Infrared Finiteness and Forward Scattering

TL;DR

This paper revises the understanding of infrared divergence cancellation in quantum field theory, emphasizing the importance of including forward scattering and degenerate states for finite cross-sections.

Contribution

It demonstrates that summing over initial or final states alone suffices for infrared finiteness when forward scattering is included, refining the traditional Kinoshita-Lee-Nauenberg theorem.

Findings

In $e^+e^- o Z$, summing over certain states yields finite results.

Infrared finiteness in Compton scattering depends on indistinguishability of forward-scattered particles.

Including degenerate initial states and forward scattering clarifies $S$ matrix definition.

Abstract

Infrared divergences have long been heralded to cancel in sufficiently inclusive cross-sections, according to the famous Kinoshita-Lee-Nauenberg theorem which mandates an initial and final state sum. While well-motivated, this theorem is much weaker than necessary: for finiteness, one need only sum over initial $\textit{or}$ final states. Moreover, the cancellation generically requires the inclusion of the forward scattering process. We provide a number of examples showing the importance of this revised understanding: in $e^+e^- \to Z$ at next-to-leading order, one can sum over certain initial and final states with an arbitrary number of extra photons, or only over final states with a finite number of photons, if forward scattering is included. For Compton scattering, infrared finiteness requires the indistinguishability of hard forward-scattered electrons and photons. This implies that in addition to experimental limits on the energy and angular resolution, there must also be an experimental limit on the momentum at which electric charge can be observed. Similar considerations are required to explain why the rate for $\gamma \gamma$ to scatter into photons alone is infrared divergent but the rate for $\gamma \gamma$ to scatter into photons or charged particles is finite. This new understanding sheds light on the importance of including degenerate initial states in physical predictions, the relevance of disconnected Feynman diagrams, the importance of dressing initial or final-state charged particles, and the quest to properly define the $S$ matrix.