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 $\sqrt{t_0}/a$ and $w_0/a$ and their tree-level improvements, $\sqrt{t_{0,\text{imp}}}$ and $w_{0,\text{imp}}$, 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 $\sqrt{t_0}=0.1416(\genfrac{}{}{0ex}{}{+8}{-5})\mathrm{fm}$ and $w_0=0.1714(\genfrac{}{}{0ex}{}{+15}{-12})\mathrm{fm}$, 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 $⟨E(t)⟩$. For long flow times, the autocorrelation length of $⟨E⟩$ appears to be comparable to that of the topological charge.

Figure 1

The scale $w_0/a$ measured on individual configurations as a function of the simulation time in molecular-dynamics time units. Configuration streams generated with RHMC and RHMD are represented by solid-red and dashed-blue lines, respectively.

Figure 2

The autocorrelation function of $⟨E(t)⟩$ plotted as a function of the separation between configurations $\tau$ in molecular-dynamics time units. For both figures the flow time $t$ is fixed near the value of $w_0^2$. The top plot is for the larger-volume ensemble with $a\approx 0.12\mathrm{fm}$, $m_l^{\prime }=m_s^{\prime }/10$. For visibility, only every fifth point in $\tau$ is shown. The bottom plot is for the $a\approx 0.09\mathrm{fm}$, $m_l^{\prime }=m_s^{\prime }/10$ ensemble. Estimates of the standard error using jackknife and the formulas in Ref. [22] are present for each data point, with the latter shifted slightly right for visibility. The lower number of analyzed configurations on the finer ensemble leads to a relatively larger statistical error and step size in $\tau$. In both plots the vertical dashed lines correspond to the smallest and largest values of ${\tau }_{\text{cut}}$ considered for each ensemble (see the text).

Figure 3

The integrated autocorrelation length (in molecular-dynamics time units) as a function of flow time for ensembles with $m_l^{\prime }/m_s^{\prime }=0.1$ and different lattice spacings. The thickness of the colored regions show the full range of the $1\sigma$ errors, obtained by adding, in quadrature, the statistical error and bias due to ${\tau }_{\text{cut}}$. Dashed vertical lines denote the flow times that determine $w_{0,\text{orig}}$ on each ensemble where the color of the line matches the color of the shaded region. For early flow times on the $a\approx 0.09\mathrm{fm}$ ensemble the integrated autocorrelation length is not plotted because the smallest step in $\tau$ between configurations is insufficient to resolve the dominant contributions to the autocorrelation function.

Figure 4

Simple continuum extrapolations for $\sqrt{t_{0,\text{orig}}}{F}_{p4s}$ and $w_{0,\text{orig}}{F}_{p4s}$ over physical quark-mass ensembles only. Statistical error bars are present, but they are nearly invisible on this scale. Three fits to each data set are shown. The red, dot-dashed line is a linear fit in $a^2$ to the three finer ensembles ($a<0.15\mathrm{fm}$), the blue dashed line is a linear fit in $a^2$ to all four ensembles, and the green solid line is a quadratic fit in $a^2$ to all four ensembles. The continuum extrapolation points, calculated from $\sqrt{t_{0,\text{imp}}}$ and $w_{0,\text{imp}}$, are shown in magenta with error bars representing the sum of statistical and systematic uncertainties in quadrature.

Figure 5

Simple continuum extrapolations for the original ($\sqrt{t_{0,\text{orig}}}$ and $w_{0,\text{orig}}$) and improved ($\sqrt{t_{0,\text{imp}}}$ and $w_{0,\text{imp}}$) gradient-flow scale times $F_{p4s}$ over only physical quark-mass ensembles. Quadratic fits in $a^2$ or ${\alpha }_s{a}^2$ over all four physical-mass ensembles are shown for the original and improved scales, respectively. The continuum extrapolation points, calculated from the improved scales, are shown in black with error bars representing the sum of statistical and systematic uncertainties in quadrature.

Figure 6

The “acceptability” for the various fits considered for the $t_0$ scales ($\sqrt{t_{0,\text{orig}}}$ and $\sqrt{t_{0,\text{imp}}}$, top panel) and $w_0$ scales ($w_{0,\text{orig}}$ and $w_{0,\text{imp}}$, bottom panel). Fit acceptability is determined by the $p$ value ($x$ axis) and further illustrated by the proximity to the results from the physical-mass $a\approx 0.06\mathrm{fm}$ ensemble in units of ${\sigma }_{\text{stat}}$ ($y$ axis). The size of the points is proportional to the number of degrees of freedom. The region to the right of the black line contains fits with $0.01<p<1.0$ and a deviation of less than $2{\sigma }_{\text{stat}}$. This line determines the acceptable subset of fits considered in the subsequent analysis. The central fit chosen from this analysis is denoted by the star.

Figure 7

Histograms of the continuum extrapolations for $\sqrt{t_0}{F}_{p4s}$ (top panel) and $w_0{F}_{p4s}$ (bottom panel) for all acceptable fits (see the text). Each histogram is a stacked combination of continuum extrapolations from the original ($\sqrt{t_{0,\text{orig}}}$ and $w_{0,\text{orig}}$) and improved scales ($\sqrt{t_{0,\text{imp}}}$ and $w_{0,\text{imp}}$), represented by the red, hashed and green, solid bars, respectively. The box and error bars along the bottom denote the minimum, mean, maximum, and central 68% of the distribution. The vertical dashed line for each distribution marks the continuum result of the associated central fit.

Figure 8

The central fits to the gradient-flow scales $\sqrt{t_{0,\text{imp}}}{F}_{p4s}$ and $w_{0,\text{imp}}{F}_{p4s}$, plotted as a function of ${\alpha }_s{a}^2$. These are used to compute $\sqrt{t_0}$ (top panel) and $w_0$ (bottom panel) at physical quark masses and in the continuum, as indicated by the black stars. Only $m_s^{\prime }\approx m_s$ ensembles are plotted, but the fits include $0.25m_s<m_s^{\prime }\le m_s$ ensembles. Dashed lines represent the fit through each ensemble’s actual quark masses and lattice spacing, while the solid bands are for varying lattice spacing at fixed quark masses retuned to the physical strange-quark mass and the ratio of $m_l^{\prime }/m_s^{\prime }$ specified in the legend.

Figure 9

An alternative fit to the gradient-flow scale $w_{0,\text{imp}}{F}_{p4s}$, plotted as a function of ${\alpha }_s{a}^2$. The value of $w_0$ at physical quark masses and in the continuum estimated from this fit is indicated by the black star. Only $m_s^{\prime }\approx m_s$ ensembles are plotted, but the fit includes $0.25m_s<m_s^{\prime }\le m_s$ ensembles. Dashed lines represent the fit through each ensemble’s actual quark masses and lattice spacing, while the solid bands are for varying lattice spacing at fixed quark masses retuned to the physical strange-quark mass and the ratio of $m_l^{\prime }/m_s^{\prime }$ specified in the legend.

Figure 10

Continuum extrapolations for the original ($\sqrt{t_{0,\text{orig}}}$ and $w_{0,\text{orig}}$) and improved ($\sqrt{t_{0,\text{imp}}}$ and $w_{0,\text{imp}}$) gradient-flow scale times $F_{p4s}$ plotted for physical quark-mass ensembles only. All fits to the original, unimproved scales include the chiral expansion to NNLO, $1/m_c^2$ NLO charm-quark corrections, the wider set of priors, all four lattice spacings, and all but the three lightest $m_s^{\prime }$ ensembles. For $\sqrt{t_{0,\text{orig}}}{F}_{p4s}$ the fit is quadratic in $a^2$. For $w_{0,\text{orig}}{F}_{p4s}$ the fit is linear in $a^2$ and ${\alpha }_s{a}^2$. For the improved scales the plotted lines are from the central fits discussed in this section. The continuum-extrapolation points are shown in black with error bars representing the statistical error only.

Figure 11

The continuum mass dependence of $\sqrt{t_0}$ and $w_0$ as a function of $P=(w_0{M}_{\pi }{)}^2$ for fixed values of $K=(w_0{M}_K{)}^2$. The black points and the star illustrate the values of the pion and kaon masses that correspond to various HISQ ensembles and to the physical point, respectively. The three boxes on the plot of $w_0$ enclose the physical strange mass ensembles with different ratios of $m_l^{\prime }/m_s^{\prime }$. From the left- to the rightmost box the ratios are $m_l^{\prime }/m_s^{\prime }=1/27$, $1/10$, and $1/5$. Similar boxes are not drawn for $\sqrt{t_0}$ due to the large discretization effects separating the points that would go in each box. The lines corresponding to $K=0.1K_{\text{phys}}$ do not extend over the full domain of $P$ because we restrict $(w_0{M}_{\eta }{)}^2\approx (4K^2-P^2)/3\ge 0$.

Figure 12

The continuum values of $\sqrt{t_0}$ and $w_0$ separated by collaboration and grouped by the number of flavors. References for each collaboration’s work can be found in Table 9. Those results for collaborations marked with an asterisk are preliminary. Our results for $\sqrt{t_0}$ and $w_0$ are consistent within 2 standard deviations to all other results except the preliminary calculations from the ALPHA Collaboration.