Feynman Polytopes and the Tropical Geometry of UV and IR Divergences

Abstract

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.

12 pages, accepted version

References in corpus (13)

• Factorization constraints for soft anomalous dimensions in QCD scattering amplitudes
• Resolution of singularities for multi-loop integrals
• Geometric approach to asymptotic expansion of Feynman integrals
• Cluster algebras for Feynman integrals
• Scattering Equations: From Projective Spaces to Tropical Grassmannians
• Multiple Series Representations of $N$-fold Mellin-Barnes Integrals
• GKZ-hypergeometric systems for Feynman integrals
• Cluster Configuration Spaces of Finite Type
• Tropical fans, scattering equations and amplitudes
• Truncated cluster algebras and Feynman integrals with algebraic letters
• Positive configuration space
• Massive Fishnets
• The Hopf algebra structure of the $R^{\ast }$-operation

Cited by in corpus (13)

• Landau Discriminants
• The on-shell expansion: from Landau equations to the Newton polytope
• Constraints on Sequential Discontinuities from the Geometry of On-shell Spaces
• Unitarity Cuts of the Worldsheet
• Exposing the threshold structure of loop integrals
• Internal boundaries of the loop amplituhedron
• The Asymptotic Structure of Cosmological Integrals
• Flow-oriented perturbation theory
• GKZ-system of the 2-loop self energy with 4 propagators
• Feynman Integrals of Grassmannians
• Light-cone limits of large rectangular fishnets
• Four-dimensional differential equations for the leading divergences of dimensionally-regulated loop integrals
• Hierarchies in relative Picard-Lefschetz theory