Scale Setting and Strong Coupling Determination in the Gradient Flow Scheme for 2+1 Flavor Lattice QCD

Abstract

We report on a scale determination, scale setting, and determination of the strong coupling in the gradient flow scheme using the $N_f=2+1$ highly improved staggered quark (HISQ) ensembles generated by the HotQCD Collaboration for bare gauge couplings ranging from $β= 7.030$ to $8.400$. The gradient flow scales we obtain in this work are $\sqrt{t_0} = 0.14229(98)$~fm and $w_0 = 0.17190(140)$~fm. Using the decay constants of the kaon and $η_s$, as well as the bottomonium mass splitting from the literature, we also calculate the potential scale $r_1$, obtaining $r_1 = 0.3072(22)$~fm. We fit the flow scales to an Allton-type ansatz as a function of $β$, providing a polynomial expression that allows for the prediction of lattice spacings at new $beta$ values. As a secondary result, we make an attempt to determine $Λ_{\overline{\mathrm{MS}}}$ and use it to estimate the strong coupling in the $\overline{\mathrm{MS}}$ scheme.

Figures (16)

• Figure 1: Numerical results for the action density, $E$ (left) and the derivative of $\tau_\mathrm{F}^2 E$(right) as function of the flow time obtained using the improved discretization.
• Figure 2: The gradient flow scales $\sqrt{t_0}$ and $w_0$ obtained from the improved discretization as functions of $\beta$. The open points in blue are for the $m_s/m_l=20$ lattices while the filled points in orange are for the $m_s/m_l=5$ lattices. The solid lines (means) and bands (errors) represent the fits to Allton Ansatz of the $m_s/m_l=20$ lattices ($6.423\leq \beta\leq 7.825$). The dashed lines represent the extrapolation of the fits to the $m_s/m_l=5$ lattices, to test the robustness of the fit by comparing its extrapolated predictions with data not included in the fitting procedure.
• Figure 3: The ratio of different gradient flow scale as function of the lattice spacing together with the continuum extrapolations. The flow scales are measured using both the clover discretizations and the improved discretization (labelled as "clover" and "impr.", respectively, see Eq. (\ref{['eq:Fmunu']})-Eq. (\ref{['eq:impF']})). In these plots only the "best" fits are shown which correspond to the numbers in bold in Tab. \ref{['tab:fit-param-flow-ratio']}.
• Figure 4: Continuum extrapolation of the ratio of the flow scales to the $r_1$ scale. Left: $\sqrt{t_0}/r_1$. Similar to before, the discretization effects are significant in this case. After applying the $\mathcal{O}(a^2)$ improvement to the field strength tensor, the slope of the extrapolation (using Ansatz linear in $a^2$) changes sign. However, in the continuum limit, the two discretizations give consistent results. Right: $w_0/r_1$. "WF" stands for Wilson flow and "ZF" for Zeuthen flow. We include results from the HotQCD Collaboration (red points) HotQCD:2014kol using Wilson flow and clover-discretized action density for comparison. It can be observed that for $w_0$, the discretization effects are so mild that the two discretizations give almost the same results. Our results using Zeuthen flow are slightly smaller than the HotQCD Collaboration's previous results using Wilson flow.
• Figure 5: The ratio of the gradient flow slales $\sqrt{t_2}$ and $w_2$ to $r_2$ scale as function of the lattice spacing.