The particle content of $R^2$ gravity revisited

TL;DR

This paper investigates the particle spectrum of $R^2$ gravity, revealing that on Minkowski background the theory lacks physical particles due to gauge redundancies, but on suitable backgrounds or with added terms it propagates the expected graviton and scalar modes.

Contribution

It clarifies the particle content of $R^2$ gravity on different backgrounds and shows how gauge redundancies affect the spectrum, providing a covariant demonstration of propagating degrees of freedom.

Findings

On Minkowski background, $R^2$ gravity has no physical particle states.

On de Sitter or with Einstein-Hilbert term, it propagates two gravitons and one scalar.

Gauge redundancies on Minkowski are artifacts of linearization, not physical symmetries.

Abstract

Studying the spectrum of (pure) $R^2$ gravity on Minkowski background inevitably results into a Catch-22: any consistent interpretation of its particle dynamics dictates that no accidental gauge symmetries emerge, a requirement that cannot be fulfilled when the theory is studied on Minkowski. For the case at hand, there is an emergent gauge redundancy corresponding to a transverse-traceless shift of the graviton. This has detrimental consequences since it empties the spectrum from all particle states. Being an artifact of the linearized approximation on top of Minkowski background, the symmetry does not persist at higher orders and degrees of freedom are reintroduced via interactions, making $R^2$ gravity infinitely strongly-coupled. Provided that the theory is considered on appropriate backgrounds -- for instance de Sitter spacetime -- or is supplemented with the Einstein-Hilbert term, $R^2$ gravity propagates the two usual graviton polarizations plus one additional (massless or massive) scalar field. We explicitly demonstrate that in a fully covariant manner.