The Glueball content of $\eta_c$

We carry out the first lattice QCD derivation of the mixing energy and the mixing angle of the pseudoscalar charmonium and glueball on two gauge ensembles with $N_f=2$ degenerate dynamical charm quarks. The mixing energy is determined to be $49(6)$ MeV on the near physical charm ensemble, which seems insensitive to charm quark mass. By the assumption that $X(2370)$ is predominantly a pseudoscalar glueball, the mixing angle is determined to be approximately $4.6(6)^\circ$, which results in a $+3.9(9)$ MeV mass shift of the ground state pseudoscalar charmonium. In the mean time, the mixing can raise the total width of the pseudoscalar charmonium by 7.2(8) MeV, which explains to some extent the relative large total width of the $\eta_c$ meson. As a result, the branching fraction of $\eta_c\to \gamma\gamma$ can be understood in this $c\bar{c}$-glueball mixing framework. On the other hand, the possible discrepancy of the theoretical predictions and the experimental results of the partial width of $J/\psi\to\gamma\eta_c$ cannot be alleviated by the $c\bar{c}$-glueball mixing picture yet, which demands future precise experimental measurements of this partial width.