Coupled-channel Omnès matrix for the $D$-wave isoscalar $ππ/K\bar K$ system and its application to $J/ψ\toπ^{0}π^{0}γ,\,K_{S}K_{S}γ$

TL;DR

This work develops a unitary two-channel $D$-wave Omnès matrix for the isoscalar $\pi\pi/K\bar K$ system by implementing a minimal two-pole $K$-matrix model for the $f_2(1270)$ and $f'_2(1525)$ resonances. The resulting MO solution enforces unitarity, analyticity, and correct high-energy behavior, and is shown to differ significantly from prior Breit–Wigner-based inputs, addressing inconsistencies in the $K\bar K$ phase and eigen-phase distributions. The MO matrix is then employed in a dispersive representation of the $J=2$ electric-dipole amplitudes for the radiative decays $J/\psi\to\pi^{0}\pi^{0}\gamma$ and $J/\psi\to K_S K_S\gamma$, achieving a simultaneous and accurate description of BESIII spectra with a shared left-hand-cut structure. The results provide a robust dispersive input for tensor meson studies and can be applied to other processes sensitive to the $f_2(1270)/f'_2(1525)$ system, improving predictive power and consistency in the tensor sector.

Abstract

In this work, we construct the $D$-wave isoscalar $ππ/K\bar K$ coupled-channel Omnès matrix, formulated to satisfy unitarity, analyticity, and the appropriate asymptotic behavior. We employ a two-channel $K$-matrix model containing poles associated with the $f_{2}(1270)$ and $f_{2}'(1525)$ resonances. The resulting unitary scattering matrix, which reproduces the experimental $ππ\toππ$ and $ππ\to K\bar K$ data and PDG information, serves as input to the homogeneous two-channel Muskhelishvili-Omnès equation. We compare our Omnès matrix with previous constructions based on $ππ\to K\bar K$ phases extracted from sums of Breit-Wigner amplitudes. The Omnès matrix developed here provides a reliable dispersive input for form-factor calculations and resonance studies in the tensor-meson sector. As an application, we show that it enables a simultaneous and accurate description of the BESIII $J/ψ\toπ^{0}π^{0}γ$ and $J/ψ\to K_{S}K_{S}γ$ spectra in the $J=2$ electric-dipole (E1) partial wave.

Figures (3)

• Figure 1: Comparison of the minimal two-pole $K$-matrix fit (solid curves) with the input used in Cao et al.Cao:2025dkv (dashed curves) for the quantities entering the MO analysis: $|t_{12}|$, $\delta_1$, and $\delta_2$. The analyses of Ref. TarrusCastella:2021pld employ a very similar input as in Cao:2025dkv and are therefore not shown separately. Brookhaven II data Longacre:1986fh are shown for the quantity $20\,(p_1\,p_2)^{5}\,|t_{12}|^2/s$. The experimental data for $\delta_1$ are taken from Hyams:1973zfProtopopescu:1973sh.
• Figure 2: Real and imaginary parts of the D-wave isoscalar $\pi\pi/K\bar{K}$ Omnès matrix elements $\Omega_{ab}(s)$. Solid curves show the result obtained from the two-pole $K$-matrix model used in this work, while dashed curves correspond to the Omnès matrix of Cao et al. Cao:2025dkv
• Figure 3: Combined fit to the normalized $J=2$, E1 partial-wave spectra in the decays $J/\psi \to \pi^0\pi^0\gamma$ and $J/\psi \to K_S K_S\gamma$. The data points with error bars show the BESIII mass-independent analysis BESIII:2015rugBESIII:2018ubj, while the solid curves represent the coupled-channel Muskhelishvili-Omnès fit based on Eq.(\ref{['eq:Jpsi-MO']}). For comparison, we also show the results using Omnès matrices of Tarrús Castellà et al.TarrusCastella:2021pld (dashed) and Cao et al.Cao:2025dkv (dotdashed).