⚛️ high-energy theory

All-plus Yang--Mills and gravity wavefunctions in de Sitter space

This paper presents sewing representations for tree-level all-plus wavefunctions in four-dimensional de Sitter space, deriving explicit formulas for both Yang--Mills theory and Einstein gravity that utilize noncrossing polygon dissections and labelled sewing trees, respectively, while highlighting the absence of total-energy poles and the necessity of local boundary terms.

Original authors: Song He, Jiajie Mei, Yuyu Mo

Published 2026-10-27

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Song He, Jiajie Mei, Yuyu Mo

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Deep in the fabric of our universe, space itself is not empty but stretching, expanding at an accelerating rate. This state, known as de Sitter space, is the best description we have for the early moments of the cosmos and its distant future. In this expanding arena, particles and forces interact in ways that leave subtle fingerprints on the boundary of the universe, much like ripples on a pond that eventually reach the shore. Physicists study these ripples, called wavefunctions, to understand how the universe began and how the fundamental forces of nature, such as light and gravity, behave when space is stretching. While we have long understood how particles scatter in a static, flat universe, the rules for interacting in this expanding, curved space are far more complex and difficult to map.

A specific puzzle has long intrigued researchers: what happens when particles interact in a universe that is expanding, but all the particles involved share the same "spin" orientation? In the flat, static universe we are familiar with, a collection of four or more such particles simply cannot interact in a way that produces a measurable result; the interaction vanishes completely. However, in the expanding de Sitter space, this silence is broken. The wavefunction describing these interactions does not vanish; it contains rich, non-zero information. The challenge has been to find a clear, organized way to calculate these interactions for any number of particles, rather than just a few. Without a general rule, physicists are forced to calculate each scenario from scratch, a process that becomes impossibly messy as the number of particles grows.

In a new study, researchers have uncovered a surprisingly simple pattern that organizes these complex interactions. They discovered that for a theory describing the strong nuclear force, the interactions of any number of these same-spin particles can be built by stitching together smaller, simpler pieces. Imagine taking a polygon, like a hexagon, and drawing lines across it that do not cross each other, dividing the shape into smaller triangles and quadrilaterals. Each of these smaller shapes represents a known, simpler interaction. The researchers found that the total interaction is a sum of all possible ways to draw these non-crossing lines. Crucially, the way these pieces are connected involves a specific mathematical operation that compares the energy of the whole system to the energy of its parts, effectively filtering out a type of singularity that usually plagues these calculations. This method reveals that the complicated behavior of the entire system is entirely determined by the behavior of its smaller components, with no mysterious extra terms needed to make the math work.

The team arrived at this conclusion by comparing the physical universe to a simpler, idealized version of itself. They solved the equations for a "self-dual" version of the force, a special case where the mathematics is much cleaner, and then carefully measured the difference between this ideal case and the real physical case. This difference turned out to be the key. It acted like a set of instructions for how to sew the smaller pieces together. By following these instructions, they could reconstruct the full, complex interaction from the ground up, confirming that the messy, high-energy singularities that usually appear in these calculations cancel out perfectly, leaving behind a clean, organized structure.

When the researchers applied this same logic to gravity, the story became slightly more intricate but followed a similar theme. Gravity is far more complex than the strong nuclear force, and the simple stitching pattern of non-crossing lines was not enough on its own. The researchers found that to fully describe the gravitational interactions, they needed to add a specific, local correction term. This term acts like a boundary condition, a piece of information fixed at the edge of the universe that cannot be deduced simply by looking at the internal connections between particles. This correction arises from a specific type of geometric twist in the fabric of space-time known as a Chern-Simons term. By including this local piece, they were able to propose a general formula for gravity that works for any number of particles, just as they did for the strong force.

The findings suggest that the universe organizes these complex interactions with a deep underlying order. For the strong force, the order is purely combinatorial, relying on the ways shapes can be divided without crossing lines. For gravity, the order is a combination of these connections and a specific boundary signature. The researchers provided explicit formulas for systems with up to seven particles, demonstrating that their method works consistently. They also showed that these formulas naturally avoid a type of mathematical infinity that would occur if the total energy of the system were zero, a feature that distinguishes these expanding-space interactions from their flat-space counterparts.

This work does not just offer a new way to calculate numbers; it changes how we visualize the structure of the universe. It shows that even in the chaotic, expanding environment of the early cosmos, the interactions of fundamental particles are governed by a rigid, elegant logic. The researchers have provided a toolkit that allows physicists to predict the behavior of these systems without getting lost in an ocean of complicated equations. By breaking the problem down into manageable, interconnected pieces, they have revealed that the wavefunction of the universe, even in its most complex configurations, is built from a finite set of simple, repeating patterns. This clarity opens the door to exploring even more complex scenarios, potentially leading to a deeper understanding of how the universe evolved from its earliest moments.

Technical Summary: All-plus Yang–Mills and Gravity Wavefunctions in de Sitter Space

Problem Statement
The late-time wavefunction in de Sitter (dS) space encodes the boundary correlation functions of interacting fields. While the analytic structure of these wavefunctions connects to flat-space scattering amplitudes via total-energy singularities, extracting this information requires managing gauge constraints and boundary conditions. A specific challenge arises in "all-plus" helicity configurations. In flat space, tree-level all-plus amplitudes vanish for four or more external particles. However, in de Sitter space, the corresponding wavefunction coefficients are generically non-zero. Their partial-energy dependence and local boundary terms contain information beyond the flat-space limit. Previous work had reconstructed these coefficients for scalar QED and Yang–Mills (up to six points) using cosmological cuts. This paper addresses the organization of all-plus coefficients in Yang–Mills theory and Einstein gravity at arbitrary multiplicity, focusing on how the classical boundary-value problem structures these coefficients.

Methodology
The authors employ a "sewing" representation derived from self-dual solutions. The core strategy involves comparing the physical Dirichlet problem with a formal self-dual reference problem that shares the same linear, positive-helicity boundary source.

Self-Dual Reference: For Yang–Mills, the self-dual reference solution has a vanishing on-shell action beyond three points (due to the cubic nature of the boundary Chern–Simons term). For gravity, the self-dual Einstein action yields a non-vanishing local boundary term (gravitational Chern–Simons) that depends nonlinearly on the boundary metric source.
Green Function Factorization: The difference between the physical Green function and the self-dual reference Green function factorizes. This difference introduces a specific "sewing" operation at each endpoint of a sewn line: an energy difference operator ( $D_p = f(p) - f(-p)$ ) and a weight factor ( $1/2p$ for Yang–Mills, $2/p^3$ for gravity).
Expansion: By expanding the physical solution in terms of the difference between the two Green functions, the authors derive representations where the full $n$ -point wavefunction is expressed as a sum of products of lower-point wavefunctions connected by these sewing lines.

Key Contributions and Results

1. Yang–Mills Theory
The paper derives an arbitrary-multiplicity formula for all-plus Yang–Mills wavefunctions ( $\Psi_n$ ).

Structure: The formula is a sum over noncrossing polygon dissections (equivalently, the proper faces of an associahedron). Each dissection corresponds to a planar tree of sewn lower-point wavefunctions.
Sewing Rules: Each sewn line connects two wavefunction nodes. The line carries a weight $1/(2p)$ and applies an energy difference operator $D_p$ to the energy argument of the wavefunction at each endpoint.
Absence of Uncut Terms: The representation contains no independent uncut $n$ -point term; the entire coefficient is built from lower-point factors.
Pole Structure: The representation makes the absence of a generic total-energy pole ( $E_T = \sum k_i = 0$ ) manifest, as no term possesses such a pole. Partial-energy poles and their residues are inherited directly from the lower-point factors.
Explicit Formulas: The authors provide explicit expressions for 4, 5, and 6 points. At six points, the structure includes both chain-like sewings and "star" configurations (branching sewings).

2. Einstein Gravity
For gravity, the authors propose an analogous sewing representation, but with distinct features due to the lack of cyclic ordering and the presence of a local boundary term.

Structure: The formula is an alternating sum over labelled sewing trees (no cyclic-order restriction) plus a local gravitational Chern–Simons boundary term ( $CS_n$ ).
Sewing Rules: The sewing weight is $2/p^3$ , and energy difference operators act on the full radial-mode dependence. The sum runs over all distinct trees with labelled external legs.
Local Chern–Simons Term: Unlike Yang–Mills, the self-dual gravity action generates a non-vanishing local term at the boundary. This term, derived from the boundary connection expansion, supplies boundary data not fixed by internal-energy cuts. It is essential for the identity to hold.
Explicit Formulas: The paper provides explicit 4- and 5-point relations and describes the 6-point sewing structure. It includes a generating formula for the Chern–Simons kernels through seven points (Appendix A), expressed in dS spinor-helicity variables.

Significance and Claims
The paper claims that these sewing representations reveal a simpler analytic organization for all-plus wavefunctions than individual bulk diagrams suggest.

Manifest Absence of Total-Energy Poles: In both theories, the sewing representation explicitly demonstrates the absence of generic total-energy poles. In gravity, this is noted as a stronger statement than the vanishing of flat-space amplitudes, as it implies the cancellation of all subleading poles as well.
Factorization: The results show that partial-energy singularities and their successive residues are entirely determined by lower-point factors, with compatible cuts organizing the simultaneous singular limits.
Recursive Nature: The constructions are recursive in multiplicity, utilizing full lower-point wavefunctions (which include mixed helicities) as building blocks. The authors note that while this organizes the coefficients, it does not yet provide a closed recursive system purely within the all-plus sector, as the building blocks require mixed-helicity inputs.
Connection to Geometry: The Yang–Mills result is explicitly linked to the combinatorics of the associahedron, while the gravity result highlights the role of the gravitational Chern–Simons term arising from the self-dual Einstein action.

The authors conclude that these representations provide a systematic way to construct arbitrary-multiplicity wavefunctions, with the local boundary terms in gravity playing a crucial role in fixing data invisible to standard cutting rules.

More like this