  We investigate the strong-coupling phases that may arise in 3D Dirac and Weyl
semimetals under the effect of the long-range Coulomb interaction, considering
the many-body theory of these electron systems as a variant of the conventional
fully relativistic Quantum Electrodynamics (QED). For this purpose, we apply
two different nonperturbative approaches, consisting in the sum of ladder
diagrams and taking the limit of a large number N of fermion flavors. We
benefit from the renormalizability that the theory shows in both cases to
compute the anomalous scaling dimensions of different operators exclusively in
terms of the renormalized coupling constant, allowing us to determine the
precise location of the singularities signaling the onset of the
strong-coupling phases. We show then that the QED of 3D Dirac semimetals has
two competing effects at strong coupling. One of them is the tendency to chiral
symmetry breaking and dynamical mass generation, which are analogous to the
same phenomena arising in the conventional QED at strong coupling. This trend
is however outweighed by the strong suppression of electron quasiparticles that
takes place at large N, leading to a different type of critical point at
sufficiently large interaction strength, shared also by the 3D Weyl semimetals.
Overall, the phase diagram of the 3D Dirac semimetals turns out to be richer
than that of their 2D counterparts, displaying a transition to a phase with
non-Fermi liquid behavior which may be observed in materials hosting a
sufficiently large number of Dirac or Weyl points.
