  With a perturbative treatment based on the Keldysh Green's function
technique, we study the resonant enhancement of nonlinear interaction effects
in a four-mode optomechanical ring. In such a system, we identify five distinct
types of resonant scattering between unperturbed polariton modes, induced by
the nonlinear optomechanical interaction. By computing the cavity density of
states and optomechanical induced transparency signal, we find that the largest
nonlinear effects are induced by a decay process involving the two phonon-like
polaritons. In contrast to the conventional two-mode optomechanical system, our
proposed system can exhibit prominent nonlinear features even in the regime
when the single-photon coupling is much smaller than the cavity damping.
