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Description
Predicting the production of doubly heavy baryons in electroweak-boson decays with precision requires a systematic treatment of QCD radiative corrections, especially for guiding experimental searches for the yet-unobserved $\Xi_{bc}$ baryon in large $W$-boson datasets. In this work, we compute the next-to-leading-order (NLO) QCD corrections to the inclusive decay channels $W^+\to\Xi_{cc}+\bar c+\bar s+X$ and $W^+\to\Xi_{bc}+\bar b+\bar s+X$ within the nonrelativistic QCD (NRQCD) heavy-diquark framework. We investigate both the color-antitriplet ($\bar{\mathbf{3}}$) and color-sextet ($\mathbf{6}$) configurations contributing to the $S$-wave diquark states. Ultraviolet divergences are regularized dimensionally and removed via on-shell and $\overline{\rm MS}$ counterterms, while infrared and collinear singularities are evaluated and canceled using the two-cutoff phase-space slicing method. Setting the benchmark heavy quark masses to $m_c=1.8$ GeV and $m_b=5.1$ GeV, we find that the pure NLO corrections are positive across all channels, yielding total $K$ factors of $2.20$ for $\Xi_{cc}$ and $2.91$ for $\Xi_{bc}$. For individual channels, the $K$ factors vary from $1.88$ for $\Xi_{cc}({}^3S_1,\bar{\mathbf 3})$ up to $5.45$ for $\Xi_{bc}({}^3S_1,\mathbf 6)$, highlighting the pronounced radiative enhancement in the color-sextet channels. The differential decay widths with respect to the invariant masses $s_1, s_2, s_3$ and the baryon energy fraction $z$ show that real gluon emission significantly softens the energy spectrum and shifts the event peak toward lower values of $z$.