Three-body decay $\phi\to\pi^+\pi^-\pi^0$ with Omn\`es-type final-state interactions

We investigate the decay $\phi\to\pi^+\pi^-\pi^0$ in an effective-Lagrangian framework that keeps the dominant $\rho\pi$ mechanism and the direct three-pion term explicitly separated at the amplitude level. The resonant contribution is described with the Gounaris-Sakurai propagator, the neutral channel includes $\rho^0$-$\omega$ mixing, and the leading elastic $P$-wave $\pi\pi$ final-state interaction is incorporated through a constant on-shell Omn\`es factor $C_\Omega\equiv|\Omega_1(m_\rho^2)|$. The purpose of this approximation is not to provide a full dispersive reconstruction of $\phi\to3\pi$, but to isolate the leading rescattering effect in a transparent phenomenological setting. With this setup, we obtain $\Gamma_{\rm th}=0.6950$ MeV, about $5\%$ above the empirical estimate $\Gamma_{\rm exp}\approx0.660\pm0.020$ MeV, while the direct integrated weight is reproduced at a realistic level, $I_{\mathrm{dir}}=8.457\times10^{-3}$. The computed on-shell Omn\`es factor, $C_\Omega=4.794$, is quantitatively sizable, indicating that $\pi\pi$ rescattering provides a nontrivial enhancement in the $\rho$-dominated channel. At the same time, the $x$ and especially the $y$ Dalitz projections still exhibit visible discrepancies near the phase-space boundary, showing that the present treatment should be viewed as an intermediate phenomenological step rather than a precision amplitude analysis. These residual tensions motivate the next stage: a fully $s$-dependent Omn\`es implementation and a direct fit to the efficiency-corrected KLOE Dalitz-bin data.