The static quark potential from the gauge invariant Abelian decomposition

TL;DR

This paper explores how topological structures in gauge invariant Abelian decomposition relate to quark confinement, using lattice QCD to identify dominant contributions to the string tension from these structures.

Contribution

It introduces a novel method of defining the Abelian field in terms of Wilson Loop eigenvectors, linking topological structures to confinement mechanisms.

Findings

Restricted field strength peaks correlate with stronger Wilson Loop area law.

Topological structures dominate the restricted field strength in lattice QCD.

These structures contribute significantly to the string tension.

Abstract

We investigate the relationship between colour confinement and topological structures derived from the gauge invariant Abelian (Cho-Duan-Ge) decomposition. This Abelian decomposition is made imposing an isometry on a colour field $n$ which selects the Abelian direction; the principle novelty of our study is that we have defined this field in terms of the eigenvectors of the Wilson Loop. This allows us to establish an equivalence between the path ordered integral of the non-Abelian gauge fields with an integral over an Abelian restricted gauge field which is tractable both theoretically and numerically in lattice QCD. By using Stokes' theorem, we can relate the Wilson Loop in terms of a surface integral over a restricted field strength, and show that the restricted field strength may be dominated by topological structures, which occur when one of the parameters parametrising the colour field $n$ winds itself around a non-analyticity in the colour field. If they exist, these objects will lead to an area law scaling for the Wilson Loop and provide a mechanism for quark confinement. We search for these structures in quenched lattice QCD. We perform the Abelian decomposition, and find that the restricted field strength is dominated by peaks on the lattice. Wilson Loops containing these peaks show a stronger area-Law and thus provide the dominant contribution to the string tension.