Second-order correction to the Bloch-Nordsieck sum rule

As shown many years ago by Bloch and Nordsieck [Phys. Rev. 52, 54 (1937)] and Nordsieck [Phys. Rev. 52, 59 (1937)], the cross section for scattering with the emission of an arbitrary number of soft photons is finite and can be expressed in terms of the cross section for scattering in the absence of the radiation field. The model used by Bloch and Nordsieck included only soft-photon modes of the field, with maximum frequency ${\mathrm{\omega }}_{\mathit{s}}$. They argued that corrections to their sum rule would be expressed in terms of two small parameters, ${\mathit{r}}_0$${\mathrm{\omega }}_{\mathit{s}}$/c and ħ${\mathrm{\omega }}_{\mathit{s}}$/ν, where ${\mathit{r}}_0$ is the classical electron radius and ν is the scattering energy. Corrections of first order were obtained previously by the author [Phys. Rev. A 21, 1939 (1980)], in the context of a nonrelativistic formulation of the scattering problem. Here all corrections of second order are derived. As in the earlier versions, the corrected cross section for scattering in the radiation field, summed over final states of the field, is determined from a knowledge of the field-free cross section. Use of the unitarity property (the optical theorem) is found to be a simplifying technical device for this calculation, as indicated by the cancellation of divergences in the scattering amplitude in the forward direction. It is shown that in the dipole approximation the term in the electron-field interaction that is quadratic in the vector potential, while contributing to an energy-level shift, has no effect, to second order, on the final form of the sum rule.