Logarithmic corrections to O($a$) and O($a^2$) effects in lattice QCD with Wilson or Ginsparg-Wilson quarks

Abstract

We derive the asymptotic lattice spacing dependence $a^n[2b_0{\bar{g}}^2(1/a){]}^{{\hat{\Gamma }}_i}$ relevant for spectral quantities of lattice QCD, when using Wilson, O$(a)$ improved Wilson or Ginsparg-Wilson quarks. We give some examples for the spectra encountered for ${\hat{\Gamma }}_i$ including the partially quenched case, mixed actions and using two different discretisations for dynamical quarks. This also includes maximally twisted mass QCD relying on automatic O$(a)$ improvement. At O$(a^2)$, all cases considered have ${\min }_i{\hat{\Gamma }}_i\gtrsim -0.3$ if $N_{\mathrm{f}}\le 4$, which ensures that the leading order lattice artifacts are not severely logarithmically enhanced in contrast to the O$(3)$ non-linear sigma model [1,2]. However, we find a very dense spectrum of these leading powers, which may result in major pile-ups and cancellations. We present in detail the computational strategy employed to obtain the 1-loop anomalous dimensions already used in [3].

39 pages, 5 figures, 2 tables, Mathematica notebook as supplement; Fixed typo in indices in eq. (6.2) and adjusted the manuscript to comments of the referee. In particular, added discussion of spurionic symmetry for the massive operators

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• The strong coupling from a nonperturbative determination of the $\Lambda$ parameter in three-flavor QCD
• The puzzle of apparent linear lattice artifacts in the 2d non-linear sigma-model and Symanzik's solution
• Logarithmic corrections to O(a^2) lattice artifacts
• The asymptotic approach to the continuum of lattice QCD spectral observables
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