  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 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.

High Energy Physics - Theory;General Relativity and Quantum Cosmology