Normal Threshold Sheet Structure of Two-Particle Scattering Amplitudes

From the requirements of analyticity and unitarity, we prove that certain normal threshold branch points are absent on unphysical sheets of certain other ones. Also the existence of complex thresholds corresponding to unstable particles is proved, and they are shown to have the discontinuities that one would expect on the basis of "nuclear democracy." To obtain these results it was necessary to derive certain elementary discontinuity formulas, to explicitly construct the inverse of an elastic $n$-particle-to-$n$-particle $S$ matrix, and to prove the factorization theorem for the residues of poles in multiparticle amplitudes.