Coupled-channel Omnès matrix for the $D$-wave isoscalar $\pi \pi /K\bar{K}$ system and its application to $J/\psi \to {\pi }^0{\pi }^0\gamma ,K_S{K}_S\gamma$

Abstract

In this work, we construct the $D$-wave isoscalar $\pi \pi /K\bar{K}$ 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 $f_2^{\prime }(1525)$ 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-Omnès equation. We compare our Omnès matrix with previous constructions based on $\pi \pi \to K\bar{K}$ 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 $J/\psi \to {\pi }^0{\pi }^0\gamma$ and $J/\psi \to K_S{K}_S\gamma$ spectra in the $J=2$ electric-dipole (E1) partial wave.

Comments: 9 pages, 3 figures, journal version + a few typos corrected