Restoration of residual gauge symmetries toward color confinement due to topological defects

In this paper, we reconsider the restoration of the residual gauge symmetry (RGS) due to topological effects as a possible criterion for color confinement. Although the RGS is “spontaneously broken” in the perturbative vacuum, it must be restored in the true confining vacuum of QCD, provided that color confinement phase is a disordered phase where all of internal symmetries are unbroken. Therefore, the disappearance of the massless Nambu–Goldstone pole associated with this spontaneously breaking can be regarded as a criterion for color confinement. In the Lorenz gauge, indeed, the restoration condition was shown to agree with the Kugo–Ojima color confinement criterion at least for a special choice of the residual gauge transformation. In the previous paper, we have proposed to generalize this idea by including the topological defects. In this paper, we elaborate this scenario and obtain the criterion (i) by examining a finite large gauge transformation to properly take into account the topological effects and to specify the RGS, and (ii) by obtaining the condition for restricting the possible topological configurations so that they have a finite Euclidean action to give a nonvanishing contribution to the path integral.