From motives to differential equations for loop integrals

TL;DR

This paper explores how mixed Hodge structures can be used to derive simpler differential equations for Feynman integrals, exemplified by the two-loop sunrise graph in two dimensions.

Contribution

It introduces a novel approach using mixed Hodge structures to obtain more straightforward differential equations for loop integrals, improving upon traditional methods.

Findings

Derived a simpler differential equation for the two-loop sunrise graph

Demonstrated the effectiveness of mixed Hodge structures in Feynman integral analysis

Compared new equations with traditional integration-by-parts results

Abstract

In this talk we discuss how ideas from the theory of mixed Hodge structures can be used to find differential equations for Feynman integrals. In particular we discuss the two-loop sunrise graph in two dimensions and show that these methods lead to a differential equation which is simpler than the ones obtained from integration-by-parts.