  We study superconductivity driven by screened Coulomb repulsion in
three-dimensional Luttinger semimetals with a quadratic band touching and
strong spin-orbit coupling. In these semimetals, the Cooper pairs are formed by
spin-3/2 fermions with non-trivial wavefunctions. We numerically solve the
linear Eliashberg equation to obtain the critical temperature of a singlet
s-wave gap function as a function of doping, with account of spin-orbit and
self-energy corrections. In order to understand the underlying mechanism of
superconductivity, we compute the sensitivity of the critical temperature to
changes in the dielectric function $\epsilon(i\Omega_n,q)$. We relate our
results to the plasmon and Kohn-Luttinger mechanisms. Finally, we discuss the
validity of our approach and compare our results to the litterature. We find
good agreement with some bismuth-based half-Heuslers, such as YPtBi, and
speculate on superconductivity in the iridate Pr2Ir2O7.
