Decay widthes of $^3 P_J$ charmonium to $DD,DD^*,D^*D^*$ and corresponding mass shifts of $^3 P_J$ charmonium

In this work, we calculate the amplitudes of the processes $cbar c({^3P_J}) rightarrow DD,DD^*, D^*D^* rightarrow cbar c({^3P_J})$ in the leading order of the nonrelativistic expansion. The imaginary parts of the amplitudes are corresponding to the branch decay widthes of the charmonium $cbar c({^3P_J}) rightarrow DD,DD^*, D^*D^*$ and the real parts are corresponding to the mass shifts of the charmonium $cbar c({^3P_J})$ due to these decay channels. After absorbing the polynomial contributions which are pure real and include the UV divergences, the ratios between the branch decay widthes and the corresponding mass shifts are only dependent on the center-of-mass energy. We find the decay widthes and the mass shifts of the $^3P_2$ states are exact zero in the leading order. The ratios between the branch decay widthes and the mass shifts for the $^3P_0, {^3P_1}$ states are larger than 5 when the center-of-mass energy is above the $DD,DD^*, D^*D^*$ threshold. The dependence of the mass shifts on the center-of-mass energy is nontrivial especially when the center-of-mass energy is below the threshold. The analytic results can be extended to the $b$ quark sector directly.