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

TL;DR

The paper develops a robust single-cut discontinuity framework for cosmological correlators in de Sitter space by leveraging unitarity and hermitian analyticity. It shows that the discontinuity of an N-site tree-level cosmological correlator can be written as a sum of products of lower-point discontinuities, plus novel auxiliary contributions arising from the imaginary parts of lower-point wave-function coefficients, with the dominant piece depending on whether the interaction power n is even or odd. The authors provide explicit tree-level and loop checks for conformally coupled and massless scalars, including derivative interactions, confirming consistency with in-in calculations and extending the formalism beyond polynomial interactions. They further extract a decomposition rule for partial-energy poles and discuss implications for flat-space limits, spinning extensions, and potential connections to alternative kinematic formalisms, underscoring the broad applicability of unitarity-based bootstrap ideas to cosmological correlators.

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.

Figures (14)

• Figure 1:
• Figure 2: Diagrammatic representation of the single-cut rule of a $2$-site correlator as in eq.\ref{['2siteDiscFinal Result']}. Red circle always denotes a correlator $\mathcal{B}$, whereas the purple circle denotes the auxiliary object $\widetilde{\mathcal{B}}$. The blue shaded line implies taking Disc with respect to the modulus of momentum corresponding to that line, and the yellow shaded line implies taking $\widetilde{\text{Disc}}$ with respect to the modulus of momentum corresponding to that line.
• Figure 3: Diagrammatic representation of the single-cut rule of a $r$-site correlator as in eq.\ref{['r site correlator']}. Red circle denotes correlator $\mathcal{B}$ , the purple circle denotes the auxiliary object $\widetilde{\mathcal{B}}$ and the patterned blob represents $\mathcal{V}^\sigma_{(r-2)}$. The first term in the RHS denotes the term $\text{Disc}_{p_{r-1}}\mathcal{B}^{(r-1)}\text{Disc}_{p_{r-1}}\mathcal{B}^{(1)}$ and the second term describes the term $\widetilde{\text{Disc}}_{p_{r-1}}\widetilde{\mathcal{B}}^{(r-1)}\widetilde{\text{Disc}}_{p_{r-1}}\widetilde{\mathcal{B}}^{(1)}$ as given in eq.\ref{['r site correlator']} .
• Figure 4: Diagrammatic representation of single-cut rule for the loop integrand of a 1-loop $2$-site correlator. Each blue shaded line corresponds to taking Disc with respect to the modulus of momentum of that line. On the other hand, each yellow line corresponds to taking $\widetilde{\text{Disc}}$ with respect to the modulus of the momentum of that line.
• Figure 5: Diagrammatic representations of the discontinuity relation of a $2$-site correlator with $\phi^5$ interaction in de Sitter. LHS signifies the eq.\ref{['Disc 2site LHS for phi5']} and the RHS denotes eq.\ref{['Disc 2site RHS for phi5']}.