  The dynamical generation of a fermion gap in graphene is studied at the
infra-red Lorentz-invariant fixed point where the system is described by an
effective relativistic-like field theory: reduced QED$_{4,3}$ with $N$ four
component fermions ($N=2$ for graphene), where photons are $(3+1)$-dimensional
and mediate a fully retarded interaction among $(2+1)$-dimensional fermions. A
correspondence between reduced QED$_{4,3}$ and QED$_3$ allows us to derive an
exact gap equation for QED$_{4,3}$ up to next-to-leading order. Our results
show that a dynamical gap is generated for $\alpha > \alpha_c$ where $1.03 <
\alpha_c < 1.08$ in the case $N=2$ or for $N < N_c$ where $N_c$ is such that
$\alpha_c \to \infty$ and takes the values $3.24 < N_c < 3.36$. The striking
feature of these results is that they are in good agreement with values found
in models with instantaneous Coulomb interaction. At the fixed point: $\alpha =
1/137 \ll \alpha_c$, and the system is therefore in the semi-metallic regime in
accordance with experiments.
