On the $\beta$-Function and Conformal Anomaly of Noncommutative QED with Adjoint Matter Fields

TL;DR

This paper computes the one-loop $eta$-function and conformal anomaly of noncommutative U(1) gauge theory with adjoint matter fields, revealing asymptotic freedom for fewer than three flavors.

Contribution

It provides the first perturbative calculation of the $eta$-function and conformal anomaly for this specific noncommutative gauge theory, confirming their consistency and asymptotic freedom conditions.

Findings

The $eta$-function is computed up to one-loop order.

The conformal anomaly is derived using Fujikawa's method.

The theory is asymptotically free for $N_f<3$.

Abstract

In the first part of this work, a perturbative analysis up to one-loop order is carried out to determine the one-loop $\beta$-function of noncommutative U(1) gauge theory with matter fields in the adjoint representation. In the second part, the conformal anomaly of the same theory is calculated using the Fujikawa's path integral method. The value of the one-loop $\beta$-function calculated in both methods coincides. As it turns out, noncommutative QED with matter fields in the adjoint representation is asymptotically free for the number of flavor degrees of freedom $N_{f}<3$.