  The coherent third-order nonlinear response of monolayer transition-metal
dichalcogenide (TMD) semiconductors, such as MoSe$_2$ is dominated by the
nonlinear exciton response, as well as biexciton and trion resonances. The fact
that these resonances may be spectrally close together makes identification of
the signatures, for example in differential transmission (DT), challenging.
Instead of focusing on explaining a given set of experimental data, a
systematic study aimed at elucidating the roles of intravalley and intervalley
electron-hole (e-h) exchange on the DT spectra is presented. Previous works
have shown that the e-h exchange introduces a linear leading-order term in the
exciton dispersion. Based on a generalized Lippmann-Schwinger equation, we show
that the presence of this linear dispersion term can reduce the biexciton
binding energy to zero, contrary to the conventional situation of quadratic
dispersion where an arbitrarily weak (well-behaved) attractive interaction
always supports bound state(s). The effects of spin-scattering and the
spin-orbit interaction caused by e-h exchange is also clarified, and the DT
lineshape at the exciton and trion resonance are studied as a function of e-h
exchange strength. In particular, as the exciton lineshape is determined by the
interplay of linear exciton susceptibility and the bound-state two-exciton
resonance in the T-matrix, the lineshape at the trion is similarly determined
by the interplay of the linear trion susceptibility and the bound-state
exciton-trion resonance in the T-matrix.
