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 $Z_{c\bar{c}}:X_{{\bar{D}}^0{D}^{0\ast }}:X_{D^{+}{D}^{-\ast }}:X_{{\bar{D}}^{\ast }{D}^{\ast }}$ $= 1:(2.67\sim 8.85):(0.45\sim 0.46):0.04$\mathrm{.}Furthermore,itsdecaysto$\pi^0$ and a $P$-wave charmonium ${\chi }_{cJ}$ state with $J=0,1$, or $2$, $J\mathrm{/}\psi {\pi }^{+}{\pi }^{-}$, and $J\mathrm{/}\psi {\pi }^{+}{\pi }^{-}{\pi }^0$ 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 $J\mathrm{/}\psi {\pi }^{+}{\pi }^{-}$.