  We present a many-body theory for exciton-trion-polaritons in doped
two-dimensional materials. Exciton-trion-polaritons are robust coherent hybrid
excitations involving excitons, trions, and photons. Signatures of these
polaritons have been recently seen in experiments. In these polaritons, the
2-body exciton states are coupled to the material ground state via
exciton-photon interaction and the 4-body trion states are coupled to the
exciton states via Coulomb interaction. The trion states are not directly
optically coupled to the material ground state. The energy-momentum dispersion
of these polaritons exhibit three bands. We calculate the energy band
dispersions and the compositions of polaritons at different doping densities
using Green's functions. The energy splittings between the polariton bands, as
well as the spectral weights of the polariton bands, depend on the strength of
the Coulomb coupling between the excitons and the trions and which in turn
depends on the doping density. The excitons are Coulomb coupled to both bound
and unbound trion states. The latter are exciton-electron scattering states and
their inclusion is necessary to capture the spectral weight transfer among the
polariton bands as a function of the doping density.
