Feynman diagrams and their algebraic lattices

TL;DR

This paper explores the algebraic lattice structure underlying Feynman diagram renormalization in quantum field theories, providing explicit formulas for counterterms and analyzing the properties of these lattices.

Contribution

It introduces the lattice structure of Feynman diagrams' renormalization and applies it to derive explicit counterterm expressions in zero-dimensional QFTs.

Findings

Lattice structures encapsulate nestedness of Feynman diagrams.

Explicit counterterm formulas derived using lattice-Moebius functions.

Analysis of semimodular lattices in tadpole-free quotients.

Abstract

We present the lattice structure of Feynman diagram renormalization in physical QFTs from the viewpoint of Dyson-Schwinger-Equations and the core Hopf algebra of Feynman diagrams. The lattice structure encapsules the nestedness of diagrams. This structure can be used to give explicit expressions for the counterterms in zero-dimensional QFTs using the lattice-Moebius function. Different applications for the tadpole-free quotient, in which all appearing elements correspond to semimodular lattices, are discussed.