Analytic Representations of Yang-Mills Amplitudes

Abstract

Scattering amplitudes in Yang-Mills theory can be represented in the formalism of Cachazo, He and Yuan (CHY) as integrals over an auxiliary projective space---fully localized on the support of the scattering equations. Because solving the scattering equations is difficult and summing over the solutions algebraically complex, a method of directly integrating the terms that appear in this representation has long been sought. We solve this important open problem by first rewriting the terms in a manifestly Mobius-invariant form and then using monodromy relations (inspired by analogy to string theory) to decompose terms into those for which combinatorial rules of integration are known. The result is a systematic procedure to obtain analytic, covariant forms of Yang-Mills tree-amplitudes for any number of external legs and in any number of dimensions. As examples, we provide compact analytic expressions for amplitudes involving up to six gluons of arbitrary helicities.

29 pages, 43 figures; also included is a Mathematica notebook with explicit formulae. v2: citations added, and several (important) typos fixed

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• New Relations for Gauge-Theory Amplitudes
• Minimal Basis for Gauge Theory Amplitudes
• Gravity and Yang-Mills Amplitude Relations
• A Proof of the Explicit Minimal-basis Expansion of Tree Amplitudes in Gauge Field Theory
• Six Gluon Open Superstring Disk Amplitude, Multiple Hypergeometric Series and Euler-Zagier Sums
• Scattering Equations and String Theory Amplitudes
• Cross-ratio Identities and Higher-order Poles of CHY-integrand
• Fundamental BCJ Relation in N=4 SYM From The Connected Formulation
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Cited by in corpus (36)

• Aspects of Scattering Amplitudes and Moduli Space Localization
• Two-Loop Scattering Amplitudes from the Riemann Sphere
• BCJ numerators from reduced Pfaffian
• Expansion of Einstein-Yang-Mills Amplitude
• Expanding Einstein-Yang-Mills by Yang-Mills in CHY frame
• Covariant Color-Kinematics Duality
• Post-Minkowskian Radial Action from Soft Limits and Velocity Cuts
• Cross-ratio Identities and Higher-order Poles of CHY-integrand
• Ambitwistor formulations of $R^2$ gravity and $(DF{)}^2$ gauge theories
• Collinear limits beyond the leading order from the scattering equations
• Scalar-Graviton Amplitudes
• 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
• Scattering of Gravitons and Spinning Massive States from Compact Numerators
• Expansion of Einstein-Yang-Mills theory by Differential Operators
• Kinematic numerators from the worldsheet: cubic trees from labelled trees
• One-Loop Yang-Mills Integrands from Scattering Equations
• A Combinatoric Shortcut to Evaluate CHY-forms
• Direct Evaluation of $n$-point single-trace MHV amplitudes in 4d Einstein-Yang-Mills theory using the CHY Formalism
• Hidden Conformal Symmetry in Tree-Level Graviton Scattering
• String-Like Dual Models for Scalar Theories
• One-loop CHY-Integrand of Bi-adjoint Scalar Theory
• A differential operator for integrating one-loop scattering equations
• Scattering Equations and a new Factorization for Amplitudes I: Gauge Theories
• New Factorization Relations for Non-Linear Sigma Model Amplitudes
• Derivation of Feynman Rules for Higher Order Poles Using Cross-ratio Identities in CHY Construction
• New Factorization Relations for Yang Mills Amplitudes
• A graphic approach to identities induced from multi-trace Einstein-Yang-Mills amplitudes
• Scattering amplitude annihilators
• Scattering Equations and Factorization of Amplitudes II: Effective Field Theories
• UV consistency conditions for CHY integrands
• Analytic expressions of amplitudes by the cross-ratio identity method
• Characterizing the solutions to scattering equations that support tree-level ${\text{N}}^k\text{MHV}$ gauge/gravity amplitudes
• CHY Theory for Several Fields