Algebraic structure of the renormalization group in the renormalizable QFT theories

TL;DR

This paper explores the algebraic structure of the renormalization group in renormalizable quantum field theories, revealing that certain finite renormalizations form a Lie algebra isomorphic to the Witt algebra, and provides explicit generators and their actions.

Contribution

It demonstrates that the generators of finite renormalizations form a Lie algebra isomorphic to the Witt algebra and constructs explicit generators for the renormalization group actions.

Findings

Generators of finite renormalizations satisfy Witt algebra relations.

Explicit expressions for renormalization group generators are provided.

Finite changes in beta-functions and anomalous dimensions are derived.

Abstract

We consider the group formed by finite renormalizations as an infinite-dimensional Lie group. It is demonstrated that for the finite renormalization of the gauge coupling constant its generators $\hat L_n$ with $n\ge 1$ satisfy the commutation relations of the Witt algebra and, therefore, form its subalgebra. The commutation relations are also written for the more general case when finite renormalizations are made for both the coupling constant and matter fields. We also construct the generator of the Abelian subgroup corresponding to the changes of the renormalization scale. The explicit expressions for the renormalization group generators are written in the case when they act on the $\beta$-function and the anomalous dimension. It is explained how the finite changes of these functions under the finite renormalizations can be obtained with the help of the exponential map.