⚛️ nuclear theory

Harmonic-dependent geometry-to-flow transfer in AMPT Ru+Ru and Zr+Zr isobar collisions

This study utilizes fixed-participant-number AMPT simulations to demonstrate that while elliptic flow in Ru+Ru and Zr+Zr isobar collisions scales accurately with initial geometric eccentricity, triangular flow exhibits a distinct harmonic-dependent response that retains a residual component beyond the initial geometry ratio.

Original authors: Murad Badshah, Muhammad Ajaz

Published 2026-07-15

📖 4 min read🧠 Deep dive

CC BY 4.0

Original authors: Murad Badshah, Muhammad Ajaz

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

Imagine two giant, squishy balls of nuclear dough crashing into each other at nearly the speed of light. This is what happens when physicists smash together two special types of atoms: Ruthenium (Ru) and Zirconium (Zr). These two atoms are "isobars," which is a fancy way of saying they have the exact same number of ingredients (96 nucleons), but they are shaped differently. One is a bit more squashed (deformed), and the other has a slightly different "skin" of particles on the outside.

Scientists want to know: When these two different-shaped balls smash, does the resulting explosion of particles look different just because the balls were shaped differently? Or does the "soup" created by the crash change the way the particles fly in a way that the original shape can't explain?

To find out, the authors of this paper ran a massive computer simulation called AMPT. Think of this simulation as a super-advanced video game engine that plays out the crash millions of times, tracking every single particle. They didn't just look at the crash; they measured two specific things:

The Shape of the Crash (Eccentricity): How oval or triangular the initial overlap of the two nuclei looks.
The Flow of the Particles (Flow): How the particles fly out in specific patterns after the crash.

They created a special "scorecard" called a double ratio. Imagine you are comparing two runners. You divide Runner A's speed by Runner B's speed. Then, you divide that by how much taller Runner A is than Runner B. If the result is exactly 1, it means the speed difference is only because of the height difference. If the result is not 1, something else is influencing the speed.

Here is what the simulation found:

The Oval Race (Elliptic Flow)
When the scientists looked at the oval-shaped patterns (the "elliptic" flow), the scorecard came out to be almost exactly 1.

What this means: The difference in how the particles flew out in an oval shape was completely explained by the difference in the initial shapes of the nuclei. The "soup" didn't add any extra twist to the oval pattern. If the nuclei were slightly more oval, the particles were slightly more oval, and that was it.

The Triangular Race (Triangular Flow)
But when they looked at the triangular patterns, the scorecard was greater than 1.

What this means: The particles flew out in a triangular shape more than you would expect just from the initial shape of the nuclei. Even after accounting for the fact that the nuclei were different, the simulation showed that the "soup" itself added an extra boost to the triangular flow. The initial shape wasn't the whole story; the way the particles interacted during the crash made the triangles stand out even more.

The "Skin" Question
The researchers also tested if the "neutron skin" (a layer of extra neutrons on the outside of the Zirconium nucleus) changed the results. They ran the simulation with the skin turned on and turned off.

The Result: The extra boost for the triangular flow happened in both cases. Whether the nucleus had a skin or not, the triangular flow was still stronger than the initial shape predicted. This suggests that while the skin is interesting, it isn't the main reason for this extra triangular boost.

How sure are they?
It is important to remember that this is a simulation, not a direct measurement from a real experiment yet. The authors ran about 325,582 simulated collisions for each setup to get a clear picture. They tested their results by changing the rules of the simulation slightly—like looking at different ranges of particle speeds or different numbers of collisions—to make sure the result wasn't just a glitch. Every time they changed the rules, the pattern stayed the same: the oval flow matched the shape perfectly, but the triangular flow always had that extra "oomph."

So, in the world of these computer crashes, the message is clear: The initial shape of the nucleus perfectly predicts the oval flow, but for the triangular flow, the chaotic dance of the particles inside the crash adds a little extra magic that the initial shape alone can't explain.

Technical Summary: Harmonic-dependent geometry-to-flow transfer in AMPT Ru+Ru and Zr+Zr isobar collisions

Problem and Motivation
In relativistic heavy-ion collisions, the final-state anisotropic flow ( $v_n$ ) is linked to the initial-state spatial eccentricity ( $\varepsilon_n$ ) via a response coefficient ( $\kappa_n$ ), typically approximated as $v_n \sim \kappa_n \varepsilon_n$ . While elliptic flow ( $n=2$ ) is dominated by the average overlap geometry, triangular flow ( $n=3$ ) is driven by event-by-event density fluctuations and higher multipole components. The RHIC isobar program, comparing $^{96}\text{Ru} + ^{96}\text{Ru}$ and $^{96}\text{Zr} + ^{96}\text{Zr}$ collisions at $\sqrt{s_{NN}} = 200$ GeV, offers a unique opportunity to study nuclear structure (deformation and neutron skin) by comparing systems with the same nucleon number but different internal structures.

Existing literature often analyzes the ratios of final-state flow ( $v_n^{\text{Ru}}/v_n^{\text{Zr}}$ ) and initial-state eccentricity ( $\varepsilon_n^{\text{Ru}}/\varepsilon_n^{\text{Zr}}$ ) separately. This paper addresses the question of whether the initial-geometry differences alone are sufficient to explain the final-state flow differences. Specifically, it investigates if the response coefficient $\kappa_n$ is harmonic-independent or if a "residual response" exists after normalizing out the initial eccentricity ratio.

Methodology
The authors utilize the string-melting version of the AMPT (A Multi-Phase Transport) model, which incorporates fluctuating initial conditions from HIJING, parton scattering, quark coalescence, and hadronic rescattering. To ensure a controlled comparison, the HIJING initialization was modified to sample deformed Woods–Saxon nuclei.

The study analyzes four distinct nuclear configurations:

Deformation-only Ru+Ru: Includes quadrupole deformation ( $\beta_2$ ) but assumes a common proton-neutron density.
Deformation-only Zr+Zr: Includes quadrupole ( $\beta_2$ ) and octupole ( $\beta_3$ ) deformation with a common density.
Deformation-plus-skin Ru+Ru: Includes deformation plus separate proton and neutron Woods–Saxon radii and diffuseness parameters (neutron skin effect).
Deformation-plus-skin Zr+Zr: Includes deformation plus separate proton and neutron parameters.

The analysis employs a fixed- $N_{\text{part}}$ approach. Instead of comparing centrality percentiles, events are binned by the number of participating nucleons ( $N_{\text{part}}$ ) to ensure participant-matched ensembles. The primary observable is the eccentricity-normalized isobar response double ratio:
$D_n(C) = \frac{v_n^{\text{Ru}}(C) / v_n^{\text{Zr}}(C)}{\varepsilon_n^{\text{Ru}}(C) / \varepsilon_n^{\text{Zr}}(C)}$
where $C$ denotes a common $N_{\text{part}}$ bin. If the flow ratio is fully determined by the eccentricity ratio, $D_n$ should equal unity. Deviations from unity indicate a harmonic-dependent response.

The study includes approximately $3.26 \times 10^5$ events per configuration. Systematic variations were performed regarding $N_{\text{part}}$ binning, peripheral bin removal, transverse momentum ( $p_T$ ) ranges, and rapidity/pseudorapidity acceptance. Null tests (split-half and label shuffling) were conducted to validate statistical stability.

Key Contributions

Introduction of the Double Ratio Observable: The paper introduces $D_n$ as a diagnostic tool to isolate the "response" component of the geometry-to-flow transfer, effectively removing the leading-order geometric scaling.
Modified AMPT Initialization: The implementation of separate proton and neutron Woods–Saxon parameters within the AMPT/HIJING framework to explicitly model the neutron skin effect alongside nuclear deformation.
Harmonic-Dependent Response Analysis: A systematic comparison of $n=2$ (elliptic) and $n=3$ (triangular) responses across different nuclear configurations and analysis cuts.

Results

Elliptic Flow ( $n=2$ ): The double ratio $D_2$ is found to be approximately unity ( $D_2 \simeq 1$ ) across all configurations and systematic variations. This indicates that the difference in elliptic flow between Ru and Zr is almost entirely explained by the difference in their initial elliptic eccentricities. The response coefficient $\kappa_2$ is effectively the same for both isobars once the initial geometry is normalized.
Triangular Flow ( $n=3$ ): The double ratio $D_3$ is consistently greater than unity ( $D_3 > 1$ ) for both deformation-only and deformation-plus-skin configurations. This enhancement persists across different $N_{\text{part}}$ binning schemes, $p_T$ ranges, and acceptance definitions. The deviation from unity is statistically significant (approximately 2 standard deviations in the standard systematic spread).
Neutron Skin Effect: While the neutron skin is present in the "deformation-plus-skin" samples, the $D_3 > 1$ trend is observed in both the "deformation-only" and "deformation-plus-skin" cases. This suggests that the residual triangular response is not dominated solely by the neutron skin but is a more general feature of the triangular flow channel in AMPT.
Fourth Harmonic: Due to the lack of explicit $\varepsilon_4$ records in the current AMPT output, direct $D_4$ calculation is not performed. Instead, nonlinear diagnostics (e.g., $v_4/\varepsilon_2^2$ ) are provided, but the primary conclusions are restricted to $n=2$ and $n=3$ .

Significance and Claims
The paper claims to provide a model-resolved statement regarding the AMPT response to isobar collisions. The central finding is a harmonic-dependent response:

Elliptic flow follows leading eccentricity scaling to high accuracy, implying that the medium's transport properties translate the average geometric deformation similarly for both Ru and Zr.
Triangular flow retains a residual response component after the triangularity ratio is divided out. This suggests that the conversion of initial triangularity to final flow is sensitive to details beyond the mean eccentricity ratio, such as fluctuation-driven correlations, finite-size effects, or nonlinear transport dynamics specific to the triangular channel.

The authors emphasize that this result is robust against systematic variations in analysis cuts and binning. They conclude that while the initial geometry difference is sufficient to describe the elliptic flow difference, it is insufficient to fully describe the triangular flow difference in the AMPT framework. The paper positions this as a quantitative, internally stable prediction for isobar response studies, highlighting the need for future comparisons with independent transport or hydrodynamic frameworks to determine the universality of this residual triangular response.

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