  We develop a theory for electron-electron interaction-induced many-body
effects in three dimensional (3D) Weyl or Dirac semimetals, including
interaction corrections to the polarizability, electron self-energy, and vertex
function, up to second order in the effective fine structure constant of the
Dirac material. These results are used to derive the higher-order ultraviolet
renormalization of the Fermi velocity, effective coupling, and quasiparticle
residue, revealing that the corrections to the renormalization group (RG) flows
of both the velocity and coupling counteract the leading-order tendencies of
velocity enhancement and coupling suppression at low energies. This in turn
leads to the emergence of a critical coupling above which the interaction
strength grows with decreasing energy scale. In addition, we identify a range
of coupling strengths below the critical point in which the Fermi velocity
varies non-monotonically as the low-energy, non-interacting fixed point is
approached. Furthermore, we find that while the higher-order correction to the
flow of the coupling is generally small compared to the leading order, the
corresponding correction to the velocity flow carries an additional factor of
the Dirac cone flavor number (the multiplicity of electron species, e.g.
ground-state valley degeneracy arising from the band structure) relative to the
leading-order result. Thus, for materials with a larger multiplicity, the
regime of velocity non-monotonicity is reached for modest values of the
coupling strength.
