  In two-dimensional Dirac semimetals, Cooper pairing instability occurs only
when the attractive interaction strength $|u|$ is larger than some critical
value $|u_{c}|$ because the density of states vanishes at Dirac points.
Disorders enhance the low-energy density of states but meanwhile shorten the
lifetime of fermions, which tend to promote and suppress superconductivity,
respectively. To determine which of the two competing effects wins, we study
the interplay of Cooper pairing interaction and disorder scattering by means of
renormalization group method. We consider three types of disorders, including
random mass, random gauge potential, and random chemical potential, and show
that the first two suppress superconductivity. In particular, the critical BCS
coupling $|u_{c}|$ is increased to certain larger value if the system contains
only random mass or random gauge potential, which makes the onset of
superconductivity more difficult. In the case of random chemical potential, the
effective disorder parameter flows to the strong coupling regime, where the
perturbation expansion breaks down and cannot provide a clear answer concerning
the fate of superconductivity. When different types of disorder coexist in one
system, their strength parameters all flow to strong couplings. In the strong
coupling regime, the perturbative renormalization group method becomes invalid,
and one needs to employ other methods to treat the disorder effects. We perform
a simple gap equation analysis of the impact of random chemical potential on
superconductivity by using the Abrikosov-Gorkov diagrammatic approach, and also
briefly discuss the possible generalization of this approach.
