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Updated Cross Section Measurement of $e^+ e^- \to K^+ K^- J/ψ$ and $K_S^0K_S^0J/ψ$ via Initial State Radiation at Belle

The cross sections of the processes $e^+ e^- \to K^+ K^- J/ψ$ and $K_S^0K_S^0J/ψ$ are measured via initial state radiation at center-of-mass energies between the threshold and 6.0~GeV using a data sample of 980~fb$^{-1}$ collected with the Belle detector on or near the $Υ(nS)$ resonances, where $n=$1, 2, ..., 5. The cross sections for $e^+ e^- \to K^+ K^- J/ψ$ are at a few pb level and the average cross section for $e^+ e^- \to K_S^0K_S^0J/ψ$ is $1.8\pm 0.6 (\rm stat.)\pm 0.3 (\rm syst.)$~pb between 4.4 and 5.2~GeV. All of them are consistent with previously published results with improved precision. A search for resonant structures and associated intermediate states in the cross section of the process $e^+ e^- \to K^+ K^- J/ψ$ is performed.