  We develop the theoretical formalism and study the formation of valley trions
in transition metal dichalcogenide (TMDC) monolayers within the framework of a
nonrelativistic potential model using the method of hyperspherical harmonics
(HH) in four-dimensional space. We present the solution of the three-body
Schr\"{o}dinger equation with the Rytova-Keldysh (RK) potential by expanding
the wave function of a trion in terms of the HH. The antisymmetrization of
trions wave function is based on the electron and hole spin and valley indices.
  We consider a long-range approximation when the RK potential is approximated
by the Coulomb potential and a short-range limit when this potential is
approximated by the logarithmic potential. In a diagonal approximation, the
coupled system of differential equations for the hyperradial functions is
decoupled in both limits. Our approach yields the analytical solution for
binding energy and wave function of trions in the diagonal approximation for
these two limiting cases: the Coulomb and logarithmic potentials. We obtain
exact analytical expressions for eigenvalues and eigenfunctions for negatively
and positively charged trions. The corresponding energy eigenvalues can be
considered as the lower and upper limits for the trions binding energies.
  The proposed theoretical approach can describe trions in TMDCs and address
the energy difference between the binding energies of $X^{-}$ and $X^{+}$ in
TMDC. Results of numerical calculations for the ground state energies with the
RK potential are in good agreement with similar calculations and in reasonable
agreement with experimental measurements of trion binding energies.
