Bottomonium production and polarization in the NRQCD with $k_T$-factorization. III: $\Upsilon(1S)$ and $\chi_b(1P)$ mesons

The $\Upsilon(1S)$ meson production and polarization at high energies is studied in the framework of the $k_T$-factorization approach. Our consideration is based on the non-relativistic QCD formalism for a bound states formation and off-shell production amplitudes for hard partonic subprocesses. The direct production mechanism, feed-down contributions from radiative $\chi_b(mP)$ decays and contributions from $\Upsilon(3S)$ and $\Upsilon(2S)$ decays are taken into account. The transverse momentum dependent (TMD) gluon densities in a proton were derived from the Ciafaloni-Catani-Fiorani-Marchesini evolution equation and the Kimber-Martin-Ryskin prescription. Treating the non-perturbative color octet transitions in terms of multipole radiation theory, we extract the corresponding non-perturbative matrix elements for $\Upsilon(1S)$ and $\chi_b(1P)$ mesons from a combined fit to transverse momenta distributions measured at various LHC experiments. Then we apply the extracted values to investigate the polarization parameters $\lambda_\theta$, $\lambda_\phi$ and $\lambda_{\theta\phi}$, which determine the $\Upsilon(1S)$ spin density matrix. Our predictions have a reasonably good agreement with the currently available Tevatron and LHC data within the theoretical and experimental uncertainties.