Constrain the ${\chi }_{cJ}\to D^{(\ast )}{\bar{D}}^{(\ast )}$ effective couplings via the $X(3872)\to {\pi }^0{\chi }_{cJ}$ decays

Abstract

The hidden-charm decays serve as irreplaceable platforms for probing the structures of charmonium-like states, such as $X(3872)$, $Y(4260)$, $Z_c(3900)$, and their heavy-quark-symmetry partners. In the hadronic molecular scenario, these hidden-charm decays are denominated by intermediate meson loops (IMLs), and the couplings of ${\chi }_{cJ}\to D^{(\ast )}{\bar{D}}^{(\ast )}$ are building blocks of the amplitudes for the pionic and radiative transitions of the charmonium-like states to the ${\chi }_{cJ}$ and $h_c$ states, e.g., $X(3872)\to {\pi }^0{\chi }_{cJ},\pi \pi {\chi }_{cJ},\gamma {\chi }_{cJ}$ and $Y(4260)\to {\pi }^0{h}_c,\eta h_c$. These couplings can not be extracted from the partial decay widths of the ${\chi }_{cJ}$ directly and only have estimated values from the vector meson dominance (VMD) model. Utilizing the recent precise determination of the pole position and the isospin breaking properties of the $X(3872)$, we give an estimation on the upper bounds of the absolute values of the ${\chi }_{cJ}\to D^{(\ast )}{\bar{D}}^{(\ast )}$ couplings. Our results show that the VMD model may over estimate the ${\chi }_{cJ}\to D^{(\ast )}{\bar{D}}^{(\ast )}$ couplings considering the $X(3872)$ as a $D{\bar{D}}^{\ast }$ hadronic molecule with a binding energy about tens of keV. These upper limits can be used and tested in other hidden-charm transitions of the charmonium-like states to the ${\chi }_{cJ}$ and $h_c$.

Comments: 12 pages, 3 figures. Comments are welcome