Generalized Planar Feynman Diagrams: Collections

Abstract

Tree-level Feynman diagrams in a cubic scalar theory can be given a metric such that each edge has a length. The space of metric trees is made out of orthants joined where a tree degenerates. Here we restrict to planar trees since each degeneration of a tree leads to a single planar neighbor. Amplitudes are computed as an integral over the space of metrics where edge lengths are Schwinger parameters. In this work we propose that a natural generalization of Feynman diagrams is provided by what are known as metric tree arrangements. These are collections of metric trees subject to a compatibility condition on the metrics. We introduce the notion of planar collections of Feynman diagrams and argue that using planarity one can generate all planar collections starting from any one. Moreover, we identify a canonical initial collection for all $n$. Generalized $k=3$ biadjoint amplitudes, introduced by Early, Guevara, Mizera, and one of the authors, are easily computed as an integral over the space of metrics of planar collections of Feynman diagrams.

arXiv comment27 pages, 11 figures

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Cited by in corpus (9)

• Singularities of eight- and nine-particle amplitudes from cluster algebras and tropical geometry
• Smoothly Splitting Amplitudes and Semi-Locality
• Tropical fans, scattering equations and amplitudes
• Positive configuration space
• Algebraic branch points at all loop orders from positive kinematics and wall crossing
• Planar Matrices and Arrays of Feynman Diagrams
• Planar Matrices and Arrays of Feynman Diagrams: Poles for Higher $k$
• One-loop integrand from generalised scattering equations
• Double Soft Theorem for Generalised Biadjoint Scalar Amplitudes