# Searching for a $C$-function on the three-dimensional sphere

## TL;DR

This paper investigates potential $C$-functions on the three-dimensional sphere for free massive Dirac and scalar fields, introducing a new candidate called the modified effective action and analyzing their properties and suitability.

## Contribution

The study provides a detailed analysis of existing $C$-function candidates and proposes a new modified effective action for scalar fields on the 3D sphere.

## Key findings

- Effective action, Rènyi entanglement entropy, and renormalized entanglement entropy partially fulfill $C$-function conditions.
- The modified effective action shows promise as a $C$-function for scalar fields.
- Existing candidates are insufficient for scalar fields, motivating the new proposal.

## Abstract

We present a detailed analytic study on the three-dimensional sphere of the most popular candidates for $C$-functions, both for Dirac and scalar free massive fields. We discuss to which extent the effective action, the R\'enyi entanglement entropy and the renormalized entanglement entropy fulfill the conditions expected from $C$-functions. In view of the absence of a good candidate in the case of the scalar field, we introduce a new candidate, which we call the modified effective action, and analyze its pros and cons.