  We theoretically study the stability of three dimensional Dirac semimetals
against short-range electron-electron interaction and quenched time-reversal
symmetric disorder (but excluding mass disorder). First we focus on the clean
interacting and the noninteracting dirty Dirac semimetal separately, and show
that they support two distinct quantum critical points. Using renormalization
group techniques, we find that while interaction driven quantum critical points
are \emph{Gaussian} (mean-field) in nature, describing quantum phase
transitions into various broken symmetry phases, the ones controlled by
disorder are \emph{non-Gaussian}, capturing the transition to a metallic phase.
We classify such diffusive quantum critical points based on the transformation
of disorder vertices under a \emph{continuous} chiral rotation. Our wek
coupling renormalization group analysis suggests that two distinct quantum
critical points are stable in an interacting dirty Dirac semimetal (with chiral
symmetric randomness), and a multicritical point (at finite interaction and
disorder) results from their interplay. By contrast, the chiral symmetry
breaking disorder driven critical point is unstable against weak interactions.
Effects of weak disorder on the ordering tendencies in Dirac semimetal are
analyzed. The clean interacting critical points, however, satisfy the
\emph{Harris criterion}, and are therefore expected to be unstable against bond
disorder. Although our weak coupling analysis is inadequate to establish the
ultimate stability of these fixed points in the strong coupling regime (when
both interaction and disorder are strong), they can still govern crossover
behaviors in Dirac semimetals over a large length scale, when either
interaction or randomness is sufficiently weak.
