An Algebraic Approach to the Scattering Equations

Abstract

We employ the so-called companion matrix method from computational algebraic geometry, tailored for zero-dimensional ideals, to study the scattering equations. The method renders the CHY-integrand of scattering amplitudes computable using simple linear algebra and is amenable to an algorithmic approach. Certain identities in the amplitudes as well as rationality of the final integrand become immediate in this formalism.

34 pages, 1 figure

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Cited by in corpus (37)

• One-loop amplitudes on the Riemann sphere
• Aspects of Scattering Amplitudes and Moduli Space Localization
• Manifesting Color-Kinematics Duality in the Scattering Equation Formalism
• Gluons and gravitons at one loop from ambitwistor strings
• Cross-ratio Identities and Higher-order Poles of CHY-integrand
• 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
• Comments on the evaluation of massless scattering
• CHY Loop Integrands from Holomorphic Forms
• CHY-Graphs on a Torus
• The Polynomial Form of the Scattering Equations is an H-Basis
• Quadratic Feynman Loop Integrands From Massless Scattering Equations
• Polynomial reduction and evaluation of tree- and loop-level CHY amplitudes
• Understanding the Cancelation of Double Poles in the Pfaffian of CHY-formulism
• One-loop Parke-Taylor factors for quadratic propagators from massless scattering equations
• Non-planar one-loop Parke-Taylor factors in the CHY approach for quadratic propagators
• Feynman Rules of Higher-order Poles in CHY Construction
• The Role of Möbius Constants and Scattering Functions in CHY Scalar Amplitudes
• Direct Evaluation of $n$-point single-trace MHV amplitudes in 4d Einstein-Yang-Mills theory using the CHY Formalism
• A Combinatoric Shortcut to Evaluate CHY-forms
• One-loop diagrams with quadratic propagators from the worldsheet
• CHY formula and MHV amplitudes
• A differential operator for integrating one-loop scattering equations
• Derivation of Feynman Rules for Higher Order Poles Using Cross-ratio Identities in CHY Construction
• Bootstrapping solutions of scattering equations
• Analytical amplitudes from numerical solutions of the scattering equations
• Pushforwards via Scattering Equations with Applications to Positive Geometries
• Note on the Labelled tree graphs
• Note on solutions of scattering equations
• Analytic expressions of amplitudes by the cross-ratio identity method
• Scattering and Strebel graphs
• CHY-construction of Planar Loop Integrands of Cubic Scalar Theory
• CHY Theory for Several Fields