# Palatini formulation of pure $R^2$ gravity yields Einstein gravity with no massless scalar

## TL;DR

This paper demonstrates that the Palatini formulation of pure $R^2$ gravity results in Einstein gravity without a massless scalar, contrasting with the metric formulation, and explores implications for standard model fields and Weyl invariance.

## Contribution

It shows that Palatini $R^2$ gravity is equivalent to Einstein gravity without a massless scalar, unlike the metric formulation, and incorporates standard model fields maintaining Weyl invariance.

## Key findings

- Palatini $R^2$ gravity yields Einstein gravity with no massless scalar.
- Inclusion of Higgs and fermions compatible with Weyl invariance.
- Contrast with metric $R^2$ gravity where a scalar field appears.

## Abstract

Pure $R^2$ gravity has been shown to be equivalent to Einstein gravity with non-zero cosmological constant and a massless scalar field. We show that the Palatini formulation of pure $R^2$ gravity is equivalent to Einstein gravity with non-zero cosmological constant as before but with no massless scalar field. This is an important new development because the massless scalar field is not readily identifiable with any known particle in nature or unknown particles like cold dark matter which are expected to be massive. We then include a non-minimally coupled Higgs field as well as fermions to discuss how the rest of the standard model fields fit into this paradigm. With Higgs field, Weyl invariance is maintained by using a hybrid formalism that includes both the Palatini curvature scalar $\mathcal{R}$ and the usual Ricci scalar $R$