Constrain the χ₂₉ D^ (*) D^ (*) effective couplings via the X (3872) π⁰χ₂₉ decays

The hidden-charm decays serve as irreplaceable platforms for probing the structures of charmonium-like states, such as X (3872), Y (4260), Zc (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 χ₂₉ D^ () D^ () are building blocks of the amplitudes for the pionic and radiative transitions of the charmonium-like states to the χ₂₉ and hc states, e. g. , X (3872) π⁰χ₂₉, \, ππχ₂₉, \, γχ₂₉ and Y (4260) π⁰ hc, \, ηhc. These couplings can not be extracted from the partial decay widths of the χ₂₉ 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 χ₂₉ D^ () D^ () couplings. Our results show that the VMD model may over estimate the χ₂₉ D^ () D^ () couplings considering the X (3872) as a DD^* 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 χ₂₉ and hc.