N-photon bremsstrahlung in the soft-photon approximation

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Abstract
An approximation is derived for the amplitude for spontaneous emission of an arbitrary number of low-frequency photons during the collision of a charged particle with a (short-range) center of force. The result correctly reproduces the first four terms in an expansion of the amplitude in powers of the frequency. The evaluation of the first two terms requires a knowledge of the physical (on-shell) amplitude for radiationless scattering. At this level the approximation represents a generalization of Low’s theorem, reducing to his result for the case of single-photon bremsstrahlung. The next two terms involve as additional input the physical one-photon emission amplitude. The calculation is based on the Bloch-Nordsieck method for summing those contributions arising from the emission of soft photons in either the initial or final state of the scattering process. The derivation of the soft-photon approximation is facilitated by the introduction of a transformation to the length gauge. The possibility of extending this approach to allow for a more general class of scattering systems is briefly discussed.