Chiral Symmetry Breaking,  and  History: Gell-Mann and Levy, Nambu, Adler and Weinberg laid the foundations in the 1960s. Many theorists realised the  amplitude contains a pole but its mass was poorly known until the 1990s. Then Pelaez, Oller and Oset developed ideas from Chiral Perturbation Theory. Crucially, the  is NOT a Breit-Wigner of constant width; its phase does NOT go through 90 at 500 MeV. BES 2 data

hep-ph/ s A = 0.41 m  2

The formulae may need updating in two respects: (i) I got my parameters for  by fitting Crystal Barrel data on pp -> 3  0 (600K events in both S- and P-state annihilation). Oller and collaborators now have predictions for the  contributions from  a 2  and a 1  These need to be compared with what I got. (ii) Work with Rupp and van Beveren using their model showed that it is very likely that  and f 0 (980) mix with a mixing angle ~25 0. This is likely since they both go through 90 0 very close to the KK threshold. Formulae for the mixing are available.

Why is the mass of the  pole much lower than the mass where the phase goes through 90 0 ? Amplitudes for complex functions of a complex variable obey the Cauchy-Riemann relations (analyticity): d (Re f)/d (Re s) = d (Im f)/d(Im s) d (Im f)/ d (Re s) = - d(Re f)/d(Im s). Im f varies nearly linearly along the real s axis. Re f has the reverse variation as one moves off the real s-axis to complex s For elastic  scattering, unitarity requires zero phase at threshold. BUT this constraint disappears as one moves into the complex plane. So the entire real part moves bodily to the left for negative Im s. For the  the pole is quite close to the  threshold; the phase of the  reaches only ~55 0 at ~1.5 GeV, but there is still a pole of large width ~650 MeV in close agreement with Moussallam’s prediction from ChPT.