  We investigate the interplay of Coulomb interactions and
short-range-correlated disorder in three dimensional systems where absent
disorder the non-interacting band structure hosts a quadratic band crossing.
Though the clean Coulomb problem is believed to host a 'non-Fermi liquid'
phase, disorder and Coulomb interactions have the same scaling dimension in a
renormalization group (RG) sense, and thus should be treated on an equal
footing. We therefore implement a controlled $\epsilon$-expansion and apply it
at leading order to derive RG flow equations valid when disorder and
interactions are both weak. We find that the non-Fermi liquid fixed point is
unstable to disorder, and demonstrate that the problem inevitably flows to
strong coupling, outside the regime of applicability of the perturbative RG. An
examination of the flow to strong coupling suggests that disorder is
asymptotically more important than interactions, so that the low energy
behavior of the system can be described by a non-interacting sigma model in the
appropriate symmetry class (which depends on whether exact particle-hole
symmetry is imposed on the problem). We close with a discussion of general
principles unveiled by our analysis that dictate the interplay of disorder and
Coulomb interactions in gapless semiconductors, and of connections to many-body
localized systems with long-range interactions.
