⚛️ nuclear theory

Driver and damping of the directed-flow response in heavy-ion collisions

Using an improved string-melting AMPT model and novel scaling observables across various collision systems at $\sqrt{s_{\mathrm{NN}}} = 200$ ~GeV, this study disentangles the initial-state driving and viscous damping effects on directed flow, identifying a collective behavior threshold at mass number $A \approx 35$ and determining a shear viscosity to entropy density ratio ( $\eta/s$ ) between 0.10 and 0.20.

Original authors: Kishora Nayak, Vipul Bairathi

Published 2026-09-14

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CC BY 4.0

Original authors: Kishora Nayak, Vipul Bairathi

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 heart of the universe's most violent moments, where atomic nuclei smash together at nearly the speed of light, matter behaves in ways that defy our everyday experience. When these tiny, dense cores collide, they melt into a seething, super-hot soup of their fundamental building blocks, a state of matter known as the quark-gluon plasma. For decades, scientists have studied this plasma in massive collisions, observing how it expands and flows like a nearly perfect liquid. However, a new question has emerged: just how small can a system be before it stops acting like this fluid and starts behaving more like a collection of individual, non-interacting particles? This boundary is crucial because it defines the minimum size required for the laws of fluid dynamics to apply to the universe's most extreme environments. Understanding where this line is drawn helps physicists determine whether the tiny droplets of matter created in smaller collisions are truly the same substance as the vast oceans of plasma seen in larger crashes, or if they represent a different kind of physics entirely.

To answer this, researchers turned their attention to a specific signal called directed flow, which acts like a subtle tilt in the way particles spray out from the collision point. Imagine the debris from a crash not just flying outward, but also leaning slightly in one direction, much like a wave cresting on a beach. This tilt is driven by the initial shape of the collision, which is often lopsided, but it is also dampened, or slowed down, by the internal friction of the fluid itself. The team behind this study, using a sophisticated computer simulation that tracks the journey of particles from the moment of impact to their final detection, investigated how this tilt changes as they varied the size of the colliding nuclei. They simulated collisions ranging from very light oxygen nuclei up to the heavy uranium nuclei, covering a wide spectrum of system sizes to see where the behavior shifts.

The researchers developed a clever method to separate the two forces at play: the initial push that creates the tilt and the internal friction that tries to smooth it out. By analyzing the data across their different simulated systems, they discovered a clear threshold. In the center of the collisions, where the overlap of the nuclei is greatest, the fluid-like behavior begins to emerge consistently once the system reaches a mass number of about 35. Below this size, the matter does not fully thermalize or behave as a cohesive fluid; above it, the collective motion takes over. This finding suggests that the transition from a chaotic collection of particles to an organized, flowing fluid happens quite abruptly as the system grows, rather than gradually. The study also revealed that this threshold is not simply a matter of geometric size; the number of particles involved and the energy density are the true governing factors, while simple size scaling fails to capture the full picture.

A key insight from the work is that the directed flow is far more sensitive to the fluid's internal friction than other types of flow previously studied. The team calculated that the point at which the fluid's resistance cuts the flow signal in half occurs at a specific scale of interaction that is roughly two and a half times smaller than what is seen in other flow patterns. This means that directed flow is a more delicate probe, one that reveals the limits of fluid behavior sooner than other measurements. By mapping these interactions, the scientists were able to estimate the ratio of the fluid's internal friction to its disorder, a value that falls between 0.10 and 0.20. This number is significant because it aligns with the idea that the quark-gluon plasma is one of the most perfect fluids known, operating near the theoretical limit of how little friction a substance can have.

The study did not stop at identifying the threshold; it also clarified what drives the flow and what stops it. The initial tilt, caused by the lopsided geometry of the colliding nuclei, grows stronger in smaller systems, but the fluid's ability to transmit this tilt is heavily dependent on how often the particles collide with one another. In the smallest systems, the particles do not collide enough to build up the collective motion, and the flow signal is suppressed significantly. In the largest systems, the collisions are frequent enough that the fluid transmits the initial push efficiently, preserving most of the flow. The researchers found that the damping effect, which reduces the flow, becomes the dominant factor only when the system is very small, while the initial geometric driver is the primary reason the flow exists in the first place.

Ultimately, this work provides a data-driven roadmap for understanding the transition between different states of matter in high-energy physics. It establishes that there is a specific, measurable size below which the matter produced in these collisions cannot be described as a hydrodynamic fluid. This threshold, found to be around a mass number of 35 for central collisions, offers a concrete target for future experiments. As scientists continue to collide nuclei of different sizes at facilities around the world, they can use these findings to test whether real-world data matches these simulations. The ability to distinguish between the initial push and the final damping of the flow offers a new lens through which to view the early universe, revealing exactly how and when the chaotic debris of a collision organizes itself into the flowing liquid that characterizes the quark-gluon plasma.

Technical Summary: Driver and Damping of the Directed-Flow Response in Heavy-Ion Collisions

Problem Statement
The formation of a thermalized, hydrodynamically expanding quark-gluon plasma (QGP) in small collision systems (e.g., p+p, O+O) remains a subject of active investigation. While collective flow is well-established in large systems (Au+Au, Pb+Pb), the minimum system size required for hydrodynamic behavior to emerge is not definitively known. Traditional geometric scaling of flow harmonics by nuclear mass number ( $A^{1/3}$ ) has been shown to fail for directed flow ( $v_1$ ) and its slope ( $dv_1/d\eta$ ) across a range of systems from O+O to U+U at $\sqrt{s_{NN}} = 200$ GeV. This violation of simple scaling suggests that the dynamics of the medium vary with system size, necessitating a method to distinguish between the initial-state geometric drivers of flow and the final-state viscous damping effects that suppress it.

Methodology
The authors utilized an improved version of the string-melting multi-phase transport (AMPT-SM) model, incorporating a new quark coalescence mechanism, to simulate collisions of five symmetric systems: O+O, Cu+Cu, Ru+Ru, Au+Au, and U+U at $\sqrt{s_{NN}} = 200$ GeV. The study focused on charged-hadron directed flow slopes ( $F = dv_1/d\eta|_{\eta=0}$ ) in the transverse momentum range $0.2 < p_T < 2.0$ GeV/c.

The methodology involved three primary analytical steps:

Scaling Observables: Three scaling observables were formulated based on entropy density ($dS/dy$), the mean number of participants ( $\langle N_{part} \rangle$ ), and the nuclear mass number ( $A^{1/3}$ ). A dimensionless ratio, $\xi(A)$ , was constructed by normalizing these observables against the O+O system to extract a threshold mass number ( $A_{th}$ ) where collective behavior sets in.
Threshold Extraction: The ratio $\xi(A)$ was fitted using a family of S-curves (Tanh, Error, and Gompertz functions) to identify the half-maximum point, defined as $A_{th}$ .
Knudsen-Number Mapping: A kinetic-theory Knudsen number ( $Kn$ ) map was generated to separate the contributions of the initial-state driver and final-state viscous damping. The scaled directed flow was plotted against $Kn$ to fit a function combining a power-law driver term ( $\propto Kn^\kappa$ ) and a damping term ( $\propto 1/(1 + Kn/Kn_0)$ ).

Key Results

Threshold Mass Number: For central collisions (0–40%), the entropy and participant scaling approaches yielded a consistent threshold mass number of $A_{th} \approx 35$ . This indicates the onset of collective, hydrodynamic-like behavior for systems larger than this threshold. In contrast, geometric scaling ( $A^{1/3}$ ) yielded significantly higher $A_{th}$ values (e.g., $\approx 46$ in central collisions, rising to $\approx 147$ in peripheral collisions), confirming that simple geometric scaling is insufficient to describe the transition.
Driver vs. Damping: The analysis of the scaled directed flow as a function of $Kn$ revealed that the system-size dependence is dominated by the initial-state driver rather than viscous damping alone. The driver component scales as $Kn^\kappa$ with $\kappa \approx 2$ , implying the initial push increases quadratically with $Kn$ (i.e., stronger per-unit-entropy tilt in smaller systems).
Damping Scale: The characteristic damping scale was determined to be $Kn_0 \approx 0.27$ for directed flow. This value is approximately 2.5 times smaller than the $Kn_0 \approx 0.7$ extracted for elliptic flow ( $v_2$ ), suggesting that directed flow is more sensitive to incomplete thermalization and dissipative effects than elliptic flow.
Viscosity Estimate: Using the extracted damping scale $Kn_0$ and kinetic-theory relations, the authors estimated the effective shear viscosity to entropy density ratio to be $(\eta/s)_{eff} \approx 0.15 \pm 0.05$ . This method provides an estimate independent of fitting flow harmonics directly.

Significance and Claims
The paper claims to provide a quantitative approach to differentiate between the driving mechanisms (initial geometry) and damping mechanisms (viscous response) of directed flow. By establishing a threshold mass number ( $A \approx 35$ ) for the onset of collectivity, the study offers a data-driven criterion for distinguishing between hadronic-dominated and hydrodynamic regimes. The construction of the $Kn$ map allows for the separation of these effects without relying on specific model assumptions about the equation of state. Furthermore, the determination of $(\eta/s)$ via the damping scale $Kn_0$ offers an alternative method to constrain the transport properties of the QGP, complementing traditional harmonic flow analyses. The authors suggest that this framework can be applied to experimental data from RHIC and LHC to further constrain the transition between transport and hydrodynamic regimes.

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