The Renormalization of Non-Commutative Field Theories in the Limit of Large Non-Commutativity

TL;DR

This paper investigates the renormalization of non-commutative scalar field theories at large non-commutativity, revealing that non-planar divergences persist and require special treatment, leading to additional couplings not present in commutative theories.

Contribution

It demonstrates that non-commutative theories do not simplify to their planar sector at large non-commutativity and introduces a consistent renormalization approach accounting for non-planar divergences.

Findings

Non-planar divergences are UV divergences needing non-local counterterms.

In 4D, an extra relevant non-planar coupling appears.

The non-planar coupling is evanescent but essential for UV finiteness.

Abstract

We show that renormalized non-commutative scalar field theories do not reduce to their planar sector in the limit of large non-commutativity. This follows from the fact that the RG equation of the Wilson-Polchinski type which describes the genus zero sector of non-commutative field theories couples generic planar amplitudes with non-planar amplitudes at exceptional values of the external momenta. We prove that the renormalization problem can be consistently restricted to this set of amplitudes. In the resulting renormalized theory non-planar divergences are treated as UV divergences requiring appropriate non-local counterterms. In 4 dimensions the model turns out to have one more relevant (non-planar) coupling than its commutative counterpart. This non-planar coupling is ``evanescent'': although in the massive (but not in the massless) case its contribution to planar amplitudes vanishes when the floating cut-off equals the renormalization scale, this coupling is needed to make the Wilsonian effective action UV finite at all values of the floating cut-off.