  Monolayer transition-metal dichalcogenides (ML-TMDs) offer exciting
opportunities to test the manifestations of many-body interactions through
changes in the charge density. Tuning the charge density by a gate voltage
leads to profound changes in the optical spectra of excitons in ML-TMDs. We
review the band-gap renormalization and dynamical screening as a function of
charge density, and then incorporate these effects through various
approximations that model long-wavelength charge excitations in the
Bethe-Salpeter Equation (BSE). We then show that coupling between excitons and
shortwave charge excitations is essential to resolve several experimental
puzzles.
  Unlike ubiquitous and well-studied plasmons, driven by collective
oscillations of the background charge density in the long-wavelength limit, we
discuss the emergence of shortwave plasmons that originate from the short-range
Coulomb interaction through which electrons transition between the $\mathbf{K}$
and $-\mathbf{K}$ valleys. We study the coupling between the shortwave plasmons
and the neutral exciton through the self-energy of the latter. We then
elucidate how this coupling as well as the spin ordering in the conduction band
give rise to an experimentally observed optical sideband in electron-doped
W-based MLs, conspicuously absent in electron-doped Mo-based MLs or any
hole-doped ML-TMDs. While the focus of this review is on the optical
manifestations of many-body effects in ML-TMDs, a systematic description of the
dynamical screening and its various approximations allow one to revisit other
phenomena, such as nonequilibrium transport or superconducting pairing, where
the use of the BSE or the emergence of shortwave plasmons can play an important
role.
