⚛️ nuclear theory

First flow harmonic of net baryon from directed diffusion of stopped baryons

The authors propose a new baryon deposition model incorporating baryon junctions and directed diffusion within a relativistic hydrodynamic framework, which successfully reproduces the observed double sign change in the directed flow slopes of net protons and net lambdas in Au+Au collisions at 7.7–200 GeV, thereby establishing a crucial noncritical baseline for the search for the QCD critical point.

Original authors: Tribhuban Parida, Sandeep Chatterjee

Published 2026-09-11

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CC BY 4.0

Original authors: Tribhuban Parida, Sandeep Chatterjee

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

In the subatomic world, the heaviest elements are not found in the periodic table but are forged in the fleeting, violent collisions of atomic nuclei. When physicists smash heavy atoms together at nearly the speed of light, they create a state of matter so hot and dense that the protons and neutrons inside them melt into a soup of their constituent parts, known as quarks and gluons. This primordial fluid, called the quark-gluon plasma, is believed to have filled the universe microseconds after the Big Bang. To understand how this fluid behaves, scientists study how it flows. Just as wind patterns reveal the shape of a landscape, the way particles stream out of these collisions reveals the hidden forces and initial conditions of the explosion. One specific pattern of this flow, known as directed flow, acts like a subtle tilt in the expanding fireball, pushing particles slightly to one side rather than straight out. This tilt is a sensitive probe of how the original atomic nuclei stop and deposit their matter during the collision, offering a window into the fundamental rules that govern the strong force holding matter together.

Researchers at the Indian Institute of Science Education and Research Berhampur and the AGH University of Krakow have proposed a new way to model this initial deposition of matter, specifically focusing on how protons and other baryons (particles made of three quarks) come to a halt and spread out. In their study, they simulated collisions of gold nuclei at various energies, ranging from 7.7 to 200 billion electron volts, a range explored by the Relativistic Heavy Ion Collider. Previous models often assumed that the matter left behind by the colliding nuclei was distributed simply in proportion to the number of nucleons that directly hit each other. However, the authors found that this simple picture failed to explain a puzzling feature observed in experiments: a double reversal in the direction of flow for certain particles as the collision energy changed. To fix this, they introduced a more complex picture based on the "baryon junction," a theoretical concept where the quantum numbers of protons are carried by specific connections between quarks.

In this new framework, the researchers suggested that the stopped matter comes from two distinct sources. The first source comes from the nucleons that directly participate in the collision, while the second comes from the violent interactions between pairs of nucleons, known as binary collisions. By blending these two sources, the model creates a tilted distribution of matter that is not perfectly aligned with the energy of the collision. This misalignment, or tilt, is crucial. It creates a pressure difference that pushes the fluid in a specific direction. Furthermore, the team included the effect of diffusion, a process where particles spread out from areas of high concentration to low concentration, driven by gradients in the chemical potential of the baryons. In their simulations, this diffusion acts as a counter-force to the general flow of the fluid, significantly altering the final direction in which protons and anti-protons move.

The results of these simulations were striking. When the researchers compared their model to real data collected by the STAR Collaboration at the Relativistic Heavy Ion Collider, they found that their approach successfully reproduced the complex behavior of the directed flow. Most notably, the model captured a "double sign change" in the flow slope for net protons and net lambda particles (a type of heavier baryon) as the collision energy varied between 7.7 and 39 billion electron volts. This double reversal had been a major challenge for previous theories to explain. The study showed that this phenomenon arises naturally from the competition between the fluid being pushed by pressure gradients and the baryons diffusing through the medium. The model achieved this without needing to invoke a first-order phase transition, a specific type of dramatic change in the state of matter that some theories had proposed as the cause for the sign change. Instead, the simulations used a smooth crossover equation of state, suggesting that the observed flow patterns are driven by the transport properties of the baryons themselves.

By successfully describing the flow of both protons and anti-protons, as well as strange particles like lambda baryons, the study provides a robust baseline for understanding the early stages of heavy-ion collisions. The authors demonstrated that the interplay between the initial tilt of the matter distribution and the subsequent diffusion of baryons is sufficient to explain the intricate flow patterns seen in nature. This work does not just fit the data; it offers a coherent physical mechanism that links the microscopic stopping of baryons to the macroscopic flow of the fireball. The findings suggest that the complex dance of particles in these collisions is governed by the subtle balance of advection and diffusion, providing a clearer path for future searches for the critical point of quantum chromodynamics and the effects of strong electromagnetic fields in these extreme environments.

Technical Summary: First Flow Harmonic of Net Baryon from Directed Diffusion of Stopped Baryons

Problem Statement
The directed flow ( $v_1$ ) of identified hadrons in relativistic heavy-ion collisions serves as a sensitive probe of initial longitudinal matter deposition and the QCD equation of state (EoS). While hydrodynamic models have successfully described the $v_1$ of charged hadrons, capturing the $v_1$ splitting between particle-antiparticle pairs (specifically baryons and anti-baryons) remains a significant challenge. Previous approaches, often relying on participant scaling or simple parametric models for baryon deposition, fail to simultaneously describe the $v_1$ of both baryons and anti-baryons across a wide range of collision energies ( $\sqrt{s_{NN}} = 7.7 - 200$ GeV). Furthermore, existing models struggle to reproduce the observed "double sign change" in the rapidity slopes of the net-proton and net- $\Lambda$ directed flow between $\sqrt{s_{NN}} = 7.7$ and $39$ GeV. A first-principles understanding of the baryon stopping mechanism and the resulting pre-equilibrium dynamics in the presence of finite baryon density is currently lacking.

Methodology
The authors propose a new ansatz for baryon deposition that integrates features from the baryon junction picture into a relativistic hydrodynamic framework. The core of the methodology involves:

Two-Component Baryon Deposition: The initial net-baryon density profile ( $n_B$ ) is constructed as a superposition of two sources:
- Participant Sources: Associated with single baryon junctions, proportional to the participant nucleon distribution ( $N_+$ and $N_-$ for forward/backward moving sources).
- Binary Collision Sources: Associated with double baryon junctions (one from the projectile, one from the target), proportional to the binary collision distribution ( $N_{bin}$ ).
- The total profile is governed by a tilt parameter $\omega$ , which controls the relative contribution of these two components.
Tilted Initial Conditions (TIC): The baryon density profile is coupled to a tilted initial energy density profile (controlled by parameter $\eta_m$ ). The relative tilt between the energy density and net-baryon density profiles is a critical input for generating asymmetric flow.
Hydrodynamic Evolution with Diffusion: The system evolves using the MUSIC code, incorporating dissipative hydrodynamics with baryon diffusion. The model employs the NEoS-BQS equation of state (based on lattice QCD crossover) and enforces local strangeness neutrality ( $n_S=0$ ) and a fixed baryon-to-charge ratio ( $n_Q = 0.4 n_B$ ).
- Crucially, the model includes a temperature and baryon chemical potential-dependent diffusion coefficient.
- The evolution accounts for both baryon advection (transport with fluid velocity) and baryon diffusion (driven by gradients in $\mu_B/T$ ).
Hadronic Afterburner: The hydrodynamic phase is followed by particlization (iSS code) and a hadronic transport stage (UrQMD).
Parameter Tuning: Model parameters, including tilt parameters, were tuned independently at each collision energy to reproduce experimental data for charged particle yields, net-proton yields, and the directed flow $v_1(y)$ of $\pi^+$ , protons, and anti-protons.

Key Results

Baryon-Antibaryon Splitting: The model successfully reproduces the splitting in $v_1$ between protons and anti-protons, as well as $\Lambda$ and $\bar{\Lambda}$ . The analysis reveals that while meson and anti-baryon flows are dominated by the pressure gradient ( $\langle -\partial_x P \rangle_\epsilon$ ), the baryon directed flow is significantly influenced by the gradient of the baryon chemical potential over temperature ( $\langle -\partial_x (\mu_B/T) \rangle_{n_B}$ ).
Role of Diffusion: The inclusion of baryon diffusion is shown to be essential. In the presence of diffusion, the flow of protons follows the $\mu_B/T$ gradient, leading to a change in the sign of the rapidity slope compared to the advection-dominated case.
Double Sign Change: The model successfully describes the experimentally observed double sign change in the mid-rapidity slope ( $dv_1/dy$ ) of both net-protons and net- $\Lambda$ baryons within the energy range $\sqrt{s_{NN}} = 7.7 - 39$ GeV. This is achieved by simultaneously capturing the yields and $v_1$ of both baryons and anti-baryons, as the net-baryon observable is a weighted difference of the two.
Equation of State Implications: The model achieves these results using a crossover EoS. This suggests that the sign change in $v_1$ slopes, previously hypothesized as a signature of a first-order phase transition, can instead be attributed to the significant influence of baryon diffusion at low collision energies.

Significance and Claims
The paper claims to provide a "noncritical baryonic baseline" that is crucial for ongoing searches for the QCD critical point and initial strong electromagnetic fields. By successfully modeling the directed flow of identified hadrons without invoking a first-order phase transition, the authors argue that their framework offers a robust reference for interpreting future experimental results.

Specifically, the work demonstrates that:

A two-component baryon stopping scheme (participant + binary collision) based on the baryon junction picture is necessary to describe initial conditions.
The interplay between baryon advection and diffusion, driven by the relative tilt of energy and baryon density profiles, is the dominant mechanism for generating the observed $v_1$ patterns.
The observed complex structures in net-baryon flow (such as the double sign change) can be explained by transport phenomena (diffusion) within a crossover EoS, rather than requiring exotic phase transition signatures.

The authors conclude that this model provides a valuable framework for constraining fundamental transport coefficients, such as the baryon diffusion coefficient, and serves as a necessary baseline for distinguishing critical point signals from standard hydrodynamic evolution in the RHIC Beam Energy Scan.

Technical Summary: First Flow Harmonic of Net Baryon from Directed Diffusion of Stopped Baryons

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