A Symbol of Uniqueness: The Cluster Bootstrap for the 3-Loop MHV Heptagon

Abstract

Seven-particle scattering amplitudes in planar super-Yang-Mills theory are believed to belong to a special class of generalised polylogarithm functions called heptagon functions. These are functions with physical branch cuts whose symbols may be written in terms of the 42 cluster A-coordinates on Gr(4,7). Motivated by the success of the hexagon bootstrap programme for constructing six-particle amplitudes we initiate the systematic study of the symbols of heptagon functions. We find that there is exactly one such symbol of weight six which satisfies the MHV last-entry condition and is finite in the $7\parallel 6$ collinear limit. This unique symbol is both dihedral and parity-symmetric, and remarkably its collinear limit is exactly the symbol of the three-loop six-particle MHV amplitude, although none of these properties were assumed a priori. It must therefore be the symbol of the three-loop seven-particle MHV amplitude. The simplicity of its construction suggests that the n-gon bootstrap may be surprisingly powerful for n>6.

30 pages, 3 ancillary files, v3: minor corrections, including a typo in (33)

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• Elliptic Feynman integrals and pure functions
• Cluster adjacency properties of scattering amplitudes
• Bootstrapping the QCD soft anomalous dimension
• The Cosmic Galois Group and Extended Steinmann Relations for Planar $\mathcal{N}=4$ SYM Amplitudes
• Elliptic symbol calculus: from elliptic polylogarithms to iterated integrals of Eisenstein series
• Gluing Ladders into Fishnets
• Evidence for a Nonplanar Amplituhedron
• Multi-Regge kinematics and the moduli space of Riemann spheres with marked points
• Elliptic polylogarithms and Feynman parameter integrals
• Bootstrapping pentagon functions
• Prescriptive Unitarity
• Cluster adjacency and the four-loop NMHV heptagon
• Cluster algebras for Feynman integrals
• Embedding Feynman Integral (Calabi-Yau) Geometries in Weighted Projective Space
• One-loop amplitudes in $Ad{S}_5\times S^5$ supergravity from $\mathcal{N}=4$ SYM at strong coupling
• The Double Pentaladder Integral to All Orders
• A Three-Point Form Factor Through Five Loops
• Rationalizing Loop Integration
• Cluster adjacency beyond MHV
• The Origin of the Six-Gluon Amplitude in Planar $\mathcal{N}=4$ SYM
• OPE for all Helicity Amplitudes II. Form Factors and Data Analysis
• Sequential Discontinuities of Feynman Integrals and the Monodromy Group
• How tropical are seven- and eight-particle amplitudes?
• Gravity On-shell Diagrams
• Landau Singularities and Symbology: One- and Two-loop MHV Amplitudes in SYM Theory
• Landau Singularities from the Amplituhedron
• Bootstrapping a Stress-Tensor Form Factor through Eight Loops
• Multi-Loop Positivity of the Planar $\mathcal{N}=4$ SYM Six-Point Amplitude
• Lifting Heptagon Symbols to Functions
• Two-loop Octagons, Algebraic Letters and $\bar{Q}$ Equations
• Tropical Grassmannians, cluster algebras and scattering amplitudes
• Bootstrapping two-loop Feynman integrals for planar N=4 sYM
• Stratifying On-Shell Cluster Varieties: the Geometry of Non-Planar On-Shell Diagrams
• All-loop singularities of scattering amplitudes in massless planar theories
• All-order amplitudes at any multiplicity in the multi-Regge limit
• MPL - a program for computations with iterated integrals on moduli spaces of curves of genus zero
• Singularities of eight- and nine-particle amplitudes from cluster algebras and tropical geometry
• Hexagon OPE Resummation and Multi-Regge Kinematics
• Prescriptive Unitarity for Non-Planar Six-Particle Amplitudes at Two Loops
• Boundaries of Amplituhedra and NMHV Symbol Alphabets at Two Loops
• Antipodal Self-Duality for a Four-Particle Form Factor
• Notes on cluster algebras and some all-loop Feynman integrals
• The symbol and alphabet of two-loop NMHV amplitudes from $\bar{Q}$ equations
• Tropical fans, scattering equations and amplitudes
• The seven-gluon amplitude in multi-Regge kinematics beyond leading logarithmic accuracy
• Truncated cluster algebras and Feynman integrals with algebraic letters
• On factorization of multiparticle pentagons
• Manifestly Dual-Conformal Loop Integration
• The Sklyanin Bracket and Cluster Adjacency at All Multiplicity
• All-Helicity Symbol Alphabets from Unwound Amplituhedra
• Symbology for elliptic multiple polylogarithms and the symbol prime
• Symbol Alphabets from Tensor Diagrams
• Cluster Algebras and the Subalgebra Constructibility of the Seven-Particle Remainder Function
• Yangian Invariants and Cluster Adjacency in N=4 Yang-Mills
• Schubert Problems, Positivity and Symbol Letters
• Algebraic branch points at all loop orders from positive kinematics and wall crossing
• Bootstrapping elliptic Feynman integrals using Schubert analysis
• A Note on Letters of Yangian Invariants
• Feynman Integrals and Scattering Amplitudes from Wilson Loops
• Towards single-valued polylogarithms in two variables for the seven-point remainder function in multi-Regge-kinematics
• Bootstrapping a Two-Loop Four-Point Form Factor
• The Wilson-loop $d\log$ representation for Feynman integrals
• Intersection theory rules symbology
• The Two-Loop Remainder Function for Eight and Nine Particles
• Cross-Order Integral Relations from Maximal Cuts
• The Two-Loop Symbol of all Multi-Regge Regions
• Hedgehog Bases for A_n Cluster Polylogarithms and An Application to Six-Point Amplitudes
• The Three-loop MHV Octagon from $\bar{Q}$ equations
• Rooting Out Letters: Octagonal Symbol Alphabets and Algebraic Number Theory
• Integrated negative geometries in ABJM
• The Full-Color Two-Loop Four-Gluon Amplitude in $\mathcal{N}=2$ Super-QCD
• Nonperturbative enhancement of superloop at strong coupling
• Cosmic Wheels: From integrability to the Galois coaction
• The recursive structure of Baikov representations I: generics and application to symbology
• Towards NMHV amplitudes at strong coupling
• Gravity loop integrands from the ultraviolet
• Comments on all-loop constraints for scattering amplitudes and Feynman integrals
• The SAGEX Review on Scattering Amplitudes, Chapter 15: The Multi-Regge Limit
• The SAGEX Review on Scattering Amplitudes, Chapter 5: Analytic Bootstraps for Scattering Amplitudes and Beyond
• On the scattering of gluons in the GKP string
• All two-loop MHV remainder functions in multi-Regge kinematics
• Periods and Feynman amplitudes
• The two-loop eight-point amplitude in ABJM theory
• Conformally Regulated Direct Integration of the Two-Loop Heptagon Remainder
• Non-planar BCFW Grassmannian Geometries
• Non-Planar On-Shell Diagrams
• Bootstrapping octagons in reduced kinematics from $A_2$ cluster algebras
• An Origin Story for Amplitudes
• Systematics of the Multi-Regge Three-Loop Symbol
• All orders results for self-crossing Wilson loops mimicking double parton scattering
• A nice two-loop next-to-next-to-MHV amplitude in $\mathcal{N}=4$ super-Yang-Mills
• Polylogarithm identities, cluster algebras and the N=4 supersymmetric theory
• Deciphering the Maximal Transcendentality Principle via Bootstrap
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• A Two-Loop Four-Point Form Factor at Function Level
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• Locally Finite Observables in sYM
• Exploring Reggeon bound states in strongly-coupled $\mathcal{N}=4$ super Yang-Mills
• Cluster Alphabets from Generalized Worldsheets: A Geometric Approach to Finite Types
• Analytic Four-Point Lightlike Form Factors and OPE of Null-Wrapped Polygons
• Hexagon Bootstrap in the Double Scaling Limit
• Cluster Functions and Scattering Amplitudes for Six and Seven Points