Motivic geometry of two-loop Feynman integrals

TL;DR

This paper explores the geometric and Hodge-theoretic structures of two-loop Feynman integrals, revealing their decomposition into mixed Tate and elliptic or hyperelliptic components, with specific results for certain graph families.

Contribution

It provides a detailed geometric and Hodge-theoretic analysis of two-loop Feynman integrals, identifying their underlying algebraic varieties and motives, including elliptic curves and K3 surfaces.

Findings

Decomposition of Hodge structures into mixed Tate and elliptic components.

Explicit description of elliptic curves in double box integrals.

Identification of K3 surface motives in non-planar graphs.

Abstract

We study the geometry and Hodge theory of the cubic hypersurfaces attached to two-loop Feynman integrals for generic physical parameters. We show that the Hodge structure attached to planar two-loop Feynman graphs decomposes into mixed Tate pieces and the Hodge structures of families of hyperelliptic, elliptic, or rational curves depending on the space-time dimension. For two-loop graphs with a small number of edges, we give more precise results. In particular, we recover a result of Bloch arXiv:2105.06132 that in the well-known double box example, there is an underlying family of elliptic curves, and we give a concrete description of these elliptic curves. We argue that the motive for the non-planar two-loop tardigrade graph is that of a K3 surface of Picard number 11 and determine the generic lattice polarization. Lastly, we show that generic members of the ice cream cone family of graph hypersurfaces correspond to pairs of sunset Calabi--Yau varieties.