Annihilation diagram contribution to charmonium masses

In this work, we generate gauge configurations with \begin{document}$ N_f = 2 $\end{document} dynamical charm quarks on anisotropic lattices. The mass shift of \begin{document}$ 1S $\end{document} and \begin{document}$ 1P $\end{document} charmonia owing to the charm quark annihilation effect can be investigated directly in a manner of unitary theory. The distillation method is adopted to treat the charm quark annihilation diagrams at a very precise level. For \begin{document}$ 1S $\end{document} charmonia, the charm quark annihilation effect almost does not change the \begin{document}$ J/\psi $\end{document} mass, but lifts the \begin{document}$ \eta_c $\end{document} mass by approximately 3-4 MeV. For \begin{document}$ 1P $\end{document} charmonia, this effect results in positive mass shifts of approximately 1 MeV for \begin{document}$ \chi_{c1} $\end{document} and \begin{document}$ h_c $\end{document} , but decreases the \begin{document}$ \chi_{c2} $\end{document} mass by approximately 3 MeV. We have not obtain a reliable result for the mass shift of \begin{document}$ \chi_{c0} $\end{document} . In addition, it is observed that the spin averaged mass of the spin-triplet \begin{document}$ 1P $\end{document} charmonia is in a good agreement with the \begin{document}$ h_c $\end{document} , as expected by the non-relativistic quark model and measured by experiments.