The one-loop six-dimensional hexagon integral with three massive corners

Abstract

We compute the six-dimensional hexagon integral with three non-adjacent external masses analytically. After a simple rescaling, it is given by a function of six dual conformally invariant cross-ratios. The result can be expressed as a sum of 24 terms involving only one basic function, which is a simple linear combination of logarithms, dilogarithms, and trilogarithms of uniform degree three transcendentality. Our method uses differential equations to determine the symbol of the function, and an algorithm to reconstruct the latter from its symbol. It is known that six-dimensional hexagon integrals are closely related to scattering amplitudes in N=4 super Yang-Mills theory, and we therefore expect our result to be helpful for understanding the structure of scattering amplitudes in this theory, in particular at two loops.

15 pages, 2 figures

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• Classical Polylogarithms for Amplitudes and Wilson Loops
• Local Integrals for Planar Scattering Amplitudes
• Hexagon Wilson loop = six-gluon MHV amplitude
• The Two-Loop Six-Gluon MHV Amplitude in Maximally Supersymmetric Yang-Mills Theory
• New differential equations for on-shell loop integrals
• The one-loop six-dimensional hexagon integral and its relation to MHV amplitudes in N=4 SYM
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• Cluster algebras for Feynman integrals
• Mellin Amplitudes for Dual Conformal Integrals
• Sequential Discontinuities of Feynman Integrals and the Monodromy Group
• Yangian Bootstrap for Conformal Feynman Integrals
• Three Point Energy Correlators in the Collinear Limit: Symmetries, Dualities and Analytic Results
• Lightcone Bootstrap at higher points
• Dual Conformal Symmetry, Integration-by-Parts Reduction, Differential Equations and the Nonplanar Sector
• Notes on cluster algebras and some all-loop Feynman integrals
• All-Mass $n$-gon Integrals in $n$ Dimensions
• The Wilson-loop $d\log$ representation for Feynman integrals
• Symbols of One-Loop Integrals From Mixed Tate Motives
• The Epsilon Expansion of Feynman Diagrams via Hypergeometric Functions and Differential Reduction
• Two-loop D-dimensional unitarity and dual conformal symmetry
• A Probabilistic Angle on One Loop Scalar Integrals