Towards a Basis for Planar Two-Loop Integrals

Abstract

The existence of a finite basis of algebraically independent one-loop integrals has underpinned important developments in the computation of one-loop amplitudes in field theories and gauge theories in particular. We give an explicit construction reducing integrals to a finite basis for planar integrals at two loops, both to all orders in the dimensional regulator e, and also when all integrals are truncated to O(e). We show how to reorganize integration-by-parts equations to obtain elements of the first basis efficiently, and how to use Gram determinants to obtain additional linear relations reducing this all-orders basis to the second one. The techniques we present should apply to non-planar integrals, to integrals with massive propagators, and beyond two loops as well.

55 pages, 8 figures, Mathematica file for some formulae; final journal version, typos corrected

References in corpus (28)

• Reducing full one-loop amplitudes to scalar integrals at the integrand level
• Algorithm FIRE -- Feynman Integral REduction
• 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
• The Two-Loop Six-Gluon MHV Amplitude in Maximally Supersymmetric Yang-Mills Theory
• Direct extraction of one-loop integral coefficients
• Assault on the NLO Wishlist: pp -> tt bb
• Maximally Supersymmetric Planar Yang-Mills Amplitudes at Five Loops
• D-dimensional unitarity cut method
• Scattering AMplitudes from Unitarity-based Reduction Algorithm at the Integrand-level
• Sector Decomposition
• Resolution of singularities for multi-loop integrals
• Manifest Ultraviolet Behavior for the Three-Loop Four-Point Amplitude of N=8 Supergravity
• On the Resolution of Singularities of Multiple Mellin-Barnes Integrals
• AMBRE - a Mathematica package for the construction of Mellin-Barnes representations for Feynman integrals
• Bootstrapping One-Loop QCD Amplitudes with General Helicities
• Direct Extraction Of One Loop Rational Terms
• W+3 jet production at the Tevatron
• Numerical integration of one-loop Feynman diagrams for N-photon amplitudes
• Two-Loop Iteration of Five-Point N=4 Super-Yang-Mills Amplitudes
• On the Numerical Evaluation of One-Loop Amplitudes: the Gluonic Case
• Group structure of the integration-by-part identities and its application to the reduction of multiloop integrals
• Integral Coefficients for One-Loop Amplitudes
• All One-loop Maximally Helicity Violating Gluonic Amplitudes in QCD
• Leading Singularities of the Two-Loop Six-Particle MHV Amplitude
• A complete reduction of one-loop tensor 5- and 6-point integrals
• An Algorithm to Construct Groebner Bases for Solving Integration by Parts Relations
• News on Ambre and CSectors

Cited by in corpus (138)

• Integral Reduction with Kira 2.0 and Finite Field Methods
• Reduze 2 - Distributed Feynman Integral Reduction
• AMFlow: a Mathematica package for Feynman integrals computation via Auxiliary Mass Flow
• FiniteFlow: multivariate functional reconstruction using finite fields and dataflow graphs
• One-loop calculations in quantum field theory: from Feynman diagrams to unitarity cuts
• 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
• Cusp and collinear anomalous dimensions in four-loop QCD from form factors
• Magnus and Dyson Series for Master Integrals
• A Two-Loop Five-Gluon Helicity Amplitude in QCD
• Pentagon functions for massless planar scattering amplitudes
• Ultraviolet Properties of N = 8 Supergravity at Five Loops
• Collider Physics at the Precision Frontier
• The two-loop five-point amplitude in $\mathcal{N}=4$ super-Yang-Mills theory
• Hepta-Cuts of Two-Loop Scattering Amplitudes
• Planar Two-Loop Five-Gluon Amplitudes from Numerical Unitarity
• Scattering Amplitudes from Multivariate Polynomial Division
• On the Integrand-Reduction Method for Two-Loop Scattering Amplitudes
• $Apart: A Generalized Mathematica Apart Function
• Maximal Unitarity at Two Loops
• The Five-Loop Four-Point Integrand of N=8 Supergravity as a Generalized Double Copy
• 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
• Two-Loop Four-Gluon Amplitudes with the Numerical Unitarity Method
• Planar Two-Loop Five-Parton Amplitudes from Numerical Unitarity
• Loop amplitudes in gauge theories: modern analytic approaches
• Transforming differential equations of multi-loop Feynman integrals into canonical form
• Multiloop corrections for collider processes using auxiliary mass flow
• Azurite: An algebraic geometry based package for finding bases of loop integrals
• Generic multiloop methods and application to N=4 super-Yang-Mills
• Recursion Relations for AdS/CFT Correlators
• Prescriptive Unitarity
• Complete integration-by-parts reductions of the non-planar hexagon-box via module intersections
• Analytic result for the nonplanar hexa-box integrals
• Determining Feynman integrals with only input from linear algebra
• Two-loop QCD corrections to $Wb\bar{b}$ production at hadron colliders
• Complete sets of logarithmic vector fields for integration-by-parts identities of Feynman integrals
• The two-loop five-point amplitude in $\mathcal{N}=8$ supergravity
• Leading-Color Two-Loop Amplitudes for Four Partons and a W Boson in QCD
• The Soft-Collinear Bootstrap: N=4 Yang-Mills Amplitudes at Six and Seven Loops
• Numerical Evaluation of Tensor Feynman Integrals in Euclidean Kinematics
• Differential equations from unitarity cuts: nonplanar hexa-box integrals
• Two-loop helicity amplitudes for diphoton plus jet production in full color
• Adaptive Integrand Decomposition in parallel and orthogonal space
• Quark and gluon form factors to four loop order in QCD: the $N_f^3$ contributions
• Two-loop leading-colour QCD helicity amplitudes for Higgs boson production in association with a bottom-quark pair at the LHC
• Determine Arbitrary Feynman Integrals by Vacuum Integrals
• Integrand Reduction for Two-Loop Scattering Amplitudes through Multivariate Polynomial Division
• Three-loop gluon scattering in QCD and the gluon Regge trajectory
• Four-loop collinear anomalous dimensions in QCD and $\mathcal{N}=4$ super Yang-Mills
• Isolated photon production in association with a jet pair through next-to-next-to-leading order in QCD
• Two-Loop Maximal Unitarity with External Masses
• The SAGEX Review on Scattering Amplitudes, Chapter 3: Mathematical structures in Feynman integrals
• On Genera of Curves from High-loop Generalized Unitarity Cuts
• Integration-by-parts reductions of Feynman integrals using Singular and GPI-Space
• Complete reduction of integrals in two-loop five-light-parton scattering amplitudes
• Di-photon amplitudes in three-loop Quantum Chromodynamics
• Direct Solution of Integration-by-Parts Systems
• Quark and gluon form factors in four-loop QCD
• Two-loop leading colour QCD corrections to $q\bar{q}\to \gamma \gamma g$ and $qg\to \gamma \gamma q$
• Caravel: A C++ Framework for the Computation of Multi-Loop Amplitudes with Numerical Unitarity
• FIRE 6.5: Feynman Integral Reduction with New Simplification Library
• The ice cone family and iterated integrals for Calabi-Yau varieties
• Dual Conformal Symmetry, Integration-by-Parts Reduction, Differential Equations and the Nonplanar Sector
• Subleading Poles in the Numerical Unitarity Method at Two Loops
• Susy Theories and QCD: Numerical Approaches
• Two-loop five-point massless QCD amplitudes within the IBP approach
• Three-loop helicity amplitudes for diphoton production in gluon fusion
• Two-loop helicity amplitudes for $gg\to ZZ$ with full top-quark mass effects
• The classification of two-loop integrand basis in pure four-dimension
• The Two-Loop Four-Graviton Scattering Amplitudes
• Two-Loop QCD Corrections for Three-Photon Production at Hadron Colliders
• Multivariate Residues and Maximal Unitarity
• Integrands of loop amplitudes within loop-tree duality
• Differential equations on unitarity cut surfaces
• Differential equations for loop integrals in Baikov representation
• IBP reduction coefficients made simple
• Three-loop helicity amplitudes for quark-gluon scattering in QCD
• Ansätze for Scattering Amplitudes from $p$-adic Numbers and Algebraic Geometry
• Counting master integrals: integration by parts vs. differential reduction
• Maximal Unitarity for the Four-Mass Double Box
• Relations of loop partial amplitudes in gauge theory by Unitarity cut method
• Numerical multi-loop integrals and applications
• Manifesting enhanced cancellations in supergravity: integrands versus integrals
• Automatic computation of Feynman integrals containing linear propagators via auxiliary mass flow
• Building Bases of Loop Integrands
• On one-loop corrections to the Bunch-Davies wavefunction of the universe
• Two-Loop Five-Point Two-Mass Planar Integrals and Double Lagrangian Insertions in a Wilson Loop
• Full-colour double-virtual amplitudes for associated production of a Higgs boson with a bottom-quark pair at the LHC
• Cross-Order Integral Relations from Maximal Cuts
• Numerical evaluation of two-loop QCD helicity amplitudes for $gg\to t\bar{t}g$ at leading colour
• Review of AdS/CFT Integrability, Chapter V.1: Scattering Amplitudes - a Brief Introduction
• The SAGEX Review on Scattering Amplitudes, Chapter 4: Multi-loop Feynman Integrals
• Global Poles of the Two-Loop Six-Point N=4 SYM integrand
• An Overview of Maximal Unitarity at Two Loops
• Tree-Loop Duality Relation beyond simple poles
• PV-Reduction of Sunset Topology with Auxiliary Vector
• Factorization of denominators in integration-by-parts reductions
• Two-loop Feynman integrals for leading colour $t\bar{t}W$ production at hadron colliders
• Computational challenges for multi-loop collider phenomenology
• Collinear and Soft Limits of Multi-Loop Integrands in N=4 Yang-Mills
• Module Intersection and Uniform Formula for Iterative Reduction of One-loop Integrals
• Multi-loop Integrand Reduction with Computational Algebraic Geometry
• Maximal transcendental weight contribution of scattering amplitudes
• Three-loop helicity amplitudes for photon+jet production
• Feynman integral reduction using Gröbner bases
• Integration-by-parts identities from the viewpoint of differential geometry
• Double virtual QCD corrections to $t\bar{t}+$jet production at the LHC
• Renormalized $\epsilon$-finite master integrals and their virtues: the three-loop self energy case
• Virtual QCD corrections to gluon-initiated diphoton plus jet production at hadron colliders
• On the finite basis topologies for multi-loop high-multiplicity Feynman integrals
• Scaling up to Multivariate Rational Function Reconstruction
• Reduction to master integrals and transverse integration identities
• Two-loop all-plus helicity amplitudes for self-dual Higgs boson with gluons via unitarity cut constraints
• Integral Reduction by Unitarity Method for Two-loop Amplitudes: A Case Study
• Four-dimensional differential equations for the leading divergences of dimensionally-regulated loop integrals
• On the two-loop amplitude for $gg\to ZZ$ production with full top-mass dependence
• Evaluation of three-loop self-energy master integrals with four or five propagators
• CALICO: Computing Annihilators from Linear Identities Constraining (differential) Operators
• Fermionic corrections to quark and gluon form factors in four-loop QCD
• Rational Function Simplification for Integration-by-Parts Reduction and Beyond
• The HiggsTools Handbook: Concepts and observables for deciphering the Nature of the Higgs Sector
• Automated computation of scattering amplitudes
• Three-loop helicity amplitudes for four-quark scattering in massless QCD
• Accelerating Berends-Giele recursion for gluons in arbitrary dimensions over finite fields
• A new method for finding more symmetry relations of Feynman integrals
• pfd-parallel, a Singular/GPI-Space package for massively parallel multivariate partial fractioning
• Top-Yukawa contributions to $pp\to b\bar{b}H$: two-loop leading-colour amplitudes
• The spectrum of Feynman-integral geometries at two loops
• Two-loop D-dimensional unitarity and dual conformal symmetry
• One-Loop QCD Corrections to $\bar{u}d\to t\bar{t}W$ at $\mathcal{O}({\epsilon }^2)$
• IBP reduction via Gröbner bases in a rational double-shift algebra
• Recent new methods and applications of the differential equation approach to master integrals
• Three-loop four-particle QCD amplitudes
• Two-loop leading-color QCD corrections for Higgs plus two-jet production in the heavy-top limit
• The integrand form of infrared singularities of two-loop QCD scattering amplitudes