A Feynman integral via higher normal functions. (6th August 2015)

• Record Type: Journal Article
• Authors: Bloch, Spencer; Kerr, Matt; Vanhove, Pierre
• Is Part Of: Compositio mathematica. Volume 151:Number 12(2015)
• Page Start: 2329
• Page End: 2375
• Publication Date: 2015-08-06
• Subjects: 81T18 (primary), -- 19F27, -- 14J28, -- 14D07 (secondary); Feynman graph, -- variation of mixed Hodge structures, -- motives, -- elliptic trilogarithm, -- Eisenstein series, -- higher normal function; Mathematics -- Periodicals; 510
• DOI: 10.1112/S0010437X15007472
• Languages: English
• ISSNs: 0010-437X

Abstract

Abstract : 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.