⚛️ phenomenology

Multi-Energy Analysis of Baryon Stopping: Extracting Non-Equilibrium Dynamics via Stochastic Rapidity Diffusion

This paper utilizes a stochastic diffusion framework to analyze net-proton spectra across a broad energy range, extracting key transport properties that reveal a dramatic transition from inertial, advection-dominated baryon stopping at lower energies to thermal, diffusion-dominated nuclear transparency at higher energies, thereby establishing a critical baseline for future high-density QCD research.

Original authors: Dhananjaya Thakur

Published 2026-10-22✓ Author reviewed ⓘ

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Dhananjaya Thakur

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 by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

In the most violent collisions imaginable, where atomic nuclei smash together at nearly the speed of light, a fundamental question has long puzzled physicists: what happens to the matter that makes up the protons and neutrons inside those nuclei? When two heavy nuclei collide, they do not simply pass through one another like ghosts, nor do they always stick together like clay. Instead, the protons and neutrons, which carry a specific type of charge known as baryon number, must slow down, stop, or pass through depending on how hard they hit. This process, known as baryon stopping, is a crucial clue to understanding the extreme states of matter that existed just moments after the Big Bang. By tracking how these particles lose their forward momentum and scatter, scientists can map the friction and resistance of the hot, dense soup created in the crash. This research is not just about the collision itself; it is about deciphering the rules of the strong force that holds the universe together, revealing whether the building blocks of matter behave like individual particles or something more complex and interconnected.

A researcher has now provided a detailed map of this stopping process, analyzing data from heavy-ion collisions across a wide range of energies, from relatively gentle impacts to some of the most energetic crashes ever recorded. By examining the sideways motion and the forward speed of protons and anti-protons produced in these collisions, the researcher constructed a comprehensive picture of how matter slows down. They treated the collision zone as a fluid that expands and cools, using the patterns of particle movement to calculate how much friction the protons experience and how much they scatter randomly. The result is a clear, data-driven story of how the universe transitions from a state where matter is completely stopped and piled up in the center, to a state where matter flows through almost unimpeded.

The researcher focused on six distinct collision energies, ranging from 2.0 A GeV up to 62.4 GeV. At the lowest energies, the collisions are like two trucks crashing head-on and coming to a dead stop; the protons lose almost all their forward momentum and pile up in the middle. As the energy increases, the behavior changes dramatically. The researcher found that at the highest energies, the protons behave more like light passing through glass; they barely slow down at all, and the center of the collision becomes nearly empty of the original matter. To measure this transition, the scientist calculated a specific ratio that compares the force of the forward motion against the tendency of the particles to scatter randomly. At the lowest energy, this ratio was high, indicating that the forward motion was dominant and the particles were being dragged to a halt. At the highest energy, this ratio dropped by a factor of roughly 163, signaling that the random scattering had taken over and the forward motion was no longer the main driver.

This massive shift in behavior points to a specific mechanism for how baryon number is transported. For decades, the standard view was that the protons and neutrons were made of smaller particles called valence quarks, and that these quarks were the ones carrying the matter through the collision. However, the new analysis suggests a different picture. The data aligns with a model where the baryon number is carried by a Y-shaped configuration of gluon fields, known as a baryon junction. Unlike the valence quarks, which might carry a significant portion of the forward momentum, this junction carries very little. Because it carries so little momentum, it is easily knocked off course by the dense environment of the collision, leading to the rapid stopping seen at low energies. At high energies, as the collision becomes more transparent, this junction decouples from the rest of the matter, allowing it to diffuse randomly rather than being dragged forward.

The study also measured how long it takes for the incoming protons to slow down, a timescale known as the relaxation time. At the lowest energy, this process happens incredibly fast, taking less than a trillionth of a trillionth of a second. As the energy increases, this time stretches out significantly, reaching nearly four times longer at the highest energy. This elongation confirms that the mechanism for stopping changes fundamentally as the energy rises. The researcher also calculated how much disorder, or entropy, is created during this process. At low energies, where the friction is high and the matter is forced to stop, a tremendous amount of entropy is generated. At high energies, where the matter flows through with little resistance, the creation of entropy drops sharply. This simultaneous drop in the stopping force and the entropy production provides a unified thermodynamic picture of the transition.

The findings offer a solid baseline for future experiments planned at major facilities around the world, including FAIR in Germany, NICA in Russia, and the CEE in China. These upcoming facilities will probe even denser regions of the matter phase diagram, where the behavior of baryon junctions is expected to be even more critical. The researcher notes that their analysis stops at 62.4 GeV because going higher would introduce mathematical instability without changing the physical conclusion; the transition to the transparent, diffusion-dominated state is already complete at this energy. By establishing this clear boundary between the world of stopped matter and the world of transparent matter, the study provides a reliable reference point for understanding the fundamental dynamics of nuclear collisions. The work does not just describe what happens; it quantifies the exact point where the rules of the game change, offering a new lens through which to view the most energetic events in the universe.

Technical Summary: Multi-Energy Analysis of Baryon Stopping via Stochastic Rapidity Diffusion

Problem Statement
Understanding the early-stage dynamics of ultrarelativistic heavy-ion collisions is essential for mapping the Quantum Chromodynamics (QCD) phase diagram. A critical observable in this regime is baryon stopping—the longitudinal deceleration of incident nucleons. While the net-proton rapidity distribution provides a record of energy deposition, existing analytical frameworks often treat bulk thermodynamics (transverse momentum spectra) and longitudinal transport (rapidity distributions) in isolation. Furthermore, current literature lacks an integrated, automated framework capable of simultaneously fitting multi-energy invariant yields, deriving local variance scalings, and extracting unified macroscopic transport variables (such as diffusion coefficients and Reynolds numbers) with coupled systematic error propagation. This gap hinders a precise characterization of the transition from inertial, advection-dominated stopping to thermal, diffusion-dominated transparency.

Methodology
The author presents a comprehensive, end-to-end analysis pipeline that integrates invariant kinematics with non-equilibrium stochastic diffusion. The methodology proceeds through the following steps:

Data Integration and Blast-Wave Fitting: Experimental net-proton transverse momentum ( $p_T$ ) spectra from central collisions ( $\le$ 10% centrality) across six beam energies ( $E_{lab} = 2.0, 4.0, 6.0, 8.0, 158.0$ A GeV, and $\sqrt{s_{NN}} = 62.4$ GeV) are analyzed. The author employs the Schnedermann-Sollfrank-Heinz (SSH) blast-wave model to fit these spectra. To ensure robustness, fits are conducted over three independent $p_T$ windows to quantify systematic uncertainties.
Profile Extraction: From the blast-wave fits, the author extracts the continuous rapidity density profile, $\rho_{net}(y)$ , and the transverse momentum broadening profile, $\langle p_T^2 \rangle_{net}(y)$ . These profiles are parameterized using a symmetric double-Gaussian function for density and a $\text{sech}^2$ scaling for variance.
Stochastic Transport Framework: The extracted profiles are fed into a stochastic diffusion framework governed by relativistic Fokker-Planck dynamics. By coupling the local transverse energy variance density to the accumulating net-baryon flux, the author analytically derives the longitudinal friction coefficient ( $\alpha_{bw}$ ).
Derivation of Transport Coefficients: Using the friction coefficient and a standard phenomenological estimate for transverse momentum diffusion ( $D_{pT}$ $D_{pT}$ ), the framework calculates:
- The rapidity diffusion coefficient ( $D_y$ ) via the Einstein fluctuation-dissipation theorem.
- The longitudinal relaxation time ( $\tau_{relax}$ ).
- The dimensionless longitudinal Reynolds number ( $Re_L$ ), defined as the ratio of inertial advection to diffusive transport.
- The local entropy production rate ( $dS_B/dy d\tau$ ).

Key Contributions

Unified Framework: The paper introduces a computational pipeline that bridges the gap between static hydrodynamic modeling of $p_T$ spectra and phenomenological diffusion models of rapidity distributions, allowing for the simultaneous extraction of thermodynamic and transport parameters.
Systematic Error Propagation: The study rigorously propagates uncertainties from fit covariance matrices and $p_T$ window variations into the final transport observables, providing a calibrated baseline for future comparisons.
Macroscopic Baseline: The work establishes a data-driven baseline for the crossover from baryon-rich, stopping-dominated environments to transparent, diffusion-dominated regimes.

Results
The analysis yields several critical findings across the energy scan:

Transport Coefficients: The longitudinal relaxation time ( $\tau_{relax}$ ) increases monotonically from $0.566 \pm 0.045$ fm/c at 2.70 GeV to $4.429 \pm 0.354$ fm/c at 62.4 GeV. Conversely, the rapidity diffusion coefficient ( $D_y$ ) increases from $0.181$ to $1.417$ rapidity $^2$ /fm, indicating a shift from frictional deceleration to stochastic momentum kicks.
Reynolds Number Crossover: The most significant result is the behavior of the longitudinal Reynolds number ( $Re_L$ ). It drops by a factor of $\sim$ 163, from $Re_L \approx 4.88$ at $\sqrt{s_{NN}} = 2.70$ GeV to $Re_L \approx 0.03$ at 62.4 GeV. This quantifies the transition from an advection-dominated regime ( $Re_L > 1$ ) to a diffusion-dominated regime ( $Re_L < 1$ ).
Entropy Production: The entropy production rate exhibits a sharp decline with increasing energy, mirroring the drop in $Re_L$ . This reflects the transition from high-dissipation frictional stopping at low energies to minimal thermodynamic dissipation in the transparent high-energy regime.
Stopping Fractions: The extracted central stopping fractions ( $f_{stop}$ ) show qualitative consistency with previous two-source thermal calculations, though slight quantitative differences are observed at low energies due to the inclusion of collective radial flow and continuous stochastic diffusion broadening in the current model.

Significance and Claims
The paper claims that the observed sharp crossover in $Re_L$ and the associated rise in $\tau_{relax}$ provide macroscopic evidence supporting the topological baryon junction paradigm. In this model, baryon number is carried by a Y-shaped configuration of neutral gluon fields rather than valence quarks. The data suggests that as collision energy increases, the baryon junction decouples from the valence quarks, leading to catastrophic momentum loss and enhanced mid-rapidity densities at low energies, followed by transparency at high energies.

The author asserts that this unified dataset provides a solid, falsifiable baseline for upcoming high-density experiments at facilities such as FAIR (CBM), NICA (MPD), and the CEE. They posit that future measurements can test the baryon junction mechanism by checking for the predicted suppression of the diffusion coefficient and extension of the relaxation time in the high-baryon chemical potential ( $\mu_B$ ) region. The study deliberately bounds its analysis at 62.4 GeV to avoid mathematical instabilities associated with vanishing net-baryon densities at higher energies, arguing that the physical transition to the diffusion-dominated regime is already fully established at this energy.

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