Higher-order poles and mass-shell singularities in electron-hydrogen scattering

Abstract

The forward scattering amplitude for electron-hydrogen scattering is expected to be an analytic function of the energy. However, when calculated within the framework of nonrelativistic quantum mechanics and the Born approximation, the exchange part has double and triple left-hand poles. From the viewpoint of field theory, such higher-order singularities are unexpected: One normally encounters only simple poles and branch points. A simple nonrelativistic model, which interpolates between binding by short-range and long-range forces, is used to show that these singularities are a direct consequence of the fact that the target system is bound by Coulomb forces: If the binding is by short-range forces there is only a simple pole. The double and triple poles emerge from a coalescence of simple poles and logarithmic branch points, in the limit of long-range binding. In the same framework, the direct amplitude has both simple and double poles, regarded as a function of the squared momentum transfer. Now even the simple pole is unexpected because the H atom is neutral. Again, the existence of these singularities is shown to be a consequence of Coulomb binding. The singularities are then studied within the framework of field theory. I show that if the H atom is treated as if it were an elementary particle, described by a field ${\ensuremath{\varphi}}_{\mathrm{H}}$ coupled to electron and proton fields ${\ensuremath{\varphi}}_{e}$ and ${\ensuremath{\varphi}}_{p}$ by a Lagrangian density ${L}_{I}=\ensuremath{-}{G}_{0}({\ensuremath{\varphi}}_{e}{\ensuremath{\varphi}}_{p}{)}^{\ifmmode\dagger\else\textdagger\fi{}}{\ensuremath{\varphi}}_{\mathrm{H}}+\mathrm{H}.\mathrm{c}.,$ the location of the singularity of the exchange amplitude is immediately obtained from a tree diagram, without the need to carry out any integrations. However, the nature of the singularity found does not agree with the nonrelativistic theory: There is no free lunch. Further analysis indicates that in full-fledged quantum electrodynamics, the vertex function \ensuremath{\Gamma} which describes the virtual decomposition of the H atom into its constituents must itself be singular when all particles are on the mass shell.