  Weyl semimetals have been intensely studied as a three dimensional
realization of a Dirac-like excitation spectrum where the conduction bands and
valence bands touch at isolated Weyl points in momentum space. Like in
graphene, this property entails various peculiar electronic properties.
However, recent theoretical studies have suggested that resonant scattering
from rare regions can give rise to a non-zero density of states even at charge
neutrality. Here, we give a detailed account of this effect and demonstrate how
the semimetallic nature is suppressed at the lowest scales. To this end, we
develop a self-consistent T-matrix approach to investigate the density of
states beyond the limit of weak disorder. Our results show a nonvanishing
density of states at the Weyl point which exhibits a non-analytic dependence on
the impurity density. This unusually strong effect of rare regions leads to a
revised estimate for the conductivity close to the Weyl point and emphasizes
possible deviations from semimetallic behavior in dirty Weyl semimetals at
charge neutrality even with very low impurity concentration.
