Double-winding Wilson loop in $SU(N)$ Yang-Mills theory: A criterion for testing the confinement models

TL;DR

This paper investigates how double-winding Wilson loops behave in $SU(N)$ Yang-Mills theory, deriving exact relations and revealing new area law behaviors that depend on the number of colors, providing tests for confinement models.

Contribution

It derives exact operator relations for double-winding Wilson loops in $SU(N)$, revealing new area law behaviors and excluding certain previously claimed laws for $N eq 2$.

Findings

Excludes the difference-of-areas law for $N eq 2$ under Casimir scaling.

Proposes a novel area law $(N - 3)A_1/(N-1)+A_2$ for distinct loops.

Confirms the new law in two-dimensional $SU(N)$ Yang-Mills theory.

Abstract

We examine how the average of double-winding Wilson loops depends on the number of color $N$ in the $SU(N)$ Yang-Mills theory. In the case where the two loops $C_1$ and $C_2$ are identical, we derive the exact operator relation which relates the double-winding Wilson loop operator in the fundamental representation to that in the higher dimensional representations depending on $N$. By taking the average of the relation, we find that the difference-of-areas law for the area law falloff recently claimed for $N=2$ is excluded for $N \geq 3$, provided that the string tension obeys the Casimir scaling for the higher representations. In the case where the two loops are distinct, we argue that the area law follows a novel law $(N - 3)A_1/(N-1)+A_2$ with $A_1$ and $A_2 (A_1<A_2)$ being the minimal areas spanned respectively by the loops $C_1$ and $C_2$, which is neither sum-of-areas ($A_1+A_2$) nor difference-of-areas ($A_2 - A_1$) law when ($N\geq3$). Indeed, this behavior can be confirmed in the two-dimensional $SU(N)$ Yang-Mills theory exactly.