Nonflow Subtraction Beyond Two-Particle Correlations

This paper presents a general framework for subtracting nonflow effects from multi-particle cumulants in small collision systems by leveraging $1/N^{m-1}$ scaling and dipolar flow estimators, thereby enabling the systematic quantification of collective flow at particle multiplicities previously inaccessible due to uncontrolled nonflow residuals.

The Gist

Imagine you are trying to listen to a beautiful, complex symphony (the "collective flow" of a quark-gluon plasma) playing in a crowded concert hall. However, the audience is making a lot of noise: people coughing, chairs scraping, and friends whispering to each other. This background noise is what physicists call "nonflow."

For a long time, scientists have been very good at silencing this noise when listening to just two instruments playing together (two-particle correlations). They figured out that the noise gets quieter as the crowd gets bigger, following a predictable rule: if you double the crowd size, the noise from any single pair of friends drops by half.

But here is the problem: The symphony's true beauty isn't just in pairs; it's in how groups of three, four, or more instruments play together (multi-particle correlations). When scientists tried to listen to these larger groups, they found that the old noise-canceling tricks weren't working perfectly. The "whispers" (nonflow) were still leaking through, and they didn't know exactly how much.

This paper is like a new, advanced noise-canceling headset designed specifically for listening to groups of instruments, not just pairs.

The Core Idea: The "Independent Source" Rule

The authors realized that the background noise in these particle collisions comes from many independent sources (like individual jets of particles or decaying atoms). They found a simple rule for how this noise behaves:

• For a pair of particles, the noise drops by 1/N (where N is the number of particles).
• For a group of three particles, the noise drops by 1/N².
• For a group of four particles, the noise drops by 1/N³.

Think of it like a game of "telephone." If you have a group of 100 people, the chance that three specific people are all whispering the same secret by accident is much, much smaller than the chance that just two people are. The larger the group, the harder it is for random noise to mimic a coordinated signal.

The New Toolkit: Using "Dipole" Signals as a Ruler

To subtract the noise, the scientists needed a ruler to measure exactly how much noise was left. They discovered a clever trick: use a specific type of signal called $v_1$ (a dipole flow) as their ruler.

Why? Because in the real "symphony" (the actual flow of the plasma), this specific signal almost completely cancels itself out when you look at the whole picture. It's like a wave that goes up and down so perfectly that the net height is zero. However, the noise (nonflow) does show up clearly in this signal.

So, the team uses the "noise-only" signal ($v_1$) to measure how loud the background noise is, and then uses that measurement to subtract the noise from the complex group signals they actually care about.

The Hidden Trap: The "Crowd Weighting" Factor

The paper also uncovers a subtle mistake that scientists have been making for years.

Imagine you are trying to estimate the average noise level of a concert by looking at a photo of the audience.

• The Mistake: You just count the total number of people in the photo and divide by that number.
• The Reality: In a large crowd, a few very rowdy sections (high-multiplicity events) produce way more pairs of whispering friends than the quiet sections. If you just take a simple average, you miss the fact that the "rowdy" sections dominate the noise statistics.

The authors introduce a "Multiplicity-Reweighting" factor. It's like realizing that you can't just count heads; you have to weigh the noise based on how many possible pairs (or triplets) exist in each section of the crowd. If you ignore this weighting, your noise subtraction fails, especially for larger groups (like 4-particle correlations). The paper shows that without this correction, you might think you've removed the noise, but you've actually left almost all of it behind.

What They Tested

To prove their new headset works, they didn't use real data immediately (because real data is messy and we don't know the "true" answer yet). Instead, they used a computer simulation called HIJING.

• The Simulation: This computer program creates a "concert" that has only noise (jets and decays) and no symphony (no collective flow).
• The Test: They applied their new subtraction method. Since the simulation has no real flow, the result should be exactly zero.
• The Result: Their method worked very well. For most cases, they managed to remove 70–80% of the noise, leaving only a small, manageable amount of "residual" noise (about 20–30%). They also found that using the $v_1$ ruler was often better than the old simple counting methods.

The Takeaway

This paper provides a new, systematic way to clean up the "static" in high-energy physics experiments when looking at groups of particles.

1. It extends the successful noise-canceling techniques from pairs to larger groups.
2. It identifies a specific mathematical correction (the reweighting factor) that fixes a long-standing error in how scientists calculate noise.
3. It offers a way to quantify the remaining uncertainty, allowing scientists to be more confident when they claim to have found evidence of the "quark-gluon plasma" in tiny collision systems.

In short, they built a better filter to hear the music of the universe, even when the crowd is making a lot of noise.

Technical Summary

Technical Summary: Nonflow Subtraction Beyond Two-Particle Correlations

Problem Statement
The establishment of collective flow in small collision systems (pp, p+A, O+O) is critical for determining the minimum conditions for quark-gluon plasma (QGP) formation. While nonflow contributions (arising from jets, resonance decays, and HBT effects) in two-particle correlations (2PC) have been successfully controlled and subtracted using pseudorapidity gaps and scaling methods, a comparable framework for multi-particle cumulants (MPC) is lacking. Current MPC analyses rely on the subevent method to suppress nonflow without explicit subtraction. This approach has two primary limitations: (1) it prevents the quantification of residual nonflow systematic uncertainties, making it difficult to distinguish residual nonflow from flow decorrelation or to extend measurements to low multiplicities where nonflow dominates; and (2) it incurs a significant statistical penalty (e.g., a factor of ~11 loss for four-particle cumulants) by restricting particle combinations to separated subevents. Consequently, the multi-particle nature of collectivity remains unestablished in the low-multiplicity regime accessible to current 2PC flow measurements.

Methodology
The authors develop a general nonflow subtraction framework for $m$-particle cumulants, extending the philosophy of 2PC subtraction to higher orders. The core of the methodology rests on three pillars:

1. Scaling Principle: Based on the independent-source picture, nonflow contributions to an $m$-particle cumulant scale approximately as $1/N^{m-1}$, where $N$ is the event multiplicity. This generalizes the $1/N$ scaling known for 2PC.
2. Nonflow Estimators: The framework utilizes correlators containing the dipolar flow coefficient $v_1$ as clean nonflow estimators. Since the $p_T$-integrated genuine dipolar flow nearly vanishes due to sign changes around $\langle p_T \rangle$, and long-range nonflow projects onto a dipolar shape via global momentum conservation (GMC), these observables are dominated by nonflow and scale as $1/N^{m-1}$. Specific estimators include $\langle v_1^2 \rangle$, $\langle v_1^2 \delta p_T \rangle$, and $c_1\{4\}$.
3. General Subtraction Formula: A general subtraction formula is derived:
   $$O\{m\} = O\{m\}_{\text{obs}} - f_{E}^{O} \times O\{m\}_{\text{obs}}^{\text{pp}}$$
   where $f_{E}^{O}$ is a scaling factor derived from a reference estimator $E\{k\}$. Crucially, the authors introduce a multiplicity-reweighting factor ($W_{m,k}$) to account for the fact that target and reference observables are weighted by different numbers of multiplets ($N^m$ vs $N^k$) when using broad minimum-bias reference samples (like pp). This factor, previously overlooked in 2PC, scales as $(N_{m,\text{pp}}/N_{k,\text{pp}}) \cdot (N_k/N_m)$ and enters the subtraction with a power of $(m-1)/(k-1)$.

The study validates this framework using the HIJING event generator, which contains nonflow sources but no genuine collective flow. This allows for a controlled test of the scaling behavior and subtraction performance. The analysis focuses on three target observables relevant to small-system collectivity: $\langle v_2^2 \rangle$, $\langle v_2^2 \delta p_T \rangle$, and $c_2\{4\}$, across O+O collisions at $\sqrt{s_{NN}} = 5.36$ TeV and 200 GeV, and d+Au at 200 GeV.

Key Contributions and Results

• Validation of Scaling: The study confirms that nonflow in HIJING follows the $1/N^{m-1}$ scaling for both target and reference observables. However, deviations from perfect scaling (20–30%) are observed, varying by observable and collision system.
• Performance of Estimators: The optimal nonflow estimator depends on the target observable:
  • For $\langle v_2^2 \rangle$, the $c_1$-scaling (using $\langle v_1^2 \rangle$) performs best, leaving residual nonflow fractions within ~15%.
  • For $\langle v_2^2 \delta p_T \rangle$, the $c_0$-scaling (pure $1/N$) is superior, with residuals below 20%.
  • For $c_2\{4\}$, the $\langle v_1^2 \rangle$-scaling yields the smallest residuals (~10–20%).
  • In all cases, using a $v_1$-containing estimator generally reduces residuals compared to naive $1/N$ scaling, though the best choice is target-dependent.
• Critical Role of Multiplicity Reweighting: A major finding is that neglecting the multiplicity-reweighting factor $W_{m,k}$ leads to severe subtraction failures, particularly for higher-order cumulants. The error grows as a power of the correlator order. For example, in $c_2\{4\}$, omitting $W$ results in a residual fraction near unity (no subtraction), whereas including it drives the residual close to zero.
• Resolution of Long-standing 2PC Issues: The framework provides a direct explanation for the known "undersubtraction" of the naive $c_0$ method in 2PC. The error arises because the naive method uses the mean multiplicity $\langle N \rangle_{\text{pp}}$ instead of the effective pair multiplicity $N_{2,\text{pp}}$ (which is larger due to the broadness of the pp multiplicity distribution).
• Systematic Uncertainty Estimation: The authors propose using the "non-closure" (residual fraction) observed in HIJING as a multiplicative correction factor ($K_O^E$) to estimate the systematic uncertainty in real data. This mirrors the procedure used by the STAR collaboration for 2PC analyses.

Significance and Claims
The paper claims to provide a systematic route for nonflow subtraction beyond two-particle correlations, thereby broadening the class of multi-particle observables accessible to the small-system flow program. By enabling explicit subtraction, the framework allows for:

1. Quantification of residual nonflow systematic uncertainties.
2. Separation of nonflow from flow decorrelation.
3. Extension of flow measurements to lower multiplicities where nonflow previously dominated.
4. Recovery of statistical power by allowing the use of standard or two-subevent methods instead of the more restrictive four-subevent method.

The authors emphasize that while the framework is validated in HIJING, the absolute nonflow magnitude in HIJING is not a direct predictor for data (due to multiplicity overestimation). Instead, the scaling behavior and the derived correction factors serve as a robust, data-driven method to constrain systematic uncertainties. The work lays the groundwork for applying these techniques to existing RHIC and LHC datasets to more rigorously establish the multi-particle nature of collectivity in small systems.