Nonperturbative Anomalous Thresholds

TL;DR

This paper explains the origin of anomalous thresholds in Feynman diagrams as a consequence of fundamental S-matrix principles, providing explicit formulas and testing them in a solvable model.

Contribution

It derives nonperturbative formulas for anomalous thresholds based on unitarity and analyticity, linking them to established principles.

Findings

Explicit nonperturbative formulas for anomalous thresholds

Validation against Coleman-Thun poles in E8 model

Clarification of anomalous thresholds' origin

Abstract

Feynman diagrams (notably the triangle diagram) involving heavy enough particles contain branch cuts on the physical sheet - anomalous thresholds - which, unlike normal thresholds and bound-state poles, do not correspond to any asymptotic $n$-particle state. ``Who ordered that?" We show that anomalous thresholds arise as a consequence of established S-matrix principles and two reasonable assumptions: unitarity below the physical region and analyticity in the mass. We find explicit nonperturbative formulas for the anomalous threshold singularity and test them against the Coleman-Thun poles of the exactly solvable $E_8$ integrable model.