Extension of the Bloch-Nordsieck model

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Abstract
An analysis of infrared radiation in the potential scattering of a Dirac electron was presented by Bloch and Nordsieck [Phys. Rev. 52, 54 (1937)], in their now classic work, to illustrate how divergence difficulties are removed by nonperturbative treatment of the radiation of soft photons. It is shown here how an improved low-frequency approximation can be obtained which provides a small correction to the Bloch-Nordsieck sum rule for the scattering cross section, while still requiring as input only physical (on-shell) values of the amplitude for scattering in the absence of the radiation field. The calculation is based on a variational determination of the radiative amplitude with trial functions chosen in accordance with the Bloch-Nordsieck approximation. An extension of the model scattering problem is introduced which allows for potentials with a Coulomb tail. The applicability of the approximation procedure to the analogous problem of relativistic scattering in a low-frequency external field is pointed out.