NLO Rutherford Scattering and the Kinoshita-Lee-Nauenberg Theorem

Abstract

We calculate to next-to-leading order accuracy the high-energy elastic scattering cross section for an electron off of a classical point source. We use the $\overline{\mathrm{MS}}$ renormalization scheme to tame the ultraviolet divergences while the infrared singularities are dealt with using the well known Kinoshita-Lee-Nauenberg theorem. We show for the first time how to correctly apply the Kinoshita-Lee-Nauenberg theorem diagrammatically in a next-to-leading order scattering process. We improve on previous works by including all initial and final state soft radiative processes, including absorption and an infinite sum of partially disconnected amplitudes. Crucially, we exploit the Monotone Convergence Theorem to prove that our delicate rearrangement of this formally divergent series is uniquely correct. This rearrangement yields a factorization of the infinite contribution from the initial state soft photons that then cancels in the physically observable cross section. Since we use the $\overline{\mathrm{MS}}$ renormalization scheme, our result is valid up to arbitrarily large momentum transfers between the source and the scattered electron as long as $α\log(1/δ)\ll 1$ and $α\log(1/δ)\log(Δ/E)\ll 1$, where $Δ$ and $δ$ are the experimental energy and angular resolutions, respectively, and $E$ is the energy of the scattered electron. Our work aims at computing the NLO corrections to the energy loss of a high energetic parton propagating in a quark-gluon plasma.