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The Interplay Between Electromagnetic Fields and Baryon Stopping in a Hydrodynamic Model for Charged Flow

This paper presents a semi-analytic hydrodynamic model demonstrating that the observed centrality-dependent sign change in charge-dependent directed flow splitting in relativistic heavy-ion collisions arises from the competition between positive contributions from baryon stopping and negative contributions from spectator-induced electromagnetic fields.

Original authors: Tuna Demircik, Dmitri E. Kharzeev, Krishna Rajagopal, Raimond Snellings

Published 2026-09-16

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CC BY 4.0

Original authors: Tuna Demircik, Dmitri E. Kharzeev, Krishna Rajagopal, Raimond Snellings

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

When two heavy atomic nuclei smash into each other at nearly the speed of light, they create a fleeting, super-hot droplet of matter so dense that protons and neutrons melt into a soup of their constituent parts. This state of matter, known as a quark-gluon plasma, exists for only a fraction of a second before cooling down and reforming into ordinary particles. In these collisions, the electrically charged protons that miss the direct hit race past the collision zone, creating an incredibly powerful magnetic field that is trillions of times stronger than anything found on Earth. Scientists have long suspected that this intense magnetic field should push positively and negatively charged particles in opposite directions, creating a subtle imbalance in how they fly out of the collision. However, recent experiments have shown a puzzling pattern: the direction of this imbalance flips depending on how head-on the collision is. In the most central crashes, the imbalance points one way, but in more glancing blows, it points the other. This flip suggests that the magnetic field is not the only force at play; something else must be pushing back.

A team of researchers has now built a new theoretical model to explain this flip, showing that it arises from a tug-of-war between the magnetic field and the stopping power of the collision itself. In a head-on collision, the nuclei stop dead in their tracks, depositing a cloud of stopped protons into the center of the fireball. Because of the geometry of the crash, these stopped protons are not distributed evenly; they pile up slightly more on one side of the collision zone than the other. As the fireball expands outward, this uneven pile of protons gets pushed sideways, creating a flow that favors one direction. This effect, driven by the stopping of the protons, pushes the imbalance in one direction. Meanwhile, the magnetic field generated by the passing spectators tries to push the charged particles in the opposite direction. The researchers found that in central collisions, the stopping effect is stronger, while in peripheral collisions, where the magnetic field is more intense, the magnetic push wins out. The result is a natural explanation for why the direction of the flow changes as the collision becomes more glancing.

To test this idea, the researchers developed a semi-analytic framework that combines the physics of the expanding fireball with the effects of the magnetic field and the stopped protons. They used a simplified mathematical description of the fireball's expansion, which allowed them to calculate the forces acting on the particles without needing a supercomputer to simulate every single interaction. They modeled the stopped protons using a standard method for predicting how nuclear matter overlaps during a collision, and they calculated the magnetic field generated by the speeding spectators. By running these calculations for different types of collisions, from very central to very peripheral, they could see how the two competing forces changed relative to each other. The model successfully reproduced the experimental data from the STAR experiment at the Relativistic Heavy Ion Collider, which had previously observed this mysterious sign change in the flow of protons and antiprotons.

The study reveals that the magnetic field alone cannot explain the observations. If only the magnetic field were acting, the flow imbalance would always point in the same direction, regardless of how central the collision was. The researchers showed that the stopping of the protons provides a crucial counter-force that is dominant in central collisions but fades away in peripheral ones. Conversely, the magnetic force is weak in central collisions but becomes the dominant player in peripheral ones. This competition creates a smooth transition where the net effect flips from positive to negative as the collision moves from central to peripheral. The model also predicted that this flip should happen at a specific centrality, matching the point where the experimental data showed the change. Furthermore, the researchers found that the two effects leave different fingerprints on the particles depending on how fast they are moving away from the center of the collision. The stopping effect tends to dominate at higher speeds, while the magnetic effect is strongest near the center, offering a way for future experiments to separate the two influences by looking at particles moving at different velocities.

While the model provides a clear and transparent explanation for the observed phenomenon, the authors are careful to note its limitations. They used a simplified, symmetric description of the fireball that does not capture all the complex details of a real, asymmetric collision. Additionally, they assumed that the electrical conductivity of the plasma remains constant, whereas in reality, it likely changes as the plasma cools. Because of these simplifications, the model is not intended to provide precise numerical predictions for experimental data but rather to offer a qualitative understanding of the underlying physics. It serves as a proof of concept that shows how the interplay between baryon stopping and electromagnetic fields can naturally produce the complex patterns seen in nature. The work suggests that future, more detailed simulations will be needed to make exact predictions, but the current study has successfully identified the two main actors in this cosmic drama and explained how their competition shapes the final outcome. By isolating these two mechanisms, the researchers have provided a solid foundation for understanding how the early universe, recreated in these tiny collisions, responds to the extreme forces of electromagnetism and the inertia of matter.

Technical Summary: The Interplay Between Electromagnetic Fields and Baryon Stopping in a Hydrodynamic Model for Charged Flow

Problem Statement
Charge-dependent directed flow ( $v_1$ ) in relativistic heavy-ion collisions serves as a sensitive probe for early-time electromagnetic (EM) fields and baryon stopping. Recent STAR measurements at $\sqrt{s_{NN}} = 200$ GeV reveal a centrality-dependent sign change in the directed flow splitting ( $\Delta v_1 \equiv v_1(p) - v_1(\bar{p})$ ). While theoretical models based solely on spectator-induced electromagnetic fields predict a negative slope for $\Delta v_1$ across all centralities, experimental data show a positive slope in central/mid-central collisions and a negative slope in peripheral collisions. This discrepancy suggests that electromagnetic effects alone are insufficient; the contribution from baryon stopping—specifically the component of the stopped proton distribution that is odd in rapidity and odd under reflection in the impact parameter direction—must be included to explain the observed phenomenology.

Methodology
The authors develop a semi-analytic hydrodynamic framework to unify these two competing mechanisms within a single model. The approach relies on the following components:

Hydrodynamic Background: The model utilizes the analytic Gubser solution for relativistic viscous hydrodynamics of a conformal fluid. While the Gubser solution is azimuthally symmetric and lacks explicit impact parameter dependence, the authors mimic centrality dependence by varying the initial temperature scale ( $\hat{T}_0$ ) and effective transverse size of the fireball for different centrality bins (5% to 70%).
Baryon Stopping Implementation: Stopped baryons are modeled using a Glauber-based participant density profile. The spatial distribution of stopped baryons is constructed to be odd in rapidity ( $\eta$ ) and odd under reflection in the impact parameter axis ( $x$ ). This distribution generates a spacetime-dependent baryon chemical potential ( $\mu_B$ ) on the freezeout hypersurface. In the Cooper-Frye freezeout prescription, $\mu_B$ acts as a fugacity factor, weighting the emission of protons versus antiprotons. The radial expansion of the fluid converts this rapidity-odd and reflection-odd density asymmetry into a directed flow signal.
Electromagnetic Field Construction: The EM fields are generated by charged spectator protons. The model solves Maxwell's equations in a conducting medium with a constant electrical conductivity ( $\sigma$ ). The relevant fields are the spectator-induced magnetic field ( $B_y$ ) and the resulting electric fields ( $E_x$ ) via Faraday and Coulomb mechanisms, as well as the Lorentz force acting on the expanding plasma.
Flow Calculation: The charge-dependent directed flow is computed by treating the EM-induced drift as a perturbation to the background hydrodynamic flow. A force-balance equation is solved in the local fluid rest frame to determine the drift velocity of charged constituents, which is then boosted back to the lab frame and inserted into the Cooper-Frye distribution.

Key Contributions

Unified Framework: The paper provides the first semi-analytic model that consistently incorporates both spectator-induced electromagnetic fields and baryon stopping to calculate charge-dependent directed flow.
Mechanism Isolation: By separating the contributions, the model demonstrates that baryon stopping yields a positive contribution to the mid-rapidity slope of $\Delta v_1$ , while electromagnetic fields yield a negative contribution.
Rapidity Dependence: The model distinguishes the rapidity profiles of the two effects: the baryon stopping contribution extends to larger rapidities ( $|Y| \gtrsim 2.6$ ), whereas the electromagnetic contribution peaks closer to mid-rapidity ( $|Y| \approx 1.8$ ) and decreases at higher rapidities due to cancellations between Faraday/Coulomb and Lorentz terms.

Results
For Au+Au collisions at $\sqrt{s_{NN}} = 200$ GeV, the model reproduces the qualitative trends observed in STAR data:

Centrality Dependence: In central collisions (e.g., 30%), the positive contribution from baryon stopping dominates, resulting in a positive slope for $d\Delta v_1/dY$ . In peripheral collisions (e.g., 70%), the negative contribution from electromagnetic fields dominates, resulting in a negative slope. In the mid-central region (around 50%), the two effects nearly cancel, reproducing the observed sign change.
U+U Collisions: The model predicts the centrality dependence for U+U collisions at $\sqrt{s_{NN}} = 193$ GeV. Due to the larger charge and atomic number of Uranium, the electromagnetic contribution is enhanced relative to Au+Au, shifting the sign-change point slightly toward more peripheral collisions. This trend aligns with preliminary STAR data.
Rapidity Profiles: The combined model successfully describes the rapidity dependence of $\Delta v_1$ in the 50–80% centrality interval, where electromagnetic effects dominate.

Significance and Limitations
The authors claim that their model provides a "transparent explanation" for the apparent mismatch between previous electromagnetic-only predictions and experimental measurements. It demonstrates that the competition between baryon stopping and electromagnetic forces naturally accounts for the centrality-dependent sign change in the directed flow splitting.

However, the paper is modest regarding quantitative precision. The authors explicitly state that the model is not intended as a quantitative prediction for experimental data due to several simplifying assumptions:

The use of the azimuthally symmetric Gubser solution cannot fully capture the geometry of non-central collisions.
The assumption of constant electrical conductivity ignores the expected temperature dependence.
The model does not include event-by-event fluctuations, which are significant in very central and very peripheral collisions.

Consequently, the work is presented as a qualitative tool to understand the interplay of these mechanisms. The authors note that future quantitative predictions will require state-of-the-art viscous hydrodynamic simulations with realistic geometries and temperature-dependent conductivity.

Technical Summary: The Interplay Between Electromagnetic Fields and Baryon Stopping in a Hydrodynamic Model for Charged Flow

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