  We study the Quantum Electrodynamics of 2D and 3D Dirac semimetals by means
of a self-consistent resolution of the Schwinger-Dyson equations, aiming to
obtain the respective phase diagrams in terms of the relative strength of the
Coulomb interaction and the number N of Dirac fermions. In this framework, 2D
Dirac semimetals have just a strong-coupling instability characterized by
exciton condensation (and dynamical generation of mass) that we find at a
critical coupling well above previous theoretical estimates, thus explaining
the absence of that instability in free-standing graphene samples. On the other
hand, we show that 3D Dirac semimetals have a richer phase diagram, with a
strong-coupling instability leading to dynamical mass generation up to N = 4
and a line of critical points for larger values of N characterized by the
vanishing of the electron quasiparticle weight in the low-energy limit. Such a
critical behavior signals the transition to a strongly correlated liquid,
characterized by noninteger scaling dimensions that imply the absence of a pole
in the electron propagator and are the signature of non-Fermi liquid behavior
with no stable electron quasiparticles.
