We present a complete evaluation for the prompt $eta_c$ production at the LHC at next-to-leading order in $alpha_s$ in nonrelativistic QCD. By assuming heavy quark spin symmetry, the recently observed $eta_c$ production data by LHCb results in a very
strong constraint on the upper bound of the color-octet long distance matrix element $1S0$ of $J/psi$. We find this upper bound is consistent with our previous study of the $J/psi$ yield and polarization and can give good descriptions for the measurements, but inconsistent with some other theoretical estimates. This may provide important information for understanding the nonrelativistic QCD factorization formulism.
Prompt $chi_c$ production at hadron colliders may provide a unique test for the color-octet mechanism in nonrelativistic QCD. We present an analysis for the polarization observables of $chi_{c1}$ and $chi_{c2}$ at next-to-leading order in $alpha_S$,
and propose to measure them at the LHC, which is expected to be important for testing the validity of NRQCD.
We calculate the annihilation decay widths of spin-singlet heavy quarkonia $h_c, h_b$ and $eta_b$} into light hadrons with both QCD and relativistic corrections at order $O(alpha_{s}v^{2})$ in nonrelativistic QCD. With appropriate estimates for the l
ong-distance matrix elements by using the potential model and operator evolution method, we find that our predictions of these decay widths are consistent with recent experimental measurements. We also find that the $O(alpha_{s}v^{2})$ corrections are small for $bbar{b}$ states but substantial for $cbar{c}$ states. In particular, the negative contribution of $O(alpha_{s}v^{2})$ correction to the $h_{c}$ decay can lower the decay width, as compared with previous predictions without the $O(alpha_{s}v^{2})$ correction, and thus result in a good agreement with the recent BESIII measurement.
