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

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.