Predictions of baryon directed flow in heavy-ion collisions at high baryon density

Using a three-fluid dynamics model with a crossover equation of state, this paper predicts a non-monotonic evolution of proton directed flow in Au+Au collisions between 4.5 and 7.7 GeV, characterized by a sign change and minimum at 7.2 GeV that signals the onset of the transition to quark-gluon plasma.

The Gist

Imagine you are trying to figure out what happens inside a car crash, but instead of metal and glass, you are smashing two giant atomic nuclei (like gold atoms) together at nearly the speed of light. When they collide, they create a tiny, super-hot drop of "primordial soup" called Quark-Gluon Plasma (QGP). This is the state of matter that existed just microseconds after the Big Bang.

The big question scientists have is: How does this soup behave? Does it flow like water, or does it act like a thick, sticky gel? And most importantly, does it suddenly change its nature (a "phase transition") like ice melting into water, or does it melt gradually?

This paper is a prediction about how protons (the building blocks of atomic nuclei) behave when they are smashed together at specific energies. Here is the breakdown using simple analogies:

1. The Experiment: The "Crash Test"

Scientists are crashing gold atoms together at different speeds (energies). They are looking at a specific angle of the crash called "directed flow."

• The Analogy: Imagine two cars crashing head-on. Usually, the debris flies straight forward or backward. But if the crash is slightly off-center (a "semicentral" collision), the debris gets pushed sideways, like a splash of water hitting a wall at an angle.
• The Goal: By measuring exactly how much the protons are pushed sideways, scientists can deduce the "stiffness" of the matter inside the crash.

2. The Three Theories (The Equations of State)

The author used a computer model (called 3FD) to simulate these crashes using three different "rulebooks" for how the matter behaves:

1. The "Hard" Rulebook (Hadronic): The matter stays as normal nuclear stuff (like a stiff rubber ball).
2. The "Explosive" Rulebook (First-Order Phase Transition): The matter suddenly snaps from normal stuff to plasma, like water suddenly boiling into steam. This creates a "soft spot" where the matter gets squishy and slows down.
3. The "Smooth" Rulebook (Crossover): The matter melts gradually, like chocolate softening in your hand. There is no sudden snap, just a smooth transition.

3. The Prediction: The "Wiggle"

The paper predicts what happens to the sideways flow of protons as we increase the crash energy from 4.5 to 7.7 GeV (a specific high-energy range that hasn't been fully tested yet).

• The "Normal" Flow: At lower energies, the protons flow one way.
• The "Anti-Flow" (The Wiggle): The author predicts that at 7.2 GeV, something weird happens. The protons will briefly flow in the opposite direction (negative flow).
  • Why? If the matter hits that "soft spot" (the phase transition), it acts like a sponge. It absorbs the energy of the crash, expands slowly, and pushes the protons back the other way for a split second.
• The Return: By 7.7 GeV, the crash is so energetic that the matter zooms past the "soft spot" too quickly to get stuck. The flow returns to normal, which matches what the STAR experiment has already seen at 7.7 GeV.

4. The Big Reveal: Smooth vs. Sudden

The paper compares the "Explosive" (First-Order) and "Smooth" (Crossover) rulebooks.

• The "Explosive" Scenario: If the transition were sudden, the "wiggle" (the anti-flow) would be huge and dramatic.
• The "Smooth" Scenario: If the transition is gradual, the "wiggle" is still there, but it's a tiny, subtle dip.

The Conclusion:
The author predicts that the "wiggle" at 7.2 GeV will be small and subtle.

• What this means: The data suggests that the transition from normal matter to Quark-Gluon Plasma is not a sudden explosion, but a smooth crossover. It's more like melting butter than boiling water.

5. Why Protons?

You might ask, "Why look at protons and not other particles?"

• The Analogy: Imagine a crowded party. If you want to know how the room is moving, you should watch the people who stay in the room the whole time (the protons).
• Other particles (like pions or kaons) are like people who leave the party early or get pushed around by the crowd later on. They get confused by "after-party" effects (called the "afterburner"). Protons are the most reliable witnesses because they stick around and tell the true story of the crash.

Summary

This paper is a crystal ball for physicists. It says:

"If you look at proton crashes at 7.2 GeV, you will see a tiny, subtle 'backwards' flow. This proves that the matter inside isn't snapping suddenly from solid to plasma; it's melting smoothly. If the backwards flow were huge, it would mean a sudden snap. Since we expect a small wiggle, we are looking for a smooth transition."

This helps scientists understand the fundamental rules of the universe and how the matter inside neutron stars or the early universe behaves under extreme pressure.

Technical Summary

1. Problem Statement

The paper addresses the challenge of determining the Equation of State (EoS) of strongly interacting matter at high baryon densities, specifically investigating the nature of the transition to the Quark-Gluon Plasma (QGP).

• Context: Heavy-ion collisions at intermediate energies ($\sqrt{s_{NN}} \approx 3\text{--}12$ GeV) create high baryon density environments. This energy range is a primary focus of the Beam Energy Scan (BES) at RHIC and future facilities like NICA and FAIR.
• The "Directed Flow" Puzzle: The directed flow ($v_1$), specifically the slope of proton $v_1$ at midrapidity ($dv_1/dy|_{y=0}$), is a sensitive probe of the EoS. However, previous data showed conflicting interpretations:
  • Data at $\sqrt{s_{NN}} = 3$ GeV suggested a "stiff" EoS.
  • Data at 4.5 GeV suggested a "soft" EoS (indicative of a phase transition).
  • Data above 7.7 GeV showed a sign change in the slope, which some attributed to a first-order phase transition, while others argued it was a kinematic effect of baryon stopping.
• The Gap: There is a critical experimental gap between 4.5 GeV and 7.7 GeV where the onset of deconfinement is expected to produce the most dramatic signals. Existing data does not cover this region, leaving the nature of the phase transition (strong first-order vs. smooth crossover) uncertain.

2. Methodology

The author employs the Three-Fluid Dynamics (3FD) model to simulate the collision evolution.

• The 3FD Model: This model treats the collision system as three interacting fluids:
  1. Two counter-streaming baryon-rich fluids (projectile and target remnants).
  2. A third "fireball" fluid representing newly produced particles in the midrapidity region.
  • These fluids evolve via hydrodynamic equations with friction terms describing energy-momentum exchange.
• The THESEUS Generator: To account for non-equilibrium effects at freeze-out (specifically the "afterburner" stage where spectators shadow participant expansion), the 3FD simulation is followed by the THESEUS event generator, which incorporates the UrQMD (Ultrarelativistic Quantum Molecular Dynamics) model for final-state interactions.
• Equations of State (EoS): Three distinct EoS scenarios are tested to compare their impact on observables:
  1. Hadronic EoS: A purely hadronic scenario with incompressibility $K=190$ MeV.
  2. 1PT EoS: A scenario with a strong first-order phase transition to QGP, featuring a mixed phase and a "softest point" (minimum speed of sound).
  3. Crossover EoS: A scenario with a smooth crossover transition to QGP.
• Simulation Parameters: Calculations focus on semi-central Au+Au collisions ($b \approx 6$ fm) across energies $\sqrt{s_{NN}} = 3, 3.2, 3.5, 3.9, 4.5, 5.2, 6.2, 7.2, 7.7,$ and $11.5$ GeV.

3. Key Contributions

• Bridging the Energy Gap: The paper provides the first theoretical predictions for proton directed flow in the unexplored 4.5–7.7 GeV energy range.
• Differentiation of Phase Transitions: It quantitatively compares the signatures of a strong first-order transition versus a smooth crossover, demonstrating that while both produce non-monotonic behavior, the amplitude of the effect differs significantly.
• Validation of Proton $v_1$: It reinforces the argument that proton directed flow is the most robust observable for EoS studies because it is minimally affected by the afterburner stage and spectator shadowing compared to pions or kaons.
• Mechanism Clarification: The paper disentangles two distinct mechanisms causing sign changes in the flow slope:
  1. The onset of the phase transition (softening of EoS).
  2. The interplay between incomplete baryon stopping and transverse expansion (dominant at higher energies).

4. Key Results

• Low Energy Consistency (3–4.5 GeV): All three EoS scenarios (Hadronic, 1PT, Crossover) reproduce experimental data well at 3 GeV. At 3.5–3.9 GeV, the Crossover EoS shows a slight preference in the midrapidity region, though differences remain small.
• The "Wiggle" Prediction (4.5–7.7 GeV):
  • 1PT Scenario: Predicts a strong "flow-antiflow-flow" pattern. At 7.2 GeV, the system exhibits a strong antiflow (negative slope) due to the system getting trapped in the mixed-phase "softest point" for a significant duration.
  • Crossover Scenario: Also predicts a non-monotonic evolution with a dip at 7.2 GeV, but the amplitude of the antiflow is much smaller than in the 1PT case.
  • Hadronic Scenario: Fails to reproduce the data at 7.7 GeV, predicting a monotonic decrease without the observed dip/antiflow.
• Comparison with STAR Data (7.7 GeV):
  • The Crossover EoS predictions align well with STAR data at 7.7 GeV, where the flow returns to a normal pattern (positive slope).
  • The 1PT EoS predicts a strong antiflow at 7.7 GeV which is not supported by the data.
• The 7.2 GeV Prediction: The model predicts a weak antiflow (negative slope) at 7.2 GeV for the crossover scenario. This "wiggle" is the signature of the phase transition onset.
• High Energy Behavior (>9 GeV): Above 9 GeV, both 1PT and Crossover scenarios converge to similar flow patterns. The second sign change observed around 10 GeV is attributed not to the phase transition, but to the interplay between incomplete baryon stopping and transverse expansion.

5. Significance and Conclusion

• Nature of the Phase Transition: The available data, combined with these predictions, exclude a strong first-order phase transition in this energy range. Instead, the data favors a weak first-order or smooth crossover transition to the QGP.
• Experimental Verification: The paper identifies 7.2 GeV as a critical energy point. Experimental observation of a weak antiflow (negative slope) at this specific energy would confirm the onset of a crossover transition.
• Methodological Impact: The study validates the 3FD model with a crossover EoS as a reliable tool for describing bulk observables and directed flow across the entire BES energy range. It highlights that while the "softest point" effect is present in both scenarios, its magnitude is the key discriminator between a strong first-order transition and a crossover.

In summary, the paper argues that the proton directed flow excitation function acts as a "wiggle" detector: a large wiggle implies a strong first-order transition, while a small wiggle (as predicted by the crossover model and consistent with current data) implies a smooth transition to the QGP.