A double integral of dlog forms which is not polylogarithmic

Abstract

Feynman integrals are central to all calculations in perturbative Quantum Field Theory. They often give rise to iterated integrals of $d\log$-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 $d\log$-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.