A second-order differential equation for the two-loop sunrise graph with arbitrary masses

TL;DR

This paper derives a second-order differential equation for the two-loop sunrise Feynman integral with arbitrary masses in two dimensions, using algebraic geometry and Hodge theory, providing a more efficient approach than traditional IBP methods.

Contribution

It introduces a novel geometric approach to derive a second-order differential equation for the sunrise integral, improving upon the standard IBP-based methods.

Findings

The differential equation is of second order, simplifying analysis.

The approach uses the variation of mixed Hodge structures.

It applies to arbitrary masses in the two-loop sunrise graph.

Abstract

We derive a second-order differential equation for the two-loop sunrise graph in two dimensions with arbitrary masses. The differential equation is obtained by viewing the Feynman integral as a period of a variation of a mixed Hodge structure, where the variation is with respect to the external momentum squared. The fibre is the complement of an elliptic curve. From the fact that the first cohomology group of this elliptic curve is two-dimensional we obtain a second-order differential equation. This is an improvement compared to the usual way of deriving differential equations: Integration-by-parts identities lead only to a coupled system of four first-order differential equations.