How tropical are seven- and eight-particle amplitudes?

Abstract

We study tropical Grassmanians Tr$(k,n)$ in relation to cluster algebras, and assess their applicability to $n$-particle amplitudes for $n=7,8$. In $\mathcal{N}=4$ super Yang-Mills theory, we first show that while the totally positive part of Tr$(4,7)$ may encompass the iterated discontinuity structure of the seven-point Maximally Helicity Violating (MHV) amplitude, it is too small for the Next-to-MHV helicity configuration. Then, using Tr$(4,8)$ we propose a finite set of 356 cluster $\mathcal{A}$-coordinates expected to contain the rational symbol letters of the eight-particle MHV amplitude, and discuss how the remaining square-root letters may be obtained from limits of infinite mutation sequences. Finally, we use a triangulation of the totally positive part of Tr$(3,8)$ to obtain the associated generalised biadjoint scalar amplitude in a form containing a near-minimal amount of spurious poles.

arXiv comment47 pages, 7 tables, 14 figures, 2 ancillary files; v2: Minor typos corrected and clarifications added, improved presentation of dihedral symmetry in sec.4 preamble, version to appear in JHEP

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Cited by in corpus (32)

• Cluster algebras for Feynman integrals
• Landau Discriminants
• Lifting Heptagon Symbols to Functions
• Singularities of eight- and nine-particle amplitudes from cluster algebras and tropical geometry
• The elliptic double box and symbology beyond polylogarithms
• Antipodal Self-Duality for a Four-Particle Form Factor
• Tropical fans, scattering equations and amplitudes
• The symbol and alphabet of two-loop NMHV amplitudes from $\bar{Q}$ equations
• Symbol Alphabets from Plabic Graphs
• Notes on cluster algebras and some all-loop Feynman integrals
• Truncated cluster algebras and Feynman integrals with algebraic letters
• Positive configuration space
• Symbol Alphabets from Tensor Diagrams
• Symbology for elliptic multiple polylogarithms and the symbol prime
• Constraints on Sequential Discontinuities from the Geometry of On-shell Spaces
• Schubert Problems, Positivity and Symbol Letters
• Algebraic branch points at all loop orders from positive kinematics and wall crossing
• A Note on Letters of Yangian Invariants
• Bootstrapping elliptic Feynman integrals using Schubert analysis
• The Two-Loop Remainder Function for Eight and Nine Particles
• Symbol Alphabets from Plabic Graphs II: Rational Letters
• The Three-loop MHV Octagon from $\bar{Q}$ equations
• Planar Matrices and Arrays of Feynman Diagrams
• Comments on all-loop constraints for scattering amplitudes and Feynman integrals
• The SAGEX Review on Scattering Amplitudes, Chapter 5: Analytic Bootstraps for Scattering Amplitudes and Beyond
• Planar Matrices and Arrays of Feynman Diagrams: Poles for Higher $k$
• Bootstrapping octagons in reduced kinematics from $A_2$ cluster algebras
• An Origin Story for Amplitudes
• A nice two-loop next-to-next-to-MHV amplitude in $\mathcal{N}=4$ super-Yang-Mills
• A Novel Algorithm for Nested Summation and Hypergeometric Expansions
• Antipodal Symmetry of Two-Loop MHV Amplitudes
• A Note on One-loop Cluster Adjacency in N = 4 SYM