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

Abstract

In this paper we analyze the one-loop renormalization of the $\theta$-expanded $\mathrm{SU}(\mathrm{N})$ 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 $bh{\theta }^{\mu \nu }{\hat{F}}_{\mu \nu }\star {\hat{F}}_{\rho \sigma }\star {\hat{F}}^{\rho \sigma }$. 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.

arXiv comment16 pages, no figures

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