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

TL;DR

This work proposes a method to determine the neutral and charged components of the $X(3872)$, treated as a weakly bound $S$-wave molecule of $D^{*0}\bar{D}^{0}$ and $D^{*+}D^{-}$, by analyzing the spin density matrix element $\rho^{D^*}_{00}$ in the transition $X(3872) \to \bar{D}^{*0}D^0$. The approach uses one-loop triangle diagrams with neutral and charged intermediate states and constrains model parameters via the measured branching ratio. A distinct near-threshold structure at the $D^{*+}D^{-}$ threshold emerges in $\rho^{D^*}_{00}$ when the charged component is sizable, with $\rho^{D^*}_{00}$ at the threshold ranging from $0.34$ to $0.48$ as the neutral/charged mix varies. This observable, insensitive to the Breit-Wigner peak, offers a robust handle on the composition and motivates high-precision measurements at BESIII and Belle to clarify the molecular content of $X(3872)$.

Abstract

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 $ρ^{D^*}_{00}$. Hence the line shape of $ρ^{D^*}_{00}$ 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.

Figures (5)

• Figure 1: The Feynman diagrams for reaction $X(3872) \to \bar{D}^{*0} D^0$.
• Figure 2: The constraint on the values of $\text{cos}^2\theta$ (neutral portion of $X(3872)$) and $\Lambda$, from the branching ratio of $X\to \bar{D}^{*0} D^0$. Note that $\text{cos}^2\theta$ does NOT physically depend on $\Lambda$.
• Figure 3: The SDME $\rho^{{D^*}}_{00}$ for the final state $\bar{D}^{*0}$ in the reaction $X(3872) \to \bar{D}^{*0} D$ with different $\text{cos}^2\theta$ values. The vertical line denotes the position of the $D^{*+}D^-$ threshold ($3.880$ GeV). The two critical values: $\cos^2\theta=0.50$ corresponds to a pure isospin singlet $X(3872)$, while $\cos^2\theta=0.94$ corresponds to the point where $g_n=g_c$ exactly holds.
• Figure 4: The results of $\rho^{{D^*}}_{00}$ at $\cos^2\theta=0$ with the cutoff from Eq. \ref{['Eq:X3872']} ($1.09$ GeV) versus a smaller value $\Lambda=0.35$ GeV. The constant values $\rho^{{D^*}}_{00}=1/3(1)$ with only ${\cal M}_{b1}({\cal M}_{b2})$ switched on are also shown here as markers.
• Figure 5: The modulus square of the ${\cal M}_b$ amplitude. The contribution from the interference term between the ${\cal M}_{b1}$ and ${\cal M}_{b2}$ amplitudes is also shown with the green dot-dashed line.