One-loop $\beta$ functions of a translation-invariant renormalizable noncommutative scalar model

TL;DR

This paper computes the one-loop beta and gamma functions for a new translation-invariant, renormalizable noncommutative scalar field model on Moyal space, revealing similarities to the commutative phi^4 theory.

Contribution

It provides the first calculation of beta functions for a novel noncommutative scalar model with a specific momentum-dependent term, demonstrating its renormalization properties.

Findings

The coupling constant beta function behaves like the commutative phi^4 model.

The beta function for the new parameter a is explicitly calculated.

The model remains perturbatively renormalizable with these features.

Abstract

Recently, a new type of renormalizable $\phi^{\star 4}_{4}$ scalar model on the Moyal space was proved to be perturbatively renormalizable. It is translation-invariant and introduces in the action a $a/(\theta^2p^2)$ term. We calculate here the $\beta$ and $\gamma$ functions at one-loop level for this model. The coupling constant $\beta_\lambda$ function is proved to have the same behaviour as the one of the $\phi^4$ model on the commutative $\mathbb{R}^4$. The $\beta_a$ function of the new parameter $a$ is also calculated. Some interpretation of these results are done.