Perturbative Wilson loop in two dimensional non-commutative Yang Mills theory

We perform a perturbative O(g4) Wilson loop calculation for the U(N) Yang–Mills theory defined on non-commutative one space–one time dimensions. We choose the light-cone gauge and compare the results obtained when using the Wu–Mandelstam–Leibbrandt (WML) and the Cauchy principal value (PV) prescription for the vector propagator. In the WML case the θ-dependent term is well- defined and regular in the limit θ → 0, where the commutative theory is recovered. In the PV case, unexpectedly, the result differs from the WML one only by the addition of two singular terms with a trivial θ-dependence. We find this feature intriguing, when remembering that, in ordinary theories on compact manifolds, the difference between the two cases can be traced back to the contribution of topological excitations. Exponentiation (at O(g4)) does not occur, signalling a difficulty of the theory with respect to (perturbative) unitarity.