Rescattering effects and the σ pole in hadronic decays

The σ resonance was observed as a conspicuous π+π− peak in hadronic decays like J/ψ→π+π−ω or D+→π+π−π+. The phase of the σ→π+π− amplitude, extracted from production data within the conventional isobar model, is assumed to coincide with that in ππ elastic scattering. We check the validity of this assumption by using Lehmann-Symanzik-Zimmermann (LSZ) reduction and unitarity. The rescattering effects in the final three-particle states are shown to generate a correction to the phase given by a naive application of Watson theorem. We briefly discuss the implications of this result for the pole determination from production data.