Noncommutative QED+QCD and $\beta$-function for QED

TL;DR

This paper investigates a noncommutative version of QED+QCD, demonstrating its one-loop renormalizability and the tunability of its beta function through a mixing parameter, potentially aligning with experimental observations.

Contribution

It introduces a noncommutative gauge group framework for QED+QCD that is one-loop renormalizable and allows the beta function to be adjusted via a free parameter.

Findings

QED in noncommutative space can be one-loop renormalizable.

The beta function can be positive, negative, or zero depending on the mixing parameter.

The model can reproduce the beta function of ordinary QED for specific parameter values.

Abstract

QED based on $\theta$-unexpanded noncomutative space-time in contrast with the noncommutative QED based on $\theta$-expanded U(1) gauge theory via the Seiberg-Witten map, is one-loop renormalizable. Meanwhile it suffers from asymptotic freedom that is not in agreement with the experiment. We show that QED part of $U_\star(3)\times U_\star(1)$ gauge group as an appropriate gauge group for the noncommutative QED+QCD, is not only one-loop renormalizable but also has a $\beta$ function that can be positive, negative and even zero. In fact the $\beta$ function depends on the mixing parameter $\delta_{13}$ as a free parameter and it will be equal to its counterpart in the ordinary QED for $\delta_{13}=0.367\pi$.