Aspects of $\chi$ and $\psi$ Production in Polarized Proton-Proton   Collisions
  Several topics of relevance to low transverse momentum $\psi$ and $\chi_{1,2}(c\overline{c})$ production in polarized proton-proton collisions are discussed. The leading ${\cal O}(\alpha_S^3)$ contributions to the low $p_T$ $\chi_1$ production cross-sections via $gg$, $qg$, and $q\overline{q}$ initial states are calculated as well as the corresponding spin-spin asymmetries. We find that $\chi_1$ production increases relative to direct $\psi$ and $\chi_2$ production, providing up to $25\%$ of the observable $e^+e^-$ pairs arising from $\psi$ decays in $pp$ collisions at $\sqrt{s} = 500\,GeV$. The spin-dependence of $\chi_1$ production, however, is much smaller than for either direct $\psi$ or $\chi_2$ production and so will likely be far less useful than either process in probing the polarized gluon structure function of the proton. A subset of the ${\cal O}(\alpha_S^3)$ radiative corrections to $\chi_2$ production involving initial state quarks are also performed and compared to leading order $gg \rightarrow \chi_2$ predictions. 
M. A. Doncheski and R. W. Robinett