New differential equations for on-shell loop integrals

Abstract

We present a novel type of differential equations for on-shell loop integrals. The equations are second-order and importantly, they reduce the loop level by one, so that they can be solved iteratively in the loop order. We present several infinite series of integrals satisfying such iterative differential equations. The differential operators we use are best written using momentum twistor space. The use of the latter was advocated in recent papers discussing loop integrals in N=4 super Yang-Mills. One of our motivations is to provide a tool for deriving analytical results for scattering amplitudes in this theory. We show that the integrals needed for planar MHV amplitudes up to two loops can be thought of as deriving from a single master topology. The master integral satisfies our differential equations, and so do most of the reduced integrals. A consequence of the differential equations is that the integrals we discuss are not arbitrarily complicated transcendental functions. For two specific two-loop integrals we give the full analytic solution. The simplicity of the integrals appearing in the scattering amplitudes in planar N=4 super Yang-Mills is strongly suggestive of a relation to the conjectured underlying integrability of the theory. We expect these differential equations to be relevant for all planar MHV and non-MHV amplitudes. We also discuss possible extensions of our method to more general classes of integrals.

39 pages, 8 figures; v2: typos corrected, definition of harmonic polylogarithms added

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Cited by in corpus (61)

• Multiloop integrals in dimensional regularization made simple
• Bootstrapping the three-loop hexagon
• Analytic result for the two-loop six-point NMHV amplitude in N=4 super Yang-Mills theory
• Hexagon functions and the three-loop remainder function
• Analytic form of the two-loop planar five-gluon all-plus-helicity amplitude in QCD
• The four-loop remainder function and multi-Regge behavior at NNLLA in planar N=4 super-Yang-Mills theory
• Pulling the straps of polygons
• Iterative structure of finite loop integrals
• Cluster adjacency properties of scattering amplitudes
• Evaluating single-scale and/or non-planar diagrams by differential equations
• The three-loop cusp anomalous dimension in QCD and its supersymmetric extensions
• The three-loop form factor in N=4 super Yang-Mills
• Generic multiloop methods and application to N=4 super-Yang-Mills
• BFKL approach and six-particle MHV amplitude in N=4 super Yang-Mills
• The Double Pentaladder Integral to All Orders
• A Three-Point Form Factor Through Five Loops
• Rationalizing Loop Integration
• Mellin Amplitudes for Dual Conformal Integrals
• Simple loop integrals and amplitudes in N=4 SYM
• Sequential Discontinuities of Feynman Integrals and the Monodromy Group
• Tropical Monte Carlo quadrature for Feynman integrals
• The one-loop six-dimensional hexagon integral and its relation to MHV amplitudes in N=4 SYM
• Three Point Energy Correlators in the Collinear Limit: Symmetries, Dualities and Analytic Results
• Bootstrapping a Stress-Tensor Form Factor through Eight Loops
• Form factors and scattering amplitudes in N=4 SYM in dimensional and massive regularizations
• Star Integrals, Convolutions and Simplices
• Generalised ladders and single-valued polylogarithms
• Dual conformal symmetry at loop level; massive regularization
• Bootstrapping two-loop Feynman integrals for planar N=4 sYM
• Nonperturbative Negative Geometries: Amplitudes at Strong Coupling and the Amplituhedron
• Logarithmic forms and differential equations for Feynman integrals
• The one-loop six-dimensional hexagon integral with three massive corners
• Notes on cluster algebras and some all-loop Feynman integrals
• Holomorphic Linking, Loop Equations and Scattering Amplitudes in Twistor Space
• Truncated cluster algebras and Feynman integrals with algebraic letters
• Conformal anomaly of generalized form factors and finite loop integrals
• Review of AdS/CFT Integrability, Chapter V.2: Dual Superconformal Symmetry
• Cluster Algebras and the Subalgebra Constructibility of the Seven-Particle Remainder Function
• Schubert Problems, Positivity and Symbol Letters
• Building Bases of Loop Integrands
• The One-Loop One-Mass Hexagon Integral in D=6 Dimensions
• Bootstrapping elliptic Feynman integrals using Schubert analysis
• Feynman Integrals and Scattering Amplitudes from Wilson Loops
• The Wilson-loop $d\log$ representation for Feynman integrals
• A first look at the function space for planar two-loop six-particle Feynman integrals
• Symbols of One-Loop Integrals From Mixed Tate Motives
• Higher loop mixed correlators in N=4 SYM
• An Infinite Family of Elliptic Ladder Integrals
• Comments on all-loop constraints for scattering amplitudes and Feynman integrals
• The SAGEX Review on Scattering Amplitudes, Chapter 5: Analytic Bootstraps for Scattering Amplitudes and Beyond
• Notes on conformal integrals: Coulomb branch amplitudes, magic identities and bootstrap
• Conformal Four-Point Integrals: Recursive Structure, Toda Equations and Double Copy
• A Novel Algorithm for Nested Summation and Hypergeometric Expansions
• Two-loop Doubly Massive Four-Point Amplitude Involving a half-BPS and Konishi Operator
• Illustrations of Integrand-Basis Building at Two Loops
• All Two-Loop, Color-Dressed, Six-Point Amplitude Integrands in sYM
• Yangian symmetry of light-like Wilson loops
• Differential equations for multi-loop integrals and two-dimensional kinematics
• The thermal representation of conformal ladder integrals
• On Solving Dual Conformal Integrals in Coulomb-branch Amplitudes and Their Periods
• The Stratification of Rigidity