The elliptic dilogarithm for the sunset graph

2013/09/30 by Spencer Bloch, Pierre Vanhove · 254 citations
Computer Science · Mathematics · Physics and Astronomy · #Algebraic Geometry and Number Theory #Astronomy #Coding theory and cryptography #Combinatorics #Finite Group Theory Research #Graph #Mathematics #Physics #Sunset #hep-th #math-ph #math.AG #math.MP

paper · pdf · doi:10.1016/j.jnt.2014.09.032

published in Journal of Number Theory 148, 328-364 (Elsevier BV) · 3 figures, 43 pages. v2: minor corrections. version to be published in The Journal of Number Theory

arxiv created 2014/10/24 · openalex publication_date 2014/11/04 · arxiv updated 2016/03/14 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05

Abstract

We study the sunset graph defined as the scalar two-point self-energy at two-loop order. We evaluate the sunset integral for all identical internal masses in two dimensions. We give two calculations for the sunset amplitude; one based on an interpretation of the amplitude as an inhomogeneous solution of a classical Picard-Fuchs differential equation, and the other using arithmetic algebraic geometry, motivic cohomology, and Eisenstein series. Both methods use the rather special fact that the amplitude in this case is a family of periods associated to the universal family of elliptic curves over the modular curve X₁(6). We show that the integral is given by an elliptic dilogarithm evaluated at a sixth root of unity modulo periods. We explain as well how this elliptic dilogarithm value is related to the regulator of a class in the motivic cohomology of the universal elliptic family.

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