Unique factorization in perturbative QFT

Over recent years, the Lie and Hopf algebra structures of Feynman graphs have been firmly established in collaboration with Alain Connes and David Broadhurst [1–11]. Hopefully, they find their way now into the algorithmic toolkit of the practitioner of QFT. They directly address the computational practice of momentum space Feynman integrals, but can be equivalently formulated in coordinate space. They take into account faithfully the self-similarity of physics at different scales and the iteration of Green functions in terms of itself by their quantum equations of motions, the Dyson–Schwinger equations. We will review quickly these Lie and Hopf algebra structures and then report in the next section on factorization properties which exist in the formal series over graphs contributing to a given Green function. We motivate this factorization in comparison with the factorization of integers leading to an Euler product for ζ-functions, and comment on the fate of such factorizations in gauge theories. Results reported here can also be found in [12], which in particular contains a more extended discussion of factorization properties and their implications on perturbative expansions with regard to the question if the factorization commutes with the application of the Feynman rules. We start our summary with the pre-Lie structure.