Measurement of correlations between elliptic flow and mean transverse momentum in pp, p-Pb, and Pb-Pb collisions at the LHC

Using the full LHC Run 2 dataset, this ALICE study reports the first measurements of event-by-event correlations between elliptic flow and mean transverse momentum in pp, p-Pb, and Pb-Pb collisions, revealing non-monotonic trends and system-consistent values at low multiplicity that challenge current theoretical models and provide new constraints on the origin of collectivity in small collision systems.

The Gist

The Big Picture: The "Tiny Fireball" Mystery

Imagine smashing two tiny marbles (protons) together at nearly the speed of light. In the old days, physicists thought these collisions were just messy, random splatters of particles—like two cars crashing and scattering debris in all directions with no pattern.

However, in recent years, scientists discovered something weird: even in these tiny crashes, the debris seems to move in a coordinated, fluid-like way. It's as if the marbles didn't just break; they briefly turned into a super-hot, super-dense drop of liquid (called a Quark-Gluon Plasma or QGP) that flowed together before cooling down.

The big question is: Is this "flow" real, or is it just a coincidence?

To solve this, the ALICE team at CERN looked at a specific "fingerprint" of this fluid behavior: the relationship between how fast the particles are spinning (Elliptic Flow) and how fast they are flying outward (Mean Transverse Momentum).

The Analogy: The Dance Floor and the DJ

Think of a crowded dance floor at a party.

1. The Shape (Geometry): If the dance floor is shaped like an oval (not a perfect circle), people naturally tend to move more easily along the long axis than the short one. This is like the Elliptic Flow ($v_2$). It's determined by the shape of the room when the party starts.
2. The Energy (Momentum): If the DJ turns the music up and the lights flash, everyone gets more excited and moves faster. This is like the Mean Transverse Momentum ($[p_T]$). It's determined by the energy or size of the party.

The Mystery:
In a normal, chaotic crowd, the shape of the room and the energy of the party shouldn't be related. A square room doesn't make people run faster just because it's square.

But, if the crowd is a fluid (like a liquid), the shape of the room does affect how the fluid moves. A specific shape might channel the energy in a specific way.

The New Discovery: The "Correlation Coefficient" ($\rho$)

The scientists invented a special math tool (called the modified Pearson correlation coefficient, $\rho$) to measure the link between the Shape and the Speed.

• If they are unrelated: The number is zero (random chaos).
• If they are linked: The number goes up or down, telling us how the shape influences the speed.

They tested this in three different scenarios:

1. Pb-Pb (Lead-Lead): Two heavy nuclei smashing together (a huge, crowded party).
2. p-Pb (Proton-Lead): A small marble hitting a big nucleus (a small party in a big hall).
3. pp (Proton-Proton): Two tiny marbles smashing (a tiny, intimate gathering).

What They Found

1. The "Small Systems" Surprise: They found that even in the tiniest collisions (pp and p-Pb), there is a strong link between the shape and the speed. This suggests that even in these tiny systems, a tiny drop of "liquid" is forming.
2. The "U-Turn" in Heavy Collisions: In the big Lead-Lead collisions, the relationship changed as the crowd got bigger. It went down, then went back up. It's like a dance floor where the rules of movement change depending on how packed it is.
3. The "Small Systems" Consistency: In the tiny collisions (pp and p-Pb), as the crowd got smaller, the link between shape and speed got stronger.

The "Detective Work": Testing the Theories

The scientists then asked: "Which theory explains this?" They ran the data against three different computer models:

• Model A (PYTHIA): This model assumes no fluid. It thinks the particles are just independent billiard balls bouncing around.
  • Result: It failed completely. It couldn't explain why the shape and speed were linked. This proves the "fluid" theory is likely right.
• Model B (AMPT): This model simulates particles interacting like a gas or fluid.
  • Result: It did a decent job for the big collisions but got the tiny ones wrong. It predicted the link would go one way, but the data showed it going the other way.
• Model C (IP-Glasma + MUSIC + UrQMD): This is the "State-of-the-Art" model. It includes complex physics about how the particles start (initial conditions) and how they flow.
  • Result: It got the big collisions mostly right but struggled with the tiny ones. It predicted the link would flip signs (go from positive to negative) in small systems, but the data showed it staying positive.

The Conclusion: We Need New Rules

The paper concludes that current theories are missing something.

The fact that the "tiny" collisions behave so consistently with the "huge" ones suggests that the initial conditions (how the particles are arranged before they even hit) are crucial.

The Metaphor:
Imagine trying to predict how water splashes out of a cup.

• Old Theories assumed the water was just a bunch of independent raindrops.
• New Theories assumed it was a fluid.
• The Reality: It's a fluid, but the shape of the cup (the initial geometry) and the way the water was poured (initial momentum correlations) are interacting in a way our current "fluid" models haven't figured out yet.

Why This Matters

This isn't just about smashing rocks. It's about understanding the very first split-second of the universe.

• If we can understand how a tiny drop of "liquid" forms in a proton-proton collision, we learn how the universe behaved fractions of a second after the Big Bang.
• The fact that our best computer models can't explain these new measurements means we need to rewrite the rulebook on how matter behaves at the smallest scales.

In short: The universe is stranger than we thought. Even the tiniest collisions create a coordinated fluid dance, and our current understanding of physics isn't quite ready to explain the choreography.

Technical Summary

1. Problem Statement and Motivation

The existence of collective behavior (hydrodynamic flow) in small collision systems (proton-proton and proton-nucleus) remains a central question in high-energy nuclear physics. While signatures like the "ridge" and azimuthal anisotropies ($v_n$) have been observed in small systems, their origin is debated. Two primary mechanisms are proposed:

1. Hydrodynamic Response: Initial spatial geometry fluctuations (eccentricity, $\varepsilon_2$) drive a collective expansion, converting spatial anisotropy into momentum anisotropy ($v_2$).
2. Initial Momentum Anisotropy (CGC): The Color Glass Condensate (CGC) framework predicts intrinsic momentum correlations in the initial state, independent of final-state interactions.

Standard observables like $v_2$ are sensitive to both initial geometry and final-state evolution, making it difficult to disentangle the contributions of the initial state (specifically the CGC effect) from the hydrodynamic response. The paper addresses the need for an observable that is uniquely sensitive to initial-state fluctuations (specifically the correlation between the initial shape/size and momentum anisotropy) while being less affected by complex dynamic evolution.

2. Methodology

The analysis utilizes the full LHC Run 2 dataset recorded by the ALICE detector:

• Collision Systems:

  • Proton-Proton (pp) at $\sqrt{s} = 13$ TeV.
  • Proton-Lead (p–Pb) at $\sqrt{s_{NN}} = 5.02$ TeV.
  • Lead-Lead (Pb–Pb) at $\sqrt{s_{NN}} = 5.02$ TeV.

• Observable: The modified Pearson correlation coefficient, $\rho(v_2^2, [p_T])$, defined as:
  $$\rho(v_2^2, [p_T]) = \frac{\text{cov}(v_2^2, [p_T])}{\sqrt{\text{var}(v_2^2)}\sqrt{\text{var}([p_T])}}$$
  Where:

  • $v_2^2$ is the event-wise squared elliptic flow coefficient, extracted from two-particle correlations.
  • $[p_T]$ is the event-by-event mean transverse momentum.
  • The covariance measures the correlation between the magnitude of flow and the mean transverse momentum on an event-by-event basis.

• Experimental Setup:

  • Event Selection: Minimum Bias (MB) triggers for p–Pb and Pb–Pb; High-Multiplicity (HM) triggers for pp to select the top 0.1% of events.
  • Track Selection: Primary tracks reconstructed using the Inner Tracking System (ITS) and Time Projection Chamber (TPC) with strict quality cuts (TPC clusters, $\chi^2$, DCA to primary vertex).
  • Subevent Method: To suppress non-flow effects (short-range correlations), the measurement uses a subevent configuration:
    • $[p_T]$ measured in subevent A ($|\eta| < 0.4$).
    • $v_2$ extracted from correlations between subevents B ($-0.8 < \eta < -0.4$) and C ($0.4 < \eta < 0.8$) with a pseudorapidity gap $|\Delta\eta| > 0.8$.
  • Kinematic Range: $0.2 < p_T < 3.0$ GeV/c. The lower $p_T$ cut is chosen to minimize hard jet contamination and maximize sensitivity to CGC effects, which diminish at large pseudorapidity separations.

3. Key Contributions

• First Measurement in Small Systems: This is the first measurement of $\rho(v_2^2, [p_T])$ in pp and p–Pb collisions using the ALICE detector.
• Systematic Comparison: The study provides a unified comparison of this observable across three distinct collision systems (pp, p–Pb, Pb–Pb) over a wide multiplicity range.
• Theoretical Discrimination: The paper rigorously tests state-of-the-art models (IP-Glasma + MUSIC + UrQMD, AMPT, and PYTHIA) to determine if they can reproduce the observed correlations, specifically probing the role of the Color Glass Condensate (CGC) and initial momentum anisotropy (IMA).
• Non-Monotonic Behavior: It identifies a non-monotonic dependence of the correlation on charged-particle multiplicity ($N_{ch}$) in Pb–Pb collisions, a feature not previously resolved with this precision in small systems.

4. Results

A. Experimental Observations

• Pb–Pb Collisions: The $\rho(v_2^2, [p_T])$ exhibits a non-monotonic dependence on multiplicity. It decreases as multiplicity increases from low values, reaches a minimum around $N_{ch} \approx 80$, and then increases again at higher multiplicities.
• pp and p–Pb Collisions: Both systems show a clear increasing trend of $\rho(v_2^2, [p_T])$ as multiplicity decreases (i.e., the correlation becomes stronger in lower multiplicity events).
• Consistency at Low Multiplicity: For $N_{ch} \lesssim 80$, the measurements in all three systems (pp, p–Pb, Pb–Pb) are consistent within uncertainties. This suggests that in this regime, the system size and shape are dominated by initial energy density fluctuations, making the shape-size correlations similar regardless of the colliding system.
• Sign of Correlation: The measured values are predominantly positive across the multiplicity range in all systems.

B. Comparison with Theoretical Models

• PYTHIA (Non-Collective): PYTHIA 8 and Angantyr (which do not generate collectivity) fail to reproduce the data. They predict weak or positive correlations that do not match the magnitude or the strong multiplicity dependence observed, confirming that the observed effects have a collective origin.
• AMPT (Transport Model):
  • Describes the Pb–Pb and p–Pb data qualitatively in the intermediate-to-high multiplicity range.
  • Fails in pp collisions: AMPT predicts a negative correlation in pp collisions, which contradicts the positive values measured by ALICE.
• IP-Glasma + MUSIC + UrQMD (Hydrodynamic + CGC):
  • High Multiplicity ($N_{ch} > 150$): Qualitatively describes the Pb–Pb data.
  • Low Multiplicity ($N_{ch} < 150$): Significant deviations are observed.
    • The default model (with IMA) predicts a sign change to negative values at low multiplicity in p–Pb and Pb–Pb, which is not observed in the data (data remains positive).
    • Without IMA (CGC effects), the model predicts a monotonic decrease or negative values, deviating further from the data.
  • Hydrodynamic Start Time ($\tau_0$): Reducing the hydrodynamic start time from 0.4 fm/c to 0.1 fm/c (enhancing early-stage interactions) improves the description of the non-monotonic trend in Pb–Pb but still fails to fully reproduce the positive sign in small systems.

5. Significance and Conclusion

• Challenge to Current Models: The new measurements pose a significant challenge to state-of-the-art theoretical models. Neither pure transport models (AMPT) nor hybrid hydrodynamic models (IP-Glasma + MUSIC + UrQMD) can simultaneously describe the magnitude, sign, and multiplicity dependence of $\rho(v_2^2, [p_T])$ across all three collision systems.
• Insight into Initial State: The persistence of positive correlations in small systems (pp, p–Pb) at low multiplicity, contrary to model predictions of negative correlations, suggests that current implementations of initial conditions (specifically the interplay between geometric eccentricity and initial momentum anisotropy from CGC) are incomplete.
• Constraints on CGC: The results provide strong constraints on the Color Glass Condensate framework. The data implies that the initial momentum anisotropy predicted by CGC must be tuned differently or that the conversion of initial fluctuations to final-state flow in small systems is more complex than currently modeled.
• Future Directions: The findings necessitate refined treatments of initial-state geometry, size correlations, and the hydrodynamic start time ($\tau_0$) in theoretical models to explain the collective phenomena observed in small systems at the LHC.

In summary, this paper establishes $\rho(v_2^2, [p_T])$ as a powerful probe of initial-state fluctuations in small systems, revealing discrepancies between experimental data and current theoretical paradigms that drive the need for improved models of the early stages of high-energy collisions.