Effects of Geometric configuration in relativistic isobaric collisions at $\sqrt{s_{NN}}=200$ GeV

This study utilizes the HYDJET++ model to investigate how nuclear deformation parameters ($\beta_2$, $\beta_3$) and surface diffuseness ($a$) influence charged hadron multiplicity and elliptic flow in symmetric isobaric ${}^{96}\mathrm{Ru}+{}^{96}\mathrm{Ru}$ and ${}^{96}\mathrm{Zr}+{}^{96}\mathrm{Zr}$ collisions at $\sqrt{s_{NN}}=200$ GeV, revealing distinct dependencies on collision geometry (tip-tip vs. body-body) that are compared with STAR experimental data.

The Gist

Imagine two massive, spinning balls of dough (atomic nuclei) smashing into each other at nearly the speed of light. Scientists at the Relativistic Heavy-Ion Collider (RHIC) have been doing this with two specific types of "dough": one made of Ruthenium (Ru) and one made of Zirconium (Zr).

Here is the simple story of what this paper investigates, using everyday analogies.

The Big Mystery: Why Do They Crash Differently?

Scientists wanted to use these crashes to find a very rare, mysterious signal called the "Chiral Magnetic Effect" (a clue about why our universe is made of matter instead of antimatter). To do this, they needed a perfect control group. Since Ru and Zr have the same total weight (mass number), they thought the crashes would be identical, differing only in their electrical charge.

However, the data came back with a surprise: the crashes weren't identical. The number of particles created and the way they flowed out were different. The paper asks: Why?

The answer lies in the shape of the nuclei. They aren't perfect spheres like billiard balls. They are lumpy, stretched, or even slightly pear-shaped.

The Ingredients: The "Lumps" and the "Crust"

The authors used a computer simulation (a digital crash-test lab called HYDJET++) to figure out how the shape affects the crash. They focused on three specific features:

1. The Stretch (Quadrupole Deformation, $\beta_2$): Imagine a rugby ball. It's stretched out at the ends. Ru is more like a rugby ball, while Zr is closer to a sphere.
2. The Pear-Shape (Octupole Deformation, $\beta_3$): Imagine a pear or a balloon with a bulge on one side. Zr has this "pear" shape, while Ru does not.
3. The Fuzzy Edge (Surface Diffuseness, $a$): Imagine the edge of a marshmallow. Is it sharp and hard, or soft and fuzzy? This parameter controls how "fuzzy" the edge of the nucleus is.

The Crash Scenarios: Head-on vs. Side-by-Side

To test these shapes, the scientists simulated two extreme ways the nuclei could hit each other:

• Tip-Tip (The "Needle" Crash): Imagine two rugby balls hitting each other end-to-end. This is the "tip-tip" collision.
• Body-Body (The "Side-by-Side" Crash): Imagine two rugby balls hitting each other along their long sides. This is the "body-body" collision.

What They Found

By running these simulations, the authors discovered how the "lumps" and "fuzziness" change the outcome:

1. The Number of Particles (Multiplicity)
Think of the crash as a crowd of people spilling out of a room.

• The Fuzzy Edge Matters: If the nuclei have a "fuzzier" edge (higher surface diffuseness), the crash zone is slightly larger, creating more particles.
• The Shape Matters:
  • In Tip-Tip crashes, the "pear" shape of Zirconium (the $\beta_3$ effect) actually reduced the number of particles in peripheral (glancing) crashes because the bulge made the overlap area smaller.
  • In Body-Body crashes, the "fuzziness" of Zirconium's edge helped create more particles, but the "pear" shape sometimes got in the way, reducing the count.

2. The Flow (Elliptic Flow, $v_2$)
When the nuclei crash, the debris doesn't fly out in a perfect circle; it flows more in one direction, like water squeezing through a narrow gap. This is called "elliptic flow."

• The "Roundness" Effect: If the nuclei are very stretched (like a rugby ball) and hit tip-to-tip, the resulting fireball looks more like a sphere. A sphere doesn't squeeze water as well, so the flow is weaker.
• The Zirconium Surprise: The "pear" shape (octupole deformation) in Zirconium actually made the flow stronger in side-by-side (body-body) crashes. It's as if the bulge on the pear helped squeeze the debris out more efficiently in that specific orientation.

The Main Conclusion

The paper concludes that you cannot treat these atomic nuclei as simple, perfect spheres.

• Orientation is Key: Whether the nuclei hit "tip-to-tip" or "side-by-side" changes the result dramatically.
• Shape Dictates Outcome: The specific "lumps" (deformation) and "fuzziness" (diffuseness) of the nuclei are the main reasons why the Ruthenium and Zirconium crashes produced different numbers of particles and different flow patterns.

Why does this matter for the scientists?
Before they can find the rare "Chiral Magnetic Effect" signal they are hunting for, they must perfectly understand and subtract the "background noise" caused by these weird shapes. If they don't account for the fact that Zirconium is a "pear" and Ruthenium is a "rugby ball," they might mistake a shape-induced effect for the new physics they are looking for.

In short: To find the hidden signal, you first have to understand exactly how the shapes of the crashing balls distort the mess they create.

Technical Summary

Technical Summary: Effects of Geometric Configurations in Relativistic Isobaric Collisions at $\sqrt{s_{NN}} = 200$ GeV

Problem Statement
The primary motivation for isobaric collisions ($^{96}_{44}\text{Ru} + ^{96}_{44}\text{Ru}$ and $^{96}_{40}\text{Zr} + ^{96}_{40}\text{Zr}$) at the Relativistic Heavy-Ion Collider (RHIC) was to isolate the Chiral Magnetic Effect (CME), a signal of local parity and charge-parity violation. The experimental strategy relied on the assumption that isobars, having the same mass number, would produce similar backgrounds, while their differing proton numbers would yield distinct magnetic field strengths and CME signals. However, recent blind analyses by the STAR collaboration revealed that the two systems exhibit different background signals in charged-particle multiplicity ($N_{ch}$) and flow harmonics ($v_n$). These discrepancies are attributed to differences in nuclear shapes—specifically, Ru possesses a larger quadrupole deformation ($\beta_2$), while Zr exhibits a stronger octupole component ($\beta_3$). This paper addresses the need to systematically understand how nuclear deformation and surface diffuseness influence final-state observables to disentangle geometry-driven backgrounds from potential CME signals.

Methodology
The study utilizes the HYDJET++ Monte Carlo event generator to simulate relativistic heavy-ion collisions at $\sqrt{s_{NN}} = 200$ GeV. The model superimposes a soft hydrodynamic state (based on Bjorken's solution) and a hard partonic state (modeled via PYTHIA and PYQUEN).

Key methodological features include:

• Nuclear Density Profiles: The standard Woods-Saxon distribution was modified to include nuclear deformation parameters ($\beta_2$ for quadrupole, $\beta_3$ for octupole) and surface diffuseness ($a$). The density function was transformed from spherical to cylindrical coordinates to accommodate specific collision orientations.
• Geometric Configurations: The analysis focuses on two extreme limiting configurations: tip–tip and body–body collisions. These are chosen to systematically probe the sensitivity of observables to initial nuclear geometry.
• Parameter Sets: Three distinct parameter sets (Case-1, Case-2, and Case-3) derived from literature and Density Functional Theory (DFT) calculations were employed to vary $\beta_2$, $\beta_3$, and $a$.
• Centrality Determination: To ensure a controlled comparison, centrality classes were determined independently for each geometric configuration based on the charged-hadron multiplicity distribution, rather than using a fixed impact parameter range.

Key Contributions and Results

1. Multiplicity ($N_{ch}$) and Pseudorapidity Distributions:

   • The study finds that quadrupole deformation ($\beta_2$) and surface diffuseness ($a$) produce orientation-dependent modifications to multiplicity.
   • In body–body collisions, the inclusion of octupole deformation ($\beta_3$) in Zr introduces asymmetry that reduces effective overlap, leading to lower charged-hadron production in peripheral collisions (50–60% centrality) compared to spherical or less asymmetric cases.
   • In tip–tip collisions, the effect of $\beta_3$ is configuration-dependent: it shows a positive impact on production in the most central (0–5%) collisions but a negative impact in peripheral collisions, where the shape asymmetry reduces the effective overlap volume.
   • The pseudorapidity distributions ($dN_{ch}/d\eta$) indicate that deformation effects primarily alter the overall particle yield rather than the longitudinal shape of the distribution.

2. Elliptic Flow ($v_2$):

   • Variations in $v_2$ are found to be modest but configuration-dependent.
   • In tip–tip collisions, the larger $\beta_2$ of Ru leads to a more spherical fireball, reducing the final-state momentum anisotropy ($v_2$) relative to Zr, resulting in a $v_2(\text{Ru})/v_2(\text{Zr})$ ratio below unity.
   • The inclusion of octupole deformation ($\beta_3$) significantly enhances the elliptic flow in body–body Zr+Zr collisions, with the effect being more pronounced in this configuration than in tip–tip collisions.

3. Comparison with Experimental Data:

   • The simulated results for $N_{ch}$ distributions and $v_2$ ratios are compared with STAR blind-analysis data. The study demonstrates that the observed differences between Ru and Zr in experimental data can be largely explained by the interplay of nuclear deformation and surface diffuseness across different collision geometries.

Significance and Claims
The paper claims that a proper understanding of orientation-dependent geometric effects is essential for interpreting CME-sensitive measurements. By isolating specific geometric configurations (tip–tip and body–body), the authors demonstrate that:

• Nuclear deformation and surface diffuseness are the primary drivers of the observed differences in $N_{ch}$ and $v_2$ between Ru and Zr isobars.
• The octupole deformation ($\beta_3$) of Zr plays a non-trivial role, introducing shape asymmetry that can either suppress or enhance particle production and flow depending on the collision centrality and orientation.
• While realistic collisions involve random orientations (averaging these effects), the sizable differences observed between the extreme limits suggest that residual deformation-induced effects remain non-negligible.

Consequently, the study argues that configuration-dependent analyses provide a structured framework to quantify deformation-induced backgrounds, offering a complementary perspective to standard event-by-event analyses to refine the interpretation of isobaric collision data.