A Zassenhaus-type algorithm solves the Bogoliubov recursion

TL;DR

This paper presents a novel Lie-theoretic Zassenhaus-type algorithm for computing counterterms in perturbative renormalization, offering a non-linear, well-behaved recursion within the Hopf algebra framework, extending previous Lie algebra methods.

Contribution

It introduces a Zassenhaus-inspired algorithm for the Bogoliubov recursion that operates inside the Hopf algebra of Feynman amplitudes, generalizing prior Lie algebra approaches.

Findings

The new method aligns with the Connes-Kreimer beta-function structure.

It generalizes previous Lie idempotent-based renormalization techniques.

The approach is compatible with dimensional regularization and minimal subtraction.

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 beta-function. This further extends previous work of ours (together with Jose M. Gracia-Bondia) on the connection between Lie idempotents and renormalization procedures, where we constructed the Connes-Kreimer beta-function by means of the classical Dynkin idempotent.