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

Understand

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.