The renormalization group equations revisited

TL;DR

This paper revisits the derivation and interpretation of renormalization group equations, emphasizing the role of different scales and providing a two-loop calculation in $^4$ theory, challenging some standard assumptions.

Contribution

It offers a detailed analysis of RG equations with two scales, clarifies the role of the regularization scale, and performs a two-loop calculation in $^4$ theory.

Findings

The regularization scale should be dimensionless in minimal subtraction.

Bare parameters are not necessarily independent of the regularization scale.

Two-loop coefficients match standard results.

Abstract

Starting from a well defined local Lagrangian, we analyze the renormalization group equations in terms of the two different arbitrary scales associated with the regularization procedure and with the physical renormalization of the bare parameters, respectively. We apply our formalism to the minimal subtraction scheme using dimensional regularization. We first argue that the relevant regularization scale in this case should be dimensionless. By relating bare and renormalized parameters to physical observables, we calculate the coefficients of the renormalization group equation up to two loop order in the $\phi^4$ theory. We show that the usual assumption, considering the bare parameters to be independent of the regularization scale, is not a direct consequence of any physical argument. The coefficients that we find in our two-loop calculation are identical to the standard practice. We finally comment on the decoupling properties of the renormalized coupling constant.