  Due to strong spin-orbit coupling, charge carriers in three-dimensional
quadratic band touching Luttinger semimetals have non-trivial wavefunctions
characterized by a pseudospin of 3/2. We compute the dielectric permittivity of
such semimetals at finite doping, within the random phase approximation.
Because of interband coupling, the dielectric screening shows a reduced plasma
frequency and an increased spectral weight at small wavevectors, compared to a
regular quadratic band. This weakens the effective Coulomb potential, modifying
the single-particle self-energy for which we present both analytical and
numerical results. At a low carrier density, the quasiparticle properties of
this model strongly deviate from that of a single quadratic band. We compare
our findings with experimental results for alpha-Sn, HgSe, HgTe, YPtBi and
Pr2Ir2O7.
