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In this article, we study in detail the double-$J/\psi$ yield through $Z$decay at the next-to-leading-order (NLO) QCD accuracy within thenonrelativistic QCD factorization. At the tree level, the pure QCD diagramspredict a branching ratio of $\mathcal{B}_{Z \to J/\psi+J/\psi} \sim 10^{-12}$;however, the inclusion of the QED diagrams would augment this prediction byapproximately 2-3 orders of magnitude. After incorporating the QCD corrections,the QCD results exhibit a considerable increase, whereas the QED resultsundergo a substantial reduction. Combining the QCD and QED contributions at NLOin $\alpha_s$, it is observed that the prediction of $\mathcal{B}_{Z \toJ/\psi+J/\psi}=(1.110^{+0.334+0.054}_{-0.241-0.001})\times 10^{-10}$, whichdisplays a fairly steady dependence on the renormalization scale, issignificantly lower than the upper limits released by CMS.

Analysis of double-$J/\psi$ production in $Z$ decay at next-to-leading-order QCD accuracy