Some rigorous analytic properties of transition amplitudes

Abstract

The causality and mass-spectral conditions are used to derive analytic properties of two-particle transition amplitudes as functions of energy and momentum transfer. While the complete double-dispersion representation cannot be proved from these postulates alone, it is shown that, within a certain domain in the space of two complex variables, the only singularities are the expected poles and cuts along the real axis. This domain surrounds the low-energy physical region. The proof is restricted to the scattering of the lightest particles in the theory. As long as no attempt is made to find the largest possible domain, the calculations are not very difficult. The domain of analyticity can then be extended by the unitarity condition. The magnitude of the extension is not very large, but the new domain differs from the old in including part of the region in which the two-dimensional spectral functions are non-zero. We only use the unitarity condition below the threshold for inelastic processes; if such processes could also be treated and the unitarity condition used at higher energies, it seems possible that the domain could be extended arbitrarily far. In the present results, the boundary of the domain of analyticity is sufficiently far from the low energy region to justify the analytic properties assumed in applications of the double-dispersion representation. Other results that can be proved are that partial-wave amplitudes are analytic functions of the energy in a certain region, that the determination of coupling constants by extrapolation in the angle is valid if the energy is not too high, and that ordinary dispersion relations are true over a slightly larger range of momentum transfer than had previously been established.