$J/\psi$ Polarization and $p_T$ distribution in $c\ \!\bar{c}$ associated hadroproduction at $\mathcal{O}(\alpha_s^5)$

The quarkonium production in association with a heavy-quark pair of the same flavor was found plays an important role in various production schemes, let alone it provides a distinctive signature that could be studied in experiment. Within the NRQCD framework, we perform the first complete calculation of the hadroproduction process $gg\to J/\psi+c\bar{c}$ at the next-to-leading order (NLO) in the expansion of strong coupling constant $\alpha_s$. The result tells that the NLO corrections substantially enhance the color-singlet yield and alter the predicted polarization patterns, and hence lead to a better agreement with experimental data across the $p_T$ spectrum. In contrast to charm-quark fragmentation, the $J/\psi$ and $c\bar{c}$ associated production channel displays a markedly different kinematic behavior, underscoring its distinct role in the production mechanism. While the inclusion of this process greatly reduces the discrepancy between color-singlet theoretical prediction and experimental measurement, a residual gap of approximately one order of magnitude still remains.