Sifting out the neutral and charged components of $X(3872)$ from a spin observable

In this work, we propose a new method to detect the composition of the $X(3872)$ state. Based on a widely accepted interpretation that $X(3872)$ is a weakly bound $S$-wave molecule of the $D^{*0} \bar{D}^{0}$ (neutral) and $D^{*+} D^{-}$ (charged) configurations, the process $X(3872) \to \bar{D}^{*0} D^0$ is described by one-loop triangle diagrams with the corresponding components as intermediate particles, and with the parameters constrained by the experimental branching ratio. Unlike the mild behavior of the neutral component, the charged component manifests itself as a distinct structure near the $D^{*+} D^{-}$ threshold at $3.880$ GeV, in the $M_{\bar{D}^{*0} D^0}$ invariant mass distribution of the spin density matrix element $\rho^{D^*}_{00}$. Hence the line shape of $\rho^{D^*}_{00}$ near this structure depends sensitively on the proportions of the two configurations. We propose high-precision measurements of this matrix element by future BESIII and Belle experiments to elucidate how a molecular $X(3872)$ state is composed.