A double integral of dlog forms which is not polylogarithmic

TL;DR

This paper presents a specific double integral of dlog-forms that evaluates to a period of a cusp form, demonstrating it cannot be expressed as a multiple polylogarithm, thus challenging common assumptions in quantum field theory calculations.

Contribution

It provides a concrete example of a double iterated integral of dlog-forms that evaluates to a non-polylogarithmic period, linking quantum field theory integrals to elliptic motives and cusp forms.

Findings

The integral evaluates to a period of a cusp form.

Motivic versions are algebraically independent from polylogarithms.

The geometric setup involves a modular elliptic curve and torsion points.

Abstract

Feynman integrals are central to all calculations in perturbative Quantum Field Theory. They often give rise to iterated integrals of dlog-forms with algebraic arguments, which in many cases can be evaluated in terms of multiple polylogarithms. This has led to certain folklore beliefs in the community stating that all such integrals evaluate to polylogarithms. Here we discuss a concrete example of a double iterated integral of two dlog-forms that evaluates to a period of a cusp form. The motivic versions of these integrals are shown to be algebraically independent from all multiple polylogarithms evaluated at algebraic arguments. From a mathematical perspective, we study a mixed elliptic Hodge structure arising from a simple geometric configuration in $\mathbb{P}^2$, consisting of a modular plane elliptic curve and a set of lines which meet it at torsion points, which may provide an interesting worked example from the point of view of periods, extensions of motives, and L-functions.