  We aim to explore the production rate of the pseudoscalar glueball in
$J/\psi$ radiative decay by lattice QCD in quenched approximation. The
calculation is performed on three anisotropic lattices with the spatial lattice
spacing ranging from 0.222(2) fm to 0.110(1) fm. As a calibration of some
systematical 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 agreement
with 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 production
rate 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 $J/\psi
X\gamma$ are similar where $X$ stands for $\eta$ states and the pseudoscalar
glueball. Possibly, the $U_{A}(1)$ anomaly plays an important role for the
large couplings of gluons to the flavor singlet $\eta$ states in $J/\psi$
radiative decays.
