Adaptive Integrand Decomposition in parallel and orthogonal space

Abstract

We present the integrand decomposition of multiloop scattering amplitudes in parallel and orthogonal space-time dimensions, $d=d_{\parallel }+d_{\perp }$, being $d_{\parallel }$ the dimension of the parallel space spanned by the legs of the diagrams. When the number $n$ of external legs is $n\le 4$, the corresponding representation of the multiloop integrals exposes a subset of integration variables which can be easily integrated away by means of Gegenbauer polynomials orthogonality condition. By decomposing the integration momenta along parallel and orthogonal directions, the polynomial division algorithm is drastically simplified. Moreover, the orthogonality conditions of Gegenbauer polynomials can be suitably applied to integrate the decomposed integrand, yielding the systematic annihilation of spurious terms. Consequently, multiloop amplitudes are expressed in terms of integrals corresponding to irreducible scalar products of loop momenta and external momenta. We revisit the one-loop decomposition, which turns out to be controlled by the maximum-cut theorem in different dimensions, and we discuss the integrand reduction of two-loop planar and non-planar integrals up to $n=8$ legs, for arbitrary external and internal kinematics. The proposed algorithm extends to all orders in perturbation theory.

arXiv comment64 pages, 4 figures, 8 tables

References in corpus (37)

• Generalized Unitarity and One-Loop Amplitudes in N=4 Super-Yang-Mills
• Reducing full one-loop amplitudes to scalar integrals at the integrand level
• CutTools: a program implementing the OPP reduction method to compute one-loop amplitudes
• Connection between Feynman integrals having different values of the space-time dimension
• The All-Loop Integrand For Scattering Amplitudes in Planar N=4 SYM
• Local Integrals for Planar Scattering Amplitudes
• Massive Loop Amplitudes from Unitarity
• A novel approach to integration by parts reduction
• Higgs boson pair production in gluon fusion at NLO with full top-quark mass dependence
• A Numerical Unitarity Formalism for Evaluating One-Loop Amplitudes
• On the Rational Terms of the one-loop amplitudes
• One-loop calculations in quantum field theory: from Feynman diagrams to unitarity cuts
• Towards a Basis for Planar Two-Loop Integrals
• Scattering AMplitudes from Unitarity-based Reduction Algorithm at the Integrand-level
• Integration-by-parts reductions from unitarity cuts and algebraic geometry
• Masses, fermions and generalized D-dimensional unitarity
• Space-time dimensionality D as complex variable: calculating loop integrals using dimensional recurrence relation and analytical properties with respect to D
• Explicit solutions of the multi-loop integral recurrence relations and its application
• Ninja: Automated Integrand Reduction via Laurent Expansion for One-Loop Amplitudes
• A Two-Loop Five-Gluon Helicity Amplitude in QCD
• Hepta-Cuts of Two-Loop Scattering Amplitudes
• On the Integrand-Reduction Method for Two-Loop Scattering Amplitudes
• Scattering Amplitudes from Multivariate Polynomial Division
• Recursive numerical calculus of one-loop tensor integrals
• S@M, a Mathematica Implementation of the Spinor-Helicity Formalism
• Integrand-Level Reduction of Loop Amplitudes by Computational Algebraic Geometry Methods
• A Complete Two-Loop, Five-Gluon Helicity Amplitude in Yang-Mills Theory
• Numerical Evaluation of Six-Photon Amplitudes
• Integrand reduction of one-loop scattering amplitudes through Laurent series expansion
• Tensorial Reconstruction at the Integrand Level
• Optimizing the Reduction of One-Loop Amplitudes
• Integrand Reduction for Two-Loop Scattering Amplitudes through Multivariate Polynomial Division
• Multiloop Integrand Reduction for Dimensionally Regulated Amplitudes
• Tensor integrand reduction via Laurent expansion
• The classification of two-loop integrand basis in pure four-dimension
• Finding Linear Dependencies in Integration-By-Parts Equations: A Monte Carlo Approach
• New representation of the two-loop crossed vertex function

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• Maximal Cuts in Arbitrary Dimension
• Determine Arbitrary Feynman Integrals by Vacuum Integrals
• Direct Solution of Integration-by-Parts Systems
• Hadronic corrections to $\mu$-$e$ scattering at NNLO with space-like data
• Dual Conformal Symmetry, Integration-by-Parts Reduction, Differential Equations and the Nonplanar Sector
• Scattering Amplitudes in Quantum Field Theory
• Mathematical properties of nested residues and their application to multi-loop scattering amplitudes
• Local Unitarity: a representation of differential cross-sections that is locally free of infrared singularities at any order
• Muon-electron scattering at NNLO
• Geometrical approach to causality in multi-loop amplitudes
• Two-loop leading colour QCD helicity amplitudes for top quark pair production in the gluon fusion channel
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• Multi-loop calculations: numerical methods and applications
• Two-loop scattering amplitude for heavy-quark pair production through light-quark annihilation in QCD
• The spectrum of Feynman-integral geometries at two loops
• Co-homology of Differential Forms and Feynman diagrams
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• Recent Developments in Higher-Order Calculations: Hard Functions at NLO with GoSam