  Material science methods aim at developing efficient computational schemes
for describing complex many-body effects and how they are revealed in
experimentally measurable properties. Bethe-Salpeter equation in the
self-consistent Hartree-Fock basis is often used for this purpose, and in this
paper we employ the real-frequency diagrammatic Monte Carlo framework for
solving the ladder-type Bethe-Salpeter equation for the 3-point vertex function
(and, ultimately, for the system's polarization) to study the effect of
electron-hole Coulomb scattering on Landau damping in the homogeneous electron
gas. We establish how this damping mechanism depends on the Coulomb parameter
$r_s$ and changes with temperature between the correlated liquid and thermal
gas regimes. In a broader context of dielectric response in metals, we also
present the full polarization and the typical dependence of the
exchange-correlation kernel on frequency at finite momentum and temperature
within the same computational framework.
