Self-Consistent Dispersion-Theoretic Electron-Positron Scattering Amplitude

Abstract

A method is outlined for the construction of dispersion-theoretic electromagnetic scattering amplitudes for processes in which bound states may appear; in particular, the electron-positron system is examined. A Lorentz-covariant spinor amplitude is exhibited which is analytic, cutoff independent, and crossing symmetric, which has the correct double-spectral functions through second order in the fine-structure constant $\alpha$, and which reduces to the usual Born approximation in lowest order. Moreover, the amplitude displays Regge asymptotic behavior and the positronium Regge poles. Self-consistency requires that the positronium poles appear at the correct position and with the proper residue, and that the amplitude possesses a well-defined Jacob-Wick expansion. It is found that this second-order amplitude provides a suitable basis for iterative calculation of the higher-order terms using the same procedure.