Understanding the cross section of $e^+e^-\to \eta J/\psi$ process via nonrelativistic QCD

Motivated by the large cross section of $e^+e^-\to \eta J/\psi$ process measured by the BESIII and Belle Collaborations recently, we evaluate this process at $\mathcal{O}(\alpha_s^4)$ accuracy in the framework of nonrelativistic QCD. We find that the cross section at the center-of-mass energy $\sqrt{s}=4.009$ {GeV} is $34.6\mathrm{pb}$, which is consistent with the BESIII data. The comparison with the Belle data in the region from $4.0\ \mathrm{GeV}$ to $5.3\ \mathrm{GeV}$ is also presented. Concerning the $\eta$ and $\eta^\prime$ mixing and the potential gluonium component of $\eta^\prime$, we also estimate the rate of $e^+e^-\to \eta^\prime J/\psi$ process, which can be checked within the updated Belle data.