Coupled-channel Omn\`es matrix for the $D$-wave isoscalar $\pi\pi/K\bar K$ system and its application to $J/\psi\to\pi^{0}\pi^{0}\gamma,\,K_{S}K_{S}\gamma$
In this work, we construct the $D$-wave isoscalar $\pi\pi/K\bar K$ coupled-channel Omn\`es 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 $\pi\pi\to\pi\pi$ and $\pi\pi\to K\bar K$ data and PDG information, serves as input to the homogeneous two-channel Muskhelishvili-Omn\`es equation. We compare our Omn\`es matrix with previous constructions based on $\pi\pi\to K\bar K$ phases extracted from sums of Breit-Wigner amplitudes. The Omn\`es 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 $J/\psi\to\pi^{0}\pi^{0}\gamma$ and $J/\psi\to K_{S}K_{S}\gamma$ spectra in the $J=2$ electric-dipole (E1) partial wave.