Renormalization Conditions and the Sliding Scale in the Implicit Regularization Scheme: A Simple Connection

TL;DR

This paper explains how a sliding scale naturally arises in implicit regularization of quantum field theories through renormalization conditions, demonstrated with QED and phi^4 theories, and can be extended to higher loops.

Contribution

It introduces a straightforward connection between renormalization conditions and the sliding scale in implicit regularization, simplifying higher-loop generalizations.

Findings

Derived one-loop beta-functions for QED and phi^4 theories.

Showed no regulator is needed at intermediate steps.

Established a formalism adaptable to higher-loop calculations.

Abstract

We describe in detail how a sliding scale is introduced in the renormalization of a QFT according to integer-dimensional implicit regularization scheme. We show that since no regulator needs to be specified at intermediate steps of the calculation, the introduction of a mass scale is a direct consequence of a set of renormalization conditions. As an illustration the one loop beta-function for QED and lambda*phi^4 theories are derived. They are given in terms of derivatives of appropriately systematized functions (related to definited parts of the amplitudes) with respect to a mass scale mu. Our formal scheme can be easily generalized to higher loop calculations.