TL;DR
This paper explores the mathematical foundations of Cutkosky rules, connecting them to geometric group theory and monodromy of Feynman amplitudes, offering a systematic approach to thresholds and dispersion relations.
Contribution
It links Cutkosky rules to cubical chain complexes and monodromy, providing a new systematic framework for analyzing Feynman amplitudes and related phenomena.
Findings
Operates on cubical chain complexes in geometric group theory
Relates internal line operations to monodromy of amplitudes
Provides a systematic approach to thresholds and dispersion relations
Abstract
We overview recent results on the mathematical foundations of Cutkosky rules. We emphasize that the two operations of shrinking an internal edge or putting internal lines on the mass-shell are natural operation on the cubical chain complex studied in the context of geometric group theory. This together with Cutkosky's theorem regarded as a theorem which informs us about variations connected to the monodromy of Feynman amplitudes allows for a systematic approach to normal and anomalous thresholds, dispersion relations and the optical theorem. In this report we follow [1] closely.
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