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

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.