Yoctosecond imaging of the ground state of $^{129}$Xe at the Large Hadron Collider

By combining Bayesian inference with hydrodynamic simulations and Large Hadron Collider data from Xe-Xe and Pb-Pb collisions, researchers successfully reconstructed the nearly maximally triaxial ground-state shape of the $^{129}$Xe nucleus, thereby establishing high-energy collider experiments as a quantitative tool for probing proton-neutron correlations driven by quantum chromodynamics.

The Gist

The Big Idea: Taking a "Group Photo" of an Atom's Core

Imagine you want to take a picture of a spinning, wobbly balloon. If you snap a single photo, you only see one specific angle. You can't tell if the balloon is perfectly round, slightly squashed, or shaped like a peanut. To understand its true shape, you need to take thousands of photos from different angles and look for patterns in how the light hits it.

This is exactly what the scientists at CERN did, but instead of a balloon, they were looking at the nucleus of a Xenon-129 atom.

The Challenge: You Can't See the Invisible

Atoms are incredibly small. You can't put a Xenon atom under a microscope and take a picture of its protons and neutrons (the "constituents") because the rules of quantum mechanics say you can't know exactly where they are at any single moment. It's like trying to photograph a swarm of bees in a dark room with a camera that only takes one picture per second; you'd just get a blur.

To "see" the shape of the nucleus, the scientists needed a different approach. They realized that if they could smash two Xenon atoms together at nearly the speed of light, the collision would act like a high-speed camera flash.

The Experiment: The "Yoctosecond" Snapshot

The paper describes a collision that happens in a yoctosecond (that's $10^{-24}$ seconds).

• The Freeze-Frame: Because the collision is so fast, the protons and neutrons inside the atoms don't have time to move. They are "frozen" in whatever random arrangement they were in at that exact moment.
• The Explosion: When they smash together, they create a tiny, super-hot soup of energy called a Quark-Gluon Plasma (QGP). Think of this as a drop of water hitting a hot pan and instantly turning into steam.
• The Flow: This "steam" expands outward. Crucially, the shape of the explosion depends on the shape of the atoms that smashed together. If the atoms were round, the explosion is round. If they were egg-shaped, the explosion stretches out like a rugby ball.

The Detective Work: Reading the Debris

The scientists didn't just watch the explosion; they measured the particles flying out of it. They looked at two main things:

1. How fast the particles are moving (Transverse Momentum).
2. How "oval" the explosion is (Elliptic Flow).

They found a clever trick: The size of the explosion and its shape are linked.

• If the atoms are shaped like a long egg (prolate) and they hit "side-on," the explosion is big and very oval.
• If they hit "end-on," the explosion is small and very round.
• By measuring thousands of these collisions, they could work backward to figure out the original shape of the Xenon nucleus.

The Discovery: The "Kiwi" Shape

Using a powerful computer method called Bayesian Inference (which is like a super-smart detective piecing together clues to solve a mystery), they analyzed data from the Large Hadron Collider (LHC).

They discovered that the Xenon-129 nucleus is not a perfect sphere, nor a simple egg.

• They describe it as a "triaxial" shape.
• The Analogy: Imagine a kiwi fruit or a slightly squashed rugby ball that has three different lengths: long, medium, and short. It's not just flat or long; it's lumpy in three different directions.
• This shape is "nearly maximally triaxial," meaning it is very distinct and not just a slight wobble.

Why This Matters

Before this, scientists had to guess the shape of these nuclei using complex math theories (like "mean-field calculations"). This paper is the first time they have experimentally measured the shape and the internal correlations of protons and neutrons in a Xenon nucleus using a particle collider.

They essentially proved that colliders can act as microscopes for the quantum world. By smashing atoms together, they can "image" the invisible arrangement of particles inside, confirming that the nucleus of Xenon-129 is a complex, three-dimensional object that looks a bit like a kiwi fruit.

Summary

• The Problem: You can't take a single photo of a quantum nucleus.
• The Solution: Smash thousands of them together and look at the pattern of the debris.
• The Result: The Xenon-129 nucleus is shaped like a triaxial ellipsoid (a kiwi fruit), not a sphere.
• The Takeaway: Particle colliders are now powerful enough to "photograph" the internal structure of atomic nuclei, providing new data to help physicists understand how matter is built.

Technical Summary

Technical Summary: Yoctosecond Imaging of the Ground State of $^{129}$Xe at the Large Hadron Collider

Problem Statement
Imaging quantum many-body systems requires resolving the coordinates of constituents across large event ensembles to measure correlation functions. While high-energy nuclear collisions offer a unique opportunity to probe nuclear deformation and constituent correlations on the femtometer scale, a quantitative extraction of these correlation properties from experimental data has historically been lacking. Specifically, there is a need to determine whether collider experiments can serve as quantitative probes of the ground-state wave functions of atomic nuclei, particularly for isotopes like Xenon-129 ($^{129}$Xe) where mean-field theories predict complex triaxial shapes.

Methodology
The authors employ a combined global Bayesian analysis of Large Hadron Collider (LHC) data from Xenon-Xenon (Xe-Xe) and Lead-Lead (Pb-Pb) collisions. The methodology integrates three primary components:

1. Theoretical Framework (Rotor Model): The nuclear ground state is modeled using a semiclassical deformed-rotor description. The many-body probability density is constructed as an orientation average of a deformed intrinsic density (parameterized by a Woods-Saxon profile with quadrupole $\beta_2$ and triaxiality $\gamma$ parameters). This approach encodes long-range angular correlations among nucleons. In this model, nucleons are sampled independently from a rotated intrinsic distribution, with correlations emerging from the averaging over Euler angles.
2. Hydrodynamic Response: The evolution of the collision is simulated using the Trajectum framework. This includes initial-state construction, pre-equilibrium dynamics, second-order viscous hydrodynamics, and hadronic transport (SMASH). The model posits that the initial geometric anisotropy of the colliding nuclei (imprinted by the rotor shape) drives the anisotropic flow ($v_n$) and radial flow (quantified by mean transverse momentum $\langle p_T \rangle$) of the resulting Quark-Gluon Plasma (QGP).
3. Bayesian Inference: A rigorous Bayesian parameter estimation is performed to constrain the nuclear structure parameters ($\beta_2, \gamma$) and QGP transport coefficients. The analysis utilizes a sensitivity matrix to identify observables most sensitive to nuclear shape, specifically focusing on the ratio of Xe-Xe to Pb-Pb data to suppress dependencies on QGP medium properties. Key observables include anisotropic flow coefficients ($v_n\{2\}$) and the linear correlation between squared elliptic flow and mean transverse momentum, $\rho(v_2\{2\}^2, \langle p_T \rangle)$.

Key Contributions

• Quantitative Extraction of Nuclear Shape: The paper demonstrates the first rigorous extraction of the triaxiality parameter $\gamma$ and quadrupole deformation $\beta_2$ for $^{129}$Xe using collider data, establishing a direct link between final-state particle correlations and ground-state nuclear structure.
• Correlation Function Measurement: By inferring the shape parameters, the authors calculate and report the first experimental determinations of two- and three-body angular correlation operators in the ground state of $^{129}$Xe. These include the mean-squared nuclear eccentricity and the covariance between nuclear radius and anisotropy.
• Validation of Mean-Field Predictions: The extracted shape parameters are compared against ab initio and mean-field calculations (Projected Generator Coordinate Method and Brussels-Skyrme-on-a-Grid), showing strong alignment with theoretical predictions for xenon isotopes away from shell closure.
• Octupole Deformation in $^{208}$Pb: The analysis also yields a robust determination of octupole deformation fluctuations in Lead-208 ($^{208}$Pb), finding a significant standard deviation $\sigma(\beta_{Pb,3}) \approx 0.127$, which aligns with recent Coulomb excitation measurements.

Results

• Shape of $^{129}$Xe: The posterior distribution indicates a well-deformed nucleus with $\beta_2 \approx 0.20$ and a triaxiality parameter $\gamma \approx 40^\circ$ (slightly oblate, but consistent with a maximally triaxial shape within uncertainties).
• Correlation Observables:
  • The two-body operator expectation value (mean-squared eccentricity) is determined to be $\langle E_2 E_{-2} \rangle_2 / \langle R^2 \rangle_1^2 = 0.0131(1)_{\text{stat}}(19)_{\text{syst}}$.
  • The three-body operator expectation (covariance of radius and anisotropy) is found to be $0.01(1)_{\text{stat}}(30)_{\text{syst}} \times 10^{-3}$, a value close to zero as expected for a triaxial rotor.
• Model Performance: The global fit reproduces experimental data (ALICE and ATLAS) with high precision, with theoretical curves lying within $\pm 5\%$ of experimental values across a broad centrality range. The analysis confirms that the correlation $\rho(v_2\{2\}^2, \langle p_T \rangle)$ is the primary observable sensitive to the triaxiality $\gamma$ in central collisions.

Significance and Claims
The paper claims to fill a critical gap in nuclear physics by providing a robust Bayesian framework for extracting nuclear-structure information from high-energy collisions with reliable uncertainty quantification. The authors assert that their work establishes collider experiments as a means of quantifying proton and neutron correlations arising from residual quantum chromodynamic forces.

The significance of these results lies in their potential to serve as new benchmarks for ab initio nuclear theory, enabling a program where Bayesian constraints from collider data are coupled with low-energy effective field theory studies. Furthermore, the authors suggest that these direct constraints on many-body correlations could reduce theoretical uncertainties in nuclear matrix elements relevant for neutrinoless double-beta decay and improve precision in dark matter and neutrino searches involving nuclear form factors. The work positions the LHC not merely as a particle discovery machine, but as a precision instrument for imaging strongly correlated nuclear systems.