English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Tools annual

Global: Zendy Free Plan

Global: Zendy Tools monthly

Global: Zendy Plus monthly

We introduce a class of polytopes that concisely capture the structure of UVand IR divergences of general Feynman integrals in Schwinger parameter space,treating them in a unified way as worldline segments shrinking and expanding atdifferent relative rates. While these polytopes conventionally arise as convexhulls - via Newton polytopes of Symanzik polynomials - we show that they alsohave a remarkably simple dual description as cut out by linear inequalitiesdefining the facets. It is this dual definition that makes it possible totransparently understand and efficiently compute leading UV and IR divergencesfor any Feynman integral. In the case of the UV, this provides a transparentgeometric understanding of the familiar nested and overlapping divergences. Inthe IR, the polytope exposes a new perspective on soft/collinear singularitiesand their intricate generalizations. Tropical geometry furnishes a simpleframework for calculating the leading UV/IR divergences of any Feynmanintegral, associating them with the volumes of certain dual cones. As concreteapplications, we generalize Weinberg's theorem to include a characterization ofIR divergences, and classify space-time dimensions in which general IRdivergences (logarithmic as well as power-law) can occur. We also compute theleading IR divergence of rectangular fishnet diagrams at all loop orders, whichturn out to have a surprisingly simple combinatorial description.

Feynman Polytopes and the Tropical Geometry of UV and IR Divergences