Quantum gauge theories on noncommutative three-dimensional space

Abstract

We consider a class of gauge-invariant models on the noncommutative space ${\mathbb{R}}_{\ensuremath{\lambda}}^{3}$, a deformation of the algebra of functions on ${\mathbb{R}}^{3}$. Focusing on massless models with no linear ${A}_{i}$ dependence, we obtain noncommutative gauge models for which the computation of the propagator can be done in a convenient gauge. We find that the infrared singularity of the massless propagator disappears in the computation of the correlation functions. We show that massless gauge-invariant models on ${\mathbb{R}}_{\ensuremath{\lambda}}^{3}$ have quantum instabilities of the vacuum, signaled by the occurrence of nonvanishing one-point functions for some but not all of the components of the gauge potential. The tadpole contribution to the effective action cannot be interpreted as a standard $\ensuremath{\sigma}$ term. Its global symmetry does not fit with the one of the classical action, reminiscent of an explicit global symmetry breaking term.