Confinement in Covariant Gauges

TL;DR

This paper investigates the behavior of Euclidean SU(n) gauge theory in covariant gauges, revealing spontaneous symmetry breaking and area-law Wilson loops, suggesting a confinement mechanism driven by Goldstone bosons in the weak coupling limit.

Contribution

It introduces an equivariant BRST construction for continuum gauge theories on a finite torus and analyzes the resulting symmetry breaking and confinement phenomena in covariant gauges.

Findings

Global SU(n) symmetry breaks to U(1)^{n-1} in certain gauges.

Wilson loops exhibit area-law decay indicating confinement.

Goldstone bosons dominate long-range gauge boson propagators.

Abstract

We examine the weak coupling limit of Euclidean SU(n) gauge theory in covariant gauges. Following an earlier suggestion, an equivariant BRST-construction is used to define the continuum theory on a finite torus. The equivariant gauge fixing introduces constant ghost fields as moduli of the model. We study the parameter- and moduli- space perturbatively. For $n_f \leq n$ quark flavors, the moduli flow to a non-trivial fixed point in certain critical covariant gauges and the one-loop effective potential indicates that the global SU(n) color symmetry of the gauge fixed model is spontaneously broken to $U(1)^{n-1}$. Ward identities and renormalization group arguments imply that the longitudinal gauge boson propagator at long range is dominated by $n(n-1)$ Goldstone bosons in these critical covariant gauges. In the large $n$ limit, we derive a nonlinear integral equation for the expectation value of large Wilson loops assuming that the exchange of Goldstone bosons dominates the interaction at long range in critical covariant gauges. We find numerically that the expectation value of large circular Wilson loops decreases exponentially with the enclosed area in the absence of dynamical fermions. The gauge invariance of this mechanism for confinement in critical covariant gauges is discussed.