Relation between Scattering and Production Amplitudes: Concerning Particle in System

In a series of previous papers the present authors and collaborators have shown strong evidence for existence of the long-sought sigma-particle by analyzing the pi pi-scattering and pi pi-production processes, applying, respectively, the IA method and the VMW method. In this paper we examine the relation between the scattering amplitude T and the production amplitude F from the viewpoint of the unitarity and the final state interaction (FSI) theorem by using a simple field theoretical model. As a result it is shown that the amplitudes in the physical state representation are directly represented through the Breit-Wigner amplitudes of the relevant resonances, and the respective forms of T and F coincide with those in the IA and VMW methods, justifying our methods of analyses