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Effects of light-cluster degrees of freedom on collective flows in heavy-ion collisions at FOPI energies

Using a lattice Boltzmann-Uehling-Uhlenbeck transport model with a kinetic approach for light-cluster formation, this study demonstrates that explicitly including dynamical light-cluster degrees of freedom significantly modifies proton collective flows at beam energies below 600 A MeV and successfully reproduces the nucleon-number scaling of light nuclei flows, thereby highlighting the necessity of dynamical cluster treatment for interpreting heavy-ion collision data in this energy regime.

Original authors: Xin Li, Si-Pei Wang, Rui Wang, Zhen Zhang, Jie Pu, Chun-Wang Ma, Lie-Wen Chen

Published 2026-08-04

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CC BY 4.0

Original authors: Xin Li, Si-Pei Wang, Rui Wang, Zhen Zhang, Jie Pu, Chun-Wang Ma, Lie-Wen Chen

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine a universe where the smallest building blocks of matter, protons and neutrons, don't just sit still like bricks in a wall. Instead, they are like a chaotic, high-speed dance party. When scientists smash heavy atoms together at incredible speeds, they create a tiny, super-hot fireball of this "nuclear matter." It's a place so dense and energetic that the rules of normal physics get a little fuzzy. To understand how this fireball behaves, scientists look at "collective flow." Think of this like watching a crowd at a concert. If the crowd is just a bunch of individuals bumping into each other randomly, that's one thing. But if the whole crowd starts swaying together in a wave, or pushing out in a specific shape, that's a "flow." This flow tells us about the pressure and the "stickiness" of the nuclear matter inside the fireball.

For a long time, scientists mostly watched the solo dancers—the individual protons and neutrons—to understand this flow. But in the middle of this nuclear dance floor, something interesting happens: the dancers sometimes grab hands and form little groups. These groups are called "light clusters," like tiny atoms made of just two or three particles (deuterons, tritons, helium). The big question is: does the fact that these dancers are holding hands change how the whole crowd sways? Do these little groups act like independent entities, or do they just drag the solo dancers around with them? Understanding this is crucial because it helps us map out the "equation of state" of nuclear matter—basically, the rulebook for how matter behaves under extreme pressure, which is also what happens inside neutron stars.

In this study, a team of researchers decided to put the "hand-holding" groups into their computer simulations to see if it changed the dance. They used a sophisticated model called the Lattice Boltzmann-Uehling-Uhlenbeck (LBUU) transport model, which acts like a high-tech video game engine for nuclear physics. They simulated smashing gold atoms (Au) into other gold atoms at speeds ranging from 120 to 1500 A MeV (a unit of energy per particle). They ran two versions of the simulation: one where the light clusters were treated as real, active characters that could form, break apart, and bounce around, and another where these clusters were ignored, and everything was just a bunch of solo particles.

The results revealed a fascinating story that depends entirely on how hard they smashed the atoms together. At the lower energy levels, specifically between 120 and 150 A MeV, the presence of these light clusters made a huge difference. When the clusters were allowed to form dynamically, the flow of the protons changed significantly. It was as if the formation of these groups rearranged the entire dance floor, altering the direction and shape of the crowd's movement. The researchers found that this effect was most dramatic for the "directed flow" (how particles move forward or backward) and "elliptic flow" (how they spread out sideways).

However, as they increased the energy of the collision, the story changed. Around 250 to 400 A MeV, the effect of the clusters was still visible but started to fade. By the time they reached energies of 600 A MeV and above, the difference between the simulation with clusters and the one without them became almost invisible. The paper suggests that this is because at higher energies, the nuclear fireball is so hot and violent that the little groups can't survive for long; they get smashed apart before they can really influence the flow. The abundance of these clusters drops off, so their ability to reshape the proton dance floor disappears.

The researchers also looked at how the light clusters themselves moved. They compared their computer results with real data from the FOPI experiment. At the lower energies (120–250 A MeV), their model tended to predict that the clusters would flow a bit more strongly than what was actually observed in the lab. This suggests that while the model captures the general idea, there might be some missing pieces, like how heavier fragments or the "spectator" parts of the atoms (the parts that didn't collide) might be interfering. But at higher energies (400 A MeV and up), the model did a much better job of matching the real-world data.

Finally, the team checked a specific rule called "nucleon-number scaling." This is a simple idea: if a cluster is just a bunch of nucleons stuck together, its flow should be exactly the sum of the flow of its parts. For example, a deuteron (2 particles) should flow twice as much as a single proton. The simulations showed that at lower energies, this simple rule broke down. The clusters didn't just act like a sum of their parts; their formation and breakup dynamics created a more complex flow pattern. At higher energies, the scaling got closer to the simple rule, but even then, the fastest-moving clusters still showed some deviations, hinting that they were formed early in the collision and escaped quickly, carrying information about the very first moments of the crash.

In short, the paper suggests that if you want to understand how nuclear matter flows at lower collision energies (below 600 A MeV), you absolutely cannot ignore the light clusters. They are active participants that reshape the flow of protons. But if you crank up the energy high enough, the clusters get destroyed so fast that they stop mattering for the overall flow of the protons. The study highlights that a "dynamic" treatment—where clusters are allowed to form and break in real-time—is essential for interpreting these collisions correctly, especially in the lower energy regime where the nuclear matter is cooler and the clusters have a better chance to survive.

Problem Statement
Heavy-ion collisions at intermediate energies (Fermi to GeV region) serve as a laboratory for studying strongly interacting nuclear matter under nonequilibrium conditions. A significant complication in interpreting collective flow observables in this regime is the abundant production of light nuclei (deuterons, tritons, $^3$ He, and $^4$ He). While transport models have successfully described free nucleon and pion flows, the treatment of light clusters has often been limited. In many analyses, light clusters are not propagated as explicit degrees of freedom but are instead reconstructed from final-state nucleon distributions using post-processing methods like coalescence or minimum spanning tree (MST) algorithms. This omission neglects the dynamical redistribution of baryon number between free nucleons and bound states, which can alter the phase-space distribution of free nucleons and their collective flows. Furthermore, the extent to which explicit light-cluster degrees of freedom modify collective-flow observables, particularly the beam-energy dependence of these effects, remains insufficiently quantified.

Methodology
The authors investigate these effects using a lattice Boltzmann-Uehling-Uhlenbeck (LBUU) transport model coupled with a kinetic approach for light-cluster formation.

Transport Framework: The model evolves the Wigner functions of protons, neutrons, and light clusters ( $d, t, h, \alpha$ ) dynamically. The single-particle energy is derived from a density-, momentum-, and isospin-dependent N5LO Skyrme pseudopotential, with parameters constrained by Bayesian inference on proton flow data at HADES energies.
Kinetic Cluster Formation: Light clusters are treated as active degrees of freedom coupled to the transport evolution. The collision integral includes elastic and inelastic scattering channels, including nucleon-induced catalytic reactions (e.g., $NNN \leftrightarrow Nd$ ) and pion-catalyzed production/dissociation mediated by $\Delta$ resonances.
In-Medium Effects: The model incorporates the Mott effect (dissolution of bound states in dense matter) via a phase-space excluded-volume approach. Cluster formation is permitted only when the average phase-space occupation of surrounding nucleons is below a specific cutoff, determined via Bayesian inference on light-nucleus yields.
Simulation Setup: The study covers Au+Au collisions at beam energies ( $E_{beam}$ ) ranging from 120 to 1500 A MeV. Two sets of calculations are performed: one including explicit light-cluster degrees of freedom ("with LN") and one where these degrees of freedom and related channels are switched off ("w/o LN").
Observables: The authors analyze directed ( $v_1$ ), elliptic ( $v_2$ ), triangular ( $v_3$ ), and quadrangular ( $v_4$ ) flows for protons and light nuclei in mid-central collisions ( $0.25 < b_0 < 0.45$ ). They also examine nucleon-number scaling of elliptic flow ( $v_2/A$ ).

Key Contributions and Results

Benchmarking Yields: The kinetic approach provides a reasonable description of light-nucleus yields across the 120–1500 A MeV range. However, the model systematically overestimates yields at the lowest energies ( $E_{beam} = 120$ and $150$ A MeV), particularly for tritons and $\alpha$ particles. The authors attribute this to the absence of heavier fragments, residual nuclei, and inter-cluster correlations in the current model, as well as uncertainties in in-medium suppression parameters. Agreement improves significantly at $E_{beam} \ge 400$ A MeV.
Impact on Proton Flows: The inclusion of dynamical light-cluster formation significantly modifies proton collective flows, with a strong beam-energy dependence:
- Low Energy (120–150 A MeV): The effect is most pronounced. The "with LN" calculation shows a significant enhancement in the magnitude of $v_1$ and drives $v_2$ to more negative values (enhancing the squeeze-out component) compared to the "w/o LN" case. This indicates a strong redistribution of baryon number between free nucleons and bound clusters.
- Intermediate Energy (250–400 A MeV): The cluster effect remains visible but weaker, representing a transition region.
- High Energy ( $\gtrsim 600$ A MeV): The difference between calculations with and without light clusters gradually weakens and becomes negligible above 800 A MeV. This trend correlates with the decreasing abundance and survival probability of light clusters at higher energies.
Light Nucleus Flows: The kinetic approach captures the overall beam-energy dependence of light-nucleus flow data. However, at low energies ( $120–250$ A MeV), the model generally overpredicts the flow magnitudes of light nuclei. At $E_{beam} \ge 400$ A MeV, the description becomes more satisfactory, though discrepancies remain at specific kinematic regions (e.g., low $u_{t0}$ for $v_1$ and large $y_0$ /high $u_{t0}$ for $v_2$ ).
Nucleon-Number Scaling: The study examines the scaling of elliptic flow ( $v_2/A$ $v_{2} / A$ ) with nucleon number.
- At low energies ( $120–400$ A MeV), clear scaling violations are observed; light nuclei do not follow a simple coalescence-like scaling with protons, especially at large scaled transverse velocities ( $u_{t0}$ ).
- At higher energies ( $600–1500$ A MeV), the scaling behavior improves, with proton, deuteron, and triton flows converging at low and intermediate $u_{t0}$ . However, deviations persist at large $u_{t0}$ . The authors suggest this is due to high-momentum clusters being produced earlier, experiencing weaker Mott suppression, and escaping the medium before undergoing significant rescattering, thus retaining flow information from an earlier dynamical stage.

Significance
The paper concludes that a dynamical treatment of light-cluster formation is crucial for the consistent interpretation of collective-flow observables in intermediate-energy heavy-ion collisions, particularly below 600 A MeV where cluster production is abundant. The study demonstrates that neglecting explicit cluster degrees of freedom leads to significant misinterpretations of proton flow data in this energy regime. While the effects on proton flows diminish at higher energies ( $E_{beam} \gtrsim 600$ A MeV), the analysis of light-nucleus flows and nucleon-number scaling violations provides essential constraints on the formation mechanisms, in-medium propagation, and interaction dynamics of light clusters. The results highlight that simple kinematic coalescence pictures are insufficient to describe the full differential behavior of collective flows in this energy range.

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