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We calculate to next-to-leading order accuracy the high-energy elasticscattering cross section for an electron off of a classical point source. Weuse the $\overline{\mathrm{MS}}$ renormalization scheme to tame the ultravioletdivergences while the infrared singularities are dealt with using the wellknown Kinoshita-Lee-Nauenberg theorem. We show for the first time how tocorrectly apply the Kinoshita-Lee-Nauenberg theorem diagrammatically in anext-to-leading order scattering process. We improve on previous works byincluding all initial and final state soft radiative processes, includingabsorption and an infinite sum of partially disconnected amplitudes. Crucially,we exploit the Monotone Convergence Theorem to prove that our delicaterearrangement of this formally divergent series is uniquely correct. Thisrearrangement yields a factorization of the infinite contribution from theinitial state soft photons that then cancels in the physically observable crosssection. Since we use the $\overline{\mathrm{MS}}$ renormalization scheme, ourresult is valid up to arbitrarily large momentum transfers between the sourceand the scattered electron as long as $\alpha \log(1/\delta)\ll 1$ and $\alpha\log(1/\delta)\log(\Delta/E)\ll 1$, where $\Delta$ and $\delta$ are theexperimental energy and angular resolutions, respectively, and $E$ is theenergy of the scattered electron. Our work aims at computing the NLOcorrections to the energy loss of a high energetic parton propagating in aquark-gluon plasma.

NLO Rutherford Scattering and the Kinoshita-Lee-Nauenberg Theorem