Integration Rules for Scattering Equations

Abstract

As described by Cachazo, He and Yuan, scattering amplitudes in many quantum field theories can be represented as integrals that are fully localized on solutions to the so-called scattering equations. Because the number of solutions to the scattering equations grows quite rapidly, the contour of integration involves contributions from many isolated components. In this paper, we provide a simple, combinatorial rule that immediately provides the result of integration against the scattering equation constraints for any Möbius-invariant integrand involving only simple poles. These rules have a simple diagrammatic interpretation that makes the evaluation of any such integrand immediate. Finally, we explain how these rules are related to the computation of amplitudes in the field theory limit of string theory.

30 pages, 29 figures

References in corpus (6)

• Minimal Basis for Gauge Theory Amplitudes
• Scattering Equations and Matrices: From Einstein To Yang-Mills, DBI and NLSM
• Einstein-Yang-Mills Scattering Amplitudes From Scattering Equations
• Scattering equations and BCJ relations for gauge and gravitational amplitudes with massive scalar particles
• CHY representations for gauge theory and gravity amplitudes with up to three massive particles
• Scattering equations, generating functions and all massless five point tree amplitudes

Cited by in corpus (56)

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• Labelled tree graphs, Feynman diagrams and disk integrals
• Comments on the evaluation of massless scattering
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• CHY-Graphs on a Torus
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• Propagators, BCFW Recursion and New Scattering Equations at One Loop
• Polynomial reduction and evaluation of tree- and loop-level CHY amplitudes
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• Non-planar one-loop Parke-Taylor factors in the CHY approach for quadratic propagators
• Scattering of Gravitons and Spinning Massive States from Compact Numerators
• Feynman Rules of Higher-order Poles in CHY Construction
• Unified web for expansions of amplitudes
• One-Loop Yang-Mills Integrands from Scattering Equations
• The Role of Möbius Constants and Scattering Functions in CHY Scalar Amplitudes
• A Combinatoric Shortcut to Evaluate CHY-forms
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• One-loop CHY-Integrand of Bi-adjoint Scalar Theory
• A differential operator for integrating one-loop scattering equations
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• Derivation of Feynman Rules for Higher Order Poles Using Cross-ratio Identities in CHY Construction
• New Factorization Relations for Non-Linear Sigma Model Amplitudes
• New Factorization Relations for Yang Mills Amplitudes
• BCJ Numerators from Differential Operator of Multidimensional Residue
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• Pfaffian Diagrams for Gluon Tree Amplitudes
• Geometric Recursion from Polytope Triangulations and Twisted Homology
• Note on the Labelled tree graphs
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• Form factors from string amplitudes
• Compatible Cycles and CHY Integrals
• Note on solutions of scattering equations
• Analytic expressions of amplitudes by the cross-ratio identity method
• CHY Theory for Several Fields
• Sudakov Representation of the Cachazo-He-Yuan Scattering Equations Formalism