Running of the Number of Degrees of Freedom in Quantum Conformal Gravity

TL;DR

This paper investigates how the degrees of freedom in Weyl conformal gravity change with energy scale, showing a monotonic decrease from six in the UV, using two different theoretical approaches.

Contribution

It introduces a novel comparison between the Fradkin--Tseytlin prescription and $a$- and $c$-function methods for counting degrees of freedom in conformal gravity.

Findings

Degrees of freedom decrease monotonically from six in the UV.

The decrease is independent of background choices.

A relation between two counting methods is established.

Abstract

We study how the number of degrees of freedom in Weyl conformal gravity runs with the energy scale from the UV fixed point. To this end we employ two approaches. First, we utilize the Fradkin--Tseytlin prescription for the number of degrees of freedom and demonstrate that the one-loop results are highly dependent on the selected background. We then employ the counting methodology based on the $a$- and $c$-functions, which are typically used to characterize the trace anomaly of conformal field theory in four dimensions. We compute these in the enhanced one-loop approximation and demonstrate that the degrees of freedom decrease monotonically from six degrees in the UV regime. This behavior is independent of the backgrounds considered. Finally, we show how to relate the Fradkin--Tseytlin prescription to counting based on the $a$- and $c$-functions.