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Delta-Baryon Electromagnetic Form Factors in Lattice QCD

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Delta-Baryon Electromagnetic Form Factors in Lattice QCD delta-baryon electromagnetic form factors in lattice QCD The MIT Faculty has made this article openly available. Please share how this access benefits you. Your story matters. Citation Alexandrou, C. et al. "delta-baryon electromagnetic form factors in lattice QCD." Physical Review D 79.1 (2009): 014507. As Published http://dx.doi.org/10.1103/PhysRevD.79.014507 Publisher American Physical Society Version Final published version Citable link http://hdl.handle.net/1721.1/52708 Terms of Use Article is made available in accordance with the publisher's policy and may be subject to US copyright law. Please refer to the publisher's site for terms of use. PHYSICAL REVIEW D 79, 014507 (2009) Á-baryon electromagnetic form factors in lattice QCD C. Alexandrou,1 T. Korzec,1 G. Koutsou,1 Th. Leontiou,1 C. Lorce´,2 J. W. Negele,3 V. Pascalutsa,2 A. Tsapalis,4 and M. Vanderhaeghen2 1Department of Physics, University of Cyprus, P.O. Box 20537, 1678 Nicosia, Cyprus 2Institut fu¨r Kernphysik, Johannes Gutenberg-Universita¨t, D-55099 Mainz, Germany 3Center for Theoretical Physics, Laboratory for Nuclear Science and Department of Physics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA 4Institute of Accelerating Systems and Applications, University of Athens P.O. Box 17214, GR-10024, Athens, Greece (Received 22 October 2008; published 21 January 2009) We develop techniques to calculate the four Á electromagnetic form factors using lattice QCD, with particular emphasis on the subdominant electric quadrupole form factor that probes deformation of the Á. Results are presented for pion masses down to approximately 350 MeV for three cases: quenched QCD, two flavors of dynamical Wilson quarks, and three flavors of quarks described by a mixed action combining domain-wall valence quarks and dynamical staggered sea quarks. The magnetic moment of the Á is compared with chiral effective field theory calculations and the Á charge density distributions are discussed. DOI: 10.1103/PhysRevD.79.014507 PACS numbers: 11.15.Ha, 12.38.Aw, 12.38.Gc I. INTRODUCTION without the construction of an optimized source for the sequential propagator it can not be extracted with the Lattice quantum chromodynamics (QCD) provides a required precision. We note that this increases the compu- well-defined framework to directly calculate hadron form tational cost since additional inversions are needed. Our factors from the fundamental theory of strong interactions. techniques are first tested in quenched QCD [12]. We then Form factors characterize the internal structure of hadrons, calculate form factors using two degenerate flavors of including their magnetic moment, their size, and their dynamical Wilson fermions, denoted by N ¼ 2, with charge density distribution. Since the Áð1232Þ decays F pion masses in the range of 700 MeV to 380 MeV strongly, experiments [1,2] to measure its form factors [13,14]. Finally, we use a mixed action with chirally sym- are harder and yield less precise results than for nucleons metric domain-wall valence quarks and staggered sea [3,4]. In this work, we compute Á form factors using lattice quarks with two degenerate light flavors and one strange QCD more accurately than can be currently obtained from flavor [15], denoted by N ¼ 2 þ 1, at a pion mass of experiment. F 353 MeV. Using the results obtained with dynamical A primary motivation for this work is to understand the quarks, we extrapolate the magnetic moment to the physi- role of deformation in baryon structure: whether any of the cal point. We extract the quark charge distributions in the low-lying baryons have deformed intrinsic states and if so, Á, and discuss their quadrupole moment. why. Thus, a major achievement of this work is the devel- opment of lattice methods with sufficient precision to show, for the first time, that the electric quadrupole form II. LATTICE EVALUATION factor is nonzero and hence the Á has a nonvanishing The Á matrix element of the EM current jEM, quadrupole moment and an associated deformed shape. hÁðp ;s Þjj jÁðp ;sÞi, can be parametrized in terms of Unlike the Á, the spin-1=2 nucleon cannot have a quadru- f f EM i i pole moment, so the experiment of choice to explore its four multipole form factors that depend only on the mo- mentum transfer q2 ÀQ2 ¼ðp À p Þ2 [16] The de- deformation has been measurement of the nucleon to Á f i ÃÁÁ electric and Coulomb quadrupole transition form factors. composition for the on shell matrix element is Major experiments [5–7] have shown that these transition given by form factors are indeed nonzero, confirming the presence hÁðp ;s Þjj jÁðp ;sÞi ¼ Au ðp ;s ÞOu ðp ;sÞ Á f f EM i i f f i i of deformation in either the nucleon , or both [8,9], and 2 a2ðq Þ lattice QCD yields comparable nonzero results [10,11]. O 2 ¼g a1ðq Þ þ ðpf þ pi Þ Our new calculation of the Á quadrupole form factor, 2mÁ coupled with the nucleon to Á transition form factors, 2 q q 2 c2ðq Þ should in turn shed light on the deformation of the nucleon. À 2 c1ðq Þ þ ðpf þ pi Þ ; (1) 4m 2mÁ In order to evaluate the Á electromagnetic (EM) form Á 2 2 2 2 factors to the required accuracy, we isolate the two domi- where a1ðq Þ, a2ðq Þ, c1ðq Þ, and c2ðq Þ are known linear 2 nant form factors and the subdominant electric quadrupole combinations of the electric charge form factor GE0ðq Þ, 2 form factor. This is particularly crucial for the latter since the magnetic dipole form factor GM1ðq Þ, the electric 1550-7998=2009=79(1)=014507(5) 014507-1 Ó 2009 The American Physical Society C. ALEXANDROU et al. PHYSICAL REVIEW D 79, 014507 (2009) 2 quadrupole form factor GE2ðq Þ, and the magnetic octu- The connected part of each combination of three-point 2 pole form factor GM3ðq Þ [17], and A is a known factor functions can be calculated efficiently using the method depending on the normalization of hadron states. These of sequential inversions [18]. These yield the isovector form factors can be extracted from correlation functions form factors. At present, it is not yet computationally calculated in lattice QCD [17]. We calculate in Euclidean feasible to calculate the contributions arising from discon- time the two- and three-point correlation functions in a nected diagrams. A calculation of disconnected contribu- frame where the final state Á is at rest: tions in the case of the electromagnetic form factors have X X3 shown that these are consistent with zero [19]. We note that 4 Àix~fÁq~ À 0 these disconnected contributions are particularly hard to Gðt; q~Þ¼ e hJjðxfÞJjð Þi 1 (2) calculate not just because they require the all-to-all propa- x~f j¼ X gators but also because they are noise dominated [20]. We À ix~Áq~ À 0 G ð ;t;q~Þ¼ e hJ ðxfÞj ðxÞJ ð Þi; expect that, like in the case of nucleon electromagnetic x~fx~ form factors, the disconnected contributions to the electro- Á where j is the electromagnetic current on the lattice, J magnetic form factors are small. The known kinematical Á and J are the Áþ interpolating fields constructed from coefficients K1, K2, K3, K4 are functions of the mass and 4 1 4 k energy as well as of and q~. The combinations above are smeared quarks [12], À ¼ 4 ð1 þ Þ, and À ¼ À4 5 k chosen such that all possible directions of and q~ con- i . The form factors can then be extracted from 2 ratios of three- and two-point functions in which unknown tribute symmetrically to the form factors at a given Q [21]. The over-constrained system of Eqs. (4)–(6) is solved normalization constants and the leading time dependence 2 cancel by a least-squares analysis, and GE2ðQ Þ can also be iso- vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi lated separately from Eq. (6). u The details of the simulations are summarized in Table I. G ðÀ;t;q~Þ u Gðt À t; p~ ÞGðt; 0~ÞGðt ; 0~Þ R ¼ t f i f : In each case, the separation between the final and initial 0~ 0~ Gðtf; Þ Gðtf À t; ÞGðt; p~ iÞGðtf; p~ iÞ time is tf À ti * 1fmand Gaussian smearing is applied to (3) both source and sink to produce adequate plateaus by suppressing contamination from higher states having the For sufficiently large tf À t and t À ti, this ratio exhibits a quantum numbers of the Áð1232Þ. For the mixed-action plateau RðÀ;t;q~Þ!ÅðÀ; q~Þ, from which the form factors calculation, the domain-wall valence quark mass was are extracted, and we use the particular combinations chosen to reproduce the lightest pion mass obtained using 2 1 X3 NF ¼ þ improved staggered quarks [21,23]. Å À4 2 2 k kð ; q~Þ¼K1GE0ðQ ÞþK2GE2ðQ Þ; (4) k¼1 III. RESULTS 2 X3 The results for GE0ðQ Þ are shown in Fig. 1 as a function Å À4 2 2 jkl j kð ; q~Þ¼K3GM1ðQ Þ; (5) of Q at the lightest pion mass for each of the three actions. j;k;l¼1 For Wilson fermions, we use the conserved lattice current requiring no renormalization. The local current is used for 3 X the mixed action, and the renormalization constant ZV ¼ Å 4 Àj 2 jkl j kð ; q~Þ¼K4GE2ðQ Þ: (6) 1:0992ð32Þ is determined by the condition that GE0ð0Þ j;k;l¼1 equals the charge of the Á in units of e.
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