Understanding $X(3872)$ and its decays in the extended Friedrichs scheme

We present that the $X(3872)$ could be represented as a dynamically generated state in the extended Friedrichs scheme, in which the ratio of "elementariness" and "compositeness" of the different components in the $X(3872)$ is about $Z_{c\bar c}:X_{\bar D^0 D^{0*}}: X_{ D^+ D^{-*}}: X_{\bar D^* D^*}$ $= 1:(2.67\sim 8.85):(0.45\sim 0.46):0.04$. Furthermore, its decays to $\pi^0$ and a $P$-wave charmonium $\chi_{cJ}$ state with $J=0,1$, or $2$, $J/\psi\pi^+\pi^-$, and $J/\psi\pi^+\pi^-\pi^0$ could be calculated out with the help of Barnes-Swanson model. The isospin breaking effects is easily understood in this scheme. This calculation also shows that the decay rate of $X(3872)$ to $\chi_{c1}\pi^0$ is much smaller than its decay rate to $J/\psi\pi^+\pi^-$.