Both spontaneous chiral symmetry breaking (SCSB) and confinement are intrinsic features of the strong interaction between quarks and gluons, although their relationship is the subject of debate. Lattice QCD has confirmed the restoration of chiral symmetry and certain aspects of deconfinement at high temperatures [1]. However, there are also studies arguing that the strong interaction between quarks still exists, and therefore, deconfinement is incomplete (e.g. the recent N_f = 2 study [2]). These discussions are based on the spatial correlation function of meson interpolators, which differ from the standard hadron mass and will be sensitive to flavors (or other systematics) [3].

All the above lattice QCD calculations focus on the infinite volume limit m_πL ≫ 1, as the finite volume effect on the physical observable can be significant when m_πL is not large enough [4]. As evidence, in a Lattice QCD simulation at a ∼ 0.11 fm with m_πL ∼ 1 using near-physical light quark mass and L ∼ 1.8 fm, an effective restoration of chiral symmetry in the mesonic two-point functions was observed [5]. A system with m_πL ≤ 1 is referred to as the ε-regime of the chiral perturbation theory (χPT) [4], and reasonable leading-order low-energy constants of χPT can be extracted using the above lattice calculations [5].

The finite temperature lattice QCD simulation requires a smaller temporal size to reach higher temperatures. Therefore, one cannot extract the standard ground-state hadron mass based on the long-distance temperature correlation function, and the use of the spatial correlation function would be unavoidable. However, chiral symmetry restoration can also be observed with a long enough temporal size in the ε-regime, allowing reliable extraction of the ground-state hadron mass. In this work, we present a preliminary study on the masses of iso-vector mesons and nucleons at different L, using the clover fermion on the highly improved staggered quark (HISQ) ensembles with near-physical quark masses. We find that the mass gap between the nucleon and pion masses remains, even when L is as small as 0.2 fm.

The ideal choice for the fermion action in the ε-regime simulation is the overlap fermion [6, 7], which preserves the exact chiral symmetry on the lattice, as demonstrated in the previous JLQCD study [5]. However, the overlap fermion can be ${ \mathcal O }(10-100)$ times more expensive than the usual clover fermion used by the CLQCD ensembles [8]. On the other hand, the clover fermion action suffers from additive quark mass renormalization, and therefore requires iterative tuning during the configuration generation to approach the chiral limit. In addition, the HISQ [9] fermion action is a widely used choice in the finite temperature study of QCD, as it is even cheaper than the clover fermion and preserves part of the chiral symmetry. However, residual taste-breaking effects can make the construction of the baryon correlation function highly non-trivial.

A possible efficient solution to take advantage of different fermion actions is to use the HISQ fermion action to generate the configuration, but to employ other fermion actions without taste-breaking effects to calculate the valence quark propagator and hadron correlators. Such a setup suffers from the so-called mixed-action effect, but a recent study [10] suggests that this effect is ${ \mathcal O }({a}^{4})$ and therefore becomes negligible at relatively small lattice spacing. Additionally, the additive chiral symmetry breaking effect of the clover fermion has been shown to be suppressed to a few percent level when non-perturbative renormalization and continuum extrapolation are applied, making it suitable for exploratory studies.

Thus, we chose the HISQ fermion action and one-loop improved Symanzik gauge action used by the milC collaboration to generate the configurations. The bare coupling and quark masses were interpolated to those at a = 0.052 fm based on the parameters used by the MILC configurations [11, 12], as such a lattice spacing is used by the CLQCD ensemble H48P32 [8]. The lattice spacing determined through the Wilson flow [13] is approximately 0.055 fm and close to the target value.

The corresponding effective mass of the pion, η_s, and η_c using this lattice spacing are shown in figure 1. Based on the constant fit at a relatively large t, we can extract m_π = 0.157(8) GeV, which is quite close to the experimental value 0.135 GeV. The η_s mass ${m}_{{\eta }_{s}}=0.681(2)$ GeV is also close to the most precise lattice determination of 0.689 63(18) MeV [14], and the η_c mass ${m}_{{\eta }_{c}}=2.824(1)$ GeV is only 5% lower than the experimental value of 2.98 GeV [15].

For the valence quark, we use the tadpole improved clover fermion action with the hypercubic (HYP) smeared gauge field, and tune the corresponding pion mass to be about 230 MeV on the 48⁴ lattice, which is close to the unitary pion mass and corresponds to m_πL ∼ 3. Using even smaller valence pion mass would make the finite volume effect non-negligible at the largest L we used.

In this work, we choose the ratio of the scalar and pseudoscalar 2-point correlators as a criterion for SCSB, since this ratio will be 1 up to the finite quark mass effect after chiral symmetry is restored. Additionally, we consider the mass difference between the nucleon mass m_N and 3/2 times the pion mass m_π as a typical example of the QCD mass gap. This difference is around 700 MeV with the physical pion mass and should also be non-zero in the chiral and infinite volume limit, but should approach zero in the deconfined phase of QCD. The factor 3/2 in front of m_π aims to cancel the quark number contribution in the nucleon, which can be important when the finite volume effects become significant, as will be shown in the following calculation. We use the pion instead of other mesons, since it is the only light meson that will not decay with the strong interaction and will be degenerate with a₀ without SCSB. The difference m_N − 3/2m_H for other hadrons H, such as the ρ meson or a₁, will be investigated in future studies.

The ensembles we generated for this work are summarized in table 1.

We calculate the hadron temporal two-point correlators, including pseudoscalar (P), scalar (S) and proton (N), with the clover valence mass m_u = − 0.0414 at the near-physical point to extract hadron masses according to the formulawith the P,S interpolators defined asand nucleon two-point functions defined aswith Γ_e = (1 + γ₄)/2. When computing quark propagators, Coulomb wall sources with random positions in the temporal direction are used to reduce autocorrelation effects between configurations. The values for the pion masses are extracted from fitting correlators by a one-state fit with the fitting range selected to ensure acceptable χ²/d.o.f.∼1,with the fitting parameters A and m_π. The result is shown in figure 2. We observe clear plateaus in the effective mass m_eff(t) for all values of L, which suggests that ${{ \mathcal C }}_{{\rm{P}}}(t)$ exhibits exponential decay, regardless of the spatial volume when T is large enough. When L ∼ 0.22 fm, m_π becomes massive around 5 GeV. The m_π rapidly decreases as L increases, and stabilizes at around 230 MeV when L > 1.8 fm.

For proton mass, due to a poor signal at large t, we employ a two-state fit for ${{ \mathcal C }}_{{\rm{N}}}(t)$ at a relatively smaller t in the ensembles with L = 0.22, 0.44, 0.55, 0.66 and 1.76 fm to eliminate the excited state contamination,where m_N, Δm and c_1,2 are fitting parameters. The cases with the other L are fitted with the single state (c₂ = 0). As shown in figure 3, it is revealed that proton masses also follow similar patterns with changes in volume to pions: m_N decreases when spatial extent increases, and tends to about 1 GeV at infinite volume limit. Since pions and nucleons have different numbers of quarks, we define the combination m_N − 1.5m_π to examine the mass gap between the proton and pion based on their quark contents. The result in figure 4 shows the mass gap between the proton and pion remains above 0.7 GeV and even increases at small volume, which suggests that the QCD confinement remains even at very small L. From another perspective shown in figure 5, m_πL has a lower limit (around 2) and tends towards 6 ∼ 2π at small volumes, which implies that the epsilon regime cannot be reached simply by reducing the spatial volume and QCD is not completely dominated by the possible massless pion field at small L. In addition, the variation in the pion and iso-vector scalar meson a₀ two-point function can reflect the nature of chiral symmetry. If chiral symmetry is effectively restored, the quark propagator ${ \mathcal S }(x,y)=\psi (x)\bar{\psi }(y)$ on each individual configuration with negligible quark mass will anti-commutate with γ₅, $\{{\gamma }_{5},{ \mathcal S }\}\sim 0$. Thus, we haveand then ${{ \mathcal C }}_{{\rm{P}}}(t)$ and ${{ \mathcal C }}_{{\rm{S}}}(t)$ will be degenerate. The meson two-point functions are shown in figure 6. When L < 0.7 fm, ${{ \mathcal C }}_{{\rm{P}}}(t)=-{{ \mathcal C }}_{{\rm{S}}}(t)$ holds effectively for all time slices, except t = 0, which may indicate the effective restoration of chiral symmetry. Based on the data shown in the sub-panel of figure 7, the ratio C_P/C_S is away from 1 by 1%, which could be the finite quark mass effect. When L reaches 0.9 fm, ${{ \mathcal C }}_{{\rm{P}}}(t)$ begins to slightly deviate from $-{{ \mathcal C }}_{{\rm{S}}}(t)$ and significantly deviates as L increases to 1.1 fm, including an unexpected sign change in the latter. The ${{ \mathcal C }}_{{\rm{S}}}(t)$ returns to negative values at L ∼ 1.8 fm and shows faster exponential decay at relatively small t, which corresponds to the physical a₀ state (or KK state) in the realistic QCD. The ${{ \mathcal C }}_{{\rm{S}}}(t)$ exhibits irregular behaviour as the volume continually increases, possibly due to the contamination from the other 0⁺⁺ states.

The mass splitting between the a₀ and pion also serves as an indicator of the chiral symmetry breaking. We determine ${m}_{{a}_{0}}$ using the same formula and fitting patterns as in the nucleon case. As depicted in figures 8 and 9, while the mass degeneration of the two mesons occurs naturally as ${{ \mathcal C }}_{{\rm{P}}}(t)=-{{ \mathcal C }}_{{\rm{S}}}(t)$ at L < 0.7 fm, the mass splits are still small with L ∈ (0.9, 1.5) fm, even though an obvious difference has been observed in the two-point functions above 1.1 fm. The ${m}_{{a}_{0}}$ increases to about 0.8 GeV at L ∼ 1.8 fm if we fit ${{ \mathcal C }}_{{\rm{S}}}(t)$ at small t, and then the uncertainty of ${{ \mathcal C }}_{{\rm{S}}}(t)$ increases rapidly with even larger L. This observation may indicate that the chiral phase transition using L as the order parameter would be a crossover, similar to that using T at finite temperature.

The results of hadron masses are summarized in figure 9, showing m_π and m_N stabilization for L > 1.8 fm, suggesting minimal finite volume effects, whereas ${m}_{{a}_{0}}$ exhibits fluctuations due to complex 0⁺-state contamination. More precisely, the ‘decay’ of a₀ on the lattice implies the presence of many (multi-particle) states below the a₀ mass with the same J^PC, and then the uncertainty of C_S will be larger when the weights of the under-threshold states become larger. As depicted in figure 9, the a₀ (more precisely, the ground state in the scalar channel) effective mass at 1.1 fm is similar to the corresponding pion mass and then lower than the ππ state; thus, the ‘decay’ cannot occur and the signal of C_S is good, as shown in figure 8 (blue diamonds). When L = 1.76 fm, the π(0)π(0) state is lower than the a₀ threshold. Therefore, in figure 8, we can observe a pseudo-plateau (green triangles) at small t, which is around the physical a₀ mass and also the KK threshold, but the uncertainty increases at large t due to contamination from the π(0)π(0) state. The weights of the π(0)π(0) state increase with even larger L, resulting in a huge uncertainty in the effective mass of C_S at L = 2.64 fm, as shown in figure 8 with purple triangles.

We also show an additional case on a 32 × 8² × 96 lattice for comparison in figure 9. The hadron masses ${m}_{\pi ,N,{a}_{0}}$ are quite close to those on the 8³ × 96 lattice. It suggests that QCD properties will be dominated by the spatial direction to a shorter extent in those cases.

We studied the hadron spectrum in the finite spatial volume with near-physical quark masses with different L, using the clover valence fermion on the HISQ sea configurations at a = 0.05 fm. The results show that the pion mass receives a huge 1/L correction at small L, leading to an increase in the corresponding m_πL when L ≤ 1.2 fm, where the chiral symmetry breaking between the scalar and pseudoscalar correlation functions is effectively restored. At the same time, the mass gap m_N − 3m_π/2 is always larger than 700 MeV in the entire L ∈ (0.4, 2.5) fm range we studied. Such an observation suggests that the confinement of QCD can exist when the SCSB is absent.

Our study also suggests that the ε-regime, where m_πL ≤ 1, cannot be reached solely by decreasing L, as the pion mass can increase significantly with decreasing L. Since the pion mass has been saturated at L > 1.5 fm, it is possible to further reduce the bare quark mass of the clover fermion to suppress m_π on the 48⁴ lattice without much finite volume effect, and achieve a smaller m_πL.

Since the valence clover fermion action we used breaks the chiral symmetry additively, it would be non-trivial to further investigate the origin of the restoration of the chiral symmetry and the remaining mass gap. The overlap fermion action would be more suitable for this task, and the study is ongoing. The mixed-action effect on the above observations would also be a concern, and further study at multiple lattice spacings is also essential.