Perturbative QCD for $J/\psi$ Inclusive Production Via Initial State Radiation at $e^+e^-$ collider

Up to the next-leading order (NLO) of quantum chromodynamics (QCD), the process $e^+e^-\to J/\psi +X$ with the center-of-mass (CM) energy range from 3.7 to 10.6 GeV is calculated. At 10.6 GeV, the results is consistent with the experiment results at the Belle. However, the predictions are much smaller than the measurement at BESIII at low CM energy range from 3.7 to 4.6 GeV. This indicates that the convergence of QCD perturbative expansion becomes worse as the CM energy becomes lower and closer to the inclusive $J/\psi$ production threshold. For a further study of the QCD mechanism on $J/\psi$ production at $e^+e^-$ collider with different CM energy, the initial state radiation effect of $e^+e^-\to J/\psi+gg$ and $e^+e^-\to J/\psi+c \bar{c}$ are calculated at the QCD NLO. The results are plotted and the numbers of events for different CM energy bins are provided for the designed SuperKEKB. This provides a method to precisely test the validity of perturbative prediction on $J/\psi$ production in future measurements.