On the consistency of (partially-)massless matter couplings in de Sitter space

We study the consistency of the cubic couplings of a (partially-)massless spinning field to two scalars in $\left(d+1\right)$-dimensional de Sitter space. Gauge invariance of observables with external (partially)-massless spinning fields translates into Ward-Takahashi identities on the boundary. Using the Mellin-Barnes representation for boundary correlators in momentum space, we give a systematic study of Ward-Takahashi identities for tree-level 3- and 4-point processes involving a single external (partially-)massless field of arbitrary integer spin-$J$. 3-point Ward-Takahashi identities constrain the mass of the scalar fields to which a (partially-)massless spin-$J$ field can couple. 4-point Ward-Takahashi identities then constrain the corresponding cubic couplings. For massless spinning fields, we show that Weinberg's flat space results carry over to $\left(d+1\right)$-dimensional de Sitter space: For spins $J=1,2$ gauge-invariance implies charge-conservation and the equivalence principle while, assuming locality, higher-spins $J>2$ cannot couple consistently to scalar matter. This result also applies to anti-de Sitter space. For partially-massless fields, restricting for simplicity to those of depth-2, we show that there is no consistent coupling to scalar matter in local theories. Along the way we give a detailed account of how contact amplitudes with and without derivatives are represented in the Mellin-Barnes representation. Various new explicit expressions for 3- and 4-point functions involving (partially-)massless fields and conformally coupled scalars in dS$_4$ are given.