Radiative decays of the isoscalar $S$-wave $D\bar{D}$ molecule

Abstract

In the hadronic molecular picture, an isoscalar $S$-wave $D\bar{D}$ molecule is naturally expected as the spin-0 partner of $X(3872)$. Assuming this state, denoted by $X_0$, to be a pure $D\bar{D}$ molecule with quantum numbers $J^{PC}=0^{++}$, we investigate its radiative decays $X_0\to \gamma V(V={\rho }^0,\omega )$ within an effective Lagrangian framework. The decay amplitudes are generated by intermediate charmed-meson loops, with electromagnetic gauge invariance consistently maintained throughout the calculation. We evaluate the partial decay widths and examine their dependence on the $X_0$ mass and the cutoff. Although the individual decay widths exhibit a sizable dependence on the cutoff, their ratio is remarkably stable. In particular, we obtain that the ratio for the decays $X_0\to \gamma {\rho }^0$ and $X_0\to \gamma \omega$ is approximately 2.26, which is insensitive to the variation of the cutoff. This robust ratio provides a useful model-insensitive signature of the molecular structure of $X_0$.

Comments: 9 pages, 4 figures