Logarithmic forms and differential equations for Feynman integrals

Abstract

We describe how a dlog representation of Feynman integrals leads to simple differential equations. We derive these differential equations directly in loop momentum or embedding space making use of a localization trick and generalized unitarity. For the examples we study, the alphabet of the differential equation is related to special points in kinematic space, described by certain cut equations which encode the geometry of the Feynman integral. At one loop, we reproduce the motivic formulae described by Goncharov \cite{Goncharov:1996tate} that reappeared in the context of Feynman parameter integrals in \cite{Spradlin:2011wp,Arkani-Hamed:2017ahv}. The dlog representation allows us to generalize the differential equations to higher loops and motivates the study of certain mixed-dimension integrals.

arXiv comment34 pages, journal version

References in corpus (47)

• Fusing Gauge Theory Tree Amplitudes Into Loop Amplitudes
• Multiloop integrals in dimensional regularization made simple
• Differential Equations for Two-Loop Four-Point Functions
• Generalized Unitarity and One-Loop Amplitudes in N=4 Super-Yang-Mills
• Differential Equations for Feynman Graph Amplitudes
• Classical Polylogarithms for Amplitudes and Wilson Loops
• Dimensionally Regulated Pentagon Integrals
• The Amplituhedron
• Magic identities for conformal four-point integrals
• Connection between Feynman integrals having different values of the space-time dimension
• One-Loop Amplitudes for e^+ e^- to Four Partons
• Local Integrals for Planar Scattering Amplitudes
• Dimensionally Regulated One-Loop Integrals
• Lectures on differential equations for Feynman integrals
• Direct extraction of one-loop integral coefficients
• From polygons and symbols to polylogarithmic functions
• Bootstrapping the three-loop hexagon
• Hopf algebras, coproducts and symbols: an application to Higgs boson amplitudes
• Maximally Supersymmetric Planar Yang-Mills Amplitudes at Five Loops
• Six-dimensional Methods for Four-dimensional Conformal Field Theories
• Feynman Integrals and Intersection Theory
• Jumpstarting the all-loop S-matrix of planar N=4 super Yang-Mills
• Hexagon functions and the three-loop remainder function
• Vector Space of Feynman Integrals and Multivariate Intersection Numbers
• Scattering Amplitudes from Intersection Theory
• Singular terms of helicity amplitudes at one-loop in QCD and the soft limit of the cross sections of multi-parton processes
• Iterative structure of finite loop integrals
• Decomposition of Feynman Integrals on the Maximal Cut by Intersection Numbers
• On the Singularity Structure of Maximally Supersymmetric Scattering Amplitudes
• Feynman integrals and hyperlogarithms
• Gluing Ladders into Fishnets
• Two-Loop Amplitudes of Gluons and Octa-Cuts in N=4 Super Yang-Mills
• Evidence for a Nonplanar Amplituhedron
• A geometrical angle on Feynman integrals
• New differential equations for on-shell loop integrals
• An Approach Toward the Numerical Evaluation of Multi-Loop Feynman Diagrams
• Cuts from residues: the one-loop case
• The algebraic structure of cut Feynman integrals and the diagrammatic coaction
• The Double Pentaladder Integral to All Orders
• The one-loop six-dimensional hexagon integral and its relation to MHV amplitudes in N=4 SYM
• Star Integrals, Convolutions and Simplices
• The massless hexagon integral in D = 6 dimensions
• Bootstrapping two-loop Feynman integrals for planar N=4 sYM
• Exponential suppression with four legs and an infinity of loops
• Symbols of One-Loop Integrals From Mixed Tate Motives
• Standard and hypergeometric representations for loop diagrams and the photon-photon scattering
• One-Loop Integrals from Spherical Projections of Planes and Quadrics

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• Deriving canonical differential equations for Feynman integrals from a single uniform weight integral
• Duals of Feynman Integrals, I: Differential Equations
• Duals of Feynman Integrals, 2: Generalized Unitarity
• Baikov representations, intersection theory, and canonical Feynman integrals
• Constructing Canonical Feynman Integrals with Intersection Theory
• Massive Conformal Symmetry and Integrability for Feynman Integrals
• The symbol and alphabet of two-loop NMHV amplitudes from $\bar{Q}$ equations
• On Epsilon Factorized Differential Equations for Elliptic Feynman Integrals
• Truncated cluster algebras and Feynman integrals with algebraic letters
• Multi-Regge Limit of the Two-Loop Five-Point Amplitudes in $\mathcal{N}=4$ Super Yang-Mills and $\mathcal{N}=8$ Supergravity
• All-Mass $n$-gon Integrals in $n$ Dimensions
• Positive geometry, local triangulations, and the dual of the Amplituhedron
• Schubert Problems, Positivity and Symbol Letters
• Yangian Bootstrap for Massive Feynman Integrals
• Feynman Integrals and Scattering Amplitudes from Wilson Loops
• What can we learn about QCD and collider physics from N=4 super Yang-Mills?
• The Wilson-loop $d\log$ representation for Feynman integrals
• Cutting the traintracks: Cauchy, Schubert and Calabi-Yau
• Comments on all-loop constraints for scattering amplitudes and Feynman integrals
• Alphabet of one-loop Feynman integrals
• Maximal transcendental weight contribution of scattering amplitudes
• All Two-Loop, Color-Dressed, Six-Point Amplitude Integrands in sYM
• Integrands of Less-Supersymmetric Yang-Mills at One Loop