The one-loop six-dimensional hexagon integral and its relation to MHV amplitudes in N=4 SYM

Abstract

We provide an analytic formula for the (rescaled) one-loop scalar hexagon integral ${\tilde{\Phi }}_6$ with all external legs massless, in terms of classical polylogarithms. We show that this integral is closely connected to two integrals appearing in one- and two-loop amplitudes in planar $mathcalN=4$ super-Yang-Mills theory, ${\Omega }^{(1)}$ and ${\Omega }^{(2)}$. The derivative of ${\Omega }^{(2)}$ with respect to one of the conformal invariants yields ${\tilde{\Phi }}_6$, while another first-order differential operator applied to ${\tilde{\Phi }}_6$ yields ${\Omega }^{(1)}$. We also introduce some kinematic variables that rationalize the arguments of the polylogarithms, making it easy to verify the latter differential equation. We also give a further example of a six-dimensional integral relevant for amplitudes in $mathcalN=4$ super-Yang-Mills.

18 pages, 2 figures

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• Jumpstarting the all-loop S-matrix of planar N=4 super Yang-Mills
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• The four-loop remainder function and multi-Regge behavior at NNLLA in planar N=4 super-Yang-Mills theory
• Iterative structure of finite loop integrals
• The Cosmic Galois Group and Extended Steinmann Relations for Planar $\mathcal{N}=4$ SYM Amplitudes
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• Multiple polylogarithms with algebraic arguments and the two-loop EW-QCD Drell-Yan master integrals
• Bootstrapping pentagon functions
• Diagrammatic Hopf algebra of cut Feynman integrals: the one-loop case
• The Double Pentaladder Integral to All Orders
• Mellin Amplitudes for Dual Conformal Integrals
• Yangian Bootstrap for Conformal Feynman Integrals
• Star Integrals, Convolutions and Simplices
• Form factors and scattering amplitudes in N=4 SYM in dimensional and massive regularizations
• Logarithmic forms and differential equations for Feynman integrals
• Tree-Level Formalism
• The one-loop six-dimensional hexagon integral with three massive corners
• Conformal anomaly of generalized form factors and finite loop integrals
• All-Mass $n$-gon Integrals in $n$ Dimensions
• Amplitudes from superconformal Ward identities
• The One-Loop One-Mass Hexagon Integral in D=6 Dimensions
• Symbols of One-Loop Integrals From Mixed Tate Motives
• One-loop hexagon integral to higher orders in the dimensional regulator
• The Epsilon Expansion of Feynman Diagrams via Hypergeometric Functions and Differential Reduction
• One-Loop Integrals from Spherical Projections of Planes and Quadrics
• The SAGEX Review on Scattering Amplitudes, Chapter 5: Analytic Bootstraps for Scattering Amplitudes and Beyond
• All two-loop MHV remainder functions in multi-Regge kinematics
• Amplitudes from anomalous superconformal symmetry
• The Steinmann Cluster Bootstrap for N=4 Super Yang-Mills Amplitudes
• Differential equations for multi-loop integrals and two-dimensional kinematics
• Hexagon Bootstrap in the Double Scaling Limit
• A Probabilistic Angle on One Loop Scalar Integrals