Remark on the subtractive renormalization of quadratically divergent scalar mass

TL;DR

This paper re-examines the subtractive renormalization of quadratically divergent scalar masses, proposing a scheme that interpolates between different regularization methods and introduces a novel renormalization group approach.

Contribution

It introduces an unconventional scheme for subtractive renormalization that incorporates a renormalization point and transforms the RG equation to be homogeneous.

Findings

Renormalized scalar mass has two components with different scaling behaviors.

The scheme interpolates between dimensional regularization and un-subtracted quadratic divergences.

The RG equation becomes inhomogeneous but is transformed to a homogeneous form.

Abstract

The quadratically divergent scalar mass is subtractively renormalized unlike other divergences which are multiplicatively renormalized. We re-examine some technical aspects of the subtractive renormalization, in particular, the mass independent renormalization of massive $\lambda\phi^{4}$ theory with higher derivative regularization. We then discuss an unconventional scheme to introduce the notion of renormalization point $\mu$ to the subtractive renormalization in a theory defined by a large fixed cut-off $M$. The resulting renormalization group equation generally becomes inhomogeneous but it is transformed to be homogeneous. The renormalized scalar mass consists of two components in this scheme, one with the ordinary anomalous dimension and the other which is proportional to the renormalization scale $\mu$. This scheme interpolates between the theory defined by dimensional regularization and the theory with un-subtracted quadratic divergences.