Nuclear Geometry-Driven Hydrodynamic Flow

• Nuclear geometry-driven hydrodynamic flow is the process by which intrinsic nuclear deformations and clustering convert spatial anisotropies into anisotropic particle momentum distributions in ultra-relativistic collisions.
• It utilizes detailed hydrodynamic models to map initial geometric eccentricities of colliding nuclei to measurable flow coefficients (v₂, v₃), offering quantitative insights into QGP properties.
• Experimental data from collider experiments combined with advanced theoretical modeling enable precise nuclear imaging and improve our understanding of fluctuation-driven and non-linear hydrodynamic responses.

Nuclear geometry-driven hydrodynamic flow refers to the conversion of spatial anisotropies inherent to the geometric structure and configuration of colliding nuclei into anisotropies of the final-state particle momentum distributions, mediated by collective expansion (hydrodynamics) in ultra-relativistic nuclear collisions. This paradigm provides a direct connection between femtometer-scale nuclear structure and macroscopic collective observables such as anisotropic flow coefficients ($v_n$) and event-by-event fluctuations, which are key to interpreting results from modern collider experiments at RHIC and the LHC. In recent years, advances in theory, modeling, and experiment have solidified the understanding that nuclear shape, deformation, clustering, and fluctuations can be imaged through hydrodynamic flow, blending nuclear structure physics with the paper of the quark–gluon plasma (QGP).

1. Nuclear Geometry: Deformation, Clustering, and Fluctuations

The initial state of a nuclear collision is dictated not only by the impact parameter and random participant placement, but also by the intrinsic structure of the colliding nuclei. Even-even nuclei such as $^{16}$O and $^{20}$Ne, and heavy nuclei like $^{238}$U, possess non-spherical shapes parameterized by multipole deformations ($\beta_2$ for quadrupole/prolate-oblate, $\gamma$ for triaxiality, $\beta_3$ for octupole/pear-shape, etc.) (Collaboration, 12 Jan 2024, Collaboration, 21 Jun 2025, Zhang et al., 21 Apr 2025, Giacalone et al., 30 May 2024, Ma et al., 2022). Light nuclei may exhibit pronounced clustering, such as the tetrahedral $\alpha$-particle arrangements in $^{16}$O or triangular structures in $^{12}$C (Behera et al., 2023, R et al., 4 Jul 2024, Shafi et al., 16 May 2025, R et al., 28 May 2025, Collaboration, 8 Sep 2025).

The most commonly encountered nuclear density distributions are either mean-field Woods–Saxon (smooth, spherically averaged) or cluster models, which introduce intrinsic inhomogeneities. The experimental realization of collisions between nuclei with well-characterized geometric shapes (prolate, oblate, triaxial, clustered) has enabled direct studies of how these details propagate into observables.

Event-by-event, further spatial fluctuations arise from nucleon and sub-nucleon configurations, leading to complex initial energy density profiles that seed higher-order geometric eccentricities ($\epsilon_n$) and local hotspots (Alver et al., 2010, Hippert et al., 2020, Chen et al., 4 Feb 2024).

2. Geometric Anisotropies and Their Hydrodynamic Evolution

The hydrodynamic evolution is initialized with an energy (or entropy) density distribution that reflects the geometric characteristics of the participant zone. Key geometric quantities include the eccentricity and higher moments: $$\epsilon_n = \frac{\sqrt{\langle r^n \cos(n \phi) \rangle^2 + \langle r^n \sin(n \phi) \rangle^2}}{\langle r^n \rangle},$$ where the averages $\langle\cdots\rangle$ are taken with respect to the energy density profile in the transverse plane (Shafi et al., 16 May 2025, Behera et al., 2023).

These spatial anisotropies result in anisotropic pressure gradients that, under rapid thermalization and hydrodynamic expansion, generate anisotropic collective flow in the produced QGP. The final-state momentum anisotropies are captured by the Fourier coefficients ($v_n$) of the single-particle azimuthal distribution: $$\frac{dN}{d\phi} \propto 1 + 2 \sum_{n=1}^\infty v_n \cos[n(\phi - \Psi_n)],$$ where $v_2$ is termed elliptic flow, $v_3$ triangular flow, and $\Psi_n$ is the $n$-th order event-plane angle (0901.4588, Collaboration, 12 Jan 2024, Chen et al., 4 Feb 2024).

The mapping between $\epsilon_n$ and $v_n$ is sensitive to properties like viscosity ($\eta/s$), system size, initial fluctuations, and non-linear hydrodynamic response (0901.4588, Kurkela et al., 2020, Hippert et al., 2020, Ma et al., 2022, Chen et al., 4 Feb 2024).

3. Experimental Evidence and Observables

Recent measurements by the ALICE experiment in $^{16}$O+$^{16}$O and $^{20}$Ne+$^{20}$Ne collisions at $\sqrt{s_{NN}}=5.36$ TeV (Collaboration, 8 Sep 2025) show clear enhancement of $v_2$ (elliptic flow) in central Ne–Ne compared to O–O, directly attributable to the intrinsic prolate ("bowling pin") deformation of $^{20}$Ne versus the near-tetrahedral structure of $^{16}$O. Multiparticle correlation techniques (cumulant and event-plane methods) provide robust extraction of $v_n$, including $v_2\{2\}$ and $v_2\{4\}$, as well as higher order cumulants and mixed-harmonic (symmetric/asymmetric) cumulants (Chen et al., 4 Feb 2024, Shafi et al., 16 May 2025). Hydrodynamic model calculations with ab initio nuclear structure inputs (e.g., projected generator coordinate method wavefunctions for $^{16}$O, deformation parameters for $^{20}$Ne and $^{238}$U) quantitatively reproduce the trends and magnitude of observed flow (Collaboration, 8 Sep 2025, Giacalone et al., 30 May 2024, Collaboration, 21 Jun 2025, Zhang et al., 21 Apr 2025).

Key observables sensitively probing the geometry-driven flow include:

• The enhancement of $v_2$ in systems with pronounced quadrupole deformation ($\beta_2$).
• Differences in $v_3$ for nuclei with octupole deformation ($\beta_3$) or strong clustering, particularly visible in ultra-central events (Zhang et al., 21 Apr 2025, Collaboration, 21 Jun 2025).
• Anti-correlation of $\epsilon_2$ and $\epsilon_3$ (and thus $v_2$ and $v_3$) as captured by normalized symmetric cumulants NSC(2,3), with negative values indicative of cluster-driven geometry (Behera et al., 2023, Shafi et al., 16 May 2025).
• The non-linear hydrodynamic response, e.g., quadratic $v_4 \sim v_2^2$ coupling enhancements in systems with larger $v_2$ (Giacalone et al., 30 May 2024).
• Rapidity-even dipolar flow $v_1^{\text{even}}$, which, while more subtle, can be enhanced for clustered or asymmetric geometries (Shafi et al., 16 May 2025).

4. Theoretical Modeling and Response Hierarchies

Modern event-by-event hydrodynamic frameworks (e.g., IP-Glasma+MUSIC+UrQMD, AMPT, TrENTO-based models) incorporate the nuclear geometry at the nucleon or sub-nucleonic level and propagate the full three-dimensional, fluctuating initial state through pre-equilibrium, hydrodynamic, and hadronic transport evolution (Hippert et al., 2020, Chen et al., 4 Feb 2024, R et al., 28 May 2025, Behera et al., 2023).

Non-linear response coefficients and mode-mode coupling become important, particularly as the system transitions from the free-streaming to hydrodynamic regime (Kurkela et al., 2020). For instance, the quadrupole flow ($v_2$) and its covariance with mean transverse momentum ($\delta p_T$) depend not only on $\beta_2$ but also on $\gamma$ (triaxiality) via terms like $\beta_2^3 \cos(3\gamma)$ (Collaboration, 12 Jan 2024, Collaboration, 21 Jun 2025). The magnitude and hierarchy of $v_n$ are non-uniform away from the fluid-dynamic limit and encode real-time information about both the early QGP and the underlying nuclear structure.

Subleading flow modes (second principal components in a PCA decomposition) are shown to be especially sensitive to higher-order cumulants of the initial geometry—thereby probing small-scale structure and fluctuations otherwise inaccessible via leading flow harmonics (Hippert et al., 2020).

5. System Size, Shape, and Fluctuation Effects

The impact of nuclear geometry is modulated by system size and collision configuration. In large nuclei (e.g., Pb, Au, U), average geometry dominates, and the hydrodynamic response is robust and quasi-linear: $v_2 \approx \langle S \epsilon_2 \rangle$, where $S$ is the proportionality factor depending on viscosity and system lifetime. In smaller systems, such as O–O or Ne–Ne, initial state fluctuations and sub-nucleonic geometry take on an outsized role (Chen et al., 4 Feb 2024, Behera et al., 2023, Collaboration, 8 Sep 2025, R et al., 28 May 2025). Purely fluctuation-driven $v_2$ and $v_3$ can dominate central or small-system collisions, and the effects of hadronization, non-linear hydrodynamic response, and the hadronic afterburner become more pronounced, reducing the one-to-one mapping between $\epsilon_n$ and $v_n$. The ratio $v_2\{4\}/v_2\{2\}$, along with its comparison to $\epsilon_2\{4\}/\epsilon_2\{2\}$, quantifies the fluctuation strength and is used to trace the deviation from geometric-driven to fluctuation-driven behavior (Chen et al., 4 Feb 2024).

6. Nuclear Imaging and Implications for QGP Physics

The collective flow–assisted nuclear shape imaging technique leverages the extreme temporal resolution of high-energy collisions (exposure timescales $\sim$0.1 fm/$c$) to provide a "snapshot" of the nuclear many-body distribution, effectively freezing the intrinsic deformation or clustering pattern (Collaboration, 12 Jan 2024, Collaboration, 21 Jun 2025, Zhang et al., 21 Apr 2025). By extracting deformation parameters $(\beta_2, \gamma, \beta_3)$, triaxiality, or clustering configurations via comparative analysis of collisional systems (e.g., U+U vs Au+Au, O+O vs Ne+Ne) and their collective observables, it is possible to quantitatively connect low-energy nuclear structure and high-density QCD phenomenology. The method is particularly discriminating in ultra-central collisions, where geometry-imposed eccentricities are maximal and statistics allow for multi-particle cumulant and fluctuation analyses at high precision.

These insights refine constraints on the initial state in hydrodynamic modeling, directly reducing uncertainties in the extraction of QGP transport coefficients (e.g., $\eta/s$) and mapping of the QCD phase diagram, as well as opening a new avenue for investigating exotic and higher-order deformations inaccessible by traditional spectroscopic means.

7. Future Directions and Open Questions

Ongoing RHIC and LHC campaigns with light ions and isobaric systems, together with advances in nuclear structure theory (e.g., ab initio no-core shell model, PGCM, NLEFT approaches), offer prospects for quantitatively mapping nuclear geometry effects across collision species and energies (Giacalone et al., 30 May 2024, Collaboration, 8 Sep 2025, R et al., 28 May 2025). The continued integration of high-precision experimental flow measurements, advanced fluctuation/cumulant observables, and multi-scale theoretical modeling will enhance the capability to disentangle nuclear geometry–driven signatures from final-state QGP dynamics and hadronization effects.

Critical open questions include: the precise limits of hydrodynamic behavior in small systems, the role of sub-nucleonic correlations, the interplay between higher-order and non-trivial geometric deformations (such as triaxiality or octupole moments), and the full exploitation of multi-observable constraints to achieve nuclear "imaging-by-smashing" with quantitative fidelity.

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In summary, nuclear geometry-driven hydrodynamic flow is a foundational link between the femtometer-scale properties of nuclei and the properties of the hottest, densest matter known, as revealed through the collective evolution and final-state signatures of high-energy nuclear collisions.