Impact of ${\eta }_c$ hadroproduction data on charmonium production and polarization within NRQCD framework

With the recent LHCb data on ${\eta }_c$ production and based on heavy quark spin symmetry, we obtain the long-distance matrix elements for both ${\eta }_c$ and $J\mathrm{/}\psi$ productions, among which, the color-singlet one for ${\eta }_c$ is obtained directly by the fit of experiment for the first time. Using our long-distance matrix elements, we can provide good description of the ${\eta }_c$ and $J\mathrm{/}\psi$ hadroproduction measurements. Our predictions on $J\mathrm{/}\psi$ polarization are in good agreement with the LHCb data and pass through the two sets of CDF measurements in medium $p_t$ region. Considering all the possible uncertainties carefully, we obtained quite narrow bands of the $J\mathrm{/}\psi$ polarization curves.