Simultaneous Dalitz-plot decomposition of the $e^+ e^- \to J/ψ\, π\, π\, (K \bar{K})$ processes in the 4.13-4.36 GeV region using dispersive final-state interactions

Abstract: We present a joint analysis of the processes $e^{+e^-} \to J/ψπ<sup>+π<sup>-$ and $e^{+e^-} \to J/ψK<sup>+K<sup>-$ at center-of-mass energies from 4.13 to 4.36 GeV. The amplitudes are constructed using the Dalitz-plot decomposition formalism, with the $e<sup>+e<sup>-$ energy dependence encoded through the $Y(4220)$ and $Y(4320)$ resonant structures together with a non-resonant production mechanism. The scalar $ππ/K\bar K$ final-state interaction is treated dispersively using a coupled-channel Omnès representation. This allows us to describe the measured total cross sections and one-dimensional invariant-mass distributions with a single set of energy-independent parameters. We find that a purely resonant description of the BESIII data is insufficient, requiring a non-resonant term at the amplitude level which undergoes $ππ/K\bar{K}$ rescattering. Within the present isobar model, we extract Breit-Wigner parameters for the $Z_c(3900)$, $Y(4220)$, and $Y(4320)$ states, and determine the corresponding subprocess cross sections.

• The paper presents a joint Dalitz-plot amplitude analysis that disentangles resonant and non-resonant contributions in e⁺e⁻ → J/ψ ππ(KK).
• It implements a dispersive coupled-channel Omnès approach to accurately model scalar ππ/KK final-state interactions.
• Global fits yield precise resonance parameters for Z_c(3900), Y(4220), and Y(4320), highlighting the necessity of including non-resonant backgrounds.

Simultaneous Dalitz-Plot Analysis of $e^+e^- \to J/\psi\,\pi\pi\,(K\bar{K})$ with Dispersive Final-State Interactions

Introduction

The structure of the spectrum in hadron spectroscopy above open-charm threshold has manifested complexities that are not accommodated by the conventional quark model. The $e^+e^- \to J/\psi\,\pi^+\pi^-$ and $e^+e^- \to J/\psi\,K^+K^-$ reactions in the 4.13-4.36 GeV region showcase the interplay of exotic vector states ($Y$), charged intermediates ($Z_c$), and strong final-state interactions (FSI). This work (2606.07225) presents a joint Dalitz-plot based amplitude analysis of these processes, treating the scalar $\pi\pi/K\bar{K}$ FSI via a coupled-channel dispersive (Omnès) approach, with $e^+e^-$ energy dependence encoded by resonant ($Y(4220)$, $Y(4320)$) and non-resonant production mechanisms. The comprehensive analysis evaluates differential cross-section data and parameterizes the resonance content, offering quantitative insights into the production and decay dynamics of these multi-body systems.

Theoretical Framework

The amplitude construction follows the Dalitz-plot decomposition (DPD) formalism, as developed in prior works, which allows explicit factorization of the angular and dynamical structure in three-body decays. Each amplitude is represented in terms of partial waves for isobar subchannels in a manner that exactly maps onto field-theoretic (Lagrangian) models. Resonant and non-resonant contributions are separated at the amplitude level, enabling global fits to cross-section data with a restricted set of energy-independent parameters.

Figure 1: Diagrams illustrating the considered contributions to the $e^+ e^- \to J/\psi \, \pi \, \pi \, (K \bar{K})$ processes.

The virtual-photon induced three-body system receives contributions from:

• Intermediate vector resonances ($e^+e^- \to J/\psi\,\pi^+\pi^-$0, $e^+e^- \to J/\psi\,\pi^+\pi^-$1) coupling to the $e^+e^- \to J/\psi\,\pi^+\pi^-$2 final states,
• Scalar and tensor resonances in the $e^+e^- \to J/\psi\,\pi^+\pi^-$3 and $e^+e^- \to J/\psi\,\pi^+\pi^-$4 subchannels ($e^+e^- \to J/\psi\,\pi^+\pi^-$5, $e^+e^- \to J/\psi\,\pi^+\pi^-$6, $e^+e^- \to J/\psi\,\pi^+\pi^-$7),
• The $e^+e^- \to J/\psi\,\pi^+\pi^-$8 in $e^+e^- \to J/\psi\,\pi^+\pi^-$9 channels,
• Non-resonant (background) amplitudes subject to dispersive FSI.

The energy dependence is implemented by extracting the $e^+e^- \to J/\psi\,K^+K^-$0 structure at the production vertex, assigning Breit-Wigner propagators to $e^+e^- \to J/\psi\,K^+K^-$1 states, and the $e^+e^- \to J/\psi\,K^+K^-$2 dependence for subchannel resonances. The non-resonant components are coupled to FSI via a data-driven coupled-channel Omnès matrix, which incorporates $e^+e^- \to J/\psi\,K^+K^-$3 S-wave amplitudes constrained by Roy and Roy-Steiner analyses.

Fit Strategy and Data Reproduction

The data set comprises BESIII measurements of total cross sections and one-dimensional invariant-mass distributions for both $e^+e^- \to J/\psi\,K^+K^-$4 and $e^+e^- \to J/\psi\,K^+K^-$5 final states. The fits proceed in two stages:

Minimal Fit

Restricting to $e^+e^- \to J/\psi\,K^+K^-$6 GeV, only the $e^+e^- \to J/\psi\,K^+K^-$7, $e^+e^- \to J/\psi\,K^+K^-$8, and the scalar channel are considered, omitting $e^+e^- \to J/\psi\,K^+K^-$9 and tensor contributions. The model describes the total and differential cross sections up to moderate energies with $Y$0, but deviations become significant at higher $Y$1. Crucially, a purely resonant scenario fails; non-resonant background amplitudes, iterated through $Y$2 FSI, are necessary for data consistency.

Figure 2: Total cross section of the process $Y$3 in the minimal fit, showing the necessity of background terms.

Full/Energy-Global Fit

Extending to $Y$4 GeV, the $Y$5 and $Y$6 are added. Here, the best-fit parameters provide a $Y$7 and yield masses and widths for $Y$8, $Y$9, and $Z_c$0, consistent with previous PWA determinations. The non-resonant background is reduced in the full fit but remains essential.

Figure 3: Total cross section illustrating distinct contributions ($Z_c$1, $Z_c$2, background) in the energy-global fit.

Figure 4: Subprocess cross sections for $Z_c$3, quantifying the $Z_c$4 and scalar contributions' strength and interference.

Analysis of Invariant Mass Distributions

The detailed comparison with one-dimensional invariant-mass distributions at various center-of-mass energies reveals that:

• The $Z_c$5 is required to account for the $Z_c$6 peaks,
• The coupled-channel FSI dynamically generates the $Z_c$7 shape and induces observable $Z_c$8 threshold effects,
• Pronounced interference between background and resonant amplitudes is evident, especially in the scalar channel, making the extraction of resonance parameters highly correlated with the modeling of the non-resonant piece.

Figure 5: Total fit to the invariant mass distributions at low $Z_c$9 in $\pi\pi/K\bar{K}$0. Data and the different theoretical contributions are shown.

Figure 6: Representative high-energy invariant mass distributions, showing improved fit quality with the inclusion of additional resonances at higher $\pi\pi/K\bar{K}$1.

Resonance Parameters and Model Implications

The extracted resonance parameters in the isobar model are as follows:

• $\pi\pi/K\bar{K}$2: $\pi\pi/K\bar{K}$3 MeV, $\pi\pi/K\bar{K}$4 MeV,
• $\pi\pi/K\bar{K}$5: $\pi\pi/K\bar{K}$6 MeV, $\pi\pi/K\bar{K}$7 MeV,
• $\pi\pi/K\bar{K}$8: $\pi\pi/K\bar{K}$9 MeV, $e^+e^-$0 MeV.

These values are compatible with, and in some cases more precise than, previous BESIII PWA extractions for the same processes. The higher-mass vector structure is phenomenological and reflects channel-dependent line shapes rather than a universal PDG mass assignment.

A significant statement from the fit is that a purely resonant treatment of BESIII data fails. The inclusion of non-resonant background, coupled into strong FSI, is mandatory to simultaneously reproduce total and differential cross sections across the full energy regime.

Practical and Theoretical Implications

The analysis underscores the necessity of dispersive coupled-channel techniques when interpreting amplitude structures in three-body production above open charm threshold. Model-independent parameterizations such as the DPD, combined with rigorous FSI treatment, guarantee that resonance parameters extracted are minimally sensitive to ad hoc background assumptions. This framework can be systematically extended to higher energies and other final states as more precise data become available.

The non-resonant component's persistence indicates that the underlying physics includes contributions from non-$e^+e^-$1 or non-exotic production mechanisms—possibly, multi-channel meson loops, triangle singularities, or other molecular effects. Future refinements should incorporate explicit open-charm thresholds and explore the analytic properties of background terms in greater detail.

Conclusion

This study provides a simultaneous Dalitz-plot-based amplitude analysis of $e^+e^-$2 in the 4.13–4.36 GeV range, with rigorous inclusion of FSI and explicit resonance and background parameterization (2606.07225). The fits yield precise resonance parameters and demonstrate that non-resonant scalar amplitudes are essential for a consistent data description. These results set a benchmark for future multi-body amplitude analyses in hadron spectroscopy and illustrate the utility of dispersive approaches in disentangling exotic and non-resonant contributions in $e^+e^-$3 annihilation channels.