  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.
