Letters in Mathematical Physics

, Volume 106, Issue 7, pp 879–911 | Cite as

Algebraic Lattices in QFT Renormalization

  • Michael BorinskyEmail author


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.


quantum field theory renormalization Hopf algebra of Feynman diagrams algebraic lattices zero-dimensional QFT 

Mathematics Subject Classification

81T18 81T15 81T16 06B99 


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© Springer Science+Business Media Dordrecht 2016

Authors and Affiliations

  1. 1.Institute of PhysicsHumboldt UniversityBerlinGermany

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