Charmonium mass shifts in an unquenched quark model

In this paper, we performed a coupled-channel calculation and evaluated the mass shifts for all $1S$, $2S$, $1P$, $2P$ and $1D$ charmonium valence states below 4 GeV, by incorporating the four-quark components ($D$, $D^*$, $D_s$ and $D_s^*$ meson pairs) into the quark model. The valence-continuum coupling is provided by the $^3P_0$ quark-pair creation model. The induced mass shifts appear to be large and negative with the original transition operator in $^3P_0$ model, which raised up challenges for the valence quark model. More QCD-motivated models should be employed for the quark-pair creation Hamiltonian. So herein, we recalculated the mass shifts with the improved $^3P_0$ transition operator introduced in our previous work and the mass shifts are reduced by $75\%$ averagely. Besides, as a exercise, we adjust the confinement parameter $\Delta$ and recalculate the spectrum of the charmonium states. The masses of some charmonium states are reproduced well.