Reduction of order, resummation, and radiation reaction

The Landau-Lifshitz equation is the first in an infinite series of approximations to the Lorentz-Abraham-Dirac equation obtained from “reduction of order.” We show that this series is divergent, predicting wildly different dynamics at successive perturbative orders. Iterating reduction of order ad infinitum in a constant crossed field, we obtain an equation of motion which is free of the erratic behavior of perturbation theory. We show that Borel-Padé resummation of the divergent series accurately reproduces the dynamics of this equation, using as little as two perturbative coefficients. Comparing with the Lorentz-Abraham-Dirac equation, our results show that for large times the optimal order of truncation typically amounts to using the Landau-Lifshitz equation, but that this fails to capture the resummed dynamics over short times.