From prompt to direct J/$\psi$ production: new insights on the $\chi_{c1}$ and $\chi_{c2}$ polarizations and feed-down contributions from a global-fit analysis of mid-rapidity LHC data

While the prompt $\mathrm{J}/\psi $ cross section and polarization have been measured with good precision as a function of transverse momentum, $p_\mathrm{T}$, those of the directly produced $\mathrm{J}/\psi $ are practically unknown, given that the cross sections and polarizations of the $\chi _{{c}1} $ and $\chi _{{c}2} $ mesons, large indirect contributors to $\mathrm{J}/\psi $ production, are only known with rather poor accuracy. The lack of precise measurements of the $\chi _{{c}J} $ polarizations induces large uncertainties in the level of their feed-down contributions to the prompt $\mathrm{J}/\psi $ yield, because of the polarization-dependent acceptance corrections. The experimental panorama of charmonium production can be significantly improved through a consistent and model-independent global analysis of existing measurements of $\mathrm{J}/\psi $, $\psi \mathrm{(2S)}$ and $\chi _{c}$ cross sections and polarizations, faithfully respecting all the correlations and uncertainties. In particular, it is seen that the $\chi _{{c}J} $ polarizations and feed-down fractions to $\mathrm{J}/\psi $ production have a negligible dependence on the $\mathrm{J}/\psi p_\mathrm{T}$, with average values $\lambda _\vartheta ^{\chi _{{c}1}} = 0.55 \pm 0.23$, $\lambda _\vartheta ^{\chi _{{c}2}} = -0.39 \pm 0.22$, $R^{\chi _{{c}1}} = (18.8 \pm 1.4)\%$ and $R^{\chi _{{c}2}} = (6.5 \pm 0.5)\%$. The analysis also shows that $(67.2 \pm 1.9)$% of the prompt $\mathrm{J}/\psi $ yield is due to directly-produced mesons, of polarization constrained to remarkably small values, $\lambda _\vartheta ^{\mathrm{J}/\psi } = 0.04 \pm 0.06$.