Abstract
The contributions from the color-octet components $|(c\overline{b}{)}_{\mathbf{8}}(^{1}S_{0})g⟩$ and $|(c\overline{b}{)}_{\mathbf{8}}(^{3}S_{1})g⟩$ to the ${h}_{{B}_{c}}$ or ${\ensuremath{\chi}}_{{B}_{c}}^{J}$ (the $P$-wave ${B}_{c}$ meson) hadroproduction are estimated in terms of the complete $\mathcal{O}({\ensuremath{\alpha}}_{s}^{4})$ calculation. As necessary inputs in the estimate, we take the values of the octet matrix elements according to the nonrelativistic quantum chromodynamics scaling rules, and as a result, we have found that the contributions to the $P$-wave production may be the same in order of magnitude as those from the color-singlet ones, $|(c\overline{b}{)}_{\mathbf{1}}(^{1}P_{1})⟩$ and $|(c\overline{b}{)}_{\mathbf{1}}(^{3}P_{J})⟩$ ($J=1,2,3$). Especially, our result indicates that the observation of the color-octet contributions to the $P$-wave production in the low transverse momentum region is not very far from the present experimental capability at Tevatron and LHC.
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