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2022

Feynman Polytopes and the Tropical Geometry of UV and IR Divergences

Arkani-Hamed, Nima, Hillman, Aaron, Mizera, Sebastian

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We introduce a class of polytopes that concisely capture the structure of UV and IR divergences of general Feynman integrals in Schwinger parameter space, treating them in a unified way as worldline segments shrinking and expanding at different relative rates.

While these polytopes conventionally arise as convex hulls - via Newton polytopes of Symanzik polynomials - we show that they also have a remarkably simple dual description as cut out by linear inequalities defining the facets.
It is this dual definition that makes it possible to transparently understand and efficiently compute leading UV and IR divergences for any Feynman integral.
In the case of the UV, this provides a transparent geometric understanding of the familiar nested and overlapping divergences.

Feynman Polytopes and the Tropical Geometry of UV and IR Divergences · Around

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