Annihilation diagram contribution to charmonium masses

In this work, we generate gauge configurations with $N_f=2$ dynamical charm quarks on anisotropic lattices. The mass shift of $1S$ and $1P$ 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 $1S$ charmonia, the charm quark annihilation effect almost does not change the $J/ψ$ mass, but lifts the $η_c$ mass by approximately 3-4 MeV. For $1P$ charmonia, this effect results in positive mass shifts of approximately 1 MeV for $χ_{c1}$ and $h_c$, but decreases the $χ_{c2}$ mass by approximately 3 MeV. We have not obtain a reliable result for the mass shift of $χ_{c0}$. In addition, it is observed that the spin averaged mass of the spin-triplet $1P$ charmonia is in a good agreement with the $h_c$, as expected by the non-relativistic quark model and measured by experiments.

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