Consistent Interactions of the 2+1 Dimensional Noncommutative Chern-Simons Field

TL;DR

This paper investigates the quantum consistency of 2+1 dimensional noncommutative Chern-Simons models coupled to matter fields, identifying conditions under which infrared divergences are absent or manageable.

Contribution

It demonstrates that fermionic models in the fundamental representation are infrared divergence-free at one loop, and extends the analysis to supersymmetric models showing finiteness under certain conditions.

Findings

Fermionic fundamental representation models are free of nonintegrable infrared divergences.

Adjoint representation models exhibit dangerous infrared divergences.

Supersymmetric models are finite in the fundamental representation, with gauge-dependent finiteness in the adjoint case.

Abstract

We consider 2+1 dimensional noncommutative models of scalar and fermionic fields coupled to the Chern-Simons field. We show that, at least up to one loop, the model containing only a fermionic field in the fundamental representation minimally coupled to the Chern-Simons field is consistent in the sense that there are no nonintegrable infrared divergences. By contrast, dangerous infrared divergences occur if the fermion field belongs to the adjoint representation or if the coupling of scalar matter is considered instead. The superfield formulation of the supersymmetric Chern-Simons model is also analyzed and shown to be free of nonintegrable infrared singularities and actually finite if the matter field belongs to the fundamental representation of the supergauge group. In the case of the adjoint representation this only happens in a particular gauge.