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 $J\mathrm{/}\psi {\pi }^{+}{\pi }^{-}$ and $J\mathrm{/}\psi {\pi }^{+}{\pi }^{-}{\pi }^0$. By Considering its dominant continuum parts coupling to $J\mathrm{/}\psi \rho$ and $J\mathrm{/}\psi \omega$ 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}$(X(3872)\rightarrow J/\psi\pi^+\pi^-)\simeq (0.58\sim 0.92)$\mathrm{.}Itisalsoshownthatthe$\bar D D^*$ invariant mass distributions in the $B\to \bar{D}{D}^{\ast }K$ 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.