# From Faddeev-Kulish to LSZ. Towards a non-perturbative description of colliding electrons

## TL;DR

This paper develops a non-perturbative framework for electron scattering in a simplified quantum field theory, reformulating Faddeev-Kulish and LSZ approaches to address infrared divergences.

## Contribution

It introduces a Faddeev-Kulish type formula within LSZ formalism for a low-energy Yukawa model, highlighting the role of photon clouds in infrared divergence cancellation.

## Key findings

- Infrared divergences are canceled by real and virtual photon clouds.
- Original Faddeev-Kulish approach omits real photon clouds, leading to ill-defined scattering matrices.
- Comparison with Pizzo's non-perturbative construction supports the proposed formulation.

## Abstract

In a low energy approximation of the massless Yukawa theory (Nelson model) we derive a Faddeev-Kulish type formula for the scattering matrix of $N$ electrons and reformulate it in LSZ terms. To this end, we perform a decomposition of the infrared finite Dollard modifier into clouds of real and virtual photons, whose infrared divergencies mutually cancel. We point out that in the original work of Faddeev and Kulish the clouds of real photons are omitted, and consequently their scattering matrix is ill-defined on the Fock space of free electrons. To support our observations, we compare our final LSZ expression for $N=1$ with a rigorous non-perturbative construction due to Pizzo. While our discussion contains some heuristic steps, they can be formulated as clear-cut mathematical conjectures.

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Source: https://tomesphere.com/paper/1706.09057