Mass spectroscopy of excited light mesons using truncated overlap fermions

We study excited light mesons by quenched lattice quantum chromodynamics (QCD) simulations with a truncated overlap fermion formalism based on domain wall fermions. Truncated overlap fermions satisfy lattice chiral symmetry instead of chiral symmetry in continuum field theory, as for domain wall fermions, but offer lower simulation costs. Our results show good agreement with the experimental values for the excited state of a 1, ρ, and π mesons, and demonstrate that a 1(1260) and a 1(1640) are simple two-quark states, whereas a 1(1420) may have a more complicated structure. The results are similar to those of previous dynamical studies using clover-Wilson fermions or chirally improved fermions, even though our lattice QCD calculations are performed with the quenched approximation. The study shows that lattice QCD simulations using truncated overlap fermions are essential in lattice studies of excited states.


Introduction
The spectra of mesons and baryons show a high degree of regularity. The organizational principle that best categorizes this regularity is encoded in the quark model. The quark model assumes that mesons are bounds state of quark (q)-antiquark (q) pairs. The lightest mesons are built from u and d quarks, andqq spins are coupled together to give a certain total spin S, which is then coupled to the angular momentum L to obtain the total observed angular momentum J. The suggested spectroscopic notation for meson states is shown in table 1. Recently, a large number of light and heavy mesons have been experimentally observed, and the variety is much richer than that predicted by the quark model. Reconstructing hadron mass spectra from first principles is an important aspect of lattice quantum chromodynamics (QCD). However, analyzing excited mesons using lattice QCD has been challenging because the masses of the excited states have to be extracted from sub-leading exponentials based on spectral decomposition data using the correlation functions for meson states. The early work on excited mesons in lattice QCD has been reviewed by Fodor and Hoelbling [2].
Recently, the restoration of chiral symmetry in excited hadrons has been discussed [3][4][5][6] and studies into this phenomenon suggest that it is important to use chiral fermions in lattice studies of excited states. The Ginsparg-Wilson (GW) relation [7], aD D 2 , 1 5 5 5 where D, γ 5 , and a are the fermion matrix, the gamma matrix, and the lattice spacing, respectively, is equivalent to the lattice version of chiral symmetry transformation [8]. Lattice fermions that satisfy the GW relation are called lattice chiral fermions. Presently, there is only one explicit formulation for lattice chiral fermions, for which the overlap fermion operator strictly satisfies the GW relation [9,10]. Lattice QCD simulations of overlap fermions are much more expensive than for Wilson-like fermions [11]. Therefore, we consider lattice chiral fermions, for which numerical calculations are realistic. The chirally improved (CI) fermion operator gives an approximate solution to the GW relation for fermions obeying chiral symmetry in a lattice [12,13]. The general Dirac operator can be expanded on the basis of operators on the lattice, and the expansion coefficients can be obtained by solving the GW equation for the operators. The CI fermionic operators can control deviations from the GW relation. However, this approach is numerically a necessary-conditional formulation of the GW relational formula, and satisfying the sufficient conditions for the GW relational formula has not been considered.
The domain wall fermion (DWF) operator [14,15] is an extension of the four-dimensional Wilson fermion operator to five dimensions N 5 . If we take N 5 to be infinite, the DWF operator is consistent with the overlap fermion operator. The overlap fermion operator is described as a sign function with a kernel of the Dirac operator, so it is not realistic for determining all eigenmodes in numerical simulations. Lattice QCD simulations using DWF have produced valuable results with N 5 set to ( )  10 [16]. The DWF has a smaller simulation cost than the overlap fermion operator. However, the computational cost is about ( )~ N N 100 5 5 greater than that for the Wilson fermion operator for calculations using the DWF operator. Therefore, Wilson-like fermions are more likely to be selected in simulations than lattice chiral fermions.
Previous reports have focused on exciting light mesons based on quenched calculations [17,18] and dynamical calculations [19,20]. The Bern-Graz-Regensburg (BGR) collaboration has presented results using CI fermion action [20], while the present study only uses lattice chiral fermions. Some preliminary results for quenched calculations using CI fermion action have been presented in [21]. No other studies have adopted lattice chiral fermions. Our ultimate goal is to perform a dynamical simulation of excited light mesons using a lattice chiral fermion operator. We therefore employ the truncated overlap fermion (TOF) operator based on the DWF formalism [16,22]. Simulations with the TOF can be faster than those with other lattice chiral fermions.
The aim of this work was to simulate pseudoscalar, vector, and axial-vector ground states along with radially excited states from lattice QCD using a quenched simulation (i.e., omitting quark loop effects) with the TOF. This study represents the first step towards using the TOF to explore these states from first principles and presents the initial mass spectroscopy results for light excited mesons based on lattice QCD in association with lattice chiral symmetry. This work also determined the effects of chiral fermions on the mass spectra of light mesons based on simulations using quenched calculations. Note that preliminary results regarding a 1 mesons have already been published [23].

Truncated overlap fermion
The fermion action of the truncated overlap fermion (TOF) is defined as where ψ andȳ are four-dimensional fermion fields and D TOF is the fermion matrix for the TOF [16,22]. D TOF is defined as where D DWF is the domain wall fermion operator, P is a five-dimensional projection operator, and ò is a fivedimensional unit vector defined by d =  x x 1, 5 5 . The indices represent the five-dimensional lattice sites defined in where N 5 is the five-dimensional lattice size. The Pauli-Villars matrix D PV is given by D PV = D DWF (m f a = 1) where m f and a correspond to the bare quark mass and the lattice spacing, respectively. The five-dimensional projection operator P is constructed from the four-dimensional projection operators P R/L = (1 ± γ 5 )/2 as Table 1. Isospin I = 1 mesons and their quantum numbers with spectroscopic notation 2S+1 L J . Isospin I = 1 mesons are constructed from u and d quarks to produceūd,ūd and (¯¯)dd uu 1 2 forms These three mesons have a positive, negative, and neutral charge, respectively. In the notation 2S+1 L J , S is the total spin of the quark and anti-quark, L is the angular momentum between the quark and anti-quark, and J is the total angular momentum [1].

Radial ground states of I = 1 mesons
Radial excited states of I = 1 mesons The domain wall fermion operator D DWF [14,15] is defined as where D WF is the Wilson fermion operator. This operator is written as where U μ (x) and M 5 are a link variable and the height of the domain wall, respectively. Within the N 5 → ∞ and m f a → 0 limits, the fermion matrix for TOF satisfies the Ginsparg-Wilson relation [7], Therefore, the TOF action is invariant in the case of lattice chiral symmetry:

Algorithm
There are two approaches to construct the fermion matrix for the TOF and its inverses. One option is to prepare fiveand four-dimensional fermion fields and to use the conjugate gradient (CG) method for five-dimensional matrixes. Using this technique, the four-dimensional fermion field, ψ, can be projected to the five-dimensional fermion field, Ψ, via the projection operator P as Solving large-scale linear equations with the CG method in five dimensions as gives the TOF action¯( The alternative is to prepare only four-dimensional fermion fields and employ the CG method with fourdimensional matrixes. The fermion matrix associated with the TOF can be rewritten as By solving the linear equation for ξ, we obtain the TOF action Using this second approach, it is not necessary to prepare five-dimensional operators, and the matrix solved using the CG method is N 5 times smaller than that in the first option described above. However, formulation (15) for the CG method is more complex than formulation (10). Therefore, the present work examined which method is more economical for calculating meson propagators. A TOF code was developed based on the Lattice QCD Tool Kit in Fortran 90 (LTKf90) [24], which is an open-source code for standard lattice QCD calculations. Test calculations for propagators of the π, ρ, and a 1 mesons were performed with a 4 3 × 8 lattice, N 5 = 4, M 5 = 1.65, and m f a = 0.20 in conjunction with a random gauge configuration on SX-ACE and SQUID at RCNP and at the Cybermedia Center, Osaka University as shown in table 2. The results of these computations indicated that the first method described above was approximately seven times faster than the second approach, and so the first algorithm was used for the subsequent calculations, as described below.

Simulation setup and lattice QCD results
The masses of π, ρ, and a 1 mesons were calculated using an 8 3 × 24 quenched lattice with plaquette gauge action and a lattice coupling β = 5.7. Gauge configurations were generated employing the pseudo-heat-bath method and, after 20,000 thermalization iterations, gauge configurations were saved every 1000 sweeps. The fermion parameters for the TOF were set to N 5 = 32, M 5 = 1.65, and m f a = 0.04-0.08. Previous work by our group confirmed that N 5 = 32 is sufficiently large to be considered infinite when using these parameters [25].
Values for the meson propagator, G(t), were calculated in association with the TOF using gauge configurations of 7864, 3600, 3000, 3000, and 3000 for m f a = 0.04, 0.05, 0.06, 0.07, and 0.08, respectively. Aqq point operator was employed as the source and sink for the meson propagator. The results of these calculations are summarized in figure 1. During these calculations, statistical errors were estimated using the jackknife method and the assumption of periodic boundary conditions in the temporal direction improved the statistical accuracy based on the inclusion of temporal symmetry.
The effective masses m eff (t) for π, ρ, and a 1 were derived by fitting with Next, this work also examined the extraction of meson masses for the excited states with the double-pole fitting form, such as in the case of where m 0 and m 1 correspond to the meson masses for the ground and excited states, respectively. We obtained m 0 and m 1 by fitting the meson propagators with (18). The meson masses for each quark mass are summarized in table 4. Figure 3 summarizes the squared pion mass dependence of the meson masses. This work employed  four combinations of single-pole and double-pole fitting forms for the π, ρ, and a 1 mesons. Specifically, singlepole fitting was used for all mesons in figure 3(a), single-pole fitting was used for the π and ρ mesons and doublepole fitting for the a 1 meson in figure 3(b), single-pole fitting was used for the π meson and double-pole fitting for the ρ and a 1 mesons in figure 3(c), and double-pole fitting was used for all mesons in figure 3(d). In figure 4, the meson masses determined in the present study are compared with the experimental values in the particle data group (PDG) [1] and with previously reported lattice data [17][18][19][20]. Our data in figure 4 correspond to the data in table 5. These results demonstrate that the meson masses were unaffected by the fitting combinations of single-pole and double-pole forms.
In the case of the a 1 mesons, the present results are seen to be consistent with the experimental values of a 1 (1260) and a 1 (1640). These simulations were performed in conjunction with the quenched approximation and theqq source and sink, meaning that virtual intermediate states such as¯qqqq states were highly suppressed, and thus these results suggest that a 1 (1260) and a 1 (1640) are simpleqq states whereas a 1 (1420) may have a more complicated structure than theqq state. It would be of interest to perform calculations of the a 1 (1420) mass using other operators such as the¯qqqq operators.  The results for the excited ρ meson exceed the experimental value for ρ(1700), similar to the findings of previous dynamical studies [19,20]. Calculating the ρ(1450) mass may require using the variational method with other operators coupled with ρ(1450), as has been previously performed [20].
The result obtained in this work for the excited π meson lies between the experimental π(1300) and π(1800) values, which would be expected based on the degenerate masses of the two excited π mesons. Our lattice data are also equivalent to or slightly larger than the values obtained from previous dynamical studies. The effect of   Table 5. Masses of a 1 , excited ρ, and excited π mesons for each fitting combination. the quenched approximation in which sea quarks are infinitely heavy is apparent in the slightly large value for the excited π meson. It should also be noted that the present results for the a 1 , excited ρ, and excited π mesons using chiral fermions are in reasonably good agreement with the values from a previous dynamical study using clover-Wilson fermions or chirally improved fermions, even though our calculations were performed with the quenched approximation.

Conclusion
This work investigated mass spectroscopy data obtained with quenched lattice QCD using truncated overlap fermion action based on the formalisms of domain wall fermions. The double-pole fitting form was used to obtain masses for the ground and first excited states of a 1 , ρ, and π mesons. The results for a 1 mesons coincided with the experimental values of a 1 (1260) and a 1 (1640). Since our simulations were performed with the quenched approximation and the two-quark source and sink, the results suggest that a 1 (1260) and a 1 (1640) are simple two-quark states, whereas a 1 (1420) may have a more complicated structure. The results for the a 1 , excited ρ and excited π mesons using truncated overlap fermions with the quenched approximation (i.e., omitting quark loop effects) were in good agreement with those obtained in prior dynamical studies (i.e., including quark loop effects) using clover-Wilson fermions or chirally improved fermions. Our work supports Glozman's suggestion that chiral fermions are essential in lattice studies of excited states [4]. In order to reconstruct the experimental values of the a 1 (1420) mass, ρ(1450) mass, and the degenerate masses of the two excited π mesons, π(1300) and π(1800), lattice QCD calculations may be required using the variational method with other operators such asqq or¯qqqq operators. Based on this, we expect that dynamical calculations using the truncated overlap fermion with the variational method will be helpful in future research into the mass spectroscopy of light excited mesons.