Understanding the $0^{++}$ and $2^{++}$ charmonium(-like) states near 3.9 GeV

We propose that the $X(3915)$ observed in the $J\mathrm{/}\psi \omega$ channel is the same state as the ${\chi }_{c2}(3930)$, and the $X(3960)$, observed in the $D_s^{+}{D}_s^{-}$ channel, is an $S$-wave $D_s^{+}{D}_s^{-}$ hadronic molecule. In addition, the $J^{PC}=0^{++}$ {component in the $B^{+}\to D^{+}{D}^{-}{K}^{+}$} assigned to the $X(3915)$ in the current {\it Review of Particle Physics} has the same origin as the $X(3960)$, which has a mass around 3.94~GeV. To check the proposal, the available data in the $D\bar{D}$ and $D_s^{+}{D}_s^{-}$ channels from both $B$ decays and $\gamma \gamma$ fusion reaction are analyzed considering both the $D\bar{D}$-$D_s{\bar{D}}_s$-$D^{\ast }{\bar{D}}^{\ast }$-$D_s^{\ast }{\bar{D}}_s^{\ast }$ coupled channels with $0^{++}$ and a $2^{++}$ state introduced additionally. It is found that all the data in different processes can be simultaneously well reproduced, and the coupled-channel dynamics produce four hidden-charm scalar molecular states with masses around 3.73, 3.94, 3.99 and 4.23~GeV, respectively. The results may deepen our understanding of the spectrum of charmonia as well as of the interactions between charmed hadrons.