How crossing affects the analytic relationship between low energy ππ amplitudes and their asymptotic behaviour

Abstract

Different physical assumptions about the asymptotic behaviour of ππ amplitudes are realised in the different number of substractions involved in fixed t dispersion relations for the various amplitudes and their inverses. The fact that each new dispersion relation must be consistent with s - t crossing leads to a number of conditions relating low energy ππ amplitudes to their high energy behaviour. These are discussed in detail. Such relationships supplement finite energy sum rules with which they are compared. The dispersive sum rules, crossing conditions, and finite energy sum rules we discuss are applied to recent phenomenological solutions to Roy's equations and shown not to narrow the presently accepted range of threshold parameters. These results are in marked contrast to the conclusions of other recent studies. To complete the study of finite energy sum rules we consider the behaviour of the isospin zero t-channel amplitude and estimate the asymptotic ππ total cross-section. We present evidence to suggest that the pomeron is late-developing in meson-meson scattering.