We study theoretically the Coulomb interaction between excitons in transition metal dichalcogenide (TMD) monolayers. We calculate direct and exchange interaction for both ground and excited states of excitons. The screening of the Coulomb interaction, specific to monolayer structures, leads to the unique behavior of the exciton-exciton scattering for excited states, characterized by the non-monotonic dependence of the interaction as function of the transferred momentum. We find that the nontrivial screening enables the description of TMD exciton interaction strength by approximate formula which includes exciton binding parameters. The influence of screening and dielectric environment on the exciton-exciton interaction was studied, showing qualitatively different behavior for ground state and excited states of excitons. Furthermore, we consider exciton-electron interaction, which for the excited states is governed by the dominant attractive contribution of the exchange component, which increases with the excitation number. The results provide a quantitative description of the exciton-exciton and exciton-electron scattering in transition metal dichalcogenides, and are of interest for the design of perspective nonlinear optical devices based on TMD monolayers.
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