A Complete Diagrammatic Implementation of the Kinoshita-Lee-Nauenberg Theorem at Next-to-Leading Order

We show for the first time in over 50 years 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 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. We derive the first complete next-to-leading order, high-energy Rutherford elastic scattering cross section in the $\overline{\mathrm{MS}}$ renormalization scheme as an explicit example of our procedure.