⚛️ nuclear theory

Elliptic flow in an expanding and rotating fireball

This paper investigates how initial orbital angular momentum influences elliptic flow in off-central heavy-ion collisions by modeling a simultaneously expanding and rotating fireball, finding that while rotation alone is insufficient, the combination of anisotropic expansion and rotation enhances elliptic flow by 2–10% depending on collision energy.

Original authors: Ashutosh Dwibedi, Anupam Panja, Sabyasachi Ghosh

Published 2026-09-30

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CC BY 4.0

Original authors: Ashutosh Dwibedi, Anupam Panja, Sabyasachi Ghosh

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 heavy atomic nuclei smash into each other at nearly the speed of light, they create a fleeting, super-hot soup of particles known as the quark-gluon plasma. This state of matter existed only microseconds after the Big Bang, and physicists study it to understand how the universe built itself from the ground up. In the most powerful collisions, scientists do not aim the beams perfectly head-on; they often miss slightly, causing the nuclei to graze past one another. This glancing blow creates a massive amount of spin, or orbital angular momentum, much like a figure skater spinning faster when they pull their arms in, though here the spin is generated by the collision geometry itself. This spin is deposited into the hot soup, causing the entire fireball of matter to rotate as it expands and cools. For decades, researchers have focused on how this spinning matter polarizes individual particles, but a new question has emerged: does this rotation also change the way the particles fly apart, specifically their tendency to move in an oval shape rather than a perfect circle?

A team of physicists at the Indian Institute of Technology Bhilai has taken a fresh look at this problem by building a new mathematical model of the collision. Instead of treating the expanding fireball as a simple, non-rotating balloon, they created a model where the fireball both expands and spins simultaneously. They tested two different shapes for this spinning fireball: a perfect sphere and a flattened, egg-like shape called a spheroid. Within each shape, they simulated two ways the spin could behave: either the entire fireball rotates at the same speed, or different layers rotate at different speeds, similar to how the atmosphere of a planet might spin faster at the equator than at the poles. By running these simulations with realistic values for the speed of rotation, they could calculate exactly how the particles would be distributed when the fireball finally freezes into a solid state of matter.

The researchers found that the rotation of the fireball has a distinct and measurable effect on the final pattern of particles, though the size of this effect depends heavily on the shape of the fireball and the energy of the collision. When they modeled a perfectly spherical fireball that was only spinning and not expanding unevenly, the rotation did create a slight oval pattern in the particle flow. However, the strength of this pattern was far too weak to match what experiments actually observe in the real world. This suggests that a spinning sphere alone cannot explain the data; the shape of the explosion itself must be uneven.

In contrast, when the team modeled a flattened, spheroidal fireball that was both expanding unevenly and spinning, the results changed dramatically. The combination of the uneven expansion and the rotation worked together to boost the oval pattern of the particle flow. At the lower collision energies tested, which correspond to 7.7 billion electron volts, the rotation added about ten percent to the strength of this oval flow compared to a non-rotating version of the same fireball. As the collision energy increased to 39 billion electron volts, this extra boost from rotation dropped to about two percent. This indicates that the spin of the fireball is a real, physical contributor to the final shape of the particle spray, but it is not the main driver. The primary cause remains the uneven expansion of the fireball, while the rotation provides a significant, non-negligible correction that becomes more important at lower energies.

The study also looked at how this effect varies for different types of particles, such as pions, kaons, and protons. The simulations showed that the percentage increase in the oval flow due to rotation was nearly the same for all these particles, regardless of their mass. This consistency helps confirm that the effect is a property of the collective motion of the fireball rather than a quirk of a specific particle type. The researchers used data from transport simulations to estimate the speed of rotation at the moment the fireball stops interacting, finding that the spin is strongest at lower collision energies. This aligns with their finding that the rotational contribution to the flow is most visible at those lower energies.

Ultimately, this work provides a clearer picture of how the initial spin of a heavy-ion collision leaves its mark on the final debris. By separating the effects of shape and spin, the authors demonstrated that while the uneven expansion of the fireball is the dominant force shaping the particle flow, the initial orbital angular momentum is not just a spectator. It actively participates, adding a measurable layer of complexity to the way matter behaves in these extreme conditions. The findings suggest that to fully understand the dynamics of these cosmic-scale collisions, scientists must account for both the shape of the explosion and the spin of the fireball, treating them as complementary forces that together dictate the final outcome.

Technical Summary: Elliptic Flow in an Expanding and Rotating Fireball

Problem Statement
In off-central heavy-ion collisions (HICs), the colliding nuclei carry substantial initial orbital angular momentum (OAM), a fraction of which is deposited into the reaction zone and transferred to the produced Quark-Gluon Plasma (QGP). While the effects of this OAM on global and local polarization (vorticity) are well-studied, its specific role in generating azimuthal anisotropies—specifically elliptic flow ( $v_2$ )—in the final-state hadron momentum spectra remains less explored within simple phenomenological models. Conventional Blast-Wave (BW) models and standard hydrodynamic descriptions typically assume purely expanding flows that lack net OAM, thereby failing to explicitly account for the centrifugal effects induced by the initial rotation. This paper addresses the question of whether initial OAM can generate measurable azimuthal anisotropies in hadron spectra and how to quantify this effect within a modified BW framework.

Methodology
The authors extend the traditional expanding fireball model to include a simultaneous expansion and rotation component, explicitly incorporating the initial OAM. The study employs a modified Blast-Wave formalism where the fluid four-velocity includes both longitudinal expansion (Bjorken profile) and transverse rotational components.

Key methodological elements include:

Geometries: Two distinct fireball geometries are analyzed to disentangle the effects of shape and rotation:
1. Spherical Fireball: Represents an isotropic expansion where any elliptic flow must arise solely from rotation.
2. Spheroidal Fireball: Represents an anisotropically expanding system, allowing for the separation of anisotropy generated by pressure gradients (geometric expansion) versus rotation.
Flow Profiles: For each geometry, two velocity profiles are considered:
- Profile I (Differential Rotation): Angular velocity depends on position (scaled by a boost factor), mimicking a relativistic quasi-rigid rotor.
- Profile II (Uniform Rotation): Uniform angular velocity across the fireball.
Parameters: The models utilize realistic kinetic freeze-out parameters (temperature, expansion rates) derived from previous analyses and transport simulations (AMPT). Crucially, the angular velocity ( $\Omega$ ) at freeze-out is constrained by values estimated from volume-averaged local vorticity in transport simulations, which are energy-dependent (higher at lower $\sqrt{s_{NN}}$ ).
Observables: The invariant momentum distributions are calculated using the Cooper-Frye prescription with a local thermal distribution function modified by the rotating flow. The study computes transverse momentum ( $p_T$ ) spectra and the second Fourier coefficient ( $v_2$ ) for identified hadrons (pions, kaons, protons) across RHIC Beam Energy Scan (BES) energies ( $\sqrt{s_{NN}} = 7.7 - 39$ GeV).

Key Contributions and Results

Model Development: The paper presents the first BW-like model that explicitly incorporates the initial OAM as a rotational component of the fluid velocity, moving beyond purely expanding scenarios.
Sensitivity of $p_T$ Spectra: The transverse momentum spectra at midrapidity are found to be largely insensitive to the inclusion of realistic angular velocities and the specific flow profiles (spherical vs. spheroidal, uniform vs. differential). Significant deviations in the high- $p_T$ region of the spectra only appear when the angular velocity is artificially magnified (e.g., $5\times$ the realistic freeze-out value).
Enhancement of Elliptic Flow ( $v_2$ ): In contrast to the spectra, $v_2$ $v_{2}$ is strongly sensitive to rotation.
- Spherical Case: A spherical fireball generates $v_2$ solely through rotation-induced momentum anisotropy. However, for realistic angular velocities, the magnitude of this flow remains significantly below experimental observations.
- Spheroidal Case: An anisotropically expanding and rotating spheroidal fireball exhibits a clear enhancement in $v_2$ . The rotation provides a complementary source of anisotropy to the geometric expansion.
Quantitative Impact: The study quantifies the contribution of vorticity to the observed elliptic flow:
- At $\sqrt{s_{NN}} = 7.7$ GeV, the rotating spheroidal model shows a $\sim 10\%$ enhancement in $v_2$ compared to the non-rotating counterpart.
- At $\sqrt{s_{NN}} = 39$ GeV, this enhancement reduces to $\sim 2\%$ .
- The percentage enhancement is consistent across identified hadrons (pions, kaons, protons) and their antiparticles, though the absolute magnitude of the effect is slightly smaller for heavier particles due to mass ordering in $v_2$ .
Profile Dependence: The study notes that Profile II (uniform rotation) generally yields higher $v_2$ than Profile I (differential rotation) for the same parameters, as the boost scaling in Profile I reduces the effective transverse velocity components.

Significance and Claims
The paper claims to demonstrate that while anisotropic transverse expansion remains the dominant source of elliptic flow in off-central HICs, the initial orbital angular momentum (vorticity) provides a non-negligible, additional contribution. Specifically, within the proposed fireball framework, vorticity contributes at the 2–10% level to the observed elliptic flow, with the effect being more pronounced at lower collision energies where the initial angular velocity is predicted to be higher.

The authors emphasize that this work provides a simple, physically motivated phenomenological approach to describe off-central collisions by generalizing the Blast-Wave model. They conclude that while rotation alone cannot explain the full magnitude of observed elliptic flow in a spherical geometry, the interplay between anisotropic expansion and rotation in a spheroidal geometry is consistent with experimental data and offers a mechanism to quantify the role of vorticity in the evolution of strongly interacting matter. The study suggests that rotation-sensitive observables could further constrain the rotational dynamics of the fireball.

Technical Summary: Elliptic Flow in an Expanding and Rotating Fireball

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