Charmonium spectrum and their electromagnetic transitions with higher   multipole contributions
  The charmonium spectrum is calculated with two nonrelativistic quark models, the linear potential model and the screened potential model. Using the obtained wavefunctions, we evaluate the electromagnetic transitions of charmonium states up to $4S$ multiplet. The higher multipole contributions are included by a multipole expansion of the electromagnetic interactions. Our results are in reasonable agreement with the measurements. As conventional charmonium states, the radiative decay properties of the newly observed charmonium-like states, such as $X(3823)$, $X(3872)$, $X(4140/4274)$, are discussed. The $X(3823)$ as $\psi_2(1D)$, its radiative decay properties well agree with the observations. From the radiative decay properties of $X(3872)$, one can not exclude it as a $\chi_{c1}(2P)$ dominant state. We also give discussions of possibly observing the missing charmonium states in radiative transitions, which might provide some useful references to look for them in forthcoming experiments. The higher multipole contributions to the electromagnetic transitions are analyzed as well. It is found that the higher contribution from the magnetic part could give notable corrections to some E1 dominant processes by interfering with the E1 amplitudes. Our predictions for the normalized magnetic quadrupole amplitudes $M_2$ of the $\chi_{c1,2}(1P)\to J/\psi \gamma$ processes are in good agreement with the recent CLEO measurements. 
Wei-Jun Deng, Hui Liu, Long-Cheng Gui and Xian-Hui Zhong
hep-ph hep-ex nucl-th