Communications in Number Theory and Physics

Volume 6 (2012)

Number 1

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

Pages: 203 – 222

DOI: https://dx.doi.org/10.4310/CNTP.2012.v6.n1.a5

Authors

Stefan Müller-Stach (Institut für Mathematik, Universität Mainz, Germany)

Stefan Weinzierl (Institut für Physik, Universität Mainz, Germany)

Raphael Zayadeh (Institut für Mathematik, Universität Mainz, Germany)

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.

Published 12 July 2012