Extrapolation to Cuts and the Scattering of Electrons and Positrons

Abstract

The low momentum transfer region of strong-interaction processes may be discussed in terms of one-pion-exchange graphs. This success of "extrapolation to poles" suggests, as an extension, "extrapolation to cuts" corresponding to low-mass intermediate states containing two or more particles. We find that this extrapolation may be performed in the case of electromagnetic interactions, owing to the vanishing photon mass. For small-angle scattering of relativistic electrons in a pure Coulomb field, ${sin}^4\left(\frac{\theta }{2}\right)\left(\frac{d\sigma }{d\Omega }\right)=a_0+a_{1}sin\left(\frac{\theta }{2}\right)+\cdots$, where $a_0$ is given precisely by the one-photon pole and $a_1$ (obtained exactly to all orders in $Z\alpha$) comes from the many-photon terms. For scattering by finite nuclei we further find that, for fixed momentum transfer, the ratio of second to first Born approximation decreases as ${\mathrm{}\left(\mathrm{energy}\right)\mathrm{}}^{-1\mathrm{}}$ at high energies. This leads to a simple approximate formula for the ratio $R\equiv \frac{({\sigma }_{-}-{\sigma }_{+})}{({\sigma }_{-}+{\sigma }_{+})}$ of electron and positron scattering, and to a simple method of determining the charge form factors of intermediate-$Z$ nuclei.