Color confinement and restoration of residual local gauge symmetries

Abstract All colored particles including dynamical quarks and gluons are confined if the color confinement criterion proposed by Kugo and Ojima is satisfied. The criterion was obtained under a gauge fixing of the Lorenz type. However, it was pointed out that the Kugo–Ojima criterion breaks down for the maximal Abelian gauge, which is quite strange in view of the fact that quark confinement has been verified according to the dual superconductivity caused by magnetic monopole condensations. In order to make a bridge between Kugo–Ojima color confinement and the dual superconductor picture for quark confinement, we investigate a generalization of the color confinement criterion to obtain a unified picture for confinement. We show that the restoration of the residual local gauge symmetry which was shown in the Lorenz gauge by Hata to be equivalent to the Kugo–Ojima criterion indeed occurs in the maximal Abelian gauge for the SU(N) Yang–Mills theory in two-, three-, and four-dimensional Euclidean spacetime once the singular topological configurations of gauge fields are taken into account. This result indicates that the color confinement phase is a disordered phase caused by non-trivial topological configurations irrespective of the gauge choice. As a by-product, we show that the compact U(1) gauge theory can have a disordered confinement phase, while the non-compact U(1) gauge theory has a deconfined Coulomb phase.