Complete sets of logarithmic vector fields for integration-by-parts identities of Feynman integrals

Abstract

Integration-by-parts identities between loop integrals arise from the vanishing integration of total derivatives in dimensional regularization. Generic choices of total derivatives in the Baikov or parametric representations lead to identities which involve dimension shifts. These dimension shifts can be avoided by imposing a certain constraint on the total derivatives. The solutions of this constraint turn out to be a specific type of syzygies which correspond to logarithmic vector fields along the Gram determinant formed of the independent external and loop momenta. We present an explicit generating set of solutions in Baikov representation, valid for any number of loops and external momenta, obtained from the Laplace expansion of the Gram determinant. We provide a rigorous mathematical proof that this set of solutions is complete. This proof relates the logarithmic vector fields in question to ideals of submaximal minors of the Gram matrix and makes use of classical resolutions of such ideals.

12 pages, two-column format, 3 figures. Added example with internal masses; minor corrections; journal version

References in corpus (26)

• Multiloop integrals in dimensional regularization made simple
• Differential Equations for Two-Loop Four-Point Functions
• Differential Equations for Feynman Graph Amplitudes
• Dimensionally Regulated Pentagon Integrals
• Algorithm FIRE -- Feynman Integral REduction
• FIRE5: a C++ implementation of Feynman Integral REduction
• Kira - A Feynman Integral Reduction Program
• Reduze - Feynman Integral Reduction in C++
• Automatic Integral Reduction for Higher Order Perturbative Calculations
• A novel approach to integration by parts reduction
• 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
• Explicit solutions of the 3--loop vacuum integral recurrence relations
• Analytic form of the two-loop planar five-gluon all-plus-helicity amplitude in QCD
• Calculating Three Loop Ladder and V-Topologies for Massive Operator Matrix Elements by Computer Algebra
• Planar Two-Loop Five-Gluon Amplitudes from Numerical Unitarity
• A New Algorithm For The Generation Of Unitarity-Compatible Integration By Parts Relations
• A first look at two-loop five-gluon scattering in QCD
• Azurite: An algebraic geometry based package for finding bases of loop integrals
• Calculation of master integrals by difference equations
• The Pentabox Master Integrals with the Simplified Differential Equations approach
• Simplified differential equations approach for Master Integrals
• Direct Solution of Integration-by-Parts Systems
• Dual Conformal Symmetry, Integration-by-Parts Reduction, Differential Equations and the Nonplanar Sector

Cited by in corpus (47)

• Integral Reduction with Kira 2.0 and Finite Field Methods
• Cusp and collinear anomalous dimensions in four-loop QCD from form factors
• Collider Physics at the Precision Frontier
• Analytic form of the full two-loop five-gluon all-plus helicity amplitude
• Complete integration-by-parts reductions of the non-planar hexagon-box via module intersections
• Analytic result for the nonplanar hexa-box integrals
• Differential equations from unitarity cuts: nonplanar hexa-box integrals
• Two-loop helicity amplitudes for diphoton plus jet production in full color
• 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
• 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
• Integration-by-parts reductions of Feynman integrals using Singular and GPI-Space
• Di-photon amplitudes in three-loop Quantum Chromodynamics
• Quark and gluon form factors in four-loop QCD
• Two-loop leading colour QCD corrections to $q\bar{q}\to \gamma \gamma g$ and $qg\to \gamma \gamma q$
• 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
• Three-loop helicity amplitudes for diphoton production in gluon fusion
• Two-loop helicity amplitudes for $gg\to ZZ$ with full top-quark mass effects
• Two-Loop QCD Corrections for Three-Photon Production at Hadron Colliders
• Three-loop helicity amplitudes for quark-gluon scattering in QCD
• Locally finite two-loop amplitudes for off-shell multi-photon production in electron-positron annihilation
• Kira 1.2 Release Notes
• Full-colour double-virtual amplitudes for associated production of a Higgs boson with a bottom-quark pair at the LHC
• Factorization of denominators in integration-by-parts reductions
• Computational challenges for multi-loop collider phenomenology
• Module Intersection and Uniform Formula for Iterative Reduction of One-loop Integrals
• Three-loop helicity amplitudes for photon+jet production
• Feynman integral reduction using Gröbner bases
• Virtual QCD corrections to gluon-initiated diphoton plus jet production at hadron colliders
• Two-loop all-plus helicity amplitudes for self-dual Higgs boson with gluons via unitarity cut constraints
• Four-dimensional differential equations for the leading divergences of dimensionally-regulated loop integrals
• Reduction to master integrals and transverse integration identities
• Fermionic corrections to quark and gluon form factors in four-loop QCD
• On the two-loop amplitude for $gg\to ZZ$ production with full top-mass dependence
• CALICO: Computing Annihilators from Linear Identities Constraining (differential) Operators
• A new method for finding more symmetry relations of Feynman integrals
• Three-loop helicity amplitudes for four-quark scattering in massless QCD
• pfd-parallel, a Singular/GPI-Space package for massively parallel multivariate partial fractioning
• Massively parallel computations in algebraic geometry - not a contradiction
• Top-Yukawa contributions to $pp\to b\bar{b}H$: two-loop leading-colour amplitudes
• Extended Projection Method for Massive Fermion
• IBP reduction via Gröbner bases in a rational double-shift algebra
• Reduction of planar double-box diagram for single-top production via auxiliary mass flow
• A stroll through the loop-tree duality