Next-to-leading order QCD and relativistic corrections to $$Z \rightarrow J/\psi +\Upsilon (nS)$$

Abstract In this paper, we calculate the decay widths and branching fractions for the decays $$Z \rightarrow J/\psi +\Upsilon (nS)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>Z</mml:mi> <mml:mo>→</mml:mo> <mml:mi>J</mml:mi> <mml:mo>/</mml:mo> <mml:mi>ψ</mml:mi> <mml:mo>+</mml:mo> <mml:mi>Υ</mml:mi> <mml:mo>(</mml:mo> <mml:mi>n</mml:mi> <mml:mi>S</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> ( $$n=1,2,3$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>n</mml:mi> <mml:mo>=</mml:mo> <mml:mn>1</mml:mn> <mml:mo>,</mml:mo> <mml:mn>2</mml:mn> <mml:mo>,</mml:mo> <mml:mn>3</mml:mn> </mml:mrow> </mml:math> ) 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.