Gradient flow and scale setting on MILC HISQ ensembles

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 root t(0)/a and their tree-level improvements, root t(0,imp) and root w(0,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 latticespacing and strong-coupling dependence, the results are simultaneously extrapolated to the continuum and interpolated to physical quark masses. We determine the scales root t(0) = 0.171416((18)(-5)) fm, and w(0) =.174 ((-15)(-12)), where the errors are sums, in quadrature, of statistical and all systematic errors. The precision of w(0) and root t(0) is comparable to or more precise than the best previous estimates, respectively. We then find the continuum mass dependence of root t(0) and w(0), which will be useful for estimating the scales of new ensembles. We also estimate the integrated autocorrelation length of < E(t)>. For long flow times, the autocorrelation length of < E > appears to be comparable to that of the topological charge.