Proof of Dispersion Relations for the Production of Pions by Real and Virtual Photons and for Related Processes

Abstract

It is shown that the amplitudes for the production of pions by photons and electrons (virtual photons), as well as for elastic photon-proton and photon-deuteron scattering, have certain analytic properties as functions of energy and momentum transfer. These properties are proven on the basis of the axioms of field theory, especially local commutativity and the spectral conditions. They guarantee the validity of the usual dispersion relations for restricted values of the invariant momentum transfer. In the construction of these dispersion formulas the electromagnetic interaction is treated in lowest order. The residua of the poles arising from the single-particle intermediate states are related to the corresponding vertex functions. For fixed values of the total energy the absorptive parts of the amplitudes are analytic functions of momentum transfer; they are regular inside certain ellipses. These properties make it possible to continue the absorptive parts into the "unphysical region" appearing in the nonforward dispersion relations by means of partial-wave expansions.