Algebraic lattices in QFT renormalization

TL;DR

This paper explores the algebraic lattice structures of subdivergences in Feynman diagrams within quantum field theory, revealing their properties and implications for renormalization, especially in the Standard Model.

Contribution

It demonstrates that subdivergences form algebraic lattices and that these lattices are semimodular in certain renormalization schemes, linking lattice theory to QFT renormalization.

Findings

Subdivergences form algebraic lattices in specific QFTs.

Lattices are semimodular in kinematic renormalization schemes.

Provides a formula for counter terms in zero-dimensional QFT.

Abstract

The structure of overlapping subdivergences, which appear in the perturbative expansions of quantum field theory, is analyzed using algebraic lattice theory. It is shown that for specific QFTs the sets of subdivergences of Feynman diagrams form algebraic lattices. This class of QFTs includes the Standard model. In kinematic renormalization schemes, in which tadpole diagrams vanish, these lattices are semimodular. This implies that the Hopf algebra of Feynman diagrams is graded by the coradical degree or equivalently that every maximal forest has the same length in the scope of BPHZ renormalization. As an application of this framework a formula for the counter terms in zero-dimensional QFT is given together with some examples of the enumeration of primitive or skeleton diagrams.