One-loop renormalization of the QCD Schr\"odinger functional

TL;DR

This paper performs a one-loop perturbative analysis of the Schr"odinger functional in lattice QCD, identifying divergences and renormalizations, including boundary field renormalization, confirming theoretical expectations.

Contribution

It provides the first one-loop renormalization calculation of the Schr"odinger functional in Wilson's lattice QCD, including boundary counterterms.

Findings

One-loop divergences are identified and partially canceled by standard renormalizations.

Boundary fields require additional multiplicative renormalization.

Results confirm the theoretical expectation of local polynomial boundary counterterms.

Abstract

In a previous publication, we have constructed the Schr\"odinger functional in Wilson's lattice QCD. It was found that the naive continuum limit leads to a well-defined classical continuum theory. Starting from the latter, a formal continuum definition of the Schr\"odinger functional is given and its saddle point expansion is carried out to one-loop order of perturbation theory. Dimensional regularization and heat kernel techniques are used to determine the one-loop divergences. These are partly canceled by the usual renormalizations of the quark mass and the coupling constant in QCD. An additional divergence can be absorbed in a multiplicative renormalization of the quark boundary fields. The corresponding boundary counterterm is a local polynomial in the fields, so that we confirm a general expectation of Symanzik.