Mass spectroscopy of excited light mesons using truncated overlap fermions

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($\bar{q}$) pairs. The lightest mesons are built from u and d quarks, and $q\bar{q}$ 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 [3–6] 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],

$$\begin{eqnarray}&&{\gamma }_{5}D+D{\gamma }_{5}=2{aD}{\gamma }_{5}D,\end{eqnarray} \tag{ 1 }$$

where D, γ₅, 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 N₅. If we take N₅ 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 N₅ set to ${ \mathcal O }(10)$ [16]. The DWF has a smaller simulation cost than the overlap fermion operator. However, the computational cost is about ${N}_{5}\times {N}_{5}\sim { \mathcal O }(100)$ 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 a₁ mesons have already been published [23].

The fermion action of the truncated overlap fermion (TOF) is defined as

$$\begin{eqnarray}&&{S}_{{TOF}}=\bar{\psi }{D}_{{TOF}}\psi ,\end{eqnarray} \tag{ 2 }$$

where ψ and $\bar{\psi }$ are four-dimensional fermion fields and D_TOF is the fermion matrix for the TOF [16, 22]. D_TOF is defined as

$$\begin{eqnarray}&&{D}_{{TOF}}={\epsilon }^{\dagger }{P}^{\dagger }{D}_{{PV}}^{-1}{D}_{{DWF}}P\epsilon ,\end{eqnarray} \tag{ 3 }$$

where D_DWF is the domain wall fermion operator, P is a five-dimensional projection operator, and is a five-dimensional unit vector defined by ${\epsilon }_{{x}_{5}}={\delta }_{1,{x}_{5}}$. The indices represent the five-dimensional lattice sites defined in x₅ ∈ [1, N₅] where N₅ 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 ± γ₅)/2 as

$$\begin{eqnarray}&&{P}_{{x}_{5},{y}_{5}}={P}_{L}{\delta }_{{x}_{5},{y}_{5}}+{P}_{R}{\delta }_{{x}_{5}+1,{y}_{5}}+{P}_{R}{\delta }_{{x}_{5},{N}_{5}}{\delta }_{{y}_{5},1}.\end{eqnarray} \tag{ 4 }$$

The domain wall fermion operator D_DWF [14, 15] is defined as

$$\begin{eqnarray}\begin{array}{rcl}{D}_{{DWF}\,{x}_{5},{y}_{5}}(x,y) & = & {D}_{{WF}}(x,y){\delta }_{{x}_{5},{y}_{5}}-({P}_{L}{\delta }_{{x}_{5}+1,{y}_{5}}+{P}_{R}{\delta }_{{x}_{5}-1,{y}_{5}}){\delta }_{x,y}\\ & & +{\delta }_{x,y}{\delta }_{{x}_{5},{y}_{5}}+{m}_{f}\,a({P}_{R}{\delta }_{{x}_{5},1}{\delta }_{{y}_{5},{N}_{5}}+{P}_{L}{\delta }_{{x}_{5},{N}_{5}}{\delta }_{{y}_{5},1}){\delta }_{x,y},\ \ \ \ \end{array}\end{eqnarray} \tag{ 5 }$$

where D_WF is the Wilson fermion operator. This operator is written as

$$\begin{eqnarray}&&{D}_{{WF}}(x,y)=(4-{M}_{5}){\delta }_{x,y}-\displaystyle \frac{1}{2}\displaystyle \sum _{\mu =\pm 1}^{\pm 4}(1-{\gamma }_{\mu }){U}_{\mu }(x){\delta }_{x+\hat{\mu },y},\end{eqnarray} \tag{ 6 }$$

where U_μ (x) and M₅ are a link variable and the height of the domain wall, respectively. Within the N₅ → ∞ and m_f a → 0 limits, the fermion matrix for TOF satisfies the Ginsparg–Wilson relation [7],

$$\begin{eqnarray}&&{\gamma }_{5}{D}_{{TOF}}+{D}_{{TOF}}{\gamma }_{5}=2{{aD}}_{{TOF}}{\gamma }_{5}{D}_{{TOF}}\ \ \ ({N}_{5}=\infty \ \mathrm{and}\ {m}_{f}\,a=0).\end{eqnarray} \tag{ 7 }$$

Therefore, the TOF action is invariant in the case of lattice chiral symmetry:

$$\begin{eqnarray}\left\{\begin{array}{ccc}\psi (x) & \to & {e}^{i\theta {\gamma }_{5}(1-{{aD}}_{{TOF}})}\psi (x)\\ \bar{\psi }(x) & \to & \bar{\psi }(x)\ {e}^{i\theta (1-{{aD}}_{{TOF}}){\gamma }_{5}}\end{array}\right..\end{eqnarray} \tag{ 8 }$$

There are two approaches to construct the fermion matrix for the TOF and its inverses. One option is to prepare five- and 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

$$\begin{eqnarray}&&{\rm{\Psi }}=P\epsilon \psi .\end{eqnarray} \tag{ 9 }$$

Solving large-scale linear equations with the CG method in five dimensions as

$$\begin{eqnarray}&&{D}_{{PV}}\xi ={D}_{{DWF}}{\rm{\Psi }}\ \end{eqnarray} \tag{ 10 }$$

gives the TOF action

$$\begin{eqnarray}&&{S}_{{TOF}}=\bar{\psi }{\epsilon }^{\dagger }{P}^{\dagger }\xi ,\end{eqnarray} \tag{ 11 }$$

where ξ is a five-dimensional matrix.

The alternative is to prepare only four-dimensional fermion fields and employ the CG method with four-dimensional matrixes. The fermion matrix associated with the TOF can be rewritten as

$$\begin{eqnarray}&&{D}_{{TOF}}=\displaystyle \frac{1+{m}_{f}\,a}{2}+\displaystyle \frac{1-{m}_{f}\,a}{2}{\gamma }_{5}\displaystyle \frac{{H}_{W+}^{{N}_{5}}-{H}_{W-}^{{N}_{5}}}{{H}_{W+}^{{N}_{5}}+{H}_{W-}^{{N}_{5}}},\end{eqnarray} \tag{ 12 }$$

where H_W+ and H_W− are defined based on the Hermitian matrix H_W as

$$\begin{eqnarray}&&{H}_{W+}=1+{H}_{W},\quad {H}_{W-}=1-{H}_{W},\quad {H}_{W}={\gamma }_{5}\displaystyle \frac{{D}_{{WF}}}{{D}_{{WF}}+2}.\end{eqnarray} \tag{ 13 }$$

From the formulation with the limits N₅ → ∞ and m_f a → 0, we obtain

$$\begin{eqnarray}&&{D}_{{TOF}}=\displaystyle \frac{1}{2}\left[1+{\gamma }_{5}\mathrm{sgn}\left({H}_{W}\right)\right],\end{eqnarray} \tag{ 14 }$$

demonstrating that the TOF action satisfies the Ginsparg–Wilson relationship (7). Because this approach does not include five-dimensional operators, the CG method can be used in ordinary four-dimensional space as

$$\begin{eqnarray}&&\left[\displaystyle \frac{1+{m}_{f}\,a}{2}\left({H}_{W+}^{{N}_{5}}+{H}_{W-}^{{N}_{5}}\right){\gamma }_{5}+\displaystyle \frac{1-{m}_{f}\,a}{2}\left({H}_{W+}^{{N}_{5}}-{H}_{W-}^{{N}_{5}}\right)\right]\xi =\left({H}_{W+}^{{N}_{5}}+{H}_{W-}^{{N}_{5}}\right){\gamma }_{5}\psi .\end{eqnarray} \tag{ 15 }$$

By solving the linear equation for ξ, we obtain the TOF action

$$\begin{eqnarray}&&{S}_{{TOF}}=\bar{\psi }\xi .\end{eqnarray} \tag{ 16 }$$

Using this second approach, it is not necessary to prepare five-dimensional operators, and the matrix solved using the CG method is N₅ 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 a₁ mesons were performed with a 4³ × 8 lattice, N₅ = 4, M₅ = 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.

The masses of π, ρ, and a₁ mesons were calculated using an 8³ × 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 N₅ = 32, M₅ = 1.65, and m_f a = 0.04–0.08. Previous work by our group confirmed that N₅ = 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. A $\bar{q}q$ 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.

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The effective masses m_eff(t) for π, ρ, and a₁ were derived by fitting with

$$\begin{eqnarray}&&\displaystyle \frac{G(t+1)}{G(t)}=\displaystyle \frac{\cosh \left\{{m}_{\mathrm{eff}}(t)\,\left(\tfrac{T}{2}-(t+1)\right)\right\}}{\cosh \left\{{m}_{\mathrm{eff}}(t)\,\left(\tfrac{T}{2}-t\right)\right\}},\end{eqnarray} \tag{ 17 }$$

where T is the temporal lattice size. The variations in m_eff a over time are plotted in figure 2. Meson masses were obtained using a single-pole fitting form based on the plateaus in these m_eff a plots. The resulting meson masses are provided in table 3 and also indicated by the solid lines in figure 2.

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Table 3. Masses of the π, ρ, and a₁ mesons (as obtained from fitting with the single-pole form), mass ratios, m_π /m_ρ and number of configurations (Confs.) for each quark mass.

mf a	mπ a	mρ a	${m}_{{a}_{1}}\,a$	mπ /mρ	Confs.
0.08	0.6675(10)	0.9508(33)	1.395(6)	0.702(3)	3000
0.07	0.6288(12)	0.9260(27)	1.361(10)	0.679(3)	3000
0.06	0.5900(7)	0.9052(30)	1.351(9)	0.652(3)	3000
0.05	0.5482(8)	0.8824(26)	1.319(8)	0.621(3)	3600
0.04	0.5034(7)	0.8606(16)	1.281(9)	0.585(2)	7864

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

$$\begin{eqnarray}&&G(t)={Z}_{0}\cosh \left\{{m}_{0}\left(\displaystyle \frac{T}{2}-t\right)\right\}+{Z}_{1}\cosh \left\{{m}_{1}\left(\displaystyle \frac{T}{2}-t\right)\right\},\end{eqnarray} \tag{ 18 }$$

where m₀ and m₁ correspond to the meson masses for the ground and excited states, respectively. We obtained m₀ and m₁ 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 a₁ mesons. Specifically, single-pole fitting was used for all mesons in figure 3(a), single-pole fitting was used for the π and ρ mesons and double-pole fitting for the a₁ meson in figure 3(b), single-pole fitting was used for the π meson and double-pole fitting for the ρ and a₁ mesons in figure 3(c), and double-pole fitting was used for all mesons in figure 3(d). A linear extrapolation of the meson masses to the chiral limit, ${\left({m}_{\pi }\,a\right)}^{2}=0$, was also performed. By tuning the ρ meson mass in the chiral limit to m_ρ = 775.26 MeV, lattice spacings a = 0.1890(13), 0.1890(13), 0.1870(8), and 0.1870(8) fm were obtained for each combination, as shown in plots (a), (b), (c), and (d) in figure 3, respectively. In the chiral limit, the quark mass does not become zero but reaches a residual mass ${m}_{f}\,a\to -{m}_{\mathrm{res}}\,a$ because N₅ is finite. The residual masses in the present work were determined to be ${m}_{\mathrm{res}}\,a=13.5(6)$, 13.5(6), 13.7(5), and 13.5(2) MeV for plots (a), (b), (c), and (d), respectively. The meson masses in the chiral limit are summarized in table 5.

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Table 4. Masses of π, ρ, and a₁ mesons for each quark mass as obtained from fitting with double-pole form.

mf a	mπ a	mπ a	mρ a	mρ a	${m}_{{a}_{1}}\,a$	${m}_{{a}_{1}}\,a$
	Ground	Excited	Ground	Excited	Ground	Excited
0.08	0.6649(2)	1.673(12)	0.9353(12)	1.791(13)	1.375(8)	2.156(69)
0.07	0.6265(5)	1.642(31)	0.9111(12)	1.785(12)	1.337(12)	2.082(79)
0.06	0.5882(2)	1.654(15)	0.8900(14)	1.771(15)	1.322(10)	2.038(60)
0.05	0.5467(3)	1.615(29)	0.8678(13)	1.766(14)	1.288(10)	1.974(48)
0.04	0.5019(3)	1.560(40)	0.8507(15)	1.809(17)	1.258(26)	1.912(102)

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 [17–20]. 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.

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In the case of the a₁ mesons, the present results are seen to be consistent with the experimental values of a₁(1260) and a₁(1640). These simulations were performed in conjunction with the quenched approximation and the $\bar{q}q$ source and sink, meaning that virtual intermediate states such as $\bar{q}\bar{q}{qq}$ states were highly suppressed, and thus these results suggest that a₁(1260) and a₁(1640) are simple $\bar{q}q$ states whereas a₁(1420) may have a more complicated structure than the $\bar{q}q$ state. It would be of interest to perform calculations of the a₁(1420) mass using other operators such as the $\bar{q}\bar{q}{qq}$ 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 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 a₁, 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.

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 a₁, ρ, and π mesons. The results for a₁ mesons coincided with the experimental values of a₁(1260) and a₁(1640). Since our simulations were performed with the quenched approximation and the two-quark source and sink, the results suggest that a₁(1260) and a₁(1640) are simple two-quark states, whereas a₁(1420) may have a more complicated structure. The results for the a₁, 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 a₁(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 as $\bar{q}q$ or $\bar{q}\bar{q}{qq}$ 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.

The authors wish to thank S Date for helpful advice regarding the lattice code and A Nakamura for valuable discussions. The simulations reported in this paper were performed using the SX-ACE and SQUID supercomputers at the Cybermedia Center of Osaka University, with the support of the Research Center for Nuclear Physics of Osaka University, and using the SX-Aurora TSUBASA system at Kokushikan University. This work was also supported by the Joint Usage/Research Center for Interdisciplinary Large-scale Information Infrastructures, Japan (project IDs jh210032-NAH and jh220024).

The data that support the findings of this study are available upon reasonable request from the authors.