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

• Published in: The journal of high energy physics 2016-08, Vol.2016 (8), p.1-29, Article 143
• Main Author: Zlotnikov, Michael
• Format: Article
• Language: English
• Subjects: Amplitudes; Classical and Quantum Gravitation; Differential and Algebraic Geometry; Elementary Particles; Field Theories in Higher Dimensions; High energy physics; Mathematical analysis; MATHEMATICS AND COMPUTING; Physics; Physics and Astronomy; PHYSICS OF ELEMENTARY PARTICLES AND FIELDS; Polynomials; Quantum Field Theories; Quantum Field Theory; Quantum Physics; Reduction; Regular Article - Theoretical Physics; Relativity Theory; Residues; Scattering; Scattering Amplitudes; String Theory; Trees
• ISSN: 1029-8479; 1029-8479
• DOI: 10.1007/JHEP08(2016)143

A bstract We develop a polynomial reduction procedure that transforms any gauge fixed CHY amplitude integrand for n scattering particles into a σ -moduli multivariate polynomial of what we call the standard form . We show that a standard form polynomial must have a specific 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.