Unitarity sum rules and soft-pion amplitudes

We examine unitarity bounds on scattering amplitudes by using « unitarity sum rules » at high energy. Threshold conditions emerge when certain mass parameters approach zero. For ππ scattering a bound similar to (but somewhat weaker than) the Adler condition is shown to be needed in order to avoid a dangerous mass (infra-red) singularity. A different argument, which uses crucially the pseudoscalar nature of the pion and a few very plausible dynamical assumptions, yields a stronger bound as well as the possibility of extension to general pion amplitudes. We finally discuss our results as well as the possibility of understanding spontaneously broken symmetries from general S-matrix principles.