Nonperturbative anomalous thresholds

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 <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"><a:mi>n</a:mi></a:math>-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 <c:math xmlns:c="http://www.w3.org/1998/Math/MathML" display="inline"><c:msub><c:mi>E</c:mi><c:mn>8</c:mn></c:msub></c:math> integrable model.