Motivic Amplitudes and Cluster Coordinates

Abstract

In this paper we study motivic amplitudes--objects which contain all of the essential mathematical content of scattering amplitudes in planar SYM theory in a completely canonical way, free from the ambiguities inherent in any attempt to choose particular functional representatives. We find that the cluster structure on the kinematic configuration space Conf_n(P^3) underlies the structure of motivic amplitudes. Specifically, we compute explicitly the coproduct of the two-loop seven-particle MHV motivic amplitude A_{7,2} and find that like the previously known six-particle amplitude, it depends only on certain preferred coordinates known in the mathematics literature as cluster X-coordinates on Conf_n(P^3). We also find intriguing relations between motivic amplitudes and the geometry of generalized associahedrons, to which cluster coordinates have a natural combinatoric connection. For example, the obstruction to A_{7,2} being expressible in terms of classical polylogarithms is most naturally represented by certain quadrilateral faces of the appropriate associahedron. We also find and prove the first known functional equation for the trilogarithm in which all 40 arguments are cluster X-coordinates of a single algebra. In this respect it is similar to Abel's 5-term dilogarithm identity.

61 pages, 10 figures; v2: minor changes and corrections

References in corpus (17)

• Multiloop integrals in dimensional regularization made simple
• Gluon scattering amplitudes at strong coupling
• Classical Polylogarithms for Amplitudes and Wilson Loops
• Conformal properties of four-gluon planar amplitudes and Wilson loops
• The All-Loop Integrand For Scattering Amplitudes in Planar N=4 SYM
• The Four-Loop Planar Amplitude and Cusp Anomalous Dimension in Maximally Supersymmetric Yang-Mills Theory
• Yangian symmetry of scattering amplitudes in N=4 super Yang-Mills theory
• Hexagon Wilson loop = six-gluon MHV amplitude
• The Two-Loop Six-Gluon MHV Amplitude in Maximally Supersymmetric Yang-Mills Theory
• An Operator Product Expansion for Polygonal null Wilson Loops
• Lectures on Twistor Strings and Perturbative Yang-Mills Theory
• Harmonic R-matrices for Scattering Amplitudes and Spectral Regularization
• Analytic Results for MHV Wilson Loops
• Null Polygonal Wilson Loops in Full N=4 Superspace
• On the Geometry of Null Polygons in Full N=4 Superspace
• Top quark pairs at two loops and Reduze 2
• Collinear and Regge behavior of 2 -> 4 MHV amplitude in N = 4 super Yang-Mills theory

Cited by in corpus (148)

• The Amplituhedron
• Lectures on differential equations for Feynman integrals
• PolyLogTools - Polylogs for the masses
• A Symbol of Uniqueness: The Cluster Bootstrap for the 3-Loop MHV Heptagon
• Heptagons from the Steinmann Cluster Bootstrap
• Space-time S-matrix and Flux tube S-matrix II. Extracting and Matching Data
• Analytic form of the two-loop planar five-gluon all-plus-helicity amplitude in QCD
• The four-loop remainder function and multi-Regge behavior at NNLLA in planar N=4 super-Yang-Mills theory
• Six-Gluon Amplitudes in Planar $\mathcal{N}=4$ Super-Yang-Mills Theory at Six and Seven Loops
• Combinatorics and Topology of Kawai-Lewellen-Tye Relations
• The Elliptic Double-Box Integral: Massless Amplitudes Beyond Polylogarithms
• Space-time S-matrix and Flux-tube S-matrix III. The two-particle contributions
• Unwinding the Amplituhedron in Binary
• Cluster adjacency properties of scattering amplitudes
• The Cosmic Galois Group and Extended Steinmann Relations for Planar $\mathcal{N}=4$ SYM Amplitudes
• Space-time S-matrix and Flux-tube S-matrix IV. Gluons and Fusion
• Closed Superstring Amplitudes, Single-Valued Multiple Zeta Values and Deligne Associator
• Evidence for a Nonplanar Amplituhedron
• Analytic result for a two-loop five-particle amplitude
• Multi-Regge kinematics and the moduli space of Riemann spheres with marked points
• Bootstrapping pentagon functions
• Prescriptive Unitarity
• Cluster adjacency and the four-loop NMHV heptagon
• Cluster algebras for Feynman integrals
• Local Integrand Representations of All Two-Loop Amplitudes in Planar SYM
• The Double Pentaladder Integral to All Orders
• A Cluster Bootstrap for Two-Loop MHV Amplitudes
• Rationalizing Loop Integration
• Cluster adjacency beyond MHV
• Cluster algebras from dualities of 2d N=(2,2) quiver gauge theories
• Landau Discriminants
• Spectral Parameters for Scattering Amplitudes in N=4 Super Yang-Mills Theory
• Light-quark two-loop corrections to heavy-quark pair production in the gluon fusion channel
• Hexagon Wilson Loop OPE and Harmonic Polylogarithms
• The two-loop five-particle amplitude in $\mathcal{N}=8$ supergravity
• 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
• An analytic result for the two-loop seven-point MHV amplitude in N=4 SYM
• Lifting Heptagon Symbols to Functions
• Two-loop Octagons, Algebraic Letters and $\bar{Q}$ Equations
• Tropical Grassmannians, cluster algebras and scattering amplitudes
• All-loop singularities of scattering amplitudes in massless planar theories
• Bootstrapping two-loop Feynman integrals for planar N=4 sYM
• Stratifying On-Shell Cluster Varieties: the Geometry of Non-Planar On-Shell Diagrams
• Singularities of eight- and nine-particle amplitudes from cluster algebras and tropical geometry
• The Differential of All Two-Loop MHV Amplitudes in N=4 Yang-Mills Theory
• Analytic structure of the $n=7$ scattering amplitude in $\mathcal{N}=4$ SYM theory at multi-Regge kinematics: Conformal Regge pole contribution
• Massive planar and non-planar double box integrals for light Nf contributions to gg->tt
• The elliptic double box and symbology beyond polylogarithms
• Hexagon OPE Resummation and Multi-Regge Kinematics
• Implications of nonplanar dual conformal symmetry
• 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
• Symbol Alphabets from Plabic Graphs
• Tropical fans, scattering equations and amplitudes
• Symbol Alphabets from the Landau Singular Locus
• Truncated cluster algebras and Feynman integrals with algebraic letters
• The seven-gluon amplitude in multi-Regge kinematics beyond leading logarithmic accuracy
• The Complete Two-Loop Integrated Jet Thrust Distribution In Soft-Collinear Effective Theory
• Manifestly Dual-Conformal Loop Integration
• Positive configuration space
• All-Mass $n$-gon Integrals in $n$ Dimensions
• 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
• Parity duality for the amplituhedron
• Cluster Adjacency for m=2 Yangian Invariants
• First look at the evaluation of three-loop non-planar Feynman diagrams for Higgs plus jet production
• Constraints on Sequential Discontinuities from the Geometry of On-shell Spaces
• 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
• A study of Feynman integrals with uniform transcendental weights and the symbology from dual conformal symmetry
• Algebraic branch points at all loop orders from positive kinematics and wall crossing
• Bootstrapping elliptic Feynman integrals using Schubert analysis
• What can we learn about QCD and collider physics from N=4 super Yang-Mills?
• A Note on Letters of Yangian Invariants
• The Two-Loop Remainder Function for Eight and Nine Particles
• A first look at the function space for planar two-loop six-particle Feynman integrals
• Intersection theory rules symbology
• Gauge Theories and Dessins d'Enfants: Beyond the Torus
• Symbol Alphabets from Plabic Graphs II: Rational Letters
• Cross-Order Integral Relations from Maximal Cuts
• The Two-Loop Symbol of all Multi-Regge Regions
• Black hole and cosmos with multiple horizons and multiple singularities in vector-tensor theories
• Scattering Amplitudes and Toric Geometry
• Hedgehog Bases for A_n Cluster Polylogarithms and An Application to Six-Point Amplitudes
• Cluster Polylogarithms for Scattering Amplitudes
• The SAGEX Review on Scattering Amplitudes, Chapter 4: Multi-loop Feynman Integrals
• The Three-loop MHV Octagon from $\bar{Q}$ equations
• Rooting Out Letters: Octagonal Symbol Alphabets and Algebraic Number Theory
• One-loop hexagon integral to higher orders in the dimensional regulator
• Donaldson-Thomas trasnsformations of moduli spaces of G-local systems
• Analytic structure of the $n=7$ scattering amplitude in $\mathcal{N}=4$ SYM theory in multi-Regge kinematics: Conformal Regge cut contribution
• Cluster Algebras and the Positive Grassmannian
• The recursive structure of Baikov representations I: generics and application to symbology
• Towards Analytic Structure of Feynman Parameter Integrals with Rational Curves
• Comments on all-loop constraints for scattering amplitudes and Feynman integrals
• Multiple polylogarithms in weight 4
• 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
• All two-loop MHV remainder functions in multi-Regge kinematics
• Motivic correlators, cluster varieties and Zagier's conjecture on zeta(F,4)
• The four-loop six-gluon NMHV ratio function
• Conformally Regulated Direct Integration of the Two-Loop Heptagon Remainder
• Non-planar BCFW Grassmannian Geometries
• Triangulations and Canonical Forms of Amplituhedra: a fiber-based approach beyond polytopes
• The m=2 amplituhedron and the hypersimplex: signs, clusters, triangulations, Eulerian numbers
• Cluster Algebras: Network Science and Machine Learning
• Bootstrapping octagons in reduced kinematics from $A_2$ cluster algebras
• Non-Planar On-Shell Diagrams
• Cluster algebras in scattering amplitudes with special 2D kinematics
• Periods and Superstring Amplitudes
• An Origin Story for Amplitudes
• Gauge Theories, Tessellations & Riemann Surfaces
• Donaldson-Thomas Transformation of Grassmannian
• Anatomy of the Amplituhedron
• One-loop integrand from generalised scattering equations
• Continuous Quivers of Type A (III) Embeddings of Cluster Theories
• A Novel Algorithm for Nested Summation and Hypergeometric Expansions
• Antipodal Symmetry of Two-Loop MHV Amplitudes
• A Two-Loop Four-Point Form Factor at Function Level
• Heptagon Functions and Seven-Gluon Amplitudes in Multi-Regge Kinematics
• A walk on sunset boulevard
• Periods and motives in the spectral action of Robertson-Walker spacetimes
• The Steinmann Cluster Bootstrap for N=4 Super Yang-Mills Amplitudes
• A Note on One-loop Cluster Adjacency in N = 4 SYM
• Logarithmic Singularities and Maximally Supersymmetric Amplitudes
• The long road from Regge poles to the LHC
• BCFW tilings and cluster adjacency for the amplituhedron
• Polylogarithms from the bound state S-matrix
• Cluster Alphabets from Generalized Worldsheets: A Geometric Approach to Finite Types
• Holomorphic polylogarithms and Bloch complexes
• A tale of two Regge limits
• Continuous Quivers of Type A (IV) Continuous Mutation and Geometric Models of $\mathbf{E}$-clusters
• Studying Quantum Field Theory
• On webs, polylogarithms and cluster algebras
• Hexagon Bootstrap in the Double Scaling Limit
• Combinatorics of $\mathcal{X}$-variables in finite type cluster algebras
• Cluster Functions and Scattering Amplitudes for Six and Seven Points
• Progrès récents sur la conjecture de Zagier et le programme de Goncharov [d'après Goncharov, Rudenko, Gangl, ...]