Integration-by-parts reductions of Feynman integrals using Singular and GPI-Space

Abstract

We introduce an algebro-geometrically motived integration-by-parts (IBP) reduction method for multi-loop and multi-scale Feynman integrals, using a framework for massively parallel computations in computer algebra. This framework combines the computer algebra system Singular with the workflow management system GPI-Space, which is being developed at the Fraunhofer Institute for Industrial Mathematics (ITWM). In our approach, the IBP relations are first trimmed by modern algebraic geometry tools and then solved by sparse linear algebra and our new interpolation methods. These steps are efficiently automatized and automatically parallelized by modeling the algorithm in GPI-Space using the language of Petri-nets. We demonstrate the potential of our method at the nontrivial example of reducing two-loop five-point nonplanar double-pentagon integrals. We also use GPI-Space to convert the basis of IBP reductions, and discuss the possible simplification of IBP coefficients in a uniformly transcendental basis.

minor corrections; We encourage researchers in high energy community to send us IBP reduction problems (mailto: [email protected])

References in corpus (66)

• Multiloop integrals in dimensional regularization made simple
• Algorithm FIRE -- Feynman Integral REduction
• FIRE5: a C++ implementation of Feynman Integral REduction
• Kira - A Feynman Integral Reduction Program
• Local Integrals for Planar Scattering Amplitudes
• FIRE6: Feynman Integral REduction with Modular Arithmetic
• Lectures on differential equations for Feynman integrals
• A novel approach to integration by parts reduction
• Reduze 2 - Distributed Feynman Integral Reduction
• Bootstrapping the three-loop hexagon
• Scattering amplitudes over finite fields and multivariate functional reconstruction
• FiniteFlow: multivariate functional reconstruction using finite fields and dataflow graphs
• Bootstrapping a Five-Loop Amplitude Using Steinmann Relations
• Towards a Basis for Planar Two-Loop Integrals
• A Systematic and Efficient Method to Compute Multi-loop Master Integrals
• Two-loop Integrand Decomposition into Master Integrals and Surface Terms
• Integration-by-parts reductions from unitarity cuts and algebraic geometry
• Bootstrapping an NMHV amplitude through three loops
• Reconstructing Rational Functions with $\text{FireFly}$
• Feynman Integrals and Intersection Theory
• Heptagons from the Steinmann Cluster Bootstrap
• Explicit solutions of the 3--loop vacuum integral recurrence relations
• Hexagon functions and the three-loop remainder function
• Analytic form of the two-loop planar five-gluon all-plus-helicity amplitude in QCD
• Vector Space of Feynman Integrals and Multivariate Intersection Numbers
• A Two-Loop Five-Gluon Helicity Amplitude in QCD
• Pentagon functions for massless planar scattering amplitudes
• Critical points and number of master integrals
• The two-loop five-point amplitude in $\mathcal{N}=4$ super-Yang-Mills theory
• Analytic Form of the Planar Two-Loop Five-Parton Scattering Amplitudes in QCD
• Six-Gluon Amplitudes in Planar $\mathcal{N}=4$ Super-Yang-Mills Theory at Six and Seven Loops
• All master integrals for three-jet production at NNLO
• Analytic Form of the Planar Two-Loop Five-Gluon Scattering Amplitudes in QCD
• Planar Two-Loop Five-Gluon Amplitudes from Numerical Unitarity
• Scattering Amplitudes from Multivariate Polynomial Division
• A New Algorithm For The Generation Of Unitarity-Compatible Integration By Parts Relations
• A first look at two-loop five-gluon scattering in QCD
• Integrand-Level Reduction of Loop Amplitudes by Computational Algebraic Geometry Methods
• Decomposition of Feynman Integrals on the Maximal Cut by Intersection Numbers
• Analytic form of the full two-loop five-gluon all-plus helicity amplitude
• Two-Loop Four-Gluon Amplitudes with the Numerical Unitarity Method
• Planar Two-Loop Five-Parton Amplitudes from Numerical Unitarity
• Analytic result for a two-loop five-particle amplitude
• 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
• Bootstrapping pentagon functions
• Complete integration-by-parts reductions of the non-planar hexagon-box via module intersections
• Analytic result for the nonplanar hexa-box integrals
• Complete sets of logarithmic vector fields for integration-by-parts identities of Feynman integrals
• Feynman integral relations from parametric annihilators
• The two-loop five-point amplitude in $\mathcal{N}=8$ supergravity
• Determine Arbitrary Feynman Integrals by Vacuum Integrals
• The two-loop five-particle amplitude in $\mathcal{N}=8$ supergravity
• Direct Solution of Integration-by-Parts Systems
• Dual Conformal Symmetry, Integration-by-Parts Reduction, Differential Equations and the Nonplanar Sector
• Two-loop five-point massless QCD amplitudes within the IBP approach
• Lecture Notes on Multi-loop Integral Reduction and Applied Algebraic Geometry
• Direct reduction of multiloop multiscale scattering amplitudes
• Kira 1.2 Release Notes
• Efficient integrand reduction for particles with spin
• Algebraic geometry and Bethe ansatz (I) the quotient ring for BAE
• Modern techniques of multiloop calculations
• The four-loop six-gluon NMHV ratio function
• Towards Massively Parallel Computations in Algebraic Geometry
• A new reduction strategy for special negative sectors of planar two-loop integrals without Laporta algorithm
• Torus partition function of the six-vertex model from algebraic geometry

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• Les Houches 2021: Physics at TeV Colliders: Report on the Standard Model Precision Wishlist
• FIRE 6.5: Feynman Integral Reduction with New Simplification Library
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• Mathematical properties of nested residues and their application to multi-loop scattering amplitudes
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• Multi-Regge Limit of the Two-Loop Five-Point Amplitudes in $\mathcal{N}=4$ Super Yang-Mills and $\mathcal{N}=8$ Supergravity
• The recursive structure of Baikov representations I: generics and application to symbology
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• Module Intersection and Uniform Formula for Iterative Reduction of One-loop Integrals
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• Feynman integral reduction using Gröbner bases
• The recursive structure of Baikov representations II: the top-down reduction with intersection theory
• Four-dimensional differential equations for the leading divergences of dimensionally-regulated loop integrals
• Rational Function Simplification for Integration-by-Parts Reduction and Beyond
• CALICO: Computing Annihilators from Linear Identities Constraining (differential) Operators
• pfd-parallel, a Singular/GPI-Space package for massively parallel multivariate partial fractioning
• Weak Pontryagin's Maximum Principle for Optimal Control Problems Involving a General Analytic Kernel
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