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We compute the two-loop fermion self-energy in massless reduced quantumelectrodynamics for an arbitrary gauge using the method of integration byparts. Focusing on the limit where the photon field is four-dimensional, ourformula involves only recursively one-loop integrals and can therefore beevaluated exactly. From this formula, we deduce the anomalous scaling dimensionof the fermion field as well as the renormalized fermion propagator up to twoloops. The results are then applied to the ultra-relativistic limit of grapheneand compared with similar results obtained for four-dimensional andthree-dimensional quantum electrodynamics.

Two-loop fermion self-energy in reduced quantum electrodynamics and application to the ultra-relativistic limit of graphene