Annihilation diagram contribution to charmonium masses

TL;DR

This study investigates the charm quark annihilation effects on charmonium masses using lattice QCD with dynamical charm quarks, revealing small but specific mass shifts consistent with theoretical expectations.

Contribution

It introduces a precise lattice QCD approach to directly measure charm quark annihilation effects on charmonium masses, including the use of the distillation method.

Findings

The charm quark annihilation effect minimally impacts the $J/eta$ mass.

The $ ext{eta}_c$ mass increases by about 3-4 MeV due to annihilation.

The mass shifts for $1P$ states vary, with some increasing and one decreasing by a few MeV.

Abstract

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 $J/\psi$ mass, but lifts the $\eta_c$ mass by approximately 3-4 MeV. For $1P$ charmonia, this effect results in positive mass shifts of approximately 1 MeV for $\chi_{c1}$ and $h_c$, but decreases the $\chi_{c2}$ mass by approximately 3 MeV. We have not obtain a reliable result for the mass shift of $\chi_{c0}$. 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.