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_{mathrm{min}}}[{bar{g}}^2(a^{-1})]^{hat{gamma }_1} sim a^{n_{mathrm{min}}}left[ frac{1}{-log (aLambda )}right] ^{hat{gamma }_1}. For spectral quantities, {n_{mathrm{min}}}=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_{mathrm{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_{mathrm{min}}}=1 case of Wilson fermions with perturbative mathrm{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}}} behavior with rather small logarithmic corrections – in contrast to the two-dimensional O(3) sigma model.
Highlights
Lattice regularizations provide a definition of quantum field theories beyond perturbation theory
While we cannot predict the relative contribution of the two powers γ1, γ2 because they depend on the non-perturbative matrix elements MRGI, their mixture is the same for any of the three different actions
Since we have seen that the one-loop anomalous dimension of Ob vanishes, this is equivalent to the form used by the ALPHA collaboration recently [43,50]
Summary
Lattice regularizations provide a definition of quantum field theories beyond perturbation theory. Since the work of [9], continuum extrapolations are routinely performed in order to obtain quantitative numbers for continuum field theory observables They have been carried out with just powers2of a, implicitly assuming that γis small. An explicit example is provided by the seminal work of Balog, Niedermayer and Weisz [7,8] It concerns the 2-d O(3) sigma model where the leading term is γ = −3 and the logarithmic corrections change the naive a2 behavior to a shape which numerically looks like a in a broad range of a [7,8]. As a first step, we do carry out the program in the pure Yang–Mills (YM) theory as well as in Wilson’s lattice QCD without non-perturbative O(a) improvement The latter case is rather simple and basically given by results in the literature.
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