Next-to-leading order QCD and relativistic corrections to $Z\to J\mathrm{/}\psi +\mathrm{{\rm Y}}(nS)$

In this paper, we calculate the decay widths and branching fractions for the decays $Z\to J\mathrm{/}\psi +\mathrm{{\rm Y}}(nS)$ ($n=1,2,3$) at future super $Z$ factory and at the CEPC/FCC-ee, including both the relativistic and QCD corrections within the framework of nonrelativistic QCD. Both the relativistic and QCD corrections are found to be large and negative. Compared to the leading-order results, the decay widths are significantly reduced by the higher-order corrections. Therefore, it is essential to take these corrections into account for a reliable estimation. For a high-luminosity electron positron collider running around the $Z$-pole, sizable event rates could be produced from these rare decay channels due to the $Z$-boson resonance effect.