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Semi-analytical Proof of Abelian Dominance on Confinement in the Maximally Abelian Gauge

We study abelian dominance for confinement in terms of the local gluon properties in the maximally abelian (MA) gauge in a semi-analytical manner with the help of the lattice QCD. The global Weyl symmetry persistently remains as the relic of SU($N_c$) in the MA gauge, and provides the ambiguity on the electric and magnetic charges. We derive the criterion on the SU($N_c$)-gauge invariance in terms of the residual symmetry in the abelian gauge. In the lattice QCD, we find microscopic abelian dominance on the link variable for the whole region of $β$ in the MA gauge. The off-diagonal angle variable, which is not constrained by the MA-gauge fixing condition, tends to be random besides the residual gauge degrees of freedom. Within the random-variable approximation for the off-diagonal angle variable, we prove that off-diagonal gluon contribution to the Wilson loop obeys the perimeter law in the MA gauge, and show exact abelian dominance for the string tension, although small deviation is brought by the finite size effect of the Wilson loop in the actual lattice QCD simulation.