  Luttinger semimetals include materials like gray tin ($\alpha$-Sn) and
mercury telluride, which are three-dimensional gapless semiconductors having a
quadratic band crossing point (QBCP). Due to a growing interest in QBCPs and
new experimental efforts, it is essential to study the finite-temperature
properties of such systems. In this paper, we investigate the emergence of
plasmons in the presence of Coulomb interactions in isotropic Luttinger
semimetals, for zero as well as generic nonzero temperatures. When the Fermi
level lies right at the QBCP, which is the point where twofold degenerate
conduction and valence bands touch each other quadratically, we find that
plasmons cannot appear at zero temperature. However, for nonzero temperatures,
thermal plasmons are generated. Whether they are long-lived or not depends on
the values of temperature, effective electron mass and effective fine-structure
constant, and the number of fermion flavors. We also numerically estimate the
behavior of the inelastic scattering rate at nonzero temperatures, as a
function of energy, where the signatures of the QBCP thermal plasmons show up
as a sharp peak. Our results will thus serve as a guide to experimental probes
on these systems.
