Correlation and localization properties of topological charge density and the pseudoscalar glueball mass in SU(3) lattice Yang-Mills theory

In order to extract continuum properties, we study the topological charge density correlator (TCDC) and the inverse participation ratio (IPR) for the topological charge density $q(x)$ in SU(3) lattice Yang-Mills theory for relatively small lattice spacings, including some smaller than those explored before. With the help of a recently proposed open boundary condition, it is possible to compute observables at a smaller lattice spacing since the trapping problem is absent. On the other hand, the reference energy scale provided by the Wilson flow allows us to study their scaling behavior, in contrast to previously proposed smearing techniques. The behavior of the TCDC for different lattice spacings at a fixed hypercubic smearing level shows apparent scaling violations. In contrast, at a particular Wilson flow time $t$ for all of the lattice spacings investigated (except the largest one), the TCDC data show universal behavior within our statistical uncertainties. The continuum properties of the TCDC are studied by investigating the small-flow-time behavior. We also extract the pseudoscalar glueball mass from the TCDC, which appears to be insensitive to the lattice spacings ($0.0345\mathrm{fm}\le a\le 0.0667\mathrm{fm}$) and agrees with the value extracted using anisotropic lattices, within statistical errors. Further, we study the localization property of $q(x)$ using the IPR (whose continuum behavior can be probed using small values of the Wilson flow time) and observe a decrease of the IPR with decreasing Wilson flow time. A detailed study of $q(x)$ under the Wilson flow time reveals that as the Wilson flow time decreases, the proximity of the regions of positive and negative charge densities of large magnitude increases, and the charge density appears to be more delocalized, resulting in the observed behavior of the IPR.

Figure 1

$C(r)$ versus $r$ at three HYP smearing steps for ensembles $P_1$, $P_2$, and $P_3$.

Figure 2

$C(r)$ versus $r$ at Wilson flow time $\sqrt{8t}=0.14\mathrm{fm}$ for ense$$mbles $O_1$, $O_2$, $O_3$, and $O_4$.

Figure 3

Plot of $C(r)$ versus $r$ at various values of the Wilson flow time $\sqrt{8t}$ at $\beta =6.42$ and $\beta =6.59$ for ensembles $P_2$ (filled triangle) and $P_3$ (filled square), respectively.

Figure 4

Plot of the topological charge density correlator $C(r)$ versus $r$ at various values of the Wilson flow time $\sqrt{8t}$ at $\beta =6.42$ and lattice volume $32^3\times 64$ for ensemble $P_2$.

Figure 5

Comparison of $C(r)$ versus $r$ at various values of the Wilson flow time $\sqrt{8t}$ at $\beta =6.42$ and lattice volume $32^3\times 64$ for ensembles $P_2$ (filled symbols) and $O_2$ (open symbols).

Figure 6

Plot of $r_c$ versus Wilson flow time $\sqrt{8t}$ for the ensembles $P_2$ (filled symbols) and $O_2$ (open symbols).

Figure 7

Plot of $C(r)$ versus $r$ for ensembles $O_1$, $O_2$, $O_3$, and $O_4$ without smoothing of the gauge fields.

Figure 8

$C(r/a)$ versus $r$ for the ensemble $O_4$ at Wilson flow time $\sqrt{8t}=0.14\mathrm{fm}$. Also shown is the the fit used to extract the pseudoscalar glueball mass.

Figure 9

Sensitivity to the starting point of the fit range for a fixed end point of the pseudoscalar glueball mass, extracted using the asymptotic formula for ensembles $P_2$ (left) and $O_3$ (right).

Figure 10

Plot of the lowest pseudoscalar glueball mass versus $a^2$ for both open (open symbols) and periodic (filled symbols) boundary conditions at Wilson flow time $\sqrt{8t}=0.14\mathrm{fm}$. Also shown is the fit to the data.

Figure 11

Plot of the configuration average of the IPR ($⟨\mathrm{IPR}⟩$) versus Wilson flow time ($t/a^2$) for the ensembles $P_1$, $P_2$, and $P_3$.

Figure 12

$⟨\mathrm{IPR}⟩$ versus $\sqrt{8t}$ for ensembles $P_1$, $P_2$, and $P_3$.

Figure 13

Plot of $⟨\mathrm{IPR}⟩$ versus HYP sweeps using two definitions of the topological charge density (clover and ten link) for ensembles $P_1$, $P_2$, and $P_3$.

Figure 14

The behavior of the topological charge density distribution $q(x)$ under the Wilson flow, as a function of $x_0$ and $x_1$ at $x_2=x_3=24$ for a typical configuration belonging to the ensemble $O_3$. The plots (a)–(f) correspond to the flow times $\sqrt{8t}=0.14$, 0.19, 0.25, 0.3, 0.38, and 0.47 fm, respectively.