Nodal mechanism for the suppressed $D\bar{D}$ decay of $\psi (4040)$ in the Bethe--Salpeter framework

The strong decay $\psi (4040)\to D\bar{D}$ is anomalously suppressed despite ample phase space, whereas the $D{\bar{D}}^{\ast }$ and $D_s{\bar{D}}_s$ channels remain sizable. In this work, we study this suppression and the associated open-charm hierarchy in the framework of the instantaneous Bethe–Salpeter equation combined with the relativistic ${}^3{P}_0$ model, with the pair-creation strength fixed independently from $\psi (3770)\to D\bar{D}$. Within this framework, we show that the suppressed $D\bar{D}$ mode can be understood as a consequence of node-induced cancellations in the relativistic decay amplitude. The $D\bar{D}$ amplitude is strongly reduced because the corresponding overlap integral receives comparable positive and negative contributions from different momentum regions, whereas the $D{\bar{D}}^{\ast }$ and $D_s{\bar{D}}_s$ channels do not undergo the same strong cancellation. This interpretation is further supported by the pronounced sensitivity of the $D\bar{D}$ width to the initial mass, the charged-neutral $D$-meson mass splitting, and the dip structure in the mass dependence of the partial width. Our results provide a dynamical explanation of the suppressed $D\bar{D}$ mode and the core open-charm hierarchy of $\psi (4040)$ within a conventional $3{}^3{S}_1$ charmonium picture, while the precise value of the near-vanishing $D\bar{D}$ width remains model dependent.