2014/06/30 by Spencer Bloch, Matt Kerr, Pierre Vanhove · 137 citations
Mathematics · Physics and Astronomy · #Advanced Algebra and Geometry #Algebraic geometry #Algebraic number #Cohomology #Conjecture #Daniell integral #Feynman diagram #Feynman integral #Geometric Analysis and Curvature Flows #Geometry and complex manifolds #Motivic cohomology #Scalar (mathematics) #hep-ph #hep-th #math-ph #math.AG #math.MP
paper · pdf · doi:10.1112/s0010437x15007472
published in Compositio Mathematica 151(12), 2329-2375 (Cambridge University Press) · Latex. 70 pages. 3 figures. v3: minor changes and clarifications. Version to appear in Compositio Mathematica
arxiv created 2015/03/26 · openalex publication_date 2015/08/06 · arxiv updated 2015/12/23 · openalex created_date 2016/06/24 · openalex updated_date 2026/08/06
We study the Feynman integral for the three-banana graph defined as the scalar two-point self-energy at three-loop order. The Feynman integral is evaluated for all identical internal masses in two space-time dimensions. Two calculations are given for the Feynman integral: one based on an interpretation of the integral 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 Feynman integral is a family of regulator periods associated to a family of K3 surfaces. We show that the integral is given by a sum of elliptic trilogarithms evaluated at sixth roots of unity. This elliptic trilogarithm value is related to the regulator of a class in the motivic cohomology of the K3 family. We prove a conjecture by David Broadhurst which states that at a special kinematical point the Feynman integral is given by a critical value of the Hasse–Weil L -function of the K3 surface. This result is shown to be a particular case of Deligne’s conjectures relating values of L -functions inside the critical strip to periods.