Asymptotic behavior of cutoff effects in Yang–Mills theory and in Wilson’s lattice QCD

Abstract

AbstractDiscretization 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 $$\sim a^{n_{\mathrm{min}}}[{\bar{g}}^2(a^{-1})]^{\hat{\gamma }_1} \sim a^{n_{\mathrm{min}}}\left[ \frac{1}{-\log (a\Lambda )}\right] ^{\hat{\gamma }_1}$$∼anmin[g¯2(a-1)]γ^1∼anmin1-log(aΛ)γ^1. For spectral quantities, $${n_{\mathrm{min}}}=d$$nmin=d is given in terms of the (lowest) canonical dimension, $$d+4$$d+4, of the operators in the local effective Lagrangian and $$\hat{\gamma }_1$$γ^1 is proportional to the leading eigenvalue of their one-loop anomalous dimension matrix $$\gamma ^{(0)}$$γ(0). We determine $$\gamma ^{(0)}$$γ(0) for Yang–Mills theory ($${n_{\mathrm{min}}}=2$$nmin=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 $${n_{\mathrm{min}}}=1$$nmin=1 case of Wilson fermions with perturbative $$\mathrm{O}(a)$$O(a) improvement and the discretization effects specific to the flavor currents. In all cases known so far, the discretization effects are found to vanish faster than the naive $$\sim a^{n_{\mathrm{min}}}$$∼anmin behavior with rather small logarithmic corrections – in contrast to the two-dimensional O(3) sigma model.