Picard-Fuchs equations for Feynman integrals

TL;DR

This paper introduces a systematic method to derive Fuchsian differential equations for Feynman integrals with respect to external variables, applicable in fixed and dimensional regularisation, revealing interesting factorisation properties.

Contribution

The paper presents a new, systematic approach to derive Picard-Fuchs differential equations for Feynman integrals, including their factorisation properties in specific dimensions.

Findings

Method works within fixed and dimensional regularisation

Differential equations are of Fuchsian type

Factorisation properties observed at integer dimensions

Abstract

We present a systematic method to derive an ordinary differential equation for any Feynman integral, where the differentiation is with respect to an external variable. The resulting differential equation is of Fuchsian type. The method can be used within fixed integer space-time dimensions as well as within dimensional regularisation. We show that finding the differential equation is equivalent to solving a linear system of equations. We observe interesting factorisation properties of the D-dimensional Picard-Fuchs operator when D is specialised to integer dimensions.