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We give a gauge-independent definition of Abelian dominance in the Wilsonloop operator and a constructive proof of the Abelian dominance through anon-Abelian Stokes theorem via lattice regularization. We obtain a necessaryand sufficient condition for the Abelian dominance in the Wilson loop operatorin the fundamental representation. In the continuum limit, the gauge field isdecomposed such that the Abelian dominance is given as an exact operatorrelation, leading to the exact (100%) Abelian dominance. On a lattice, weestimate the deviation from the exact Abelian dominance due to non-zero latticespacing. In order to obtain the best Abelian dominance on a lattice byminimizing the deviation, we discuss how to decompose the gauge field variableinto the dominant part and the remaining one to be decoupled on a lattice.

Proving Abelian dominance in the Wilson loop operator