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

TL;DR

This paper explains the anomalous suppression of the $ ext{D}ar{ ext{D}}$ decay mode of $ ext{psi}(4040)$ using a relativistic Bethe--Salpeter and $^3P_0$ model, attributing it to node-induced cancellations in decay amplitudes.

Contribution

It provides a dynamical explanation for the suppressed $ ext{D}ar{ ext{D}}$ decay of $ ext{psi}(4040)$ within a conventional charmonium framework, highlighting the role of node effects.

Findings

The suppressed $ ext{D}ar{ ext{D}}$ decay is due to cancellations in the decay amplitude.

The $ ext{D}ar{ ext{D}}$ width is highly sensitive to initial mass and $D$-meson mass splitting.

The model reproduces the open-charm hierarchy observed in $ ext{psi}(4040)$ decays.

Abstract

The strong decay $\psi(4040)\to D\bar D$ is anomalously suppressed despite ample phase space, whereas the $D\bar D^*$ 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 $^3P_0$ 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 $D\bar D^*$ 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 $3\,{}^3S_1$ charmonium picture, while the precise value of the near-vanishing $D\bar D$ width remains model dependent.