Next-to-leading order QCD and relativistic corrections to Z → J/ψ +Υ (nS)

In this paper, we calculate the decay widths and branching fractions for the decays Z → J/ψ +Υ (nS) Z → J / ψ + Υ ( n S ) ( $$n=1,2,3$$ 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 due to significant numerical cancellations. Despite the resulting large theoretical uncertainties, sizable event rates from these rare decay channels could still be anticipated at future high-luminosity electron-positron colliders running around the Z-pole. Ultimately, our results provide an essential theoretical baseline, highlighting the necessity of incorporating even higher-order corrections and resummation techniques for future precision phenomenological studies.