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

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 group U(∞). We perform an explicit perturbative calculation of such a loop up to O(g6); 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 θ = ∞. We present arguments in favour of the persistence of such a scaling at any perturbative order and succeed in summing the related perturbative series. DFPD 02/TH/12, UTF 448. PACS numbers: 11.15.Bt, 02.40.Gh