Measurement of the elliptic flow of $^3$He and $^3_\Lambda$H in Pb-Pb collisions at $\sqrt{s_{\rm NN}} = 5.36$ TeV

Using a large dataset of approximately five billion Pb-Pb collisions at $\sqrt{s_{\rm NN}} = 5.36$ TeV collected by the ALICE detector during LHC Run 3, this paper presents the first measurement of the elliptic flow for (anti)hypertriton ($^3_\Lambda$H) and a study of that for helium-3 ($^3$He), providing critical constraints on hadronization models and insights into the production mechanisms of light (hyper)nuclei.

The Gist

Imagine a massive, high-speed collision between two lead nuclei (the cores of lead atoms) at the Large Hadron Collider. It's like smashing two complex, spinning clocks together at nearly the speed of light. When they hit, they don't just shatter; they melt into a super-hot, super-dense soup of fundamental particles called Quark-Gluon Plasma (QGP). This soup behaves like a perfect fluid, expanding and cooling down incredibly fast before freezing into the particles we can detect.

This paper is about studying how that "soup" flows and how tiny, rare particles form from it. Specifically, the ALICE collaboration looked at two very special, rare particles: Helium-3 (a light version of helium with two protons and one neutron) and the Hypertriton (a "strange" cousin of Helium-3 that contains a hyperon, a particle with a "strange" quark).

Here is the breakdown of what they found, using some everyday analogies:

1. The "Oval" Shape of the Explosion

When these lead nuclei collide, they don't always hit head-on. Often, they glance off each other, creating an overlap region that looks more like a football or an oval than a perfect circle.

• The Analogy: Imagine squeezing a water balloon that is already slightly oval-shaped. The pressure inside pushes out harder in the short direction of the oval than the long direction.
• The Result: The particles flying out of this "soup" aren't scattered randomly. They prefer to fly out along the short axis of the oval. This directional preference is called Elliptic Flow ($v_2$). It's the universe's way of saying, "We are expanding, and we are expanding unevenly."

2. The Mystery of the "Lego Bricks"

The big question in physics is: How do these rare particles (Helium-3 and Hypertriton) form?
There are two main theories:

• Theory A (The Hydrodynamic Soup): The particles are just carried along by the flow of the soup, like leaves floating down a river. If this is true, their flow depends mostly on their mass (how heavy they are). Since Helium-3 and Hypertriton have almost the same mass, they should flow the same way.
• Theory B (The Lego Coalescence): The particles form when smaller pieces (protons, neutrons, and hyperons) happen to bump into each other and stick together, like Lego bricks snapping together. This depends on how close the bricks are to each other in space and speed.

What the paper found:
The researchers measured the "flow" of these particles. They found that both Helium-3 and Hypertriton flowed in almost the exact same way, despite the Hypertriton being much "fluffier" and larger (like a loosely held bunch of balloons) compared to the tight Helium-3 (like a solid rock).

• The Takeaway: This suggests that the "Lego Coalescence" theory is correct. The particles form by snapping together at the very end of the explosion. The fact that they flow the same way tells us that the "soup" they are forming in is uniform enough that the size of the Lego bricks doesn't matter much; it's all about how the pieces are moving when they snap together.

3. The "Twist" in the Pattern (Higher Harmonics)

Here is the most surprising part. When they looked closely at the Helium-3 particles, they noticed something weird at high speeds.

• The Analogy: Imagine a spinning top. Usually, we describe its wobble with a simple back-and-forth motion (like a sine wave). But at very high speeds, the Helium-3 particles started wiggling in a more complex way, adding a "fourth-order" twist to their movement.
• Why it matters: Standard physics models usually stop looking after the second wiggle. But because Helium-3 is made of three particles sticking together, the math gets complicated. When three particles combine, their individual movements get "mixed up" in a non-linear way, creating these extra twists.
• The Discovery: This is the first time scientists saw this specific "extra twist" in a nucleus. It proves that the process of these particles snapping together (coalescence) is complex and introduces new patterns that simple models miss.

4. Why This Matters

Think of the early universe (a fraction of a second after the Big Bang) as this same hot soup.

• The "Recipe": By understanding how Helium-3 and Hypertriton form and flow, scientists are essentially reverse-engineering the recipe of the early universe.
• The "Source": The fact that these particles flow the way they do tells us about the "shape" and "texture" of the fireball created in the collision. It helps us understand how matter transitions from a hot soup into solid particles.

Summary

In simple terms, this paper is like watching a massive, high-speed car crash and analyzing the specific way two very rare, fragile toys (Helium-3 and Hypertriton) are formed from the debris.

1. They found that these toys form by snapping together from smaller pieces (Lego bricks) rather than just floating out of the soup.
2. They discovered that at high speeds, these toys spin in a more complex, "twisted" pattern than anyone expected, revealing the complex math behind how they stick together.
3. This helps physicists understand the fundamental rules of how matter is built in the most extreme conditions in the universe.

Technical Summary

1. Problem and Motivation

The paper addresses the production mechanisms and phase-space distributions of light (hyper)nuclei in ultrarelativistic heavy-ion collisions. Specifically, it investigates the elliptic flow ($v_2$) of two A=3 nuclei:

• Helium-3 ($^3$He): A nucleus composed of two protons and one neutron.
• Hypertriton ($^3_\Lambda$H): A hypernucleus composed of a proton, a neutron, and a $\Lambda$ hyperon.

Key Scientific Questions:

• Coalescence vs. Hydrodynamics: Do these composite particles behave according to hydrodynamic models (where flow is driven by mass and collective expansion) or coalescence models (where flow depends on the phase-space correlations of constituent particles)?
• Higher-Order Harmonics: At high transverse momentum ($p_T$), the standard second-order Fourier decomposition of azimuthal distributions may be insufficient. The paper investigates whether the non-linear mapping from constituent nucleons to composite nuclei introduces significant higher-order harmonics (e.g., $v_4$) that distort the measured $v_2$.
• Structural Differences: Despite having similar masses ($m_{^3\text{He}} \approx 2.81$ GeV/$c^2$ vs. $m_{^3_\Lambda\text{H}} \approx 2.99$ GeV/$c^2$), the hypertriton has a much larger spatial extent and weaker binding energy. The study aims to determine if these structural differences manifest in distinct flow behaviors.

2. Methodology

Data Sample:

• Experiment: ALICE detector at the CERN LHC.
• Collision System: Lead-Lead (Pb–Pb) collisions.
• Energy: $\sqrt{s_{NN}} = 5.36$ TeV (Run 3, 2023 data).
• Statistics: Approximately $5 \times 10^9$ minimum-bias events.

Particle Identification and Reconstruction:

• $^3$He: Identified directly via specific energy loss ($dE/dx$) in the Time Projection Chamber (TPC) and cluster size in the Inner Tracking System (ITS). The sample is extremely pure (>99%).
• $^3_\Lambda$H: Reconstructed via its weak decay channel: $^3_\Lambda$H $\to$ $^3$He + $\pi^-$. A V0-like secondary vertex finder was used. Backgrounds were suppressed using topological cuts (decay length, pointing angle) and particle identification of daughter tracks.
• Centrality: Events were classified into centrality intervals (0–10% to 40–60%) based on forward multiplicity measured by the FT0 detector.

Flow Measurement Technique:

• Method: The Scalar Product (SP) method was used to measure $v_2$.
• Reference Flow: The event plane was determined using energy deposits in the Forward Timing Zero (FT0) detectors and tracks in the TPC.
• Gap: A pseudorapidity gap of $|\Delta\eta| > 1.3$ was enforced between the (hyper)nucleus and reference particles to suppress non-flow effects (e.g., resonance decays, jets).
• Signal Extraction:
  • For $^3$He, $v_2$ was taken as the mean of the distribution.
  • For $^3_\Lambda$H, the measured $v_2$ is a mixture of signal and combinatorial background. A simultaneous fit of the invariant mass spectrum and the $v_2$ vs. mass distribution was performed to extract the signal $v_2$.

Systematic Uncertainties:

• Evaluated by varying track selection criteria (PID, quality cuts), background fit functions (polynomial order, exponential), and event plane determination methods (twisting corrections).

3. Key Contributions

1. First Measurement of $^3_\Lambda$H Elliptic Flow: This is the first experimental determination of the elliptic flow coefficient for the hypertriton at LHC energies.
2. High-Precision $^3$He Flow: Provides the most precise measurement of $^3$He $v_2$ to date, utilizing the large Run 3 dataset.
3. Investigation of Higher-Order Harmonics: The paper explicitly tests the validity of the standard Fourier truncation at $n=2$ for nuclei at high $p_T$. It demonstrates that for $^3$He, higher-order terms (specifically the 4th harmonic) are necessary to describe the azimuthal distribution.
4. Model Comparison: Extensive comparison with state-of-the-art theoretical models, specifically a multi-stage hybrid approach (MUSIC + UrQMD + Coalescence).

4. Results

Elliptic Flow ($v_2$) Trends:

• Both $^3$He and $^3_\Lambda$H exhibit a significant $v_2$ that increases with $p_T$ at low-to-intermediate momenta and decreases at high $p_T$.
• The magnitude of $v_2$ increases from central to peripheral collisions, consistent with the larger initial geometric eccentricity in peripheral events.
• Agreement between Species: Despite the large difference in binding energy and spatial size, the $v_2$ of $^3$He and $^3_\Lambda$H are consistent with each other within uncertainties.

Higher-Order Harmonics in $^3$He:

• In peripheral collisions at high $p_T$ ($>6$ GeV/$c$), the measured $v_2$ for $^3$He exceeds the physical limit of 0.5 (if only the second harmonic is considered), implying negative particle counts in certain azimuthal angles.
• Resolution: Fitting the azimuthal distribution $(\phi - \Psi_2)$ with a function including the 4th harmonic ($v_4$) provides an excellent description of the data, whereas the 2nd-order-only fit fails.
• Interpretation: This suggests that the coalescence mechanism, which binds nucleons with large individual $v_2$, naturally introduces non-linear modulations (higher harmonics) in the composite particle's distribution.

Model Comparison:

• The data is well-described by the MUSIC + UrQMD + Coalescence model.
• Models without a coalescence afterburner (where nuclei are produced directly at hadronization) fail to reproduce the data, particularly the high $v_2$ values at intermediate $p_T$.
• The consistency between $^3$He and $^3_\Lambda$H flow suggests that the anisotropy of the emission source (spatial/momentum correlations) is the dominant factor, rather than the intrinsic size or binding energy of the nucleus.

5. Significance

• Constraints on Hadronization: The results provide critical constraints on hadronization models, confirming that the coalescence mechanism is essential for describing the production and flow of light nuclei in heavy-ion collisions.
• Validation of Hydrodynamics + Coalescence: The success of the hybrid model (hydrodynamics for the bulk medium + coalescence for nuclei formation) reinforces the picture of a thermalized Quark-Gluon Plasma (QGP) that expands collectively before freezing out.
• Implications for Flow Analysis: The discovery of significant higher-order harmonics in $^3$He flow implies that standard flow analyses for composite particles at high $p_T$ must account for non-linear effects arising from the coalescence of constituents. Ignoring these terms can lead to unphysical interpretations of flow coefficients.
• Future Directions: The measurement of hypertriton flow opens the door to measuring its global polarization (spin alignment), which is a crucial input for understanding the spin properties of hypernuclei and the vorticity of the QGP.

In summary, this paper establishes that the elliptic flow of A=3 nuclei is governed by the collective motion of the medium and the phase-space correlations of their constituents, with the coalescence mechanism playing a pivotal role in shaping their azimuthal anisotropy, including the generation of higher-order harmonics.