Revisiting the line shape of $e^+e^-\to \jpsi \eta$ cross section

We calculate the cross sections for the processes $e^+e^-\to \jpsi \eta$ and $e^+e^-\to \jpsi \eta^\prime$ at various CM energies $\sqrt{s}$. We first predict these cross sections by combining NRQCD with LC factorization. The predicted cross sections are on the order of several femtobarns for $e^+e^-\to \jpsi \eta^\prime$, and less than 1 fb for $e^+e^-\to \jpsi \eta$, which are significantly smaller than the experimental measurements. It is anticipated that the cross sections are dominated by resonant contributions when $\sqrt{s}$ is close to the resonance mass. In this study, we employ the Vector Meson Dominance (VMD) model to predict these resonant contributions. The effective coupling constants between the photon and the resonance, as well as between the resonance and $J/\psi\eta$ are extracted from the data either provided by the latest PDG or predicted by theoretical calculations. Taking the predictions from the factorization calculation as the continuum contribution, we predict the cross section of $e^+e^-\to \jpsi \eta$ through a coherent sum of contributions from various resonances and the continuum. The relative phase angles between these contributions are determined through a least-$\chi^2$ fit to the experimental data. We then compare our theoretical predictions with the experimental data. Additionally, we find our theoretical prediction for the cross section of $e^+e^-\to \jpsi \eta$ is significantly larger than those for $e^+e^-\to \jpsi \eta^\prime$ measured by the BESIII collaboration.