⚛️ high-energy theory

Gluon Scattering in Expanding de Sitter Universe

This paper constructs an observer-centric S-matrix for pure Yang-Mills theory in the expanding Poincaré patch of four-dimensional de Sitter space, defining asymptotic states along an eternal comoving observer's worldline to compute tree-level gluon scattering amplitudes that exhibit structures similar to cosmological correlators and satisfy de Sitter isometry Ward identities.

Original authors: Audrey Lindsay, Tomasz R. Taylor

Published 2026-10-16

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CC BY 4.0

Original authors: Audrey Lindsay, Tomasz R. Taylor

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

In the vast, quiet expanse of the universe, physicists have long relied on a specific set of rules to predict how particles collide and scatter. These rules, known as scattering amplitudes, work beautifully in a flat, empty universe where time flows in a single, unchanging direction. In such a setting, scientists can define a clear "before" and "after," allowing them to calculate the odds of one particle turning into another with great precision. However, our actual universe is not flat and static; it is expanding, stretching like a balloon being blown up, a state described by a geometry called de Sitter space. In this expanding reality, the familiar rules break down. There is no single, universal clock that ticks the same for everyone, and the boundaries of the observable universe are not fixed points in time but rather horizons that shift as space stretches. This creates a profound puzzle: if the standard tools for predicting particle collisions rely on a static backdrop, how can we describe what happens when the stage itself is moving?

A team of researchers has now constructed a new way to answer this question, creating a mathematical framework that allows them to calculate particle collisions within this expanding universe. They focused on a specific type of particle interaction involving gluons, the force carriers that bind quarks together inside protons and neutrons. The challenge was to define what it means for particles to "collide" when the universe is constantly growing and when no single observer can see the entire event. To solve this, the researchers turned their attention to a special kind of observer: one who remains at a fixed position relative to the expanding space, drifting along with the cosmic flow. They realized that while many paths through this universe eventually cross a boundary and disappear from view, only these specific, "eternal" observers remain within the observable patch for all time, from the distant past to the far future.

By anchoring their calculations to the perspective of one of these eternal observers, the team built a new version of the scattering matrix, a tool that connects the state of particles before a collision to their state after. This new tool does not depend on which specific eternal observer is chosen; the results are the same regardless of where the observer sits in the expanding space. The researchers applied this method to a theory describing the strong nuclear force, known as Yang-Mills theory, and successfully calculated the probabilities for three and four gluons to interact. They found that these interactions follow a structure very similar to the patterns seen in the large-scale distribution of matter in the universe, yet they are distinct in how they handle the exchange of particles between the colliding gluons.

The study reveals that while the expanding universe complicates the definition of energy and time, it does not make particle physics impossible to calculate. The researchers showed that the symmetries of this expanding space impose strict rules on how these particles can behave, much like the conservation laws in a flat universe. They also discovered that as the expansion rate of the universe slows down and the space becomes flatter, their new calculations smoothly transition back into the familiar formulas used by physicists today. This work provides a concrete bridge between the physics of the early universe, where space was expanding rapidly, and the static physics we observe in laboratories on Earth. It demonstrates that even in a universe without a global clock, the fundamental interactions of matter remain consistent and calculable, provided one looks at the problem from the right perspective.

Technical Summary: Gluon Scattering in Expanding de Sitter Universe

Problem Statement
The construction of a scattering matrix (S-matrix) in de Sitter (dS) space presents fundamental challenges absent in flat spacetime. Unlike Minkowski space, dS space lacks a global timelike Killing vector, meaning there is no conserved energy for the spacetime as a whole, no preferred time coordinate for defining asymptotic in- and out-states, and the Hamiltonian is generally time-dependent. Furthermore, the presence of cosmological horizons and the spacelike nature of the asymptotic boundaries ( $I^\pm$ ) complicate the standard LSZ reduction procedure. While various approaches exist—such as the static patch (thermal description) or the closed slicing (global S-matrix)—a canonical S-matrix within the expanding Poincaré patch (the chart relevant for cosmology and inflation) has remained elusive. In this patch, the observables are typically late-time correlation functions or the wavefunction of the universe, rather than S-matrix elements.

Methodology
The authors construct an observer-centric S-matrix within the four-dimensional expanding Poincaré patch of de Sitter space, applying the technique previously used for global dS space [25] to pure Yang–Mills theory. The methodology proceeds through the following steps:

Selection of Observers: The authors identify "eternal observers" as the only geodesic observers that remain within the expanding patch for all proper time. These correspond to comoving worldlines ( $x = \text{const}$ ) in the patch. For any such observer, there exists a specific Killing vector (a linear combination of the dilation and translation generators) that generates translations in their proper time.
Hamiltonian Construction: The conserved charge associated with this Killing vector is promoted to the free Hamiltonian ( $H_0$ ). Unlike the global time translation in flat space, this Hamiltonian generates a flow that rescales momenta rather than leaving them invariant. The full Hamiltonian includes interaction terms derived from the conserved current of the same Killing vector contracted with the interacting energy-momentum tensor.
Gauge Fixing and Quantization: The theory is quantized in Coulomb gauge ( $\partial_i A^a_i = 0$ ). Because the patch is conformally flat and Yang–Mills is classically conformally invariant, the free mode functions are identical to those in Minkowski space. However, the action of the dS isometries (specifically boosts) does not preserve the Coulomb gauge. Consequently, the symmetry generators acting on the fields must include a compensating gauge transformation to restore the gauge slice.
S-matrix Definition: The S-matrix is defined as the overlap between asymptotic in- and out-states evolved by the full Hamiltonian relative to the free Hamiltonian along the eternal observer's worldline. The authors demonstrate that while the definition of asymptotic states depends on the specific observer chosen, the resulting S-matrix operator is independent of this choice, differing only by a phase factor that vanishes upon imposing momentum conservation.
Amplitude Computation: The authors compute tree-level three- and four-gluon scattering amplitudes. They express these results using spinor-helicity variables adapted to de Sitter space [29].

Key Results

Observer Independence: The constructed S-matrix is shown to be independent of the specific eternal observer chosen. The dependence on the observer's position $a$ is a unitary conjugation that reduces to momentum conservation in the momentum basis.
Amplitude Structure:
- Contact Terms: The contact contributions to the three- and four-gluon amplitudes agree structurally with the corresponding coefficients of the late-time wavefunction of the universe (cosmological correlators), differing only by the normalization of external states.
- Exchange Terms: The exchange contributions differ from the wavefunction coefficients. This discrepancy arises because the S-matrix utilizes the Feynman propagator, whereas the wavefunction coefficients utilize a bulk-to-bulk propagator that vanishes at the future boundary $I^+$ .
- Explicit Forms: The three-gluon amplitude is derived in spinor-helicity variables for all helicity configurations. The four-gluon amplitude is derived including quartic vertices, instantaneous Coulomb interactions, and gluon exchange diagrams, with color factors organized via trace identities.
Ward Identities: The paper derives Ward identities associated with the de Sitter isometry group $SO(1,4)$ $S O (1, 4)$ .
- For translations, rotations, and dilations (which preserve the Coulomb gauge), the Ward identities are homogeneous.
- For boosts, which do not preserve the gauge, the Ward identities contain an inhomogeneous term arising from the compensating gauge transformation. This term is identified as a boundary contribution at $I^+$ . The authors explicitly verify this at the three-point level, showing it corresponds to the momentum-space special conformal generator, and demonstrate its non-vanishing nature at the four-point level.
Flat Limit: The paper analyzes the flat limit ( $H \to 0$ ). Since the theory is conformal, the de Sitter radius drops out, and the limit is taken by examining the behavior of the time integral. The integral over conformal time $\eta$ produces a pole at the total energy $k_T = 0$ . The flat-space amplitudes (including the three-gluon and four-gluon MHV amplitudes) are recovered as residues of this total-energy pole. The authors confirm that the boost Ward identity remains consistent with this flat limit.

Significance and Claims
The paper claims to provide a rigorous construction of an S-matrix in the expanding Poincaré patch of de Sitter space, a setting previously dominated by correlation function approaches. By anchoring the scattering process to the worldline of an eternal observer, the authors define a consistent asymptotic framework despite the lack of a global time translation.

The work establishes that:

An observer-centric S-matrix is well-defined and unique (up to observer choice) in the expanding patch.
The resulting amplitudes for pure Yang–Mills theory share structural similarities with cosmological correlators but possess distinct features in the exchange channels due to different propagator prescriptions.
The symmetries of de Sitter space impose specific constraints (Ward identities) on these amplitudes, including non-trivial inhomogeneous terms for boosts that mirror constraints found in boundary conformal field theories.
The formalism correctly reproduces standard flat-space scattering amplitudes in the appropriate limit, validating the construction as a consistent generalization of flat-space QFT to expanding spacetimes.

The authors position this work as a step toward understanding scattering in cosmological backgrounds, bridging the gap between the "in-in" formalism of cosmological correlators and the "in-out" formalism of scattering amplitudes.

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