Two-loop computation of the Schrödinger functional in pure SU(3) lattice gauge theory

Abstract

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.

Two-loop computation of the Schrödinger functional in lattice QCD

“…[8], [14] respectively (and for the case of SU(2) in [26]); we note that in these papers the coefficients p i0 were denoted by m i . The one-loop coefficient p 11 was computed in ref.…”

“…This has been described in detail in previous publications [7], [26], [14]. In particular we use the gauge fixing procedure as in ref.…”

“…The numerical computation of the gluon propagator is described in Section 2 of ref. [14], and here in Appendix B we give some technical details on how we numerically computed the fermion propagator S. An enumeration of the Feynman diagrams contributing to the Schrödinger functional to two-loop order reveals that the coefficients p ij depend on the improvement coefficients and critical mass in the following way:…”

“…All the coefficients coming from diagrams not involving the quarks have been computed in [14]. Here we recall that only two of the coefficients can be given in closed form:…”

“…[9], [10] and [11]. In this paper we perform the second step, the relationship of α SF to α 0 for N f > 0 flavors (the case N f = 0 has already been presented in [12]- [14]). The final result between the continuum quantities (1.1) is of course regularization independent, but the intermediate relations in both steps depend on the details of the lattice action employed.…”