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.
Based on a general form of the effective vertex functions for the decays of P-wave charmonia $chicj$, angular distribution formulas for the subsequent decays $chicjrightarrow jpsi gamma$ and $jpsi to mu^+mu^-$ are derived. The formulas are the same a
s those obtained in a different approach in the literature. Our formulas are expressed in a more general form, including parity violation effects and the full angular dependence of $jpsi$ and muon in the cascade decay $chicjtojpsigammatomu^+mu^-gamma$. The $chicj$ polarization observables are expressed in terms of rational functions of the spin density matrix elements of $chicj$ production. Generalized rotation-invariant relations for arbitrary integer-spin particles are also derived and their expressions in terms of observable angular distribution parameters are given in the $chi_{c1}$ and $chi_{c2}$. To complement our previous direct-$jpsi$ polarization result, we also discuss the impact on the observable prompt-$jpsi$ polarization. As an illustrative application of our angular distribution formulas, we present the angular distributions in terms of the tree-level spin density matrix elements of $chi_{c1}$ and $chi_{c2}$ production in several different frames at the Large Hadron Collider. Moreover, a reweighting method is also proposed to determine the entire set of the production spin density matrix elements of the $chi_{c2}$, some of which disappear or are suppressed for vanishing higher-order multipole effects making the complete extraction difficult experimentally.
