  In this paper, we preformed a further research on the exclusive productions
of double charmonium via $Z$-boson decay by using nonrelativistic QCD
factorizations approach, where the single-photon fragmentation topologies of
the 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 interference
terms show a significantly phenomenological effect due to the addition of the
newly calculated NLO QCD corrections. After adding together all contributions,
the branching fraction $\mathcal{B}_{Z\to J/\psi J/\psi}$ still undershoots the
CMS collaboration data obviously. In addition, we simultaneously complete the
next-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 QED
and cross terms will have obvious effective on the total decay widths.
