Radiative Corrections for Electron Scattering in an External Field—A New Method of Calculation

The lowest-order radiative corrections for electron scattering in an external electromagnetic field are considered, using Schwinger's source theory. Spectral forms are employed, and the range of soft photons is invariantly specified by the final-state mass resolution ($\delta M$; $\delta M\ll m=\mathrm{electron}\mathrm{mass}$). The differential cross section is dealt with directly, and elastic and inelastic contributions are not separated, thereby avoiding some of the usual infrared divergences. The resulting differential cross section appears as a factor ($1-\delta$) multiplying the lowest-order (Born) differential cross section plus a contribution referring to the magnetic form factor; $\delta$ is given by $\delta =\frac{\alpha }{2\pi }\frac{q^2}{m^2}\int 0^1\mathrm{dv}\frac{1}{1+\left(\frac{q^2}{4m^2}\right)(1-v^2)}\left[(1+v^2)\ln \left(\frac{4m}{\delta M}\frac{v^2}{{(1-v^2)}^{\frac{3}{2}}}\right)-1-\frac{5}{2}{v}^2+\frac{1}{3}{v}^4\right],$ where $q$ is the momentum transfer. Also presented is a brief review of certain aspects of source theory.