Correspondence between Unstable Particles and Poles inS-Matrix Theory

The correspondence (unstable particle) → (pole of the $S$ matrix) is discussed. In particular, it is explicitly shown that $S$ matrices, satisfying all requirements of analyticity, unitarity, and threshold and asymptotic behavior in energy, can be constructed in such a way that a sharp isolated resonance is produced without an accompanying pole in the unphysical sheet. As a matter of fact, in two of the examples which we consider, no singularities, except kinematical and dynamical branch lines, are found in any sheet of the energy Riemann surface.