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 χ${}_{c1}$(2P) and other states and that its prompt hadroproduction predominately proceeds via its χ${}_{c1}$(2P) component, we calculate the prompt-X(3872) polarization at the CERN LHC through next-to-leading order in α${}_s$ within the factorization formalism of nonrelativistic QCD, including both the color-singlet ${}^3$P${}_1^{\left[1\right]}$ and color-octet ${}^3$S${}_1^{\left[8\right]}$ c$\overline{c}$ Fock states. We also consider the polarization of the J/ψ 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/ψ 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.