Non-commutative SU(N) gauge theories and asymptotic freedom

TL;DR

This paper investigates the one-loop renormalization of non-commutative SU(N) gauge theories, revealing conditions for renormalizability and demonstrating asymptotic freedom of the deformation parameter in specific cases.

Contribution

It identifies the role of a freedom parameter in renormalization, determines its fixed values, and shows that certain non-commutative gauge theories are one-loop renormalizable and asymptotically free.

Findings

Allowed values of the parameter a are 1 or 3.

Non-commutative SU(N) gauge theory is one-loop renormalizable at a=3.

The deformation parameter h is asymptotically free in the adjoint case.

Abstract

In this paper we analyze the one-loop renormalization of the $\theta$-expanded $\rm SU(N)$ Yang-Mills theory. We show that the {\it freedom parameter} $a$, key to renormalization, originates from higher order non-commutative gauge interaction, represented by a higher derivative term $ b h \theta^{\mu\nu}\hat F_{\mu\nu}\star\hat F_{\rho\sigma}\star\hat F^{\rho\sigma}$. The renormalization condition fixes the allowed values of the parameter $a$ to one of the two solutions: $a=1$ or $a=3$, i.e. to $b=0$ or to $b=1/2$, respectively. When the higher order interaction is switched on, ($a=3$), pure non-commutative SU(N) gauge theory at first order in $\theta$-expansion becomes one-loop renormalizable for various representations of the gauge group. We also show that, in the case $a=3$ and the adjoint representation of the gauge fields, the non-commutative deformation parameter $h$ has to be renormalized and it is asymptotically free.