Soft Factorisation and Exponentiation from Schwinger-Space Geometry

TL;DR

This paper introduces a novel geometric approach using Schwinger parametrization and tropical geometry to analyze infrared divergences in quantum field theories, leading to systematic insights into soft factorization and exponentiation.

Contribution

It develops a new framework employing Schwinger-space geometry and tropical limits to understand IR divergences and soft factorization in quantum field theories.

Findings

Demonstrates soft-hard factorization for a broad class of diagrams.

Shows topologically distinct diagrams asymptote to the same integrand in Schwinger space.

Reveals how ladder diagrams in QED exponentiate to produce the soft anomalous dimension.

Abstract

Infrared divergences in Quantum Field Theory govern the low-energy dynamics of many physical theories, and their understanding is a crucial ingredient in predicting the outcomes of collider experiments. We present a novel approach to deriving the structure of these divergences by employing the Schwinger parametrization of Feynman integrals. After using tropical geometry to identify divergent limits, we study the all-orders asymptotic properties of Feynman diagrams via matrix manipulations of graph Laplacians, which allows us to analyse their IR behaviour systematically. We explicitly demonstrate the soft-hard factorization of the integrand for a broad class of diagrams, and reveal that when written in terms of "worldline distances", topologically distinct diagrams asymptote to the same integrand. In particular, for the case of Quantum Electrodynamics, we use this fact to show how ladder-type diagrams combine in Schwinger-parameter space to yield the correct exponentiated soft anomalous dimension. This framework provides a foundation for extending these methods to more complex theories like Quantum Chromodynamics and offers a pathway towards a systematic understanding of infrared divergences in perturbative amplitudes.