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Using the semi-classical method of Nikishov-Ritus (NR), the derivation of thetransition rate of the beamstrahlung process is reviewed. This method uses theBound Interaction Picture and the exact solutions of the Dirac equation in theexternal field potential. For future linear colliders, the nominal machineparameters are such that this external field can be considered to be a constantcrossed electromagnetic field. The Dirac equation solutions can be Fouriertransformed such that they are functions of Dirac gamma matrices, Airyfunctions and the usual non-external field solution. The resultant analyticform for the transition rate is the same as that obtained by the QuasiclassicalOperator (QO) method of Baier-Katkov which sets a limit of ultra-relativisticelectron and vanishingly small radiation angle. The NR calculation however alsoexhibits a pole in the radiation angle for back-radiated photons. The removalof this pole requires a further study of IR divergences within the BoundInteraction Picture.

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