How to extract the "Abelian" part of double-winding Wilson loop

TL;DR

This paper discusses a redefined 'Abelian' Wilson loop that accurately captures the behavior of double-winding Wilson loops, addressing limitations of the naive Abelian projection in non-Abelian gauge theories.

Contribution

It proposes a new redefinition of the 'Abelian' double-winding Wilson loop that correctly reproduces expected behaviors, improving upon previous approaches.

Findings

Redefinition of 'Abelian' Wilson loop improves accuracy

Proposed method aligns with theoretical expectations

Addresses limitations of naive Abelian projection

Abstract

It is known that the naive Abelian Wilson loop defined by the Abelian projection cannot reproduce the correct behavior of the double-winding Wilson loop. It is also known that the naive Abelian Wilson loop cannot reproduce the correct behavior of the Wilson loops in higher representations, but this problem was recently solved by using the redefined "Abelian" Wilson loop. In this talk, we will give another reason why this redefined "Abelian" Wilson loop behaves correctly, and by following the same line of the argument, we will propose redefined "Abelian" double-winding Wilson loop which is considered to behave correctly.