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

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 ∼ a n min [ g ¯ 2 ( a - 1 ) ] γ ^ 1 ∼ a n min 1 - log ( a Λ ) γ ^ 1 . For spectral quantities, n min = d is given in terms of the (lowest) canonical dimension, d + 4 , of the operators in the local effective Lagrangian and γ ^ 1 is proportional to the leading eigenvalue of their one-loop anomalous dimension matrix γ ( 0 ) . We determine γ ( 0 ) for Yang–Mills theory ( n min = 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 min = 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 vanish faster than the naive ∼ a n min behavior with rather small logarithmic corrections – in contrast to the two-dimensional O(3) sigma model.