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

Charlotte Sleight, Massimo Taronna

TL;DR

The paper extends Weinberg-like constraints for massless higher-spin couplings to curved de Sitter space by formulating gauge invariance as Ward-Takahashi identities on the boundary and employing a Mellin-Barnes framework to analyze tree-level 3- and 4-point functions. It shows that for massless spins J=1,2, gauge invariance enforces charge conservation and the equivalence principle, while higher spins cannot couple locally to scalar matter in dS_{d+1} (also true in AdS). For partially-massless fields, depth-2 couplings to scalars are forbidden in local theories, with depth-1 couplings subject to specific scalar-dimension constraints. The work also provides new explicit 3- and 4-point expressions involving (partially-)massless fields and conformally coupled scalars in dS_4 and clarifies how contact terms and improvements map within the Mellin-Barnes representation, offering a robust framework for checking consistency of higher-spin interactions in de Sitter space.

Abstract

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.

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

TL;DR

The paper extends Weinberg-like constraints for massless higher-spin couplings to curved de Sitter space by formulating gauge invariance as Ward-Takahashi identities on the boundary and employing a Mellin-Barnes framework to analyze tree-level 3- and 4-point functions. It shows that for massless spins J=1,2, gauge invariance enforces charge conservation and the equivalence principle, while higher spins cannot couple locally to scalar matter in dS_{d+1} (also true in AdS). For partially-massless fields, depth-2 couplings to scalars are forbidden in local theories, with depth-1 couplings subject to specific scalar-dimension constraints. The work also provides new explicit 3- and 4-point expressions involving (partially-)massless fields and conformally coupled scalars in dS_4 and clarifies how contact terms and improvements map within the Mellin-Barnes representation, offering a robust framework for checking consistency of higher-spin interactions in de Sitter space.

Abstract

We study the consistency of the cubic couplings of a (partially-)massless spinning field to two scalars in -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-. 3-point Ward-Takahashi identities constrain the mass of the scalar fields to which a (partially-)massless spin- 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 -dimensional de Sitter space: For spins gauge-invariance implies charge-conservation and the equivalence principle while, assuming locality, higher-spins 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 are given.

Paper Structure

This paper contains 28 sections, 189 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Notation and conventions.
  3. Three-point functions
  4. Mellin-Barnes representation
  5. Example 1: Three scalars.
  6. Example 2: two scalars and a spin $J$.
  7. Weight shifting operators
  8. Ward-Takahashi identities
  9. Massless fields
  10. Partially-massless fields
  11. Partially-massless of depth $r=1$.
  12. Partially-massless of depth $r=2$.
  13. On improvement terms
  14. Special case: Conformally coupled scalars
  15. Massless spinning field and two conformally coupled scalars.
  16. ...and 13 more sections

Figures (2)

  • Figure 1: Boundary three-point function of two scalars $\phi_{1,2}$ and a spin-$J$ partially massless field $\varphi_J$ of depth-$r$ in de Sitter space, with coupling $g^{\left(J,r\right)}_{12}$.
  • Figure 2: ${\sf s}$-, ${\sf t}$- and ${\sf u}$-channel exchange of a scalar $\phi_0$ between scalars $\phi_{2,3,4}$ and a single partially-massless spin-$J$ field of depth-$r$ in de Sitter space.