Multivariate Residues and Maximal Unitarity

Abstract

We extend the maximal unitarity method to amplitude contributions whose cuts define multidimensional algebraic varieties. The technique is valid to all orders and is explicitly demonstrated at three loops in gauge theories with any number of fermions and scalars in the adjoint representation. Deca-cuts realized by replacement of real slice integration contours by higher-dimensional tori encircling the global poles are used to factorize the planar triple box onto a product of trees. We apply computational algebraic geometry and multivariate complex analysis to derive unique projectors for all master integral coefficients and obtain compact analytic formulae in terms of tree-level data.

34 pages, 3 figures

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Cited by in corpus (28)

• Ultraviolet Properties of N = 8 Supergravity at Five Loops
• Cuts of Feynman Integrals in Baikov representation
• The Five-Loop Four-Point Integrand of N=8 Supergravity as a Generalized Double Copy
• A Complete Two-Loop, Five-Gluon Helicity Amplitude in Yang-Mills Theory
• Azurite: An algebraic geometry based package for finding bases of loop integrals
• Local Integrand Representations of All Two-Loop Amplitudes in Planar SYM
• Maximal Cuts in Arbitrary Dimension
• Dual Conformal Symmetry, Integration-by-Parts Reduction, Differential Equations and the Nonplanar Sector
• Scattering Equations and Global Duality of Residues
• Elliptic Functions and Maximal Unitarity
• Local integrands for two-loop all-plus Yang-Mills amplitudes
• Getting more flavour out of one-flavour QCD
• Full Colour for Loop Amplitudes in Yang-Mills Theory
• The Polynomial Form of the Scattering Equations is an H-Basis
• Unitarity Cuts of Integrals with Doubled Propagators
• MultivariateResidues: a Mathematica package for computing multivariate residues
• A first look at the function space for planar two-loop six-particle Feynman integrals
• Cross-Order Integral Relations from Maximal Cuts
• On-shell diagrammatics and the perturbative structure of planar gauge theories
• Global Structure of Curves from Generalized Unitarity Cut of Three-loop Diagrams
• Massive Nonplanar Two-Loop Maximal Unitarity
• Maximal transcendental weight contribution of scattering amplitudes
• Two-loop Integral Reduction from Elliptic and Hyperelliptic Curves
• Integration-by-parts identities from the viewpoint of differential geometry
• Integral Reduction by Unitarity Method for Two-loop Amplitudes: A Case Study
• The HiggsTools Handbook: Concepts and observables for deciphering the Nature of the Higgs Sector
• MultivariateResidues - a Mathematica package for computing multivariate residues
• Co-homology of Differential Forms and Feynman diagrams