The Schr\"odinger Functional - a Renormalizable Probe for Non-Abelian Gauge Theories

TL;DR

This paper demonstrates that the Schr"odinger functional in lattice gauge theories without matter fields is well-defined in the continuum limit and can be used as a numerical tool to study the renormalized gauge coupling's evolution across energy scales.

Contribution

It establishes the continuum limit of the Schr"odinger functional in non-Abelian gauge theories and performs a 1-loop perturbative analysis to facilitate numerical studies of gauge coupling evolution.

Findings

The Schr"odinger functional has a well-defined continuum limit without extra counter terms.

It is accessible to numerical simulations for studying gauge theories.

Perturbative analysis up to 1-loop order supports its use in renormalization studies.

Abstract

Following Symanzik we argue that the Schr\"odinger functional in lattice gauge theories without matter fields has a well-defined continuum limit. Due to gauge invariance no extra counter terms are required. The Schr\"odinger functional is, moreover, accessible to numerical simulations. It may hence be used to study the scaling properties of the theory and in particular the evolution of the renormalized gauge coupling from low to high energies. A concrete proposition along this line is made and the necessary perturbative analysis of the Schr\"odinger functional is carried through to 1-loop order.