Scaling properties of the perturbative Wilson loop in two-dimensional non-commutative Yang-Mills theory

TL;DR

This paper investigates the scaling behavior of Wilson loops in two-dimensional non-commutative Yang-Mills theory, revealing that a specific scaling property emerges at large parameters and persists through all orders of perturbation theory.

Contribution

The authors perform a detailed perturbative analysis up to order g^6 and demonstrate the persistence of a non-trivial scaling in non-commutative Wilson loops at large parameters, summing the series.

Findings

Scaling occurs for large n and N at maximal non-commutativity

Perturbative series related to crossed graphs can be summed

Scaling behavior likely persists at all perturbative orders

Abstract

Commutative Yang-Mills theories in 1+1 dimensions exhibit an interesting interplay between geometrical properties and U(N) gauge structures: in the exact expression of a Wilson loop with $n$ windings a non trivial scaling intertwines $n$ and $N$. In the non-commutative case the interplay becomes tighter owing to the merging of space-time and ``internal'' symmetries in a larger gauge group $U(\infty)$. We perform an explicit perturbative calculation of such a loop up to ${\cal O}(g^6)$; rather surprisingly, we find that in the contribution from the crossed graphs (the genuine non-commutative terms) the scaling we mentioned occurs for large $n$ and $N$ in the limit of maximal non-commutativity $\theta=\infty$. We present arguments in favour of the persistence of such a scaling at any perturbative order and succeed in summing the related perturbative series.