Decays of $X(3872)$ to $χ_{cJ}π^0$ and $J/ψπ^+π^-$

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

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