Polynomial reduction and evaluation of tree- and loop-level CHY amplitudes

Abstract

We develop a polynomial reduction procedure that transforms any gauge fixed CHY amplitude integrand for $n$ scattering particles into a $\sigma$-moduli multivariate polynomial of what we call the $\text{standard form}$. We show that a standard form polynomial must have a specific $\text{ladder type}$ monomial structure, which has finite size at any $n$, with highest multivariate degree given by $(n-3)(n-4)/2$. This set of monomials spans a complete basis for polynomials with rational coefficients in kinematic data on the support of scattering equations. Subsequently, at tree and one-loop level, we employ the global residue theorem to derive a prescription that evaluates any CHY amplitude by means of collecting simple residues at infinity only. The prescription is then applied explicitly to some tree and one-loop amplitude examples.

28 pages, published version

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Cited by in corpus (21)

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• Manifesting Color-Kinematics Duality in the Scattering Equation Formalism
• Cross-ratio Identities and Higher-order Poles of CHY-integrand
• Analytic Representations of Yang-Mills Amplitudes
• CHY Loop Integrands from Holomorphic Forms
• CHY-Graphs on a Torus
• Quadratic Feynman Loop Integrands From Massless Scattering Equations
• 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
• Asymptotic Four Point Functions
• 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 CHY-Integrand of Bi-adjoint Scalar Theory
• 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
• Note on solutions of scattering equations
• CHY Theory for Several Fields
• Characterizing the solutions to scattering equations that support tree-level ${\text{N}}^k\text{MHV}$ gauge/gravity amplitudes
• Analytic expressions of amplitudes by the cross-ratio identity method