Many-body interactions in monolayer transition-metal dichalcogenides are strongly affected by their unique band structure. We study these interactions by measuring the energy shift of neutral excitons (bound electron-hole pairs) in gated WSe$_2$ and MoSe$_2$. Surprisingly, while the blueshift of the neutral exciton, $X^0$, in electron-doped samples can be more than 10~meV, the blueshift in hole-doped samples is nearly absent. Taking into account dynamical screening and local-field effects, we present a transparent and analytical model that elucidates the crucial role played by intervalley plasmons in electron-doped conditions. The energy shift of $X^0$ as a function of charge density is computed showing agreement with experiment, where the renormalization of $X^0$ by intervalley plasmons yields a stronger blueshift in MoSe$_2$ than in WSe$_2$ due to differences in their band ordering.
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