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

We derive the asymptotic lattice spacing dependence $a^n[2b_0{\bar{g}}^2(1\mathrm{/}a){]}^{{\widehat{\mathrm{\Gamma }}}_i}$ 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 ${\widehat{\mathrm{\Gamma }}}_i$ 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 ${\min }_i{\widehat{\mathrm{\Gamma }}}_i\gtrsim -0.3$ 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].