Three-body decay $\varphi \to {\pi }^{+}{\pi }^{-}{\pi }^0$ with Omnès-type final-state interactions

Abstract

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 ${\rho }^0$-$\omega$ mixing, and the leading elastic $I=1$, $P$-wave $\pi \pi$ final-state interaction is incorporated through the complex, $s$-dependent Omnès function ${\Omega }_1(s+i0)$. The Omnès factor is applied channel by channel to the three two-pion subsystems, while the direct amplitude is dressed by the averaged factor ${\Omega }_{\mathrm{dir}}=({\Omega }_0+{\Omega }_{+}+{\Omega }_{-})/3$. To avoid double-counting the elastic rescattering already contained in the physical $\rho \to \pi \pi$ coupling, we introduce $g_{\rho \pi \pi }^{\mathrm{bare}}=g_{\rho \pi \pi }^{\mathrm{phys}}/|{\Omega }_1(m_{\rho }^2)|$ in the Omnès-dressed $\rho$-exchange amplitude. With the final parameter set used in this work, we obtain ${\Gamma }_{\mathrm{th}}=0.6915\mathrm{MeV}$, slightly above the empirical estimate ${\Gamma }_{\exp }\approx 0.660\pm 0.020\mathrm{MeV}$, while the direct integrated weight is reproduced at the KLOE 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 ${\rho }^0$-$\omega$ 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.

Comments: 13 pages, 4 figures, accepted for publication in PRD