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 [ḡ 2 (a −1 )]γ 1 ∼ a n min 1 − log(aΛ) 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 andγ 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 disappear faster than the naive ∼ a n min and the log-corrections are a rather weak modification -in contrast to the two-dimensional O(3) sigma model.

We investigate the SU(3) Yang Mills theory at small gradient flow time and at short distances. Lattice spacings down to a = 0.015 fm are simulated with open boundary conditions to allow topology to flow in and out. We study the behaviour of the action density E(t) close to the boundaries, the feasibility of the small flow-time expansion and the extraction of the Λ-parameter from the static force at small distances. For the latter, significant deviations from the 4-loop perturbative β-function are visible at α ≈ 0.2 . We still can extrapolate to extract r 0 Λ.Speaker,

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We derive the asymptotic lattice spacing dependence a n [2b 0 ḡ2 (1/a)] Γi 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 Γi 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 Γi −0.3 if N f ≤ 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].