Speaker
Description
Vector-meson decays into three pions offer a clean and stringent testing ground for dispersion theory. In this talk I present a unified Khuri-Treiman analysis of $\omega\to3\pi$, $\phi\to3\pi$ and $J/\psi\to3\pi$, together with the associated $\omega\pi^0$, $\phi\pi^0$ and $J/\psi\pi^0$ transition form factors. The approach implements analyticity, unitarity and crossing through once-subtracted dispersion relations, taking the $\pi\pi$ $P$-wave phase shift as the main input. I show that the BESIII ($\omega$, $J/\psi$) and KLOE ($\phi$) data are well reproduced, and that the transition form factors, tied to the $3\pi$ amplitudes at low energies, are simultaneously described where data exist (MAMI, NA60, KLOE) and predicted otherwise. A key role is played by the subtraction constant: comparing it with its sum-rule value reveals a striking pattern, since for $\phi$ and $J/\psi$ it stays close to the sum rule whereas for $\omega$ it deviates markedly, driven by the high-energy form-factor data. I discuss the origin of these intriguing differences and how forthcoming measurements could help to disentangle them.