We compute the Schrödinger functional (SF) for the case of lattice QCD with Wilson fermions (with and without SW improvement) at two-loop order in lattice perturbation theory. This allows us to extract the three-loop β-function in the SF-scheme. These results are required to compute the running coupling, the Λ-parameter and quark masses by finite size techniques with negligible systematic errors. In addition our results enable the implementation of two-loop O(a) improvement in SF-simulations. This article is based on the revised version of ref. [11].

We compute the Schrödinger functional (SF) for the case of pure SU(3) gauge theory at two-loop order in lattice perturbation theory. This allows us to extract the three-loop β-function in the SF-scheme. These results are required to compute the running coupling, the Λ-parameter and quark masses by finite size techniques with negligible systematic errors. In addition, we can now implement two-loop O(a) improvement in simulations and extend and study series in alternative ("tadpole-improved") bare couplings.

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We calculate, to 3 loops in perturbation theory, the bare β-function of QCD, formulated on the lattice with the clover fermionic action. The dependence of our result on the number of colors N , the number of fermionic flavors N f , as well as the clover parameter c SW , is shown explicitly.A direct outcome of our calculation is the two-loop relation between the bare coupling constant g 0 and the one renormalized in the MS scheme.Further, we can immediately derive the three-loop correction to the relation between the lattice Λ-parameter and g 0 , which is important in checks of asymptotic scaling. For typical values of c SW , this correction is found to be very pronounced.

We investigate monopoles in four-dimensional compact U(1) with Wilson action. We focus our attention on monopole clusters as they can be identified unambiguously contrary to monopole loops. We locate the clusters and determine their properties near the U(1) phase transition. The Coulomb phase is characterized by several small clusters, whereas in the confined phase the small clusters coalesce to one large cluster filling up the whole system. We find that clusters winding around the periodic lattice are absent within both phases and during the transition. However, within the confined phase, we observe periodically closed monopole loops if cooling is applied.