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

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.