Planar Matrices and Arrays of Feynman Diagrams: Poles for Higher $k$

Abstract

Planar arrays of tree diagrams were introduced as a generalization of Feynman diagrams that enables the computation biadjoint amplitudes $m_n^{(k)}$ for $k>2$ . In this follow-up work we investigate the poles of $m_n^{(k)}$ from the perspective of such arrays. For general $k$ we characterize the underlying polytope as a Flag Complex and propose a computation of the amplitude based solely on the knowledge of poles, which number is drastically less than the number of full arrays. As an example we first provide all the poles for the cases $(k,n)=(3,7),(3,8),(4,8)$ and $(4,9)$ in terms of their generalized Feynman diagrams. We then implement a simple compatibility criteria together with an addition operation between arrays, and recover the full collections/arrays recently presented for such cases. Along the way we implement hard and soft kinematical limits, which provide a map between poles in kinematic space and their combinatoric arrays. We use the operation to give a proof of a previously conjectured combinatorial duality for arrays in $(k,n)$ and $(n-k,n)$. We also outline the relation to boundary maps of the hypersimplex ${\Delta }_{k,n}$ and rays in the tropical Grassmannian $\text{Tr}(k,n)$.

comment: v2: 21 pages, 2 figures, 2 tables, references added, missing ancillary files in the Dropbox link are recovered. Planar arrays of Feynman diagrams for (k,n)=(4,9) are reconstructed just from their poles while those for (3,10) are predicted for the first time

References in corpus (11)

• 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
• Smoothly Splitting Amplitudes and Semi-Locality
• Symbol Alphabets from Tensor Diagrams
• Planar kinematic invariants, matroid subdivisions and generalized Feynman diagrams
• Stringy canonical forms and binary geometries from associahedra, cyclohedra and generalized permutohedra
• The tropical totally positive Grassmannian
• Generalized Color Orderings: CEGM Integrands and Decoupling Identities
• Factorization for Generalized Biadjoint Scalar Amplitudes via Matroid Subdivisions
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Cited by in corpus (4)

• Generalized Color Orderings: CEGM Integrands and Decoupling Identities
• Notes on Worldsheet-Like Variables for Cluster Configuration Spaces
• One-loop integrand from generalised scattering equations
• Color-Dressed Generalized Biadjoint Scalar Amplitudes: Local Planarity