Transverse vertex and gauge technique in quantum electrodynamics

It is known that the Schwinger-Dyson equation for the electron propagator $S\left(p\right)\mathrm{}$ in quantum electrodynamics is linear if the full vertex in this equation is approximated by a special form (the longitudinal vertex) which satisfies the Ward identity and which yields exact results in the infrared regime. However, the approximate equation cannot be multiplicatively renormalized (nor is it properly gauge covariant) using only the longitudinal vertex. In the present work, we construct a transverse (i.e., identically conserved) vertex which, when added to the longitudinal vertex, yields an equation for $S\left(p\right)\mathrm{}$ which remains linear and exact in the infrared, but which is multiplicatively renormalizable and gauge covariant. In the ultraviolet regime, the equation gives the known results of renormalization-group-improved perturbation theory. The essential difficulty which is overcome by the present analysis is that of overlapping divergences, which are mishandled if only the longitudinal vertex is kept.