Deciphering the $X(3872)$ via its polarization in prompt production at the CERN LHC

Based on the hypothesis that the $X(3872)$ exotic hadron is a mixture of $\chi_{c1}(2P)$ and other states and that its prompt hadroproduction predominately proceeds via its $\chi_{c1}(2P)$ component, we calculate the prompt-$X(3872)$ polarization at the CERN LHC through next-to-leading order in $\alpha_s$ within the factorization formalism of nonrelativistic QCD, including both the color-singlet $^3\!P_1^{[1]}$ and color-octet $^3\!S_1^{[8]}$ $c\bar c$ Fock states. We also consider the polarization of the $J/\psi$ produced by the subsequent $X(3872)$ decay. We predict that, under ATLAS, CMS, and LHCb experimental conditions, the $X(3872)$ is largely longitudinally polarized, while the $J/\psi$ is largely transversely polarized. We propose that the LHC experiments perform such polarization measurements to pin down the nature of the $X(3872)$ and other $X$, $Y$, $Z$ exotic states with non-zero spin.