N ov 2 00 4 The phase of the σ → ππ amplitude in J / Ψ → ωπ + π −

The phase variation of the σ → ππ amplitude is accurately determined as a function of mass from BES II data for J/Ψ → ωπ + π −. The determination arises from interference with the strong b 1 (1235)π amplitude. The observed phase variation agrees within errors with that in ππ elastic scattering. The σ pole appears as a conspicuous π + π − peak in BES II data for J/Ψ → ωπ + π − [1]. This peak is absent from data on ππ S-wave elastic scattering. The connection between these two processes is a question which is explored here. For both processes, the partial wave amplitude f (s) may be written f (s) = N(s)/D(s), (1) where N(s) has only left-hand cuts and D(s) has only right-hand cuts. The N function can be different for the two processes. We pursue the hypothesis that N(s) for ππ elastic scattering contains an Adler zero, which is absent from the production process. The phase variation above the ππ threshold arises from the right-hand cut. The D function should be the same for all processes if only a single resonance contributes. The question is whether BES data and ππ elastic scattering data are consistent with this hypothesis. The Dalitz plot for J/Ψ → ωπ + π − is shown in Fig. 1. The σ pole appears as a diagonal band at the upper right-hand edge of this plot. There are also strong vertical and horizontal bands due to b ± 1 (1235) → ωπ ±. These two bands account for 41% of the data; the σ pole accounts for 19% and f 2 (1270) for most of the remaining intensity. There is strong interference between the b 1 (1235) bands and the σ amplitude; this interference provides an accurate determination of the phase δ σ of the σ as a function of ππ mass.