Scattering Amplitudes from Multivariate Polynomial Division

Abstract

We show that the evaluation of scattering amplitudes can be formulated as a problem of multivariate polynomial division, with the components of the integration-momenta as indeterminates. We present a recurrence relation which, independently of the number of loops, leads to the multi-particle pole decomposition of the integrands of the scattering amplitudes. The recursive algorithm is based on the Weak Nullstellensatz Theorem and on the division modulo the Groebner basis associated to all possible multi-particle cuts. We apply it to dimensionally regulated one-loop amplitudes, recovering the well-known integrand-decomposition formula. Finally, we focus on the maximum-cut, defined as a system of on-shell conditions constraining the components of all the integration-momenta. By means of the Finiteness Theorem and of the Shape Lemma, we prove that the residue at the maximum-cut is parametrised by a number of coefficients equal to the number of solutions of the cut itself.

6 pages, 1 figure; Published version. References added

References in corpus (11)

• 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
• 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
• S@M, a Mathematica Implementation of the Spinor-Helicity Formalism
• Integrand-Level Reduction of Loop Amplitudes by Computational Algebraic Geometry Methods
• On the Numerical Evaluation of One-Loop Amplitudes: the Gluonic Case
• Numerical Evaluation of Six-Photon Amplitudes
• Optimizing the Reduction of One-Loop Amplitudes

Cited by in corpus (83)

• QCD and strongly coupled gauge theories: challenges and perspectives
• Lectures on differential equations for Feynman integrals
• GoSam-2.0: a tool for automated one-loop calculations within the Standard Model and beyond
• Scattering amplitudes over finite fields and multivariate functional reconstruction
• FiniteFlow: multivariate functional reconstruction using finite fields and dataflow graphs
• Two-loop Integrand Decomposition into Master Integrals and Surface Terms
• Les Houches 2015: Physics at TeV Colliders Standard Model Working Group Report
• Ninja: Automated Integrand Reduction via Laurent Expansion for One-Loop Amplitudes
• On the maximal cut of Feynman integrals and the solution of their differential equations
• A Two-Loop Five-Gluon Helicity Amplitude in QCD
• To $d$, or not to $d$: Recent developments and comparisons of regularization schemes
• Collider Physics at the Precision Frontier
• Analytic Form of the Planar Two-Loop Five-Parton Scattering Amplitudes in QCD
• Analytic Form of the Planar Two-Loop Five-Gluon Scattering Amplitudes in QCD
• Planar Two-Loop Five-Gluon Amplitudes from Numerical Unitarity
• A first look at two-loop five-gluon scattering in QCD
• Constructing d-log integrands and computing master integrals for three-loop four-particle scattering
• Two-Loop Four-Gluon Amplitudes with the Numerical Unitarity Method
• Decomposition of Feynman Integrals by Multivariate Intersection Numbers
• A Complete Two-Loop, Five-Gluon Helicity Amplitude in Yang-Mills Theory
• Analytic helicity amplitudes for two-loop five-gluon scattering: the single-minus case
• Azurite: An algebraic geometry based package for finding bases of loop integrals
• NLO QCD corrections to Higgs boson production plus three jets in gluon fusion
• Prescriptive Unitarity
• Complete integration-by-parts reductions of the non-planar hexagon-box via module intersections
• NLO QCD corrections to the production of Higgs plus two jets at the LHC
• Adaptive Integrand Decomposition in parallel and orthogonal space
• Physical projectors for multi-leg helicity amplitudes
• Multi-leg One-loop Massive Amplitudes from Integrand Reduction via Laurent Expansion
• Three-loop master integrals for ladder-box diagrams with one massive leg
• Two-loop five-point all plus helicity Yang-Mills amplitude
• Integrand Reduction for Two-Loop Scattering Amplitudes through Multivariate Polynomial Division
• Two-Loop Maximal Unitarity with External Masses
• Integration-by-parts reductions of Feynman integrals using Singular and GPI-Space
• On Genera of Curves from High-loop Generalized Unitarity Cuts
• Multiloop Integrand Reduction for Dimensionally Regulated Amplitudes
• Caravel: A C++ Framework for the Computation of Multi-Loop Amplitudes with Numerical Unitarity
• Dual Conformal Symmetry, Integration-by-Parts Reduction, Differential Equations and the Nonplanar Sector
• Local integrands for two-loop all-plus Yang-Mills amplitudes
• Mathematical properties of nested residues and their application to multi-loop scattering amplitudes
• Subleading Poles in the Numerical Unitarity Method at Two Loops
• Gauge invariance at work in FDR: H -> gamma gamma
• The classification of two-loop integrand basis in pure four-dimension
• Multi-Quark Colour Decompositions from Unitarity
• Geometrical approach to causality in multi-loop amplitudes
• Multivariate Residues and Maximal Unitarity
• Automating QCD amplitudes with on-shell methods
• Quantum algorithm for Feynman loop integrals
• Ansätze for Scattering Amplitudes from $p$-adic Numbers and Algebraic Geometry
• Full Colour for Loop Amplitudes in Yang-Mills Theory
• Counting to one: reducibility of one- and two-loop amplitudes at the integrand level
• Lecture Notes on Multi-loop Integral Reduction and Applied Algebraic Geometry
• Numerical multi-loop integrals and applications
• FDR, an easier way to NNLO calculations: a two-loop case study
• Maximal Unitarity for the Four-Mass Double Box
• Unitarity Cuts of Integrals with Doubled Propagators
• Building Bases of Loop Integrands
• Global Residues and Two-Loop Hepta-Cuts
• BCJ Identities and $d$-Dimensional Generalized Unitarity
• Computation of multi-leg amplitudes with NJet
• Cross-Order Integral Relations from Maximal Cuts
• The Q-cut Representation of One-loop Integrands and Unitarity Cut Method
• Radiative Classical Gravitational Observables at $\mathcal{O}(G^3)$ from Scattering Amplitudes
• An Overview of Maximal Unitarity at Two Loops
• Global Structure of Curves from Generalized Unitarity Cut of Three-loop Diagrams
• Massive Nonplanar Two-Loop Maximal Unitarity
• Multi-loop Integrand Reduction with Computational Algebraic Geometry
• Maximal transcendental weight contribution of scattering amplitudes
• Integration-by-parts identities from the viewpoint of differential geometry
• Virtual QCD corrections to gluon-initiated diphoton plus jet production at hadron colliders
• The Integrand Reduction of One- and Two-Loop Scattering Amplitudes
• Integral Reduction by Unitarity Method for Two-loop Amplitudes: A Case Study
• NNLO QED contribution to the $\mu e\to \mu e$ elastic scattering
• Analytic structure of one-loop coefficients
• Automated computation of scattering amplitudes
• The HiggsTools Handbook: Concepts and observables for deciphering the Nature of the Higgs Sector
• The $\gamma \gamma$ decay of the Higgs boson in FDR
• Scattering Amplitudes in Gauge Theories
• Universal Decomposition of Phase-Space Integrands
• QCD Amplitudes: new perspectives on Feynman integral calculus
• A stroll through the loop-tree duality
• Quantum field theory in four dimensions
• Co-homology of Differential Forms and Feynman diagrams