The Role of Möbius Constants and Scattering Functions in CHY Scalar Amplitudes

Abstract

The integrations leading to the Cachazo-He-Yuan (CHY) double-color $n$-point massless scalar amplitude are carried out one integral at a time. Möbius invariance dictates the final amplitude to be independent of the three Möbius constants ${\sigma }_r,{\sigma }_s,{\sigma }_t$, but their choice affects integrations and the intermediate results. The effect of the Möbius constants, the two colors, and the scattering functions on each integration is investigated. A systematic way to carry out the $n-3$ integrations is explained, each exposing one of the $n-3$ propagators of the Feynman diagrams. Two detailed examples are shown to illustrate the procedure, one a five-point amplitude, and the other a nine-point amplitude.

Version to appear in Physical Review D

References in corpus (9)

• Einstein-Yang-Mills Scattering Amplitudes From Scattering Equations
• Scattering equations and BCJ relations for gauge and gravitational amplitudes with massive scalar particles
• Scattering Equations and String Theory Amplitudes
• CHY representations for gauge theory and gravity amplitudes with up to three massive particles
• Elimination and recursions in the scattering equations
• On the solutions of the scattering equations
• Scattering equations, generating functions and all massless five point tree amplitudes
• Permutation Symmetry of the Scattering Equations
• The Scattering Variety

Cited by in corpus (14)

• Berends-Giele recursion for double-color-ordered amplitudes
• Cross-ratio Identities and Higher-order Poles of CHY-integrand
• The Polynomial Form of the Scattering Equations is an H-Basis
• Polynomial reduction and evaluation of tree- and loop-level CHY amplitudes
• Understanding the Cancelation of Double Poles in the Pfaffian of CHY-formulism
• Feynman Rules of Higher-order Poles in CHY Construction
• 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
• Bootstrapping solutions of scattering equations
• Pfaffian Diagrams for Gluon Tree Amplitudes
• Off-Shell Yang-Mills Amplitude in the CHY Formalism
• Permutation in the CHY-Formulation
• Note on solutions of scattering equations
• CHY Theory for Several Fields