The topological susceptibility is an interesting quantity that characterizes how many topological excitations are created in the QCD vacuum. Witten [1] and Veneziano [2] estimated in the large- (number of colors) limit and showed that it is proportional to the square of the meson mass. In real QCD with and light dynamical quarks, however, the argument of Witten and Veneziano is no longer valid. It is not the meson but the (zero-momentum mode of the) pion that controls the topological susceptibility.

According to the prediction of chiral perturbation theory (ChPT) at leading order (LO), is expected to be proportional to the quark mass , when the up and down quark masses are degenerate. At one-loop, the quark mass dependence is predicted as [3–5] (1) where denotes the chiral condensate, is a combination of the low-energy constants at next-to-leading order (NLO) [6], and MeV) and MeV) are the physical values of the pion mass and decay constant, respectively. Here are renormalized at . In the formula, we have assumed that the strange quark is decoupled from the theory and chiral perturbation theory works. In other words, the strange quark mass dependence is assumed to be absorbed in the low-energy constants.

By taking a ratio with the ChPT predictions for the pion mass and decay constant (let us denote them by and ), one can eliminate the chiral logarithm in Eq. (1): (2) where is again a combination of the NLO low-energy constants, which is independent of the renormalization scheme and scale at this order. This ratio also cancels possible finite volume effects at NLO. Moreover, the chiral limit of the ratio, 1/4, is protected from the strange sea quark effects (see Appendix A for details), as they always enter as a function of the ratio , which can be absorbed into the (finite) renormalization of . We can therefore precisely estimate the topological susceptibility at the physical point by measuring , , and at each simulation point.

It has been a challenging task for lattice QCD to compute , since it is sensitive to the discretization effects and the violation of chiral symmetry [7–9] in particular. This is partly because the quark mass dependence of is due to sea quarks, or a small quantum mechanical effect suppressed by , to which the discretization error is relatively large. Even if we could simulate QCD on a sufficiently fine lattice, the global topological charge would become frozen along the Monte Carlo history [10]. Due to these difficulties, the study of the quark mass dependence and its comparison with the ChPT formula of has been very limited, and only some pilot works with dynamical chiral fermions on rather small or coarse lattices [11–19] are available.

In this work, we improve the computation of in two ways. One is to employ the domain-wall fermion [20,21] with an improvement by Refs. [22,23], known as the Möbius domain-wall fermion, for the dynamical quarks, which enables us to precisely preserve chiral symmetry. Even on our coarsest lattice, the residual mass, related to the violation of the chiral symmetry, is kept at the order of MeV. As will be shown below, our results show only a mild dependence of on the lattice spacing, up to fm. The use of the domain-wall fermion allows us to sample configurations in different topological sectors, which is also an advantage over the simulation with the overlap fermion where we fixed the global topological charge in Ref. [12].

Another improvement comes from the use of sub-volumes of the simulated lattices. Since the correlation length of QCD is limited, at most by , there is essentially no need to use the global topological charge to compute . The use of sub-volume was tested in our previous simulations with overlap quarks [12,16] (see also Refs. [24,25]), where the signal was extracted from finite volume effects, which have some sensitivity to [26,27]. In this work, we utilize a different method, which was originally proposed by Bietenholz et al. [28,29] (similar methods were proposed in Refs. [30] and [31]). The method is based on a correlator, which gives a positive finite value even in the thermodynamical limit, and thus is less noisy than our recent attempt in Ref. [32]¹. We confirm that 30%–50% sub-volumes of the whole lattice, whose size is fm, are sufficient to extract . Moreover, the new definition shows more frequent fluctuation than that of the global topological charge on our finest lattice.

We also employ a modern technique, the Yang–Mills (YM) gradient flow [34–36], in order to make the global topological charge close to integers, to remove the UV divergences, and to reduce the statistical noise. With these improvements, we achieve good enough statistical precision to investigate the dependence of on the sea quark mass. In fact, our data of the topological susceptibility are consistent with the ChPT prediction (1), from which the values of chiral condensate and are extracted.

The same set of data was also used to calculate the meson mass [37], which was extracted from the shorter distance region of the correlator of the topological charge density. These two results show a nontrivial double-scale structure of topological fluctuation of the gauge field: it creates the meson at short distances, while describing the vacuum mode of the pion (or the lowest mode, which is constant over space-time) at long distances.

The rest of this paper is organized as follows. First, we describe our lattice setup in Sect. 2. We then explain the method to extract the topological susceptibility from the slab sub-volume in Sect. 3. Our results at lower are presented in Sect. 4. Comparing the data with those obtained from global topology, we examine the validity of our sub-volume method. The results at higher are shown in Sect. 5, and how we estimate the statistical errors is explained in Sect. 6. Finally, we estimate the chiral and continuum limits in Sect. 7 and give a summary in Sect. 8.

In the numerical simulation² of QCD, we use the Symanzik gauge action and the Möbius domain-wall fermion action for gauge ensemble generations [38–41]. We apply three steps of stout smearing of the gauge links [42] for the computation of the Dirac operator. Our main runs of -flavor lattice QCD simulations are performed with two different lattice sizes and , for which we set = 4.17 and 4.35, respectively. The inverse lattice spacing is estimated to be 2.453(4) GeV (for ) and 3.610(9) GeV (for ), using the input fm [43] where we use the reference YM gradient flow–time , defined by [34] with the energy density of the gluon field. The two lattices share a similar physical size fm. For the quark masses, we choose two values of the strange quark mass around its physical point, and 3–4 values of the up and down quark masses for each . Since our data at the lightest pion mass around 230 MeV ( = 0.0035 at = 4.17) may contain significant finite size effects, we simulate a larger lattice with the same set of parameters to check if the finite volume systematics is small enough. We also perform a simulation on a finer lattice (at [ GeV] and 285 MeV). For each ensemble, 500–1000 gauge configurations are sampled from 5000 molecular dynamics (MD) time. The ensembles used in this work are listed in Table 1.

Table 1.

Parameters of the JLQCD gauge ensembles used in this work. Pion masses are rounded to the nearest MeV.

In this setup, we confirm that the violation of the chiral symmetry in the Möbius domain-wall fermion formalism is well under control. The residual mass is MeV [44] by choosing the lattice size in the fifth direction at and less than 0.2 MeV with at (and 4.47).

On generated configurations, we perform 500–1640 steps of the YM gradient flow (using the conventional Wilson gauge action) with a step-size . At every 200–400 steps (depending on the parameters) we store the configuration of the topological charge density. The two-point correlator is measured using the fast Fourier transform (FFT) technique to average source and sink points over whole lattice sites.

In the following analysis, we measure the integrated auto-correlation time of every quantity, following the method proposed in Refs. [10,45]. The statistical error is estimated by the jackknife method (without binning) multiplied by the square root of auto-correlation time normalized by the MD time interval of the configuration samples. We will discuss more details about the auto-correlation time of topological fluctuations in Sect. 6.

The pion mass and decay constant are computed combining the pseudoscalar correlators with local and smeared source operators. Details of the computation are presented in a separate article [46].

We use the conventional gluonic definition of the topological charge density , the so-called clover construction [9]. Since the YM gradient flow smooths the gauge field in the range of fm of the lattice, a simple summation over the whole sites gives values close to integers, as shown in Fig. 1.

As is well known, the global topological charge suffers from long auto-correlation time in lattice simulations, especially when the lattice spacing is small. This is also true in our simulations, as shown in Fig. 2. At the highest , drifts very slowly with an auto-correlation time of possibly . It is, therefore, not feasible to estimate without performing much longer runs. The details of the auto-correlation time of the topological charge and its density operator will be discussed in Sect. 6.

Instead of using the global topological charge , we attempt to extract the topological susceptibility from a sub-volume of the whole lattice . Since the correlation length of QCD is limited by at most , the sub-volume should contain sufficient information to extract , provided that its size is larger than . One can then effectively increase the statistics by , since each piece of sub-lattices may be considered as an uncorrelated sample. Moreover, there is no potential barrier among topological sectors: the instantons and anti-instantons freely come in and go out of the sub-volume, which should make the auto-correlation time of the observable shorter than that of the global topological charge.

There are various ways of cutting the whole lattice into sub-volumes and computing the correlation functions in them. After some trial and error, we find that the so-called “slab” method, proposed by Bietenholz et al. [28], is efficient for the purpose of computing . The idea is to sum up the two-point correlators of the topological charge density, over and in the same sub-volume: (3)

Here the integration over and in the spatial directions runs in the whole spatial volume (since the YM gradient flow is already performed, there is no divergence from the points of ), while the temporal sum is restricted to the region , which is called a “slab”. Here is an arbitrary reference time. Due to the translational invariance, slabs sharing the same thickness are physically equivalent, and one can average over . This method is statistically more stable than the other sub-volume method that we applied in Refs. [12,16] because is guaranteed to be always positive.

If we sample large statistics on a large enough lattice volume, should be just of . Namely, should be a linear function in . Its leading finite volume correction can be estimated using the formula in Ref. [27]: (4) where is an unknown constant, and is the mass of the first excited state, the meson³. Note that for , the formula gives a simple linear function in plus a constant. Also, note that in the limit of , .

Assuming the linearity in , one can extract the topological susceptibility through (5) with two reference thicknesses and . In our numerical analysis, is averaged over the temporal direction. Since the data at and are not independent, we choose and in a range 1.6 fm . In the numerical analysis, we replace by to cancel a possible bias due to the long auto-correlation of the global topology.

The original proposal in Ref. [28] mainly used the correlator in a fixed topological sector. The formula corresponding to Eq. (4) then contains a subtraction of the contribution from the global topology. We find that the statistical noise is larger with this choice while the results from different topological sectors are consistent. In the following analysis, we use Eq. (4) after summing over the topological sectors.

We find that the signal using this slab method is less noisy than the previous attempts in Refs. [12,16]. Moreover, as shown in Fig. 2 and discussed in detail later, the new definition shows more frequent fluctuation than that of the global topological charge on our finest lattice.

At , which corresponds to the lattice spacing fm, both the global topological charge and fluctuate well, as shown in the top panel of Fig. 2. The data on this lattice, therefore, provide a good testing ground to examine the validity of the slab sub-volume method, compared with the naive definition of the topological susceptibility with .

In Fig. 3, observed at the lightest sea quark mass , on two different volumes and is plotted as a function of . The data converge to a linear plus constant function given in Eq. (4) at , which corresponds to fm. The slope, or , is consistent with that from global topology, shown by solid and dotted lines for the and lattices, respectively. We also observe the consistency between the and data, which suggests that the systematics due to the finite volume is well under control.

The “linear constant” behavior is also seen in ensembles with heavier quark masses, as presented in Fig. 4.

The extracted values of the topological susceptibility from the slope show a good agreement with the ChPT prediction, as shown in Fig. 5 by open and filled squares. The leading-order ChPT formula, , with (solid line) is shown to guide the eye. In the same plot, we also plot the estimate for obtained from the global topological charge by circles, which again agrees with the results, validating the slab method. The values of are listed in Table 2. How we estimate their error bars is explained in the following two sections.

Table 2.

Results for the pion mass , decay constant , , , and . for is estimated from the first zero-crossing point of . All the data are shown in lattice units. For , the first error denotes the statistical error, while the second shows the systematic error due to the effect of freezing the global topological charge.

At higher values, we still find a reasonable slope at the lightest quark mass for each and , as shown in Fig. 6. For heavier masses, however, some curvature is seen. We consider this curvature to be an effect from the bias of the global topological charge. This observation is consistent with previous works (see, e.g., Ref. [10]), which reported that heavier pion mass ensembles show the longer auto-correlation of the topological charge, and the larger deviation of from zero. We determine the reference fm using data at the lightest quark mass and always choose fm. In order to estimate the systematic errors due to nonlinear behavior, we compare the results with 1) those obtained from different reference times , and , and 2) those obtained without the subtraction of in the definition of the topological charge density. The larger deviation is treated as a systematic error. More details are presented in Appendix B.

Our results are summarized in Fig. 7 (see also Fig. 8 for a comparison with Refs. [8] and [9]). Although the data at higher are rather scattered compared to those at , they can be used to estimate the chiral condensate , assuming the linear suppression around the chiral limit. Before going to the details, we discuss the auto-correlation of and show how we estimate the statistical errors in the next section.

Gauge configurations generated by a Markov chain are generally not independent but have auto-correlations. How much they are correlated depends on the observables. We therefore need to carefully measure the auto-correlation of the target observable : (6) where denotes the Monte Carlo time, and the average is taken over .

When the Monte Carlo trajectory is long enough, compared to the auto-correlation time of any observables, one can estimate the so-called integrated auto-correlation time by (7) where the upper end of the summation window is chosen to where becomes consistent with zero within the error. The above formula assumes that converges to a single exponential function well below .

If the observables suffer from long auto-correlation, and the Monte Carlo trajectory is not long enough, on the other hand, the above procedure may underestimate the auto-correlation time, since some very slow decay modes can be hidden in the error of . This problem is similar to that of hadron spectroscopy with a short temporal extension, where one does not have a long enough fitting range to disentangle the ground state from excited states, which leads to over-estimation of the mass.

The ALPHA Collaboration [10] carefully studied the effect of slow modes, and proposed an improved estimate of the auto-correlation time, (8) where is the same summation as Eq. (7) but with a smaller upper bound where becomes lower than 3/2 standard deviations. is the auto-correlation of the slowest mode. The proposal is equivalent to considering a continuation of at to the slowest possible exponential function .

In lattice QCD simulations, it is natural to assume that is equal to the auto-correlation of the global topological charge. In our simulations, is estimated by of , except for where we choose MD time by hand (and assuming 100% error for it), which is a rough order estimate from the first zero-crossing point of . Then we compute the auto-correlation of our target observable by to estimate the error. The results for , , and are summarized in Table 2 and the auto-correlation function at three different with a similar pion mass MeV is shown in Fig. 9. At the highest , it is clear that has a shorter auto-correlation time than that of the global topological charge.

With the measured improved auto-correlation time , we estimate the statistical errors of by multiplying to the naive error estimates, where is the standard deviation of and denotes the interval trajectory between samples. The results, as well as the systematic error from the choice of reference points, are listed in the last column of Table 2.

Figure 7 presents our data for from all ensembles plotted in physical units. The horizontal axis, the quark mass defined in the scheme at 2 GeV, is (9) where the renormalization factor is nonperturbatively computed in Ref. [48]: , and for , and , respectively. In contrast to the results by other groups with non-chiral fermions, we find no strong dependence on .

First, we compare our results directly to the ChPT formula (1). We perform a two-parameter ( and ) fit to the data at (solid curve in Fig. 7) and (dashed curve) separately⁴. The results for and are listed in Table 3. Here we also perform the same fit but omitting the heaviest two points, and take the difference as an estimate for the systematic error in the chiral extrapolation. Since the heaviest points have several problems—1) a strong bias is seen in the global topology, 2) ChPT is less reliable, and 3) there is mismatch between different —we take the result without them as our central values. Note, however, that this inclusion/elimination affects but is stable against the change in the fit-range. Namely, the chiral condensate is determined by the low quark mass data. We then estimate the continuum limit by a constant fit, as shown in the top two panels in Fig. 10. Comparing our result from the constant fit with linear extrapolation of the central values, we take the difference as an estimate for the systematic error in the continuum limit. In the plots in Fig. 10, all these errors are added in quadrature.

Table 3.

Our results for , , , and at the physical point. The first error denotes the the statistic fluctuation at each simulation point, including the effect of long auto-correlation of global topology. The second is the systematic error in chiral extrapolation, and the third error denotes that in the continuum limit estimates. See the main text for details.

Next, using our data for the pion mass and decay constant together with , obtained from each ensemble, we take the ratio given in Eq. (2). By a linear one-parameter fit, we determine and the ratio at the physical point. In the same way as the determination of and , we take the chiral and continuum limits of both quantities. Note that the fixed chiral limit at of the ratio helps us to determine these quantities.

Finally let us discuss other possible systematic effects. In our analysis, the ensembles satisfying 3.9$]]> are used and we do not expect any sizable finite volume effects. In particular, our lightest mass point has . We have used configurations at the YM gradient flow–time around fm. We confirm that the flow–time dependence is negligible in the range fm fm. We conclude that all these systematic effects are negligibly small compared to the statistical and systematic errors given above.

With dynamical Möbius domain-wall fermions and the new method using a sub-volume of the simulated lattice, we have computed the topological susceptibility of QCD. Its quark mass dependence is consistent with the ChPT prediction, from which we have obtained (10) (11) where the first error comes from the statistical uncertainty at each simulation point, including the effect of freezing topology. The second and third errors represent the systematics in the chiral and continuum limits, respectively. The value of is consistent with our recent determination through the Dirac spectrum [49]. We have also estimated the NLO coefficient (12) (13) where is renormalized at the physical pion mass, while is renormalization invariant. It is interesting to note that and include a combination of the coefficients , which are supposed to be unphysical in ChPT unless dependence is considered. These are important for possible couplings of QCD to axions [50].