Nonperturbative $O(a)$ improvement of the Wilson quark action with the RG-improved gauge action using the Schrödinger functional method

TL;DR

The paper tackles discretization errors in Wilson quark actions by nonperturbatively determining the $O(a)$ improvement coefficient $c_{ m SW}$ and the critical hopping parameter $\, frac{1}{2} abla M_c$ using the Schrödinger functional with an RG-improved gauge action. It identifies sizable finite-volume ($O(a/L)$) effects and introduces a fixed physical length scale $L^*$ to remove residual $O(a)$ scaling violations, converting results between $L/a$ and $L^*/a$ with perturbative corrections. The authors produce interpolation formulas for $c_{ m SW}$ and $\, frac{1}{2} abla abla_c$ across $N_f=3,2,0$, facilitating accurate continuum extrapolations in dynamical QCD simulations. They also assess systematic uncertainties from the scale-setting procedure and perturbative corrections, concluding these are controlled within their chosen framework. Overall, the work enables more reliable, $O(a)$-improved lattice QCD with RG-improved gauge actions in realistic quark scenarios.

Abstract

We perform a nonperturbative determination of the $O(a)$-improvement coefficient $c_{\rm SW}$ and the critical hopping parameter $κ_c$ for $N_f$=3, 2, 0 flavor QCD with the RG-improved gauge action using the Schrödinger functional method. In order to interpolate $c_{\rm SW}$ and $κ_c$ as a function of the bare coupling, a wide range of $β$ from the weak coupling region to the moderately strong coupling points used in large-scale simulations is studied. Corrections at finite lattice size of $O(a/L)$ turned out to be large for the RG-improved gauge action, and hence we make the determination at a size fixed in physical units using a modified improvement condition. This enables us to avoid $O(a)$ scaling violations which would remain in physical observables if $c_{\rm SW}$ determined for a fixed lattice size $L/a$ is used in numerical simulations.

Figures (5)

• Figure 1: $aM$ dependence of $a\Delta M$ in three-flavor QCD.
• Figure 2: Same as Fig. \ref{['fig:MandDM_3f']}, but in two-flavor QCD.
• Figure 3: Same as Fig. \ref{['fig:MandDM_3f']}, but in quenched QCD.
• Figure 4: $g^{2}_{0}$ dependence of $c^{\rm NP}_{\rm SW}(g^{2}_{0},L^*/a)$ in $N_f$=3, 2, and 0 flavor QCD from top to bottom. Filled symbols are used for curve fitting.
• Figure 5: $g^{2}_{0}$ dependence of $\kappa_c(g^{2}_{0},L^*/a)$ in $N_f$=3, 2, and 0 flavor QCD from top to bottom.