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Reconstructing cosmological correlators via dispersion: from cutting to dressing rules

Shibam Das, Debanjan Karan, Babli Khatun, Nilay Kundu

TL;DR

The work develops a momentum-space dispersion framework to reconstruct cosmological correlators in de Sitter space from the discontinuities of their tree-level Witten diagrams. By applying successive single-cut discontinuities and dispersion integrals, higher-site correlators are expressed entirely in terms of 1-site data and their discontinuities, up to contact-term ambiguities. A universal set of diagrammatic dressing rules is introduced, mapping flat-space diagrams to late-time de Sitter correlators via color-coded vertices and auxiliary propagators, grounded in unitarity and the cosmological cutting rules. The framework is validated through explicit checks for conformally coupled and massless scalar theories with polynomial interactions, spanning IR-convergent and IR-divergent cases, and connects to existing results from the shadow formalism, offering a versatile tool for analytic control of inflationary correlators.

Abstract

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.

Reconstructing cosmological correlators via dispersion: from cutting to dressing rules

TL;DR

The work develops a momentum-space dispersion framework to reconstruct cosmological correlators in de Sitter space from the discontinuities of their tree-level Witten diagrams. By applying successive single-cut discontinuities and dispersion integrals, higher-site correlators are expressed entirely in terms of 1-site data and their discontinuities, up to contact-term ambiguities. A universal set of diagrammatic dressing rules is introduced, mapping flat-space diagrams to late-time de Sitter correlators via color-coded vertices and auxiliary propagators, grounded in unitarity and the cosmological cutting rules. The framework is validated through explicit checks for conformally coupled and massless scalar theories with polynomial interactions, spanning IR-convergent and IR-divergent cases, and connects to existing results from the shadow formalism, offering a versatile tool for analytic control of inflationary correlators.

Abstract

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.
Paper Structure (28 sections, 149 equations, 5 figures)

This paper contains 28 sections, 149 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Review of the background material
  3. Reconstructing cosmological correlators via dispersion integration
  4. $2$-site correlator
  5. $3$-site correlator
  6. Reconstructing $r$-site correlator via induction method
  7. Diagrammatic dressing rules for cosmological correlators
  8. General statement of the diagrammatic dressing rules
  9. Applying dressing rules: Conformally coupled scalars with $\lambda \phi^n$ interaction
  10. IR convergent cases: conformally coupled $\lambda \phi^n$ with $n \ge 4$
  11. IR divergent case: conformally coupled $\lambda\phi^3$ interaction
  12. Applying dressing rules: Massless scalars with $\lambda\chi^3$ interaction
  13. Explicit checks of the diagrammatic dressing rules
  14. Explicit checks for conformally coupled $\lambda\phi^4$ theory
  15. Explicit checks for conformally coupled $\lambda\phi^3$ theory
  16. ...and 13 more sections

Figures (5)

  • Figure 1: Diagrammatic representation of a $2$-site dS correlator for conformally coupled $\phi^4$
  • Figure 2: Diagrammatic representation of a $2$-site dS correlator for conformally coupled $\phi^3$
  • Figure 3: Diagrammatic representation of a $3$-site dS correlator for conformally coupled $\phi^3$
  • Figure 4: Diagrammatic representation of a $3$-site dS correlator for conformally coupled $\phi^5$
  • Figure 5: The left and right contours are related by contour deformation. The point $O$ is $k^2 = p^2 + i\epsilon$.