# Infrared Finiteness and Forward Scattering

**Authors:** Christopher Frye, Holmfridur Hannesdottir, Nisarga Paul, Matthew D. Schwartz, Kai Yan  
**arXiv:** [1810.10022](https://arxiv.org/abs/1810.10022)  
**Date:** 2018-10-23  
**Summary page:** https://www.summarizepaper.com/en/paper/1810.10022v1/

## Key points

- Infrared divergences cancel out in inclusive cross-sections in quantum field theory calculations.
- The Kinoshita-Lee-Nauenberg theorem states that the sum of initial and final states should eliminate these divergences.
- Recent research suggests that the theorem is weaker than previously thought, and it is sufficient to sum over either initial or final states for finiteness.
- The cancellation of infrared divergences typically requires the inclusion of forward scattering processes.
- Examples are provided to illustrate this revised understanding, such as $e^+e^- \to Z$ at next-to-leading order.
- Experimental limits on energy and angular resolution must include a limit on the momentum at which electric charge can be observed for Compton scattering to achieve infrared finiteness.
- Including degenerate initial states is important, as well as considering disconnected Feynman diagrams.
- Dressing initial or final-state charged particles is significant in properly defining the $S$ matrix in quantum field theory.

## Summary

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In quantum field theory calculations, it has long been believed that <kw>Infrared divergences</kw> cancel out in inclusive cross-sections. This is based on the Kinoshita-Lee-Nauenberg theorem, which states that the sum of initial and final states should eliminate these divergences. However, recent research suggests that this theorem is actually weaker than previously thought. The revised understanding is that for finiteness, it is sufficient to sum over either initial or final states, rather than both. Additionally, the cancellation of infrared divergences typically requires the inclusion of forward scattering processes. To illustrate this revised understanding, several examples are provided. For instance, in $e^+e^- \to Z$ at next-to-leading order, it is possible to 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 taken into account. Similarly, for Compton scattering to achieve infrared finiteness, hard forward-scattered electrons and photons must be indistinguishable. This implies that experimental limits on energy and angular resolution must also include a limit on the momentum at which electric charge can be observed. This new understanding sheds light on various aspects of physical predictions. It emphasizes the importance of including degenerate initial states and highlights the relevance of disconnected Feynman diagrams. Furthermore, it underscores the significance of dressing initial or final-state charged particles and contributes to ongoing efforts aimed at properly defining the $S$ matrix in quantum field theory.