Precision computation of a low-energy reference scale in quenched lattice QCD

TL;DR

This work delivers a precision determination of the low-energy reference scale $r_0$ in SU(3) lattice gauge theory over $5.7\leq\beta\leq6.57$, achieving $0.3$–$0.6\%$ relative errors in $r_0/a$ through variance-reduced Wilson-loop measurements and a variational extraction of the static potential. A uniquely defined, tree-level improved lattice force is used to set $r_0$ via $r_0^2 F(r_0)=1.65$, with $F(r)$ interpolated for continuum matching, enabling smooth dependence of $r_0/a$ on the bare coupling. The authors provide a cubic polynomial parametrization of $\ln(a/r_0)$ in terms of $(\beta-6)$, achieving high-precision interpolation across the studied range, and apply $r_0$ to continuum extrapolations of the ground-state potential gap $\Delta$ and of the scale ratio $L_{\max}/r_0$ in the Schrödinger functional scheme, obtaining $r_0\Delta|_{r=r_0}=3.3(1)$ and $(L_{\max}/r_0)_{a=0}=0.718(16)$. These results yield a robust, gluonic reference scale for nonperturbative scale setting and running-coupling analyses in quenched QCD, and they highlight the importance of precise force definitions and higher-order improvement in controlling lattice artifacts for gluonic observables.

Abstract

We present results for the reference scale r_0 in SU(3) Lattice Gauge Theory for beta = 6/g_0^2 in the range 5.7 <= beta <= 6.57. The high relative accuracy of 0.3-0.6% in r_0/a was achieved through good statistics, the application of a multi-hit procedure and a variational approach in the computation of Wilson loops. A precise definition of the force used to extract r_0 has been used throughout the calculation which guarantees that r_0/a is a smooth function of the bare coupling and that subsequent continuum extrapolations are possible. The results are applied to the continuum extrapolations of the energy gap Delta in the static quark potential and the scale L_max/r_0 used in the calculation of the running coupling constant.

Figures (5)

• Figure 1: Estimates $r_0/a$ for various minimum time separations, $t_{\rm min}$, in the fits eq. (\ref{['e_potfits']}). $\beta$ increases from bottom to top and the values for different $\beta$ have been shifted relative to each other for clarity. The $x$-coordinate corresponds to the slowest varying finite-$t$ correction term (as a function of $t_{\rm min}$). For the gap $\Delta$ the value $\Delta \approx 3.3/r_0$ has been used. Dashed error bands denote our final estimates.
• Figure 2: Finite size dependence of the force $F(r)$.
• Figure 3: The data for $\ln(a/r_0)$ (circles) and their representation as a polynomial in $\beta$ (solid line).
• Figure 4: Continuum extrapolation of $r_0\,\Delta$.
• Figure 5: Continuum extrapolation of $L_{\rm max}/r_0$. The two curves show the fit eq. (\ref{['e_cont_extr_ratio']}) who's value in the continuum limit is indicated by the square.