The Polynomial Form of the Scattering Equations

Abstract

The scattering equations, recently proposed by Cachazo, He and Yuan as providing a kinematic basis for describing tree amplitudes for massless particles in arbitrary space-time dimension (including scalars, gauge bosons and gravitons), are reformulated in polynomial form. The scattering equations for $N$ particles are shown to be equivalent to a Moebius invariant system of $N-3$ equations, ${\tilde{h}}_m=0$, $2\le m\le N-2$, in $N$ variables, where ${\tilde{h}}_m$ is a homogeneous polynomial of degree m, with the exceptional property of being linear in each variable taken separately. Fixing the Moebius invariance appropriately, yields polynomial equations $h_m=0$, $1\le m\le N-3$, in $N-3$ variables, where $h_m$ has degree $m$. The linearity of the equations in the individual variables facilitates computation, e.g the elimination of variables to obtain single variable equations determining the solutions. Expressions are given for the tree amplitudes in terms of the ${\tilde{h}}_m$ and $h_m$. The extension to the massive case for scalar particles is described and the special case of four dimensional space-time is discussed.

24 pages

References in corpus (3)

• Proof of the Formula of Cachazo, He and Yuan for Yang-Mills Tree Amplitudes in Arbitrary Dimension
• From Twistor String Theory To Recursion Relations
• Massless scattering at special kinematics as Jacobi polynomials

Cited by in corpus (68)

• Scattering Equations and Matrices: From Einstein To Yang-Mills, DBI and NLSM
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• Celestial amplitudes and conformal soft theorems
• Black Hole Formation and Classicalization in Ultra-Planckian 2 -> N Scattering
• Aspects of Scattering Amplitudes and Moduli Space Localization
• Scattering equations and BCJ relations for gauge and gravitational amplitudes with massive scalar particles
• One-Loop Corrections from Higher Dimensional Tree Amplitudes
• One-loop Scattering Equations and Amplitudes from Forward Limit
• Prescriptive Unitarity
• Manifesting Color-Kinematics Duality in the Scattering Equation Formalism
• Integration Rules for Loop Scattering Equations
• CHY representations for gauge theory and gravity amplitudes with up to three massive particles
• Cross-ratio Identities and Higher-order Poles of CHY-integrand
• Computation of Contour Integrals on ${\mathcal{M}}_{0,n}$
• Scattering Equations: From Projective Spaces to Tropical Grassmannians
• An Algebraic Approach to the Scattering Equations
• Scattering equations and virtuous kinematic numerators and dual-trace functions
• Scattering Equations and Global Duality of Residues
• Elliptic scattering equations
• Elimination and recursions in the scattering equations
• Relations for Einstein-Yang-Mills amplitudes from the CHY representation
• $\Lambda$ Scattering Equations
• General Solution of the Scattering Equations
• Scattering equations, generating functions and all massless five point tree amplitudes
• Comments on the evaluation of massless scattering
• CHY Loop Integrands from Holomorphic Forms
• Collinear limits beyond the leading order from the scattering equations
• A note on connected formula for form factors
• Scattering Equations: Real Solutions and Particles on a Line
• The Polynomial Form of the Scattering Equations is an H-Basis
• CHY-Graphs on a Torus
• Quadratic Feynman Loop Integrands From Massless Scattering Equations
• Polynomial reduction and evaluation of tree- and loop-level CHY amplitudes
• One-loop Parke-Taylor factors for quadratic propagators from massless scattering equations
• Understanding the Cancelation of Double Poles in the Pfaffian of CHY-formulism
• Non-planar one-loop Parke-Taylor factors in the CHY approach for quadratic propagators
• Scattering of Gravitons and Spinning Massive States from Compact Numerators
• Permutation Symmetry of the Scattering Equations
• Amplitudes for massive vector and scalar bosons in spontaneously-broken gauge theory from the CHY representation
• Feynman Rules of Higher-order Poles in CHY Construction
• The Scattering Variety
• The Role of Möbius Constants and Scattering Functions in CHY Scalar Amplitudes
• Planar Matrices and Arrays of Feynman Diagrams
• A Combinatoric Shortcut to Evaluate CHY-forms
• The CHY representation of tree-level primitive QCD amplitudes
• One-loop CHY-Integrand of Bi-adjoint Scalar Theory
• All-multiplicity amplitudes with four massive quarks and identical-helicity gluons
• A differential operator for integrating one-loop scattering equations
• A Monte Carlo Approach to the 4D Scattering Equations
• Ambitwistor Strings and the Scattering Equations on AdS${}_3\times$S${}^3$
• Lorentzian contours for tree-level string amplitudes
• Bootstrapping solutions of scattering equations
• BCJ Numerators from Differential Operator of Multidimensional Residue
• Off-Shell Yang-Mills Amplitude in the CHY Formalism
• Analytical amplitudes from numerical solutions of the scattering equations
• Massive Color-Kinematics Duality and Double-Copy for Kaluza-Klein Scattering Amplitudes
• Color-Dressed Generalized Biadjoint Scalar Amplitudes: Local Planarity
• Infrared behaviour of the one-loop scattering equations and supergravity integrands
• Pushforwards via Scattering Equations with Applications to Positive Geometries
• Note on the Labelled tree graphs
• Multi-Regge kinematics and the scattering equations
• Note on solutions of scattering equations
• CHY-construction of Planar Loop Integrands of Cubic Scalar Theory
• Fermions and the scattering equations
• CHY Theory for Several Fields
• Off-Shell CHY Amplitudes and Feynman Graphs
• Sudakov Representation of the Cachazo-He-Yuan Scattering Equations Formalism
• Ambitwistor strings and amplitudes in curved backgrounds