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

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 the nonrelativistic effective field theory power-counting rule, we demonstrate that box diagrams dominate these dipion transition processes. Branching ratios are calculated as functions of the mixing angle $θ$, which parametrizes 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 the orders of $10^{-4}$, $10^{-3}$, 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 future experimental measurements.

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