Charge-Dependent Directed Flow in Symmetric Nuclear Collisions

This study utilizes the string-melting AMPT model to demonstrate that in symmetric nuclear collisions at 200 GeV, charge-dependent directed flow splitting exhibits a distinct baryon-meson dichotomy driven by transported quarks during the partonic phase, with significant system-size dependence at high transverse momentum that establishes a baseline for interpreting electromagnetic field effects.

The Gist

Imagine a massive, high-speed collision between two heavy nuclei (like gold or uranium atoms) as a chaotic, high-energy dance floor. When these nuclei smash into each other at nearly the speed of light, they create a tiny, super-hot soup of fundamental particles called quarks and gluons. This soup is so hot that protons and neutrons melt apart, existing only as a free-flowing fluid for a split second before cooling down and reforming into new particles.

Physicists want to know: How does this "soup" move? specifically, does it flow more to the left or the right?

This paper investigates a specific type of movement called "Directed Flow" ($v_1$). Think of it like a crowd of people at a concert. If the stage is tilted, the crowd might naturally drift to one side. In nuclear collisions, the "tilt" comes from the shape of the collision and the magnetic forces involved.

Here is the breakdown of what the researchers found, using simple analogies:

1. The Players: Particles vs. Antiparticles

The researchers looked at different "dancers" in this soup:

• Mesons (Pions and Kaons): These are like lightweight, nimble dancers.
• Baryons (Protons and Lambda particles): These are the heavier, more robust dancers.
• Antiparticles: These are the "mirror images" of the dancers (like a left-handed version of a right-handed person).

The study compared how the "regular" dancers moved compared to their "mirror" partners.

2. The Main Discovery: The "Baryon-Meson Dichotomy"

The most striking finding is a split personality in how these particles behave:

• The Mesons (Lightweights): When you compare a positive pion ($\pi^+$) to a negative pion ($\pi^-$), they move almost exactly the same way. They are like two identical twins walking side-by-side; there is almost no difference in their direction.
• The Baryons (Heavyweights): When you compare a proton ($p$) to an anti-proton ($\bar{p}$), they move in very different directions. The difference is huge and gets even bigger as the collision gets larger (like going from a small room to a giant stadium).

The Analogy: Imagine a river.

• The Mesons are like leaves floating on the surface; the current pushes them all the same way, regardless of their color.
• The Baryons are like swimmers. The "regular" swimmers (protons) are being pushed by a strong current from the starting point (the initial collision), while the "mirror" swimmers (anti-protons) are being pushed the opposite way. The bigger the river (the larger the nucleus), the stronger this separation becomes.

3. Why Does This Happen? (The "Transported" vs. "New" Quarks)

The paper explains this using the concept of Quark Coalescence (particles sticking together to form new ones).

• The "Tourists" (Transported Quarks): Some quarks come from the original nuclei that smashed together. They carry a "memory" of where they started. These are mostly found in the heavy baryons (protons). Because they started at the edge and moved inward, they have a strong directional bias.
• The "Locals" (Produced Quarks): Other quarks are created fresh from the energy of the collision. They are born in the middle and don't have that same directional memory. These dominate the lighter mesons.

The Result: Since protons are made of "Tourist" quarks, they feel a strong push in one direction. Anti-protons, made of "Anti-Tourist" quarks, feel a push in the opposite direction. This creates a massive split in their movement.

4. The "Soft" vs. "Hard" Speeds

The researchers looked at slow particles (low momentum) and fast particles (high momentum).

• Slow particles generally flow with the bulk of the soup.
• Fast particles (the "hard" ones) act differently. In large collisions, they actually flow in the opposite direction of the slow ones.
• The Analogy: Imagine a crowded hallway. The slow walkers (low energy) are swept along by the crowd's flow. But the sprinters (high energy) are so fast and aggressive that they push against the crowd, moving in the opposite direction. This "Hard-Soft Asymmetry" was clearly seen in the data.

5. The Magnetic Field Mystery

There is a famous theory that the collision creates a massive magnetic field (like a giant magnet) that should push positive and negative particles in opposite directions.

• The Twist: The computer model used in this paper (AMPT) did not include this magnetic field. It only included the movement of quarks.
• The Finding: Even without the magnetic field, the model still predicted a huge split between protons and anti-protons.
• Conclusion: This means the "split" we see in experiments isn't just because of the magnetic field. A huge part of it comes from the baryon transport (the "Tourist" quarks). The magnetic field is an extra layer on top of this, but the main driver is the movement of the heavy particles from the initial collision.

6. Why Small Systems Matter

The study looked at collisions ranging from tiny Oxygen nuclei to huge Uranium nuclei.

• In the tiny systems (Oxygen), the "soup" is so small and short-lived that it doesn't have enough time to develop these complex flow patterns. The fast and slow particles behave similarly.
• In the huge systems (Uranium), the soup is thick and long-lasting, allowing these complex directional flows to develop fully.

Summary for the General Audience

This paper is like a detective story solving a mystery about how matter moves after a nuclear crash.

1. The Mystery: Why do protons and anti-protons move in opposite directions, while pions and anti-pions move together?
2. The Clue: The heavy particles (protons) carry "tourist" quarks from the original crash, while light particles (pions) are made of "local" quarks created in the crash.
3. The Solution: The "Tourist" quarks create a strong directional flow that separates protons from anti-protons. This effect gets stronger in bigger collisions.
4. The Takeaway: Scientists previously thought the magnetic field was the main reason for this separation. This study shows that the movement of the quarks themselves is actually the biggest factor. The magnetic field is just the cherry on top.

This helps physicists understand the very early moments of the universe, right after the Big Bang, when matter was in this hot, fluid state.

Technical Summary

1. Problem Statement and Motivation

The directed flow ($v_1$) of particles in relativistic heavy-ion collisions is a critical observable for understanding the early-time dynamics of the Quark-Gluon Plasma (QGP), including pressure gradients and electromagnetic (EM) fields generated by spectator protons. Recent experimental data from the STAR collaboration at RHIC (Au+Au collisions at $\sqrt{s_{NN}} = 200$ GeV) revealed a charge-dependent splitting in the mid-rapidity slope of directed flow ($\Delta dv_1/dy$) between particles and anti-particles.

• The Puzzle: A sign change in $\Delta dv_1/dy$ for protons vs. anti-protons was observed from central to peripheral collisions, attributed to strong EM fields. However, the baseline contribution from baryon transport (the movement of initial-state quarks vs. newly produced quarks) remains to be disentangled from EM effects.
• The Gap: There is a lack of systematic theoretical understanding regarding how this charge-dependent splitting evolves with system size (from small O+O to large U+U) and transverse momentum ($p_T$), specifically distinguishing between mesons and baryons.

2. Methodology

The authors employed the AMPT (A Multi-Phase Transport) model with String Melting (SM) and an improved quark coalescence mechanism.

• Collision Systems: Simulations were performed for symmetric nuclear collisions: O+O, Cu+Cu, Ru+Ru, Au+Au, and U+U.
• Energy: Center-of-mass energy $\sqrt{s_{NN}} = 200$ GeV.
• Key Model Features:
  • Initial Conditions: Derived from the HIJING model using Wood-Saxon distributions, including nuclear deformation parameters ($\beta_2$) for Ru and U.
  • Partonic Phase: Uses Zhang's parton cascade with a cross-section of 1.5 mb.
  • Hadronization: Uses a quark coalescence model (improved version) rather than standard string fragmentation.
  • Hadronic Phase: Two scenarios were tested:
    1. Default: $t_{max} = 0.4$ fm/c (effectively disabling final-state hadronic rescattering).
    2. Full Interaction: $t_{max} = 30$ fm/c (enabling full hadronic cascade via the ART model).
• Observables:
  • Directed flow $v_1(y)$ fitted to a linear function $v_1(y) = Fy + C$ to extract the slope $F \equiv dv_1/dy|_{y=0}$.
  • Charge-dependent splitting: $\Delta F = F_{particle} - F_{antiparticle}$.
  • Analysis performed for identified hadrons ($\pi^\pm, K^\pm, p, \bar{p}, \phi, \Lambda, \bar{\Lambda}$) in low-$p_T$ (0.2–2.0 GeV/c) and high-$p_T$ (2.0–5.0 GeV/c) regions.
  • Geometric scaling tested by dividing $\Delta F$ by $A^{1/3}$.

3. Key Results

A. System Size Dependence of $v_1$ Slope ($F$)

• Low-$p_T$: The slope $F$ shows weak dependence on the nuclear mass number ($A$). Mesons and baryons (except anti-baryons) exhibit positive slopes, indicating flow aligned with the bulk medium. Anti-baryons ($\bar{p}, \bar{\Lambda}$) show negative slopes.
• High-$p_T$: A strong system-size dependence is observed. As $A$ increases, the slope becomes increasingly negative.
  • Interpretation: This reflects a hard-soft asymmetry. In large systems, high-$p_T$ particles flow opposite to the bulk medium, whereas in small systems (O+O), the medium is not opaque enough to reverse the flow direction of high-$p_T$ particles.

B. Charge-Dependent Splitting ($\Delta F$)

The study reveals a striking baryon-meson dichotomy:

• Mesons ($\pi^+ - \pi^-$, $K^+ - K^-$): Exhibit minimal splitting ($\Delta F \approx 0$) that remains nearly constant across all system sizes and centralities. This suggests mesons are primarily formed from produced quarks with symmetric transport.
• Baryons ($p - \bar{p}$, $\Lambda - \bar{\Lambda}$): Exhibit significant splitting that grows with system size and saturates for large nuclei (Au, U).
  • Mechanism: This arises from the transport of initial-state quarks (carrying net baryon number) which coalesce into baryons but not anti-baryons.
  • $\Lambda - \bar{\Lambda}$ Significance: Since $\Lambda$ and $\bar{\Lambda}$ are electrically neutral, their splitting in the AMPT model (which lacks EM fields) is solely due to baryon number transport. This provides a unique baseline to isolate EM effects in future experiments.

C. Centrality Dependence

• Baryons: $\Delta F$ increases from central to mid-central collisions and decreases toward peripheral collisions.
• Mesons: $\Delta F$ remains nearly constant.
• Scaling: When scaled by $A^{1/3}$, the splitting for all species shows similar centrality profiles, suggesting the mechanism is governed by geometric factors in the absence of EM fields.

D. Comparison with Experiment (STAR)

• The AMPT-SM model qualitatively reproduces the hierarchy (large baryon splitting, small meson splitting) but underestimates the magnitude of $\Delta F$ for protons in peripheral Au+Au collisions.
• Conclusion: The AMPT results establish a baseline contribution from transported quark dynamics. The discrepancy with experimental data in peripheral collisions is attributed to the electromagnetic field, which is not included in the AMPT model but is known to be strong in peripheral collisions.

E. Hadronic Interaction Effects

• Comparing $t_{max} = 0.4$ fm/c and $t_{max} = 30$ fm/c showed negligible differences in the $v_1$ slope.
• Implication: The directed flow slope and its charge-dependent splitting are established during the partonic phase and coalescence process, not modified significantly by late-stage hadronic rescattering.

4. Significance and Contributions

1. Disentangling Mechanisms: The paper successfully isolates the contribution of baryon transport from electromagnetic effects. The $\Lambda - \bar{\Lambda}$ splitting is proposed as a critical observable for future experiments to measure the "pure" transport contribution without EM contamination.
2. System Size Evolution: It provides the first comprehensive theoretical prediction for charge-dependent directed flow across a wide range of system sizes (O to U), highlighting the transition from soft-dominated dynamics in small systems to hard-soft asymmetry in large systems.
3. Baseline for EM Studies: By establishing the AMPT-SM results as a baseline, the authors provide a necessary reference for quantifying the strength of the electromagnetic field in heavy-ion collisions when compared to future experimental data.
4. Small System Predictions: The predictions for O+O and Cu+Cu collisions serve as benchmarks for upcoming experimental campaigns (e.g., at RHIC and LHC) to test the limits of QGP formation and collective flow.

Conclusion

The study confirms that charge-dependent directed flow splitting is primarily driven by the transport of initial-state quarks during the partonic phase, manifesting strongly in baryons but not mesons. While the AMPT model captures the geometric and transport baseline, the full experimental magnitude in peripheral collisions requires the superposition of strong electromagnetic field effects. This work lays the groundwork for precise measurements of EM fields and baryon transport in future heavy-ion collision experiments.