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

By describing the $X(3872)$ using the extended Friedrichs scheme, in which $D{\overline{D}}^{*}$ is considered as 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 the rate of its decay to $J/\psi {\pi }^{+}{\pi }^{-}$ with the help of the Barnes-Swanson model, where ${\pi }^{+}{\pi }^{-}$ are assumed to be produced via an intermediate $\rho$ state. This calculation shows that the decay rate of $X(3872)$ to ${\chi }_{c1}{\pi }^0$ is 1 order of magnitude smaller than its decay rate to $J/\psi {\pi }^{+}{\pi }^{-}$ and the decay widths of $X(3872)\to {\chi }_{cJ}{\pi }^0$ for $J=0$, 1, 2 are of the same order.

Figure 1

$S$-wave term (solid) and $D$-wave one (dashed) of coupling form factors for $D^0{\overline{D}}^{0*}$ components.

Figure 2

The scattering amplitudes without the phase space factors of $D^0{\overline{D}}^{0*}\to {\chi }_{c0}{\pi }^0$, ${\chi }_{c1}{\pi }^0$, ${\chi }_{c2}{\pi }^0$. The right one shows the prior (solid) and the post (dashed) contributions to the amplitudes. The left one shows the averaged amplitudes.

Figure 3

Comparison of the integrands $\frac{f_{ls}{\mathcal{M}}_{l^{\prime }{s}^{\prime },ls}}{(z_X-E)}$ for $D^0{\overline{D}}^{0*}\to {\chi }_{c1}{\pi }^0$ (solid) and $D^{+}{\overline{D}}^{-*}\to {\chi }_{c1}{\pi }^0$ (dashed), when $z_{X(3872)}$ is chosen at 3.8716 GeV as an example.