Quarkonium production and polarization: where do we stand?

We review the current status of heavy quarkonium production phenomenology based on nonrelativistic effective field theories, focusing on spin-triplet $S$-wave states such as $J/\psi$, $\psi(2S)$, and $\Upsilon$. We present some representative examples for heavy quarkonium production mechanisms proposed in the literature, which vary significantly depending on the choice of data employed in analyses. We then discuss the r\^{o}le of polarization in discriminating between the different possible scenarios for quarkonium production. Other observables that may be useful in pinpointing the production mechanism are also introduced, such as the $\eta_c$ production, associated production of $J/\psi$ plus a gauge boson, and $J/\psi$ production at the Electron-Ion Collider.