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 $end{document} charmonia due to the charm quark annihilation effect can be investigated directly in a manner of unitary theory. The distillation method is adopted to treat the charm quark annihilation diagrams at a very precise level. For begin{document}$ 1S $end{document} charmonia, the charm quark annihilation effect barely changes the begin{document}$ 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} $end{document} mass by approximately 3 MeV. We did not obtain a reliable result for the mass shift of begin{document}$ chi_{c0} $end{document}. In addition, we observed that the spin averaged mass of the spin-triplet begin{document}$ 1P $end{document} charmonia is in good agreement with begin{document}$ h_c $end{document}, as expected by the non-relativistic quark model and measured by experiments.