Superconvergence Relations, FESR and Duality

Abstract

This chapter is on the very important concept of Duality. It starts with superconvergence relations for amplitudes that fall off sufficiently fast at high energies. These relate scattering amplitudes at high and low energies, with many consequences. Igi showed that by splitting amplitudes into two parts one of which had the right high energy behaviour, the applicability could be extended. The next important development was the work by Dolen, Horn and Schmid(DHS), who derived the famous Finite Energy Sum Rules(FESR) . The DHS construction is explained in minute detail. This laid a quantitative foundation for the deep concept of Duality. Some important applications of FESR for the pion-nucleon charge exchange reaction are discussed. Then, some serious alarm bells raised by Mandula, Slansky and Fujisaki regarding FESR are presented. This was that no finite number of Regge trajectories could ever satisfy FESR as there is one such for each value of the Mandelstam invariant t. These points are critically examined. The chapter ends with some very recent applications of a phenomenological nature.