Bootstrapping the three-loop hexagon

Abstract

We consider the hexagonal Wilson loop dual to the six-point MHV amplitude in planar N=4 super Yang-Mills theory. We apply constraints from the operator product expansion in the near-collinear limit to the symbol of the remainder function at three loops. Using these constraints, and assuming a natural ansatz for the symbol's entries, we determine the symbol up to just two undetermined constants. In the multi-Regge limit, both constants drop out from the symbol, enabling us to make a non-trivial confirmation of the BFKL prediction for the leading-log approximation. This result provides a strong consistency check of both our ansatz for the symbol and the duality between Wilson loops and MHV amplitudes. Furthermore, we predict the form of the full three-loop remainder function in the multi-Regge limit, beyond the leading-log approximation, up to a few constants representing terms not detected by the symbol. Our results confirm an all-loop prediction for the real part of the remainder function in multi-Regge 3-->3 scattering. In the multi-Regge limit, our result for the remainder function can be expressed entirely in terms of classical polylogarithms. For generic six-point kinematics other functions are required.

36 pages, 1 figure, plus 8 ancillary files containing symbols of functions; v2 minor typo corrections

References in corpus (23)

• Gluon scattering amplitudes at strong coupling
• Classical Polylogarithms for Amplitudes and Wilson Loops
• Conformal properties of four-gluon planar amplitudes and Wilson loops
• MHV Amplitudes in N=4 Super Yang-Mills 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
• Maximally Supersymmetric Planar Yang-Mills Amplitudes at Five Loops
• Dual Superconformal Symmetry from AdS5 x S5 Superstring Integrability
• An Operator Product Expansion for Polygonal null Wilson Loops
• Scattering Amplitudes, Wilson Loops and the String/Gauge Theory Correspondence
• The hexagon Wilson loop and the BDS ansatz for the six-gluon amplitude
• BFKL Pomeron, Reggeized gluons and Bern-Dixon-Smirnov amplitudes
• Spinor Helicity and Dual Conformal Symmetry in Ten Dimensions
• New differential equations for on-shell loop integrals
• Dual Conformal Properties of Six-Dimensional Maximal Super Yang-Mills Amplitudes
• BFKL approach and six-particle MHV amplitude in N=4 super Yang-Mills
• Iterated amplitudes in the high-energy limit
• Analytic Results for MHV Wilson Loops
• Mandelstam cuts and light-like Wilson loops in N=4 SUSY
• Collinear and Regge behavior of 2 -> 4 MHV amplitude in N = 4 super Yang-Mills theory

Cited by in corpus (182)

• From polygons and symbols to polylogarithmic functions
• Space-time S-matrix and Flux-tube S-matrix at Finite Coupling
• Hopf algebras, coproducts and symbols: an application to Higgs boson amplitudes
• Motivic Amplitudes and Cluster Coordinates
• Bootstrapping a Five-Loop Amplitude Using Steinmann Relations
• Bootstrapping rapidity anomalous dimension for transverse-momentum resummation
• Analytic result for the two-loop six-point NMHV amplitude in N=4 super Yang-Mills theory
• Bootstrapping an NMHV amplitude through three loops
• Uniqueness of two-loop master contours
• 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
• 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
• Quantum Gravity from Conformal Field Theory
• Collider Physics at the Precision Frontier
• The two-loop five-point amplitude in $\mathcal{N}=4$ super-Yang-Mills theory
• Single-valued harmonic polylogarithms and the multi-Regge limit
• Six-Gluon Amplitudes in Planar $\mathcal{N}=4$ Super-Yang-Mills Theory at Six and Seven Loops
• Analytic two-loop form factors in N=4 SYM
• Space-time S-matrix and Flux-tube S-matrix III. The two-particle contributions
• Elliptic Feynman integrals and pure functions
• N >= 4 Supergravity Amplitudes from Gauge Theory at Two Loops
• Pentagon Functions for Scattering of Five Massless Particles
• 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
• Space-time S-matrix and Flux-tube S-matrix IV. Gluons and Fusion
• Gluing Ladders into Fishnets
• Simplicity from Recoil: The Three-Loop Soft Function and Factorization for the Energy-Energy Correlation
• 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
• Elliptic polylogarithms and Feynman parameter integrals
• Two-Loop Integrals for Planar Five-Point One-Mass Processes
• BFKL equation for the adjoint representation of the gauge group in the next-to-leading approximation at N=4 SUSY
• Bootstrapping pentagon functions
• Prescriptive Unitarity
• OPE for all Helicity Amplitudes
• Cluster adjacency and the four-loop NMHV heptagon
• Cluster algebras for Feynman integrals
• The two-loop five-point amplitude in $\mathcal{N}=8$ supergravity
• The last of the simple remainders
• From multiple unitarity cuts to the coproduct of Feynman integrals
• Local Integrand Representations of All Two-Loop Amplitudes in Planar SYM
• Subleading Power Resummation of Rapidity Logarithms: The Energy-Energy Correlator in $\mathcal{N}=4$ SYM
• The Double Pentaladder Integral to All Orders
• A Three-Point Form Factor Through Five Loops
• The Origin of the Six-Gluon Amplitude in Planar $\mathcal{N}=4$ SYM
• Cluster adjacency beyond MHV
• Rationalizing Loop Integration
• Mellin Amplitudes for Dual Conformal Integrals
• Hexagonal Wilson Loops in Planar $\mathcal{N}=4$ SYM Theory at Finite Coupling
• OPE for all Helicity Amplitudes II. Form Factors and Data Analysis
• Nonsinglet pentagons and NMHV amplitudes
• Sequential Discontinuities of Feynman Integrals and the Monodromy Group
• Subleading Power Factorization with Radiative Functions
• Light-quark two-loop corrections to heavy-quark pair production in the gluon fusion channel
• Integration-by-parts reductions of Feynman integrals using Singular and GPI-Space
• Y-system for form factors at strong coupling in AdS5 and with multi-operator insertions in AdS3
• Hexagon Wilson Loop OPE and Harmonic Polylogarithms
• How tropical are seven- and eight-particle amplitudes?
• BFKL approach and 2->5 MHV amplitude
• Three Point Energy Correlators in the Collinear Limit: Symmetries, Dualities and Analytic Results
• Landau Singularities and Symbology: One- and Two-loop MHV Amplitudes in SYM Theory
• Landau Singularities from the Amplituhedron
• Multi-Loop Positivity of the Planar $\mathcal{N}=4$ SYM Six-Point Amplitude
• Bootstrapping a Stress-Tensor Form Factor through Eight Loops
• An analytic result for the two-loop seven-point MHV amplitude in N=4 SYM
• Lifting Heptagon Symbols to Functions
• Lightcone Bootstrap at higher points
• Form factors and scattering amplitudes in N=4 SYM in dimensional and massive regularizations
• Star Integrals, Convolutions and Simplices
• Two-loop Octagons, Algebraic Letters and $\bar{Q}$ Equations
• Generalised ladders and single-valued polylogarithms
• The colour octet representation of the non-forward BFKL Green function
• From the Holomorphic Wilson Loop to `d log' Loop-Integrands of Super-Yang-Mills Amplitudes
• All-loop singularities of scattering amplitudes in massless planar theories
• Soft gluon emission at two loops in full color
• Stratifying On-Shell Cluster Varieties: the Geometry of Non-Planar On-Shell Diagrams
• Bootstrapping two-loop Feynman integrals for planar N=4 sYM
• All-order amplitudes at any multiplicity in the multi-Regge limit
• Logarithmic forms and differential equations for Feynman integrals
• An Operator Product Expansion for Form Factors
• Superstring amplitudes and the associator
• Singularities of eight- and nine-particle amplitudes from cluster algebras and tropical geometry
• The Multi-Regge limit of NMHV Amplitudes in N=4 SYM Theory
• Massive planar and non-planar double box integrals for light Nf contributions to gg->tt
• Analytic structure of the $n=7$ scattering amplitude in $\mathcal{N}=4$ SYM theory at multi-Regge kinematics: Conformal Regge pole contribution
• Systematics of the cusp anomalous dimension
• Hexagon OPE Resummation and Multi-Regge Kinematics
• The Principle of Maximal Transcendentality and the Four-Loop Collinear Anomalous Dimension
• Antipodal Self-Duality for a Four-Particle Form Factor
• Boundaries of Amplituhedra and NMHV Symbol Alphabets at Two Loops
• Tropical fans, scattering equations and amplitudes
• The symbol and alphabet of two-loop NMHV amplitudes from $\bar{Q}$ equations
• Looking for a bulk point
• Notes on cluster algebras and some all-loop Feynman integrals
• Truncated cluster algebras and Feynman integrals with algebraic letters
• The Complete Two-Loop Integrated Jet Thrust Distribution In Soft-Collinear Effective Theory
• Multi-Regge Limit of the Two-Loop Five-Point Amplitudes in $\mathcal{N}=4$ Super Yang-Mills and $\mathcal{N}=8$ Supergravity
• The seven-gluon amplitude in multi-Regge kinematics beyond leading logarithmic accuracy
• Manifestly Dual-Conformal Loop Integration
• The role of leading twist operators in the Regge and Lorentzian OPE limits
• Wilson loop OPE, analytic continuation and multi-Regge limit
• Amplitudes from superconformal Ward identities
• Symbol Alphabets from Tensor Diagrams
• Symbology for elliptic multiple polylogarithms and the symbol prime
• The six-point remainder function to all loop orders in the multi-Regge limit
• From Polygon Wilson Loops to Spin Chains and Back
• Three-loop octagons and n-gons in maximally supersymmetric Yang-Mills theory
• Multi-Regge Limit of the n-Gluon Bubble Ansatz
• All Two-Loop MHV Amplitudes in Multi-Regge Kinematics From Applied Symbology
• Wilson Loops @ 3-Loops in Special Kinematics
• Schubert Problems, Positivity and Symbol Letters
• Fermionic Glauber Operators and Quark Reggeization
• Bootstrapping six-gluon scattering in planar $\mathcal{N}=4$ super-Yang-Mills theory
• The Wilson Loop -- Large Spin OPE Dictionary
• A study of Feynman integrals with uniform transcendental weights and the symbology from dual conformal symmetry
• What can we learn about QCD and collider physics from N=4 super Yang-Mills?
• A Note on Letters of Yangian Invariants
• OPE for null Wilson loops and open spin chains
• Feynman Integrals and Scattering Amplitudes from Wilson Loops
• Bootstrapping elliptic Feynman integrals using Schubert analysis
• Bootstrapping a Two-Loop Four-Point Form Factor
• The Wilson-loop $d\log$ representation for Feynman integrals
• Intersection theory rules symbology
• Heptagon Amplitude in the Multi-Regge Regime
• A first look at the function space for planar two-loop six-particle Feynman integrals
• The Two-Loop Symbol of all Multi-Regge Regions
• Cluster Polylogarithms for Scattering Amplitudes
• Hedgehog Bases for A_n Cluster Polylogarithms and An Application to Six-Point Amplitudes
• The Three-loop MHV Octagon from $\bar{Q}$ equations
• Integrated negative geometries in ABJM
• The Full-Color Two-Loop Four-Gluon Amplitude in $\mathcal{N}=2$ Super-QCD
• An Infinite Family of Elliptic Ladder Integrals
• Analytic structure of the $n=7$ scattering amplitude in $\mathcal{N}=4$ SYM theory in multi-Regge kinematics: Conformal Regge cut contribution
• Cosmic Wheels: From integrability to the Galois coaction
• Transforming the Bootstrap: Using Transformers to Compute Scattering Amplitudes in Planar N = 4 Super Yang-Mills Theory
• The recursive structure of Baikov representations I: generics and application to symbology
• Comments on all-loop constraints for scattering amplitudes and Feynman integrals
• Wilson Loop Duality and OPE for Super Form Factors of Half-BPS Operators
• Gravity loop integrands from the ultraviolet
• The SAGEX Review on Scattering Amplitudes, Chapter 15: The Multi-Regge Limit
• All two-loop MHV remainder functions in multi-Regge kinematics
• The SAGEX Review on Scattering Amplitudes, Chapter 5: Analytic Bootstraps for Scattering Amplitudes and Beyond
• The two-loop eight-point amplitude in ABJM theory
• Uplifting Amplitudes in Special Kinematics
• Hexagon remainder function in the limit of self-crossing up to three loops
• The four-loop six-gluon NMHV ratio function
• Non-Planar On-Shell Diagrams
• Moebius invariant BFKL equation for the adjoint representation in N=4 SUSY
• Bootstrapping octagons in reduced kinematics from $A_2$ cluster algebras
• Regge meets collinear in strongly-coupled $\mathcal{N}=4$ super Yang-Mills
• Amplitudes from anomalous superconformal symmetry
• A nice two-loop next-to-next-to-MHV amplitude in $\mathcal{N}=4$ super-Yang-Mills
• Systematics of the Multi-Regge Three-Loop Symbol
• All orders results for self-crossing Wilson loops mimicking double parton scattering
• Polylogarithm identities, cluster algebras and the N=4 supersymmetric theory
• A Two-Loop Four-Point Form Factor at Function Level
• Gluing Hexagons at Three Loops
• Deciphering the Maximal Transcendentality Principle via Bootstrap
• SymBuild: a package for the computation of integrable symbols in scattering amplitudes
• The Steinmann Cluster Bootstrap for N=4 Super Yang-Mills Amplitudes
• ABJM flux-tube and scattering amplitudes
• The Multi-Regge Limit from the Wilson Loop OPE
• All Two-Loop, Color-Dressed, Six-Point Amplitude Integrands in sYM
• Differential equations for multi-loop integrals and two-dimensional kinematics
• Perturbative Methods for Superconformal Quantum Field Theories in String - Gauge Theory Dualities
• Integrated Unitarity for Scattering Amplitudes
• Exploring Reggeon bound states in strongly-coupled $\mathcal{N}=4$ super Yang-Mills
• Cluster Alphabets from Generalized Worldsheets: A Geometric Approach to Finite Types
• Antenna Operator Product Expansion of ABJ(M) Lightlike Polygon Wilson Loop
• $\mathcal{N}=4$ polygonal Wilson loops: fermions
• Analytic Four-Point Lightlike Form Factors and OPE of Null-Wrapped Polygons
• Quark-Gluon Backscattering in the Regge Limit at One-Loop
• Integrands of Less-Supersymmetric Yang-Mills at One Loop
• Cluster Functions and Scattering Amplitudes for Six and Seven Points
• Hexagon Bootstrap in the Double Scaling Limit
• Universal transcendentality limit of BFKL eigenvalue