Algebraic lattices in QFT renormalization. (English) Zbl 1355.81117
Summary: 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.
MSC:
| 81T18 | Feynman diagrams |
| 81T15 | Perturbative methods of renormalization applied to problems in quantum field theory |
| 81T16 | Nonperturbative methods of renormalization applied to problems in quantum field theory |
| 06B99 | Lattices |
| 16T05 | Hopf algebras and their applications |
Keywords:
quantum field theory; renormalization; Hopf algebra of Feynman diagrams; algebraic lattices; zero-dimensional QFTReferences:
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