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Inclusive $P$-wave charmonia production in hadronic collisions at highenergies is discussed in the framework of non-relativistic QCD and$k_T$-factorization formalism. We present two consistent approches to merge theusual leading order $k_T$-factorization calculations with tree-levelnext-to-leading order off-shell amplitudes. Using these prescriptions, weextracted long-distance matrix elements for $\chi_c$ mesons from a combined fitto available Tevatron and LHC data. In contrast to previous (leading order)calculations, our fits do not contradict equal color singlet wave functions of$\chi_{c1}$ and $\chi_{c2}$ states. The extracted values of long-distancematrix elements are employed to analyse the $\chi_c$ polarization data reportedrecently by the CMS Collaboration. Our predictions are in a reasonably goodagreement with the Tevatron and LHC measurements within the theoretical andexperimental uncertainties.

Towards higher-order calculations of quarkonia production with $k_T$-factorization: $P$-wave charmonia