On the unphysical region in dispersion relations

Abstract

Conditions on the masses of four particles are written down such that present methods can prove inelastic dispersion relations for processes of the type A+B → C+D. One necessary condition is that both elastic relations A+B → A′+B′ and C+D → C′+D′ can be proved for forward scattering at least. The masses of p n π− π0 satisfy the conditions, and also photomeson production, and the model K-meson systemK=3Π/2, Λ ≏N (where these are the masses of the particles). Using dispersion relations and the unitarity condition for the unphysical region of this model system, integral equations for the unphysical region in KN → Aπ relations are obtained; these equations have solutions in terms of physicalS matrix elements. It is also possible to set up a model KN → KN system such that the usual dispersion relations can be proved by present methods, and also all the inelastic relations which the method of this paper needs in order to interpret the unphysical region. However, a model can be constructed such that dispersion relations can be proved, but such that their unphysical region cannot be interpreted by present methods.