Unstable particles in s-matrix theory

The consequences of the existence of complex poles in scattering amplitudes, corresponding to resonances or unstable particles, are investigated. Specific examples show that corresponding to such poles are normal threshold cuts lying near the physical region and which cause "wooly cusps" in scattering cross sections. The discontinuity across such a cut is expressed by a unitarity-like relation in terms of unphysical amplitudes with unstable external particles defined by the residues of the complex poles. More generally, it is shown that the Landau equations for the singularities must be extended to include all unstable as well as stable particles. The Cutkosky formulas specify the corresponding discontinuities in terms of the physical and unphysical amplitudes.