;625, 196 Zwicky, R., 625, 225