Polarizations of $chi_{c1}$ and $chi_{c2}$ in prompt production at the LHC

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.