BCJ numerators from reduced Pfaffian

Abstract

By expanding the reduced Pfaffian in the tree level Cachazo-He-Yuan (CHY) integrands for Yang-Mills (YM) and nonlinear sigma model (NLSM), we can get the Bern-Carrasco-Johansson (BCJ) numerators in Del Duca-Dixon-Maltoni (DDM) form for arbitrary number of particles in any spacetime dimensions. In this work, we give a set of very straightforward graphic rules based on spanning trees for a direct evaluation of the BCJ numerators for YM and NLSM. Such rules can be derived from the Laplace expansion of the corresponding reduced Pfaffian. For YM, the each one of the $(n-2)!$ DDM form BCJ numerators contains exactly $(n-1)!$ terms, corresponding to the increasing trees with respect to the color order. For NLSM, the number of nonzero numerators is at most $(n-2)!-(n-3)!$, less than those of several previous constructions.

29 pages, 3 captioned figures

References in corpus (8)

• The Momentum Kernel of Gauge and Gravity Theories
• The Kinematic Algebra From the Self-Dual Sector
• Einstein-Yang-Mills Scattering Amplitudes From Scattering Equations
• Scattering amplitudes in N=2 Maxwell-Einstein and Yang-Mills/Einstein supergravity
• Symmetry and Action for Flavor-Kinematics Duality
• New Identities among Gauge Theory Amplitudes
• Manifesting Color-Kinematics Duality in the Scattering Equation Formalism
• Scattering equations and virtuous kinematic numerators and dual-trace functions

Cited by in corpus (57)

• Ultraviolet Properties of N = 8 Supergravity at Five Loops
• Type D Spacetimes and the Weyl Double Copy
• The Five-Loop Four-Point Integrand of N=8 Supergravity as a Generalized Double Copy
• A new gauge-invariant double copy for heavy-mass effective theory
• On the kinematic algebra for BCJ numerators beyond the MHV sector
• Covariant Color-Kinematics Duality
• Kinematic Hopf Algebra for BCJ Numerators in Heavy-Mass Effective Field Theory and Yang-Mills Theory
• Kinematic Jacobi Identity is a Residue Theorem: Geometry of Color-Kinematics Duality for Gauge and Gravity Amplitudes
• Pions as Gluons in Higher Dimensions
• Expansion of All Multitrace Tree Level EYM Amplitudes
• Off-Shell Color-Kinematics Duality for Chern-Simons
• Classical Double Copy of Worldline Quantum Field Theory
• Unraveling conformal gravity amplitudes
• Double Field Theory as the Double Copy of Yang-Mills
• New heavenly double copies
• String amplitudes from field-theory amplitudes and vice versa
• 10D Super-Yang-Mills Scattering Amplitudes From Its Pure Spinor Action
• Two-Loop Scattering Amplitudes: Double-Forward Limit and Colour-Kinematics Duality
• Amplitudes, Hopf algebras and the colour-kinematics duality
• A Vertex Operator Algebra Construction of the Colour-Kinematics Dual numerator
• Properties of scattering forms and their relation to associahedra
• Scattering Forms, Worldsheet Forms and Amplitudes from Subspaces
• Scattering of Gravitons and Spinning Massive States from Compact Numerators
• Expansion of Einstein-Yang-Mills theory by Differential Operators
• Kinematic numerators from the worldsheet: cubic trees from labelled trees
• Color-kinematics duality, double copy and the unitarity method for higher-derivative QCD and quadratic gravity
• Unified web for expansions of amplitudes
• Off-shell extended graphic rule and the expansion of Berends-Giele currents in Yang-Mills theory
• One-Loop Yang-Mills Integrands from Scattering Equations
• Universal expansions of scattering amplitudes for gravitons, gluons and Goldstone particles
• A graphic approach to gauge invariance induced identity
• One-loop diagrams with quadratic propagators from the worldsheet
• One-loop Bern-Carrasco-Johansson Numerators on Quadratic Propagators from the Worldsheet
• Flavour-kinematics duality for Goldstone modes
• Gauge invariance induced relations and the equivalence between distinct approaches to NLSM amplitudes
• Covariant color-kinematics duality, Hopf algebras and permutohedra
• Expansion of tree amplitudes for EM and other theories
• Evaluating EYM amplitudes in four dimensions by refined graphic expansion
• BCJ Numerators from Differential Operator of Multidimensional Residue
• Transmutation operators and expansions for $1$-loop Feynman integrands
• Off-Shell Yang-Mills Amplitude in the CHY Formalism
• Pfaffian Diagrams for Gluon Tree Amplitudes
• The Algebraic Structure of the KLT Relations for Gauge and Gravity Tree Amplitudes
• Next-to-MHV Yang-Mills kinematic algebra
• Note on graph-based BCJ relation for Berends-Giele currents
• New relations for tree-level form factors and scattering amplitudes
• Transmuting off-shell CHY integrals in the double-cover framework
• Permutation in the CHY-Formulation
• BCJ Amplitude Relations for Anti-de Sitter Boundary Correlators in Embedding Space
• Note on scalar-graviton and scalar-photon-graviton amplitudes
• Tree and $1$-loop fundamental BCJ relations from soft theorems
• A note on multi-trace EYM amplitudes in four dimensions
• The off-shell expansion relation of the Yang-Mills scalar theory
• Algebraic Consistency and Explicit Construction of One-Loop BCJ Numerators of Yang-Mills and Related Theories
• A Novel On-Shell Recursive Relation
• Leading singularities in higher-derivative Yang-Mills theory and quadratic gravity
• Color-Kinematics Relation from the Feynman Diagram Perspective