# How to extract the dominant part of the Wilson loop average in higher representations

## TL;DR

This paper introduces gauge-invariant operators based on the non-Abelian Stokes theorem to accurately extract the dominant part of Wilson loop averages in higher representations, improving understanding of quark confinement.

## Contribution

It proposes a new gauge-invariant method to extract the dominant Wilson loop contribution in higher representations, addressing limitations of previous Abelian projection approaches.

## Key findings

- Proposed operators reproduce the original Wilson loop behavior in higher representations.
- Lattice simulations confirm the linear potential and correct string tension.
- Overcomes issues with naive Abelian projection in higher representations.

## Abstract

In previous works, we have proposed a new formulation of Yang-Mills theory on the lattice so that the so-called restricted field obtained from the gauge-covariant decomposition plays the dominant role in quark confinement. This framework improves the Abelian projection in the gauge-independent manner. For quarks in the fundamental representation, we have demonstrated some numerical evidences for the restricted field dominance in the string tension, which means that the string tension extracted from the restricted part of the Wilson loop reproduces the string tension extracted from the original Wilson loop. However, it is known that the restricted field dominance is not observed for the Wilson loop in higher representations if the restricted part of the Wilson loop is extracted by adopting the Abelian projection or the field decomposition naively in the same way as in the fundamental representation. In this paper, therefore, we focus on confinement of quarks in higher representations. By virtue of the non-Abelian Stokes theorem for the Wilson loop operator, we propose suitable gauge-invariant operators constructed from the restricted field to reproduce the correct behavior of the original Wilson loop averages for higher representations. Moreover, we perform lattice simulations to measure the static potential for quarks in higher representations using the proposed operators. We find that the proposed operators well reproduce the behavior of the original Wilson loop average, namely, the linear part of the static potential with the correct value of the string tension, which overcomes the problem that occurs in naively applying Abelian-projection to the Wilson loop operator for higher representations.