Constrain the $χ_{cJ}\to D^{(*)}\bar{D}^{(*)}$ effective couplings via the $X(3872)\to π^0χ_{cJ}$ decays

TL;DR

The paper addresses how to constrain the effective couplings $g_1$ for $\chi_{cJ}\to D^{(*)}\bar{D}^{(*)}$ in the hadronic-molecular picture of XYZ states by exploiting the isospin-breaking decay $X(3872)\to \pi^0\chi_{cJ}$. It combines an effective Lagrangian/HG$\chi$PT framework with intermediate meson loop dynamics and uses the precise $X(3872)$ pole position, compositeness, and the isospin-breaking amplitude ratio $R_X$ to relate the neutral/charged content to the mixing angle $\theta$, thereby deriving upper bounds on $g_1^2$. The key results show: (i) $\theta=0.16^{+0.07}_{-0.09}$ for $BE_n$ in $[2,180]$ keV, indicating a dominant $D^0\bar{D}^{*0}$ component; (ii) $g_1^2\lesssim 0.28^{+1.36}_{-0.14}\rm{\,GeV^{-1}}$, which is about an order of magnitude smaller than the VMD estimate $g_1^2(\mathrm{VMD})\approx 3.13^{+7.54}_{-1.25}\rm{\,GeV^{-1}}$; and (iii) predicted upper bounds $\mathrm{Br}(X(3872)\to \pi^0\chi_{c1})\lesssim 6\%$ and $\mathrm{Br}(X(3872)\to \pi^0\chi_{c2})\lesssim 5\%$, consistent with data. These bounds provide a quantitatively improved understanding of hidden-charm transitions and can be tested in other processes like $X(3872)\to \pi\pi\chi_{cJ}$ and $Y(4260)\to \pi h_c$.

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 $χ_{cJ}\to D^{(\ast)}\bar{D}^{(\ast)}$ are building blocks of the amplitudes for the pionic and radiative transitions of the charmonium-like states to the $χ_{cJ}$ and $h_c$ states, e.g., $X(3872)\to π^0χ_{cJ},\,ππχ_{cJ},\,γχ_{cJ}$ and $Y(4260)\to π^0 h_c,\,ηh_c$. These couplings can not be extracted from the partial decay widths of the $χ_{cJ}$ 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 $χ_{cJ}\to D^{(\ast)}\bar{D}^{(\ast)}$ couplings. Our results show that the VMD model may over estimate the $χ_{cJ}\to D^{(\ast)}\bar{D}^{(\ast)}$ couplings considering the $X(3872)$ as a $D\bar{D}^{*}$ 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 $χ_{cJ}$ and $h_c$.

Figures (3)

• Figure 1: Feynman diagrams for $X(3872) \to \pi^0\chi_{cJ}$. The double lines represent vector mesons ($D^*, \bar{D}^*$), the single solid lines represent scalar mesons ($D, \bar{D}$), and the dashed lines represent the scalar meson $\pi^0$.
• Figure 2: The mixing angle $\theta$ describing the proportion of neutral and charged constituents in $X(3872)$ (a), and the upper limits of the coupling constant square $g_1^2$ of $\chi_{cJ}$ to charmed mesons (b) as a function of the binding energy $BE_n$. The red and blue bands show the uncertainties of $\theta$ and $g_1^2$ determined from the $X(3872)$ properties propagated from the uncertainty of $R_X$, and the orange band gives the uncertainty of $g_1^2$ from the VMD model propagated from the uncertainty of the $\chi_{c0}$ decay constant $f_{\chi_{c0}}$.
• Figure 3: Branching ratios of $X(3872) \to \pi^0\chi_{c1}$ (a) and $X(3872) \to \pi^0\chi_{c2}$ (b) as functions of the binding energy $BE_n$ calculated by utilizing the $g_1^2$ determined from the $X(3872) \to \pi^0\chi_{c0}$ decay. The uncertainties are propagated from that of $R_X$.