Charmonium dissociation in the context of Bethe-Salpeter equation at finite temperature

We derive the spin-dependent Bethe-Salpeter equation for $q\bar{q}$ bound state at finite temperature in the ladder and cross box diagram approximations. The imaginary time formalism is adopted in the adiabatic approximation where the flux tube thermal excitation is separated from the thermal excitation of the constituent quarks. The thermal excitation of the flux tube is implemented by using a temperature dependent potential inferred from the lattice gauge calculations. The thermal excitations of the constituent quarks are evaluated by summing Matsubara frequencies. The resultant equation is used to study the thermal charmonium spontaneous dissociation mechanism at temperatures below the critical one of the phase transition to the quark gluon plasma. Our results show that when the thermal excitation of the constituent quarks is considered beside the flux tube thermal excitation, the spontaneous dissociation unlikely takes place in the hadronic phase. In contrary, when the thermal excitation of the flux tube is considered only in the calculations, the spontaneous dissociation likely takes place just below the critical temperature.