The remaining parts for the long-standing J/psi polarization puzzle

Based on the non-relativistic quantum chromodynamics factorization formalism, the polarization parameters ${\lambda }_{\theta \varphi }$ and ${\lambda }_{\varphi }$ of $J\mathrm{/}\psi$ hadroproduction are analyzed in helicity frame and calculated at QCD next-to-leading order for the first time. For prompt $J\mathrm{/}\psi$ production, we take into account the feeddown contributions from ${\chi }_{cJ}$ and $\psi (2S)$ decays. The theoretical predictions for the polarization parameters ${\lambda }_{\theta \varphi }$ and ${\lambda }_{\varphi }$ of $J\mathrm{/}\psi$ are presented. With the theoretical results we have done the fit to the experimental measurements on yield and polarization for $J\mathrm{/}\psi$ hadroproduction simultaneously, and found that the results are coincide with the experimental measurements at the LHC quite well.