Bulg. J. Phys. vol.35 no.s1 (2008), pp. 303-315

A Zassenhaus-Type Algorithm Solves the Bogoliubov Recursion

Kurusch Ebrahimi-Fard1, Frédéric Patras2
1Laboratoire MIA, Université de Haute Alsace, 4 rue des Frères Lumière, 68093 Mulhouse, France
2Laboratoire J.-A. Dieudonné UMR 6621, CNRS, Parc Valrose, 06108 Nice Cedex 02, France
Abstract. This paper introduces a new Lie-theoretic approach to the computation of counterterms in perturbative renormalization. Contrary to the usual approach, the devised version of the Bogoliubov recursion does not follow a linear induction on the number of loops. It is well-behaved with respect to the Connes-Kreimer approach: that is, the recursion takes place inside the group of Hopf algebra characters with values in regularized Feynman amplitudes. (Paradigmatically, we use dimensional regularization in the minimal subtraction scheme, although our procedure is generalizable to other schemes.) The new method is related to Zassenhaus' approach to the Baker-Campbell-Hausdorff formula for computing products of exponentials. The decomposition of counterterms is parametrized by a family of Lie idempotents known as the Zassenhaus idempotents. It is shown, inter alia, that the corresponding Feynman rules generate the same algebra as the graded components of the Connes-Kreimer β-function. This further extends previous work of ours (together with José M. Gracia-Bondía) on the connection between Lie idempotents and renormalization procedures, where we constructed the Connes-Kreimer β-function by means of the classical Dynkin idempotent.

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