Symbols of One-Loop Integrals From Mixed Tate Motives

Abstract

We use a result on mixed Tate motives due to Goncharov (arXiv:alg-geom/9601021) to show that the symbol of an arbitrary one-loop 2m-gon integral in 2m dimensions may be read off directly from its Feynman parameterization. The algorithm proceeds via recursion in m seeded by the well-known box integrals in four dimensions. As a simple application of this method we write down the symbol of a three-mass hexagon integral in six dimensions.

13 pages, v2: minor typos corrected

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• Yangian-invariant fishnet integrals in 2 dimensions as volumes of Calabi-Yau varieties
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• Logarithmic forms and differential equations for Feynman integrals
• The Wilson-loop $d\log$ representation for Feynman integrals
• The Epsilon Expansion of Feynman Diagrams via Hypergeometric Functions and Differential Reduction
• One-Loop Integrals from Spherical Projections of Planes and Quadrics
• Out-of-equilibrium Chiral Magnetic Effect from simulations on Euclidean lattices