Magic identities for conformal four-point integrals

Abstract

We propose an iterative procedure for constructing classes of off-shell four-point conformal integrals which are identical. The proof of the identity is based on the conformal properties of a subintegral common for the whole class. The simplest example are the so-called `triple scalar box' and `tennis court' integrals. In this case we also give an independent proof using the method of Mellin--Barnes representation which can be applied in a similar way for general off-shell Feynman integrals.

13 pages, 12 figures. New proof included with neater discussion of contact terms. Typo corrected

Cited by in corpus (342)

• New Relations for Gauge-Theory Amplitudes
• Dual superconformal symmetry of scattering amplitudes in N=4 super-Yang-Mills theory
• Conformal properties of four-gluon planar amplitudes and Wilson loops
• The Amplituhedron
• 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
• Local Integrals for Planar Scattering Amplitudes
• Conformal Ward identities for Wilson loops and a test of the duality with gluon amplitudes
• On planar gluon amplitudes/Wilson loops duality
• Yangian symmetry of scattering amplitudes in N=4 super Yang-Mills theory
• Comments on gluon scattering amplitudes via AdS/CFT
• Hexagon Wilson loop = six-gluon MHV amplitude
• A Duality For The S Matrix
• The Two-Loop Six-Gluon MHV Amplitude in Maximally Supersymmetric Yang-Mills Theory
• Fermionic T-Duality, Dual Superconformal Symmetry, and the Amplitude/Wilson Loop Connection
• Bootstrapping the three-loop hexagon
• Maximally Supersymmetric Planar Yang-Mills Amplitudes at Five Loops
• Motivic Amplitudes and Cluster Coordinates
• Dual Superconformal Symmetry from AdS5 x S5 Superstring Integrability
• A note on dual superconformal symmetry of the N=4 super Yang-Mills S-matrix
• Bootstrapping a Five-Loop Amplitude Using Steinmann Relations
• Simplifying Multiloop Integrands and Ultraviolet Divergences of Gauge Theory and Gravity Amplitudes
• Y-system for Scattering Amplitudes
• Towards a Basis for Planar Two-Loop Integrals
• Scattering into the fifth dimension of N=4 super Yang-Mills
• Five-Point Amplitudes in N=4 Super-Yang-Mills Theory and N=8 Supergravity
• Analytic result for the two-loop six-point NMHV amplitude in N=4 super Yang-Mills theory
• The Grassmannian Origin Of Dual Superconformal Invariance
• Bootstrapping an NMHV amplitude through three loops
• Scattering Amplitudes, Wilson Loops and the String/Gauge Theory Correspondence
• The hexagon Wilson loop and the BDS ansatz for the six-gluon amplitude
• Manifest Ultraviolet Behavior for the Three-Loop Four-Point Amplitude of N=8 Supergravity
• A Symbol of Uniqueness: The Cluster Bootstrap for the 3-Loop MHV Heptagon
• The Bulk Dual of SYK: Cubic Couplings
• Notes on the scattering amplitude / Wilson loop duality
• Heptagons from the Steinmann Cluster Bootstrap
• On the Resolution of Singularities of Multiple Mellin-Barnes Integrals
• The S-Matrix in Twistor Space
• Jumpstarting the all-loop S-matrix of planar N=4 super Yang-Mills
• Hexagon functions and the three-loop remainder function
• The four-loop remainder function and multi-Regge behavior at NNLLA in planar N=4 super-Yang-Mills theory
• Scalar 3-point functions in CFT: renormalisation, beta functions and anomalies
• All tree-level amplitudes in N=4 SYM
• Unification of Residues and Grassmannian Dualities
• The super-correlator/super-amplitude duality: Part I
• N=4 supersymmetric Yang Mills scattering amplitudes at high energies: the Regge cut contribution
• From correlation functions to scattering amplitudes
• The Complete Four-Loop Four-Point Amplitude in N=4 Super-Yang-Mills Theory
• Leading singularities and off-shell conformal integrals
• 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
• Basics of Generalized Unitarity
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• One-loop four-point amplitudes in pure and matter-coupled N <= 4 supergravity
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• Exacting N=4 Superconformal Symmetry
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• Gluing Ladders into Fishnets
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• Analytic result for a two-loop five-particle amplitude
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• Spinor Helicity and Dual Conformal Symmetry in Ten Dimensions
• The three-loop form factor in N=4 super Yang-Mills
• On T-Duality and Integrability for Strings on AdS Backgrounds
• Generic multiloop methods and application to N=4 super-Yang-Mills
• New differential equations for on-shell loop integrals
• More on the duality correlators/amplitudes
• Scattering in Mass-Deformed N>=4 Chern-Simons Models
• Diagrammatic Hopf algebra of cut Feynman integrals: the one-loop case
• Prescriptive Unitarity
• A Note on Polytopes for Scattering Amplitudes
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• Symmetries and analytic properties of scattering amplitudes in N=4 SYM theory
• Dual Superconformal Symmetry of N=6 Chern-Simons Theory
• Bipartite Field Theories: from D-Brane Probes to Scattering Amplitudes
• Dual Conformal Properties of Six-Dimensional Maximal Super Yang-Mills Amplitudes
• The two-loop five-point amplitude in $\mathcal{N}=8$ supergravity
• Solution to the Ward Identities for Superamplitudes
• The super-correlator/super-amplitude duality: Part II
• Numerical Evaluation of Tensor Feynman Integrals in Euclidean Kinematics
• The Soft-Collinear Bootstrap: N=4 Yang-Mills Amplitudes at Six and Seven Loops
• Integrability of scattering amplitudes in N=4 SUSY
• Superconformal invariants for scattering amplitudes in N=4 SYM theory
• Energy-energy correlations at next-to-next-to-leading order
• Are scattering amplitudes dual to super Wilson loops?
• D = 5 maximally supersymmetric Yang-Mills theory diverges at six loops
• N >= 4 Supergravity Amplitudes from Gauge Theory at One Loop
• Leading Singularities of the Two-Loop Six-Particle MHV Amplitude
• More loops and legs in Higgs-regulated N=4 SYM amplitudes
• Local Integrand Representations of All Two-Loop Amplitudes in Planar SYM
• Perturbation Theory at Eight Loops: Novel Structures and the Breakdown of Manifest Conformality
• All three-loop four-point correlators of half-BPS operators in planar N=4 SYM
• Conformal algebra: R-matrix and star-triangle relation
• On-shell superamplitudes in N<4 SYM
• Double copy structure and the flat space limit of conformal correlators in even dimensions
• The Double Pentaladder Integral to All Orders
• Dualities for Loop Amplitudes of N=6 Chern-Simons Matter Theory
• Rationalizing Loop Integration
• Mellin Amplitudes for Dual Conformal Integrals
• Ten dimensional symmetry of N=4 SYM correlators
• Conformal Anomaly for Amplitudes in N=6 Superconformal Chern-Simons Theory
• Scattering Amplitudes/Wilson Loop Duality In ABJM Theory
• Simple loop integrals and amplitudes in N=4 SYM
• On Integrable Backgrounds Self-dual under Fermionic T-duality
• Yangian-invariant fishnet integrals in 2 dimensions as volumes of Calabi-Yau varieties
• The one-loop six-dimensional hexagon integral and its relation to MHV amplitudes in N=4 SYM
• Hexagon Wilson Loop OPE and Harmonic Polylogarithms
• Eliminating spurious poles from gauge-theoretic amplitudes
• Conformal Twists, Yang-Baxter $\sigma$-models & Holographic Noncommutativity
• How tropical are seven- and eight-particle amplitudes?
• Three-Loop Leading Singularities and BDS Ansatz for Five Particles
• Three Point Energy Correlators in the Collinear Limit: Symmetries, Dualities and Analytic Results
• On the Structure of Supersymmetric Sums in Multi-Loop Unitarity Cuts
• Gravity On-shell Diagrams
• Correlators of supersymmetric Wilson-loops, protected operators and matrix models in N=4 SYM
• Amplitudes meet Cosmology: A (Scalar) Primer
• The Amplituhedron from Momentum Twistor Diagrams
• Four-point Amplitudes in N=8 Supergravity and Wilson Loops
• Tree-Level Amplitudes in N=8 Supergravity
• New Dual Conformally Invariant Off-Shell Integrals
• Lifting Heptagon Symbols to Functions
• On Yangian Symmetry in Planar N=4 SYM
• The Classical Exchange Algebra of AdS5 x S5 String Theory
• Star Integrals, Convolutions and Simplices
• Multiple Series Representations of $N$-fold Mellin-Barnes Integrals
• Quantum mechanics of null polygonal Wilson loops
• Dual Conformal Symmetry, Integration-by-Parts Reduction, Differential Equations and the Nonplanar Sector
• Mandelstam cuts and light-like Wilson loops in N=4 SUSY
• Two-loop Octagons, Algebraic Letters and $\bar{Q}$ Equations
• The Six-Point NMHV amplitude in Maximally Supersymmetric Yang-Mills Theory
• Unraveling L_{n,k}: Grassmannian Kinematics
• Four-graviton scattering to three loops in $\mathcal{N}=8$ supergravity
• Generalised ladders and single-valued polylogarithms
• Dual conformal symmetry of 1-loop NMHV amplitudes in N=4 SYM theory
• Dual conformal symmetry at loop level; massive regularization
• From the Holomorphic Wilson Loop to `d log' Loop-Integrands of Super-Yang-Mills Amplitudes
• From dlogs to dilogs; the super Yang-Mills MHV amplitude revisited
• MHV diagrams in twistor space and the twistor action
• On Some Hypergeometric Solutions of the Conformal Ward Identities of Scalar 4-point Functions in Momentum Space
• A First Course on Twistors, Integrability and Gluon Scattering Amplitudes
• On the structure of scattering amplitudes in N=4 super Yang-Mills and N=8 supergravity
• The massless hexagon integral in D = 6 dimensions
• 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-loop singularities of scattering amplitudes in massless planar theories
• Duality between Wilson loops and gluon amplitudes
• One-Loop Amplitudes in N=4 Super Yang-Mills and Anomalous Dual Conformal Symmetry
• Nonperturbative Negative Geometries: Amplitudes at Strong Coupling and the Amplituhedron
• On-Shell Diagrams for N = 8 Supergravity Amplitudes
• Logarithmic forms and differential equations for Feynman integrals
• Singularities of eight- and nine-particle amplitudes from cluster algebras and tropical geometry
• T-Duality, Dual Conformal Symmetry and Integrability for Strings on AdS_5 x S^5
• The S Matrix of 6D Super Yang-Mills and Maximal Supergravity from Rational Maps
• Bonus Yangian Symmetry for the Planar S-Matrix of N=4 Super Yang-Mills
• Symmetries of the N=4 SYM S-matrix
• New structures in scattering amplitudes: a review
• Dual Conformal Structure Beyond the Planar Limit
• The one-loop six-dimensional hexagon integral with three massive corners
• Scattering in ABJ theories
• Implications of nonplanar dual conformal symmetry
• All-Multiplicity Non-Planar MHV Amplitudes in sYM at Two Loops
• Boundaries of Amplituhedra and NMHV Symbol Alphabets at Two Loops
• Prescriptive Unitarity for Non-Planar Six-Particle Amplitudes at Two Loops
• Antipodal Self-Duality for a Four-Particle Form Factor
• Monodromy--like Relations for Finite Loop Amplitudes
• Solvable Relativistic Hydrogenlike System in Supersymmetric Yang-Mills Theory
• All five-loop planar four-point functions of half-BPS operators in $\mathcal{N}=4$ SYM
• The symbol and alphabet of two-loop NMHV amplitudes from $\bar{Q}$ equations
• Gluon Reggeization in Yang-Mills Theories
• Truncated cluster algebras and Feynman integrals with algebraic letters
• 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
• Exponential suppression with four legs and an infinity of loops
• Manifestly Dual-Conformal Loop Integration
• The role of leading twist operators in the Regge and Lorentzian OPE limits
• Constructing $\mathcal{N}=4$ Coulomb Branch Superamplitudes
• Wilson loop OPE, analytic continuation and multi-Regge limit
• Amplitudes at Weak Coupling as Polytopes in AdS_5
• Traintrack Calabi-Yaus from Twistor Geometry
• All-Mass $n$-gon Integrals in $n$ Dimensions
• Conformal anomaly of generalized form factors and finite loop integrals
• The six-point remainder function to all loop orders in the multi-Regge limit
• Exact Superconformal and Yangian Symmetry of Scattering Amplitudes
• Hepp's bound for Feynman graphs and matroids
• Amplitudes of 3d Yang Mills Theory
• Review of AdS/CFT Integrability, Chapter V.2: Dual Superconformal Symmetry
• Dual conformal invariance for form factors
• Three-loop octagons and n-gons in maximally supersymmetric Yang-Mills theory
• Full-color three-loop three-point form factors in N=4 SYM
• Review of AdS/CFT Integrability, Chapter II.3: Sigma Model, Gauge Fixing
• Cluster Algebras and the Subalgebra Constructibility of the Seven-Particle Remainder Function
• Building Bases of Loop Integrands
• Correlation functions of local composite operators from generalized unitarity
• An all order identity between ABJM and N=4 SYM four-point amplitudes
• Celestial Superamplitudes
• The One-Loop One-Mass Hexagon Integral in D=6 Dimensions
• Analytic properties of high energy production amplitudes in N=4 SUSY
• Light-like Wilson loops in ABJM and maximal transcendentality
• Graphical functions in even dimensions
• Form factor recursion relations at loop level
• On the ABJM four-point amplitude at three loops and BDS exponentiation
• A study of Feynman integrals with uniform transcendental weights and the symbology from dual conformal symmetry
• Yangian Bootstrap for Massive Feynman Integrals
• From correlation functions to event shapes in QCD
• Feynman Integrals and Scattering Amplitudes from Wilson Loops
• Integrable spin chains and scattering amplitudes
• The SAGEX Review on Scattering Amplitudes, Chapter 7: Positive Geometry of Scattering Amplitudes
• Bootstrapping elliptic Feynman integrals using Schubert analysis
• What can we learn about QCD and collider physics from N=4 super Yang-Mills?
• The Wilson-loop $d\log$ representation for Feynman integrals
• Non-planarity through unitarity in ABJM
• A Requiem for $Ad{S}_4\times \mathbb{C}{P}^3$ Fermionic self-T-duality
• Positivity of hexagon perturbation theory
• The Two-Loop Remainder Function for Eight and Nine Particles
• A super MHV vertex expansion for N=4 SYM theory
• Konishi OPE coefficient at the five loop order
• The two-loop MHV amplitudes in N=4 supersymmetric Yang-Mills theory
• The Two-Loop Symbol of all Multi-Regge Regions
• Cross-Order Integral Relations from Maximal Cuts
• T-functions and multi-gluon scattering amplitudes
• Symbols of One-Loop Integrals From Mixed Tate Motives
• IR divergences and Regge limits of subleading-color contributions to the four-gluon amplitude in N=4 SYM Theory
• Review of AdS/CFT Integrability, Chapter V.1: Scattering Amplitudes - a Brief Introduction
• Hedgehog Bases for A_n Cluster Polylogarithms and An Application to Six-Point Amplitudes
• Scattering Amplitudes and Toric Geometry
• New four-dimensional integrals by Mellin-Barnes transform
• The Three-loop MHV Octagon from $\bar{Q}$ equations
• Null-polygonal minimal surfaces in AdS_4 from perturbed W minimal models
• Recursive computation of Feynman periods
• Operator approach to analytical evaluation of Feynman diagrams
• On-shell diagrams and the geometry of planar N < 4 SYM theories
• On-shell diagrammatics and the perturbative structure of planar gauge theories
• An Infinite Family of Elliptic Ladder Integrals
• The Effective Action of N=8 Supergravity
• Infrared properties of five-point massive amplitudes in N=4 SYM on the Coulomb branch
• Yangian Ward Identities for Fishnet Four-Point Integrals
• Dual-Conformal Regularization of Infrared Loop Divergences and the Chiral Box Expansion
• ABJM amplitudes and WL at finite N
• On Fermionic T-duality of Sigma modes on AdS backgrounds
• The Dual Conformal Box Integral in Minkowski Space
• Fourier transforms of UD integrals
• Gravity loop integrands from the ultraviolet
• Comments on all-loop constraints for scattering amplitudes and Feynman integrals
• Minimal surfaces in AdS space and Integrable systems
• Real fermionic symmetry in type II supergravity
• Transformations of triangle ladder diagrams
• Positivity Sectors and the Amplituhedron
• Dual conformal symmetry on the light-cone
• On the Classification of Residues of the Grassmannian
• The SAGEX Review on Scattering Amplitudes, Chapter 15: The Multi-Regge Limit
• Cyclic Mario Worlds -- Color-Decomposition for One-Loop QCD
• 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
• Multi-Particle Amplitudes from the Four-Point Correlator in Planar N=4 SYM
• Scattering amplitudes at strong coupling for 4K gluons
• Higher point MHV amplitudes in N=4 Supersymmetric Yang-Mills Theory
• Yangian in the Twistor String
• Conformally Regulated Direct Integration of the Two-Loop Heptagon Remainder
• Non-planar BCFW Grassmannian Geometries
• A Surprise in the Amplitude/Wilson Loop Duality
• Review of AdS/CFT Integrability, Chapter V.3: Scattering Amplitudes at Strong Coupling
• A Note on Dual MHV Diagrams in N=4 SYM
• $\mathcal{N}=7$ On-Shell Diagrams and Supergravity Amplitudes in Momentum Twistor Space
• Further results for the two-loop Lcc vertex in the Landau gauge
• A Connection between Twistors and Superstring Sigma Models on Coset Superspaces
• Conformal Four-Point Integrals: Recursive Structure, Toda Equations and Double Copy
• Notes on conformal integrals: Coulomb branch amplitudes, magic identities and bootstrap
• An Origin Story for Amplitudes
• Poles At Infinity in On-shell Diagrams
• Cluster algebras in scattering amplitudes with special 2D kinematics
• Anatomy of the Amplituhedron
• 6d Dual Conformal Symmetry and Minimal Volumes in AdS
• Supersymmetric Yang-Mills and Supergravity Amplitudes at One Loop
• All orders results for self-crossing Wilson loops mimicking double parton scattering
• Scattering amplitudes on the Coulomb branch of N=4 super Yang-Mills
• On All-loop Integrands of Scattering Amplitudes in Planar N=4 SYM
• A Two-Loop Four-Point Form Factor at Function Level
• N=8 SYM vs. N=6 Chern-Simons: Four-point amplitudes at two-loops
• Permutation Relations of Yangian Invariants, Unitarity Cuts, and Scattering Amplitudes
• Twistor and Polytope Interpretations for Subleading Color One-Loop Amplitudes
• Polylogarithm identities, cluster algebras and the N=4 supersymmetric theory
• S-duality invariant perturbation theory improved by holography
• The Steinmann Cluster Bootstrap for N=4 Super Yang-Mills Amplitudes
• Scattering Amplitudes of Massive N=2 Gauge Theories in Three Dimensions
• Logarithmic Singularities and Maximally Supersymmetric Amplitudes
• Illustrations of Integrand-Basis Building at Two Loops
• Dual conformal invariance in the Regge limit
• Five-loop massless propagator integrals
• Interactions as conformal intertwiners in 4D QFT
• Antipodal self-duality of square fishnet graphs
• D=3 N=6 superconformal symmetry of AdS_4 x CP^3 superstring
• Twistors, strings and supersymmetric gauge theories
• Quantum Wronskian approach to six-point gluon scattering amplitudes at strong coupling
• All Two-Loop, Color-Dressed, Six-Point Amplitude Integrands in sYM
• ABJM flux-tube and scattering amplitudes
• The massless supersymmetric ladder with L rungs
• Differential equations for multi-loop integrals and two-dimensional kinematics
• Minimally helicity violating, maximally simple scalar amplitudes in N=4 SYM
• Super Yangian of superstring on AdS5xS5 revisited
• Locally Finite Observables in sYM
• Supersums for all supersymmetric amplitudes
• Yangian Symmetry in Five Dimensions
• Perturbative Methods for Superconformal Quantum Field Theories in String - Gauge Theory Dualities
• Commuting Conformal and Dual Conformal Symmetries in the Regge limit
• Analytic Four-Point Lightlike Form Factors and OPE of Null-Wrapped Polygons
• Wilson Loops and Integrability
• A tale of two Regge limits
• MHV amplitudes, Wilson loops and the BDS ansatz in the superconformal N=2 QCD
• Symplectic Grassmannians, dual conformal symmetry and 4-point amplitudes in 6D
• Eikonal Approach to N=4 SYM Regge Amplitudes in the AdS/CFT Correspondence
• Relating diagrammatic expansion with conformal correlator expansion
• Studying Quantum Field Theory
• Integrands of Less-Supersymmetric Yang-Mills at One Loop
• On Solving Dual Conformal Integrals in Coulomb-branch Amplitudes and Their Periods
• A twistor string for the ABJ(M) theory
• Yangian-Like Symmetry of Probe-Brane Actions
• Duality between Wilson Loops and Scattering Amplitudes
• The canonical form has got baubles
• Twistor Parametrization of Locally BPS Super-Wilson Loops
• The Stratification of Rigidity
• On-shell Methods for Form Factors in N=4 SYM and Their Applications
• A note on scattering amplitudes on the moduli space of ABJM
• On the Integrable Structure of Super Yang-Mills Scattering Amplitudes
• Three-loop color-kinematics duality: A 24-dimensional solution space induced by new generalized gauge transformations