Bottomonia production and polarization in the NRQCD with

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, Kimber-Martin-Ryskin prescription and Parton Branching method. 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