  We develop a many-body theory for the properties of exciton-polaritons
interacting strongly with a Bose-Einstein condensate (BEC) of
exciton-polaritons in another spin state. Interactions lead to the presence of
a two-body bound state, the bi-exciton, giving rise to a Feshbach resonance in
the polariton spectrum when its energy is equal to that of two free polaritons.
Using the minimal set of terms to describe this resonance, our theory recovers
the main findings of two experiments probing interaction effects for upper and
lower polaritons in a BEC. This strongly supports that Feshbach physics has
indeed been realized, and we furthermore extract the energy and decay of
biexciton from the experimental data. The decay rate is predicted to be much
larger than that coming from its dissociation into two free polaritons
indicating that other decay channels are important.
