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 $693$, $13612$, and $346710$ collections for $(k,n)=(3,7),(3,8),$ and $(3,9)$ respectively. As applications of the second kind, we find all $90608$ and $30659424$ 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.

v3: 31 pages, 9 figures, references added, missing ancillary files in Dropbox link recovered

References in corpus (25)

• How tropical are seven- and eight-particle amplitudes?
• Scattering Equations: From Projective Spaces to Tropical Grassmannians
• Likelihood Equations and Scattering Amplitudes
• Singularities of eight- and nine-particle amplitudes from cluster algebras and tropical geometry
• Cluster Configuration Spaces of Finite Type
• Tropical fans, scattering equations and amplitudes
• Smoothly Splitting Amplitudes and Semi-Locality
• Truncated cluster algebras and Feynman integrals with algebraic letters
• Symbol Alphabets from Tensor Diagrams
• Singular Solutions in Soft Limits
• The positive tropical Grassmannian, the hypersimplex, and the m=2 amplituhedron
• Planar kinematic invariants, matroid subdivisions and generalized Feynman diagrams
• Notes on polytopes, amplitudes and boundary configurations for Grassmannian string integrals
• Stringy canonical forms and binary geometries from associahedra, cyclohedra and generalized permutohedra
• Generalized Planar Feynman Diagrams: Collections
• A Soft Theorem for the Tropical Grassmannian
• The tropical totally positive Grassmannian
• Planar Matrices and Arrays of Feynman Diagrams: Poles for Higher $k$
• Generalized Color Orderings: CEGM Integrands and Decoupling Identities
• Notes on Worldsheet-Like Variables for Cluster Configuration Spaces
• Factorization for Generalized Biadjoint Scalar Amplitudes via Matroid Subdivisions
• Minimal Kinematics: An All $k$ and $n$ Peek into ${\mathrm{Trop}}^{+}\mathrm{G}(k,n)$
• Cluster algebras III: Upper bounds and double Bruhat cells
• Double Soft Theorem for Generalised Biadjoint Scalar Amplitudes
• Color-Dressed Generalized Biadjoint Scalar Amplitudes: Local Planarity

Cited by in corpus (6)

• Algebraic branch points at all loop orders from positive kinematics and wall crossing
• Generalized Color Orderings: CEGM Integrands and Decoupling Identities
• Notes on Worldsheet-Like Variables for Cluster Configuration Spaces
• One-loop integrand from generalised scattering equations
• Minimal Kinematics: An All $k$ and $n$ Peek into ${\mathrm{Trop}}^{+}\mathrm{G}(k,n)$
• Color-Dressed Generalized Biadjoint Scalar Amplitudes: Local Planarity