How well does nonrelativistic QCD factorization work at next-to-leading order?

Abstract: We perform a thorough investigation of the universality of the long distance matrix elements (LDMEs) of nonrelativistic QCD factorization based on a next-to-leading order (NLO) fit of $J/\psi$ color octet (CO) LDMEs to high transverse momentum $p_T$ $J/\psi$ and $\eta_c$ production data at the LHC. We thereby apply a novel fit-and-predict procedure to systematically take into account scale variations, and predict various observables never studied in this context before. In particular, the LDMEs can well describe $J/\psi$ hadroproduction up to the highest measured values of $p_T$, as well as $\Upsilon(nS)$ production via potential NRQCD based relations. Furthermore, $J/\psi$ production in $\gamma \gamma$ and $\gamma p$ collisions is surprisingly reproduced down to $p_T=1$ GeV, as long as the region of large inelasticity $z$ is excluded, which may be of significance in future quarkonium studies, in particular at the EIC and the high-luminosity LHC. In addition, our summary reveals an interesting pattern as to which observables still evade a consistent description.