A single-cut discontinuity for cosmological correlators from unitarity and analyticity

Abstract

We derive discontinuity relations, also known as cutting rules, and explore the analytic properties of cosmological correlators, fundamental observables of the primordial universe. Our emphasis is on how these relations arise from unitarity and hermitian analyticity in interacting quantum field theories on de Sitter space-time. Instead of analyzing wave-function coefficients, we apply these relations directly to cosmological correlators. By studying conformally coupled and massless scalar fields with $ϕ^n$ self-interactions, we demonstrate that the discontinuity of a cosmological correlator can be expressed as a sum of products of lower-point discontinuities, stemming from a single-cut of one internal line in the corresponding tree-level exchange Witten diagram. Notably, beyond lower-point correlators, the decomposition of the discontinuities of cosmological correlators includes contributions from auxiliary elements that consist of both the real and imaginary parts of the lower-point wave-function coefficients, which have not been reported in the existing literature. Interestingly, depending on whether $n$ is even or odd in a $ϕ^n$ interaction, these different lower-point discontinuities contribute as the leading or sub-leading piece in the late-time limit to the discontinuity relations. Additionally, our single-cut discontinuity relation leads to a decomposition rule for the residue of the cosmological correlators at partial energy singularities, incorporating contributions from these auxiliary objects. Through explicit calculations in several models, we confirm that our discontinuity relations are consistent with results from the in-in formalism. While primarily developed using tree-level exchanges with polynomial interactions, we also demonstrate that our framework can be extended to include loop corrections and cases with derivative interactions.