Charmonium resonances in the 3.9 GeV/$c^2$ energy region and the $X(3915)/X(3930)$ puzzle

An interesting controversy has emerged challenging the widely accepted nature of the $X(3915)$ and the $X(3930)$ resonances, which had initially been assigned to the $\chi_{c0}(2P)$ and $\chi_{c2}(2P)$ $c\bar c$ states, respectively. To unveil their inner structure, the properties of the $J^{PC}\!\!=\!0^{++}$ and $J^{PC}\!\!=\!2^{++}$ charmonium states in the energy region of these resonances are analyzed in the framework of a constituent quark model. Together with the bare $q\bar q$ states, threshold effects due to the opening of nearby meson-meson channels are included in a coupled-channels scheme calculation. We find that the structure of both states is dominantly molecular with a probability of bare $q\bar q$ states lower than $45\%$. Our results favor the hypothesis that $X(3915)$ and $X(3930)$ resonances arise as different decay mechanisms of the same $J^{PC}\!\!=\!2^{++}$ state. Moreover we found an explanation for the recently discovered $M=3860$ MeV$/c^2$ as a $J^{PC}\!\!=\!0^{++}$ $2P$ state and rediscovery the lost $Y(3940)$ as an additional state in the $J^{PC}\!\!=\!0^{++}$ family.