Picard-Fuchs equations for Feynman integrals

TL;DR

The paper addresses the problem of obtaining minimal-order differential equations for Feynman integrals with respect to an external variable. It introduces a systematic, algebraic method that reduces the search for a Picard-Fuchs operator $L^{(r)}$ to solving a linear system, valid in arbitrary space-time dimension including dimensional regularisation. Through a Griffiths-reduction-inspired ansatz and a master equation, the approach yields explicit operators and boundary terms, demonstrated on banana graphs across various dimensions and mass configurations. A key finding is that in integer dimensions the $D$-dimensional operator often factorises, which simplifies Laurent-expansion calculations and connects Feynman integrals to the Picard-Fuchs framework from algebraic geometry.

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.

Figures (1)

• Figure 1: The three-loop banana graph.