Time exponentiation of a Wilson loop for Yang-Mills theories in 2+\epsilon dimensions

TL;DR

This paper computes the perturbative behavior of Wilson loops in Yang-Mills theories near two dimensions, revealing different exponentiation properties depending on the spacetime dimension and loop size.

Contribution

It provides a detailed perturbative analysis of Wilson loops in 1+(D-1) dimensions, highlighting the dimension-dependent exponentiation behavior and the special case at two dimensions.

Findings

For D>2, the Wilson loop exhibits Abelian-like time exponentiation.

At D=2, the result depends on the adjoint Casimir and follows a pure area law.

The behavior of Wilson loops varies significantly between two and higher dimensions.

Abstract

A rectangular Wilson loop centered at the origin, with sides parallel to space and time directions and length $2L$ and $2T$ respectively, is perturbatively evaluated ${\cal O}(g^4)$ in Feynman gauge for Yang--Mills theory in $1+(D-1)$ dimensions. When $D>2$, there is a dependence on the dimensionless ratio $L/T$, besides the area. In the limit $T \to \infty$, keeping $D>2$, the leading expression of the loop involves only the Casimir constant $C_F$ of the fundamental representation and is thereby in agreement with the expected Abelian-like time exponentiation (ALTE). At $D= 2$ the result depends also on $C_A$, the Casimir constant of the adjoint representation and a pure area law behavior is recovered, but no agreement with ALTE in the limit $T\to\infty$. Consequences of these results concerning two and higher-dimensional gauge theories are pointed out.