A Proof of the Explicit Minimal-basis Expansion of Tree Amplitudes in Gauge Field Theory

Abstract

In last couple years, an important relation (BCJ relation) between color-ordered tree-level scattering amplitudes of gauge theory has inspired many studies. This relation implies that the minimal basis for the color-ordered tree-level amplitudes is $(n-3)!$ and other amplitudes can be expanded into a particular chosen basis. In this paper we will prove the conjectured explicit minimal basis expansion. For this purpose we will write down general BCJ relation of gauge theory by taking the field theory limit of BCJ relation in string theory. Then we prove these general BCJ relations using BCFW on-shell recursion relation. Using these general BCJ relations, we prove the conjectured explicit minimal-basis expansion of gauge theory tree amplitudes inductively.

arXiv comment22 pages, no figure

References in corpus (32)

• Direct Proof Of Tree-Level Recursion Relation In Yang-Mills Theory
• New Relations for Gauge-Theory Amplitudes
• New Recursion Relations for Tree Amplitudes of Gluons
• What is the Simplest Quantum Field Theory?
• On the Relationship between Yang-Mills Theory and Gravity and its Implication for Ultraviolet Divergences
• The All-Loop Integrand For Scattering Amplitudes in Planar N=4 SYM
• New Color Decompositions for Gauge Amplitudes at Tree and Loop Level
• Minimal Basis for Gauge Theory Amplitudes
• On Tree Amplitudes in Gauge Theory and Gravity
• The Momentum Kernel of Gauge and Gravity Theories
• Gauge Amplitude Identities by On-shell Recursion Relation in S-matrix Program
• On-Shell Recurrence Relations for One-Loop QCD Amplitudes
• Open & Closed vs. Pure Open String Disk Amplitudes
• Consistency Conditions on the S-Matrix of Massless Particles
• Taming Tree Amplitudes In General Relativity
• Gravity and Yang-Mills Amplitude Relations
• Tree Level Recursion Relations In General Relativity
• Proof of Gravity and Yang-Mills Amplitude Relations
• Dual Identities inside the Gluon and the Graviton Scattering Amplitudes
• MHV-Vertices for Gravity Amplitudes
• New Identities among Gauge Theory Amplitudes
• BCFW Recursion Relation with Nonzero Boundary Contribution
• On BCFW shifts of integrands and integrals
• On-shell Recursion in String Theory
• MHV, CSW and BCFW: field theory structures in string theory amplitudes
• New Relations for Gauge-Theory Amplitudes with Matter
• BCFW Recursion Relations and String Theory
• A note on the boundary contribution with bad deformation in gauge theory
• KLT and New Relations for N=8 SUGRA and N=4 SYM
• U(1)-decoupling, KK and BCJ relations in $\mathcal{N}=4$ SYM
• Comment on the Identities of the Gluon Tree Amplitudes
• Pomerons and BCFW recursion relations for strings on D-branes

Cited by in corpus (81)

• One-loop calculations in quantum field theory: from Feynman diagrams to unitarity cuts
• Simplifying Multiloop Integrands and Ultraviolet Divergences of Gauge Theory and Gravity Amplitudes
• Five-Point Amplitudes in N=4 Super-Yang-Mills Theory and N=8 Supergravity
• Pure Gravities via Color-Kinematics Duality for Fundamental Matter
• Color-Kinematics Duality for Pure Yang-Mills and Gravity at One and Two Loops
• Scattering amplitudes in N=2 Maxwell-Einstein and Yang-Mills/Einstein supergravity
• Type D Spacetimes and the Weyl Double Copy
• The Five-Loop Four-Point Integrand of N=8 Supergravity as a Generalized Double Copy
• N >= 4 Supergravity Amplitudes from Gauge Theory at Two Loops
• Scattering equations and BCJ relations for gauge and gravitational amplitudes with massive scalar particles
• Explicit Formulae for Yang-Mills-Einstein Amplitudes from the Double Copy
• Amplitude Relations in Non-linear Sigma Model
• Berends-Giele recursions and the BCJ duality in superspace and components
• Generic multiloop methods and application to N=4 super-Yang-Mills
• Ultraviolet Cancellations in Half-Maximal Supergravity as a Consequence of the Double-Copy Structure
• Multiparticle SYM equations of motion and pure spinor BRST blocks
• Scattering amplitudes: the most perfect microscopic structures in the universe
• Construction of an effective Yang-Mills Lagrangian with manifest BCJ duality
• Expansion of Einstein-Yang-Mills Amplitude
• Two-loop supersymmetric QCD and half-maximal supergravity amplitudes
• Einstein-Yang-Mills from pure Yang-Mills amplitudes
• Double Copy from Homotopy Algebras
• Color-Kinematics Duality in One-Loop Four-Gluon Amplitudes with Matter
• BRST-Lagrangian Double Copy of Yang-Mills Theory
• Manifesting Color-Kinematics Duality in the Scattering Equation Formalism
• N >= 4 Supergravity Amplitudes from Gauge Theory at One Loop
• On the kinematic algebra for BCJ numerators beyond the MHV sector
• An algebraic approach to BCJ numerators
• BCJ Relation of Color Scalar Theory and KLT Relation of Gauge Theory
• A Color Dual Form for Gauge-Theory Amplitudes
• Color-kinematics duality and Sudakov form factor at five loops for N=4 supersymmetric Yang-Mills theory
• Expansion of All Multitrace Tree Level EYM Amplitudes
• Gauge and Gravity Amplitude Relations
• Cross-ratio Identities and Higher-order Poles of CHY-integrand
• An Introduction to On-shell Recursion Relations
• Tree-level amplitudes from the pure spinor superstring
• The Radiative Double Copy for Einstein-Yang-Mills Theory
• Berends-Giele currents in Bern-Carrasco-Johansson gauge for $F^3$- and $F^4$-deformed Yang-Mills amplitudes
• Analytic Representations of Yang-Mills Amplitudes
• All-loop group-theory constraints for color-ordered SU(N) gauge-theory amplitudes
• SU(N) group-theory constraints on color-ordered five-point amplitudes at all loop orders
• Monodromy--like Relations for Finite Loop Amplitudes
• Virtuous Trees at Five and Six Points for Yang-Mills and Gravity
• Off-shell Currents and Color-Kinematics Duality
• Relations of loop partial amplitudes in gauge theory by Unitarity cut method
• Dual-color decompositions at one-loop level in Yang-Mills theory
• A Vertex Operator Algebra Construction of the Colour-Kinematics Dual numerator
• Kaluza-Klein from Colour-Kinematics Duality for Massive Fields
• Correlation functions of local composite operators from generalized unitarity
• On Primary Relations at Tree-level in String Theory and Field Theory
• On General BCJ Relation at One-loop Level in Yang-Mills Theory
• The Complete KLT-Map Between Gravity and Gauge Theories
• KLT-type relations for QCD and bicolor amplitudes from color-factor symmetry
• Expansion of Einstein-Yang-Mills theory by Differential Operators
• A Lie bracket for the momentum kernel
• Gauge $\times$ Gauge $=$ Gravity on Homogeneous Spaces using Tensor Convolutions
• The Construction of Dual-trace Factor in Yang-Mills Theory
• Symmetric-group decomposition of SU(N) group-theory constraints on four-, five-, and six-point color-ordered amplitudes
• Note on Construction of Dual-trace Factor in Yang-Mills Theory
• One-loop SYM-supergravity relation for five-point amplitudes
• BCJ relations from a new symmetry of gauge-theory amplitudes
• A graphic approach to gauge invariance induced identity
• Recurrence relations of Kummer functions and Regge string scattering amplitudes
• Note on Permutation Sum of Color-ordered Gluon Amplitudes
• Gauge invariance induced relations and the equivalence between distinct approaches to NLSM amplitudes
• A graphic approach to identities induced from multi-trace Einstein-Yang-Mills amplitudes
• Group-theoretic relations for amplitudes in gauge theories with orthogonal and symplectic groups
• Group-theoretic relations for partial amplitudes in gauge theories with orthogonal group
• Expansion of EYM Amplitudes in Gauge Invariant Vector Space
• Recurrence Relations of Higher Spin BPST Vertex Operators for Open String
• Permutation Relations of Yangian Invariants, Unitarity Cuts, and Scattering Amplitudes
• Next-to-MHV Yang-Mills kinematic algebra
• Note on off-shell relations in nonlinear sigma model
• Note on graph-based BCJ relation for Berends-Giele currents
• Group constraint relations for five-point amplitudes in gauge theories with SO(N) and Sp(2N) groups
• Scattering amplitudes in gauge theories with and without supersymmetry
• Color-factor symmetry of the amplitudes of Yang-Mills and biadjoint scalar theory using perturbiner methods
• Note on Cyclic Sum and Combination Sum of Color-ordered Gluon Amplitudes
• Color-factor symmetry and BCJ relations for QCD amplitudes
• Recursion in multiplet bases for tree-level MHV gluon amplitudes
• A Tale Of Two Amplitudes In High Energy Physics