Decomposition of Feynman Integrals by Multivariate Intersection Numbers

Abstract

We present a detailed description of the recent idea for a direct decomposition of Feynman integrals onto a basis of master integrals by projections, as well as a direct derivation of the differential equations satisfied by the master integrals, employing multivariate intersection numbers. We discuss a recursive algorithm for the computation of multivariate intersection numbers and provide three different approaches for a direct decomposition of Feynman integrals, which we dub the straight decomposition, the bottom-up decomposition, and the top-down decomposition. These algorithms exploit the unitarity structure of Feynman integrals by computing intersection numbers supported on cuts, in various orders, thus showing the synthesis of the intersection-theory concepts with unitarity-based methods and integrand decomposition. We perform explicit computations to exemplify all of these approaches applied to Feynman integrals, paving a way towards potential applications to generic multi-loop integrals.

arXiv comment53 Pages, references added; matches published version

References in corpus (60)

• One-Loop n-Point Gauge Theory Amplitudes, Unitarity and Collinear Limits
• Fusing Gauge Theory Tree Amplitudes Into Loop Amplitudes
• Multiloop integrals in dimensional regularization made simple
• Generalized Unitarity and One-Loop Amplitudes in N=4 Super-Yang-Mills
• Reducing full one-loop amplitudes to scalar integrals at the integrand level
• JaxoDraw: A graphical user interface for drawing Feynman diagrams
• FIRE5: a C++ implementation of Feynman Integral REduction
• Kira - A Feynman Integral Reduction Program
• JaxoDraw: A graphical user interface for drawing Feynman diagrams. Version 2.0 release notes
• Direct extraction of one-loop integral coefficients
• Reduze 2 - Distributed Feynman Integral Reduction
• A Numerical Unitarity Formalism for Evaluating One-Loop Amplitudes
• D-dimensional unitarity cut method
• Presenting LiteRed: a tool for the Loop InTEgrals REDuction
• Integration-by-parts reductions from unitarity cuts and algebraic geometry
• Masses, fermions and generalized D-dimensional unitarity
• One-Loop Amplitudes Of Gluons In SQCD
• Feynman Integrals and Intersection Theory
• Magnus and Dyson Series for Master Integrals
• On the maximal cut of Feynman integrals and the solution of their differential equations
• The Cut-Constructible Part of QCD Amplitudes
• Vector Space of Feynman Integrals and Multivariate Intersection Numbers
• A Two-Loop Five-Gluon Helicity Amplitude in QCD
• Scattering Amplitudes from Intersection Theory
• Direct Extraction Of One Loop Rational Terms
• Critical points and number of master integrals
• Cuts of Feynman Integrals in Baikov representation
• On the Integrand-Reduction Method for Two-Loop Scattering Amplitudes
• Scattering Amplitudes from Multivariate Polynomial Division
• Aspects of Scattering Amplitudes and Moduli Space Localization
• Maximal Unitarity at Two Loops
• Integrand-Level Reduction of Loop Amplitudes by Computational Algebraic Geometry Methods
• Decomposition of Feynman Integrals on the Maximal Cut by Intersection Numbers
• On Triple-Cut of Scattering Amplitudes
• Double-Cut of Scattering Amplitudes and Stokes' Theorem
• Feynman integral relations from parametric annihilators
• Baikov-Lee Representations Of Cut Feynman Integrals
• From Infinity to Four Dimensions: Higher Residue Pairings and Feynman Integrals
• On the computation of intersection numbers for twisted cocycles
• Constructing Canonical Feynman Integrals with Intersection Theory
• Analytic Structure of Three-Mass Triangle Coefficients
• Twisted cycles and twisted period relations for Lauricella's hypergeometric function F_C
• Status of Intersection Theory and Feynman Integrals
• From positive geometries to a coaction on hypergeometric functions
• One-Loop Gluonic Amplitudes from Single Unitarity Cuts
• Single Cut Integration
• Solving for tadpole coefficients in one-loop amplitudes
• A note on the Drinfeld associator for genus-zero superstring amplitudes in twisted de Rham theory
• Correlation functions on the lattice and twisted cocycles
• Positive Geometries for all Scalar Theories from Twisted Intersection Theory
• Symmetric $\epsilon$- and $(\epsilon +1/2)$-forms and quadratic constraints in "elliptic" sectors
• Wrońskian factorizations and Broadhurst-Mellit determinant formulae
• Wick rotations, Eichler integrals, and multi-loop Feynman diagrams
• Quadratic relations between Bessel moments
• Quadratic relations between periods of connections
• Euler and Laplace integral representations of GKZ hypergeometric functions
• Relative twisted homology and cohomology groups associated with Lauricella's $F_D$
• Homology and cohomology intersection numbers of GKZ systems
• Intersection numbers and twisted period relations for the generalized hypergeometric function ${}_{m+1}{F}_m$
• Generalized hypergeometric functions and intersection theory for Feynman integrals

Cited by in corpus (53)

• Integral Reduction with Kira 2.0 and Finite Field Methods
• Collider Physics at the Precision Frontier
• Feynman Integrals in Dimensional Regularization and Extensions of Calabi-Yau Motives
• Duals of Feynman Integrals, I: Differential Equations
• Landau Discriminants
• Duals of Feynman Integrals, 2: Generalized Unitarity
• Baikov representations, intersection theory, and canonical Feynman integrals
• On the computation of intersection numbers for twisted cocycles
• The SAGEX Review on Scattering Amplitudes, Chapter 3: Mathematical structures in Feynman integrals
• Macaulay Matrix for Feynman Integrals: Linear Relations and Intersection Numbers
• Constructing Canonical Feynman Integrals with Intersection Theory
• Cosmology meets cohomology
• Les Houches 2021: Physics at TeV Colliders: Report on the Standard Model Precision Wishlist
• Leading singularities in Baikov representation and Feynman integrals with uniform transcendental weight
• Locally finite two-loop amplitudes for off-shell multi-photon production in electron-positron annihilation
• On Epsilon Factorized Differential Equations for Elliptic Feynman Integrals
• Intersection Numbers from Higher-order Partial Differential Equations
• Reduction to master integrals via intersection numbers and polynomial expansions
• Canonical Differential Equations Beyond Genus One
• On one-loop corrections to the Bunch-Davies wavefunction of the universe
• A study of Feynman integrals with uniform transcendental weights and the symbology from dual conformal symmetry
• Coaction and double-copy properties of configuration-space integrals at genus zero
• A first look at the function space for planar two-loop six-particle Feynman integrals
• Non-perturbative computation of lattice correlation functions by differential equations
• Generation Function For One-loop Tensor Reduction
• The recursive structure of Baikov representations I: generics and application to symbology
• A double copy from twisted (co)homology at genus one
• Three-loop banana integrals with four unequal masses
• Real time lattice correlation functions from differential equations
• Mixed QCD-EW corrections for Higgs leptonic decay via $H{W}^{+}{W}^{-}$ vertex
• Graph complexes and Feynman rules
• Module Intersection and Uniform Formula for Iterative Reduction of One-loop Integrals
• Intersection Numbers from Companion Tensor Algebra
• Twisted Riemann bilinear relations and Feynman integrals
• Recursion-free solution for two-loop vacuum integrals with "collinear" masses
• Feynman integral reduction using Gröbner bases
• Higher-order tails and RG flows due to scattering of gravitational radiation from binary inspirals
• The recursive structure of Baikov representations II: the top-down reduction with intersection theory
• Les Houches 2023 -- Physics at TeV Colliders: Report on the Standard Model Precision Wishlist
• Basis decompositions of genus-one string integrals
• General Relativity from Intersection Theory
• Nontrivial One-loop Recursive Reduction Relation
• Differential equations for tree--level cosmological correlators with massive states
• Advanced tools for basis decompositions of genus-one string integrals
• Virtual QCD corrections to gluon-initiated diphoton plus jet production at hadron colliders
• Canonical differential equations and intersection matrices
• Reduction to master integrals and transverse integration identities
• pfd-parallel, a Singular/GPI-Space package for massively parallel multivariate partial fractioning
• Evaluating master integrals in non-factorizable corrections to $t$-channel single-top production at NNLO QCD
• Vanishing cycles and analysis of singularities of Feynman diagrams
• Co-homology of Differential Forms and Feynman diagrams
• Box Singularity Conditions in Box Diagrams of Decay Processes
• The Stratification of Rigidity