The Schr\"odinger functional in QCD

TL;DR

This paper investigates the Schr"odinger functional in lattice QCD, analyzing its continuum limit and renormalization properties, including one-loop perturbative corrections and boundary field renormalizations.

Contribution

It provides a detailed one-loop perturbative analysis of the Schr"odinger functional in QCD, confirming the necessity of boundary counterterms and their local polynomial form.

Findings

The Schr"odinger functional leads to a sensible continuum theory.

Divergences are canceled by standard renormalizations and boundary counterterms.

Boundary fields require multiplicative renormalization.

Abstract

The Schr\"odinger functional in Wilson's lattice QCD leads to a sensible classical continuum theory which can be taken as starting point for a perturbative analysis. In dimensional regularization, the saddle point expansion of the Schr\"odinger functional is performed to one-loop order of perturbation theory. The divergences are partly cancelled by the usual coupling constant and quark mass renormalization. An additional divergence can be absorbed in a multiplicative renormalization of the quark boundary fields. The corresponding boundary counterterm being a local polynomial in the fields we confirm the general expectation expressed by Symanzik~\cite{Symanzik}.