  The influence of weak disorder on the superconductivity in ordinary metals
can be formally described by the Abrikosov-Gorkov diagrammatic approach. The
vertex correction is ignored in this approach because an inequality $k_F l \gg
1$, where $k_F$ is the Fermi momentum and $l$ mean free path, is satisfied in
ordinary metals with a large Fermi surface. In a Dirac semimetal that has
discrete Fermi points, this inequality may break down even for arbitrarily weak
disorder since $k_F \rightarrow 0$, and thus the vertex correction could be
important. We incorporate the vertex correction into the self-consistent
equations of the superconducting gap and the disorder scattering rate, and then
apply the generalized approach to study how $s$-wave superconductivity is
affected by random chemical potential in two- and three-dimensional Dirac
semimetals, as well as two-dimensional semi-Dirac semimetal. In the clean
limit, superconductivity is formed only when the pairing interaction strength
is greater than some critical value in these materials. Adding random chemical
potential to the system promotes superconductivity by generating a finite
fermionic density of states at the Fermi level. In three-dimensional Dirac
semimetal, the critical attraction strength is reduced by weak disorder, but
remains finite. In the other two cases, superconductivity is induced by
arbitrarily weak attraction. Including the vertex correction does not change
these qualitative results, and actually could further promote superconductivity
in the weak-attraction regime. Bilayer graphene is quite special in that its
zero-energy density of states is nonzero despite the existence of Fermi points.
Due to this peculiar property, superconductivity is always slightly suppressed
by random chemical potential, and the impact of vertex correction is nearly
negligible.
