Asymptotic behavior of cutoff effects in Yang-Mills theory and in Wilson's lattice QCD

Abstract

Discretization effects of lattice QCD are described by Symanzik's effective theory when the lattice spacing, $a$, is small. Asymptotic freedom predicts that the leading asymptotic behavior is $\sim a^n[{\bar{g}}^2(a^{-1}){]}^{{\hat{\gamma }}_1}\sim a^n{\left[\frac{1}{-\log (a\Lambda )}\right]}^{{\hat{\gamma }}_1}$. For spectral quantities, $n=d$ is given in terms of the (lowest) canonical dimension, $d+4$, of the operators in the local effective Lagrangian and ${\hat{\gamma }}_1$ is proportional to the leading eigenvalue of their one-loop anomalous dimension matrix ${\gamma }^{(0)}$. We determine ${\gamma }^{(0)}$ for Yang-Mills theory ($n=2$) and discuss consequences in general and for perturbatively improved short distance observables. With the help of results from the literature, we also discuss the $n=1$ case of Wilson fermions with perturbative O$(a)$ improvement and the discretization effects specific to the flavor currents. In all cases known so far, the discretization effects are found to disappear faster than the naive $\sim a^n$ and the log-corrections are a rather weak modification -- in contrast to the two-dimensional O(3) sigma model.