From Cosmological Cuts to Yang--Mills Wavefunctions in de Sitter Space

Abstract: We study tree-level Yang--Mills wavefunctions in four-dimensional de Sitter space using their discontinuities. Cosmological cuts factorize gluon discontinuities into lower-point wavefunctions glued by cut propagators and transverse projectors. For ray-like trees and one-loop $n$-gons, the maximal cuts take a particularly simple form: a scalar $φ^3$ discontinuity dressed by an ordered Yang--Mills numerator built from local gluing maps. We then use these cuts as reconstruction data for the four-, five-, and six-gluon wavefunctions in momentum space. The result separates into a cut-detectable part obtained from lower-point gluing and a cut-invisible completion fixed by current conservation and the flat-space limit. Through six points, the terms without longitudinal propagators follow the pole structure of color-ordered scalar $φ^3+φ^4$ wavefunctions, dressed by local Yang--Mills numerators. Longitudinal propagators collapse part of this scalar structure into contact-type contributions, with the first internal-line corrections appearing at six points. The reconstructed expressions agree with direct momentum-space Feynman-rule computations and give concrete low-point data for an all-$n$ organization of spinning de Sitter wavefunctions.

• The paper develops a novel framework to reconstruct tree-level Yang–Mills wavefunctions in de Sitter space via their energy discontinuities.
• It adapts flat-space amplitude methods to analyze the factorization and gluing of lower-point wavefunction coefficients while enforcing current conservation.
• The study provides explicit momentum-space expressions for four-, five-, and six-point sectors, laying groundwork for higher-multiplicity generalizations.

Detailed Summary of "From Cosmological Cuts to Yang--Mills Wavefunctions in de Sitter Space" (2606.25878)

Introduction and Motivation

This paper develops an explicit framework for reconstructing tree-level Yang--Mills wavefunction coefficients in four-dimensional de Sitter (dS$_4$) space using their energy discontinuities. The motivation is grounded in the cosmological optical theorem, which relates the singularity structure of the Bunch--Davies wavefunction to Hermitian-analytic discontinuities (“cosmological cuts”). These discontinuities, associated with internal energies in exchange diagrams, factorize observables into products of lower-point coefficients and thus provide recursive reconstruction data for cosmological wavefunctions.

The central question investigated is: how much of a spinning (i.e., Yang--Mills) cosmological wavefunction is fixed by its discontinuities? The approach leverages similarities with flat-space amplitude techniques (generalized unitarity cuts, leading singularities, etc.), but adapts them to the different analytic structure of cosmological wavefunctions in de Sitter space. The work is technically situated at the intersection of cosmological correlator bootstrap, amplitude-inspired organization, and combinatorial/geometric structures (e.g., cosmological polytopes).

Review of Wavefunction Coefficients and Cutting Rules

The paper starts by reviewing the structure of late-time Bunch--Davies wavefunction coefficients in momentum space. At tree-level, $\Psi[\varphi]$ admits a functional Taylor expansion that yields $n$-point coefficients obeying conformal boundary symmetry. Color decomposition and transverse polarization contraction is discussed for Yang--Mills theory: the basic object of interest is the color-ordered, contracted, transverse gluon wavefunction coefficient $\psi_n^{YM}$.

Cosmological cuts (discontinuities) are systematically defined as sign-flip differences across internal energy channels. The discontinuity of a bulk-to-bulk propagator factorizes the exchange diagram into a product of left and right lower-point coefficients, glued along the cut propagator and summed over physical states with transverse projectors. Maximal compatible cuts, where every internal line is cut, are shown to yield particularly localized expressions analogous to leading singularities in flat-space amplitudes.

Tree-level explicit examples in scalar theory ($\phi^3$) are provided, clarifying how singularity factorization operates vertex-by-vertex and how it extends to one-loop $n$-gons (cyclic diagrams), with associated energy variables. Scalar maximal cuts are then generalized to the Yang--Mills case, with tensor dressing via contracted three-gluon vertices and transverse projectors.

Yang--Mills Maximal Discontinuities and Tensor Structure

Explicit formulas are developed for the maximal discontinuity of tree-level Yang--Mills half-ladder (ray-like) diagrams and their one-loop $n$-gon analogues. The core result is that the cut expression factorizes into a scalar ($\phi^3$) maximal discontinuity and an ordered tensor numerator constructed recursively from three-point vertices and projectors.

A recursive gluing map is defined, building up open currents for chain diagrams. At tree level, the maximal cut is then an open-chain numerator $\mathcal N^{HL_n}$ contracted with the last polarization; at one loop, it closes into a cyclic trace. This structure mirrors the amplitude-style numerator/scalar organization familiar in flat-space, but with intrinsically cosmological singularity data.

Reconstruction from Discontinuities: Algorithm and Four-, Five-, Six-Point Results

The central methodological section applies dispersive reconstruction: given the discontinuity in an internal channel $k_I$, the non-analytic part is reconstructed via spectral integration over lower-point blocks, dressed by polarization sums. The approach is first demonstrated in scalar theory, then in Yang--Mills, with explicit gluing rules for three-, four-, five-, and six-point wavefunctions.

At each multiplicity, the answer is organized into:

• Cut-detectable part: sum over gluing lower-point blocks across the chord configuration of the $\Psi[\varphi]$0-gon (triangulation), reconstructing all terms with non-trivial discontinuities.
• Cut-invisible completion: rational terms not detected by cuts, constrained by current conservation (cancellation of spurious OPE poles), and the flat-space total-energy pole.

Four-Point Sector

At four points, two exchange channels ($\Psi[\varphi]$1, $\Psi[\varphi]$2) are reconstructed from gluing triangles, while a contact term is added to satisfy current conservation and reproduce the flat-space limit. The explicit expressions for $\Psi[\varphi]$3, $\Psi[\varphi]$4, and contact term $\Psi[\varphi]$5 are provided, ensuring vanishing residues at channel poles.

Five-Point Sector

Five-point reconstruction requires inclusion of two-chord (maximal cut), one-chord, and zero-chord (pentagon) sectors. The cut-detectable part is cyclically summed over gluing polygons, while the zero-chord sector (completion) cancels single-OPE residues. The recursive structure of numerators and their relation to Feynman-rule computation is made explicit.

Six-Point Sector

The six-point sector, with its higher combinatorial complexity, is efficiently organized by chord topology: ray-like maximally-cut sectors, non-ray-like variants, lower-codimension chord sectors (two-, one-chord), and the zero-chord completion (to cancel higher-order OPE poles). Representatives for each sector are written out, accompanied by their orbit sizes. The zero-chord sector is constructed from triple- and double-OPE poles, fully fixing the completion required for current conservation.

Generalization and Structural Patterns

Low-point results reveal that the scalar reference theory for organizational purposes is not pure $\Psi[\varphi]$6, but trace $\Psi[\varphi]$7 (planar scalar cubic and quartic vertices). This sets up a correspondence between scalar tubings of the $\Psi[\varphi]$8-gon and Yang--Mills numerators, with longitudinal-sector corrections manifesting as contact or internal-collapse type terms. At higher multiplicity, the completion sector is expected to consist of sparse corrections, inheriting their pole structure from scalar wavefunctions and restricted by current conservation and analyticity constraints.

Implications and Outlook

Theoretical Implications

• Factorization and Recursion: The explicit cutting rules and gluing formulas extend recursion techniques to spinning field theories in cosmological spacetimes. Maximal discontinuities isolate much of the structural content of Yang--Mills wavefunctions.
• Diagrammatic Organization: The polygonal decomposition provides a combinatorial/geometric organization for Yang--Mills wavefunctions, paralleling cosmological polytopes and positive geometry in scalar theories.
• Compatibility Constraints: Current conservation and OPE-pole cancellation act as powerful constraints, reducing the space of possible rational completions.
• All-$\Psi[\varphi]$9 pattern: The results support an inductive hypothesis that an all-$n$0 organization is possible, in terms of scalar tubings, Yang--Mills numerators, and sparse collapse corrections (potential connections to BCJ/graph relations).

Practical Implications

• Momentum-space Construction: Provides explicit momentum-space expressions for up to six-point gluon wavefunctions, suitable for concrete computation and further generalization.
• Foundation for Spinning Correlators: The reconstructed wavefunction data are a natural starting point for dispersive and dressing-rule approaches to spinning correlators.

Future Directions

• Correlation Functions: Extension to correlator construction, including application of physical cut bases and dressing-rule approaches for spinning fields.
• Gravity Double Copy and Tensor Algebra: Investigation of gravitational analogues and possible double-copy structure in cosmological observables.
• Spinor-Helicity and Representation-Theoretic Refinements: Application of spinor-helicity, particularly for all-plus helicity cases, and exploration of Grassmannian/on-shell diagram structures.
• Geometric Interpretations: Connection to cosmological polytopes, positive geometry, and possible canonical form or stratification descriptions that encode gluing/completion directly in momentum space.
• Numerical and Analytical Checks: The expressions, verified against direct Feynman-rule summations and analytic limits, provide a benchmark for future analytic or numeric approaches to higher-point cosmological amplitudes.

Conclusion

This work implements a constructive framework for reconstructing tree-level Yang--Mills wavefunctions in de Sitter space by gluing lower-point discontinuity data and enforcing current conservation. Through explicit computation at low multiplicity, a clear organizational structure emerges, potentially scalable to all multiplicities, whereby cut-detectable parts are recursively assembled, and sparse completions are fixed by local constraints. The results serve not only as consistency checks but as explicit, structured data enabling future progress both in cosmological correlator construction and in understanding the combinatorial and geometric structure of spinning observables in de Sitter space.