Elliptic flow of strange and multi-strange hadrons in isobar collisions at $\sqrt{s_{\mathrm {NN}}} = 200\mathrm{~GeV}$ at RHIC

This paper reports a systematic measurement of elliptic flow for strange and multi-strange hadrons in isobar collisions at $\sqrt{s_{\mathrm{NN}}} = 200$ GeV, revealing approximate constituent quark scaling indicative of partonic collectivity and observing a $\sim$2% deviation in flow ratios between Ru+Ru and Zr+Zr collisions that highlights differences in their nuclear structure and deformation.

The Gist

The Big Picture: Smashing Atoms to Find the "Perfect Soup"

Imagine you have two identical-looking balls of clay. They weigh exactly the same, and they are made of the same total amount of material. However, one ball is perfectly round, while the other is slightly squashed like a rugby ball.

Scientists at the STAR experiment (part of the Relativistic Heavy Ion Collider, or RHIC) decided to smash these two types of "clay balls" together at nearly the speed of light. The goal was to see if the difference in their shape would change the way they explode.

The "clay" they used was actually two types of atomic nuclei called Isobars:

1. Ruthenium (Ru): A slightly squashed (deformed) nucleus.
2. Zirconium (Zr): A rounder nucleus.

They smashed them together to create a tiny, super-hot drop of "perfect soup" called the Quark-Gluon Plasma (QGP). This is the state of matter that existed just microseconds after the Big Bang, where particles are so hot they melt into a fluid of their smallest building blocks (quarks and gluons).

The Main Question: Does Shape Matter?

The scientists wanted to know: Does the shape of the starting ball change how the "soup" flows?

To measure this, they looked at a specific type of flow called Elliptic Flow ($v_2$).

• The Analogy: Imagine dropping a handful of glitter into a spinning bowl of water. If the bowl is round, the glitter spreads out evenly. If the bowl is oval (like a rugby ball), the glitter will flow more easily along the long side of the oval than the short side.
• In the experiment, when the nuclei collide, they create an oval-shaped "soup." As this soup expands, it pushes particles out. The scientists measured how much the particles preferred to fly out in the direction of the "long side" of the oval. This preference is the Elliptic Flow.

The Special Particles: The "Strange" Messengers

The paper focuses on a specific group of particles called Strange and Multi-Strange Hadrons (like $K^0_s$, $\Lambda$, $\Xi$, and $\Omega$).

• Why them? Think of these particles as "ghosts" or "snipers." Unlike other particles that bump into everything around them, these strange particles rarely interact with their neighbors.
• Because they don't bump into things much, they escape the "soup" very early. This makes them perfect messengers. They carry information about what the soup was like right at the beginning, before it cooled down and changed.

Key Findings (The "Aha!" Moments)

1. The "Lego" Rule (Quark Scaling)

The scientists found that these strange particles follow a universal rule called Constituent Quark Scaling.

• The Analogy: Imagine you are building with Legos. Whether you build a small car (2 Legos) or a big truck (10 Legos), the way they move depends on how many bricks they have.
• The data showed that the flow of these particles depends exactly on how many "quark bricks" they are made of. This proves that even in these smaller collisions, the matter behaves like a fluid made of free-floating quarks, not just whole atoms. It's like seeing the "soup" act like a single, cohesive liquid.

2. The Shape Difference (The 2% Clue)

When they compared the "squashed" Ruthenium collisions to the "round" Zirconium collisions, they found a tiny but important difference.

• The Result: In the middle of the collision (not too head-on, not too glancing), the "squashed" Ru collisions created about 2% more flow than the round Zr collisions.
• The Meaning: This tiny 2% difference is like a fingerprint. It confirms that the Ruthenium nucleus is indeed more deformed (squashed) than the Zirconium nucleus. The shape of the starting ball directly influenced how the "soup" flowed.

3. Size Matters (System Size)

They also compared these small collisions to much bigger ones (like Gold+Gold or Uranium+Uranium).

• The Analogy: Think of a small puddle vs. a giant ocean. If you throw a stone in both, the ripples behave differently.
• They found that as the collision gets bigger (more nucleons involved), the flow gets stronger. The "soup" in bigger collisions is more "collective" and flows more smoothly, just like a larger body of water.

The "Crystal Ball" (Computer Models)

To make sure their findings were real, they compared their data to a super-computer simulation called the AMPT model.

• They fed the computer the exact shapes of the Ru and Zr nuclei.
• The Result: The computer simulation matched the real-world data almost perfectly. This gives scientists confidence that they truly understand the nuclear structure of these atoms.

Why Does This Matter?

1. Mapping the Universe: It helps us understand the "equation of state" of the universe—how matter behaves under extreme pressure and heat.
2. Nuclear Structure: It acts as a new way to "see" the shape of atomic nuclei. We can now tell if a nucleus is round or squashed just by watching how the particles fly out after a crash.
3. The Big Bang: It confirms that even in smaller collisions, we can create a fluid-like state of matter that mimics the conditions of the early universe.

Summary

In short, the STAR team smashed two slightly different atomic twins together. By watching how "ghostly" particles flew out, they proved that the shape of the starting atom changes the flow of the resulting "perfect soup." They confirmed that this soup acts like a fluid of quarks and that the size of the collision determines how strong that flow is. It's like using the splash of a water balloon to figure out the exact shape of the balloon before it popped.

Technical Summary

1. Problem Statement and Motivation

The primary goal of the 2018 RHIC isobar run was to measure the Chiral Magnetic Effect (CME) by comparing collisions of two isobars: Ruthenium-96 ($^{96}_{44}\text{Ru} + ^{96}_{44}\text{Ru}$) and Zirconium-96 ($^{96}_{40}\text{Zr} + ^{96}_{40}\text{Zr}$). While the mass numbers are identical, the proton numbers differ, leading to different magnetic fields and expected CME signals.

However, initial results revealed unexpected differences in the collective flow (elliptic flow $v_2$ and triangular flow $v_3$) between the two systems, suggesting distinct nuclear structures (deformation and density profiles) despite their isobaric nature.

• The Specific Problem: While previous studies focused on charged hadrons, there was a lack of systematic data on strange and multi-strange hadrons ($K^0_S, \Lambda, \phi, \Xi, \Omega$) in these isobar collisions.
• Scientific Gap: Strange hadrons have small hadronic interaction cross-sections and are expected to "freeze out" earlier than light hadrons, closer to the phase transition from the Quark-Gluon Plasma (QGP). Measuring their elliptic flow ($v_2$) provides a cleaner probe of the early-stage partonic dynamics and the nuclear structure effects (deformation) without the contamination of late-stage hadronic rescattering.

2. Methodology

The analysis utilized data from the STAR experiment at RHIC, collecting approximately 1.77 billion minimum bias events for Ru+Ru and 1.84 billion for Zr+Zr at $\sqrt{s_{NN}} = 200$ GeV.

A. Particle Identification and Reconstruction

• Particles Studied: $K^0_S$, $\Lambda$, $\bar{\Lambda}$, $\phi$, $\Xi^-$, $\Xi^+$, and $\Omega^- + \Omega^+$.
• Technique: Particles were reconstructed via their hadronic decay channels using the Invariant Mass technique.
  • TPC (Time Projection Chamber): Used for tracking and specific energy loss ($dE/dx$) measurements.
  • TOF (Time-of-Flight): Used for velocity ($\beta$) measurements to calculate mass squared ($m^2$) for higher momentum particles.
  • Background Suppression:
    • For $\phi$ mesons: Mixed-event technique.
    • For $K^0_S, \Lambda, \Xi, \Omega$: Rotational background technique (rotating daughter tracks by $180^\circ$).
• Selection Cuts: Strict cuts on decay topology (DCA, decay length), track quality ($N_{hits} \ge 15$), and particle identification ($|n\sigma| < 3$) were applied.

B. Flow Analysis

• Method: The $\eta$ sub-event plane method was employed to measure the second-order flow coefficient ($v_2$).
• Event Plane: Reconstructed using charged tracks in two pseudorapidity windows ($-1.0 < \eta < -0.05$ and $0.05 < \eta < 1.0$) with a gap of $\Delta\eta = 0.1$ to minimize non-flow effects.
• Resolution Correction: The raw $v_2$ was corrected for the event plane resolution ($R$), calculated via the correlation between the two sub-events.
• Centrality: Events were classified into centrality bins (0-5% to 70-80%) based on charged particle multiplicity in $|\eta| < 0.5$, calibrated using a Monte Carlo Glauber model.

C. Systematic Uncertainties

Uncertainties were evaluated by varying event selection cuts, track selection criteria, particle identification parameters, and background estimation methods (polynomial order). The total systematic uncertainty was calculated by adding individual sources in quadrature.

3. Key Contributions and Results

A. Constituent Quark Scaling ($n_q$ scaling)

• Observation: The $v_2$ of strange and multi-strange hadrons, when scaled by the number of constituent quarks ($n_q$) and plotted against transverse kinetic energy per quark ($K_{ET}/n_q$), follows a universal curve within $\sim 20\%$ uncertainty.
• Significance: This scaling holds for both Ru+Ru and Zr+Zr collisions, similar to larger systems like Au+Au and U+U. It provides strong evidence for partonic collectivity and quark coalescence as the dominant particle production mechanism in the intermediate $p_T$ range, even in these smaller isobar systems.

B. Nuclear Structure and Deformation Effects

• $v_2$ Ratio: The ratio of average $v_2$ ($\langle v_2 \rangle$) between Ru+Ru and Zr+Zr was analyzed.
  • Central Collisions (0-10%): A deviation of $\sim 2\%$ from unity was observed for $K^0_S$ and $\Lambda$. This is attributed to the larger quadrupole deformation ($\beta_2$) of the Ru nucleus ($\beta_2 \approx 0.162$) compared to Zr ($\beta_2 \approx 0.06$).
  • Mid-Central Collisions (20-50%): A similar $\sim 2\%$ deviation persists, suggesting differences in surface diffuseness and nuclear density profiles between the two isobars.
• Statistical Significance: The deviations for $K^0_S$ and $\Lambda$ are statistically significant ($3.7\sigma$ and $9.3\sigma$ respectively), confirming that nuclear structure differences directly impact collective flow.
• Multi-strange Hadrons: For $\phi$ and $\Xi$, the ratio is consistent with unity within current uncertainties, likely due to larger statistical errors or different sensitivity to the initial geometry.

C. System Size Dependence

• Comparison: The $v_2(p_T)$ of strange hadrons in isobar collisions was compared with Cu+Cu, Au+Au, and U+U collisions.
• Hierarchy: A clear system-size hierarchy was observed: $v_2^{\text{Cu}} < v_2^{\text{Ru}} \approx v_2^{\text{Zr}} < v_2^{\text{Au}} < v_2^{\text{U}}$.
• Trend: $v_2$ increases with system size, particularly at intermediate $p_T$ ($> 1.5$ GeV/c), indicating that larger systems generate stronger collective flow due to higher initial eccentricity and participant nucleon density.

D. Model Comparison

• AMPT Model: The results were compared with the AMPT (A Multi-Phase Transport) model using a string melting (SM) mode with a partonic cross-section of 3 mb.
• Configuration: The model incorporated deformed Woods-Saxon distributions for Ru and Zr nuclei (using specific $\beta_2$ and $\beta_3$ parameters).
• Outcome: The AMPT-SM model qualitatively reproduces the experimental $v_2(p_T)$ data for all studied particles in both collision systems, validating the role of nuclear deformation in the initial state.

4. Significance

This paper provides the first differential measurement of elliptic flow for strange and multi-strange hadrons in isobar collisions. Its significance lies in three main areas:

1. Validation of QGP Signatures: It confirms that even in smaller, deformed nuclei (Ru and Zr), the medium exhibits QGP-like collective behavior (partonic collectivity) evidenced by constituent quark scaling.
2. Nuclear Structure Probe: It demonstrates that elliptic flow is a sensitive probe of nuclear structure. The $\sim 2\%$ difference in flow between isobars provides independent, high-precision constraints on the deformation ($\beta_2$) and surface diffuseness of $^{96}\text{Ru}$ and $^{96}\text{Zr}$, complementing other nuclear physics measurements.
3. System Size Evolution: It establishes a clear hierarchy of collectivity across different collision systems, bridging the gap between small systems (Cu) and large systems (Au, U), and confirming that system size is a critical driver of the initial geometric anisotropy and subsequent hydrodynamic evolution.

In conclusion, the study successfully decouples the effects of nuclear structure from collision dynamics, offering a refined understanding of how initial nuclear geometry translates into final-state momentum anisotropy in relativistic heavy-ion collisions.