  We present a fully microscopic approach to the transition rate of two
exciton-photon polaritons. The non-trivial consequences of the polariton
composite nature -- here treated exactly through a development of our
composite-exciton many-body theory -- lead to results noticeably different from
the ones of the conventional approaches in which polaritons are mapped into
elementary bosons. Our work reveals an appealing fundamental scattering which
corresponds to a photon-assisted exchange -- in the absence of Coulomb process.
This scattering being dominant when one of the scattered polaritons has a
strong photon character, it should be directly accessible to experiment. In the
case of microcavity polaritons, it produces a significant enhancement of the
polariton transition rate when compared to the one coming from Coulomb
interaction. This paper also contains the crucial tools to securely tackle the
many-body physics of polaritons, in particular towards its possible BEC.
