Radiative transitions of charmonium states in a constituent quark model

We study the electromagnetic (EM) transitions of the $nS$, $nP$ ($n\leq 3$), and $nD$ ($n\leq 2$) charmonium states with a constituent quark model. We obtain a reasonable description of the EM transitions of the well-established charmonium states $J/\psi$, $\psi(2S)$, $\chi_{cJ}(1P)$, $h_c(1P)$ and $\psi(3770)$. We find that the M2 transitions give notable corrections to some E1 dominant processes by interfering with the E1 transitions. Our predictions of EM decay properties for the higher charmonium states are also presented and compared with other model predictions. In particular, we discuss the EM decay properties of some $"XYZ"$ states, such as $X(3823)$, $X(3872)$, $X(3915)$, $X(3940)$ and $X(4350)$ as conventional charmonium states. Assuming $X(3872)$ as the $\chi_{c1}(2P)$ state, our predicted ratio $\Gamma[X(3872)\to \psi(2S)\gamma]/\Gamma[X(3872)\to J/\psi \gamma]\simeq 4.0$ is consistent with BaBar's measurement.