Traintracks All the Way Down

TL;DR

This paper investigates a specific class of planar Feynman integrals called traintrack diagrams, revealing their integral representations, geometric structures like Calabi-Yau varieties, and nested differential relations that connect different loop orders.

Contribution

It introduces a systematic way to derive integral representations for traintrack diagrams and uncovers their geometric and differential properties, including their relation to Calabi-Yau manifolds.

Findings

Leading singularities correspond to integrals over Calabi-Yau varieties.

Traintrack diagrams exhibit a nested structure via differential operators.

Explicit integral representations for arbitrary loop orders are derived.

Abstract

We study the class of planar Feynman integrals that can be constructed by sequentially intersecting traintrack diagrams without forming a closed traintrack loop. After describing how to derive a $2L$-fold integral representation of any $L$-loop diagram in this class, we provide evidence that their leading singularities always give rise to integrals over $(L{-}1)$-dimensional varieties for generic external momenta, which for certain graphs we can identify as Calabi-Yau $(L{-}1)$-folds. We then show that these diagrams possess an interesting nested structure, due to the large number of second-order differential operators that map them to (products of) lower-loop integrals of the same type.