Femtoscopic signatures of unique nuclear structures in relativistic collisions

Using the AMPT model, this study demonstrates that femtoscopic source parameters of pion pairs in $^{208}$Pb+$^{20}$Ne and $^{208}$Pb+$^{16}$O collisions at $\sqrt{s_{NN}}$ = 68.5 GeV serve as robust signals for detecting unique nuclear structures and initial shape deformations, thereby expanding the scope of nuclear structure investigations to femtoscopic observables.

The Gist

The Big Picture: Taking a "Flash Photo" of the Atomic Nucleus

Imagine you are trying to figure out what a strange, invisible object looks like. You can't see it directly, but you can throw a bunch of tiny marbles at it and watch how the marbles bounce off. By studying the pattern of the bouncing marbles, you can reconstruct the shape of the object.

This is essentially what high-energy physicists do. They smash heavy atoms (like Lead) together at nearly the speed of light. When they collide, they create a tiny, super-hot "soup" of particles called a fireball.

For a long time, scientists have studied this fireball by looking at how the particles flow (like water swirling in a storm). But this paper introduces a new, sharper camera lens called Femtoscopy.

The New Tool: Femtoscopy (The "Ghostly" Ruler)

Femtoscopy is a technique that measures the size and shape of the fireball at the exact moment the particles stop interacting and fly apart. It's like taking a high-speed flash photo of the explosion.

• The Analogy: Imagine two identical twins (pions) born in the same explosion. Because they are identical quantum particles, they have a special "dance" relationship. If they are born close together, they tend to stick together in their flight path. If they are born far apart, they drift apart.
• By measuring how often these twin particles stick together versus how far apart they are, scientists can calculate the exact size and shape of the "room" (the fireball) they were born in.

The Mystery: Are Nuclei Perfect Balls?

Usually, we think of atomic nuclei (the center of an atom) as perfect, round balls. However, some nuclei might actually be weird shapes:

• The Neon Nucleus ($^{20}$Ne): The paper suggests this might look like a bowling pin or a pyramid made of four smaller clusters (alpha particles) stuck together.
• The Oxygen Nucleus ($^{16}$O): This might look like a perfect tetrahedron (a pyramid with a triangular base) made of four clusters.

The big question is: Does the initial weird shape of the nucleus survive the explosion, or does the fireball just become a round blob?

The Experiment: The "Bowling Pin" vs. The "Round Ball"

The author, Dániel Kincses, used a supercomputer simulation (called AMPT) to crash two different teams of atoms together:

1. Team A: Lead ($^{208}$Pb) + Neon ($^{20}$Ne)
2. Team B: Lead ($^{208}$Pb) + Oxygen ($^{16}$O)

He ran the simulation twice for each team:

• Scenario 1 (The Standard Model): Assuming the nuclei are perfect, round balls (Woods-Saxon shape).
• Scenario 2 (The Cluster Model): Assuming the nuclei are made of clusters, giving them the "bowling pin" or "tetrahedron" shapes.

The Discovery: The Shape Survives!

The results were exciting. When the "bowling pin" Neon nucleus was smashed, the resulting fireball didn't just become a round blob. It kept the memory of that weird shape.

• The "Eccentricity" Meter: The paper introduces a measurement called freeze-out eccentricity. Think of this as a "squishiness" meter.
  • If the fireball is a perfect circle, the meter reads 0.
  • If the fireball is an oval (stretched out), the meter reads higher.

The Finding:

• When they smashed Oxygen (which is more symmetric in its cluster shape), the fireball was fairly round, regardless of whether they assumed it was a ball or a cluster.
• When they smashed Neon (the bowling pin), the fireball became significantly more oval (eccentric) only when they assumed the bowling pin shape.

The Analogy: Imagine squeezing a round water balloon vs. a water balloon shaped like a football.

• If you squeeze the round one, it stays round.
• If you squeeze the football-shaped one, it stretches out even more.
  The paper shows that the "football" shape of the Neon nucleus survives the collision and makes the final explosion look more stretched out than the Oxygen explosion.

Why Does This Matter?

1. A New Way to See: Previously, scientists used "flow" (how particles swirl) to guess nuclear shapes. This paper shows that Femtoscopy (measuring the size of the explosion) is a second, independent way to check. It's like using both a ruler and a scale to weigh an object; if they agree, you know the measurement is real.
2. Proving the Shape: It provides strong evidence that the Neon nucleus really does have a "bowling pin" shape made of clusters, rather than being a smooth, round ball.
3. Future Experiments: This study sets the stage for real experiments at the LHCb (a detector at the Large Hadron Collider). They are planning to use a fixed-target program (SMOG2) to smash Lead and Neon/Oxygen beams together. This paper tells them exactly what to look for: If the Neon collisions show a more "stretched" fireball than Oxygen, we have proven the bowling pin shape exists.

Summary in One Sentence

This paper uses computer simulations to show that if you smash a "bowling pin-shaped" Neon nucleus, the resulting explosion stays stretched out, proving that we can use the size and shape of particle explosions to map the hidden, weird geometries of atomic nuclei.

Technical Summary

1. Problem Statement

High-energy nuclear physics has increasingly focused on imaging nuclear structures (e.g., deformation, clustering) through relativistic heavy-ion collisions. While anisotropic flow ($v_n$) has been established as a sensitive probe of initial-state nuclear geometry, femtoscopic correlation measurements (HBT interferometry) have not yet been widely utilized for this specific purpose.

The core challenge is that standard femtoscopic analyses often integrate over the azimuthal angle, losing geometric information about the fireball's shape. Furthermore, previous studies often relied on Gaussian approximations for the source function, which may fail to capture the power-law tails observed in high-energy collisions. This paper aims to determine if azimuthally sensitive femtoscopic observables can serve as a robust signal for unique nuclear structures, specifically distinguishing between spherical and clustered (deformed) initial configurations.

2. Methodology

The study utilizes the Multi-Phase Transport (AMPT) model to simulate ultra-central ($b=0$) collisions at a center-of-mass energy per nucleon pair of $\sqrt{s_{NN}} = 68.5$ GeV. Two specific collision systems were chosen to align with the LHCb SMOG2 fixed-target program:

• $^{208}\text{Pb} + ^{20}\text{Ne}$
• $^{208}\text{Pb} + ^{16}\text{O}$

Initial State Configurations:
Two distinct nucleon distributions were modeled for the light nuclei ($^{20}\text{Ne}$ and $^{16}\text{O}$):

1. Woods-Saxon: A standard spherically symmetric distribution.
2. NLEFT (Nuclear Lattice Effective Field Theory): A configuration where nucleons form $\alpha$-clusters.
   • $^{16}\text{O}$: Forms a tetrahedral shape (4 $\alpha$-clusters).
   • $^{20}\text{Ne}$: Forms a "bowling pin" shape (5 $\alpha$-clusters), introducing significant deformation.

Analysis Procedure:

• Event Selection: 100,000 events per system/configuration. Particles were selected within the LHCb pseudo-rapidity acceptance ($2 < \eta < 5$ in lab frame).
• Event Plane: The second-order event plane ($\Psi_2$) was reconstructed using charged pions, kaons, and protons.
• Femtoscopy: Same-charge pion pairs were analyzed in bins of average transverse momentum ($k_T$) and relative azimuthal angle ($\phi_{\text{pair}} - \Psi_2$).
• Source Reconstruction: The pair source distribution $D(\vec{\rho})$ was reconstructed in the Out-Side-Longitudinal (OSL) frame.
• Fitting Model: Instead of a Gaussian, the authors fitted the 3D source distribution to a symmetric Lévy-stable distribution with an elliptical contour. This involved extracting:
  • The Lévy exponent ($\alpha$), characterizing the power-law tail.
  • The $R^2$ matrix containing 6 independent Lévy-scale parameters.
• Azimuthal Dependence: The scale parameters were fitted to Fourier expansions (cosine/sine terms) to extract the zeroth-order ($R_{\mu,0}$) and second-order ($R_{\mu,2}$) harmonics.
• Key Observable: The freeze-out eccentricity ($\epsilon_F$) was calculated using the ratio of the second-order to zeroth-order terms of the side-radius:
  $$\epsilon_F = 2 \frac{R^2_{\text{side},2}}{R^2_{\text{side},0}}$$

3. Key Contributions

• First 3D Lévy Analysis: This is the first investigation to determine all six independent scale parameters of a 3D Lévy-stable pion pair source in a transport model, moving beyond previous azimuthally integrated or Gaussian analyses.
• Azimuthal Sensitivity: The study demonstrates that femtoscopic parameters, when analyzed relative to the event plane, are sensitive to the initial nuclear deformation, complementing flow measurements.
• Cluster Structure Detection: It provides a theoretical baseline showing that the "bowling pin" deformation of $^{20}\text{Ne}$ (NLEFT) leaves a distinct signature in the freeze-out eccentricity, whereas the tetrahedral $^{16}\text{O}$ does not significantly alter the elliptical asymmetry compared to the spherical case.

4. Results

• Source Shape: The 3D Lévy-stable distribution successfully described the source shape, capturing power-law tails that Gaussian approximations miss. The Lévy exponent $\alpha$ showed little dependence on the azimuthal angle or the initial configuration.
• Scale Parameters:
  • The diagonal elements ($R^2_{\text{out}}$, $R^2_{\text{side}}$, $R^2_{\text{long}}$) showed clear separation between Woods-Saxon and NLEFT configurations for $^{20}\text{Ne}$, particularly in the 'out' and 'side' directions.
  • Off-diagonal elements exhibited non-zero zeroth-order terms, attributed to collision-system asymmetry.
• Freeze-out Eccentricity ($\epsilon_F$):
  • $^{20}\text{Ne}$ (NLEFT): The bowling-pin shape resulted in a significantly higher freeze-out eccentricity compared to the spherical Woods-Saxon case. This enhancement persisted through the hadronic phase.
  • $^{16}\text{O}$ (NLEFT): The tetrahedral clustering did not significantly increase the elliptical asymmetry compared to the spherical Woods-Saxon case.
  • Comparison: In Woods-Saxon configurations, the eccentricity of Pb+Ne was systematically lower than Pb+O (likely due to system size effects). However, in the NLEFT case, the Pb+Ne eccentricity exceeded the Pb+O eccentricity.
• Ratio Observable: The ratio of freeze-out eccentricities ($\epsilon_F^{\text{Pb+Ne}} / \epsilon_F^{\text{Pb+O}}$) was found to be greater than unity for the NLEFT configuration across the transverse mass range, while it remained below unity for the Woods-Saxon configuration. This ratio serves as a robust, model-independent signal of the deformed initial shape.

5. Significance

• New Probe for Nuclear Structure: The paper establishes that azimuthally sensitive femtoscopy is a viable and robust tool for investigating nuclear deformation and clustering, offering a complementary perspective to anisotropic flow.
• SMOG2 Program Relevance: The results provide a critical baseline for the upcoming LHCb SMOG2 fixed-target program, specifically for $^{208}\text{Pb}+^{20}\text{Ne}$ and $^{208}\text{Pb}+^{16}\text{O}$ collisions, guiding experimentalists on what signatures to look for.
• Disentangling Effects: By measuring both elliptic flow and femtoscopic parameters, researchers can better disentangle geometric effects from collective flow dynamics, as these observables contain geometric parameters in different, entangled ways.
• Future Directions: The study suggests extending these analyses to non-identical particles, higher-order event planes, and hybrid hydrodynamic models to further refine the understanding of the quark-gluon plasma and nuclear structure.