\begin{document}$ \Xi_{bQ} $\end{document} baryon because a large number of its events can be collected at high energy colliders. In this paper, we perform a complete investigation of the indirect production of the \begin{document}$ \Xi_{bc} $\end{document} and \begin{document}$ \Xi_{bb} $\end{document} baryons via the Z-boson decay process \begin{document}$ Z\to \Xi_{bQ}+\bar b +\bar Q $\end{document} with a \begin{document}$ Q= (c,b) $\end{document} quark under the NRQCD factorization approach. After considering the contribution from the diquark states \begin{document}$ \langle bc\rangle[^3S_1]_{\bar 3/6} $\end{document}, \begin{document}$ \langle bc\rangle[^1S_0]_{\bar 3/6} $\end{document}, \begin{document}$ \langle bb\rangle [^1S_0]_6 $\end{document}, and \begin{document}$ \langle bb\rangle[^3S_1]_{\bar 3} $\end{document}, the calculated branching fractions are \begin{document}$ {\cal B}(Z\to\Xi_{bc}+X) = 3.595\times 10^{-5} $\end{document} and \begin{document}$ {\cal B}(Z\to\Xi_{bb}+X) = 1.213\times 10^{-6} $\end{document}. Furthermore, the \begin{document}$ \Xi_{bc} $\end{document} and \begin{document}$ \Xi_{bb} $\end{document} production events are predicted to be of the order of \begin{document}$ 10^4 $\end{document} and \begin{document}$ 10^3 $\end{document} at the LHC collider, and the order of \begin{document}$ 10^7 $\end{document} and \begin{document}$ 10^6 $\end{document} for the CEPC collider. We then estimate the production ratio for \begin{document}$ Z\to c\bar c(b\bar b)\to\Xi^{+,0}_{bc}+X $\end{document} with \begin{document}$ \Xi^{+,0}_{bc} \to \Lambda^+_c $\end{document} in Z-boson decay, that is, \begin{document}$ {\cal R}(Z\to c\bar c\to\Xi^{+,0}_{bc}+X) \sim {\cal O}(10^{-6}) $\end{document} and \begin{document}$ {\cal R}(Z\to b\bar b\to\Xi^{+,0}_{bc}+X) \sim {\cal O}(10^{-5}) $\end{document}, respectively. Finally, we present the differential decay widths of \begin{document}$ \Xi_{bc} $\end{document} and \begin{document}$ \Xi_{bb} $\end{document} with respect to \begin{document}$ s_{23} $\end{document} and z distributions and analyze the uncertainties."> Investigation of <i>Z</i>-boson decay into <inline-formula><tex-math id="M1">\begin{document}$ {\boldsymbol\Xi_{\boldsymbol bc}} $\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="//www.macurncorp.com/cpc/article/app/id/9e7d2971-fdf1-4023-8141-55c4b689f75c/CPC-2022-0485_M1.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="//www.macurncorp.com/cpc/article/app/id/9e7d2971-fdf1-4023-8141-55c4b689f75c/CPC-2022-0485_M1.png"/></alternatives></inline-formula> and <inline-formula><tex-math id="M2">\begin{document}${ \boldsymbol\Xi_{\boldsymbol bb}} $\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="//www.macurncorp.com/cpc/article/app/id/9e7d2971-fdf1-4023-8141-55c4b689f75c/CPC-2022-0485_M2.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="//www.macurncorp.com/cpc/article/app/id/9e7d2971-fdf1-4023-8141-55c4b689f75c/CPC-2022-0485_M2.png"/></alternatives></inline-formula> baryons within the NRQCD factorization approach -
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