Gradient flow and scale setting on MILC HISQ ensembles

TL;DR

This paper uses gradient-flow techniques on MILC HISQ ensembles to precisely determine lattice scales, extrapolate to the continuum, and analyze autocorrelation lengths, improving the accuracy of scale setting in lattice QCD simulations.

Contribution

It introduces a comprehensive scale setting method using gradient flow on MILC HISQ ensembles, including continuum extrapolation and autocorrelation analysis, with improved precision over previous estimates.

Findings

Determined scales .1416 fm and .1714 fm with high precision.

Established continuum mass dependence of .1416 fm and .1714 fm.

Estimated autocorrelation length of .1416 fm and .1714 fm for long flow times.

Abstract

We report on a scale determination with gradient-flow techniques on the $N_f=2+1+1$ highly improved staggered quark ensembles generated by the MILC Collaboration. The ensembles include four lattice spacings, ranging from approximately 0.15 to 0.06 fm, and both physical and unphysical values of the quark masses. The scales $\sqrt{t_0}/a$ and $w_0/a$ and their tree-level improvements, $\sqrt{t_{0,{\rm imp}}}$ and $w_{0,{\rm imp}}$, are computed on each ensemble using Symanzik flow and the cloverleaf definition of the energy density $E$. Using a combination of continuum chiral-perturbation theory and a Taylor-series ansatz for the lattice-spacing and strong-coupling dependence, the results are simultaneously extrapolated to the continuum and interpolated to physical quark masses. We determine the scales $\sqrt{t_0} = 0.1416({}_{-5}^{+8})$ fm and $w_0 = 0.1714({}_{-12}^{+15})$ fm, where the errors are sums, in quadrature, of statistical and all systematic errors. The precision of $w_0$ and $\sqrt{t_0}$ is comparable to or more precise than the best previous estimates, respectively. We then find the continuum mass dependence of $\sqrt{t_0}$ and $w_0$, which will be useful for estimating the scales of new ensembles. We also estimate the integrated autocorrelation length of $\langle E(t) \rangle$. For long flow times, the autocorrelation length of $\langle E \rangle$ appears to be comparable to that of the topological charge.