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In this paper, we calculate the total decay widths for the $W^+$-bosondecays, $W^+ \to B_c+b+\bar{s}+X$ and $W^+ \to B^*_c+b+\bar{s}+X$, up tonext-to-leading order (NLO) accuracy within the framework of thenonrelativistic QCD theory. Both the fixed-order and the fragmentationapproaches are adopted to do the calculation. Differential decay widths$d\Gamma/dz$ and $d\Gamma/ds_1$ are also given. We find that the NLOcorrections are significant in those two $W^+$ decay channels. Our numericalresults show that at the LHC, there are about $7.03\times 10^4$ $B_c$-mesonevents and $5.10\times 10^4$ $B^*_c$-meson events to be produced via the$W^+$-boson decays per operation year.

Next-to-leading order QCD corrections to the production of $B_c$ and $B_c^*$ through $W^+$-boson decays