$\mathrm{{\rm Y}}(nS)$ and ${\chi }_b(nP)$ production at hadron colliders in nonrelativistic QCD

$\mathrm{{\rm Y}}(nS)$ and ${\chi }_b(nP)$ (n=1,2,3) production at the LHC is studied at next-to-leading order in ${\alpha }_s$ in nonrelativistic QCD. Feeddown contributions from higher ${\chi }_b$ and $\mathrm{{\rm Y}}$ states are all considered for lower $\mathrm{{\rm Y}}$ cross sections and polarizations. The long distance matrix elements (LDMEs) are extracted from the yield data, and then used to make predictions for the $\mathrm{{\rm Y}}(nS)$ polarizations, which are found to be consistent with the measured polarization data within errors. In particular, the $\mathrm{{\rm Y}}(3S)$ polarization puzzle can be understood by a large feeddown contribution from ${\chi }_b(3P)$ states. Our results may provide a good description for both cross sections and polarizations of prompt $\mathrm{{\rm Y}}(nS)$ and ${\chi }_b(nP)$ production at the LHC.