Comprehending Isospin breaking effects of $X(3872)$ in a Friedrichs-model-like scheme

Recently, we have shown that the $X(3872)$ state can be naturally generated as a bound state by incorporating the hadron interactions into the Godfrey-Isgur quark model using the Friedrichs-like model combined with the QPC model, in which the wave function for the $X(3872)$ as a combination of the bare $c\bar c$ state and the continuum states can also be obtained. Under this scheme, we now investigate the isospin breaking effect of $X(3872)$ in its decays to $J/\psi\pi^+\pi^-$ and $J/\psi\pi^+\pi^-\pi^0$. By Considering its dominant continuum parts coupling to $J/\psi\rho$ and $J/\psi\omega$ through the quark rearrangement process, one could obtain the reasonable ratio of $\mathcal{B}(X(3872)\rightarrow J/\psi\pi^+\pi^-\pi^0)/\mathcal{B}(X(3872)\rightarrow J/\psi\pi^+\pi^-)\simeq (0.58\sim 0.92)$. It is also shown that the $\bar D D^*$ invariant mass distributions in the $B\rightarrow \bar D D^* K$ decays could be understood qualitatively at the same time. This scheme may provide more insight to understand the enigmatic nature of the $X(3872)$ state.