  Large exciton-polariton optical nonlinearities present a key mechanism for
photonics-based communication, ultimately in the quantum regime. Enhanced
nonlinear response from various materials hosting excitons and allowing for
their strong coupling with light is therefore the topic of intense studies,
both in theoretical and experimental domains. Reports on the scattering rates
arising due to various system's nonlinearities, such as the exciton-exciton
Coulomb interaction and the Pauli blocking that leads to the saturation of the
exciton oscillator strength, however, are contradictory. In this work, we
develop a formalism allowing to track the exciton nonlinearities appearing in
the regime of strong coupling with photons, that includes finite temperatures,
mixing of the exciton excited states, and the dark exciton contributions to
saturation self-consistently. The equilibrium path integration approach
employed here to address the polariton composite nature, leads to a transparent
hierarchy of various contributions to nonlinearity. At the same time, by taking
the simplest limit of zero temperature and so-called "rigid" excitons, through
our framework we retrieve the expressions derived in conventional approaches
for exciton interaction constants. In particular, our theory allows to clearly
show that such interaction constants cannot be used as fitting parameters
tunable in a wide range of values, as they are strictly defined by the material
properties, and that other explanations are due for large optical
nonlinearities recently reported.
