  We study quasiparticle scattering effects on the dynamics of a homogeneous
Bose-Einstein condensate of ultracold atoms coupled to a single mode of an
optical cavity. The relevant excitations, which are polariton-like mixed
excitations of photonic and atomic density-wave modes, are identified. All the
first-order correlation functions are presented by means of the Keldysh Green's
function technique. Beyond confirming the existence of the resonant enhancement
of Beliaev damping, we find a very structured spectrum of fluctuations. There
is a spectral hole burning at half of the recoil frequency reflecting the
singularity of the Beliaev scattering process. The effects of the photon-loss
dissipation channel and that of the Beliaev damping due to atom-atom collisions
can be well separated. We show that the Beliaev process does not influence the
properties of the self-organization criticality.
