The gradient-flow coupling of three-and four-dimensional QED

TL;DR

This paper computes the gradient-flow scheme coupling in three and four-dimensional QED, providing a general framework applicable to U(1) gauge theories with various interactions, and analyzes the beta functions and fixed points.

Contribution

It introduces a general method for evaluating the gradient-flow coupling in U(1) gauge theories and applies it to QED in different dimensions, analyzing beta functions and fixed points.

Findings

In four dimensions, the beta function's perturbative expansion is well-behaved.

In three dimensions, the study identifies both UV and IR fixed points in the large-flavor limit.

The method applies broadly to theories with U(1) gauge fields and arbitrary interactions.

Abstract

We evaluate the QED coupling in the gradient-flow scheme in three and four space-time dimensions. Our general result applies to any theory with a U(1) gauge field coupled to arbitary other fields via arbitrary interactions. As an example, we consider QED with $n_\text{f}$ flavors in three and four space-time dimensions and evaluate the corresponding $\beta$ functions. In four dimensions, we find that the perturbative expansion of the $\beta$ function behaves much better than the corresponding expression in QCD. In three dimensions, we recover both the ultraviolet as well as the infrared fixed points of the QED coupling in the large-$n_\text{f}$ limit.