A Feynman integral via higher normal functions

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.