The Color-Octet Contributions to $P$-wave $B_c$ Meson Hadroproduction

The contributions from the color-octet components $|(c\bar b){\bf 8}(^{1}S{0}) g>$and$|(c\bar b){\bf 8}(^{3}S{1}) g>$tothe$h_{B_c}$or$\chi_{B_c}^J$(the$P$-wave$B_c$ meson) hadroproduction are estimated in terms of the complete $\mathcal{O}({\alpha }_s^4)$ calculation. As necessary inputs in the estimate, we take the values of the octet matrix elements according to the NRQCD 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, $\mathrm{\mid }(c\bar{b}{)}_{\mathbf{1}}{(}^1{P}_1)>$ and $|(c\bar b){\bf 1}(^{3}P{J})>$($J=1,2,3$). 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.