\chi_{cJ} polarizations at the Fermilab Tevatron

We propose the measurement of $\chi_{cJ}$ polarizations at high energy hadron colliders to study heavy quarkonium production mechanism. We find that the color-singlet model in the $k_t$ factorization approach predicts very different behavior for $\chi_{cJ}$ polarizations at the Fermilab Tevatron compared with the NRQCD predictions in the collinear parton model. In the color-singlet $k_t$ factorization approach, for both $\chi_{c1}$ and $\chi_{c2}$ productions, the helicity $h=0$ states dominate over other helicity states at large $p_T$. These properties are very useful in distinguishing between the two production mechanisms which are related to the interesting issue of $J\psi$ and $\psi'$ polarizations, and may provide a crucial test for the $k_t$ factorization approach.