Palatini formulation of pure R2 gravity yields Einstein gravity with no massless scalar

Ariel Edery,Yu Nakayama

doi:10.1103/physrevd.99.124018

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 $R$

Highlights

Pure R2 gravity (i.e., R2 gravity with no extra R term) possesses a symmetry that is larger than scale symmetry and smaller than full Weyl symmetry
We show that the Palatini formulation of pure R2 gravity is equivalent to Einstein gravity with a nonzero cosmological constant as before but with no massless scalar field when the Weyl symmetry is spontaneously broken
This theory was shown to be equivalent to Einstein gravity with nonzero cosmological constant and a massless scalar field [2,3,4,5,6] The massless scalar is identified as the Nambu-Goldstone boson of the spontaneously broken restricted Weyl symmetry [6]

Summary

INTRODUCTION

Pure R2 gravity (i.e., R2 gravity with no extra R term) possesses a symmetry that is larger than scale symmetry and smaller than full Weyl symmetry. The Palatini formulation [8,9]1 of pure R2 gravity is Weyl invariant [39] and we show in this paper that when the Weyl symmetry is spontaneously broken it is equivalent to Einstein gravity with nonzero cosmological constant as before but no massless scalar field appears in contrast to the metric formulation This is an important and positive development; the presence of a massless scalar is not an issue theoretically but as far as we know, there is no evidence for such a particle in nature. To maintain Weyl invariance, we use a hybrid formalism where the action includes both the Palatini curvature scalar R and the usual Ricci scalar R We show that this action is equivalent to Einstein gravity with cosmological constant and a nonminimally coupled massive Higgs field. The use of these two connections in the action is prescribed by the principle that the action is Weyl invariant

PALATINI FORMULATION OF PURE R2 GRAVITY

INCLUSION OF HIGGS FIELD AND FERMIONS

CONCLUSION