Renormalisability of the matter determinants in noncommutative gauge theory in the enveloping-algebra formalism

TL;DR

This paper investigates the one-loop UV divergences caused by matter fields in noncommutative gauge theories using Seiberg-Witten maps, revealing that divergences are proportional to the noncommutative Yang-Mills action, supporting its inclusion in the classical action.

Contribution

It provides a comprehensive calculation of matter contributions to UV divergences in noncommutative gauge theories for arbitrary gauge groups and representations, highlighting the role of traces over matter representations.

Findings

UV divergences are proportional to the noncommutative Yang-Mills action.

Results support including trace terms in the classical gauge action.

Calculations are valid for arbitrary representations and Seiberg-Witten maps.

Abstract

We consider noncommutative gauge theory defined by means of Seiberg-Witten maps for an arbitrary semisimple gauge group. We compute the one-loop UV divergent matter contributions to the gauge field effective action to all orders in the noncommutative parameters $\theta$. We do this for Dirac fermions and complex scalars carrying arbitrary representations of the gauge group. We use path-integral methods in the framework of dimensional regularisation and consider arbitrary invertible Seiberg-Witten maps that are linear in the matter fields. Surprisingly, it turns out that the UV divergent parts of the matter contributions are proportional to the noncommutative Yang-Mills action where traces are taken over the representation of the matter fields; this result supports the need to include such traces in the classical action of the gauge sector of the noncommutative theory.