New Recursion Relations for Tree Amplitudes of Gluons

Abstract

We present new recursion relations for tree amplitudes in gauge theory that give very compact formulas. Our relations give any tree amplitude as a sum over terms constructed from products of two amplitudes of fewer particles multiplied by a Feynman propagator. The two amplitudes in each term are physical, in the sense that all particles are on-shell and momentum conservation is preserved. This is striking, since it is just like adding certain factorization limits of the original amplitude to build up the full answer. As examples, we recompute all known tree-level amplitudes of up to seven gluons and show that our recursion relations naturally give their most compact forms. We give a new result for an eight-gluon amplitude, A(1+,2-,3+,4-,5+,6-,7+,8-). We show how to build any amplitude in terms of three-gluon amplitudes only.

27 pages, harvmac. 6 figures. v2: Fixed typos in (4.1), added symmetric form of the alternating-helicity eight-gluon amplitude, added references. v3: Fixed typo in (4.1), added reference, added acknowledgment, minor clarifications

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• Hidden Simplicity of Gauge Theory Amplitudes
• Scattering in the static patch of de Sitter space
• Notes on Biadjoint Amplitudes, $\mathrm{Trop}G(3,7)$ and $X(3,7)$ Scattering Equations
• Recursion relations for scattering amplitudes with massive particles
• Amplituhedra, and Beyond
• Seven parton amplitudes from recursion relations
• On tree amplitudes with gluons coupled to gravitons
• Manifest colour-kinematics duality and double-copy in the string-based formalism
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• Two-Loop QCD Corrections to the Higgs plus three-parton amplitudes with Top Mass Correction
• Soft Graviton Emission at High and Low Energies in Yukawa and Scalar Theories
• Determination of Boundary Contributions in Recursion Relation
• Tree level amplitudes from soft theorems
• Taming higher-derivative interactions and bootstrapping gravity with soft theorems
• Spinor frame formalism for amplitudes and constrained superamplitudes of 10D SYM and 11D supergravity
• Towards the Gravituhedron: New Expressions for NMHV Gravity Amplitudes
• N=4 SYM NMHV Loop Amplitude in Superspace
• Multi-spin soft bootstrap and scalar-vector Galileon
• A manifestly MHV Lagrangian for N=4 Yang-Mills
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• On-shell constructibility of Born amplitudes in spontaneously broken gauge theories
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• Generation Function For One-loop Tensor Reduction
• Massive Quark-Gluon Scattering Amplitudes at Tree Level
• Global Structure of Curves from Generalized Unitarity Cut of Three-loop Diagrams
• Compact QED Tree-Level Amplitudes From Dressed BCFW Recursion Relations
• Gearing up for the next generation of LFV experiments, via on-shell methods
• Nonrelativistic effective field theories with enhanced symmetries and soft behavior
• Conformal anomaly of super Wilson loop
• One-loop Feynman Integral Reduction by Differential Operators
• Automated Evaluation of One-Loop Six-Point Processes for the LHC
• Direct Evaluation of $n$-point single-trace MHV amplitudes in 4d Einstein-Yang-Mills theory using the CHY Formalism
• Exploring minijets beyond the leading power
• Universal expansions of scattering amplitudes for gravitons, gluons and Goldstone particles
• Recursion relations for multi-gluon off-shell amplitudes on the light-front and Wilson lines
• Dual-Conformal Regularization of Infrared Loop Divergences and the Chiral Box Expansion
• Double-Soft Limits of Gluons and Gravitons
• Generalised Unitarity At One-Loop With Massive Fermions
• Form Factor and Boundary Contribution of Amplitude
• One-loop renormalization and the S-matrix
• First-quantized N=4 Yang-Mills
• Hidden Conformal Symmetry in Tree-Level Graviton Scattering
• Bootstrap and Amplitudes: A Hike in the Landscape of Quantum Field Theory
• All rational one-loop Einstein-Yang-Mills amplitudes at four points
• One-loop N=8 Supergravity Amplitudes from MHV Diagrams
• Weights and recursion relations for ${\varphi }^p$ tree amplitudes from the positive geometry
• CHY formula and MHV amplitudes
• All-multiplicity amplitudes with four massive quarks and identical-helicity gluons
• BCFW-type recurrent relations for tree amplitudes of D=11 supergravity
• Note on Bonus Relations for N=8 Supergravity Tree Amplitudes
• A Forward Branching Phase-Space Generator
• The Grassmannian for Celestial Superamplitudes
• A Type of Unifying Relation in (A)dS Spacetime
• A graphic approach to gauge invariance induced identity
• Wilson Loop Duality and OPE for Super Form Factors of Half-BPS Operators
• Gravity loop integrands from the ultraviolet
• Feynman Rules for the Rational Part of One-loop QCD Corrections in the MSSM
• Momentum Amplituhedron meets Kinematic Associahedron
• Tree-level amplitudes and dual superconformal symmetry
• On the Classification of Residues of the Grassmannian
• Massive Nonplanar Two-Loop Maximal Unitarity
• Tree amplitudes of noncommutative U(N) Yang-Mills Theory
• Matter Couplings and Equivalence Principles for Soft Scalars
• General Split Helicity Gluon Tree Amplitudes in Open Twistor String Theory
• On-shell recursion relations for gravity
• Positivity Sectors and the Amplituhedron
• (Non)-supersymmetric Marginal Deformations from Twistor String Theory
• An analytic superfield formalism for tree superamplitudes in D=10 and D=11
• Wilson lines in the MHV action
• The Boostless Bootstrap and BCFW Momentum Shifts
• One-loop N=8 supergravity coefficients from N=4 super Yang-Mills
• On-Shell Recursion Relations for Nonrelativistic Effective Field Theories
• On-Shell Unitarity Bootstrap for QCD Amplitudes
• BCJ Relations for One-Loop QCD Integral Coefficients
• Production of non-Abelian tensor gauge bosons. Tree amplitudes in generalized Yang-Mills theory and BCFW recursion relation
• Recursion relations, Helicity Amplitudes and Dimensional Regularization
• Diagrammatic proof of the BCFW recursion relation for gluon amplitudes in QCD
• Cyclic Mario Worlds -- Color-Decomposition for One-Loop QCD
• Consistency Conditions on S-Matrix of Spin 1 Massless Particles
• On Tensorial Spaces and BCFW Recursion Relations for Higher Spin Fields
• Non-planar cusp and transcendental anomalous dimension at four loops in N=4 supersymmetric Yang-Mills theory
• A Monte Carlo Approach to the 4D Scattering Equations
• Challenges with Internal Photons in Constructive QED
• Complete Equivalence Between Gluon Tree Amplitudes in Twistor String Theory and in Gauge Theory
• Massive On-shell Recursion Relations for n-point Amplitudes
• Non-planar BCFW Grassmannian Geometries
• The MHV Lagrangian for a spontaneously broken gauge theory
• A note on on-shell recursion relation of string amplitudes
• Conformal invariance of tensor bosons tree amplitudes
• On Multi-step BCFW Recursion Relations
• Note on Permutation Sum of Color-ordered Gluon Amplitudes
• Renormalization of effective field theories via on-shell methods: the case of axion-like particles
• Constraining the weights of Stokes Polytopes using BCFW recursions for Phi^4
• Recursion and worldsheet formulae for 6d superamplitudes
• $\Delta$-Algebra and Scattering Amplitudes
• Bipartite field theories and D-brane instantons
• Gauge invariance induced relations and the equivalence between distinct approaches to NLSM amplitudes
• The all-loop conjecture for integrands of reggeon amplitudes in N=4 SYM
• MHV amplitudes and BCFW recursion for Yang-Mills theory in the de Sitter static patch
• Note on Identities Inspired by New Soft Theorems
• Recursion relations for scattering amplitudes with massive particles II: massive vector bosons
• ABJM Amplitudes in U-gauge and a Soft Theorem
• Two-loop Integral Reduction from Elliptic and Hyperelliptic Curves
• Non-Planar On-Shell Diagrams
• Flow-oriented perturbation theory
• Gauge and Gravity Amplitudes on the Celestial Sphere
• One-loop N = 4 super Yang-Mills scattering amplitudes in d dimensions, relation to open strings and polygonal Wilson loops
• Scattering of charged tensor bosons in gauge and superstring theories
• Holonomies of gauge fields in twistor space 1: bialgebra, supersymmetry, and gluon amplitudes
• Weights, Recursion relations and Projective triangulations for Positive Geometry of scalar theories
• Selection rules for helicity amplitudes in massive gauge theories
• A graphic approach to identities induced from multi-trace Einstein-Yang-Mills amplitudes
• Leading order multi-soft behaviors of tree amplitudes in NLSM
• The SAGEX Review on Scattering Amplitudes, Chapter 12: Amplitudes and collider physics
• Grassmannian Integral for General Gauge Invariant Off-shell Amplitudes in N=4 SYM
• Recursion relations for the general tree-level amplitudes in QCD with massive dirac fields
• Celestial Berends-Giele current
• Higher-order tails and RG flows due to scattering of gravitational radiation from binary inspirals
• Emergent unitarity from the amplituhedron
• Poles At Infinity in On-shell Diagrams
• From lightcone actions to maximally supersymmetric amplitudes
• Applications of Subleading-color Amplitudes in N=4 SYM Theory
• General tree-level amplitudes by factorization limits
• Supersymmetric Yang-Mills and Supergravity Amplitudes at One Loop
• Constructing EYM amplitudes by inverse soft limit
• On All-loop Integrands of Scattering Amplitudes in Planar N=4 SYM
• Comments on scattering in massive gravity, vDVZ and BCFW
• Exploring straight infinite Wilson lines in the Self Dual and the MHV Lagrangians
• Recursive equations for arbitrary scattering processes
• Roots of Amplitudes
• Connection between non-Abelian tensor gauge fields and open strings
• A Note on Single Soft Scalar Emission of $\mathcal{N}=8$ SUGRA and $E_{7(7)}$ Symmetry
• Evaluating EYM amplitudes in four dimensions by refined graphic expansion
• Recursive Relations For Processes With n Photons Of Noncommutative QED
• α'-Expansion of Open String Disk Integrals via Mellin Transformations
• An Algebraic Classification of Exceptional EFTs Part II: Supersymmetry
• Recursive prescription for logarithmic jet rate coefficients
• Permutation Relations of Yangian Invariants, Unitarity Cuts, and Scattering Amplitudes
• Unusual identities for QCD at tree-level
• Field diffeomorphisms and the algebraic structure of perturbative expansions
• Higher Derivative Theories for Interacting Massless Gravitons in Minkowski Spacetime
• ALPs, the on-shell way
• TeV Scale Strings and Scattering Amplitudes at the LHC
• Application of the Feynman-tree theorem together with BCFW recursion relations
• Note on recursion relations for the $\mathcal{Q}$-cut representation
• On-shell Supersymmetry and higher-spin amplitudes
• Off-shell amplitudes as boundary integrals of analytically continued Wilson line slope
• Geometric Recursion from Polytope Triangulations and Twisted Homology
• Boundary Operators of BCFW Recursion Relation
• Dual conformal constraints and infrared equations from global residue theorems in N=4 SYM theory
• Helicity Amplitudes for massive gravitinos in N=1 Supergravity
• Six-meson amplitude in QCD-like theories
• Amplitudes for Astrophysicists: Known Knowns
• CSW-like Expansion for Einstein Gravity
• N=8 SYM vs. N=6 Chern-Simons: Four-point amplitudes at two-loops
• Hard QCD at hadron colliders
• Recent developments for multi-leg QCD amplitudes with massive particles
• SpinorsExtras - Mathematica implementation of massive spinor-helicity formalism
• Disk level S-matrix elements at eikonal Regge limit
• The last of the seven-parton tree amplitudes
• Leading multi-soft limits from scattering equations
• Plahte Diagrams for String Scattering Amplitudes
• Multigluon tree amplitudes with a pair of massive fermions
• Illustrations of Integrand-Basis Building at Two Loops
• Automated calculations for multi-leg processes
• A new Wilson line-based action for gluodynamics
• Spinor formalism for massive fields with half-integral spin
• Logarithmic Singularities and Maximally Supersymmetric Amplitudes
• Off-shell BCJ Relation in Nonlinear Sigma Model
• Top Quark Amplitudes with an Anomalous Magnetic Moment
• Proton structure and tensor gluons
• Augmented Recursion For One-loop Amplitudes
• Ambitwistor strings and reggeon amplitudes in N=4 SYM
• Recent theoretical progress in perturbative QCD
• Multi gluon collinear limits from MHV amplitudes
• Improving Colour Computations in MadGraph5_aMC@NLO and Exploring a 1/Nc Expansion
• One-Loop Effective Action Approach to Quantum MHV Theory
• Nonplanar On-shell Diagrams and Leading Singularities of Scattering Amplitudes
• Generating MHV super-vertices in light-cone gauge
• The Amplituhedron BCFW Triangulation
• Triangulations for ABHY Polytopes and Recursions for Tree and Loop Amplitudes
• Rational Terms in Theories with Matter
• Analytic Computations of Massive One-Loop Amplitudes
• All Two-Loop, Color-Dressed, Six-Point Amplitude Integrands in sYM
• The exact SL(K+3,C) symmetry of string theory
• BCFW construction of the Veneziano Amplitude
• Boundary Behaviors for General Off-shell Amplitudes in Yang-Mills Theory
• Recursion Relation for Boundary Contribution
• Exploring $HVV$ amplitudes with $CP$ violation by decomposition and on-shell scattering amplitude methods
• Amplitudes With Different Helicity Configurations Of Noncommutative QED
• Surface Gauge Invariance, Soft Limits and the Transmutation of Gluons into Scalars
• Note on solutions of scattering equations
• Soft gravitational radiation from multi-body collisions
• Analytic structure of one-loop coefficients
• MHV Lagrangians for Yang--Mills and QCD
• All one-loop NMHV gluon amplitudes in N=1 SYM
• Yangian symmetry of light-like Wilson loops
• The MHV lagrangian vertices and the Parke-Taylor formula
• A Note on Loop Amplitudes in QED
• BCFW Deformation and Regge Limit
• Propagator-cancelling scalar fields
• Developments in perturbative QCD
• Locally Finite Observables in sYM
• Tree and $1$-loop fundamental BCJ relations from soft theorems
• Factorizations for tree amplitudes in the double-cover framework: from gravity to other theories
• Recursive relations for a quiver gauge theory
• Scattering amplitudes as multi-particle higher-spin charges in the correspondence space
• Holonomies of gauge fields in twistor space 5: amplitudes of gluons and massive scalars
• Supersums for all supersymmetric amplitudes
• The four-dimensional on-shell three-point amplitude in spinor-helicity formalism and BCFW recursion relations
• CHY-construction of Planar Loop Integrands of Cubic Scalar Theory
• Application of BCFW-recursion relations and the Feynman-tree theorem to the four gluon amplitude with all plus helicities
• Positivity, Grassmannian Geometry and Simplex-like Structures of Scattering Amplitudes
• Perturbative Methods for Superconformal Quantum Field Theories in String - Gauge Theory Dualities
• Soft Theorems and the KLT-Relation
• Multi-Particle Processes in QCD without Feynman Diagrams
• On perturbative field theory and twistor string theory
• New N=4 SYM Path Integral
• Tree-level Split Helicity Amplitudes in Ambitwistor Space
• Generalisation of the Yang-Mills Theory
• Perturbative quantum gravity in double field theory
• Ward Identity implied recursion relations in Yang-Mills theory
• Integrands of Less-Supersymmetric Yang-Mills at One Loop
• Wilson Lines and Boundary Operators of BCFW Shifts
• Note on Cyclic Sum and Combination Sum of Color-ordered Gluon Amplitudes
• Amplitude Surrogates for Multi-Jet Processes
• Speeding up MadGraph5_aMC@NLO
• Quantum point particle approximation of spinning black holes and compact stars
• Ward Identity Implies Recursion Relation at Tree and Loop Level
• How to calculate colourful cross sections efficiently
• Deriving interaction vertices in higher derivative theories
• On-shell recursion relations for generic integrands
• QCD amplitudes with 2 initial spacelike legs via generalised BCFW recursion
• Holonomies of gauge fields in twistor space 6: incorporation of massive fermions
• The $SL(K+3,\mathbb{C})$ symmetry of string scatterings from D-branes
• Scattering Amplitudes in Gauge Theories
• Orientations of BCFW Charts on the Grassmannian
• Parity Symmetry and Soft Limit for the Cachazo-Geyer Gravity Amplitude
• Higher Codimension Singularities Constructing Yang-Mills Tree Amplitudes
• General expressions for extra-dimensional tree amplitudes and all-plus 1-loop integrands in Q-cut representaion
• Simplex-like Structures of Maximally Supersymmetric Scattering Amplitudes
• Born amplitudes in QCD from scalar diagrams
• On bound states in Quantum Field Theory
• Field theory for gravity at all scales
• Analytic Scattering Amplitudes for QCD
• Boundary Contributions of On-shell Recursion Relations With Multiple-line Deformation
• Bilocal phase operators in beta-deformed super Yang-Mills
• A new approach for amplitudes with multiple fermion lines
• Smooth Splitting and Zeros from On-Shell Recursion
• Strings from large charged fields
• Recursion in multiplet bases for tree-level MHV gluon amplitudes
• Shifts in BCFW method for QED
• Quantum Off-Shell Recursion Relation
• Equivalence of Coefficients Extraction of One-loop Master Integrals
• QCD Amplitudes: new perspectives on Feynman integral calculus
• On the Integrable Structure of Super Yang-Mills Scattering Amplitudes
• Extended DBI and its generalizations from graded soft theorems
• One-loop central-emission vertex for two gluons in $\mathcal{N}=4$ super Yang-Mills theory
• Leading singularities in higher-derivative Yang-Mills theory and quadratic gravity
• Graviton scattering amplitudes and Pure Connection Formulation of GR
• A Tale Of Two Amplitudes In High Energy Physics
• The Aharony-Bergman-Jafferis-Maldacena theory on a circle
• Equivalence Theorems and Double-Copy Structure in Scattering Amplitudes of Massive Kaluza-Klein States with Matter Interactions
• On-shell Methods for Form Factors in N=4 SYM and Their Applications
• Geometric duality between effective field theories I: scattering amplitudes
• BCFW like recursion for Deformed Associahedron
• A Novel On-Shell Recursive Relation
• Higher-Point Correlators in N=4 SYM: Ten-Dimensional Null Polygons
• String Theory from Maximal Supersymmetry
• Fourier transform from momentum space to twistor space
• A Pure Spinor Twistor Description of Ambitwistor Strings
• Complexified Ward Identity in pure Yang-Mills theory at tree-level
• Brane Tilings as On-shell Diagrams
• Exploration of the Tree-Level S-Matrix of Massless Particles
• Constructing QCD one-loop amplitudes
• Recent progress for five-particle two-loop scattering amplitudes with an off-shell leg