  The effects of gauge interactions in graphene have been analyzed up to now in
terms of effective models of Dirac fermions. However, in several cases lattice
effects play an important role and need to be taken consistently into account.
In this paper we introduce and analyze a lattice gauge theory model for
graphene, which describes tight binding electrons hopping on the honeycomb
lattice and interacting with a three-dimensional quantum U(1) gauge field. We
perform an exact Renormalization Group analysis, which leads to a renormalized
expansion that is finite at all orders. The flow of the effective parameters is
controlled thanks to Ward Identities and a careful analysis of the discrete
lattice symmetry properties of the model. We show that the Fermi velocity
increases up to the speed of light and Lorentz invariance spontaneously emerges
in the infrared. The interaction produces critical exponents in the response
functions; this removes the degeneracy present in the non interacting case and
allow us to identify the dominant excitations. Finally we add mass terms to the
Hamiltonian and derive by a variational argument the correspondent gap
equations, which have an anomalous non-BCS form, due to the non trivial effects
of the interaction.
