No unitary theory of PM spin two and gravity

We examine the generic theory of a partially-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 sign of the PM kinetic term: the former for positive sign and the latter for 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 positive sign. We conclude that there cannot exist a unitary theory of PM spin-two field coupled to Einstein gravity with a perturbatively local Lagrangian. Finally, in the case of negative sign we recover conformal gravity