Bayesian extraction of TMC-free collectivity in p+p and p+Pb collisions at the LHC

This paper develops a Bayesian inference framework to disentangle genuine collective flow from transverse momentum conservation (TMC) backgrounds in small collision systems, revealing that while $p$+Pb data aligns with standard flow measurements, $p$+$p$ collisions require this new method to accurately extract flow magnitudes that are otherwise underestimated due to competing TMC effects.

The Gist

Imagine you are at a crowded party. You want to know if the guests are dancing in a coordinated, synchronized way (like a flash mob), or if they are just bumping into each other randomly because the room is so packed.

In the world of particle physics, scientists smash tiny particles together (like protons) to see if they create a "perfect fluid" that flows together, similar to the Quark-Gluon Plasma (QGP) formed in massive collisions. But there's a problem: in these tiny collisions, it's hard to tell if the particles are actually "dancing together" (collective flow) or if they are just moving in a specific pattern because they have to balance each other out to conserve momentum (like a group of people on a small boat trying not to tip over). This balancing act is called Transverse Momentum Conservation (TMC).

This paper is like a new, high-tech detective tool that helps scientists separate the "real dancing" from the "balancing act."

The Problem: The "Noise" in the Signal

For years, scientists have seen signs of this coordinated flow in tiny collisions (proton-proton and proton-lead). However, a lot of the "signal" they see might just be the "noise" of TMC.

• The Analogy: Imagine trying to hear a specific song at a loud concert. The music is the "collective flow," but the crowd cheering and clapping in rhythm just because they are in a small room is the "TMC noise." If you don't filter out the crowd noise, you might think the band is playing a different song than they actually are.

The Solution: A Bayesian "Smart Filter"

The authors developed a Bayesian inference framework. In simple terms, this is a mathematical method that acts like a super-smart filter.

• How it works: Instead of guessing the answer, the scientists start with a theory (the TMC math) and then let the actual experimental data (from the ATLAS detector at the LHC) "teach" the model what the real numbers should be.
• The Metaphor: Think of it like tuning a radio. You have a static-filled station (the raw data with noise). You turn the dial (adjusting the parameters) until the music (the genuine flow) comes through clearly. The Bayesian method doesn't just find one setting; it calculates the probability of every possible setting to find the most accurate one, while also telling you how confident it is in that answer.

The Investigation: Two Different Parties

The scientists looked at two types of collisions:

1. p+p (Proton + Proton): A very small, tight room.
2. p+Pb (Proton + Lead): A slightly larger room.

They used four specific "clues" (mathematical measurements called cumulants) to figure out the true dance moves.

What they found:

• The Dance Moves are Similar: Once they filtered out the "noise" (TMC), they found that the actual "dance moves" (the collective flow, specifically the elliptic and triangular shapes) were surprisingly similar in both the tiny room and the slightly larger room. This suggests that even in these tiny collisions, the particles are behaving like a fluid, just like in the massive explosions.
• The Background Noise is Different: However, the "crowd noise" (TMC) was very different in the two rooms.
  • In the p+Pb collision, the standard measurements were actually quite good at ignoring the noise. The "real dance" was clearly visible.
  • In the p+p collision, the noise was overwhelming. The standard measurements were actually hiding the true dance. The scientists had to use their new filter to realize that the particles were dancing much more vigorously than the raw data suggested.

The Big Takeaway

Before this paper, if you looked at the raw data from the tiny proton collisions, you might have thought, "Oh, the flow is weak here, and strong there."

This paper says: "Wait a minute! The flow is actually about the same in both places. The difference is just how much 'background noise' (TMC) is messing up the view."

Why This Matters

This is a breakthrough because it gives physicists a reliable way to strip away the confusion. It proves that the "perfect fluid" behavior isn't just a fluke of big explosions; it happens in the smallest collisions too. But to see it, you need the right tools to separate the genuine collective motion from the simple physics of momentum conservation.

In a nutshell: The authors built a mathematical "noise-canceling headphone" that lets us hear the true "music" of particle collisions, revealing that even the smallest collisions create a synchronized, fluid-like dance.

Technical Summary

1. Problem Statement

In high-energy nuclear physics, the discovery of collective flow-like signatures (such as the "ridge" structure) in small collision systems (p+p and p+Pb) has sparked debate regarding the formation of Quark-Gluon Plasma (QGP) in these environments. A major challenge in interpreting these signals is distinguishing genuine collective flow (hydrodynamic expansion or parton escape) from non-flow correlations.

A dominant source of non-flow correlations in small systems is Transverse Momentum Conservation (TMC). TMC induces azimuthal correlations that mimic collective flow, particularly in multi-particle cumulants. Previous theoretical attempts to disentangle these effects relied on manually tuning parameters (e.g., fixing flow coefficients $v_2, v_3$ and transverse momentum scales) to reproduce specific observables. This approach lacked predictive power, failed to provide rigorous uncertainty quantification, and often required different parameter sets for different measurements, preventing a systematic extraction of "pure" collective flow.

2. Methodology

The authors developed a Bayesian inference framework integrated with a theoretical TMC formalism to extract genuine collective flow parameters directly from experimental data.

• Theoretical Framework:

  • The study utilizes a TMC-based probability distribution for $k$-particle transverse momenta in an $N$-particle system, incorporating collective flow harmonics ($v_2, v_3$).
  • They derive analytical expressions for four key observables:
    • Four-particle elliptic cumulant: $c_2\{4\}$
    • Two- and four-particle triangular cumulants: $c_3\{2\}$ and $c_3\{4\}$
    • Four-particle symmetric cumulant: $sc_{2,3}\{4\}$
  • These cumulants are decomposed into three physical components: Pure Flow, Pure TMC, and TMC-Flow Interplay.

• Bayesian Inference Strategy:

  • Parameters: The model treats the genuine elliptic flow ($v_2$), triangular flow ($v_3$), and the transverse momentum ratio ($r = p^2 / \langle p^2 \rangle_F$) as free parameters to be inferred.
  • Likelihood Function: Constructed using experimental data from the ATLAS collaboration ($c_2\{4\}, c_3\{2\}, c_3\{4\}, sc_{2,3}\{4\}$) with a Gaussian likelihood based on experimental covariance matrices.
  • Priors: Uniform priors are adopted over physically reasonable ranges.
  • Sampling: The posterior distributions are sampled using Markov Chain Monte Carlo (MCMC) methods (specifically the No-U-Turn Sampler, NUTS, via the PyMC framework).
  • Multiplicity Dependence: The analysis proceeds in two stages:
    1. Constant Parameter Model: Assumes $v_2, v_3, r$ are constant across charged-particle multiplicity ($N_{ch}$).
    2. Power-Law Model: Introduces a functional dependence $v_n = a \cdot N_{ch}^b$ to capture the evolution of collectivity with system size.

3. Key Contributions

• Data-Driven Extraction: This is the first application of a rigorous Bayesian framework to simultaneously constrain TMC and flow parameters in small systems, moving beyond manual tuning to a statistically robust inverse problem solution.
• Disentanglement of Effects: The method successfully separates the "genuine" collective flow from TMC-induced backgrounds, providing a quantitative assessment of the non-flow contamination in standard flow observables.
• Systematic Uncertainty Quantification: Unlike previous studies, this approach provides full posterior distributions and credible intervals for the extracted flow coefficients, accounting for correlations between parameters.
• Unified Description: The framework unifies the description of multiple observables ($c_2\{4\}, c_3\{2\}, c_3\{4\}, sc_{2,3}\{4\}$) within a single theoretical model for both p+p and p+Pb systems.

4. Key Results

A. Model Performance

• The Power-Law Model (where parameters depend on $N_{ch}$) significantly outperforms the constant-parameter model, yielding much lower $\chi^2$ values and accurately reproducing the multiplicity dependence of the experimental cumulants.
• The constant-parameter model failed to capture the upward trend of $c_3\{2\}$ with increasing $N_{ch}$, highlighting the necessity of multiplicity-dependent flow parameters.

B. Extracted Flow Coefficients ($v_2$ and $v_3$)

• Similarity in Magnitude: The extracted genuine $v_2$ and $v_3$ values are remarkably similar between p+p and p+Pb collisions, suggesting a common underlying mechanism (likely initial-state energy density fluctuations) for collective flow generation in small systems.
• Discrepancy with Raw Measurements:
  • In p+Pb, the extracted genuine $v_2$ aligns well with the experimental $v_2\{4\}$ (derived from $c_2\{4\}$), indicating that the four-particle cumulant effectively suppresses TMC backgrounds in this system.
  • In p+p, the extracted genuine $v_2$ and $v_3$ are systematically higher than the experimental measurements ($v_2\{4\}$ and $v_3\{2\}$). This implies that standard experimental measurements in p+p significantly underestimate the true flow due to substantial TMC contamination that is not fully suppressed by standard cumulant methods.

C. Physical Decomposition and Differences

• TMC Backgrounds: The TMC contribution is significantly stronger in p+p than in p+Pb across the entire multiplicity range.
• Interplay Effects: The interplay between TMC and flow differs between the two systems. In p+Pb, the interplay term drives the negative values of $c_3\{2\}$ at low multiplicities. In p+p, the interplay term is significant across all multiplicities, distorting the experimental signal.
• Correlations: The correlation between $v_2$ and $v_3$ (probed by $sc_{2,3}\{4\}$) is consistent with zero in p+p but significantly negative in p+Pb.
• Transverse Momentum Ratio ($r$): The inferred ratio $r = p^2 / \langle p^2 \rangle_F$ shows distinct behaviors: it saturates in p+Pb at high $N_{ch}$ but continues to increase in p+p, consistent with model calculations (PYTHIA for p+p, AMPT for p+Pb) and experimental mean transverse momentum trends.

5. Significance

This work provides a robust, data-driven methodology to isolate genuine collective flow from non-flow backgrounds in small collision systems. The findings suggest that:

1. Collective Flow Exists in p+p: Genuine collective flow is present in p+p collisions and is comparable in magnitude to that in p+Pb, supporting the hypothesis of QGP-like behavior or collective mechanisms in the smallest systems.
2. Limitations of Standard Observables: Standard experimental measurements (like $v_2\{4\}$ and $v_3\{2\}$) in p+p collisions are heavily contaminated by TMC, leading to a systematic underestimation of the true flow magnitude.
3. Theoretical Insight: The distinct TMC backgrounds and flow-TMC interplay between p+p and p+Pb reveal that while the origin of collectivity may be similar, the manifestation and background contamination are system-dependent.

The Bayesian framework established here offers a new standard for analyzing small-system collisions, enabling more reliable physical interpretations of azimuthal correlations and guiding future theoretical models of collectivity in high-energy physics.