Scattering Equations and Feynman Diagrams

Abstract

We show a direct matching between individual Feynman diagrams and integration measures in the scattering equation formalism of Cachazo, He and Yuan. The connection is most easily explained in terms of triangular graphs associated with planar Feynman diagrams in ${\varphi }^3$-theory. We also discuss the generalization to general scalar field theories with ${\varphi }^p$ interactions, corresponding to polygonal graphs involving vertices of order $p$. Finally, we describe how the same graph-theoretic language can be used to provide the precise link between individual Feynman diagrams and string theory integrands.

18 pages, 57 figures

References in corpus (2)

• Minimal Basis for Gauge Theory Amplitudes
• Scattering Equations and Matrices: From Einstein To Yang-Mills, DBI and NLSM

Cited by in corpus (59)

• One-loop amplitudes on the Riemann sphere
• Berends-Giele recursion for double-color-ordered amplitudes
• One-loop Scattering Equations and Amplitudes from Forward Limit
• Prescriptive Unitarity
• Expansion of Einstein-Yang-Mills Amplitude
• Manifesting Color-Kinematics Duality in the Scattering Equation Formalism
• Integration Rules for Loop Scattering Equations
• Post-Minkowskian Radial Action from Soft Limits and Velocity Cuts
• Cross-ratio Identities and Higher-order Poles of CHY-integrand
• Stokes Polytopes : The positive geometry for ${\varphi }^4$ interactions
• An Algebraic Approach to the Scattering Equations
• Scattering Equations and Global Duality of Residues
• Evaluation of the CHY Gauge Amplitude
• Elliptic scattering equations
• Elimination and recursions in the scattering equations
• $\Lambda$ Scattering Equations
• Labelled tree graphs, Feynman diagrams and disk integrals
• Comments on the evaluation of massless scattering
• CHY Loop Integrands from Holomorphic Forms
• On Positive Geometries of Quartic Interactions II : Stokes polytopes, Lower Forms on Associahedra and Worldsheet Forms
• The Polynomial Form of the Scattering Equations is an H-Basis
• CHY-Graphs on a Torus
• Quadratic Feynman Loop Integrands From Massless Scattering Equations
• The positive geometry for ${\varphi }^p$ interactions
• 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
• Ambitwistor Integrands from Tensionless Chiral Superstring Integrands
• 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
• The CHY representation of tree-level primitive QCD amplitudes
• Direct Evaluation of $n$-point single-trace MHV amplitudes in 4d Einstein-Yang-Mills theory using the CHY Formalism
• One-loop CHY-Integrand of Bi-adjoint Scalar Theory
• One-loop diagrams with quadratic propagators from the worldsheet
• String-Like Dual Models for Scalar Theories
• CHY formula and MHV amplitudes
• Scattering Equations and a new Factorization for Amplitudes I: Gauge Theories
• Towards Positive Geometry of Multi Scalar Field Amplitudes : Accordiohedron and Effective Field Theory
• New Factorization Relations for Non-Linear Sigma Model Amplitudes
• Derivation of Feynman Rules for Higher Order Poles Using Cross-ratio Identities in CHY Construction
• On Polytopes and Generalizations of the KLT Relations
• New Factorization Relations for Yang Mills Amplitudes
• One-loop integrand from generalised scattering equations
• Scattering Equations and Factorization of Amplitudes II: Effective Field Theories
• Pfaffian Diagrams for Gluon Tree Amplitudes
• Geometric Recursion from Polytope Triangulations and Twisted Homology
• Off-Shell Yang-Mills Amplitude in the CHY Formalism
• Note on the Labelled tree graphs
• Permutation in the CHY-Formulation
• Note on solutions of scattering equations
• Characterizing the solutions to scattering equations that support tree-level ${\text{N}}^k\text{MHV}$ gauge/gravity amplitudes
• Off-Shell CHY Amplitudes and Feynman Graphs
• 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