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

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