We investigate the decay $\varphi \to {\pi }^{+}{\pi }^{-}{\pi }^0$ in an effective-Lagrangian framework that keeps the dominant $\rho \pi$ mechanism, the direct three-pion term, and a real constant background contribution explicitly separated at the amplitude level. The resonant contribution is described with the Gounaris-Sakurai propagator, the neutral channel includes ${\rho }^0\text{−}\omega$ mixing, and the leading elastic $I=1$, $P$-wave $\pi \pi$ final-state interaction is incorporated through the complex, $s$-dependent Omnès function ${\mathrm{\Omega }}_1(s+i0)$. The Omnès factor is applied channel by channel to the three two-pion subsystems, while the direct amplitude is dressed by the averaged factor ${\mathrm{\Omega }}_{\mathrm{dir}}=({\mathrm{\Omega }}_0+{\mathrm{\Omega }}_{+}+{\mathrm{\Omega }}_{-})/3$. To avoid double counting the elastic rescattering already contained in the physical $\rho \to \pi \pi$ coupling, we introduce $g_{\rho \pi \pi }^{\mathrm{bare}}=g_{\rho \pi \pi }^{\mathrm{phys}}/|{\mathrm{\Omega }}_1(m_{\rho }^2)|$ in the Omnès-dressed $\rho$-exchange amplitude. With the final parameter set used in this work, we obtain ${\mathrm{\Gamma }}_{\mathrm{th}}=0.6915\mathrm{MeV}$, slightly above the empirical estimate ${\mathrm{\Gamma }}_{\exp }\approx 0.660\pm 0.020\mathrm{MeV}$, while the direct integrated weight is reproduced at the KLOE (K LOng Experiment) scale, $I_{\mathrm{dir}}=8.393\times {10}^{-3}$. The corresponding resonant weight is $I_{\rho \pi }=0.9619$. The resulting Dalitz distribution retains the expected $\rho \pi$-dominated three-band structure and shows the localized neutral-channel deformation induced by ${\rho }^0\text{−}\omega$ mixing. Remaining discrepancies in the $x$ and especially the $y$ projections indicate the limitations of the present minimal single-channel Omnès implementation and motivate a next-stage analysis based on a full dispersive crossed-channel treatment and a direct fit to the efficiency-corrected KLOE Dalitz-bin data.

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• Particle decays
• Strong interaction