Systematic study of flow of protons and light clusters in intermediate-energy heavy-ion collisions with momentum-dependent potentials

This study utilizes the PHQMD approach with a novel momentum-dependent potential to demonstrate that a soft, momentum-dependent nuclear equation of state best reproduces experimental flow data for protons and light clusters in intermediate-energy heavy-ion collisions, while also revealing distinct flow signatures that can help discriminate between different cluster formation mechanisms.

The Gist

Imagine you are a detective trying to figure out the rules of a very strange, invisible game played by tiny particles called protons and neutrons. These particles make up the core of every atom in the universe.

In this paper, a team of scientists acts as "cosmic architects." They build a massive digital simulation of a heavy-ion collision—essentially smashing two giant gold atoms together at nearly the speed of light. Their goal? To understand the Nuclear Equation of State (EoS).

Think of the EoS as the "stiffness recipe" for nuclear matter. Is the stuff inside an atomic nucleus like a soft, squishy marshmallow (a "soft" EoS), or is it like a hard, unyielding steel ball (a "hard" EoS)?

Here is how they cracked the case, explained simply:

1. The Big Smash and the "Traffic Jam"

When they smash these gold atoms together, they create a tiny, super-hot fireball of nuclear matter. It's like a massive traffic jam where millions of cars (protons and neutrons) are bumper-to-bumper.

• The Mystery: How do these cars move when they are squeezed together? Do they bounce off each other like billiard balls, or do they flow like a thick liquid?
• The Clue: The scientists look at how the particles "flow" out of the crash. They measure two specific types of flow:
  • Directed Flow ($v_1$): Like cars being pushed sideways out of a tunnel.
  • Elliptic Flow ($v_2$): Like cars being squeezed out of a round hole into an oval shape.

2. The Three "Stiffness" Recipes

To solve the mystery, the scientists ran their simulation three times, using three different "recipes" for how the particles interact:

1. The Soft Recipe: The nuclear matter is squishy and easy to compress.
2. The Hard Recipe: The nuclear matter is stiff and resists compression.
3. The "Smart" Soft Recipe (Momentum-Dependent): This is the new trick. It's like saying, "The material is soft, but it reacts differently depending on how fast the cars are moving." In the real world, particles don't just interact based on where they are; they interact based on how fast they are zooming past each other.

3. The "Light Clusters" (The Lego Bricks)

Usually, scientists just track individual protons. But in this crash, protons and neutrons sometimes stick together to form small groups, like deuterons (a proton + neutron) or tritons (three nucleons).

• The Analogy: Imagine the crash is a room full of people running. Sometimes, two people grab hands and run together as a pair.
• The Question: Do these pairs form because they just happened to bump into each other and stick (like magnets snapping together), or do they form because the "traffic rules" (the potential energy) naturally keep them close?

4. The Detective Work: What Did They Find?

The scientists compared their simulation results with real data from two famous experiments: HADES and FOPI.

• The "Soft" Recipe Failed: When they used the simple "soft" recipe, the simulation didn't match reality. The particles didn't flow the way they should.
• The "Hard" Recipe Was Close, But Not Quite: The "hard" recipe got the flow direction right, but it missed some details.
• The Winner: The "Smart" Soft Recipe: The recipe that included the momentum dependence (the "how fast" factor) was the champion. It matched the real-world data almost perfectly.
  • The Lesson: Nuclear matter isn't just "soft" or "hard." It's soft, but it has a "smart" reaction to speed. If you ignore how fast the particles are moving, you get the wrong answer.

5. The "Lego" Mystery Solved

The paper also solved a debate about how those "Lego pairs" (deuterons) are formed.

• Scenario A: They form naturally during the crash because the physics keeps them close (MST method).
• Scenario B: They form at the very end, just as the particles stop interacting, by randomly sticking together (Coalescence).
• The Finding: The way these pairs flow (their $v_1$ and $v_2$) is different depending on how they were made. The data suggests that the "natural formation during the crash" (Scenario A) is the dominant method. It's like realizing the pairs were formed during the traffic jam, not just when the cars finally stopped.

The Big Picture Takeaway

This paper tells us that the universe's building blocks are soft but speed-sensitive.

If you imagine the nucleus of an atom as a crowd of people:

• Old Theory: They are either a mosh pit (soft) or a rigid wall (hard).
• New Theory: They are a mosh pit, but if you run fast, they push back harder.

By understanding this "smart softness," scientists can better understand:

1. How stars explode: Supernovae depend on how stiff the core is.
2. What neutron stars are made of: These are giant atomic nuclei floating in space. Knowing the "stiffness recipe" tells us how heavy they can get before collapsing into black holes.

In short, the scientists built a digital crash test, tried different physics rules, and found the one that perfectly matches reality, proving that speed matters even in the tiniest corners of the universe.

Technical Summary

1. Problem Statement

The primary objective of nuclear physics in this energy regime is to determine the Nuclear Equation of State (EoS), specifically its density and momentum dependence. While static EoS models (characterized by compressibility modulus $K$) have been used historically, they fail to fully explain experimental flow data.

• The Gap: Previous studies established that nucleon flow is sensitive to the momentum dependence of the nucleon-nucleon potential. However, the behavior of light clusters (deuterons, tritons, $^3$He) in relation to the EoS and their production mechanisms remains debated.
• The Challenge: It is unclear whether collective flow observables ($v_1$ directed flow and $v_2$ elliptic flow) can distinguish between different theoretical models of cluster formation (e.g., coalescence vs. dynamical formation via potentials) and whether current transport models can simultaneously reproduce proton and cluster flow data with a single consistent EoS.

2. Methodology

The authors utilize the Parton-Hadron-Quantum-Molecular Dynamics (PHQMD) approach, a microscopic $n$-body transport model that combines Quantum Molecular Dynamics (QMD) with Parton-Hadron-String-Dynamics (PHSD).

Key Model Developments:

• Momentum-Dependent Potential: A novel feature of this work is the inclusion of a momentum-dependent nucleon potential in PHQMD, derived from the Schrödinger-equivalent optical potential fitted to elastic $pA$ scattering data. This is added to the standard static, density-dependent Skyrme interaction.
• Three EoS Scenarios: The study compares three distinct EoS configurations:
  1. Soft (S): Static, low compressibility ($K=200$ MeV).
  2. Hard (H): Static, high compressibility ($K=380$ MeV).
  3. Soft Momentum-Dependent (SM): Static soft potential combined with the new momentum-dependent term ($K=200$ MeV).
• Cluster Production Mechanisms: The study investigates three mechanisms for light cluster formation:
  1. MST (Minimum Spanning Tree): Clusters form dynamically during the collision based on spatial proximity and binding energy (identified via the aMST algorithm).
  2. Kinetic Mechanism: Deuterons formed via hadronic reactions (e.g., $\pi NN \leftrightarrow \pi d$) during the evolution.
  3. Coalescence: A post-freeze-out procedure where nucleons form clusters if they are close in phase space ($\Delta r, \Delta p$).
• Experimental Comparison: Calculations are benchmarked against experimental data from the HADES and FOPI collaborations for Au+Au collisions at SIS energies ($E_{kin} \approx 1.2 - 1.5$ A GeV).

3. Key Contributions

• Implementation of Momentum Dependence: Successfully integrated a momentum-dependent potential into the PHQMD framework, allowing for a direct comparison between static and momentum-dependent EoS scenarios within the same transport code.
• Unified Analysis of Protons and Clusters: Provided a systematic study of flow harmonics ($v_1, v_2$) for both free nucleons and light clusters, addressing the long-standing question of whether clusters follow the same EoS sensitivity as protons.
• Mechanism Discrimination: Demonstrated that flow observables are sensitive to the specific mechanism of cluster formation, offering a new tool to distinguish between MST/dynamical formation and coalescence.
• Scaling Violation: Investigated the scaling of elliptic flow with cluster mass number $A$ ($v_2/A$ vs. $p_T/A$), confirming its validity at low $p_T$ and its breakdown at high $p_T$.

4. Key Results

A. EoS Sensitivity in Spectra and Flow:

• Rapidity and $p_T$ Distributions: Soft (S) and Soft Momentum-Dependent (SM) EoS yield similar results for particle yields, which differ significantly from the Hard (H) EoS. Softening the EoS reduces proton yields at mid-rapidity but enhances light-cluster production.
• Directed Flow ($v_1$):
  • For protons, the SM EoS provides the best agreement with HADES and FOPI data, slightly increasing the magnitude of $v_1$ compared to the Hard EoS.
  • The static Soft (S) EoS significantly underestimates the flow slope.
  • Cluster Size Effect: The slope of $v_1(y)$ increases with cluster mass number $A$. This is reproduced by the SM EoS, suggesting clusters form in regions of high density gradients (participant-spectator interface).
• Elliptic Flow ($v_2$):
  • The SM EoS describes the proton $v_2$ data best, though it slightly underestimates the magnitude at mid-rapidity.
  • For light clusters, the SM and Hard (H) EoS produce similar $v_2$ values, while the static Soft (S) EoS fails to reproduce the data.
  • Scaling: At mid-rapidity and low transverse momentum ($p_T/A < 0.8$ GeV/c), $v_2/A$ scales with $p_T/A$ for protons, deuterons, and tritons. This scaling breaks down at higher $p_T$.

B. Sensitivity to Production Mechanisms:

• MST/Kinetic vs. Coalescence: The study found a substantial difference in flow coefficients ($v_1$ and $v_2$) between deuterons formed via the MST + Kinetic mechanisms (standard PHQMD) and those formed via Coalescence.
• Implication: MST/kinetic deuterons exhibit slightly larger flow magnitudes than coalescence deuterons. This suggests that flow harmonics can serve as an experimental discriminator to identify the dominant cluster formation mechanism in nature.

C. Comparison with Other Models:

• The PHQMD results with the SM EoS align well with UrQMD (which uses a static hard EoS), confirming the literature finding that a static hard EoS and a soft momentum-dependent EoS yield similar flow observables.
• The results also show agreement with SMASH (a BUU-type approach), despite different implementations of the momentum-dependent mean field.

5. Significance and Conclusion

• EoS Constraints: The results strongly support a soft, momentum-dependent EoS for nuclear matter at densities up to $\sim 3\rho_0$. A static soft EoS is ruled out by flow data, while a static hard EoS is only viable if momentum dependence is neglected (which is physically inconsistent with $pA$ scattering data).
• Compressibility: The data suggests a compressibility modulus slightly higher than the standard soft value ($K \approx 200$ MeV) but consistent with the soft momentum-dependent scenario.
• Cluster Formation: The study concludes that flow observables provide critical information beyond multiplicity spectra. The distinct flow patterns of MST/kinetic deuterons versus coalescence deuterons offer a pathway to experimentally resolve the debate on how light clusters form in heavy-ion collisions.
• Future Directions: The authors advocate for Bayesian analysis to precisely constrain the compressibility and in-medium cross-sections, and call for better alignment between theoretical centrality definitions and experimental event selection to reduce systematic uncertainties.

In summary, this paper establishes that momentum dependence is a non-negotiable component of the nuclear EoS for describing intermediate-energy heavy-ion collisions and demonstrates that collective flow of light clusters is a powerful, sensitive probe for both the EoS and the microscopic mechanisms of cluster formation.