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

In this work, we investigate the dipion transition processes $X(3872)\to \pi \pi \chi_{cJ} (J=0,1,2)$ 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 $\theta$, which parametrizes the neutral and charged meson compositions of the $X(3872)$. Our results indicate that the branching fractions for $X(3872)\to\pi\pi\chi_{c0}$, $X(3872)\to \pi\pi\chi_{c1}$, and $X(3872)\to \pi\pi\chi_{c2}$ are of the orders of $10^{-4}$, $10^{-3}$, and $10^{-5}$, respectively. We also predict the ratios ${\mathcal{B}[X(3872)\rightarrow \pi\pi\chi_{c0/2}]}/{\mathcal{B}[X(3872)\rightarrow \pi\pi\chi_{c1}]}$ and ${\mathcal{B}[X(3872)\rightarrow \pi^+\pi^-\chi_{cJ}]}/{\mathcal{B}[X(3872)\rightarrow \pi^0\pi^0\chi_{cJ}]}$. The latter deviates from isospin-symmetry expectations, revealing various degrees of isospin violation. By studying the $\pi^+\pi^-$ and $\pi^+\chi_{cJ}$ invariant mass spectra, we find a double-bump structure in the $\pi^ + \pi^-$ invariant mass distributions of the process $X(3872)\to \pi^+\pi^-\chi_{c1}$ and $\pi^+\chi_{c0}$ invariant mass distribution of the process $X(3872)\to \pi^+\pi^-\chi_{c0}$, which could be tested by future experimental measurements.