# Double-winding Wilson loops in the $SU(N)$ Yang-Mills theory

## TL;DR

This paper investigates the area law behavior of double- and multi-winding Wilson loops in $SU(N)$ Yang-Mills theory, revealing a novel area law that differs from previously known sum or difference-of-areas laws, especially for $N
eq 2$.

## Contribution

It demonstrates a new area law for double-winding Wilson loops in $SU(3)$ and extends the analysis to arbitrary multi-winding loops, challenging existing notions of area law behavior.

## Key findings

- Double-winding $SU(3)$ Wilson loops follow a novel area law.
- For $N
eq 2$, the area law is neither sum nor difference of areas.
- The results are explicitly derived in two-dimensional $SU(N)$ Yang-Mills theory.

## Abstract

We consider double-winding, triple-winding and multiple-winding Wilson loops in the $SU(N)$ Yang-Mills gauge theory. We examine how the area law falloff of the vacuum expectation value of a multiple-winding Wilson loop depends on the number of color $N$. In sharp contrast to the difference-of-areas law recently found for a double-winding $SU(2)$ Wilson loop average, we show irrespective of the spacetime dimensionality that a double-winding $SU(3)$ Wilson loop follows a novel area law which is neither difference-of-areas nor sum-of-areas law for the area law falloff and that the difference-of-areas law is excluded and the sum-of-areas law is allowed for $SU(N)$ ($N \ge 4$), provided that the string tension obeys the Casimir scaling for the higher representations. Moreover, we extend these results to arbitrary multi-winding Wilson loops. Finally, we argue that the area law follows a novel law, which is neither sum-of-areas nor difference-of-areas law when $N\ge 3$. In fact, such a behavior is exactly derived in the $SU(N)$ Yang-Mills theory in the two-dimensional spacetime.