$\phi \to 3\pi$ and $\phi\pi^{0}$ transition form factor from Khuri-Treiman equations

This work studies the $\phi \to 3\pi$ decay and the $\phi \to \pi^0 \gamma^\ast$ transition form factor, utilizing the Khuri-Treiman formalism to account for analyticity, crossing, and unitarity. Using once-subtracted dispersion relations, we perform a simultaneous fit to the $\phi \to 3\pi$ Dalitz plot distribution and the $\phi \to \pi^0 \gamma^\ast$ measurements from the KLOE collaboration, finding good agreement with these experimental data. These results reaffirm the applicability of the Khuri-Treiman approach in the analysis of three-body decays. An interesting result is that the subtraction constant appearing in the equations is similar to a sum rule expectation, in contrast to analogous studies of $\omega \to 3\pi$ decays and $\omega \to \pi^0 \gamma^\ast$, which shows significant deviations. Our results also provide a reasonable description of the trend of the transition form factor data from BaBar in the $e^{+}e^{-}\to\phi\pi^{0}$ scattering region. These intriguing theoretical differences between the decays of $\phi$ and $\omega$ could encourage further experimental measurements to assess the discrepancies and refine the theoretical predictions.