  We study nonlinear interactions in a strongly driven optomechanical cavity,
in regimes where the interactions give rise to resonant scattering between
optomechanical polaritons and are thus strongly enhanced. We use a Keldysh
formulation and self-consistent perturbation theory, allowing us to include
self energy diagrams at all orders in the interaction. Our main focus is
understanding how non-equilibrium effects are modified by the polariton
interactions, in particular the generation of non-zero effective polariton
temperatures from vacuum fluctuations (both in the incident cavity drive and in
the mechanical dissipation). We discuss how these effects could be observed in
the output spectrum of the cavity. Our work also provides a technical toolkit
that will be useful for studies of more complex optomechanical systems.
