Dipion transitions from $X(3872)$ to $χ_{cJ}\ (J=0,1,2)$

TL;DR

This paper analyzes dipion transitions from X(3872) to $\chi_{cJ}$ within heavy hadron chiral perturbation theory, modeling X(3872) as a $D\bar{D}^*$-molecule with a neutral/charged mixture controlled by mixing angle $\theta$. A nonrelativistic power-counting analysis shows box diagrams dominate over triangle diagrams, enabling quantitative predictions of decay rates as functions of $\theta$ and revealing significant isospin violation. The predicted branching fractions are of order $10^{-4}$ (J=0), $10^{-3}$ (J=1), and $10^{-5}$ (J=2), with ratios comparing charged to neutral channels and among $\chi_{cJ}$ showing distinctive isospin-violating patterns; invariant-mass distributions exhibit double-bump structures in certain channels, offering clear experimental signatures. These results provide concrete tests of the molecular picture of X(3872) and motivate precision measurements at BESIII and Belle II to constrain the neutral/charged content and the degree of isospin breaking.

Abstract

In this work, we investigate the dipion transition processes $X(3872)\to ππχ_{cJ} (J=0,1,2)$ within the framework of heavy hadron chiral perturbation theory, treating $X(3872)$ as a molecular state composed of $D\bar{D}^*+h.c.$ components. By analyzing the box and triangle loop diagrams with nonrelativistic effective field theory power counting rule, we demonstrate that box diagrams dominate these dipion transitions processes. Branching ratios are calculated as functions of the mixing angle $θ$, which parameterizes the neutral and charged meson compositions of the $X(3872)$. Our results indicate that the branching fractions for $X(3872)\toππχ_{c0}$, $X(3872)\to ππχ_{c1}$, and $X(3872)\to ππχ_{c2}$ are of orders $10^{-4}$, $10^{-3}$, and $10^{-5}$, respectively. We also predict the ratios ${\mathcal{B}[X(3872)\rightarrow ππχ_{c0/2}]}/{\mathcal{B}[X(3872)\rightarrow ππχ_{c1}]}$ and ${\mathcal{B}[X(3872)\rightarrow π^+π^-χ_{cJ}]}/{\mathcal{B}[X(3872)\rightarrow π^0π^0χ_{cJ}]}$. The latter deviates from isospin-symmetry expectations, revealing various degrees of isospin violation. By studying the $π^+π^-$ and $π^+χ_{cJ}$ invariant mass spectra, we find a double-bump structure in the $π^ + π^-$ invariant mass distributions of the process $X(3872)\to π^+π^-χ_{c1}$ and $π^+χ_{c0}$ invariant mass distribution of the process $X(3872)\to π^+π^-χ_{c0}$, which could be tested by the future experimental measurements.

Figures (4)

• Figure 1: Box [(a) and (b)] and triangle [(c) and (d)] diagrams contributing to the dipionic transitions from $X(3872)$ to $\chi_{cJ}$.
• Figure 2: Invariant mass distributions of $\pi^+\pi^-$ and $\pi^+\chi_{cJ}$ for the decays of $X(3872) \to \pi^+\pi^-\chi_{cJ},\ (J=0,1,2)$. The black, blue, red, and green lines correspond to $\theta=0$, $\pi/12$, $\pi/4$, and $\pi/2$, respectively.
• Figure 3: Kinematics used for calculating 4-point integrals.
• Figure 4: The cutoff dependence of the branching ratios of $X(3872) \to \pi^+ \pi^- \chi_{cJ} \ (J=0,1,2)$ when we take $\theta=\pi/12$.