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

Pure $R^2$ gravity has been shown to be equivalent to Einstein gravity with nonzero cosmological constant and a massless scalar field when restricted Weyl symmetry is spontaneously broken. We show that the Palatini formulation of pure $R^2$ 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 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 nonminimally 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 $R$.