  We study the many-body theory of graphene Dirac quasiparticles interacting
via the long-range Coulomb potential, taking as a starting point the ladder
approximation to different vertex functions. We test in this way the low-energy
behavior of the electron system beyond the simple logarithmic dependence of
electronic correlators on the high-energy cutoff, which is characteristic of
the large-N approximation. We show that the graphene many-body theory is
perfectly renormalizable in the ladder approximation, as all higher powers in
the cutoff dependence can be absorbed into the redefinition of a finite number
of parameters (namely, the Fermi velocity and the weight of the fields) that
remain free of infrared divergences even at the charge neutrality point. We
illustrate this fact in the case of the vertex for the current density, where a
complete cancellation between the cutoff dependences of vertex and electron
self-energy corrections becomes crucial for the preservation of the gauge
invariance of the theory. The other potentially divergent vertex corresponds to
the staggered (sublattice odd) charge density, which is made cutoff independent
by a redefinition in the scale of the density operator. This allows to compute
a well-defined, scale invariant anomalous dimension to all orders in the ladder
series, which becomes singular at a value of the interaction strength marking
the onset of chiral symmetry breaking (and gap opening) in the Dirac field
theory. The critical coupling we obtain in this way matches with great accuracy
the value found with a quite different method, based on the resolution of the
gap equation, thus reassuring the predictability of our renormalization
approach.
