Functional renormalization group flows and gauge consistency in QED

We consider quantum electrodynamics with chiral four-Fermi interactions in the functional renormalization group approach. In gauge theories, the functional flow equation for the effective action is accompanied by the quantum master equation that governs the underlying gauge symmetry. Beyond perturbation theory, fully gauge-consistent solutions are very difficult to obtain. In the present work, by including the lowest-order correction to the photon two-point function, we construct a RG-improved one-loop formula that gives rise dressed non-perturbative photon propagators in the flow equation as well as QME. We show that the 1PI effective action with the new formula satisfies both equations within our truncations. We also discuss the phase structure in terms of the gauge and four-Fermi couplings based on a numerical solution of the system.