Nature of the a 1 meson in lattice quantum chromodynamics studied with chiral fermions
Yuko Murakami, Shin Muroya, Atsushi Nakamura, Chiho Nonaka, Motoo Sekiguchi, Hiroaki Wada, Masayuki Wakayama
aa r X i v : . [ h e p - l a t ] D ec APS/123-QED
Nature of the a meson in lattice quantum chromodynamics studied with chiralfermions Yuko Murakami, Shin Muroya, Atsushi Nakamura,
3, 4, 5
Chiho Nonaka,
6, 7
Motoo Sekiguchi, Hiroaki Wada, and Masayuki Wakayama (SCALAR Collaboration) Research and Development Laboratory, Seikow Chemical Engineering & Machinery, LTD, Akashi 674-0093, Japan Matsumoto University, Matsumoto 390-1295, Japan School of Biomedicine, Far Eastern Federal University, 690950 Vladivostok, Russia Theoretical Research Division, Nishina Center, RIKEN, Wako 351-0198, Japan Research Center for Nuclear Physics (RCNP), Osaka University, Ibaraki, Osaka 567-0047, Japan Department of Physics, Nagoya University, Nagoya 464-8602, Japan Kobayashi Maskawa Institute, Nagoya University, Nagoya 464-8602, Japan School of Science and Engineering, Kokushikan University, Tokyo 154-8515, Japan (Dated: December 20, 2018)We study the a meson using a quenched lattice quantum chromodynamics simulation with thetruncated overlap fermions formalism based on the domain wall fermions. The obtained lightestmass of the a meson, 1272(45) MeV, is consistent with the experimental value for a (1260). Thus, a (1260) can be identified to have a simple two-body constituent-quark structure. Our quenchedsimulation result of a (1420) can not explain the experimental mass value, which suggests a (1420)is not a simple q ¯ q two quark state. PACS numbers: 11.15.Ha 12.38.Gc 14.40.Be
Introduction
In hadron spectroscopy, the fundamental ingredientsare light-meson sector, whose understanding both fromthe theoretical and experimental aspects are indispens-able. And yet, the classification of light axial-vectormesons ( a ) is a long-standing issue in the meson spec-troscopy. Recently, the resonance of a (1260) was ob-served clearly in the COMPASS experiment at CERN .Moreover, a new a meson was discovered in the f π channel with a mass of 1420 MeV and a narrow widthby the COMPASS collaboration . Currently the particledata group lists three a mesons : a (1260), a (1420),and a (1640); however, this is a richer spectrum thanthat in the usual q ¯ q mesons in a constituent quarkmodel. In the conventional constituent quark model ,the a (1260) meson is assigned to a I = 1, P state. If a (1260) is the ground state for a meson, the mass of thenext radial excitation becomes at least 1.7 GeV . Thissuggests that the radial excitation of a (1260) can notbe a (1420), but a (1640). Consequently, the structureof a (1420) cannot be understood as a simple two-quarkstate. It is a possible candidate for the exotic multi-quarkstate or the dynamical effect due to a singularity in thetriangle diagram .There have been several interpretations for the struc-ture of the a (1260) meson: i) in the Nambu-Jona-Lasinio model , the a (1260) meson is the chiral partnerof the ρ meson as q ¯ q state, ii) it could also be interpretedas the gauge boson of the hidden local symmetry ,iii) in the coupled-channel approaches based on chiraleffective theory , it is described as the dynamically gen-erated resonance in πρ scattering, and iv) Nagahiro et al. discussed the mixing properties of a (1260) of the quarkcomposite state and the hadronic composite state . In this report, we present the structure of the lightest a meson determined with lattice quantum chromody-namics (QCD), a first-principles approach. Our objectiveis to clarify the relation between the nature of the a me-son and the chiral symmetry associated with the chiralpartner of the ρ meson and dynamical chiral symmetrybreaking, as is the case for π and the chiral partner ofthe σ meson . We, therefore, employ the truncated over-lap fermion formalism by Bori¸ci based on domain wallfermions formalism , which holds good chiral symme-try. The truncated overlap fermion formalism is classifiedinto lattice chiral fermions .In the previous work , we investigated the σ mesonbased on the full QCD with dynamical Wilson quarksthat has an explicit chiral symmetry breaking term, us-ing q ¯ q interpolating operators. Our work indicates theexistence of the light σ state, whose mass is in m π 08. Open circles, triangles, and diamonds rep-resent the propagators of π meson, ρ meson, and a meson,respectively. Their obtained mass of the ground state a meson is closeto a (1420), instead of a (1260), i.e., these simulationsare inconsistent.Here, we perform quenched simulations for the a meson using the truncated overlap fermion formalismwith q ¯ q interpolating operators. We will show that thelightest a meson is the q ¯ q state composed of u and d quarks. Lattice simulation We perform quenched lattice QCD calculations usingtruncated overlap fermions with the plaquette gaugeaction. We use point sources and sinks when calculatinghadron propagators, which leads to larger masses on arelatively small lattice because of a mixture of highermass states. The masses obtained in our simulationshould thus be considered as the upper limits. The a meson propagator is more noisy than those of π and ρ mesons, and therefore more statistics are required. Sincetruncated overlap fermions are a variant of domain wallfermions, we use the same simulation parameters as thoseused by Blum et al. , except for the temporal lattice size( N t = 24 is here used, instead of N t = 32.): β = 5 . 7, thelength of the fifth dimension N = 32 for which m π isstable, the five-dimensional mass m = 1 . 65, and thethree-dimensional spatial lattice size N s = 8 .We adopt the following interpolating operator for cre-ating the a meson with I = 1 and J P C = 1 ++ , O a = ¯ qγ µ γ q , (1)where q denotes the u or d quark operator. We generategauge configurations based on the plaquette gauge ac- FIG. 2. (color online). Time dependence of the effectivemasses at m f a = 0 . 08. Open circles, triangles, and diamondsrepresent the propagators of π meson, ρ meson, and a meson,respectively. tion by using the pseudo heat-bath method. After 20000thermalization iterations, we start to save gauge config-urations every 1000 sweeps. We calculate meson propa-gators on the stored gauge configurations for each of thequark mass values, m f a = 0 . 08, 0.06, and 0.04, where a is the lattice spacing. We use 3000 (7964) configura-tions for the calculation of the meson propagators with m f a = 0 . 08 and 0.06 ( m f a = 0 . π , ρ , and a mesons for m f a = 0 . m eff a , of thesemesons are displayed in Fig. 2, which are determined as G ( t ) G ( t + 1) = e − m eff ( t ) t + e − m eff ( t )( T − t ) e − m eff ( t )( t +1) + e − m eff ( t )( T − ( t +1)) , (2)where G ( t ) represents the propagators of the mesons. Weestimate the statistical errors using the jackknife method.Thanks to the large enough statistics, we obtain veryclear propagators and effective masses for the a meson.The masses of the π , ρ , and a mesons for m f a = 0 . π and ρ massesare evaluated from effective masses in the range of 6 ≤ TABLE I. Masses of π , ρ , and a mesons, mass ratios andnumbers of configurations. m f a m π a m ρ a m π /m ρ m a /m ρ N config a a Number of configurations separated by 1000 sweeps. TABLE II. Masses of π and ρ mesons, mass ratio and num-ber of configurations reported by Blum et al. . Simulationparameters are β = 5 . 7, 8 × N = 32, and m = 1 . m f a m π a m ρ a m π /m ρ N config t/a ≤ 9. The a mass, on the other hand, is obtained inthe range of 5 ≤ t/a ≤ 8, because the effective masses of a suffer from large errors at large t .In Table I, the results for meson masses and mass ratiosare summarized, while those of Blum et al. are shownin Table II. The masses of π and ρ mesons obtained inour simulation on a small lattice show good agreementwith those on a large lattice (8 × m ρ a and m a a vary linearly with ( m π a ) .In the chiral limit, ( m π a ) = 0. Using m ρ = 775 MeVas the input, we obtain a = 0 . m f a → m f a → − m res a is negligible due to the smallness of m res a = 1 . × − , where m res is the residual mass.Therefore, we apply m f a → 0. We estimate the massratio m a /m ρ to be 1.64(6) and the mass of the a mesonto be m a = 1272(45) MeV. Our result is consistent withthe experimental value of 1230(40) MeV . FIG. 3. (color online). Dependences of ρ meson masses m ρ a (open triangles) and a meson masses m a a (open diamonds)on ( m π a ) . Lines for m ρ a and m a a show linear fits. Starsrepresent the experimental values of a (1260) and a (1420) . Conclusion and Discussion We studied the lightest a meson based on a quenchedapproximation using truncated overlap fermions. We es- timated the mass of the a meson to be 1272(45) MeV,which is in good agreement with the experimental valuefor a (1260) . The masses obtained in our simulationshould be considered as the upper limits. Our results areconsistent with those of Wingate et al. who employed afull QCD simulation without chiral symmetry. Our simu-lation used truncated overlap fermions, and thus respectschiral symmetry, but in the quench approximation.Gattringer et al. determined the mass of the a me-son using the chirally improved Dirac operator in thequenched approximation with the L¨uscher-Weisz gaugeaction . The ground state of a meson in their calcula-tion is close to a (1420). Possible reason for the differencebetween our result and theirs is the difference of statis-tics: our statistics are 30 or 80 times as large as theirs.We succeeded in obtaining the lowest state of a meson,in spite of utilizing a simple two-quark interpolator.Our lattice study and quark model analysis supportthat the simple two-body constituent-quark structure of a (1260) is consistent with the experimentally observed a (1260). Our a meson does not agree with a (1420).A quench simulation is a clean theoretical experiment inwhich virtual intermediate states such as qq ¯ q ¯ q are highlysuppressed. Therefore, a (1420) may contain an uncon-ventional state, such as qq ¯ q ¯ q . In the qq ¯ q ¯ q case, dynamicalquarks may play an essential role. Note that there havebeen arguments to consider a (1420) as a dynamical ef-fect of the triangle diagram . Also, a (1640) mightbe a radial excitation of a (1260), according to the quarkmodel analysis. We leave it to the future task to complete a meson spectroscopy with the lattice QCD simulation. acknowledgments This work could not be completed without valuable ad-vices by T. Kunihiro. This work was completed with thesupport of RSF grant 15-12-20008.It was also supported in part by the JSPS KAKENHIGrant-in-Aid for Scientific Research (S) Grant NumberJP26220707, the JSPS KAKENHI Grant-in-Aid for Sci-entific Research (C) Grant Number JP17K05438, Re-search Activity of Matsumoto University (No.14111048,No.16111048), and Scientific Research (Kakenhi) Num-bers 24340054 and 26610072. The simulation was per-formed on an NEC SX-ACE supercomputer at RCNPand the Cybermedia Center, Osaka University, and wasconducted using the Fujitsu PRIMEHPC FX10 System(Oakleaf-FX, Oakbridge-FX) at the Information Tech-nology Center, The University of Tokyo. This work wassupported by “Joint Usage/Research Center for Inter-disciplinary Large-scale Information Infrastructures” inJapan (Project ID: EX17706 and jh180053-NAJ). M. Alekseev et al. [COMPASS Collaboration], “Observa-tion of a J**PC = 1-+ exotic resonance in diffractive dis- sociation of 190-GeV/c pi- into pi- pi- pi+,” Phys. 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