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In the papers of the 1950s and 1960s, I see reference to anomalous threshold singularities. What are these? Is there a good reference that discusses this subject?

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"Anomalous" means that they're coming from quantum corrections, loops. "Threshold" means that they appear because some loop momenta approach a threshold for production at this very point – virtual particles are just becoming real. "Singularities" mean that these conditions obeyed by a loop Feynman diagram lead to a singular behavior in the resulting amplitude.

See e.g.

The last preprint mentions the anomalous threshold singularities in the context of their major application, the 1978 Coleman-Thun mechanism.

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Anomalous thresholds are branch points in the complex plane of scattering amplitudes/Greens functions that are not easily deducible from unitarity. The book "The Analytic S-Matrix" by R.J. Eden explains it well, and shows how it may be deduced non-trivilaly from unitarity considerations.

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