Effects of nonequilibrated topological charge distributions on pseudoscalar meson masses and decay constants

We study the effects of failure to equilibrate the squared topological charge $Q^2$ on lattice calculations of pseudoscalar masses and decay constants. The analysis is based on chiral perturbation theory calculations of the dependence of these quantities on the QCD vacuum angle $\theta$. For the light-light partially quenched case, we rederive the known chiral perturbation theory results of Aoki and Fukaya, but using the nonperturbatively valid chiral theory worked out by Golterman, Sharpe and Singleton, and by Sharpe and Shoresh. We then extend these calculations to heavy-light mesons. Results when staggered taste violations are important are also presented. The derived $Q^2$ dependence is compared to that of simulations using the MILC Collaboration’s ensembles of lattices with four flavors of dynamical highly improved staggered quarks. We find agreement, albeit with large statistical errors. These results can be used to correct for the leading effects of unequilibrated $Q^2$, or to make estimates of the systematic error coming from the failure to equilibrate $Q^2$. In an appendix, we show that the partially quenched chiral theory may be extended beyond a lower bound on valence masses discovered by Sharpe and Shoresh. Subtleties occurring when a sea-quark mass vanishes are discussed in another appendix.

Figure 1

Topological charge time histories for various lattice spacings; note that the vertical scale decreases as the lattice spacing decreases (top to bottom). Blue lines are for ensembles with a light sea-quark mass that is one fifth of the strange-quark mass, and the red lines are for ensembles with a light sea-quark mass at its physical value, $\approx m_s/27$. Notice the narrower distributions and shorter autocorrelation times for physical quark mass ensembles. Breaks in the traces separate multiple runs at the same couplings. The second short blue trace at $a=0.0425\mathrm{fm}$ is from a run with 3 times longer molecular dynamics trajectories than the main run.

Figure 2

Autocorrelations of the squared topological charge, where $A(\mathrm{\Delta })$ is defined in Eq. (1). For each lattice spacing, the crosses are the ensemble with $m_l=m_s/5$ and the octagons the ensemble with $m_l$ at its physical value, $\approx m_s/27$.

Figure 3

Average topological susceptibility $⟨Q^2/V⟩$ (crosses) and tunneling rate per unit volume $⟨{(\mathrm{\Delta }Q)}^2/(V\mathrm{\Delta }t)⟩$ (octagons) versus lattice spacing. Red and blue points are results for ensembles with $m_l=m_s/5$ and $m_l$ approximately physical, respectively. Similarly, the magenta and green squares are the lowest-order chiral perturbation theory predictions for the susceptibility at $m_l=m_s/5$ and $m_l$ at its physical value, $\approx m_s/27$. The leading-order $\chi \mathrm{PT}$ results shown include staggered corrections and are taken from Eq. (95) below.

Figure 4

$\frac{{\partial }^{2}M}{{\partial \theta }^2}$ on ensembles with $m_l=m_s/5$ as a function of $m_x$, with $m_y=m=m_x$. The black line is the $\mathrm{PQ}\chi \mathrm{PT}$ prediction at $a=0$, or without taste-breaking effects (no free parameters). The green and red lines show the $\mathrm{PQ}\chi \mathrm{PT}$ prediction including taste breaking [Eq. (86)] for the 0.09 and 0.06 fm ensembles. The black square marks the unitary point, with the valence-quark mass equal to the sea-quark mass.

Figure 5

$\frac{{\partial }^{2}F}{{\partial \theta }^2}$ on ensembles with $m_l=m_s/5$. The left panel shows $\frac{{\partial }^{2}F}{{\partial \theta }^2}$ as a function of one valence quark mass, $m_x$ along the line $m_y=m_s$. The black line is the $\mathrm{PQ}\chi \mathrm{PT}$ prediction (no free parameters), which vanishes for degenerate quarks. The red ($a=0.06\mathrm{fm}$) and green ($a=0.09\mathrm{fm}$) lines include the effects of taste breaking from Eq. (87). The right panel shows the quantity with $m_y$ fixed to the smallest available value. The dotted lines are $\mathrm{PQ}\chi \mathrm{PT}$ predictions ignoring taste breaking, or at $a=0$; there are three separate lines because the smallest valence-quark mass is different in each ensemble: $0.1m_s$, $0.05m_s$ and $0.037m_s$ for the 0.09, 0.06 and 0.042 fm ensembles respectively. Again, the solid red and green lines include the effects of taste breaking from Eq. (87).