Renormalization automated by Hopf algebra

It was recently shown that the renormalization of quantum field theory is o rganized by the Hopf algebra of decorated rooted trees, whose coproduct ide ntifies the divergences requiring subtraction and whose antipode achieves t his. We automate this process in a few lines of recursive symbolic code, wh ich deliver a finite renormalized expression for any Feynman diagram. We th us verify a representation of the operator product expansion, which general izes Chen's Lemma for iterated integrals. The subset of diagrams whose fore st structure entails a unique primitive subdivergence provides a representa tion of the Hopf algebra H-R of undecorated rooted trees. Our undecorated H oof algebra program is designed to process the 24 213 878 BPHZ contribution s to the renormalization of 7813 diagrams, with up to 12 loops. We consider 10 models, each in nine renormalization schemes. The two simplest models r eveal a notable feature of the subalgebra of Connes and Moscovici, correspo nding to the commutative part of the Hopf algebra H-T of the diffeomorphism group: it assigns to Feynman diagrams those weights which remove zeta valu es from the counterterms of the minimal subtraction scheme. We devise a fas t algorithm for these weights, whose squares are summed with a permutation factor, to give rational counterterms. (C) 1999 Academic Press.