The classification of two-loop integrand basis in pure four-dimension

Abstract

In this paper, we have made the attempt to classify the integrand basis of all two-loop diagrams in pure four-dimension space-time. Our classification includes the topology of two-loop diagrams which determines the structure of denominators, and the set of numerators under different kinematic configurations of external momenta by using Gröbner basis method. In our study, the variety defined by setting all propagators to on-shell has played an important role. We discuss the structure of variety and how it splits to various irreducible branches when external momenta at each corner of diagrams satisfy some special kinematic conditions. This information is crucial to the numerical or analytical fitting of coefficients for integrand basis in reduction process.

52 pages, 9 figures. v2 reference added, v3 published version

References in corpus (16)

• Reducing full one-loop amplitudes to scalar integrals at the integrand level
• Direct extraction of one-loop integral coefficients
• Full one-loop amplitudes from tree amplitudes
• A Numerical Unitarity Formalism for Evaluating One-Loop Amplitudes
• On the Rational Terms of the one-loop amplitudes
• D-dimensional unitarity cut method
• Uniqueness of two-loop master contours
• Direct Extraction Of One Loop Rational Terms
• Scattering Amplitudes from Multivariate Polynomial Division
• Integrand-Level Reduction of Loop Amplitudes by Computational Algebraic Geometry Methods
• The SM and NLO Multileg and SM MC Working Groups: Summary Report
• Two-Loop Maximal Unitarity with External Masses
• An Integrand Reconstruction Method for Three-Loop Amplitudes
• One-loop Integral Coefficients from Generalized Unitarity
• Counting to one: reducibility of one- and two-loop amplitudes at the integrand level
• Global Poles of the Two-Loop Six-Point N=4 SYM integrand

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