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In this paper, we preformed a further research on the exclusive productionsof double charmonium via $Z$-boson decay by using nonrelativistic QCDfactorizations approach, where the single-photon fragmentation topologies ofthe QED diagrams, the interference terms between the QCD and full QED diagrams,the next-to-leading-order calculations of the interference terms are preformed.For the production of $J/\psi+J/\psi$ in $Z$-boson decay, the interferenceterms show a significantly phenomenological effect due to the addition of thenewly calculated NLO QCD corrections. After adding together all contributions,the branching fraction $\mathcal{B}_{Z\to J/\psi J/\psi}$ still undershoots theCMS collaboration data obviously. In addition, we simultaneously complete thenext-to-leading-order calculations for $Z\to J/\psi+\eta_c (\chi_{cJ})$ with$J=(0,1,2)$. The calculated results show that the newly-calculated complete QEDand cross terms will have obvious effective on the total decay widths.

Next-to-leading-order QCD correction to the exclusive double charmonium production via $Z$ decays