Spectroscopy of $a_1$ mesons from lattice QCD with the truncated overlap fermions

We study the ground state and next radial excitation of the $a_1$ mesons from a quenched lattice QCD simulation with the truncated overlap fermions formalism based on domain wall fermions. Our results are consistent with the experimental values for $a_1$(1260) and $a_1$(1640).


Introduction
In hadron physics, determining the properties of excited light mesons will provide important information on the chiral dynamics of quantum chromodynamics (QCD). We study the structures of a 1 mesons determined using the truncated overlap fermion (TOF) formalism by Boriçi 1 based on the domain wall fermion formalism 2,3 . The TOF formalism is classified into lattice chiral fermions [2][3][4][5] , and exhibits good chiral symmetry. Our objective is to reveal the relationship between the nature of the a 1 meson associated with the chiral partner of the ρ meson and dynamical chiral symmetry breaking, as for the π and the chiral partner of the σ meson 6 . In the conventional constituent quark model, the σ meson and the a 1 meson are assigned excited states.
Lattice simulations of a 1 mesons have been previously conducted.
Wingate et al. were the first to measure the mass of the a 1 meson using two-flavor lattice QCD 7 . Their result agrees with the experimental value for a 1 (1260). A decade later, a study of the a 1 meson was performed with the Lüscher-Weisz gauge action and the chirally improved Dirac operator in the quenched approximation 8 . Although the ground state of the a 1 meson was improved by using various interpolators including derivative quark sources in the simulations, the obtained mass of the ground state is close to a 1 (1420), instead of a 1 (1260). Moreover, the mass of the first excited state of a 1 was observed to be above 2 GeV. Recently, Prelovsek et al. presented results for the mass of the a 1 meson and its coupling constant 9 . They performed a simulation for a full QCD lattice with clover-improved Wilson quarks. This work was continued in Ref. 10 , in which they extracted the resonance mass of the ground state of the a 1 meson m a1 = 1.435(53)( +0 −109 ) GeV and the coupling g a1πρ = 1.71(39) GeV by simulating the corresponding scattering channel πρ. Their obtained value of the a 1 meson mass is higher than the experimental result of a 1 (1260) 11,12 . In our previous work, we investigated the mass of the ground state of the a 1 meson by a quenched lattice QCD using TOF 13 . Our result is consistent with the experimental value of a 1 (1260).

Truncated overlap fermions
The TOF 1 are defined by where the five dimensional projection operator P is constructed from the four-dimensional projection operators P R/L = (1 ± γ 5 )/2. The indexes x 5 and y 5 represent the fifth-dimensional lattice sites, which are defined in x 5 , y 5 ∈ [1, N 5 ]. The domain wall fermion operator D DW F 2,3 is defined by where D W F is the Wilson fermion operator, The parameters of TOF are m f and M 5 , which correspond to the bare quark mass and the height of the domain wall, respectively. The Pauli-Villars matrix D P V is given by D P V = D DW F (m f = 1). In the N 5 → ∞ limit, the lattice chiral symmetry is exactly reproduced in TOF.

Simulation setup and Lattice QCD results
In this work, we simulate the spectroscopy of a 1 mesons on 8 3 ×24 quenched lattice with the plaquette gauge action with β = 5.7. Gauge configurations are generated with the pseudo-heat-bath method. After 20000 thermalization iterations, we start to save gauge configurations every 1000 sweeps. The propagators of the π, ρ and a 1 mesons are calculated with TOF. The fermion parameters are set to N 5 = 32, M 5 = 1.65 and m f a = 0.04-0.08. We calculate the meson propagators and their effective masses (see Fig. 1). We estimate the statistical errors using the jackknife method. By performing a single-or double-pole fit to the effective masses, we obtain the meson masses as listed in Table 1.   Fig. 2 shows the quark mass dependence of the meson masses. We linearly extrapolate the meson masses to the chiral limit, (m π a) 2 = 0. By tuning the ρ meson mass in the chiral limit to m ρ = 775 MeV, we obtain a lattice spacing of a = 0.1893(15) fm. Note that calculations for TOF have a residual mass such as m f a = −m res a in the chiral limit due to the finite N 5 effect. In our calculation with N 5 = 32, we obtain m res a = 1.29(4) × 10 −2 , that is m res = 13.4(5) MeV, which is negligible. We estimate the masses of the ground state and first excited states of the a 1 mesons to be

Conclusion
We have investigated the masses of the ground and first excited states of a 1 mesons from a quenched lattice QCD with the TOF action. We have obtained the masses of the a 1 mesons to be 1158(42) MeV and 1667(202) MeV, which are in good agreement with the experimental values of a 1 (1260) and a 1 (1640). Since our simulation was performed in the quenched approximation and theqq source and sink, in which virtual intermediate states such asqqqq states are highly suppressed, our results suggest that a 1 (1260) and a 1 (1640) are the simpleqq states whereas a 1 (1420) may have a more