Reconstructing cosmological correlators via dispersion: from cutting to dressing rules

In this work, we investigate how cosmological correlators can be reconstructed by applying the momentum-space dispersion formula to their discontinuities, treating them as functions of momentum variables associated with the corresponding de Sitter Witten diagrams. We focus on conformally coupled and massless polynomial scalar interactions (both IR-divergent and IR-convergent), and consider tree-level de Sitter Witten diagrams. We explicitly utilize the single-cut discontinuity relations, or cutting rules, involving the cosmological correlators recently constructed in arXiv:2512.20720. For diagrams with multiple interaction vertices, we apply the dispersion formula by cutting all internal lines in the diagram one by one, successively, thereby allowing us to reconstruct the full correlator using only lower-point contact-level objects and their discontinuity data, up to contact diagram ambiguities. We also rediscover how the cosmological correlators on the late-time slice of de Sitter space can be obtained from flat-space Feynman diagrams via a set of dressing rules. Our starting point, being the cutting rules for the cosmological correlators, also emphasizes how basic principles, such as unitarity for in-in correlators, can lead us to the dressing rules, which were previously derived in literature following a different method.

💡 Research Summary

This paper addresses the problem of efficiently computing tree‑level cosmological correlators in de Sitter space for scalar theories with polynomial self‑interactions. Traditional approaches—Schwinger‑Keldysh (in‑in) perturbation theory or wave‑function methods—require cumbersome time‑ordered integrals, especially for diagrams with many external legs or multiple interaction vertices. Recent developments in the cosmological bootstrap have shown that unitarity, analyticity, and de Sitter isometries constrain the analytic structure of these correlators, leading to cutting rules (the cosmological optical theorem) and, separately, “dressing rules” that map flat‑space Feynman diagrams to de Sitter correlators via auxiliary shadow fields.

The authors take a different route: they apply a momentum‑space dispersion formula directly to the discontinuities of the cosmological correlators themselves, rather than to the wave‑function coefficients. The key ingredients are:

1. Single‑cut discontinuity relations derived in a companion paper (arXiv:2512.20720). For a two‑site exchange diagram, the discontinuity across the internal momentum p can be expressed as a product of lower‑point data: a genuine one‑site contact correlator B(1) and an auxiliary counterpart e B(1), with appropriate “Disc” and “g Disc” operations separating even/odd parts under p→−p.

2. Momentum‑space dispersion integral: given Discₚ B(p), the full correlator B(p) can be reconstructed (up to contact terms) via a Cauchy‑type integral over p′. The authors perform this integral explicitly for 2‑site and 3‑site diagrams, showing that the result is a sum of products of B(1) and e B(1) evaluated at the external momenta.

3. Inductive reconstruction for r‑site trees: By cutting the right‑most internal line, applying the single‑cut rule, and then using the dispersion integral, the r‑site correlator reduces to an (r‑1)‑site object. Repeating this procedure yields a general formula that expresses any tree‑level r‑site correlator solely in terms of the elementary one‑site data, plus possible contact ambiguities.

4. Generalized dressing rules: The authors reinterpret the above reconstruction as a set of operator “dressings” acting on ordinary flat‑space Feynman propagators. Instead of invoking the shadow formalism for each interaction, they introduce a universal dressing operator that multiplies each internal line by a factor involving time‑derivatives of the de Sitter bulk‑to‑boundary propagator P(η). Symbolically, \