No de Sitter invariant vacuum in massive gravity theory with ghost

TL;DR

This paper demonstrates that massive gravity theories with ghost modes in de Sitter space cannot have a de Sitter invariant vacuum, implying potential stability issues and the need for non-covariant cutoffs to avoid instabilities.

Contribution

It shows the impossibility of a de Sitter invariant vacuum in massive gravity with ghost modes and discusses implications for stability and effective theory cutoffs.

Findings

No de Sitter invariant vacuum exists in such models.

Ghost mode quantization breaks de Sitter symmetry.

Non-covariant cutoff may stabilize the theory.

Abstract

In this letter we point out that the massive gravity theory with a graviton ghost mode in de Sitter background cannot possess a de Sitter invariant vacuum state. In order to avoid a negative norm state, we must associate the creation operator of the ghost mode with a negative-energy mode function instead of a positive-energy one as the mode function. Namely, we have to adopt a different procedure of quantization for a ghost. When a theory has a symmetry mixing a ghost mode with ordinary non-ghost modes, the choice of a ghost mode is not unique. However, quantization of a ghost is impossible without specifying a choice of ghost mode, which breaks the symmetry. For this reason, the vacuum state cannot respect the symmetry. In the massive gravity theory with a graviton ghost mode in de Sitter background, the ghost is the helicity-0 mode of the graviton. This ghost mode is mixed with the other helicity graviton modes under the action of de Sitter symmetry. Therefore, there is no de Sitter invariant vacuum in such models. This leads to an interesting possibility that non-covariant cutoff of the low energy effective theory may naturally arise. As a result, the instability due to the pair production of a ghost and normal non-ghost particles gets much milder and that the model may escape from being rejected.