The geometric $\beta$-function in curved space-time under operator regularization

TL;DR

This paper compares different regularization methods for defining the geometric $eta$-function in curved space-time, extending the analysis to scalar field theories on compact Riemannian manifolds and examining the conditions under which the $eta$-function is well-defined.

Contribution

It provides a comparison of renormalization group generators for dimensional and operator regularization and explores their applicability to scalar field theories on curved manifolds.

Findings

The geometric $eta$-function is defined on the entire manifold for scalar theories.

The geometric $eta$-function is not defined for conformal scalar-field theories on the same manifolds.

Abstract

In this paper, I compare the generators of the renormalization group flow, or the geometric $\beta$-functions for dimensional regularization and operator regularization. I then extend the analysis to show that the geometric $\beta$-function for a scalar field theory on a closed compact Riemannian manifold is defined on the entire manifold. I then extend the analysis to find the generator of the renormalization group flow for a conformal scalar-field theories on the same manifolds. The geometric $\beta$-function in this case is not defined.