Manifestly Gauge Invariant Exact Renormalization Group

TL;DR

This paper develops a gauge invariant exact renormalization group framework for SU(N) Yang-Mills theory that avoids gauge fixing, enabling non-perturbative calculations and insights into confinement and QCD.

Contribution

It introduces a novel gauge invariant ERG approach using spontaneously broken SU(N|N) symmetry, allowing calculations without gauge fixing or cutoff specification.

Findings

Computed one and two loop beta functions without fixing gauge.

Proposed a non-perturbative expansion scheme in 1/Λ_{QCD}.

Analyzed confinement as coupling diverges.

Abstract

We construct a manifestly gauge invariant Exact Renormalization Group for SU(N) Yang-Mills theory, in a form suitable for calculations without gauge fixing at any order of perturbation theory. The effective cutoff is incorporated via a manifestly realised spontaneously broken SU(N|N) gauge invariance. Diagrammatic methods are developed which allow the calculations to proceed without specifying the precise form of the cutoff structure. We confirm consistency by computing for the first time both the one and two loop beta function coefficients without fixing the gauge or specifying the details of the cutoff. We sketch how to incorporate quarks and thus compute in QCD. Finally we analyse the renormalization group behaviour as the renormalized coupling becomes large, and show that confinement is a consequence if and only if the coupling diverges in the limit that all modes are integrated out. We also investigate an expansion in the inverse square renormalized coupling, and show that under general assumptions it yields a new non-perturbative approximation scheme corresponding to expanding in 1/\Lambda_{QCD}.