  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 $\alpha \log(1/\delta)\ll 1$ and $\alpha
\log(1/\delta)\log(\Delta/E)\ll 1$, where $\Delta$ and $\delta$ 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.

High Energy Physics - Phenomenology;High Energy Physics - Theory