Algorithms for minimal Picard-Fuchs operators of Feynman integrals

TL;DR

This paper develops algorithms to derive minimal Picard-Fuchs differential operators for multi-loop Feynman integrals, revealing their geometric structures and providing explicit operators for complex integrals.

Contribution

It introduces an extended Griffiths--Dwork reduction algorithm to compute minimal Picard-Fuchs operators for multi-loop Feynman integrals, connecting them to Calabi--Yau and K3 geometries.

Findings

Order of Picard-Fuchs operator for n-1 loop sunset integral is 2^n minus a binomial coefficient.

Explicit Picard-Fuchs operator of order 11 for a five-point non-planar two-loop integral.

Operators exhibit Liouvillian or elliptic solutions, indicating diverse geometric structures.

Abstract

In even space-time dimensions the multi-loop Feynman integrals are integrals of rational function in projective space. By using an algorithm that extends the Griffiths--Dwork reduction for the case of projective hypersurfaces with singularities, we derive Fuchsian linear differential equations, the Picard--Fuchs equations, with respect to kinematic parameters for a large class of massive multi-loop Feynman integrals. With this approach we obtain the differential operator for Feynman integrals to high multiplicities and high loop orders. Using recent factorisation algorithms we give the minimal order differential operator in most of the cases studied in this paper. Amongst our results are that the order of Picard--Fuchs operator for the generic massive two-point $n-1$-loop sunset integral in two-dimensions is $2^{n}-\binom{n+1}{\left\lfloor \frac{n+1}{2}\right\rfloor }$ supporting the conjecture that the sunset Feynman integrals are relative periods of Calabi--Yau of dimensions $n-2$. We have checked this explicitly till six loops. As well, we obtain a particular Picard--Fuchs operator of order 11 for the massive five-point tardigrade non-planar two-loop integral in four dimensions for generic mass and kinematic configurations, suggesting that it arises from $K3$ surface with Picard number 11. We determine as well Picard--Fuchs operators of two-loop graphs with various multiplicities in four dimensions, finding Fuchsian differential operators with either Liouvillian or elliptic solutions.