On the unitarity of quantum gauge theories on non-commutative spaces

TL;DR

This paper investigates the perturbative unitarity of non-commutative quantum Yang-Mills theories, revealing issues when non-commutativity involves time and demonstrating that Wilson loop exponentiation persists at order g^4 despite non-commutativity effects.

Contribution

It extends unitarity analysis from scalar to gauge theories on non-commutative spaces and examines the effects of time-involving non-commutativity on unitarity and Wilson loop behavior.

Findings

Unitarity is preserved when non-commutativity does not involve time.

Extra singularities and tachyonic poles appear when time is non-commutative.

Wilson loop exponentiation persists at order g^4 despite non-commutative effects.

Abstract

We study the perturbative unitarity of non-commutative quantum Yang-Mills theories, extending previous investigations on scalar field theories to the gauge case where non-locality mingles with the presence of unphysical states. We concentrate our efforts on two different aspects of the problem. We start by discussing the analytical structure of the vacuum polarization tensor, showing how Cutkoski's rules and positivity of the spectral function are realized when non-commutativity does not affect the temporal coordinate. When instead non-commutativity involves time, we find the presence of extra troublesome singularities on the $p_0^2$-plane that seem to invalidate the perturbative unitarity of the theory. The existence of new tachyonic poles, with respect to the scalar case, is also uncovered. Then we turn our attention to a different unitarity check in the ordinary theories, namely time exponentiation of a Wilson loop. We perform a $O(g^4)$ generalization to the (spatial) non-commutative case of the familiar results in the usual Yang-Mills theory. We show that exponentiation persists at $O(g^4)$ in spite of the presence of Moyal phases reflecting non-commutativity and of the singular infrared behaviour induced by UV/IR mixing.