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We aim to explore the production rate of the pseudoscalar glueball in$J/\psi$ radiative decay by lattice QCD in quenched approximation. Thecalculation is performed on three anisotropic lattices with the spatial latticespacing ranging from 0.222(2) fm to 0.110(1) fm. As a calibration of somesystematical uncertainties, we first extract the $M1$ form factor $\hat{V}(0)$of the process $J/\psi\to\gamma\eta_{c}$ and get the result$\hat{V}(0)=1.933(41)$ in the continuum limit, which gives the partial width$\Gamma(J/\psi\to\gamma\eta_{c})=2.47(11)$ keV. These results are in agreementwith that of previous lattice studies. As for the pseudoscalar glueball$G_{0^{-+}}$, its mass is derived to be $2.395(14)$ GeV, and the form factor$\hat{V}(0)$ of the process $J/\psi\to\gamma G_{0^{-+}}$ is determined to be$\hat{V}(0)=0.0246(43)$ after continuum extrapolation. Finally, the productionrate of the pseudoscalar glueball is predicted to be $2.31(90)\times10^{-4}$,which is much smaller than that of conventional light $q\bar{q}$ $\eta$ states.After the subtraction of the phase space factor, the couplings of $J/\psiX\gamma$ are similar where $X$ stands for $\eta$ states and the pseudoscalarglueball. Possibly, the $U_{A}(1)$ anomaly plays an important role for thelarge couplings of gluons to the flavor singlet $\eta$ states in $J/\psi$radiative decays.

Study of the pseudoscalar glueball in $J/\psi$ radiative decays