Electrons in transition metal dichalcogenides (TMDs) possess valley and spin degrees of freedom, which leads to rich exciton and exciton-polariton physics with nontrivial scattering dynamics and enhanced nonlinearities, presenting a key mechanism for photonic devices. Yet, existing descriptions of bosonization and polariton interactions in TMD-based systems overlook the valley degree of freedom as well as the various particles' spins combinations. In this work, we derive a nonequilibrium field-theory approach in the path integral formalism that allows one to track all the polariton nonlinearities in the strong-coupling regime. We demonstrate that, when all the bright- and dark-exciton species are considered, the TMD monolayer-based polariton systems feature 16 different nonlinear contributions due to interactions and even more saturation-related terms. Strikingly, while the interactions of excitons within one valley are overall dominant, we show that the interaction constant corresponding to the scattering of spin-dark and bright excitons is much larger than that for bright excitons alone, indicating a possibly much higher contribution to the blueshift from spin-dark excitons than from bright intravalley excitons.

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• Exciton polariton
• Excitons
• Valley degrees of freedom
• Transition metal dichalcogenides
• Nonequilibrium Green's function
• Path-integral methods