Bottomonia production and polarization in the NRQCD with $k_T$-factorization. II: $\Upsilon(2S)$ and $\chi_b(2P)$ mesons

The $\Upsilon(2S)$ production and polarization at high energies is studied in the framework of $k_T$-factorization approach. Our consideration is based on the non-relativistic QCD formalism for bound states formation and off-shell production amplitudes for hard partonic subprocesses. The direct production mechanism, feed-down contributions from radiative $\chi_b(3P)$ and $\chi_b(2P)$ decays and contributions from $\Upsilon(3S)$ decays are taken into account. The transverse momentum dependent gluon densities in a proton were derived from the Ciafaloni-Catani-Fiorani-Marchesini evolution equation as well as from the Kimber-Martin-Ryskin prescription. Treating the non-perturbative color octet transitions in terms of the mulitpole radiation theory, we extract the corresponding non-perturbative matrix elements for $\Upsilon(2S)$ and $\chi_b(2P)$ mesons from a combined fit to $\Upsilon(2S)$ transverse momenta distributions measured by the CMS and ATLAS Collaborations at the LHC energies $\sqrt s = 7$ and $13$ TeV and from the relative production rate $R^{\chi_b(2P)}_{\Upsilon(2S)}$ measured by the LHCb Collaboration at $\sqrt s = 7$ and $8$ TeV. Then we apply the extracted values to investigate the polarization parameters $\lambda_\theta$, $\lambda_\phi$ and $\lambda_{\theta\phi}$, which determine the $\Upsilon(2S)$ spin density matrix. Our predictions have a good agreement with the currently available data within the theoretical and experimental uncertainties.