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

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 nontrivial scaling intertwines n and N. In the noncommutative case the interplay becomes tighter owing to the merging of space-time and ``internal'' symmetries in a larger gauge group $U(\ensuremath{\infty}).$ We perform an explicit perturbative calculation of such a loop up to $\mathcal{O}{(g}^{6});$ rather surprisingly, we find that in the contribution from the crossed graphs (the genuine noncommutative terms) the scaling we mentioned occurs for large n and N in the limit of maximal noncommutativity $\ensuremath{\theta}=\ensuremath{\infty}.$ We present arguments in favor of the persistence of such a scaling at any perturbative order and succeed in summing the related perturbative series.