Production Amplitudes. II. Partial-Wave Decomposition and Analytic Continuation in Total Angular Momentum

The problem of analytic continuation in total angular momentum $j$ of a production amplitude is investigated on the basis of analyticity in one of the momentum-transfer variables $z$ for definite combinations of helicity amplitudes which are free of certain kinematical singularities. On this basis also an upper bound for production amplitudes is obtained which has the same value for all $z$ as the bound of a forward-scattering elastic-scattering amplitude. Furthermore, the consistency of the analyticity in the whole cut $z$ plane is investigated. For this case the singularities in the $j$ plane extend indefinitely to the right even if only $N$ subtractions are made. These singularities are removed only if the spectral functions satisfy certain integral relations to the effect that they oscillate more rapidly for higher spin. When these conditions are satisfied the essential singularities of the Gribov-Pomeranchuk type need not be introduced. The analysis further shows the dynamical role played by the spin variable in an analytic $S$-matrix theory.