Quark--antiquark states and their radiative transitions in terms of the spectral integral equation. {\Huge II.} Charmonia

In the precedent paper of the authors (hep-ph/0510410), the $b\bar{b}$ states were treated in the framework of the spectral integral equation, together with simultaneous calculations of radiative decays of the considered bottomonia. In the present paper, such a study is carried out for the charmonium $(c\bar{c})$ states. We reconstruct the interaction in the $c\bar{c}$-sector on the basis of data for the charmonium levels with $J^{PC}=0^{-+}$, $1^{--}$, $0^{++}$, $1^{++}$, $2^{++}$, $1^{+-}$ and radiative transitions $\psi (2S)\to \gamma {\chi }_{c0}(1P)$, $\gamma {\chi }_{c1}(1P)$, $\gamma {\chi }_{c2}(1P)$, $\gamma {\eta }_c(1S)$ and ${\chi }_{c0}(1P)$, ${\chi }_{c1}(1P)$, ${\chi }_{c2}(1P)\to \gamma J\mathrm{/}\psi$. The $c\bar{c}$ levels and their wave functions are calculated for the radial excitations with $n\le 6$. Also, we determine the $c\bar{c}$ component of the photon wave function using the $e^{+}{e}^{-}$ annihilation data: $e^{+}{e}^{-}\to J\mathrm{/}\psi (3097)$, $\psi (3686)$, $\psi (3770)$, $\psi (4040)$, $\psi (4160)$, $\psi (4415)$ and perform the calculations of the partial widths of the two-photon decays for the $n=1$ states: ${\eta }_{c0}(1S)$, ${\chi }_{c0}(1P)$, ${\chi }_{c2}(1P)\to \gamma \gamma$, and $n=2$ states: ${\eta }_{c0}(2S)\to \gamma \gamma$, ${\chi }_{c0}(2P)$, $\chi_{c2}(2P)\to \gamma\gamma$\mathrm{.}Wediscussthestatusoftherecentlyobserved$c\bar c$ states X(3872) and Y(3941): according to our results, the X(3872) can be either ${\chi }_{c1}(2P)$ or ${\eta }_{c2}(1D)$, while Y(3941) is ${\chi }_{c2}(2P)$.