Threshold behaviour in quantum field theory

Abstract

The elements of the S matrix are functions of the energies and momenta of a set of incident particles. For sufficiently high relative energies of the incident particles new particles of non-zero rest mass can be created. At the thresholds for such creation processes the S matrix will have a complicated behaviour. This behaviour is investigated when the S matrix is calculated by means of renormalized quantum field theory. For a typical matrix element there are thresholds of two main types. The first is a creation threshold below which the element is zero on account of energy-momentum conservation; mathematically this is due to a Dirac S function factor. The second is an interference threshold above which a competing process has non-zero probability. Interference thresholds are closely connected with the appearance of displaced poles in the integration. It is shown that a matrix element will always contain a term having a branch point at an interference threshold; the path of analytic continuation round these branch points is obtained from the physical assumption that particles interact through their retarded fields. Between the threshold values it is shown that the S matrix elements are analytic functions of the energies and momenta of the incident particles.