Abstract
We extend the cosmological bootstrap to correlators involving massless spinning particles, focusing on spin-1 and spin-2. In de Sitter space, these correlators are constrained both by symmetries and by locality. In particular, the de Sitter isometries become conformal symmetries on the future boundary of the spacetime, which are reflected in a set of Ward identities that the boundary correlators must satisfy. We solve these Ward identities by acting with weight-shifting operators on scalar seed solutions. Using this weight-shifting approach, we derive three- and four-point correlators of massless spin-1 and spin-2 fields with conformally coupled scalars. Four-point functions arising from tree-level exchange are singular in particular kinematic configurations, and the coefficients of these singularities satisfy certain factorization properties. We show that in many cases these factorization limits fix the structure of the correlators uniquely, without having to solve the conformal Ward identities. The additional constraint of locality for massless spinning particles manifests itself as current conservation on the boundary. We find that the four-point functions only satisfy current conservation if the s, t, and u-channels are related to each other, leading to nontrivial constraints on the couplings between the conserved currents and other operators in the theory. For spin-1 currents this implies charge conservation, while for spin-2 currents we recover the equivalence principle from a purely boundary perspective. For multiple spin-1 fields, we recover the structure of Yang--Mills theory. Finally, we apply our methods to slow-roll inflation and derive a few phenomenologically relevant scalar-tensor three-point functions.
Highlights
Outline The plan of the paper is as follows: In Section 2, we introduce our main objects of study, namely boundary correlators in de Sitter space
We describe the symmetries that these correlators must satisfy, derive the corresponding conformal Ward identities, and discuss the expected singularities of their solutions
Reading guide Given the length of the paper, we provide a short reading guide: Section 2 contains mostly standard review material that can be skipped by experts, we suggest skimming it to get familiar with our notation
Summary
Long-range forces determine the essential features of the macroscopic world. The large-scale structure of the universe is shaped by the force of gravity, while the electromagnetic force plays a fundamental role on a terrestrial scale. The operators dual to massless fields are conserved currents and must satisfy Ward–Takahashi (WT) identities associated to this current conservation This implies that the structure of spinning correlators is more rigid and more likely to be completely fixed by theoretical consistency, suggesting that the bootstrap approach should be powerful. For spin-1 and spin-2 currents, this implies charge conservation and the equivalence principle, respectively, allowing us to re-discover these bulk facts from a purely boundary perspective These constraints have a deep relation to the singularity structure of cosmological correlators and we will show that the same conclusions can be reached by demanding consistency of the total energy singularity. We capitalize the Mandelstam variables to avoid confusion with s ≡ |k1 + k2|, t ≡ |k1 + k4| and u ≡ |k1 + k3|, which we employ for the exchange momenta in cosmological correlators
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