Anisotropic flows in Au+Au collisions at $\sqrt{s_{\rm{NN}}} = 2.4\,\text{GeV}$ with a Skyrme pseudopotential

Using a lattice Boltzmann-Uehling-Uhlenbeck transport model with a density-, momentum-, and isospin-dependent N$^5$LO Skyrme pseudopotential, this study analyzes proton anisotropic flows in Au+Au collisions at $\sqrt{s_{\rm{NN}}} = 2.4\,\text{GeV}$ to demonstrate their strong sensitivity to momentum-dependent mean-field potentials and the symmetric nuclear matter incompressibility, while highlighting the need to incorporate higher-order equation-of-state parameters and in-medium cross-section modifications for future Bayesian extractions of nuclear matter properties.

The Gist

Imagine you are trying to understand how a giant, invisible sponge behaves when you squeeze it. In the world of physics, this "sponge" is nuclear matter—the stuff inside the heart of an atom. Scientists want to know exactly how stiff or squishy this matter is, especially when it gets crushed under extreme pressure, like inside a neutron star or during a high-speed crash between two heavy atoms.

This paper is a detailed report on a "crash test" experiment where scientists smashed Gold atoms together at incredibly high speeds. Here is the story of what they found, explained simply.

The Setup: The Ultimate Car Crash

The researchers used a giant particle accelerator to smash two Gold nuclei (Au) together. Think of this like taking two massive, dense trucks and crashing them head-on at 99% of the speed of light.

When they crash, they don't just bounce off; they create a tiny, super-hot, super-dense fireball of nuclear matter. This fireball expands and cools down in a split second. As it expands, the particles inside (mostly protons) fly out in specific patterns.

The Clues: The "Flow" of Particles

The scientists didn't just look at how many particles flew out; they looked at where they flew. They measured something called anisotropic flow.

• The Analogy: Imagine a crowd of people in a crowded room. If you push them from the side, they don't just move forward; they get squeezed out the sides.
  • Directed Flow ($v_1$): Like a crowd being pushed sideways by a wall.
  • Elliptic Flow ($v_2$): Like a crowd squeezing out of a narrow doorway, forming an oval shape.
  • Triangular & Quadrangular Flow ($v_3, v_4$): These are more complex, wavy patterns, like ripples in a pond or the jagged edges of a shattered glass.

By measuring these flow patterns, the scientists can work backward to figure out the rules that governed the crash.

The Mystery: What Rules the Crash?

The big question is: What forces are pushing these particles around? The paper investigates four main "rules" or ingredients that might change how the crash plays out:

1. The "Speed-Dependent" Push (Momentum Dependence):

   • The Metaphor: Imagine a crowd where people move differently depending on how fast they are running. If you run fast, the crowd pushes you harder; if you walk, they push less.
   • The Finding: The scientists found that this "speed-dependent push" is crucial. If they ignored it (assuming the push was the same for everyone), their simulation failed to match the real crash data. The particles need this rule to explain why they fly out the way they do.

2. The Stiffness of the Sponge (Equation of State):

   • The Metaphor: Is the nuclear matter like a stiff rubber ball or a soft marshmallow?
   • The Finding: The "stiffness" (specifically a number called $K_0$) is the most important factor for the overall shape of the crash. A stiffer "sponge" creates a stronger explosion, sending particles flying faster and further. The data suggests the nuclear matter is moderately stiff, not too soft and not too hard.

3. The "High-Density" Flavor (Symmetry Energy):

   • The Metaphor: Nuclear matter has two types of "flavors": Protons and Neutrons. The "Symmetry Energy" is the rule that says how much the matter dislikes having too many of one flavor compared to the other.
   • The Finding: Surprisingly, changing this rule didn't change the crash results very much. The flow patterns were mostly "blind" to this specific flavor preference at these energies.

4. The "Crowded Room" Effect (In-Medium Cross Sections):

   • The Metaphor: In a normal room, people bump into each other easily. But in a super-dense, hot room, maybe they start slipping past each other more easily, or bumping less often.
   • The Finding: This effect mostly changed the sideways push (Directed Flow). It didn't change the complex oval or triangular shapes much. This is a big clue: the complex shapes are a "pure" way to measure the stiffness of the matter without the noise of the "slippery" collisions.

The Big Picture: Why This Matters

The scientists used a sophisticated computer model (like a video game engine for atoms) called the Lattice BUU model. They fed it different sets of rules (the "Skyrme pseudopotential") to see which set recreated the real crash data best.

The Takeaway:
To understand the universe's most extreme environments (like neutron stars), we need to know exactly how nuclear matter behaves. This paper tells us:

• Don't ignore speed: You must account for how fast particles are moving when calculating the forces between them.
• Stiffness matters most: The "hardness" of nuclear matter is the main driver of the explosion's shape.
• Complex shapes are clean: The wiggly, triangular, and square-shaped flows ($v_3$ and $v_4$) are the best tools to measure the stiffness of nuclear matter because they aren't confused by other messy factors like particle collisions or flavor preferences.

In short, this paper helps us tune the "physics engine" of the universe, ensuring that when we simulate the birth of stars or the death of heavy atoms, we are using the correct rules of the road.

Technical Summary

1. Problem Statement

The equation of state (EOS) of dense nuclear matter, particularly its isospin-dependent component (symmetry energy) and the behavior of nucleon single-particle potentials (mean fields) at suprasaturation densities, remains highly uncertain. While symmetric nuclear matter EOS is relatively well-constrained near saturation density ($\rho_0$) by giant monopole resonance experiments, its behavior at higher densities and the isospin dependence of the nucleon potential are less understood.

Heavy-ion collisions (HICs) at intermediate energies (specifically $\sqrt{s_{NN}} = 2.4$ GeV, corresponding to $E_{beam} = 1.23A$ GeV) provide a unique laboratory to probe these conditions. The HADES collaboration has recently provided high-precision measurements of proton anisotropic flows ($v_1, v_2, v_3, v_4$) in Au+Au collisions. However, extracting nuclear matter properties from these observables is challenging because flow coefficients are sensitive to multiple, often correlated, physical inputs:

• The stiffness of the symmetric nuclear matter (SNM) EOS.
• The momentum dependence of the nucleon mean-field potential (including the symmetry potential).
• The high-density behavior of the symmetry energy.
• In-medium modifications of nucleon-nucleon elastic cross sections.

Current theoretical frameworks often struggle to disentangle these factors due to model dependencies and the lack of flexible effective interactions capable of varying these parameters independently.

2. Methodology

The authors employ a systematic theoretical approach combining a flexible nuclear effective interaction with a microscopic transport model:

• Transport Model: The study utilizes the Lattice Boltzmann-Uehling-Uhlenbeck (LBUU) transport model. This model solves the BUU equation using a test-particle method and a lattice Hamiltonian approach for mean-field evolution, ensuring accurate treatment of collision dynamics and ground-state stability.
• Effective Interaction (Skyrme Pseudopotential): The core innovation is the use of the N5LO Skyrme pseudopotential (extended up to the 10th-order momentum derivative, N5LO) combined with density-dependent (DD) terms (DD4).
  • This interaction allows for the decoupling of correlations between different macroscopic quantities.
  • It includes 30 Skyrme parameters mapped to 30 macroscopic quantities, including:
    • SNM EOS parameters: Saturation density ($\rho_0$), binding energy ($E_0$), incompressibility ($K_0$), skewness ($J_0$), and kurtosis ($I_0$).
    • Symmetry energy parameters: $E_{sym}(\rho_0)$, slope ($L$), curvature ($K_{sym}$), skewness ($J_{sym}$), and kurtosis ($I_{sym}$).
    • Momentum dependence: Coefficients for the single-particle potential ($a_n$) and symmetry potential ($b_n$) up to $p^{10}$.
    • Effective mass splitting: Linear isospin splitting coefficient ($\Delta m^*_1$).
• Simulation Setup:
  • System: Au+Au collisions at $\sqrt{s_{NN}} = 2.4$ GeV.
  • Centrality: Mid-peripheral (20–30%), corresponding to impact parameters $b \approx 6.6 - 8.1$ fm.
  • Particles: Neutrons, protons, $\Delta$ resonances, $N^*$, $\Delta^*$, and $\pi$ mesons are included. Light nuclei formation is currently excluded.
  • Cross Sections: Both free nucleon-nucleon cross sections and in-medium modified cross sections (parameterized based on T-matrix calculations) are tested.
• Baseline and Variations: A baseline interaction (L45D03) is constructed to satisfy empirical constraints (Hama optical potential, neutron star data, nuclear saturation properties). Several variations are then created to isolate specific effects:
  • L45D03FL: Different momentum dependence (Feldmeier-Lindner potential).
  • L45Dm03: Different symmetry potential momentum dependence (negative effective mass splitting).
  • L45MID: Momentum-independent potential.
  • K0380, J0m800, I04000: Variations in SNM EOS stiffness ($K_0$), skewness ($J_0$), and kurtosis ($I_0$).
  • L35D03: Softer symmetry energy at high densities.
  • L45D03 with CS: Includes in-medium cross-section suppression.

3. Key Contributions

• Development of a Flexible Framework: The application of the N5LODD4 Skyrme pseudopotential within the LBUU model allows for the independent variation of EOS stiffness, symmetry energy behavior, and momentum dependence of mean fields, which was previously difficult due to parameter correlations in standard Skyrme forces.
• Systematic Sensitivity Analysis: The paper provides a comprehensive dissection of how specific nuclear matter properties influence different orders of anisotropic flow ($v_1$ through $v_4$) at HADES energies.
• Benchmarking against HADES Data: The study rigorously compares theoretical predictions with the latest high-precision experimental data from the HADES collaboration.

4. Key Results

• Momentum Dependence is Critical:

  • The momentum dependence of the nucleon mean-field potential is the most dominant factor.
  • The momentum-independent interaction (L45MID) systematically underpredicts all flow coefficients ($v_1, v_2, v_3, v_4$) compared to HADES data.
  • The baseline interaction (L45D03) with proper momentum dependence reproduces the data well, except for low transverse momentum ($p_t$) behavior of $v_1$.
  • Different forms of momentum dependence (e.g., Hama vs. Feldmeier-Lindner potentials, or different symmetry potential slopes) show modest but distinct effects, particularly on $v_2$ at high $p_t$.

• Stiffness of Symmetric Nuclear Matter (SNM) EOS:

  • The incompressibility coefficient ($K_0$) strongly influences all flow observables. A stiffer EOS ($K_0 = 380$ MeV) leads to significantly larger flow magnitudes due to stronger pressure gradients, while $K_0 = 230$ MeV (empirically constrained) provides the best agreement with data.
  • Higher-order coefficients ($J_0, I_0$): The skewness ($J_0$) and kurtosis ($I_0$) of the SNM EOS have a modest sensitivity, primarily affecting the transverse momentum dependence of $v_2$ at mid-rapidity and high $p_t$. Their impact is less pronounced than $K_0$ because the collision duration at very high densities is short.

• Symmetry Energy:

  • The high-density behavior of the symmetry energy (tested via interaction L35D03) has limited effects on proton anisotropic flows. Even with a significantly softer symmetry energy at $1.5\rho_0 - 3\rho_0$, the changes in $v_1, v_2, v_3, v_4$ are minimal.

• In-Medium Cross Sections:

  • In-medium modifications to nucleon-nucleon elastic cross sections (suppression of cross sections in dense matter) have a negligible effect on higher-order flows ($v_2, v_3, v_4$).
  • However, they induce a noticeable effect on the directed flow ($v_1$), specifically its transverse momentum dependence. Reduced cross sections lead to fewer scatterings, less nuclear stopping, and consequently weaker directed flow.

5. Significance and Conclusion

• Validation of Momentum Dependence: The study confirms that incorporating momentum dependence into the nucleon mean-field potential (including the symmetry potential) is essential for describing collective flow data at HADES energies.
• Probes for Nuclear Matter:
  • $v_1$ is sensitive to in-medium cross-section modifications.
  • $v_2, v_3, v_4$ (higher-order flows) are robust probes of the SNM EOS stiffness ($K_0$) and the momentum dependence of the mean field, as they are largely insensitive to the uncertainties in high-density symmetry energy and in-medium cross sections.
• Future Directions: The authors emphasize that future Bayesian transport model analyses must simultaneously account for the momentum dependence of potentials, higher-order EOS parameters ($J_0, I_0$), and in-medium cross-section corrections to accurately extract nuclear matter properties. The N5LO Skyrme framework provides the necessary flexibility for such multi-parameter analyses.

In summary, this work establishes that while the symmetry energy and in-medium cross sections play roles, the momentum dependence of the mean field and the stiffness of the symmetric EOS are the primary drivers of proton anisotropic flows at $\sqrt{s_{NN}} = 2.4$ GeV, with higher-order flows offering a cleaner window into these properties.