Bethe-Salpeter wave functions of $\eta_c$(1S, 2S) and $\psi$(1S, 2S) states: local-potential description of the charmonium system revisited

The quark potential models with an energy-independent central potential have been successful for understanding the conventional charmonium states especially below the open charm threshold. As one might consider, however, the interquark potential is in general energy-dependent, and its tendency gets stronger in higher lying states. Confirmation of whether the interquark potential is energy-independent is also important to verify the validity of the quark potential models. In this talk, we examine the energy dependence of the charmonium potential, which can be determined from the Bethe-Salpeter (BS) amplitudes of cc mesons in lattice QCD.We first calculate the BS amplitudes of radially excited charmonium states, the η c (2 S ) and ψ (2 S ) states, using the variational method and then determine both the quark kinetic mass and the charmonium potential within the HAL QCD method. Through a direct comparison of charmonium potentials determined from both the 1 S and 2 S states, we confirm that neither the central nor spin-spin potential shows visible energy dependence at least up to 2 S state.