  Motivated by the experimental detection of superconductivity in the
low-carrier density half-Heusler compound YPtBi, we study the pairing
instabilities of three-dimensional strongly spin-orbit coupled semimetals with
a quadratic band touching point. In these semimetals the electronic structure
at the Fermi energy is described by spin j=3/2 quasiparticles, which are
fundamentally different from those in ordinary metals with spin j=1/2. We
develop a general approach to analyzing pairing instabilities in j=3/2
materials by decomposing the pair scattering interaction into irreducible
channels, projecting them to the Fermi surface and deriving the corresponding
Eliashberg theory. Applying our method to a generic density-density interaction
in YPtBi we establish the following results: (i) The pairing strength in each
channel uniquely encodes the j=3/2 nature of the Fermi surface band
structure--a manifestation of the fundamental difference with ordinary metals.
In particular, this implies that Anderson's theorem, which addresses the effect
of spin-orbit coupling and disorder on pairing states of spin-1/2 electrons,
cannot be applied in this case. (ii) The leading pairing instabilities are
different for electron and hole doping. This implies that superconductivity
depends on carrier type. (iii) In the case of hole doping--relevant to YPtBi,
we find two odd-parity channels in close competition with s-wave pairing. One
of these two channels is a multicomponent pairing channel, allowing for the
possibility of time-reversal symmetry breaking. (iv) In the case of Coulomb
interactions mediated by the long-ranged electric polarization of optical
phonon modes, a significant coupling strength is generated in spite of the
extremely low density of carriers. Furthermore, non-linear response and Fermi
liquid corrections can favor non-s-wave pairing and potentially account for the
experimentally-observed Tc.
