No-Go Theorems for Unitary and Interacting Partially Massless Spin-Two Fields

TL;DR

The paper proves a no-go result for a unitary, perturbatively local theory of a partially massless spin-2 field coupled to gravity in four dimensions. Using a bottom-up Lagrangian approach, it derives the quadratic and cubic PM–gravity couplings, and analyzes the gauge and global symmetries to constrain the admissible interactions. The global symmetry on de Sitter space is shown to be $so(1,5)$ for a positive kinetic-sign and $so(2,4)$ for a negative sign, with the admissibility condition forcing the PM cubic coupling to be imaginary in the positive-sign case, thereby ruling out unitarity; the negative-sign case reduces to conformal gravity. The PM limit of massive gravity is found to lack a true PM gauge symmetry, reinforcing the conclusion that a unitary PM gauge theory with gravity does not exist within this perturbative, local framework.

Abstract

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.