  We consider the 2D Hubbard model on the honeycomb lattice, as a model for a
single layer graphene sheet in the presence of screened Coulomb interactions.
At half filling and weak enough coupling, we compute the free energy, the
ground state energy and we construct the correlation functions up to zero
temperature in terms of convergent series; analiticity is proved by making use
of constructive fermionic renormalization group methods. We show that the
interaction produces a modification of the Fermi velocity and of the wave
function renormalization without changing the asymptotic infrared properties of
the model with respect to the unperturbed non-interacting case; this rules out
the possibility of superconducting or magnetic instabilities in the ground
state. We also prove that the correlations verify a Ward Identity similar to
the one for massless Dirac fermions, up to asymptotically negligible
corrections and a renormalization of the charge velocity.
