Decays of $X(3872)$ to ${\chi }_{cJ}{\pi }^0$ and $J/\psi {\pi }^{+}{\pi }^{-}$

Abstract

By describing the $X(3872)$ using the extended Friedrichs scheme, in which $D{\bar{D}}^{\ast }$ is the dominant component, we calculate the decay rates of the $X(3872)$ to ${\pi }^0$ and a $P$-wave charmonium ${\chi }_{cJ}$ state with $J=0,1$, or $2$, and its decays to $J/\psi {\pi }^{+}{\pi }^{-}$ where ${\pi }^{+}{\pi }^{-}$ are assumed to be produced via an intermediate $\rho$ state. The decay widths of $X(3872)\to {\chi }_{cJ}{\pi }^0$ for $J=0,1,2$ are of the same order. However, this model calculation exhibits that the decay rate of $X(3872)$ to ${\chi }_{c1}{\pi }^0$ is one order of magnitude smaller than its decay rate to $J/\psi {\pi }^{+}{\pi }^{-}$.

7 pages, 3 figures; v2, some corrections and minor modification of the discussion. Final published version