Lifting Heptagon Symbols to Functions

Abstract

Seven-point amplitudes in planar $\mathcal{N}=4$ super-Yang-Mills theory have previously been constructed through four loops using the Steinmann cluster bootstrap, but only at the level of the symbol. We promote these symbols to actual functions, by specifying their first derivatives and boundary conditions on a particular two-dimensional surface. To do this, we impose branch-cut conditions and construct the entire heptagon function space through weight six. We plot the amplitudes on a few lines in the bulk Euclidean region, and explore the properties of the heptagon function space under the coaction associated with multiple polylogarithms.

arXiv comment46 pages, 5 figures, 3 tables

References in corpus (64)

• One-Loop n-Point Gauge Theory Amplitudes, Unitarity and Collinear Limits
• Harmonic Polylogarithms
• Gluon scattering amplitudes at strong coupling
• Iteration of Planar Amplitudes in Maximally Supersymmetric Yang-Mills Theory at Three Loops and Beyond
• Transcendentality and Crossing
• Classical Polylogarithms for Amplitudes and Wilson Loops
• Dual superconformal symmetry of scattering amplitudes in N=4 super-Yang-Mills theory
• Magic identities for conformal four-point integrals
• The Four-Loop Planar Amplitude and Cusp Anomalous Dimension in Maximally Supersymmetric Yang-Mills Theory
• Conformal Ward identities for Wilson loops and a test of the duality with gluon amplitudes
• Comments on gluon scattering amplitudes via AdS/CFT
• Hexagon Wilson loop = six-gluon MHV amplitude
• The Two-Loop Six-Gluon MHV Amplitude in Maximally Supersymmetric Yang-Mills Theory
• Dual Superconformal Invariance, Momentum Twistors and Grassmannians
• From polygons and symbols to polylogarithmic functions
• Space-time S-matrix and Flux-tube S-matrix at Finite Coupling
• Bootstrapping the three-loop hexagon
• PolyLogTools - Polylogs for the masses
• Hopf algebras, coproducts and symbols: an application to Higgs boson amplitudes
• Maximally Supersymmetric Planar Yang-Mills Amplitudes at Five Loops
• Motivic Amplitudes and Cluster Coordinates
• Bootstrapping a Five-Loop Amplitude Using Steinmann Relations
• Thermodynamic Bubble Ansatz
• Analytic result for the two-loop six-point NMHV amplitude in N=4 super Yang-Mills theory
• Bootstrapping an NMHV amplitude through three loops
• The hexagon Wilson loop and the BDS ansatz for the six-gluon amplitude
• BFKL Pomeron, Reggeized gluons and Bern-Dixon-Smirnov amplitudes
• Superconformal symmetry and two-loop amplitudes in planar N=4 super Yang-Mills
• The massless higher-loop two-point function
• A Symbol of Uniqueness: The Cluster Bootstrap for the 3-Loop MHV Heptagon
• Heptagons from the Steinmann Cluster Bootstrap
• Jumpstarting the all-loop S-matrix of planar N=4 super Yang-Mills
• Hexagon functions and the three-loop remainder function
• Space-time S-matrix and Flux tube S-matrix II. Extracting and Matching Data
• The four-loop remainder function and multi-Regge behavior at NNLLA in planar N=4 super-Yang-Mills theory
• Pulling the straps of polygons
• Six-Gluon Amplitudes in Planar $\mathcal{N}=4$ Super-Yang-Mills Theory at Six and Seven Loops
• The S-matrix Bootstrap II: Two Dimensional Amplitudes
• Space-time S-matrix and Flux-tube S-matrix III. The two-particle contributions
• Generalized unitarity for N=4 super-amplitudes
• Cluster adjacency properties of scattering amplitudes
• The Cosmic Galois Group and Extended Steinmann Relations for Planar $\mathcal{N}=4$ SYM Amplitudes
• Multi-Regge kinematics and the moduli space of Riemann spheres with marked points
• Cluster adjacency and the four-loop NMHV heptagon
• Adjoint BFKL at finite coupling: a short-cut from the collinear limit
• The Galois coaction on ${\varphi }^4$ periods
• The Double Pentaladder Integral to All Orders
• Flux Tube S-matrix Bootstrap
• The Origin of the Six-Gluon Amplitude in Planar $\mathcal{N}=4$ SYM
• Cluster adjacency beyond MHV
• BFKL approach and 2->5 MHV amplitude
• How tropical are seven- and eight-particle amplitudes?
• An analytic result for the two-loop seven-point MHV amplitude in N=4 SYM
• Tropical Grassmannians, cluster algebras and scattering amplitudes
• The O(N) S-matrix Monolith
• The Six-Point NMHV amplitude in Maximally Supersymmetric Yang-Mills Theory
• All-order amplitudes at any multiplicity in the multi-Regge limit
• SUSY S-matrix Bootstrap and Friends
• The seven-gluon amplitude in multi-Regge kinematics beyond leading logarithmic accuracy
• The Sklyanin Bracket and Cluster Adjacency at All Multiplicity
• Cluster Algebras and the Subalgebra Constructibility of the Seven-Particle Remainder Function
• The One-Loop One-Mass Hexagon Integral in D=6 Dimensions
• Towards single-valued polylogarithms in two variables for the seven-point remainder function in multi-Regge-kinematics
• Conformally Regulated Direct Integration of the Two-Loop Heptagon Remainder

Cited by in corpus (35)

• Pentagon Functions for Scattering of Five Massless Particles
• Cluster algebras for Feynman integrals
• A Three-Point Form Factor Through Five Loops
• Bootstrapping a Stress-Tensor Form Factor through Eight Loops
• 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
• The symbol and alphabet of two-loop NMHV amplitudes from $\bar{Q}$ equations
• Notes on cluster algebras and some all-loop Feynman integrals
• Locally finite two-loop amplitudes for off-shell multi-photon production in electron-positron annihilation
• Truncated cluster algebras and Feynman integrals with algebraic letters
• Bootstrapping $\mathcal{N}=4$ sYM correlators using integrability
• Schubert Problems, Positivity and Symbol Letters
• Bootstrapping elliptic Feynman integrals using Schubert analysis
• Feynman Integrals and Scattering Amplitudes from Wilson Loops
• Bootstrapping a Two-Loop Four-Point Form Factor
• The Two-Loop Remainder Function for Eight and Nine Particles
• Intersection theory rules symbology
• An Infinite Family of Elliptic Ladder Integrals
• 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
• The two-loop eight-point amplitude in ABJM theory
• Bootstrapping octagons in reduced kinematics from $A_2$ cluster algebras
• An Origin Story for Amplitudes
• The dimension-shift conjecture for one-loop amplitudes
• A nice two-loop next-to-next-to-MHV amplitude in $\mathcal{N}=4$ super-Yang-Mills
• A Two-Loop Four-Point Form Factor at Function Level
• Heptagon Functions and Seven-Gluon Amplitudes in Multi-Regge Kinematics
• Antipodal Symmetry of Two-Loop MHV Amplitudes
• Deciphering the Maximal Transcendentality Principle via Bootstrap
• Exploring Reggeon bound states in strongly-coupled $\mathcal{N}=4$ super Yang-Mills
• Analytic Four-Point Lightlike Form Factors and OPE of Null-Wrapped Polygons
• Integrands of Less-Supersymmetric Yang-Mills at One Loop
• Hexagon Bootstrap in the Double Scaling Limit