A double integral of d log forms which is not polylogarithmic

Conference item

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

Authors: Brown, FCS; Duhr, C

• Publication status: Published
• Peer review status: Peer reviewed
• Publisher: Sissa Medialab
• Journal: Proceedings of Science
• Article number: CERN-TH-2020-097
• Publication date: 2022-02-15
• Acceptance date: 2021-01-25
• Event title: MathemAmplitudes 2019: Intersection Theory and Feynman Integrals
• Event location: Padova
• Event website: https://indico.cern.ch/event/836413/
• Event start date: 2019-12-18
• Event end date: 2019-12-20
• DOI: 10.22323/1.383.0005
• ISSN: 1824-8039
• Language: English
• Keywords: FFR