Closed star product on noncommutative $\mathbb{R}^3$ and scalar field dynamics

TL;DR

This paper investigates noncommutative scalar field theories on a deformed three-dimensional space with a closed star product, revealing they are UV finite and free of UV/IR mixing, with the deformation parameter acting as a natural UV cutoff.

Contribution

It demonstrates the absence of UV/IR mixing and UV divergences in scalar field theories on noncommutative $\,\mathbb{R}^3_\theta$ with a closed star product, highlighting the role of the deformation parameter.

Findings

2-point functions have no IR singularities.

Scalar theories are UV finite with a natural cutoff.

UV/IR mixing is absent in these models.

Abstract

We consider the noncommutative space $\mathbb{R}^3_\theta$, a deformation of $\mathbb{R}^3$ for which the star product is closed for the trace functional. We study one-loop IR and UV properties of the 2-point function for real and complex noncommutative scalar field theories with quartic interactions and Laplacian on $\mathbb{R}^3$ as kinetic operator. We find that the 2-point functions for these noncommutative scalar field theories have no IR singularities in the external momenta, indicating the absence of UV/IR mixing. We also find that the 2-point functions are UV finite with the deformation parameter $\theta$ playing the role of a natural UV cut-off. The possible origin of the absence of UV/IR mixing in noncommutative scalar field theories on $\mathbb{R}^3_\theta$ as well as on $\mathbb{R}^3_\lambda $, another deformation of $\mathbb{R}^3$, is discussed.