  We theoretically study the single particle Green function of a three
dimensional disordered Weyl semimetal using a combination of techniques. These
include analytic $T$-matrix and renormalization group methods with
complementary regimes of validity, and an exact numerical approach based on the
kernel polynomial technique. We show that at any nonzero disorder, Weyl
excitations are not ballistic: they instead have a nonzero linewidth that for
weak short-range disorder arises from non-perturbative resonant impurity
scattering. Perturbative approaches find a quantum critical point between a
semimetal and a metal at a finite disorder strength, but this transition is
avoided due to nonperturbative effects. At moderate disorder strength and
intermediate energies the avoided quantum critical point renormalizes the
scaling of single particle properties. In this regime we compute numerically
the anomalous dimension of the fermion field and find $\eta= 0.13 \pm 0.04$,
which agrees well with a renormalization group analysis ($\eta= 0.125$). Our
predictions can be directly tested by ARPES and STM measurements in samples
dominated by neutral impurities.
