N* masses in the spin-1/2 and spin-3/2 sectors are computed using two
non-perturbative methods: lattice QCD and QCD sum rules. States with both
positive and negative parity are isolated via parity projection methods. The
basic pattern of the mass splittings is consistent with experiments. The mass
splitting within the same parity pair is directly linked to the chiral symmetry
breaking QCD.Comment: contribution to Baryon 2002, 5 pages, 3 figure

Lee, Frank X.

English

arXiv

N* masses in the spin-1/2 and spin-3/2 sectors are computed using two non-perturbative methods: lattice QCD and QCD sum rules. States with both positive and negative parity are isolated via parity projection methods. The basic pattern of the mass splittings is consistent with experiments. The mass splitting within the same parity pair is directly linked to the chiral symmetry breaking QCD

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arXiv:nucl-th/0208026 v1   14 Aug 2002August 14, 2002 21:8 WSPC/Trim Size: 9in x 6in for Proceedings fxleeEXCITED BARYONS AND CHIRAL SYMMETRYBREAKING OF QCDFRANK X. LEECenter for Nuclear Studies, George Washington University,Washington, DC20052, USA andJefferson Lab, 12000 Jefferson Avenue, Newport News, VA 23606, USAN* masses in the spin-1/2 and spin-3/2 sectors are computed using two non-perturbative methods: lattice QCD and QCD sum rules. Stateswith both positiveand negative parity are isolated via parity projection methods. The basic patternof the mass splittings is consistent with experiments. The mass splitting withinthe same parity pair is directly linked to the chiral symmetry breaking QCD.1. IntroductionThere is increasing experimental information on the baryon spectrum fromJLab and other accelerators (as tabulated in the particle data group 1),and the associated desire to understand it from first principles. The richstructure of the excited baryon spectrum provides a fertile ground for ex-ploring how the internal degrees of freedom in the nucleon are excited andhow QCD works in a wider context. One outstanding example is the par-ity splitting pattern in the low-lying N* spectrum. The splittings mustbe some manifestation of spontaneous chiral symmetry breaking of QCDbecause without it, QCD predicts parity doubling in the baryon spectrum.2. Lattice QCDLattice QCD plays an important role in understanding the N* spectrum.One can systematically study the spectrum sector by sector, with the abilityto dial the quark masses, and dissect the degrees of freedom most respon-sible. Given that state-of-the-art lattice QCD simulations have produceda ground-state spectrum that is very close to the observed values 2, it isimportant to extend the success beyond the ground states. There existalready a number of lattice studies of the N* spectrum 3,4,5,6,7,8, focusingmostly on the spin-1/2 sector. All established a clear splitting from the1August 14, 2002 21:8 WSPC/Trim Size: 9in x 6in for Proceedings fxlee2ground state. Here, we focus on calculating the excited baryon states inthe spin-3/2 sector. We consider the following interpolating field with thequantum numbers I(JP ) = 12(32+),χNµ = εabc(uTa Cγ5γµdb)γ5uc. (1)The interpolating fields for other members of the octet can be found byappropriate substitutions of quark fields. Despite having an explicit par-ity by construction, these interpolating fields couple to both positive andnegative parity states. A parity projection is needed to separate the two.In the large Euclidean time limit, the correlator with Dirichlet boundarycondition in the time direction and zero spatial momentum becomesGµν(t) =∑x< 0|χµ(x) ¯χν(0)|0 > (2)= fµν[λ2+γ4 + 12e−M+ t + λ2−−γ4 + 12e−M− t](3)where fµν is a function common to both terms. The relative sign in frontof γ4 provides the solution: by taking the trace of Gµν(t) with (1 ± γ4)/4,one can isolate M+ and M−, respectively.It is well-known that a spin-3/2 interpolating field couples to both spin-3/2 and spin-1/2 states. A spin projection can be used to isolate the in-dividual contributions in the correlation function Gµν9. Numerical testof spin projection in the spin-3/2+ channel reveals two different exponen-tials in G(t) from the spin-3/2+ and spin-1/2+ parts, with the spin-3/2+state heavier than the spin-1/2+ one, which is consistent with experiment.One would mistake the dominant spin-1/2+ state for the spin-3/2+ statewithout spin projection.Figure 1 presents preliminary results for mass ratios extracted fromthe correlation functions for the 3/2+ N* states to the nucleon groundstate as a function of (π/ρ)2. Mass ratios have minimal dependence on theuncertainties in determining the scale and the quark masses, so that a morereliable comparison with experiment can be made. These ratios appearheaded in the right direction compared to experiment where available, butmore study is needed to address the systematics. Figure 2 shows the similarplots for the 3/2− N* states.3. QCD sum rulesThe QCD Sum Rule method 10 is a time-honored method that has provenuseful in revealing a connection between hadron phenomenology and theAugust 14, 2002 21:8 WSPC/Trim Size: 9in x 6in for Proceedings fxlee3Figure 1. Mass ratios for the 3/2+ N* states as compared to experimental values whereavailable.Figure 2. Similar to Figure 1, but for the 3/2− N* states.non-perturbative nature of the QCD vacuum via only a few parameters(the vacuum condensates). It has been successfully applied to a variety ofobservables in hadron phenomenology, providing valuable insights from aunique, QCD-based perspective, and continues an active field. The methodis analytical (no path integrals!), is physically transparent (one can traceAugust 14, 2002 21:8 WSPC/Trim Size: 9in x 6in for Proceedings fxlee4back to the QCD operators responsible), and has minimal model depen-dence. The accuracy of the approach is limited due to limitations inherentin the operator-product-expansion (OPE), but well understood.One progress in this area is the use of Monte Carlo-based analysis toexplore the predictive ability of the method for N* properties 11,12,13. Theidea is to probe the entire QCD parameter space and map the error dis-tribution on the OPE side to the phenomenological side. It is found thatsome QCD sum rules are truly predictive for N* masses, while others aremarginal.Another progress is that a parity separation similar to that in latticeQCD can be performed in the QCD sum rule approach, resulting in theso-called parity-projected sum rules 14,15,16 which has the general structureA(M, w+) + B(M, w+) = λ̃2+ exp(−m2+M2), (4)A(M, w−) − B(M, w−) = λ̃2−exp(−m2−M2), (5)where M is the Borel mass parameter, (mB , λ̃2, w) are the phenomenologi-cal parameters (mass, coupling, threshold). The term B controls the masssplitting: if B = 0, then m+ = m−. Term B involves only dimension-oddcondensates, such as the quark condensate 〈qq〉 and the mixed condensate〈qgσ · Gq〉. So a direct link is established between the mass splitting ofparity pairs and dynamical chiral symmetry breaking of QCD. Fig. 3 showsa numerical confirmation in the case of nucleon. As 〈qq〉 is decreased, bothmasses decrease, but with a different rate. N 12−falls faster than N 12+. Asa result, the mass splitting decreases. In the limit that chiral-symmetry isrestored (〈qq〉=0), it is expected that N 12+ and N 12−become degenerate.In conclusion, we can compute the baryon spectrum in the spin-1/2and spin-3/2 sectors for all particle channels using two methods: latticeQCD and QCD sum rules. Parity projection further reveals that the masssplitting within the same baryon pair is directly controlled by spontaneouschiral symmetry breaking of QCD.AcknowledgmentsThis work is supported in part by U.S. Department of Energy under grantDE-FG02-95ER40907.ReferencesAugust 14, 2002 21:8 WSPC/Trim Size: 9in x 6in for Proceedings fxlee5Figure 3. Mass splitting betweenN 12−and N 12+as a function of the quark condensate.The ratio is relative to the standard value ofa = −(2π)2 〈q̄q〉 = 0.52 GeV3.1. Particle Data Group, Eur. Phys. J. C 15, 1 (2000).2. S. Aoki, et al., Phys. Rev. Lett. 84, 238 (2000).3. D.B. Leinweber, Phys. Rev. D 51, 6383 (1995).4. F.X. Lee, D.B. Leinweber, Nucl. Phys. B (Proc. Suppl.) 73, 258 (1999),5. F.X. Lee, Nucl. Phys. B (Proc. Suppl.) 94, 251 (2001).6. S. Sasaki, Nucl. Phys. B (Proc. Suppl.) 83, 206 (2000); hep-ph/0004252;T. Blum, S. Sasaki, hep-lat/0002019; S. Sasaki, T. Blum, S. Ohta, hep-lat/0102010.7. D. Richards, Nucl. Phys. B (Proc. Suppl.) 94, 269 (2001)8. M. Grökeler, R. Horseley, D. Pleiter, P.E.L. Rakow, G. Schierholz, C.M.Maynard, D.G. Richards, hep-lat/0106022.9. F.X. Lee, D. Leinweber, L. Zhou, J. Zanotti, S. Choe Nucl. Phys. B (Proc.Suppl.) 106, 248 (2002).10. M.A. Shifman, A.I. Vainshtein and Z.I. Zakharov, Nucl. Phys. B147, 385,448 (1979).11. D.B. Leinweber, Ann. of Phys. (N.Y.) 254, 328 (1997).12. F.X. Lee, Phys. Rev. D57, 1801 (1998); Phys. Rev. C57, 322 (1998)13. F.X. Lee, X. Liu, Phys. Rev. D66, 014014 (2002).14. D. Jido, N. Kodama, and M. Oka, Phys. Rev. D54, 4532 (1996).15. D. Jido and M. Oka, hep-ph/9611322.16. F.X. Lee, X. Liu, to be submitted to Phys. Rev. D.

Excited Baryons and Chiral Symmetry Breaking of QCD

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