Investigation of the shape of uranium in relativistic $^{238}$U+$^{238}$U collisions with nuclear densities from covariant density functional theory

This study utilizes state-of-the-art 3D lattice covariant density functional theory to calculate uranium densities for hydrodynamic simulations of relativistic $^{238}$U+$^{238}$U collisions, revealing a tension between elliptic flow and transverse-momentum observables regarding effective quadrupole deformation while highlighting the challenges of constraining octupole deformation due to uncertainties in reference nuclear structures.

The Gist

Imagine trying to figure out the exact shape of a squishy, invisible ball by smashing two of them together at nearly the speed of light. That is essentially what this paper is about.

The scientists are studying Uranium-238, a heavy atom that isn't perfectly round like a billiard ball. Instead, it's a bit squashed and stretched, like a rugby ball or a peanut. They want to know exactly how squashed it is and if it has any weird "pear-shaped" bumps on it.

Here is the story of their investigation, broken down into simple parts:

1. The Old Way vs. The New Way

For a long time, scientists tried to guess the shape of these atoms using a simple, standard recipe (called the "Woods-Saxon" profile). It was like trying to describe a complex, hand-carved wooden sculpture using a generic, mass-produced plastic mold. It gave a rough idea, but it wasn't precise enough.

In this study, the researchers used a super-advanced computer model called Covariant Density Functional Theory (CDFT). Think of this as using a high-resolution 3D scanner to map every tiny bump, dip, and curve of the uranium atom's "skin" (its density) before smashing it. This new map includes not just the main squish (quadrupole), but also smaller, more complex wiggles (octupole and hexadecapole deformations).

2. The Big Smash

They simulated smashing two of these uranium atoms together at the Relativistic Heavy Ion Collider (RHIC). When they smash, they create a tiny, super-hot soup of particles called a Quark-Gluon Plasma (QGP).

As this soup cools down and expands, it sprays particles out in all directions. The way these particles fly out depends entirely on the shape of the two atoms that collided.

• If the atoms were perfect spheres, the spray would be round.
• If the atoms were rugby balls, the spray would be oval.
• If they had pear-shaped bumps, the spray would have a specific triangular twist.

3. The "Gold" Problem

To make sense of the Uranium smash, the scientists needed a control group. They compared the Uranium smash to smashing two Gold atoms together. Gold is usually treated as a perfect sphere in these experiments.

However, the researchers found a major issue: The "Gold" reference wasn't actually a perfect sphere.

• When they used the old, simple "Gold" mold, their Uranium predictions were way off.
• When they adjusted the "Gold" mold to match real-world data (making it slightly squashed too), the Uranium predictions for the "oval" spray (called elliptic flow) suddenly became perfect.

The Analogy: Imagine you are trying to measure the weight of a new fruit by comparing it to an apple. If you assume the apple weighs 100 grams, but it actually weighs 120 grams, your calculation for the new fruit will be wrong. The scientists realized they had been using the wrong weight for their "apple" (Gold), which threw off their measurements of the "new fruit" (Uranium).

4. The Mystery That Remains

Here is where the plot thickens. The new, high-tech Uranium map worked perfectly to predict the oval shape of the spray. But when they looked at other details—specifically how the speed of the particles fluctuated—the new map failed.

It's like having a map that perfectly predicts the direction a car will turn, but completely fails to predict how fast the car will go.

• The Flow: The shape of the spray matched the new Uranium map.
• The Speed: The speed of the spray did not match the new Uranium map.

This creates a "tension." The scientists can't find a single version of the Uranium atom that explains both the direction and the speed of the particles at the same time.

5. The "Pear" Shape Challenge

The researchers also tried to find out if Uranium has a "pear shape" (a specific kind of bump). They looked for a triangular twist in the spray to prove this.

• The Problem: The signal for this "pear shape" is so weak that it gets easily confused by the shape of the Gold atoms.
• The Result: Because they aren't 100% sure about the exact shape of the Gold atoms, they can't be sure if the Uranium is actually pear-shaped or if it just looks that way because of the Gold. It's like trying to hear a whisper in a room where the background noise (Gold) is constantly changing volume.

The Bottom Line

This paper tells us two main things:

1. We need better maps: Using the new, high-tech 3D maps for Uranium is a huge improvement over the old, simple guesses. It solves a long-standing mystery about why the "oval" spray looked wrong in the past.
2. We need better references: To fully understand the shape of Uranium, we also need to know the exact shape of Gold. Without that, we can't be sure about the "pear" shape, and we can't explain why the particle speeds don't match our predictions.

The scientists conclude that to truly understand the shape of these atomic nuclei, we need to combine the best nuclear physics maps with the best collision simulations, and we need to stop treating the "control" atoms (Gold) as perfect spheres when they clearly aren't.

Technical Summary

Technical Summary: Investigation of the Shape of Uranium in Relativistic 238U+238U Collisions

Problem Statement
Relativistic heavy-ion collisions, specifically 238U+238U, serve as a unique probe for determining the ground-state shape deformation of uranium nuclei. While the quadrupole deformation ($\beta_2$) of 238U is well-established via low-energy nuclear data, higher-order deformations (octupole $\beta_3$ and hexadecapole $\beta_4$) remain difficult to quantify experimentally and theoretically. Previous studies relying on simplified Woods–Saxon density profiles have led to ambiguities, particularly regarding the interpretation of elliptic flow ($v_2$) ratios between U+U and Au+Au collisions (the "ultra-central $v_2$ puzzle") and the estimation of background contributions in isobar collisions. Furthermore, there is a need to determine if realistic nuclear densities derived from modern theoretical frameworks can resolve discrepancies between flow observables and transverse-momentum-related observables.

Methodology
The authors employ a state-of-the-art interdisciplinary approach combining microscopic nuclear structure calculations with macroscopic hydrodynamic simulations:

• Nuclear Density Input: The initial nuclear density distributions for 238U are calculated using three-dimensional (3D) lattice covariant density functional theory (CDFT) with pairing correlations (using the PC-PK1 functional). This framework captures equilibrium states with specific deformation parameters: a quadrupole-deformed prolate state ($\beta_{20}=0.29, \beta_{30}=0$) and an octupole-deformed state ($\beta_{20}=0.29, \beta_{30}=0.09$).
• Collision Simulation: The dynamic evolution of the quark-gluon plasma (QGP) and subsequent hadronic matter is simulated using the event-by-event (2+1)-dimensional viscous hydrodynamic model iEBE-VISHNU, coupled with the UrQMD hadron cascade model.
• Initial Conditions: The TRENTo model is used to generate initial conditions based on the CDFT-derived density profiles.
• Reference System: 197Au+197Au collisions are used as a reference to construct relative observables $R(X) = X_{UU}/X_{AuAu}$. Two distinct Woods–Saxon parameter sets for 197Au are tested: a standard set ($Au_{default}$) and a new set ($Au_{new}$) derived from electron scattering data to better account for nuclear deformation.
• Observables: The study analyzes elliptic flow ($v_2$), triangular flow ($v_3$), their squared cumulants ($v_n\{2\}^2$), and transverse-momentum correlations, specifically $\langle v_2^2 \delta p_T \rangle$ and the variance $\langle (\delta p_T)^2 \rangle$.

Key Contributions

1. Integration of CDFT Densities: The study introduces realistic 3D CDFT nuclear densities, including octupole and hexadecapole deformations, as inputs for relativistic heavy-ion simulations, moving beyond traditional Woods–Saxon approximations.
2. Resolution of the Ultra-Central $v_2$ Puzzle: The authors demonstrate that the CDFT-derived density, which inherently includes a substantial hexadecapole deformation ($\beta_{40}=0.16$), results in an effective Woods–Saxon quadrupole parameter that is smaller than the intrinsic value. This correction largely resolves the overestimation of the $v_2^2$ ratio in ultra-central collisions previously observed with simplified parameterizations.
3. Sensitivity Analysis of Reference Systems: The paper highlights that the extraction of uranium's shape from flow ratios is critically dependent on the precise nuclear structure of the reference nucleus (197Au).

Results

• Quadrupole Deformation ($v_2$): The CDFT-based simulations successfully reproduce the experimental STAR data for the ratio $R(v_2\{2\}^2)$ in ultra-central collisions, validating the linear response regime and the necessity of realistic density profiles.
• Octupole Deformation ($v_3$): While the octupole-deformed configuration ($\beta_{30}=0.09$) shows an enhancement in the triangular flow ratio $R(v_3\{2\}^2)$ compared to the prolate configuration, the absolute magnitude of this signal is highly sensitive to the choice of 197Au parameters. The $Au_{new}$ set improves agreement for central collisions but worsens it for mid-centralities, creating an ambiguity that prevents a clear extraction of the octupole deformation solely from $v_3$ comparisons.
• Transverse Momentum Observables: A significant tension emerges when examining transverse-momentum-related observables. While the CDFT density describes $v_2$ well, it fails to simultaneously describe $\langle v_2^2 \delta p_T \rangle$ and $\langle (\delta p_T)^2 \rangle$. The model systematically overestimates $\langle v_2^2 \delta p_T \rangle$ and underestimates $\langle (\delta p_T)^2 \rangle$. This discrepancy suggests that the effective quadrupole deformation encoded in the initial state does not map consistently to both flow and momentum fluctuations under current modeling assumptions.
• Gold Nucleus Uncertainty: The study identifies that uncertainties in the triaxial deformation and surface diffuseness of 197Au significantly impact the reliability of relative observables. Preliminary CDFT calculations suggest 197Au may possess a triaxial deformation ($\gamma \approx 44^\circ$) differing from the standard assumption ($\gamma = 60^\circ$), which could affect predictions for $\langle v_2^2 \delta p_T \rangle$.

Significance and Claims
The paper claims that realistic nuclear densities are essential for the quantitative modeling of relativistic heavy-ion collisions. The authors assert that their work resolves the "ultra-central $v_2$ puzzle" by demonstrating that the hexadecapole deformation inherent in CDFT calculations naturally corrects the effective quadrupole parameter used in phenomenological models.

However, the authors remain modest regarding the full characterization of nuclear deformation. They conclude that while CDFT densities improve the description of elliptic flow, a "clear mismatch" persists with transverse-momentum-related observables. This indicates a fundamental tension in how initial-state deformations are encoded in different experimental probes. Consequently, the paper argues that constraining initial-state modeling solely through flow observables is inadequate. The extraction of octupole deformation is further complicated by the sensitivity to the reference system's nuclear structure. The authors emphasize that future progress requires interdisciplinary efforts to integrate realistic nuclear structure calculations (including 197Au) with advanced heavy-ion modeling to fully characterize nuclear deformation.