Jumpstarting the all-loop S-matrix of planar N=4 super Yang-Mills

Abstract

We derive a set of first-order differential equations obeyed by the S-matrix of planar maximally supersymmetric Yang-Mills theory. The equations, based on the Yangian symmetry of the theory, involve only finite and regulator-independent quantities and uniquely determine the all-loop S-matrix. When expanded in powers of the coupling they give derivatives of amplitudes as single integrals over lower-loop, higher-point amplitudes/Wilson loops. We outline a derivation for the equations using the Operator Product Expansion for Wilson loops. We apply them on a few examples at two- and three-loops, reproducing a recent result on the two-loop NMHV hexagon and fixing previously undermined coefficients in a recent Ansatz for the three-loop MHV hexagon. In addition, we consider amplitudes restricted to a two-dimensional subspace of Minkowski space and derive a particularly simple set of partially closed equations in that case.

32 pages, 3 figures; discussion of two-dimensional kinematics corrected in version 2

References in corpus (18)

• Gluon scattering amplitudes at strong coupling
• Classical Polylogarithms for Amplitudes and Wilson Loops
• MHV Amplitudes in N=4 Super Yang-Mills and Wilson Loops
• The All-Loop Integrand For Scattering Amplitudes in Planar N=4 SYM
• Yangian symmetry of scattering amplitudes in N=4 super Yang-Mills theory
• Hexagon Wilson loop = six-gluon MHV amplitude
• Bootstrapping the three-loop hexagon
• The Complete Planar S-matrix of N=4 SYM as a Wilson Loop in Twistor Space
• Analytic result for the two-loop six-point NMHV amplitude in N=4 super Yang-Mills theory
• An Operator Product Expansion for Polygonal null Wilson Loops
• Notes on the scattering amplitude / Wilson loop duality
• Exacting N=4 Superconformal Symmetry
• The Yangian origin of the Grassmannian integral
• One-Loop Superconformal and Yangian Symmetries of Scattering Amplitudes in N=4 Super Yang-Mills
• BFKL equation for the adjoint representation of the gauge group in the next-to-leading approximation at N=4 SUSY
• MHV Diagrams in Momentum Twistor Space
• Analytic Results for MHV Wilson Loops
• Wilson Loops @ 3-Loops in Special Kinematics

Cited by in corpus (118)

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• The Structure of n-Point One-Loop Open Superstring Amplitudes
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• Yangian Symmetry for Fishnet Feynman Graphs
• Lectures on Yangian Symmetry
• The four-loop cusp anomalous dimension in N=4 super Yang-Mills and analytic integration techniques for Wilson line integrals
• A Three-Point Form Factor Through Five Loops
• A Cluster Bootstrap for Two-Loop MHV Amplitudes
• Yangian Symmetry for Bi-Scalar Loop Amplitudes
• Cluster adjacency beyond MHV
• Rationalizing Loop Integration
• Mellin Amplitudes for Dual Conformal Integrals
• Hexagonal Wilson Loops in Planar $\mathcal{N}=4$ SYM Theory at Finite Coupling
• Conformal Anomaly for Amplitudes in N=6 Superconformal Chern-Simons Theory
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• From the Holomorphic Wilson Loop to `d log' Loop-Integrands of Super-Yang-Mills Amplitudes
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• Logarithmic forms and differential equations for Feynman integrals
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• Singularities of eight- and nine-particle amplitudes from cluster algebras and tropical geometry
• Systematics of the cusp anomalous dimension
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• The symbol and alphabet of two-loop NMHV amplitudes from $\bar{Q}$ equations
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• Truncated cluster algebras and Feynman integrals with algebraic letters
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• The Sklyanin Bracket and Cluster Adjacency at All Multiplicity
• Amplitudes from superconformal Ward identities
• Conformal anomaly of generalized form factors and finite loop integrals
• Symbol Alphabets from Tensor Diagrams
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• The SAGEX Review on Scattering Amplitudes, Chapter 15: The Multi-Regge Limit
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