Bottomonia production and polarization in the NRQCD with $k_T$-factorization. I: $\Upsilon(3S)$ and $\chi_b(3P)$ mesons

The $\Upsilon(3S)$ 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 transverse momentum dependent (TMD, or unintegrated) gluon densities in a proton were derived from the Ciafaloni-Catani-Fiorani-Marchesini (CCFM) evolution equation as well as from the Kimber-Martin-Ryskin (KMR) prescription. Treating the non-perturbative color octet transitions in terms of the multipole radiation theory and taking into account feed-down contributions from radiative $\chi_b(3P)$ decays, we extract the corresponding non-perturbative matrix elements for $\Upsilon(3S)$ and $\chi_b(3P)$ mesons from a combined fit to $\Upsilon(3S)$ transverse momenta distributions measured by the CMS and ATLAS Collaborations at the~LHC energies $\sqrt s = 7$ and $13$~TeV and central rapidities. Then we apply the extracted values to describe the CDF and LHCb data on $\Upsilon(3S)$ production and to investigate the polarization parameters $\lambda_\theta$, $\lambda_\phi$ and $\lambda_{\theta\phi}$, %and $\tilde\lambda$, which determine the $\Upsilon(3S)$ spin density matrix. Our predictions have a good agreement with the currently available data within the theoretical and experimental uncertainties.