  We investigate two-photon transitions to the electron-hole scattering
continuum in monolayer transition-metal dichalcogenides, and identify two
contributions to this nonlinear optical process with opposite circularly
polarized valley selection rules. In the non-interacting limit, the competition
between the two contributions leads to a crossover of the selection rule with
the increase of the two-photon energy. With the strong Coulomb interaction
between the electron and hole, the two contributions excite electron-hole
scattering states in orthogonal angular momentum channels, while the strength
of the transition can be substantially enhanced by the interaction. Based on
this picture of the two-photon transition, the second harmonic generation (SHG)
in the electron-hole continuum is analyzed, where the Coulomb interaction is
shown to greatly alter the relative strength of different cross-circular
polarized SHG processes. Valley current injection by the quantum interference
of one-photon and two-photon transition is also investigated in the presence of
the strong Coulomb interaction, which significantly enhances the injection
rate.
