  Gradient Flow Exact Renormalization Group (GF-ERG) is a framework to define
the renormalization group flow of Wilsonian effective action utilizing
coarse-graining along the diffusion equations. We apply it for Scalar Quantum
Electrodynamics and derive flow equations for the Wilsonian effective action
with the perturbative expansion in the gauge coupling. We focus on the quantum
corrections to the correlation functions up to the second order of the gauge
coupling and discuss the gauge invariance of the GF-ERG flow. We demonstrate
that the anomalous dimension of the gauge field agrees with the standard
perturbative computation and that the mass of the photon keeps vanishing in
general spacetime dimensions. The latter is a noteworthy fact that contrasts
with the conventional Exact Renormalization Group formalism in which an
artificial photon mass proportional to a cutoff scale is induced. Our results
imply that the GF-ERG can give a gauge-invariant renormalization group flow in
a non-perturbative way.
