  We study three-dimensional Dirac fermions with weak finite-range scalar
potential disorder. In the clean system, the density of states vanishes
quadratically at the Dirac point. Disorder is known to be perturbatively
irrelevant, and previous theoretical work has assumed that the Dirac semimetal
phase, characterized by a vanishing density of states, survives at weak
disorder, with a finite disorder phase transition to a diffusive metal with a
non-vanishing density of states. In this paper we show that nonperturbative
effects from rare regions, which are missed by conventional disorder-averaged
calculations, instead give rise to a nonzero density of states for any nonzero
disorder. Thus, there is no Dirac semimetal phase at non-zero disorder. The
results are established both by a heuristic scaling argument and via a
systematic saddle point analysis. We also discuss transport near the Dirac
point. At the Dirac point, we argue that transport is diffusive, and proceeds
via hopping between rare resonances. As one moves in chemical potential away
from the Dirac point, there are interesting intermediate-energy regimes where
the rare regions produce scattering resonances that determine the DC
conductivity. We derive a scaling theory of transport near disordered 3D Dirac
points. We also discuss the interplay of disorder with attractive interactions
at the Dirac point, and the resulting granular superconducting and Bose glass
phases. Our results are relevant for all 3D systems with Dirac points,
including Weyl semimetals.
