Scattering equations and BCJ relations for gauge and gravitational amplitudes with massive scalar particles

Abstract

We generalize the scattering equations to include both massless and massive particles. We construct an expression for the tree-level n-point amplitude with n-2 gluons or gravitons and a pair of massive scalars in arbitrary spacetime dimension as a sum over the (n-3)! solutions of the scattering equations, a la Cachazo, He, and Yuan. We derive the BCJ relations obeyed by these massive amplitudes.

22 pages. v2: clarification of eqs. (5.5) and (5.6)

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• Infinite Tension Limit of the Pure Spinor Superstring
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• A Proof of the Explicit Minimal-basis Expansion of Tree Amplitudes in Gauge Field Theory
• Generic multiloop methods and application to N=4 super-Yang-Mills
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• Scattering equations and virtuous kinematic numerators and dual-trace functions
• Color-Kinematics Duality and the Regge Limit of Inelastic Amplitudes
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• Perturbative spacetimes from Yang-Mills theory
• The double copy: Bremsstrahlung and accelerating black holes
• Holomorphic Classical Limit for Spin Effects in Gravitational and Electromagnetic Scattering
• Color-Kinematics Duality for QCD Amplitudes
• The Kerr-Schild double copy in curved spacetime
• One-loop amplitudes on the Riemann sphere
• Gravitational Radiation from Color-Kinematics Duality
• Scattering amplitudes in N=2 Maxwell-Einstein and Yang-Mills/Einstein supergravity
• Aspects of Scattering Amplitudes and Moduli Space Localization
• Gravity as Gauge Theory Squared: A Ghost Story
• One-Loop Corrections from Higher Dimensional Tree Amplitudes
• Two-Loop Scattering Amplitudes from the Riemann Sphere
• One-loop Scattering Equations and Amplitudes from Forward Limit
• Double Copy from Homotopy Algebras
• Gluons and gravitons at one loop from ambitwistor strings
• Integration Rules for Scattering Equations
• Twin Supergravities from Yang-Mills Squared
• Scattering Equations and String Theory Amplitudes
• The pure BRST Einstein-Hilbert Lagrangian from the double-copy to cubic order
• CHY representations for gauge theory and gravity amplitudes with up to three massive particles
• Cosmological Scattering Equations
• A Worldsheet Theory for Supergravity
• Constraints on a Massive Double-Copy and Applications to Massive Gravity
• Generalizations of the Double-Copy: the KLT Bootstrap
• Elliptic scattering equations
• Ambitwistor formulations of $R^2$ gravity and $(DF{)}^2$ gauge theories
• Elimination and recursions in the scattering equations
• Loop-Level Double-Copy for Massive Quantum Particles
• Relations for Einstein-Yang-Mills amplitudes from the CHY representation
• Cosmological Scattering Equations at Tree-level and One-loop
• Chiral approach to massive higher spins
• General Solution of the Scattering Equations
• Scattering equations, generating functions and all massless five point tree amplitudes
• Two-Loop Scattering Amplitudes: Double-Forward Limit and Colour-Kinematics Duality
• On single and double soft behaviors in NLSM
• CHY Loop Integrands from Holomorphic Forms
• Gauge $\times$ Gauge on Spheres
• Off-shell Currents and Color-Kinematics Duality
• The Polynomial Form of the Scattering Equations is an H-Basis
• CHY-Graphs on a Torus
• Quadratic Feynman Loop Integrands From Massless Scattering Equations
• Scalar-Graviton Amplitudes
• Kaluza-Klein from Colour-Kinematics Duality for Massive Fields
• Polynomial reduction and evaluation of tree- and loop-level CHY amplitudes
• One-loop Parke-Taylor factors for quadratic propagators from massless scattering equations
• BCJ Identities and $d$-Dimensional Generalized Unitarity
• Scattering of Gravitons and Spinning Massive States from Compact Numerators
• KLT-type relations for QCD and bicolor amplitudes from color-factor symmetry
• Double copies of fermions as only gravitational interacting matter
• Amplitudes for massive vector and scalar bosons in spontaneously-broken gauge theory from the CHY representation
• The Scattering Variety
• Color-kinematics duality, double copy and the unitarity method for higher-derivative QCD and quadratic gravity
• Double Copy from Tensor Products of Metric BV${}^{■}$-algebras
• Explicit BCJ numerators of nonlinear sigma model
• The Role of Möbius Constants and Scattering Functions in CHY Scalar Amplitudes
• Worldline description of a bi-adjoint scalar and the zeroth copy
• Direct Evaluation of $n$-point single-trace MHV amplitudes in 4d Einstein-Yang-Mills theory using the CHY Formalism
• BCJ relations from a new symmetry of gauge-theory amplitudes
• New relations for graviton-matter amplitudes
• The CHY representation of tree-level primitive QCD amplitudes
• All-multiplicity amplitudes with four massive quarks and identical-helicity gluons
• CHY formula and MHV amplitudes
• A String Deformation of the Parke-Taylor Factor
• On Positive Geometry and Scattering Forms for Matter Particles
• Chiral strings, the sectorized description and their integrated vertex operators
• Open associahedra and scattering forms
• One-loop integrand from generalised scattering equations
• New Ambitwistor String Theories
• Spontaneously Broken Yang-Mills-Einstein Supergravities as Double Copies
• Massive Color-Kinematics Duality and Double-Copy for Kaluza-Klein Scattering Amplitudes
• Infrared behaviour of the one-loop scattering equations and supergravity integrands
• Off-shell BCJ Relation in Nonlinear Sigma Model
• BCJ Amplitude Relations for Anti-de Sitter Boundary Correlators in Embedding Space
• Note on off-shell relations in nonlinear sigma model
• Group constraint relations for five-point amplitudes in gauge theories with SO(N) and Sp(2N) groups
• Fermions and the scattering equations
• CHY-construction of Planar Loop Integrands of Cubic Scalar Theory
• CHY Theory for Several Fields
• Off-Shell CHY Amplitudes and Feynman Graphs
• On-shell representation and further instances of the 2-split behavior of amplitudes
• Algebraic Consistency and Explicit Construction of One-Loop BCJ Numerators of Yang-Mills and Related Theories
• Sudakov Representation of the Cachazo-He-Yuan Scattering Equations Formalism
• Relations and representations of QCD amplitudes
• Color-factor symmetry and BCJ relations for QCD amplitudes