  We theoretically study the role of the Berry curvature on neutral and charged
excitons in two-dimensional transition-metal dichalcogenides. The Berry
curvature arises due to a strong coupling between the conduction and valence
bands in these materials that can to great extent be described within the model
of massive Dirac fermions. The Berry curvature lifts the degeneracy of exciton
states with opposite angular momentum. Using an electronic interaction that
accounts for non-local screening effects, we find a Berry-curvature induced
splitting of $\sim 17$ meV between the 2$p_{-}$ and 2$p_{+}$ exciton states in
WS$_2$, consistent with experimental findings. Furthermore, we calculate the
trion binding energies in WS$_2$ and WSe$_2$ for a large variety of screening
lenghts and different dielectric constants for the environment. Our approach
indicates the prominent role played by the Berry curvature along with non-local
electronic interactions in the understanding of the energy spectra of neutral
and charged excitons in transition-metal dichalcogenides and in the the
interpretation of their optical properties.
