$\mathbf{{\textbf{K}}^{0}_{\textbf{S}}}-\mathbf{{\textbf{K}}^{0}_{\textbf{S}}}$ femtoscopy in Pb$-$Pb collisions at $\mathbf{\sqrt{\textit{s}_{\rm NN}} = 5.02}$ TeV at the LHC

This paper presents a one-dimensional femtoscopic analysis of ${\rm K}^{0}_{\rm S}-{\rm K}^{0}_{\rm S}$ correlations in Pb$-$Pb collisions at $\sqrt{s_{\mathrm{NN}}} = 5.02$ TeV using ALICE data, revealing source characteristics consistent with collective expansion and providing a high-energy baseline compatible with previous measurements and hydrokinetic model predictions.

The Gist

Imagine two massive lead balls smashing into each other at nearly the speed of light. When they collide, they don't just shatter; they create a tiny, super-hot "soup" of energy and particles that expands and cools down in a fraction of a second. This is what happens in the Large Hadron Collider (LHC) at CERN.

The ALICE collaboration, a group of scientists using a giant detector, wanted to take a "snapshot" of this soup to understand its size and how it behaves. Specifically, they looked at pairs of neutral kaons (a type of subatomic particle called $K^0_S$) that were born from the same collision.

Here is the story of what they found, explained simply:

1. The "Femtoscopy" Camera

To understand the size of this invisible explosion, the scientists used a technique called femtoscopy. Think of it like trying to guess the size of a room by listening to how two people's voices echo off the walls.

In this case, the "voices" are the particles. Because these particles are identical twins (bosons), they have a special quantum rule: they prefer to stick together or avoid each other depending on how fast they are moving relative to one another. By measuring how often these pairs stick together versus how far apart they are, the scientists can calculate the size of the "room" (the source) they came from.

2. The Experiment: A Higher Energy Crash

Previously, scientists had studied these collisions at a certain energy level (2.76 TeV). In this new paper, they cranked the energy up to 5.02 TeV (about twice as hard).

They asked two main questions:

• Does the "room" get bigger when we crash harder?
• Does the behavior of the particles change depending on how hard we look at them?

3. The Findings: A Stretching Balloon

The scientists looked at the data in two ways: by how "central" the crash was (did the balls hit dead-on or just graze each other?) and by the momentum of the particle pairs.

• The Size of the Source ($R$):

  • Central Collisions (Dead-on hits): When the lead balls hit head-on, they created a large, expanding fireball. The scientists found that the size of this fireball was consistent with what they saw at the lower energy. It's like a balloon inflating; the bigger the explosion, the bigger the balloon.
  • Peripheral Collisions (Graze hits): When the balls just grazed each other, the "balloon" was much smaller.
  • The Flow: They noticed that particles moving faster (higher momentum) seemed to come from a smaller effective area. Imagine a crowd of people running out of a stadium. The people running fastest (the high-momentum particles) are usually the ones who started near the exit and ran straight out, so they seem to come from a smaller, more focused area. The slower people are still milling about in the middle. This confirms that the "soup" is expanding collectively, like a fluid.

• The "Strength" of the Connection ($\lambda$):

  • This number tells us how "pure" the signal is. If every particle pair came directly from the explosion, the number would be 1. If many pairs came from other sources (like the decay of other unstable particles), the number drops.
  • The scientists found this number stayed roughly the same (around 0.6) regardless of the energy or how hard the crash was. This suggests that the "recipe" for making these particles didn't change much between the lower and higher energy crashes. About 60% of the pairs they saw were "primordial" (born directly in the crash), while the rest were "second-hand" (born from the decay of other particles).

4. Checking the Map: Models and Other Teams

The scientists didn't just look at their own data; they checked it against two things:

• Computer Simulations (The Hydrokinetic Model): They compared their results to a complex computer model that tries to simulate the physics of the explosion.

  • The Good News: The model worked perfectly for the big, central crashes.
  • The Bad News: The model struggled with the smaller, "grazing" crashes. It predicted the particles would flow differently than they actually did. This suggests our computer models aren't quite ready to perfectly describe the "messy" edges of these collisions yet.

• The Rival Team (CMS): Another team at the LHC (CMS) had recently measured the same thing. The ALICE team compared notes and found their results matched the CMS results very closely (within a small margin of error). This is like two different photographers taking pictures of the same event from slightly different angles and agreeing on the size of the subject.

Summary

In short, this paper confirms that when we smash lead atoms together at record-breaking energies, the resulting "soup" behaves consistently with what we saw at lower energies. It expands like a fluid, and the size of the explosion depends on how hard the atoms hit. While our computer models are great at describing the center of the explosion, they still need some work to understand the edges.

The study provides a solid, consistent baseline for future research, proving that the fundamental rules of this high-energy "soup" remain stable even as we turn up the power.

Technical Summary

1. Problem and Motivation

The primary goal of this study is to investigate the space-time geometry and dynamics of the particle-emitting source created in high-energy heavy-ion collisions. Specifically, the paper addresses the beam-energy dependence of femtoscopic observables for neutral kaons ($K^0_S$).

• Context: Femtoscopy (HBT interferometry) measures momentum correlations between identical particles to extract the source size ($R$) and correlation strength ($\lambda$). While pion and charged kaon femtoscopy are well-established, neutral kaon femtoscopy offers unique advantages:
  • They can be identified at higher momenta via topological reconstruction of their decay ($K^0_S \to \pi^+\pi^-$), allowing the study of collective expansion over a broader kinematic range.
  • They provide a complementary dataset to charged kaons, utilizing different identification methods (secondary vertex reconstruction vs. tracking) and handling final-state interactions (FSI) differently (no Coulomb force).
• Gap: Previous measurements of $K^0_S$ correlations existed at $\sqrt{s_{NN}} = 200$ GeV (STAR) and 2.76 TeV (ALICE). This work extends these measurements to the higher LHC energy of 5.02 TeV to determine if the space-time structure and collective flow evolve with collision energy.

2. Methodology

Data and Event Selection

• Dataset: 1.4 billion minimum-bias Pb–Pb collision events collected by the ALICE detector during LHC Run 2 (2015–2018) at $\sqrt{s_{NN}} = 5.02$ TeV.
• Centrality: Events are classified into four centrality classes based on charged-particle multiplicity: 0–10%, 10–30%, 30–50%, and 50–90%.
• Particle Identification:
  • $K^0_S$ candidates are reconstructed via the decay $K^0_S \to \pi^+\pi^-$.
  • Daughter pions are identified using the Time Projection Chamber (TPC) and Time-Of-Flight (TOF) detectors.
  • Strict topological cuts are applied (decay length, distance of closest approach to the primary vertex) to ensure purity. The average purity of reconstructed $K^0_S$ is 96%.
  • A specific cut on the separation of same-sign daughter tracks in the TPC (>10 cm) is applied to minimize two-track effects (merging/splitting).

Correlation Function Construction

• Variable: The analysis uses the relative momentum in the pair rest frame, $k^*$. For identical particles, $k^* = q_{inv}/2$.
• Function: The experimental correlation function is defined as $C(k^*) = A(k^*)/B(k^*)$, where $A$ is the same-event pair distribution and $B$ is the mixed-event background distribution.
• Event Mixing: Mixed events are constructed from pools of 100 events with similar centrality and primary vertex z-position (within 2 cm and 2% centrality).

Fitting Model

The correlation functions are fitted using the Lednický-Lyuboshitz (LL) analytical formula, which accounts for:

1. Quantum Statistics (Bose-Einstein): The symmetrization of the wavefunction for identical bosons.
2. Final-State Interactions (FSI): Strong interactions between the kaons, dominated by the $f_0(980)$ and $a_0(980)$ resonances.
3. Non-femtoscopic Background: Modeled by a first-order polynomial ($bk^* + c$) to account for momentum conservation and other non-interaction correlations.

The fit extracts two key parameters:

• $R$ (Source Radius): Characterizes the spatial size of the emission region.
• $\lambda$ (Correlation Strength): Represents the fraction of pairs contributing to the genuine femtoscopic correlation, reduced by experimental purity and non-primordial sources (resonance decays).

Systematic Uncertainties

Uncertainties were evaluated by varying selection criteria (mass window, DCA, PID $N_\sigma$, track merging cuts) and fitting ranges. The total systematic uncertainty is calculated by adding individual variations in quadrature.

3. Key Contributions

1. First High-Energy $K^0_S$ Femtoscopy: This is the first measurement of $K^0_S$-$K^0_S$ correlations at $\sqrt{s_{NN}} = 5.02$ TeV, extending the energy range of previous ALICE results (2.76 TeV).
2. Precision Comparison with CMS: The results are compared with recent CMS measurements at the same energy, showing compatibility within $1.3\sigma$, validating the consistency of femtoscopic measurements across different experiments and analysis techniques.
3. Hydrokinetic Model Validation: The data is compared against the Integrated Hydrokinetic Model (iHKM). The study highlights where the model succeeds (central collisions) and where it deviates (peripheral collisions), providing constraints on theoretical descriptions of the collision evolution.
4. Resonance Dilution Analysis: The paper explicitly quantifies the dilution of the $\lambda$ parameter due to long-lived resonances ($K^*(892)$ and $\phi(1020)$), estimating the "genuine" primordial $\lambda$ to be $\approx 0.65$.

4. Results

Source Radius ($R$)

• Centrality Dependence: $R$ decreases as collisions become more peripheral (from $\approx 6.4$ fm in 0–10% to $\approx 2.5$ fm in 50–90%). This reflects the reduction in the initial overlap volume of the nuclei.
• $k_T$ Dependence: In central and mid-central collisions (0–50%), $R$ decreases with increasing pair transverse momentum ($k_T$). This is a signature of collective radial flow, where faster particles are emitted from a smaller effective region due to the expansion dynamics.
• Peripheral Behavior: In the most peripheral bin (50–90%), the $R$ vs. $k_T$ dependence flattens, suggesting weaker radial flow and a transition toward behavior more similar to proton-proton ($pp$) collisions.
• Energy Dependence: Comparing 5.02 TeV with 2.76 TeV, the radii are consistent within $1.6\sigma$ for the same multiplicity classes. This indicates that the space-time structure and collective expansion are energy-independent at high multiplicities.

Correlation Strength ($\lambda$)

• Stability: The purity-corrected $\lambda$ values remain stable around 0.6 across all centralities, $k_T$ ranges, and collision energies.
• Interpretation: The stability suggests that the relative fraction of primordial $K^0_S$ pairs (those produced directly in the collision) versus those from resonance decays does not change significantly with collision energy or multiplicity.
• Model Comparison: The Hydrokinetic Model (HKM) tends to overestimate $\lambda$ at low $k_T$, suggesting potential oversimplifications in the model's treatment of source purity or resonance contributions.

Model Comparison

• Hydrokinetic Model (HKM): The model reproduces the measured radii well for central collisions (0–10%) at both energies. However, in peripheral collisions (30–50% at 5.02 TeV), the model predicts a steeper decrease in $R$ with $k_T$ than observed, deviating by up to $3.6\sigma$. This suggests the model may overestimate the strength of collective flow in low-multiplicity events.

5. Significance

• Validation of Collective Flow: The results confirm that collective radial flow is a robust feature of the quark-gluon plasma (QGP) formed in Pb–Pb collisions, persisting up to 5.02 TeV.
• Energy Independence: The finding that source sizes and correlation strengths are consistent between 2.76 TeV and 5.02 TeV implies that the hydrodynamic evolution of the system is largely determined by the initial geometry (multiplicity) rather than the collision energy itself in this regime.
• Theoretical Constraints: The deviation of the HKM in peripheral collisions provides critical feedback for theoretical models, indicating a need to better describe the transition from a hydrodynamic medium to a hadronic cascade in low-density systems.
• Baseline for Future Physics: These measurements establish a consistent baseline for future comparisons with higher-statistics data and other particle species, aiding in the precise characterization of the QGP's space-time evolution.

In conclusion, this paper successfully extends the understanding of heavy-ion collision dynamics to higher energies, demonstrating that the collective behavior of the emitting source remains consistent with previous findings while highlighting specific limitations in current hydrokinetic models for peripheral events.