Quarkonium polarization in low-$p_{\rm T}$ hadro-production: from past data to future opportunities

Several fixed-target experiments reported J/$\psi$ and $\Upsilon$ polarization measurements, as functions of Feynman $x$ ($x_{\rm F}$) and transverse momentum ($p_{\rm T}$), in three different polarization frames, using different combinations of beam particles, target nuclei and collision energies. The data form such a diverse and heterogeneous picture that, at first sight, no clear trends can be observed. A more detailed look, however, allows us to discern qualitative physical patterns that inspire and support a simple interpretation: the directly-produced quarkonia result from either gluon-gluon fusion or from quark-antiquark annihilation, with the former mesons being fully longitudinally polarized and the latter being fully transversely polarized. This hypothesis provides a reasonable quantitative description of the J/$\psi$ and $\Upsilon$(1S) polarizations measured in the $x_{\rm F} \lesssim 0.5$ kinematical domain. We provide predictions that can be experimentally tested, using proton and/or pion beams, and show that improved J/$\psi$ and $\psi$(2S) polarization measurements in pion-nucleus collisions can provide significant constraints on the poorly known parton distribution functions of the pion.