Generalized Optical Theorems and Steinmann Relations

Abstract

Generalized optical theorems and generalized Steinmann relations for 3 \ensuremath{\rightarrow} 3 processes are described. These results follow from the field-theoretic formalism of Bros, Epstein, and Glaser. The theorems express in terms of physical-region scattering functions the discontinuity of any 3 \ensuremath{\rightarrow} 3 scattering function across any basic cut, which is a channel-energy cut that runs from the lowest normal threshold to plus infinity. Previously these discontinuities had been expressed, in general, only in terms of the analytic continuations of scattering functions. A remarkable property is found: The discontinuities have the algebraic structure they would have if the basic cuts were cuts in independent variables. Moreover, a generalized version of the Steinmann relations holds: The double discontinuity across any pair of independently treated crossed basic cuts vanishes. The original Steinmann relations have a much narrower scope. The generalized Steinmann relations are used to express compactly in terms of physical-region scattering functions any single, double, or higher-order multiple discontinuity of any 3 \ensuremath{\rightarrow} 3 scattering function across any combination of basic cuts, evaluated on any specified side of each of the remaining basic cuts. All these basic discontinuities are expected to enjoy Regge asymptotic behavior. Although only basic cuts are mentioned, there is no neglect of the effects of other cuts.