  Luttinger semimetals are three-dimensional electron systems with a parabolic
band touching and an effective total spin $J=3/2$. In this paper, we present an
analytical theory of dielectric screening of inversion-symmetric Luttinger
semimetals with an arbitrary carrier density and conduction-valence effective
mass asymmetry. Assuming a spherical approximation for the single-particle
Luttinger Hamiltonian, we determine analytically the dielectric screening
function in the random phase approximation for arbitrary values of the wave
vector and frequency, the latter in the complex plane. We use this analytical
expression to calculate the dispersion relation and Landau damping of the
collective modes in the charge sector (i.e., plasmons).
