Joung, E (reprint author), CNRS, AstroParticule & Cosmol, Unite Mixte Rech 7164, 10 Rue Alice Domon & Leonie Duquet, F-75205 Paris 13, France.
; Li, Wenliang] CNRS, AstroParticule & Cosmol, Unite Mixte Rech 7164, F-75205 Paris 13, France
; [Taronna, Massimo] Max Planck Inst Gravitat Phys, Albert Einstein Inst, D-14476 Golm, Germany
; [Taronna, Massimo] Chinese Acad Sci, Kavli Inst Theoret Phys China, Beijing 100190, Peoples R China
We examine the generic theory of a paratially massless (PM) spin-two field interacting with gravity in four dimensions from a bottom-up perspective. By analyzing the most general form of the Lagrangian, we first show that if such a theory exists, its de Sitter background must admit either so (1, 5) or so (2, 4) global symmetry depending on the relative sign of the kinetic terms: the former for a positive sign the latter for a negative sign. Further analysis reveals that the coupling constant of the PM cubic self-interaction must be fixed with a purely imaginary number in the case of a positive sign. We conclude that there cannot exist a unitary theory of a PM spin-two field coupled to Einstein gravity with a perturbatively local Lagrangian. In the case of a negative sign we recover conformal gravity. As a special case of our analysis, it is shown that the PM limit of massive gravity also lacks the PM gauge symmetry.