  We compute the two-loop fermion self-energy in massless reduced quantum
electrodynamics for an arbitrary gauge using the method of integration by
parts. Focusing on the limit where the photon field is four-dimensional, our
formula involves only recursively one-loop integrals and can therefore be
evaluated exactly. From this formula, we deduce the anomalous scaling dimension
of the fermion field as well as the renormalized fermion propagator up to two
loops. The results are then applied to the ultra-relativistic limit of graphene
and compared with similar results obtained for four-dimensional and
three-dimensional quantum electrodynamics.
