Characterizing the initial state and dynamical evolution in XeXe and PbPb collisions using multiparticle cumulants

Using the CMS detector, this study presents the first measurements of correlations among mixed-order moments of two and three flow harmonics in XeXe and PbPb collisions, leveraging the distinct nuclear shapes of doubly-magic $^{208}$Pb and triaxially deformed $^{129}$Xe to probe initial-state geometry fluctuations and constrain the nonlinear hydrodynamic response of the quark-gluon plasma.

The Gist

The Big Picture: Smashing Nuclei to See the "Perfect Fluid"

Imagine you are a chef trying to figure out exactly how a new, super-hot soup behaves. You can't just look at it; you have to throw ingredients into it and watch how they swirl.

In this paper, scientists at CERN (the European Organization for Nuclear Research) did exactly that, but instead of soup, they created the Quark-Gluon Plasma (QGP). This is a state of matter that existed just microseconds after the Big Bang. It's so hot and dense that protons and neutrons melt down into a "perfect fluid" of their smallest parts (quarks and gluons).

To study this, they smashed two types of atomic nuclei together at nearly the speed of light:

1. Lead (Pb): Think of this as a perfectly round, smooth marble. It's a "doubly magic" nucleus, meaning it's very stable and spherical.
2. Xenon (Xe): Think of this as a slightly squashed, rugby-ball-shaped object. It's "triaxially deformed," meaning it's not a perfect sphere; it's a bit lumpy and stretched.

The Experiment: The "Shape-Shifting" Collision

The scientists wanted to see how the shape of the "ingredients" (the nuclei) changes the "soup" (the QGP) they create.

• The Lead Collision: When two perfect marbles smash head-on, the resulting fireball is very round.
• The Xenon Collision: When two rugby balls smash, the resulting fireball depends on how they hit. If they hit side-by-side, the fireball is very oval. If they hit end-to-end, it's rounder. Because the Xenon nuclei are lumpy, every single collision creates a slightly different shape.

The Measurement: Listening to the "Flow"

When these nuclei smash, the resulting plasma expands. Because the initial shape isn't perfectly round, the plasma expands faster in some directions than others. This is called anisotropic flow.

The scientists measured this flow using three "harmonics" (like musical notes):

• $v_2$ (Elliptic Flow): The oval shape (like a rugby ball).
• $v_3$ (Triangular Flow): A triangular shape (like a slice of pizza).
• $v_4$ (Quadrangular Flow): A four-pointed star shape.

The New Discovery: The "Group Hug" of Particles

In the past, scientists mostly looked at how particles moved in pairs. This paper is special because it looked at groups of 2, 4, 6, and even 8 particles moving together.

The Analogy: The Dance Floor

• Two-particle correlation: Imagine watching two people dance. You can see if they are holding hands.
• Multiparticle cumulants: Now imagine watching a whole group dance. Are they all moving in a synchronized wave? Or are they just bumping into each other randomly?

By looking at groups of 4, 6, or 8 particles, the scientists could filter out the "noise" (random bumps) and hear the true "music" of the collective fluid. They measured how these groups correlated with each other. For example, they asked: "If the oval shape ($v_2$) is strong, does the four-pointed star shape ($v_4$) automatically get stronger too?"

Key Findings: What the Shapes Told Them

1. The Rugby Ball Effect (Deformation)
The Xenon collisions (the rugby balls) showed much stronger "oval" flow ($v_2$) in central collisions than the Lead collisions. This confirmed that the initial shape of the nucleus matters. The lumpy Xenon nuclei create a more distorted starting point, which the fluid responds to strongly.

2. The Non-Linear Response (The Ripple Effect)
This is the most complex part. The fluid doesn't just copy the shape; it transforms it.

• The Analogy: Imagine throwing a stone into a pond. The ripples aren't just a copy of the stone; they interact with each other.
• The scientists found that the "star" shape ($v_4$) wasn't just caused by the initial star shape of the nucleus. A huge part of it was created because the "oval" flow ($v_2$) got so strong that it squashed the fluid into a star shape. This is called non-linear hydrodynamic response. The fluid is so sticky and interactive that one shape creates another.

3. The "Triangular" Surprise
The triangular flow ($v_3$) was actually stronger in the Xenon collisions than in Lead, even though Xenon is more "lumpy." This suggests that the internal "lumpiness" (fluctuations) of the Xenon nucleus creates more chaotic, triangular ripples in the fluid than the smooth Lead nucleus does.

Why Does This Matter?

Think of the QGP as a new kind of material with rules we don't fully understand yet.

• By comparing the Round Marble (Lead) vs. the Rugby Ball (Xenon), the scientists can tune their computer models.
• They found that to predict the Xenon results correctly, they had to assume the Xenon nucleus is indeed squashed (deformed) and that the fluid has a specific "stickiness" (viscosity).
• If their models didn't account for the non-linear "ripple effect" (where $v_2$ creates $v_4$), they would get the numbers wrong.

The Bottom Line

This paper is like a forensic investigation of the Big Bang. By smashing two different shaped "marbles" together and listening to how the resulting "soup" flows in complex group patterns, the scientists proved that:

1. The shape of the nucleus dictates the shape of the explosion.
2. The resulting fluid is so interactive that it creates new shapes from old ones (non-linear response).
3. We now have a much better "recipe" for how this perfect fluid behaves, helping us understand the fundamental laws of the universe.

In short: Shape matters, and the fluid is more creative than we thought.

Technical Summary

1. Problem and Motivation

The primary goal of high-energy heavy-ion physics is to understand the properties of the Quark-Gluon Plasma (QGP) and the initial conditions of nuclear collisions. While the collective behavior of the QGP (anisotropic flow, $v_n$) is well-studied in Lead-Lead (PbPb) collisions, the influence of nuclear deformation on these observables remains a critical area of investigation.

• The Specific Challenge: Most previous studies utilized spherical nuclei (like the doubly magic $^{208}\text{Pb}$). However, the Xenon nucleus ($^{129}\text{Xe}$) is triaxially deformed. Comparing collisions of these two distinct nuclear shapes allows physicists to disentangle the effects of initial-state geometry fluctuations from the dynamical evolution of the medium.
• The Gap: While single-order flow harmonics ($v_n$) and two-harmonic correlations have been measured, mixed-order cumulants (correlations between different harmonics and different orders, e.g., $v_n^k$ and $v_m^l$) had not been systematically measured in XeXe collisions. These higher-order observables are essential for probing nonlinear hydrodynamic responses and mode-coupling effects that linear models cannot capture.

2. Methodology

The analysis utilizes data collected by the CMS detector at the LHC:

• Data Samples:
  • XeXe collisions: $\sqrt{s_{NN}} = 5.44$ TeV (2017 data), integrated luminosity $3.42 \, \mu\text{b}^{-1}$.
  • PbPb collisions: $\sqrt{s_{NN}} = 5.36$ TeV (2023 data), integrated luminosity $1.92 \, \text{nb}^{-1}$.
• Event Selection: Events were selected based on energy deposits in the Hadron Forward (HF) calorimeters to ensure hadronic collisions. Primary vertices were reconstructed, and tracks were selected within $|\eta| < 2.4$ and $0.5 < p_T < 3.0$ GeV/c.
• Analysis Techniques:
  • Two-Particle Correlations: Used to extract $v_n\{2, |\Delta\eta| > 2\}$, suppressing non-flow effects (like jets) via large pseudorapidity gaps.
  • Multiparticle Cumulants (Q-cumulant method): Used to extract genuine multi-particle correlations.
    • Symmetric Cumulants (SC): Measure correlations between squared flow coefficients of different harmonics (e.g., $\langle v_n^2 v_m^2 \rangle - \langle v_n^2 \rangle \langle v_m^2 \rangle$).
    • Mixed-Harmonic Cumulants (MHC): Higher-order correlations involving three or more harmonics (e.g., 6-particle and 8-particle cumulants involving $v_2, v_3, v_4$).
  • Normalization: Observables were normalized (NSC, nMHC) to remove dependencies on the magnitude of flow coefficients, facilitating direct comparison between systems and models.
• Theoretical Models:
  • Initial State (IC): IP-GLASMA and TRENTO models.
  • Hydrodynamic Evolution: IP-GLASMA + MUSIC (viscous hydrodynamics) + URQMD (hadronic afterburner).
  • Nuclear Geometry: The Woods-Saxon distribution was used with varying deformation parameters ($\beta_2, \beta_4$) for Xe (four configurations) and Pb (one configuration).

3. Key Contributions

This paper represents the first measurement of mixed-order moments involving two or three flow harmonics in XeXe collisions. Key contributions include:

1. Systematic Comparison: A direct comparison of XeXe (deformed) and PbPb (spherical) collisions using identical analysis techniques to isolate the impact of nuclear shape.
2. Higher-Order Observables: The measurement of 6- and 8-particle mixed-harmonic cumulants (nMHC) and 3-harmonic symmetric cumulants, providing unprecedented sensitivity to nonlinear hydrodynamic mode-coupling.
3. Constraining Nuclear Deformation: Using flow observables to constrain the quadrupole ($\beta_2$) and skin depth ($a_0$) parameters of the $^{129}\text{Xe}$ nucleus.
4. Nonlinearity Quantification: Demonstrating how higher-order moments reveal the nonlinear response of the QGP to initial spatial anisotropies, particularly the coupling between $v_2$ and $v_4$.

4. Results

• Nuclear Deformation Sensitivity:
  • Elliptic Flow ($v_2$): In central collisions, $v_2$ is enhanced in XeXe compared to PbPb due to the prolate deformation of the Xe nucleus. The data is best described by the IP-GLASMA+MUSIC+URQMD model using a quadrupole deformation parameter $\beta_2 = 0.207$ and a nuclear skin depth $a_0 = 0.492$ fm.
  • Triangular Flow ($v_3$): XeXe shows larger $v_3$ values than PbPb in central collisions, indicating stronger initial-state density fluctuations in the deformed Xe system.
  • Quadrangular Flow ($v_4$): In peripheral collisions, $v_4$ is suppressed in XeXe relative to PbPb because it is dominated by the nonlinear coupling to $v_2$ (which is smaller in XeXe in this centrality range).
• Fluctuations and Ratios:
  • The ratio $v_n\{4\}/v_n\{2\}$ is smaller in XeXe than in PbPb, indicating larger event-by-event fluctuations in the deformed system.
  • The ratio $v_3\{4\}/v_3\{2\}$ follows the predicted $A^{-1/4}$ scaling (where $A$ is the atomic mass number), confirming that non-Gaussian fluctuations scale with the number of participants.
• Symmetric Cumulants (SC):
  • NSC(2,3): Shows negative correlation (anticorrelation between $\epsilon_2$ and $\epsilon_3$), consistent across systems.
  • NSC(2,4): Shows positive correlation due to nonlinear mode coupling ($v_4 \propto \epsilon_2^2$). Initial-state models (TRENTO, IP-GLASMA-IC) fail to reproduce the magnitude of NSC(2,4) in peripheral collisions, highlighting the necessity of hydrodynamic evolution.
• Mixed-Harmonic Cumulants (nMHC):
  • Sign Patterns: The paper establishes distinct sign patterns for correlations between moments of $v_2$ and $v_3$ (negative-positive-negative) versus $v_2$ and $v_4$.
  • Nonlinear Dominance: Initial-state models predict correlations between $v_2$ and $v_4$ with the wrong sign compared to data. This proves that $v_2$-$v_4$ correlations are predominantly generated dynamically during the hydrodynamic expansion, not just by initial geometry.
  • System Size: Higher-order moments of fluctuation-driven harmonics ($v_3, v_4$) are larger in the smaller XeXe system, reflecting enhanced sensitivity to fluctuations.

5. Significance

• Validation of Hydrodynamics: The results provide strong evidence that the QGP behaves as a fluid with significant nonlinear response. The failure of initial-state-only models to describe higher-order mixed cumulants confirms that hydrodynamic evolution is essential for generating observed flow correlations.
• Precision Nuclear Structure: The study demonstrates that heavy-ion collisions at the LHC can serve as a precision tool to measure nuclear deformation parameters ($\beta_2, a_0$) that are difficult to constrain via low-energy nuclear physics alone.
• Transport Properties: By constraining the initial state geometry, these measurements reduce uncertainties in extracting QGP transport coefficients, such as the specific shear viscosity to entropy density ratio ($\eta/s$).
• New Observables: The successful measurement of 6- and 8-particle mixed cumulants opens a new frontier for probing the complex, nonlinear interplay between different flow harmonics, offering a more complete picture of the QGP's evolution.

In conclusion, this work establishes that the triaxial deformation of the $^{129}\text{Xe}$ nucleus significantly alters the initial geometry and subsequent flow fluctuations compared to spherical $^{208}\text{Pb}$. Furthermore, it proves that nonlinear hydrodynamic response is the dominant mechanism driving correlations between even-order flow harmonics ($v_2$ and $v_4$), a feature that cannot be captured by static initial-state models.