Renormalization Automated by Hopf Algebra

It was recently shown that the renormalization of quantum field theory is organized by the Hopf algebra of decorated rooted trees, whose coproduct identifies the divergences requiring subtraction and whose antipode achieves this. We automate this process in a few lines of recursive symbolic code, which deliver a finite renormalized expression for any Feynman diagram. We thus verify a representation of the operator product expansion, which generalizes Chen’s Lemma for iterated integrals. The subset of diagrams whose forest structure entails a unique primitive subdivergence provides a representation of the Hopf algebra HRof undecorated rooted trees. Our undecorated Hopf algebra program is designed to process the 24 213 878 BPHZ contributions to the renormalization of 7813 diagrams, with up to 12 loops. We consider 10 models, each in nine renormalization schemes. The two simplest models reveal a notable feature of the subalgebra of Connes and Moscovici, corresponding to the commutative part of the Hopf algebra HTof the diffeomorphism group: it assigns to Feynman diagrams those weights which remove zeta values from the counterterms of the minimal subtraction scheme. We devise a fast algorithm for these weights, whose squares are summed with a permutation factor, to give rational counterterms.