Parametric representation of a translation-invariant renormalizable noncommutative model

TL;DR

This paper develops a parametric representation for a translation-invariant, renormalizable scalar field model on noncommutative Moyal space, enabling power counting and potential dimensional regularization.

Contribution

It introduces a novel parametric representation for noncommutative quantum field theory amplitudes, applicable to all graph types and facilitating dimensional regularization.

Findings

Derived explicit parametric formulas for noncommutative propagators.

Analyzed both planar and non-planar Feynman graphs.

Established a method for power counting and dimensional regularization.

Abstract

We construct here the parametric representation of a translation-invariant renormalizable scalar model on the noncommutative Moyal space of even dimension $D$. This representation of the Feynman amplitudes is based on some integral form of the noncommutative propagator. All types of graphs (planar and non-planar) are analyzed. The r\^ole played by noncommutativity is explicitly shown. This parametric representation established allows to calculate the power counting of the model. Furthermore, the space dimension $D$ is just a parameter in the formulas obtained. This paves the road for the dimensional regularization of this noncommutative model.