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 $\mathrm{prompt}\text{−}X(3872)$ 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 ${}^3{S}_1^{[8]}$ $c\overline{c}$ 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 nonzero spin.

Figure 1

The $m_{\pi \pi }$ distributions of $\mathrm{\Gamma }(X(3872)\to J/\psi {\pi }^{+}{\pi }^{-})$ measured by CMS [17] (left panel) and ATLAS [18] (right panel), normalized to unity across the experimental $m_{\pi \pi }$ ranges, are compared with our theoretical predictions based on $A_{{\mu }_1}^1$ (dashed lines), $A_{{\mu }_1}^2$ (dotted lines), and their linear combination in Eq. (4) with fitted value of $g$ (solid lines).

Figure 2

The differential cross sections of prompt $X(3872)$ production as measured by CMS [17] (left panel) and ATLAS [18] (right panel) are compared with our NLO NRQCD results based on the new fit (solid lines). The yellow bands indicate the theoretical uncertainties.

Figure 3

$p_T^X$ dependence of ${\lambda }_{\theta }^X$ (upper row) and $p_T^{\psi }$ dependence of ${\lambda }_{\theta }^{\psi }$ (lower row) in the HX frame for prompt $X(3872)$’s decaying to $J/\psi {\pi }^{+}{\pi }^{-}$ for the LHCb (left), CMS (middle), and ATLAS (right) setups. The shaded or pink bands indicate the allowed corridor for ${\lambda }_{\theta }^X$.