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

TL;DR

This paper calculates the Wilson loop in a two-dimensional non-commutative Yang-Mills theory, comparing two prescriptions for the propagator, revealing differences and regularities related to non-commutativity involving time.

Contribution

It provides a perturbative ${ m O}(g^4)$ analysis of Wilson loops in non-commutative Yang-Mills theory with a comparison of Wu-Mandelstam-Leibbrandt and principal value prescriptions.

Findings

WML prescription yields a well-defined, regular $ heta o 0$ limit.

PV prescription results differ by singular terms, with trivial $ heta$-dependence.

The difference between prescriptions relates to topological excitations in similar theories.

Abstract

We perform a perturbative ${\cal O}(g^4)$ 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 $\theta$-dependent term is well-defined and regular in the limit $\theta \to 0$, where the commutative theory is recovered; it provides a non-trivial example of a consistent calculation when non-commutativity involves the time variable. In the $PV$ case, unexpectedly, the result differs from the $WML$ one only by the addition of two singular terms with a trivial $\theta$-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.