Traintracks All the Way Down
Andrew J. McLeod, Matt von Hippel

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 -fold integral representation of any -loop diagram in this class, we provide evidence that their leading singularities always give rise to integrals over -dimensional varieties for generic external momenta, which for certain graphs we can identify as Calabi-Yau -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.
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Taxonomy
TopicsBlack Holes and Theoretical Physics · Particle physics theoretical and experimental studies · Advanced Algebra and Geometry
