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The $Z$-boson decay provides good opportunities for the research on$\Xi_{bQ'}$ baryon due to large quantity of $Z$ events that can be collected atthe high-energy colliders. We performed a completed investigation of theindirect production of the $\Xi_{bc}$ and $\Xi_{bb}$ baryon via $Z$-boson decay$Z\to \Xi_{bQ'}+\bar b +\bar Q'$ with $Q'= (c, b)$ quark according to NRQCDfactorizations approach. After considering the contribution of the diquarkstates $\langle bc\rangle [^3S_1]_{\bar 3/6}$, $\langle bc\rangle [^1S_0]_{\bar3/6}$, $\langle bb\rangle [^1S_0]_{6}$ and $\langle bb\rangle [^3S_1]_{\bar3}$, the calculated branching ratio for $Z\to\Xi_{bQ'}+X$ are ${\calB}(Z\to\Xi_{bc}+X) = 3.595\times 10^{-5}$ and ${\cal B}(Z\to\Xi_{bc}+X) =1.213\times 10^{-6}$. Moreover, the $\Xi_{bc}$ events produced are predicted tobe of the $10^4(10^7)$ order at the LHC(CEPC), while the $\Xi_{bb}$ eventsproduced are forecasted to be of the $10^3(10^6)$ order. Furthermore, we haveestimated the production ratio ${\cal R}(Z_Q\to\Xi^{+,0}_{bc})$ with four$Z$-boson decay channels. The ${\cal R}(Z_Q\to\Xi^{+,0}_{bc})$ up to $10^{-6}$for $Z\to c\bar c$ channel and $10^{-5}$ for $Z\to b\bar b$ channel,respectively. Finally, we present the differential decay widths of$\Xi_{bc}(\Xi_{bb})$ with respect to $s_{23}$ and $z$ distributions, andanalysis the uncertainties.

Investigation for $Z$-boson decay into $\Xi_{bc}$ and $\Xi_{bb}$ baryon with the NRQCD factorizations approach