Planar Matrices and Arrays of Feynman Diagrams

Abstract

Very recently planar collections of Feynman diagrams were proposed by Borges and one of the authors as the natural generalization of Feynman diagrams for the computation of $k=3$ biadjoint amplitudes. Planar collections are one-dimensional arrays of metric trees satisfying an induced planarity and compatibility condition. In this work we introduce planar matrices of Feynman diagrams as the objects that compute $k=4$ biadjoint amplitudes. These are symmetric matrices of metric trees satisfying compatibility conditions. We introduce two notions of combinatorial bootstrap techniques for finding collections from Feynman diagrams and matrices from collections. As applications of the first, we find all $693$, $13\,612$, and $346\,710$ collections for $(k,n)=(3,7), (3,8),$ and $(3,9)$ respectively. As applications of the second kind, we find all $90\, 608$ and $30\,659\,424$ planar matrices that compute $(k,n)=(4,8)$ and $(4,9)$ biadjoint amplitudes respectively. As an example of the evaluation of matrices of Feynman diagrams, we present the complete form of the $(4,8)$ and $(4,9)$ biadjoint amplitudes. We also start the study of higher dimensional arrays of Feynman diagrams, including the combinatorial version of the duality between $(k,n)$ and $(n-k,n)$ objects.

Figures (9)

• Figure 1: An example for an initial planar collection obtained by pruning a 6-point Feynman diagram. Above is the 6-point Feynman diagram to be pruned. Below is the planar collection of 5-point Feynman diagrams obtained by pruning the leaves $1,2,\cdots, 6$ of the above Feynman diagram respectively.
• Figure 2: An example for a 6-point initial planar matrix. Above we show a 6-point Feynman diagram. Below there is a symmetric matrix of 4-point Feynman diagrams obtained by pruning two leaves from the set $1,2,\cdots, 6$ at a time of the above Feynman diagram. The Feynman diagram from the $i^{\rm th}$ column and $j^{\rm th}$ row has the $i^{\rm th}$ and $j^{\rm th}$ leaves pruned.
• Figure 3: The five $k=2$ planar Feynman diagrams and their corresponding collections in $(k,n)=(3,5)$.
• Figure 4: Illustration of the second combinatorial bootstrap for obtaining planar matrices of Feynman diagrams. Here we choose ${\cal C}_{1}^{(1)}$, ${\cal C}_{2}^{(2)}$ and ${\cal C}_{1}^{(3)}$ as the first three columns and then get a symmetric planar matrix by filling in the remaining three columns with ${\cal C}_{2}^{(4)}$, ${\cal C}_{1}^{(5)}$ and ${\cal C}_{2}^{(6)}$. See also table \ref{['46matrices22']}.
• Figure 5: Bipyramidal collection for $(3,6)$

Theorems & Definitions (6)

• Definition 3.1
• Lemma 3.2
• proof
• Definition 6.1
• Lemma A.1
• proof