Dispersive analysis of the $\boldsymbol{\phi \to \gamma \pi^0 \pi^0}$ process

We present an analysis of the radiative decay $\phi \to \gamma \pi^0 \pi^0$ in a dispersive framework, where the two-pion subsystem undergoes strong final-state interactions that cover the $f_0(500)$ and $f_0(980)$ regions. We employ a coupled-channel Muskhelishvili-Omn\`es framework that allows for a consistent treatment of two scalar resonances and crossed-channel singularities induced by the Born and vector-meson exchanges. We explicitly verify the equivalence between the modified and standard Muskhelishvili-Omn\`es representations for vector-meson pole contributions when the isoscalar Omn\`es matrix is chosen asymptotically bounded, and we adopt the standard representation in decay kinematics. This yields, for the first time, a parameter-free dispersive prediction for the kaon Born rescattering, which provides a dominant contribution. To obtain a good fit to the KLOE and SND data, we employ a once-subtracted coupled-channel dispersion relation with heavier left-hand cut contributions and two unknown subtraction constants. The results demonstrate the consistency among the data for $\pi\pi$ scattering, $\gamma\gamma$ fusion, and $\phi$ radiative decay, thereby validating the underlying dispersive formalism and the input used for the hadronic Omn\`es matrix and left-hand cuts.