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 symmetries remain unbroken.
Fukushima et al. (Tue,) studied this question.