D-ideal of generic mass banana integrals in dimensional regularization

TL;DR

This paper introduces a set of differential operators that annihilate generic mass banana integrals in dimensional regularization, providing insights into their singularities and master integrals, and conjecturing a complete generating set.

Contribution

The authors propose a new set of differential operators that generate the annihilating ideal for generic mass banana integrals, advancing understanding of their singularity structure and master integrals.

Findings

Operators annihilate the integrals up to 8 loops

Holonomic rank matches the number of master integrals

Singular locus aligns with Landau singularities

Abstract

We present a set of $\ell + 3$ simple differential operators and prove that they annihilate an $\ell$-loop generic mass banana integral in dimensional regularization. We study the singular locus of the ideal generated by these operators and show that it is contained in the set of Landau singularities of the first and second type. Through the Macaulay matrix method, we calculate the corresponding holonomic rank up to $\ell = 8$, obtaining $2^{\ell +1}-1$, which agrees with the number of master integrals for the generic mass banana integrals. Based on these findings, we conjecture that the proposed operators generate the annihilating ideal for the generic mass banana integrals in dimensional regularization.