Two-loop QCD-corrections to $e^{+} e^{-} \rightarrow Z^{\ast} \rightarrow \boldsymbol{J /\psi}+\boldsymbol{J/ \psi}$

We present a next-to-next-to-leading-order calculation within the nonrelativistic QCD framework for the process of $e^{+}e^{-} \rightarrow Z^{\ast} \rightarrow J/\psi+J/\psi$ . We find that the NNLO contribution is 2-3 times larger than the next-to-leading-order contribution, which itself is already 3-5 times larger than the leading-order result. In the high-energy limit, we provide analytic expressions for the leading-power coefficients in the asymptotic expansion of the two-loop amplitudes. Our results are directly applicable to the bottomonium process $Z^{\ast} \rightarrow \Upsilon+\Upsilon$. Using the obtained hadronic amplitudes, we predict the decay width of the $Z$ boson into these rare di-charmonium and di-bottomonium final states.