Statistical properties of non-flow correlations in pp and heavy-ion collisions at RHIC energies

This study analyzes the statistical properties of two-particle cumulants in pp, d-Au, and Au-Au collisions at RHIC energies, revealing that non-flow correlations consistently produce skewed distributions with increasing skewness and kurtosis in non-QGP models, whereas the HYDJET++ model yields a Gaussian distribution that diminishes to zero at larger pseudorapidity separations.

The Gist

Imagine you are at a massive, chaotic concert. The crowd is so dense that people are bumping into each other, pushing, and moving in waves. Physicists call this "flow." They want to study how the crowd moves together as a single unit (like a wave) because that tells them if a special, super-hot state of matter called a Quark-Gluon Plasma (QGP) was created.

However, there's a problem. Not everyone in the crowd is moving in a wave. Some people are just walking in pairs because they know each other, or they are running away from a specific spot where a fight broke out. In physics terms, these are "non-flow" correlations (like jets or particle decays). They create noise that looks like a wave but isn't.

This paper is like a detective story trying to figure out how to tell the difference between the real wave (the QGP) and the fake noise (the non-flow) in different types of collisions, from small ones (like two people bumping into each other) to huge ones (like two crowds crashing together).

Here is the breakdown using simple analogies:

1. The Tool: The "Two-Person Dance"

To measure the crowd's movement, the scientists look at pairs of people (particles) and see how they are dancing relative to each other. They calculate a number called the cumulant ($c_2$).

• If the whole crowd is moving in a perfect wave, this number behaves one way.
• If it's just random chaos or small groups of friends, it behaves another way.

2. The Suspects: The Models

The researchers didn't just look at real data (which is hard to get in this specific format); they used computer simulations (models) to act as "suspects."

• The "Chaos" Models (PYTHIA, PHOJET, etc.): These simulate collisions where there is no big wave (no QGP). They are full of jets, decays, and random interactions.
• The "Wave" Model (HYDJET++): This simulates a collision where a big, smooth wave (hydrodynamic flow) does exist, like in a heavy-ion collision.

3. The Clue: The Shape of the Distribution

The scientists looked at the shape of the data graph for these "dance numbers." They used two statistical tools to describe the shape:

• Skewness (The "Tail"): Is the graph lopsided? Does it have a long tail stretching to one side?
• Kurtosis (The "Peakedness"): Is the graph pointy and stiff, or smooth and round?

The Discovery: The "Lopsided Tail"

When they looked at the Chaos Models (PYTHIA, etc.), the graph was skewed. It had a long, fat tail on the positive side.

• The Analogy: Imagine a crowd where a few people are running super fast because they saw a celebrity (a "jet"). These fast runners pull the average speed up, creating a "tail" on the graph. No matter how you slice the data, if jets are involved, you get this lopsided tail.

When they looked at the Wave Model (HYDJET++), the graph was smooth and round (Gaussian), like a perfect bell curve.

• The Analogy: In a true wave, everyone moves together smoothly. There are no sudden, fast runners pulling the average. The distribution is balanced and symmetrical.

4. The Magic Trick: The "Eta-Gap"

The scientists tried a trick to remove the noise: they put a "gap" between the two people they were watching. They only looked at pairs that were far apart in the crowd (in terms of "pseudorapidity," or $\eta$).

• In the Chaos Models: Even with the gap, the skewness (the lopsided tail) got worse as the gap got bigger. Why? Because the "jets" (the fast runners) are so powerful that they still influence the data even when you try to separate the pairs. The noise persists.
• In the Wave Model: As the gap got bigger, the skewness and kurtosis disappeared, and the graph became perfectly smooth (zero). The "wave" is a global property, so it looks the same even when you look at distant parts of the crowd.

5. The Conclusion: A New Detective Tool

The paper concludes that we can use the shape of the graph (specifically the skewness and kurtosis) to spot the "fake noise."

• If you see a lopsided tail (high skewness): It's likely just noise (jets/decays), not a Quark-Gluon Plasma.
• If you see a smooth, round hill (zero skewness/kurtosis): It's likely a real collective flow (QGP).

Why does this matter?
In small collisions (like proton-proton or proton-lead), it's very hard to prove a QGP exists because the "noise" usually drowns out the "wave." This paper suggests a new way to check: Look at the shape of the distribution. If the graph is lopsided, you probably don't have a QGP. If it's a perfect bell curve, you might!

Summary in One Sentence

The paper shows that if you look at the "shape" of particle movement data, noise creates a lopsided, jagged tail, while real collective flow creates a smooth, perfect hill, allowing scientists to filter out the noise and find the true signal of the Quark-Gluon Plasma.

Technical Summary

1. Problem Statement

In relativistic heavy-ion collisions, the formation of a Quark-Gluon Plasma (QGP) is often inferred through the measurement of elliptic flow ($v_2$), which arises from the conversion of spatial anisotropy into momentum anisotropy. The standard method to calculate $v_2$ involves the two-particle cumulant ($c_2$).

However, a significant challenge exists in distinguishing true hydrodynamic flow from non-flow correlations. Non-flow effects arise from processes unrelated to the collective expansion of the medium, such as:

• Jet correlations (back-to-back jets).
• Resonance decays.
• Mini-jets and Multi-Parton Interactions (MPIs).
• String fragmentation.

In high-multiplicity collisions (e.g., central Au-Au or Pb-Pb), these non-flow contributions average out. However, in low-multiplicity collisions (e.g., pp, d-Au, p-Pb, or peripheral Au-Au at lower energies like BES), non-flow contributions are significant and can mimic elliptic flow, leading to false conclusions about QGP formation. Current methods to suppress non-flow (e.g., $\eta$-gaps, higher-order cumulants) are effective but lack a deep statistical characterization of the non-flow distribution itself. This paper aims to analyze the statistical properties (skewness and kurtosis) of the event-by-event two-particle cumulant distribution to provide a robust alternative method for identifying and cross-checking non-flow contamination.

2. Methodology

The authors utilized Monte Carlo event generators to simulate collisions at RHIC energies (200 GeV and 39 GeV) across different systems (pp, d-Au, Au-Au).

• Models Used:

  • Non-Flow Proxies: PYTHIA (specifically Angantyr for heavy ions) and EPOS-LHC were used to simulate non-flow dominated scenarios. Additional models like PHOJET, QGSJET, and DPMJET (often used in cosmic ray physics) were employed to test model dependence. These models lack full hydrodynamic evolution.
  • Flow Proxies: HYDJET++ was used to simulate hydrodynamic flow. It was run in two configurations:
    1. Default: Includes soft thermal processes (hydrodynamics) and hard processes (jets).
    2. Jets Only: Hydrodynamics turned off, simulating only non-flow.
  • Collision Systems: pp, d-Au, and Au-Au collisions.
  • Variables: The study analyzed the event-by-event distribution of the two-particle cumulant ($c_2$) as a function of:
    • Pseudorapidity window ($\Delta\eta$).
    • Impact parameter ($b$).
    • Collision centrality.

• Statistical Analysis:
  Instead of averaging $c_2$ values, the authors treated the distribution of $c_2$ on an event-by-event basis. They calculated:

  • Skewness ($sk$): Measures the asymmetry of the distribution.
  • Excess Kurtosis ($kr$): Measures the "tailedness" or peakedness of the distribution relative to a Gaussian.

3. Key Contributions

• Event-by-Event Distribution Analysis: The paper moves beyond the standard practice of averaging cumulants, proposing that the shape of the $c_2$ distribution itself contains critical information about the underlying physics (flow vs. non-flow).
• Identification of a "Signature Skewed Tail": The authors establish that non-flow correlations consistently produce a positively skewed distribution with a long tail, regardless of the specific model used (PYTHIA, PHOJET, etc.).
• Differentiation Mechanism: They demonstrate that the statistical behavior of skewness and kurtosis with respect to $\Delta\eta$ provides a clear diagnostic tool:
  • Non-flow: Skewness and kurtosis increase as the $\Delta\eta$ window widens.
  • Hydrodynamic Flow: Skewness and kurtosis decrease and approach zero as $\Delta\eta$ widens.
• Impact Parameter Dependence: The study clarifies that a Gaussian distribution of $c_2$ (characteristic of flow) is only observed in fixed impact parameter collisions where initial spatial eccentricity fluctuations are Gaussian. Minimum-bias collisions (superposition of many impact parameters) result in a plateau-like distribution even with hydrodynamics.

4. Key Results

A. Non-Flow Models (PYTHIA, PHOJET, QGSJET, DPMJET)

• Distribution Shape: All non-QGP models produced a positively skewed distribution with a distinct tail. This tail is caused by rare events with large $c_2$ values arising from jets, decays, and string fragmentation.
• $\Delta\eta$ Dependence:
  • Skewness: Increased consistently with larger $\Delta\eta$ windows. This is attributed to the accumulation of more non-flow correlation pairs.
  • Kurtosis: Also increased with $\Delta\eta$. The authors attribute this to the stiffening of the distribution caused by the averaging of random $\cos(2\Delta\phi)$ values, which reduces the variance and sharpens the peak.
• $\eta$-Gap Effect: Introducing a pseudorapidity gap ($\eta$-gap) significantly suppressed the non-flow correlations, removing the skewed tail and making the distribution more symmetric (though not perfectly Gaussian due to residual correlations).

B. Hydrodynamic Models (HYDJET++)

• Fixed Impact Parameter: When the impact parameter was fixed, HYDJET++ (with hydrodynamics enabled) produced a smooth Gaussian distribution. This is due to the Gaussian nature of event-by-event initial spatial eccentricity fluctuations ($\epsilon_0$), which transfer to the final state momentum anisotropy.
• $\Delta\eta$ Dependence:
  • Skewness: Decreased as $\Delta\eta$ increased, eventually reaching zero.
  • Kurtosis: Decreased as $\Delta\eta$ increased, approaching zero.
  • Interpretation: As the $\Delta\eta$ window widens, the hydrodynamic flow signal dominates, and the random non-flow contributions are averaged out, resulting in a distribution that converges to a normal distribution with negligible skewness and kurtosis.
• Jets Only (HYDJET++): When hydrodynamics were turned off in HYDJET++, the model reverted to producing a positively skewed distribution similar to PYTHIA, confirming that the Gaussian shape is a direct result of hydrodynamic evolution.

C. Minimum Bias vs. Fixed Impact Parameter

• In minimum-bias collisions (a mix of all impact parameters), even HYDJET++ failed to produce a pure Gaussian distribution, instead showing a plateau-like shape. This is because the superposition of Gaussians with different means (corresponding to different impact parameters) smears the distribution.
• This highlights that observing a Gaussian $c_2$ distribution requires selecting events with a specific impact parameter (or centrality) to isolate the eccentricity fluctuations.

5. Significance and Conclusion

The paper provides a novel statistical framework for identifying non-flow contamination in small collision systems (pp, d-Au) and peripheral heavy-ion collisions.

• Diagnostic Tool: The behavior of skewness and kurtosis as a function of $\Delta\eta$ serves as a powerful cross-check. If these higher-order moments increase with $\Delta\eta$, the signal is dominated by non-flow. If they decrease toward zero, the signal is likely dominated by hydrodynamic flow.
• Model Independence: The finding that non-flow models consistently produce skewed tails, while hydrodynamic models produce Gaussians (under fixed conditions), suggests this is a universal property rather than an artifact of a specific generator.
• Implications for QGP Search: This method offers an alternative to traditional $\eta$-gaps for validating QGP formation in low-multiplicity systems. It suggests that in systems where a Gaussian $c_2$ distribution is observed (with vanishing skewness/kurtosis at large $\Delta\eta$), the collective flow signal is robust against non-flow contamination.

In summary, the authors successfully characterized the statistical "fingerprint" of non-flow correlations, proposing that the positively skewed tail and increasing kurtosis with $\Delta\eta$ are definitive signatures of non-flow, whereas a Gaussian distribution with vanishing higher moments indicates genuine hydrodynamic flow.