Baikov representations, intersection theory, and canonical Feynman integrals

Abstract

The method of canonical differential equations is an important tool in the calculation of Feynman integrals in quantum field theories. It has been realized that the canonical bases are closely related to $d$-dimensional $d\log$-form integrands. In this work, we introduce the concept of generalized loop-by-loop Baikov representation, and clarify its relation and difference with Feynman integrals using the language of intersection theory. We then utilize the generalized Baikov representation to construct $d$-dimensional $d\log$-form integrands, and discuss how to convert them to Feynman integrals. We describe the technical details of our method, in particular how to deal with the difficulties encountered in the construction procedure. Our method provides a constructive approach to the problem of finding canonical bases of Feynman integrals, and we demonstrate its applicability to complicated scattering amplitudes involving multiple physical scales.

arXiv commentv2: published version in JHEP

References in corpus (60)

• Multiloop integrals in dimensional regularization made simple
• Differential Equations for Two-Loop Four-Point Functions
• Differential Equations for Feynman Graph Amplitudes
• The Amplituhedron
• LiteRed 1.4: a powerful tool for the reduction of the multiloop integrals
• Algorithm FIRE -- Feynman Integral REduction
• Numerical evaluation of multiple polylogarithms
• Kira - A Feynman Integral Reduction Program
• Local Integrals for Planar Scattering Amplitudes
• Reduze - Feynman Integral Reduction in C++
• FIRE6: Feynman Integral REduction with Modular Arithmetic
• Integral Reduction with Kira 2.0 and Finite Field Methods
• Automatic Integral Reduction for Higher Order Perturbative Calculations
• Lectures on differential equations for Feynman integrals
• Reducing differential equations for multiloop master integrals
• Explicit solutions of the multi-loop integral recurrence relations and its application
• Feynman Integrals and Intersection Theory
• Magnus and Dyson Series for Master Integrals
• Vector Space of Feynman Integrals and Multivariate Intersection Numbers
• Algorithmic transformation of multi-loop master integrals to a canonical basis with CANONICA
• Scattering Amplitudes from Intersection Theory
• Fuchsia: a tool for reducing differential equations for Feynman master integrals to epsilon form
• Leading singularities and off-shell conformal integrals
• Critical points and number of master integrals
• Cuts of Feynman Integrals in Baikov representation
• All master integrals for three-jet production at NNLO
• Aspects of Scattering Amplitudes and Moduli Space Localization
• epsilon: A tool to find a canonical basis of master integrals
• Constructing d-log integrands and computing master integrals for three-loop four-particle scattering
• Decomposition of Feynman Integrals on the Maximal Cut by Intersection Numbers
• On the Singularity Structure of Maximally Supersymmetric Scattering Amplitudes
• Generalised power series expansions for the elliptic planar families of Higgs + jet production at two loops
• Decomposition of Feynman Integrals by Multivariate Intersection Numbers
• The Two-Loop Master Integrals for $q\bar{q}\to VV$
• Transforming differential equations of multi-loop Feynman integrals into canonical form
• Libra: a package for transformation of differential systems for multiloop integrals
• Azurite: An algebraic geometry based package for finding bases of loop integrals
• Diagrammatic Hopf algebra of cut Feynman integrals: the one-loop case
• Simplifying differential equations for multi-scale Feynman integrals beyond multiple polylogarithms
• HandyG -- rapid numerical evaluation of generalised polylogarithms in Fortran
• Cuts from residues: the one-loop case
• The algebraic structure of cut Feynman integrals and the diagrammatic coaction
• Deriving canonical differential equations for Feynman integrals from a single uniform weight integral
• Picard-Fuchs equations for Feynman integrals
• RationalizeRoots: Software Package for the Rationalization of Square Roots
• Kinematic Jacobi Identity is a Residue Theorem: Geometry of Color-Kinematics Duality for Gauge and Gravity Amplitudes
• Maximal Cuts in Arbitrary Dimension
• From Infinity to Four Dimensions: Higher Residue Pairings and Feynman Integrals
• Baikov-Lee Representations Of Cut Feynman Integrals
• On the computation of intersection numbers for twisted cocycles
• Calculating multiloop integrals using dimensional recurrence relation and D-analyticity
• Two-Loop Master Integrals for the Planar QCD Massive Corrections to Di-photon and Di-jet Hadro-production
• Constructing Canonical Feynman Integrals with Intersection Theory
• Towards a new approximation for pair-production and associated-production of the Higgs boson
• Logarithmic forms and differential equations for Feynman integrals
• Leading singularities in Baikov representation and Feynman integrals with uniform transcendental weight
• Differential equations for loop integrals in Baikov representation
• Two-loop master integrals for the leading QCD corrections to the Higgs coupling to a $W$ pair and to the triple gauge couplings $ZWW$ and ${\gamma }^{\ast }WW$
• Mixed QCD-EW corrections for Higgs leptonic decay via $H{W}^{+}{W}^{-}$ vertex
• Maximal transcendental weight contribution of scattering amplitudes

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