Generalized Planar Feynman Diagrams: Collections

TL;DR

This paper introduces a generalization of planar Feynman diagrams called metric tree arrangements, providing a new geometric framework for computing amplitudes in scalar theories.

Contribution

It proposes metric tree arrangements as a natural extension of Feynman diagrams and develops a method to generate all planar collections from a canonical initial collection.

Findings

Defines planar collections of Feynman diagrams with compatibility conditions.

Shows how to generate all planar collections from a single initial collection.

Provides a method to compute generalized biadjoint amplitudes as integrals over metric spaces.

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.