Heat kernel of non-minimal gauge field kinetic operators on Moyal plane

TL;DR

This paper extends the heat kernel asymptotic expansion techniques to noncommutative gauge theories on the Moyal plane, revealing gauge-fixing dependencies and non-local singularities affecting renormalizability.

Contribution

It generalizes the Endo formula to noncommutative spaces and computes the first three heat trace coefficients for non-minimal gauge operators on the Moyal plane.

Findings

Non-planar heat trace parts are governed by the U(1) sector.

Non-planar coefficients depend on gauge-fixing choices.

Degenerate deformation leads to non-local singularities that impair renormalizability.

Abstract

We generalize the Endo formula originally developed for the computation of the heat kernel asymptotic expansion for non-minimal operators in commutative gauge theories to the noncommutative case. In this way, the first three non-zero heat trace coefficients of the non-minimal U(N) gauge field kinetic operator on the Moyal plane taken in an arbitrary background are calculated. We show that the non-planar part of the heat trace asymptotics is determined by U(1) sector of the gauge model. The non-planar or mixed heat kernel coefficients are shown to be gauge-fixing dependent in any dimension of space-time. In the case of the degenerate deformation parameter the lowest mixed coefficients in the heat expansion produce non-local gauge-fixing dependent singularities of the one-loop effective action that destroy the renormalizability of the U(N) model at one-loop level. The twisted-gauge transformation approach is discussed.