Unified coupled-channel description for the five near-threshold structures $ψ(3770)$, $G(3900)$, $R(3760)$, $R(3780)$ and $R(3810)$ from $e^+e^-$ annihilation

TL;DR

The paper tackles the puzzle of multiple near-threshold structures in $e^+e^-$ annihilation by developing a unified coupled-channel framework that includes a charmonium core and both open- and nonopen-charm hadron channels. It treats the $s$-channel coupling between bare $c\bar{c}$ states and hadronic final states via a separable potential, solved exactly with a Lippmann-Schwinger approach, and incorporates a QPC-based transition mechanism for $c\bar{c}$ to $D^{(*)}\bar{D}^{(*)}$ channels along with a rescattering pathway to nonopen-charm hadrons and an $h_c\pi\pi$ channel. The model reproduces the asymmetric $\psi(3770)$ line shape, identifies $R(3780)$ with $\psi(3770)$, explains $R(3760)$ as $D\bar{D}\to\mathrm{nOCH}$ rescattering, and attributes $R(3810)$ to coupling between the $\psi(1D)$ and the $h_c\pi\pi$ channel, while naturalizing the large non-$D\bar{D}$ decay width of $\psi(3770)$ through near-threshold dynamics. This unquenched framework provides a coherent interpretation of near-threshold phenomena and offers a predictive path to bottomonium and future Belle II data analyses.

Abstract

The recent observation of multiple near-threshold structures in $e^+e^-$ annihilation-including $ψ(3770)$, $G(3900)$, $R(3760)$, $R(3780)$, and $R(3810)$-reveals limitations in existing models of charmonium and exotic hadrons. In this paper, we propose a unified coupled-channel description that simultaneously incorporates all five near-threshold structures using parameters constrained by hadron spectroscopy. Our model accurately reproduces the line shapes in both $D\bar{D}$ and nonopen-charm hadron (nOCH) channels, resolves the asymmetric profile of $ψ(3770)$, and produces the $G(3900)$ bump. We identify the $R(3780)$ as the dominant manifestation of $ψ(3770)$, attribute $R(3760)$ to $D\bar{D} \to \text{nOCH}$ rescattering and suggest that $R(3810)$ arises from coupling between the $ψ(1D)$ state and the $h_cππ$ channel. The significant non-$D\bar{D}$ decay of $ψ(3770)$ is explained by its near-threshold position. This analysis provides a coherent, unquenched framework centered on a charmonium $ψ(1D)$ core for near-threshold phenomena and offers a predictive approach extendable to bottomonium systems and future data from Belle II.

Figures (5)

• Figure 1: (a) Coupled-channel description of the $e^+e^-\to D\bar{D}$ cross section. The red solid line represents the full model including $\psi(2S)$, $\psi(1D)$, and $\psi(3S)$. Experimental data are from Belle Belle:2007qxm and BESIII Husken:2024hmiBESIII:2024ths. (b) Cross section when either $\psi(2S)$ or $\psi(3S)$ is removed.
• Figure 2: Coupled-channel description of (a) the $e^+e^-\to$ nonopen-charm hadron and (b) the inclusive hadronic cross sections. Vertical dashed lines indicate the $D^0\bar{D}^0$ and $D^+\bar{D}^-$ thresholds. nonopen-charm hadron data are from Ref. BESIII:2023bed, inclusive hadronic data are from Ref. BESIII:2023oql.
• Figure S1: The calculated QPC partial wave amplitudes (orange circle points) for $\gamma = 1$ and fits to them in terms of analytical function (blue lines).
• Figure S2: (a) The non-open-charm cross section for different $\Lambda_{nOCH}$, where $\Lambda_{h_c} = 40$ MeV. (b) The non-open-charm cross section for different $\Lambda_{h_c}$, where $\Lambda_{nOCH}=90$ MeV.
• Figure S3: (a) The non-open-charm cross section calculated via LS equations when including the $V_{nOCH,D\bar{D}}$ potential. (b) The $D\bar{D}$ cross section calculated via the LS equation with/without $V_{nOCH,D\bar{D}}$ potential.