Integrand Reduction for Two-Loop Scattering Amplitudes through Multivariate Polynomial Division

Abstract

We describe the application of a novel approach for the reduction of scattering amplitudes, based on multivariate polynomial division, which we have recently presented. This technique yields the complete integrand decomposition for arbitrary amplitudes, regardless of the number of loops. It allows for the determination of the residue at any multiparticle cut, whose knowledge is a mandatory prerequisite for applying the integrand-reduction procedure. By using the division modulo Groebner basis, we can derive a simple integrand recurrence relation that generates the multiparticle pole decomposition for integrands of arbitrary multiloop amplitudes. We apply the new reduction algorithm to the two-loop planar and nonplanar diagrams contributing to the five-point scattering amplitudes in N=4 super Yang-Mills and N=8 supergravity in four dimensions, whose numerator functions contain up to rank-two terms in the integration momenta. We determine all polynomial residues parametrizing the cuts of the corresponding topologies and subtopologies. We obtain the integral basis for the decomposition of each diagram from the polynomial form of the residues. Our approach is well suited for a seminumerical implementation, and its general mathematical properties provide an effective algorithm for the generalization of the integrand-reduction method to all orders in perturbation theory.

32 pages, 4 figures. v2: published version, text improved, new subsection 4.4 added

References in corpus (23)

• Reducing full one-loop amplitudes to scalar integrals at the integrand level
• An Automated Implementation of On-Shell Methods for One-Loop Amplitudes
• The Four-Loop Planar Amplitude and Cusp Anomalous Dimension in Maximally Supersymmetric Yang-Mills Theory
• Direct extraction of one-loop integral coefficients
• Full one-loop amplitudes from tree amplitudes
• Maximally Supersymmetric Planar Yang-Mills Amplitudes at Five Loops
• A Numerical Unitarity Formalism for Evaluating One-Loop Amplitudes
• On the Rational Terms of the one-loop amplitudes
• Scattering Amplitudes, Wilson Loops and the String/Gauge Theory Correspondence
• Numerical evaluation of virtual corrections to multi-jet production in massless QCD
• Direct Extraction Of One Loop Rational Terms
• S@M, a Mathematica Implementation of the Spinor-Helicity Formalism
• Sharpening The Leading Singularity
• On the Numerical Evaluation of One-Loop Amplitudes: the Gluonic Case
• Numerical Evaluation of Six-Photon Amplitudes
• Double-Cut of Scattering Amplitudes and Stokes' Theorem
• Optimizing the Reduction of One-Loop Amplitudes
• Holomorphic Anomaly Of Unitarity Cuts And One-Loop Gauge Theory Amplitudes
• Two-Loop Maximal Unitarity with External Masses
• An Integrand Reconstruction Method for Three-Loop Amplitudes
• The classification of two-loop integrand basis in pure four-dimension
• Single Cut Integration
• Counting to one: reducibility of one- and two-loop amplitudes at the integrand level

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