New Relations for Gauge-Theory Amplitudes

Abstract

We present an identity satisfied by the kinematic factors of diagrams describing the tree amplitudes of massless gauge theories. This identity is a kinematic analog of the Jacobi identity for color factors. Using this we find new relations between color-ordered partial amplitudes. We discuss applications to multi-loop calculations via the unitarity method. In particular, we illustrate the relations between different contributions to a two-loop four-point QCD amplitude. We also use this identity to reorganize gravity tree amplitudes diagram by diagram, offering new insight into the structure of the KLT relations between gauge and gravity tree amplitudes. This can be used to obtain novel relations similar to the KLT ones. We expect this to be helpful in higher-loop studies of the ultraviolet properties of gravity theories.

40 pages, 7 figures, RevTex, v2 minor corrections

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• The momentum amplituhedron of SYM and ABJM from twistor-string maps
• On Primary Relations at Tree-level in String Theory and Field Theory
• Correlation functions of local composite operators from generalized unitarity
• Weyl double copy and massless free fields in curved spacetimes
• On Cancellations of Ultraviolet Divergences in Supergravity Amplitudes
• Complete N-Point Superstring Disk Amplitude I. Pure Spinor Computation
• Two-loop five-point amplitudes of super Yang-Mills and supergravity in pure spinor superspace
• Scattering Forms, Worldsheet Forms and Amplitudes from Subspaces
• One-loop Parke-Taylor factors for quadratic propagators from massless scattering equations
• Note on New KLT relations
• Structure of Kaluza-Klein Graviton Scattering Amplitudes from Gravitational Equivalence Theorem and Double-Copy
• Effective Field Theories and Cosmological Scattering Equations
• Non-planar one-loop Parke-Taylor factors in the CHY approach for quadratic propagators
• KLT-type relations for QCD and bicolor amplitudes from color-factor symmetry
• What can we learn about QCD and collider physics from N=4 super Yang-Mills?
• Expansion of Einstein-Yang-Mills theory by Differential Operators
• Proof of a new colour decomposition for QCD amplitudes
• Exact results for scattering on ultra-short plane wave backgrounds
• Absence of U(1) Anomalous Superamplitudes in $\mathcal{N}\ge 5$ Supergravities
• Resonant resurgent asymptotics from quantum field theory
• BCJ Identities and $d$-Dimensional Generalized Unitarity
• Scattering of Gravitons and Spinning Massive States from Compact Numerators
• The $L_{\infty }$ structure of gauge theories with matter
• Spinning Black Hole Scattering at $\mathcal{O}(G^3{S}^2)$: Casimir Terms, Radial Action and Hidden Symmetry
• One-loop Amplitudes for N=2 Homogeneous Supergravities
• A Lie bracket for the momentum kernel
• First Look at Quartic-in-Spin Binary Dynamics at Third Post-Minkowskian Order
• Form Factors of Chiral Primary Operators at Two Loops in ABJ(M)
• Two-loop Sudakov Form Factor in ABJM
• Amplitude recursions with an extra marked point
• Kinematic Numerators and a Double-Copy Formula for N = 4 Super-Yang-Mills Residues
• Double copies of fermions as only gravitational interacting matter
• Analytical On-shell Calculation of Low Energy Higher Order Scattering
• Transmuting CHY formulae
• Non-planarity through unitarity in ABJM
• On the low energy limit of one loop photon-graviton amplitudes
• Kinematic numerators from the worldsheet: cubic trees from labelled trees
• Gauge $\times$ Gauge $=$ Gravity on Homogeneous Spaces using Tensor Convolutions
• Binary Dynamics from Worldline QFT for Scalar-QED
• Perturbative expansion of N<8 Supergravity
• Structure of Chern-Simons Scattering Amplitudes from Topological Equivalence Theorem and Double-Copy
• Solutions in Nonlinear Electrodynamics and their double copy regular black holes
• Gravitino Interactions from Yang-Mills Theory
• Multiparticle solutions to Einstein's equations
• A minimal approach to the scattering of physical massless bosons
• Spinor-Helicity Formalism for Massless Fields in AdS${}_4$ III: Contact Four-Point Amplitudes
• Chiral Squaring
• Classifying post-Minkowskian geometries for gravitational waves via loop-by-loop Baikov
• The Classical Double Copy for Half-Maximal Supergravities and T-duality
• Extracting Einstein from the Loop-Level Double-Copy
• Enhanced Soft Limits in de Sitter Space
• Soft Theorems from Effective Field Theory
• The classical double copy in curved spacetimes: Perturbative Yang-Mills from the bi-adjoint scalar
• On the New Factorizations of Yang-Mills Amplitudes
• Revisiting Buchdahl transformations: New static and rotating black holes in vacuum, double copy, and hairy extensions
• Soft-collinear gravity beyond the leading power
• Color-kinematics duality and dimensional reduction for graviton emission in Regge limit
• Tree-Level Color-Kinematics Duality Implies Loop-Level Color-Kinematics Duality up to Counterterms
• Color-kinematics duality, double copy and the unitarity method for higher-derivative QCD and quadratic gravity
• Color-kinematics duality from the Bern-Kosower formalism
• Double copy for Lagrangians at trilinear order
• Soft theorem for the graviton, dilaton and the Kalb-Ramond field in the bosonic string
• Amplituhedra, and Beyond
• The Color-Dual Fates of $F^3$, $R^3$, and $\mathcal{N}=4$ Supergravity
• Virtues of a symmetric-structure double copy
• Off-shell extended graphic rule and the expansion of Berends-Giele currents in Yang-Mills theory
• Manifest colour-kinematics duality and double-copy in the string-based formalism
• Review of AdS/CFT Integrability, Chapter V.1: Scattering Amplitudes - a Brief Introduction
• One-loop amplitudes in Einstein-Yang-Mills from forward limits
• Unified web for expansions of amplitudes
• Gravitational soft theorem from emergent soft gauge symmetries
• Beyond the standard model with six-dimensional spinors
• Local BCJ numerators for ten-dimensional SYM at one loop
• Double-copy towards supergravity inflation with $\alpha$-attractor models
• Frequency-domain approach to self-force in hyperbolic scattering
• Scattering Amplitudes and N-Body Post-Minkowskian Hamiltonians in General Relativity and Beyond
• Spinor-helicity formalism for massive and massless amplitudes in five dimensions
• Double Copy from Tensor Products of Metric BV${}^{■}$-algebras
• Radiative Classical Gravitational Observables at $\mathcal{O}(G^3)$ from Scattering Amplitudes
• Symmetric-group decomposition of SU(N) group-theory constraints on four-, five-, and six-point color-ordered amplitudes
• Towards the Gravituhedron: New Expressions for NMHV Gravity Amplitudes
• Gravitational Dyonic Amplitude at One-Loop and its Inconsistency with the Classical Impulse
• One-Loop Yang-Mills Integrands from Scattering Equations
• Stokes Polytopes and Intersection Theory
• Ambitwistor Strings: Worldsheet Approaches to perturbative Quantum Field Theories
• The Lauricella Functions and Exact String Scattering Amplitudes
• Bosonic sectorized strings and the $(DF{)}^2$ theory
• The Infrared Structure of Exceptional Scalar Theories
• The QED four -- photon amplitudes off-shell: part 1
• Explicit BCJ numerators of nonlinear sigma model
• Uniqueness from locality and BCFW shifts
• Multi-spin soft bootstrap and scalar-vector Galileon
• Electromagnetic Duality and D3-Brane Scattering Amplitudes Beyond Leading Order
• String theory in target space
• Unified Formalism for 6D Superamplitudes Based on a Symplectic Grassmannian
• KLT Factorization of Nonrelativistic String Amplitudes
• The QED four-photon amplitudes off-shell: part 2
• Scattering Amplitudes of Kaluza-Klein Strings and Extended Massive Double-Copy
• The Equivalence Principle in a Quantum World
• Worldline description of a bi-adjoint scalar and the zeroth copy
• Constructing Gravity Amplitudes from Real Soft and Collinear Factorisation
• A Scattering Amplitude for Massive Particles in AdS
• The two-loop six-point amplitude in ABJM theory
• The Structure of Multiloop Amplitudes in Gauge and Gravity Theories
• Regularized Weyl double copy
• Compact QED Tree-Level Amplitudes From Dressed BCFW Recursion Relations
• Effective observables for electromagnetic duality from novel amplitude decomposition
• Quantum corrections to tidal Love number for Schwarzschild black holes
• The Full-Color Two-Loop Four-Gluon Amplitude in $\mathcal{N}=2$ Super-QCD
• Tree-Level Color-Kinematics Duality from Pure Spinor Actions
• The ChPT: top-down and bottom-up
• Page Entropy of Proton System in Deep-Inelastic-Scattering at Small-$x$
• A double copy from twisted (co)homology at genus one
• KLT-Like Behaviour of Inflationary Graviton Correlators
• One-loop SYM-supergravity relation for five-point amplitudes
• ABJM amplitudes and WL at finite N
• Universal expansions of scattering amplitudes for gravitons, gluons and Goldstone particles
• Note on the asymptotic structure of Kerr-Schild form
• Hidden Conformal Symmetry in Tree-Level Graviton Scattering
• All rational one-loop Einstein-Yang-Mills amplitudes at four points
• New relations for graviton-matter amplitudes
• Bootstrap and Amplitudes: A Hike in the Landscape of Quantum Field Theory
• All-multiplicity amplitudes with four massive quarks and identical-helicity gluons
• Color kinematic symmetric (BCJ) numerators in a light-like gauge
• On KLT and SYM-supergravity relations from 5-point 1-loop amplitudes
• A scattering amplitudes approach to hard thermal loops
• BRST Symmetry and the Convolutional Double Copy
• Gravity loop integrands from the ultraviolet
• Unified non-metric (1,0) tensor-Einstein supergravity theories and (4,0) supergravity in six dimensions
• On symmetries and dynamics of exotic supermultiplets
• Gravitons, Bose-symmetry, and Bonus-scaling
• Exploring the Landscape for Soft Theorems of Nonlinear Sigma Models
• Colour-kinematics duality, double copy, and homotopy algebras
• On Positive Geometry and Scattering Forms for Matter Particles
• Symmetrization of Jordan dialgebras
• A Note on Soft Factors for Closed String Scattering
• The Magic Star of Exceptional Periodicity
• Generalized Yang-Mills Theory and Gravity
• The Mile High Magic Pyramid
• Strong Field Scattering of Black Holes: Assessing Resummation Strategies
• Aspects of High Energy Scattering
• Homotopy double copy and the Kawai-Lewellen-Tye relations for the non-abelian and tensor Navier-Stokes equations
• Minimal models of field theories: SDYM and SDGR
• Tree amplitudes of noncommutative U(N) Yang-Mills Theory
• Color structures and permutations
• Higher point MHV amplitudes in N=4 Supersymmetric Yang-Mills Theory
• Hidden Conformal Invariance of Scalar Effective Field Theories
• Non-planar cusp and transcendental anomalous dimension at four loops in N=4 supersymmetric Yang-Mills theory
• Cyclic Mario Worlds -- Color-Decomposition for One-Loop QCD
• Uniqueness Criteria for the Virasoro-Shapiro Amplitude
• Soft collinear effective theory for gravity
• The $pp\to W(\to l\nu )+\gamma$ process at next-to-next-to-leading order
• String Scattering Amplitudes and Deformed Cubic String Field Theory
• Asymptotic freedom from the two loop term of the $\beta$-function in a cubic theory
• Kerr-Schild double copy for Kundt spacetimes of any dimension
• Flat-space structure of gluon and graviton in AdS
• One-loop Bern-Carrasco-Johansson Numerators on Quadratic Propagators from the Worldsheet
• BCJ Relations for One-Loop QCD Integral Coefficients
• Color-Kinematics Duality for Sudakov Form Factor in Non-Supersymmetric Pure Yang-Mills Theory
• Kleiss-Kuijf Relations from Momentum Amplituhedron Geometry
• Non-planar BCFW Grassmannian Geometries
• Perturbative Quantum Field Theory and Homotopy Algebras
• Double Copy of Form Factors and Higgs Amplitudes: A mechanism of turning spurious poles in Yang-Mills into physical poles in gravity
• Flavour-kinematics duality for Goldstone modes
• Note on Permutation Sum of Color-ordered Gluon Amplitudes
• The double copy: from optics to quantum gravity
• Minimal Basis in Four Dimensions and Scalar Blocks
• Covariant color-kinematics duality, Hopf algebras and permutohedra
• Color-Kinematic Duality in ABJM Theory Without Amplitude Relations
• $\Delta$-Algebra and Scattering Amplitudes
• Holonomies of gauge fields in twistor space 1: bialgebra, supersymmetry, and gluon amplitudes
• Loop-level double-copy for massive fermions in the fundamental
• New Factorization Relations for Non-Linear Sigma Model Amplitudes
• Asymptotic Weyl double copy in Newman-Penrose formalism
• On-Shell Structures of MHV Amplitudes Beyond the Planar Limit
• Emergence of Unitarity and Locality from Hidden Zeros at One-Loop Order
• Note on Identities Inspired by New Soft Theorems
• MHV amplitudes and BCFW recursion for Yang-Mills theory in the de Sitter static patch
• Entanglement and the Double Copy
• Higher-order tails and RG flows due to scattering of gravitational radiation from binary inspirals
• Generalized Color Orderings: CEGM Integrands and Decoupling Identities
• Soft and Collinear Limits in $\mathcal{N}=8$ Supergravity using Double Copy Formalism
• Compactifying the Kerr-Schild Double Copy
• Waveforms from Amplitudes
• BCJ, worldsheet quantum algebra and KZ equations
• Expansion of tree amplitudes for EM and other theories
• Gravitational Equivalence Theorem and Double-Copy for Kaluza-Klein Graviton Scattering Amplitudes
• Holonomies of gauge fields in twistor space 2: Hecke algebra, diffeomorphism, and graviton amplitudes
• Bootstrapping solutions of scattering equations
• New Factorization Relations for Yang Mills Amplitudes
• Fermionic one-loop amplitudes of the RNS superstring
• Strong Gravity Approach to QCD and General Relativity
• Periods and Superstring Amplitudes
• From lightcone actions to maximally supersymmetric amplitudes
• FiniteFieldSolve: Exactly Solving Large Linear Systems in High-Energy Theory
• Applications of Subleading-color Amplitudes in N=4 SYM Theory
• Group-theoretic relations for amplitudes in gauge theories with orthogonal and symplectic groups
• Disk relations for tree amplitudes in minimal coupling theory of gauge field and gravity
• Evaluating EYM amplitudes in four dimensions by refined graphic expansion
• Proportionality of gravitational and electromagnetic radiation by an electron in an intense plane wave
• Supersymmetric Yang-Mills and Supergravity Amplitudes at One Loop
• Double copy for tree-level form factors. Part I. Foundations
• Symmetries, Spin-2 Scattering Amplitudes, and Equivalence theorems in Warped Five-Dimensional Gravitational Theories
• Spontaneously Broken Yang-Mills-Einstein Supergravities as Double Copies
• Gravito-electromagnetism, Kerr-Schild and Weyl double copies; a unified perspective
• Recent developments of the Lauricella string scattering amplitudes and their exact SL(K+3,C) Symmetry
• Basis decompositions of genus-one string integrals
• BCJ Numerators from Differential Operator of Multidimensional Residue
• New Ambitwistor String Theories
• α'-Expansion of Open String Disk Integrals via Mellin Transformations
• Holonomies of gauge fields in twistor space 3: gravity as a square of N=4 theory
• Group-theoretic relations for partial amplitudes in gauge theories with orthogonal group
• On differential operators and unifying relations for $1$-loop Feynman integrands
• Massive Color-Kinematics Duality and Double-Copy for Kaluza-Klein Scattering Amplitudes
• Twistor and Polytope Interpretations for Subleading Color One-Loop Amplitudes
• Permutation Relations of Yangian Invariants, Unitarity Cuts, and Scattering Amplitudes
• Infrared behaviour of the one-loop scattering equations and supergravity integrands
• Four-Gauge-Particle Scattering Amplitudes and Polyakov String Path Integral in the proper-time gauge
• Local integrands for the five-point amplitude in planar N=4 SYM up to five loops
• KLT Factorization of Winding String Amplitudes
• Amplitudes for Astrophysicists: Known Knowns
• Recurrence Relations of Higher Spin BPST Vertex Operators for Open String
• Expansion of EYM Amplitudes in Gauge Invariant Vector Space
• Unusual identities for QCD at tree-level
• Helicity Amplitudes for massive gravitinos in N=1 Supergravity
• Color-Kinematic Numerators for Fermion Compton Amplitudes
• Color-Dressed Generalized Biadjoint Scalar Amplitudes: Local Planarity
• Transmutation operators and expansions for $1$-loop Feynman integrands
• Cosmological Correlators, In-In Formalism and Double Copy
• One-Loop N = 4 Super Yang-Mills Scattering Amplitudes to All Orders in the Dimensional Regularization Parameter
• Superstring Amplitudes, Unitarity, and Hankel Determinants of Multiple Zeta Values
• On-shell Supersymmetry and higher-spin amplitudes
• Weyl double copy in type D spacetime in four and five dimensions
• One-loop impact factor for the emission of two gluons
• Next-to-MHV Yang-Mills kinematic algebra
• Spinning amplitudes from scalar amplitudes
• A high-precision result for a full-color three-loop three-point form factor in $\mathcal{N}=4$ SYM
• Logarithmic Singularities and Maximally Supersymmetric Amplitudes
• Non-Abelian gauged supergravities as double copies
• Properties of the Conformal Yangian in Scalar and Gauge Field Theories
• Two-loop n-point anomalous amplitudes in N=4 supergravity
• Off-shell BCJ Relation in Nonlinear Sigma Model
• Einstein Yang-Mills Amplitudes from Intersections of Twisted Forms
• Plahte Diagrams for String Scattering Amplitudes
• UV consistency conditions for CHY integrands
• The Algebraic Structure of the KLT Relations for Gauge and Gravity Tree Amplitudes
• Background ambiguity and the Gödel double copy
• Twistors, strings and supersymmetric gauge theories
• New framework for classical double copies
• Note on off-shell relations in nonlinear sigma model
• Form factors from string amplitudes
• Permutation in the CHY-Formulation
• Pure spinor formulation of the superstring and its applications
• Introductory Lectures on Resurgence: CERN Summer School 2024
• Classical gravitational scattering from a gauge-invariant double copy
• BCJ Amplitude Relations for Anti-de Sitter Boundary Correlators in Embedding Space
• Taming the 11D pure spinor b-ghost
• The origin of the KLT relations and nonlinear relations for Yang-Mills amplitudes
• Note on graph-based BCJ relation for Berends-Giele currents
• Two-loop superstring five-point amplitudes I: Construction via chiral splitting and pure spinors
• One-loop $N$-point correlators in pure gravity
• New relations for tree-level form factors and scattering amplitudes
• Five dimensional Weyl double copy
• Streamlining resummed QCD calculations using Monte Carlo integration
• Classicalization and unitarization of wee partons in QCD and Gravity: The CGC-Black Hole correspondence
• Generalised unitarity for dimensionally regulated amplitudes within FDF
• Group constraint relations for five-point amplitudes in gauge theories with SO(N) and Sp(2N) groups
• Infrared and transcendental structure of two-loop supersymmetric QCD amplitudes
• Tree and $1$-loop fundamental BCJ relations from soft theorems
• Boundary Behaviors for General Off-shell Amplitudes in Yang-Mills Theory
• Projectivity of Planar Zeros in Field and String Theory Amplitudes
• Off-shell Diagrammatics for Quantum Gravity
• Three-dimensional $\mathrm{SL}(2,\mathbb{R})$ Yang-Mills theory is three-dimensional gravity with background sources
• Note on scalar-graviton and scalar-photon-graviton amplitudes
• Classical double copy in the black hole mini-superspace
• On the non-planar beta-deformed N=4 super-Yang-Mills theory
• Analytic two-Loop four-point form factor of the stress-tensor supermultiplet in $\mathcal{N}=4$ SYM
• Topological Equivalence Theorem and Double-Copy for Chern-Simons Scattering Amplitudes
• Double-logarithms in N=8 supergravity: impact parameter description and mapping to 1-rooted ribbon graphs
• A note on multi-trace EYM amplitudes in four dimensions
• Higher-dimensional invariants in $6D$ super Yang-Mills theory
• Perturbative Methods for Superconformal Quantum Field Theories in String - Gauge Theory Dualities
• Fermions and the scattering equations
• Slowly rotating charges from Weyl double copy for Kerr black hole with Chern-Simons correction
• Scattering amplitudes in gauge theories with and without supersymmetry
• From Twistor-Particle Models to Massive Amplitudes
• Massive Covariant Colour-Kinematics in 3D
• 1-loop Color structures and sunny diagrams
• Steps Towards Generalization of Tensionless String Theory with Contact Interactions as Wilson loop of Non-Abelian Yang-Mills Theory
• Helicity Amplitudes for production of massive gravitino/goldstino
• Sudakov Representation of the Cachazo-He-Yuan Scattering Equations Formalism
• A twistor string for the ABJ(M) theory
• Field theory for gravity at all scales
• Untwisting the double copy: the zeroth copy as an optical seed
• $D$-Dimensional Modular Assembly of Higher-Derivative Four-Point Contact Amplitudes Involving Fermions
• Double soft graviton factors from the gravitational Wilson line
• Algebraic Consistency and Explicit Construction of One-Loop BCJ Numerators of Yang-Mills and Related Theories
• One-loop four-graviton string amplitude at finite ${\alpha }^{\prime }$
• Influence of a Perfectly Conducting Plate on the Uehling Potential of QED
• Systematic approach to $\ell$-loop planar integrands from the classical equation of motion
• Superstring amplitudes meet surfaceology
• Scattering Amplitudes in Gauge Theories
• Structure of Massive Gauge/Gravity Scattering Amplitudes, Equivalence Theorems, and Extended Double-Copy with Compactified Warped Space
• The Radial Action from Probe Amplitudes to All Orders
• The off-shell expansion relation of the Yang-Mills scalar theory
• Color-factor symmetry of the amplitudes of Yang-Mills and biadjoint scalar theory using perturbiner methods
• Deriving interaction vertices in higher derivative theories
• Building Momentum Kernel from Shapovalov Form
• Color-Kinematics Relation from the Feynman Diagram Perspective
• Decomposing color structure into multiplet bases
• Gravity model from (A)dS Yang-Mills theory
• Gravitational Deflection of Vector Photons via Effective Field Theory
• Equivalence Theorems and Double-Copy Structure in Scattering Amplitudes of Massive Kaluza-Klein States with Matter Interactions
• A Novel On-Shell Recursive Relation
• Cotton Double Copy for Gravitational Waves
• A Tale Of Two Amplitudes In High Energy Physics
• Superstring/Supergravity Mellin Correspondence in Grassmannian Formulation
• Weyl double copy in Lifshitz spacetimes
• Renormalization of gluonic leading-twist Operators in covariant Gauges
• Multi-centered $\mathcal{N}=2$ BPS black holes: a double copy description
• Twisted de Rham theory for string double copy in AdS
• Invariant traces of the flat space chiral higher-spin algebra as scattering amplitudes
• Entire Four-Graviton EFT from the Duality Between Color and Kinematics
• String Theory from Maximal Supersymmetry
• Leading singularities in higher-derivative Yang-Mills theory and quadratic gravity
• Ersatz gravity and black-hole thermodynamics from Manin gauge theory with noncompact gauge group
• Kerr-Schild Double Copy of the Randall-Sundrum Black String
• Color-factor symmetry and BCJ relations for QCD amplitudes
• The seeds of EFT double copy
• Properties of Yang-Mills scattering forms
• Relations and representations of QCD amplitudes
• Emergent symmetries of the Standard Model
• On-shell Correlators and Color-Kinematics Duality in Curved Symmetric Spacetimes
• Three-loop color-kinematics duality: A 24-dimensional solution space induced by new generalized gauge transformations
• Extended DBI and its generalizations from graded soft theorems
• Scalar Gravity and the Double Copy in the Framework of General Relativity
• Effective-one-body formalism for leading-order radiative effects in the post-linear framework
• Recursion in multiplet bases for tree-level MHV gluon amplitudes
• Einstein gravity as a theory with a SL(2,C) connection double copy
• Ambitwistor strings and amplitudes in curved backgrounds
• A dynamically constrained Yang-Mills theory with Lorentz symmetry group as an alternative theory of gravity
• One-loop matrix elements of effective superstring interactions: ${\alpha }^{\prime }$-expanding loop integrands
• Gravity Amplitudes from a Gaussian Matrix Model
• On-shell Methods for Form Factors in N=4 SYM and Their Applications
• New Variables For Graviton Scattering Amplitudes
• Ambitwistor Yang-Mills Theory Revisited