\begin{document}$ \Xi_{cc} $\end{document}, \begin{document}$ \Xi_{bc} $\end{document}, and \begin{document}$ \Xi_{bb} $\end{document}, within the nonrelativistic QCD framework at the Muon-Ion Collider (MuIC). It examines two production mechanisms: photon-gluon fusion (\begin{document}$ \gamma + g \to (QQ')[n] +\bar{Q} +\bar{Q'} $\end{document}) and extrinsic heavy quark channels (\begin{document}$ \gamma + Q \to (QQ')[n] + \bar{Q'} $\end{document}), where \begin{document}$ Q $\end{document} and \begin{document}$ Q' $\end{document} denote heavy quarks (\begin{document}$ c $\end{document} or \begin{document}$ b $\end{document}) and \begin{document}$ (QQ')[n] $\end{document} represents a diquark in specific spin-color configurations. The diquark fragments into \begin{document}$ \Xi_{QQ'} $\end{document} baryons with high probability. For \begin{document}$ \Xi_{cc} $\end{document} and \begin{document}$ \Xi_{bb} $\end{document}, the relevant configurations are \begin{document}$ [^1S_0]_{{6}} $\end{document} (spin-singlet and color-sextuplet) and \begin{document}$ [^3S_1]_{\bar{{3}}} $\end{document} (spin-triplet and color-antitriplet). For \begin{document}$ \Xi_{bc} $\end{document}, the configurations are \begin{document}$ [^1S_0]_{\bar{{3}}} $\end{document}, \begin{document}$ [^1S_0]_{{6}} $\end{document}, \begin{document}$ [^3S_1]_{\bar{{3}}} $\end{document}, and \begin{document}$ [^3S_1]_{{6}} $\end{document}. The study compares total and differential cross-sections for these channels, highlighting their uncertainties. The results indicate that the extrinsic heavy quark channel, particularly the \begin{document}$ [^3S_1]_{\bar{{3}}} $\end{document} configuration, dominates \begin{document}$ \Xi_{QQ'} $\end{document} production, though other diquark states also contribute significantly. Using quark masses \begin{document}$ m_c = 1.80 \pm 0.10 $\end{document} GeV and \begin{document}$ m_b = 5.1 \pm 0.20 $\end{document} GeV, the study estimates annual event yields at MuIC (\begin{document}$ \sqrt{s} = 1 $\end{document} TeV, luminosity \begin{document}$ {\mathcal L}\simeq 40 $\end{document} \begin{document}$ {\rm{fb}}^{-1} $\end{document}) of \begin{document}$ (3.67^{+1.29}_{-0.91}) \times 10^9 $\end{document} for \begin{document}$ \Xi_{cc} $\end{document}, \begin{document}$ (2.24^{+0.28}_{-0.20}) \times 10^8 $\end{document} for \begin{document}$ \Xi_{bc} $\end{document}, and \begin{document}$ (3.00^{+0.64}_{-0.56}) \times 10^6 $\end{document} for \begin{document}$ \Xi_{bb} $\end{document}. These findings suggest that MuIC will significantly enhance our understanding of doubly heavy baryons."> Production of doubly heavy baryon at the Muon-Ion collider -
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