A Feynman integral via higher normal functions

TL;DR

This paper evaluates a three-loop Feynman integral for the three-banana graph in two dimensions using differential equations and algebraic geometry, revealing its connection to elliptic trilogarithms and L-functions.

Contribution

It introduces a novel approach linking Feynman integrals to higher normal functions, motivic cohomology, and special values of L-functions, providing new insights into their geometric and arithmetic structure.

Findings

The Feynman integral is expressed via elliptic trilogarithms at roots of unity.

The integral relates to the regulator of a motivic cohomology class on a K3 surface.

Proves Broadhurst's conjecture connecting the integral to a critical L-value.

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 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.