Residual gauge symmetry and color confinement in the Yang-Mills theory

TL;DR

This paper investigates how the restoration of residual gauge symmetry in Yang-Mills theory can serve as a criterion for confinement, using instanton configurations and dimensional reduction to demonstrate the unbroken electric symmetries in the confinement phase.

Contribution

It introduces a novel confinement criterion based on residual gauge symmetry restoration, analyzed through instanton effects and dimensional reduction in Yang-Mills theory.

Findings

Restoration of residual gauge symmetry occurs via instanton condensations.

The confinement phase is characterized by unbroken internal electric symmetries.

Dimensional reduction to a 2D gauge-scalar theory facilitates analysis.

Abstract

We examine the restoration of the residual gauge symmetry in the Yang-Mills theory to be regarded as a confinement criterion. For this purpose we restrict the four-dimensional $SU(2)$ Yang-Mills instantons to those with spatial spherical symmetry $SO(3)$, which automatically causes the dimensional reduction of the four-dimensional $SU(2)$ Yang-Mills theory to the two-dimensional $U(1)$ gauge-scalar theory as implemented explicitly by the Witten transformation. In this setting, we show that the restoration of the residual gauge symmetry occurs due to condensations of instantons and anti-instantons, although the residual gauge symmetry was spontaneously broken in the perturbative vacuum. This result demonstrates that the true confinement phase is a disordered phase in which all internal electric symmetries of the gauge field remain unbroken.