Feynman Polytopes and the Tropical Geometry of UV and IR Divergences

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. In the IR, the polytope exposes a new perspective on soft/collinear singularities and their intricate generalizations. Tropical geometry furnishes a simple framework for calculating the leading UV/IR divergences of any Feynman integral, associating them with the volumes of certain dual cones. As concrete applications, we generalize Weinberg's theorem to include a characterization of IR divergences, and classify space-time dimensions in which general IR divergences (logarithmic as well as power-law) can occur. We also compute the leading IR divergence of rectangular fishnet diagrams at all loop orders, which turn out to have a surprisingly simple combinatorial description.