# On the equivalence of semi-classical methods for QED in intense external fields

## TL;DR

This paper reviews semi-classical methods for calculating QED transition rates in intense external fields, demonstrating the equivalence of the Nikishov-Ritus and Baier-Katkov approaches, and discusses IR divergence issues.

## Contribution

It shows the equivalence of the NR and QO methods for QED in intense fields and highlights the need to address IR divergences in the NR approach.

## Key findings

- The analytic transition rate matches between NR and QO methods.
- NR method exhibits a pole in back-radiated photon angles.
- Further IR divergence analysis is necessary for the NR method.

## Abstract

Using the semi-classical method of Nikishov-Ritus (NR), the derivation of the transition rate of the beamstrahlung process is reviewed. This method uses the Bound Interaction Picture and the exact solutions of the Dirac equation in the external field potential. For future linear colliders, the nominal machine parameters are such that this external field can be considered to be a constant crossed electromagnetic field. The Dirac equation solutions can be Fourier transformed such that they are functions of Dirac gamma matrices, Airy functions and the usual non-external field solution. The resultant analytic form for the transition rate is the same as that obtained by the Quasiclassical Operator (QO) method of Baier-Katkov which sets a limit of ultra-relativistic electron and vanishingly small radiation angle. The NR calculation however also exhibits a pole in the radiation angle for back-radiated photons. The removal of this pole requires a further study of IR divergences within the Bound Interaction Picture.