On the Viability of Lattice Perturbation Theory

TL;DR

The paper addresses the apparent failure of lattice perturbation theory when using the bare lattice coupling by introducing renormalized couplings defined from physical inputs, notably α_V from the heavy-quark potential, and by automatically setting the relevant scale q*. Together with mean-field tadpole improvement, this approach yields significantly improved convergence and agreement with Monte Carlo results for short-distance quantities at β as low as 5.7. The authors demonstrate that perturbative expansions in α_V(q*) reproduce MC data for a range of observables (⟨A_μ^2⟩, Wilson loops, Creutz ratios, and Wilson quark mass renormalization) far better than expansions in α_lat, and that tadpole-improved operators reduce renormalization effects, bringing lattice results closer to continuum behavior. Consequently, renormalized perturbation theory with tadpole improvement supports perturbative scaling down to modest lattice spacings, suggesting cost-effective, reliable simulations of quenched QCD on coarser lattices and guiding the design of more continuum-like lattice operators.

Abstract

In this paper we show that the apparent failure of QCD lattice perturbation theory to account for Monte Carlo measurements of perturbative quantities results from choosing the bare lattice coupling constant as the expansion parameter. Using instead ``renormalized'' coupling constants defined in terms of physical quantities, like the heavy-quark potential, greatly enhances the predictive power of lattice perturbation theory. The quality of these predictions is further enhanced by a method for automatically determining the coupling-constant scale most appropriate to a particular quantity. We present a mean-field analysis that explains the large renormalizations relating lattice quantities, like the coupling constant, to their continuum analogues. This suggests a new prescription for designing lattice operators that are more continuum-like than conventional operators. Finally, we provide evidence that the scaling of physical quantities is asymptotic or perturbative already at $β$'s as low as 5.7, provided the evolution from scale to scale is analyzed using renormalized perturbation theory. This result indicates that reliable simulations of (quenched) QCD are possible at these same low $β$'s.

Figures (10)

• Figure 1: The expectation value of the trace of a link in Landau gauge, calculated by Monte Carlo (circles), and in first order perturbation theory using for the expansion parameter $\alpha_{V}(q^*)$ (diamonds), $\alpha_{{\rm\overline{MS}}}(q^*)$ (boxes), and $\alpha_{{\rm lat}}$ (crosses).
• Figure 2: The critical quark mass $m_c$ for Wilson quarks, calculated by Monte Carlo (circles), and in first order perturbation theory using for the expansion parameter $\alpha_{V}(q^*)$ (diamonds), $\alpha_{{\rm\overline{MS}}}(q^*)$ (boxes), and $\alpha_{{\rm lat}}$ (crosses). Statistical errors in the Monte Carlo results are negligible.
• Figure 3: Results for Creutz ratio $\chi_{22}$ at different couplings $\beta$ from Monte Carlo simulations (circles), and from perturbation theory (using $\alpha_{V}(q^*)$ (diamonds), $\alpha_{{\rm\overline{MS}}}(q^*)$ (boxes), and $\alpha_{{\rm lat}}$ (crosses)). The first plot shows perturbation theory through one-loop order, and the second through two-loop order. Statistical errors in the Monte Carlo results are negligible.
• Figure 4: Results from perturbation theory (with $\alpha_V(q^*)$ (diamonds), $\alpha_{\rm\overline{MS}}(q^*)$ (boxes), and $\alpha_{\rm lat}$ (crosses)) and Monte Carlo simulations (circles) for diagonal Creutz ratios $\chi_{n,n}$ at $\beta=6.2$. Statistical errors in the Monte Carlo results are negligible.
• Figure 5: Results for the Creutz ratio $\chi_{22}$ from Monte Carlo simulations (line) and from the perturbation expansion in $\alpha_V(q)$ (diamonds) versus the scale $q$.