Elliptic Flow Coefficient in Nuclear Collisions

• Elliptic flow coefficient is defined as the amplitude of the second Fourier harmonic, measuring momentum anisotropy arising from initial spatial asymmetries.
• It plays a diagnostic role in inferring QGP transport properties and the interplay between early dynamics and pressure gradients using cumulant and event-plane methods.
• The observable is sensitive to system size and centrality, with analyses employing models from ideal hydrodynamics to advanced machine learning techniques.

The elliptic flow coefficient ($v_2$) is central to the characterization of collective dynamics in high-energy nuclear and hadronic collisions. It quantifies the second harmonic of the azimuthal distribution of final-state particles, reflecting the conversion of initial spatial anisotropies—stemming from the collision geometry or nuclear substructure—into observable momentum anisotropies. $v_2$ is both a sensitive probe of the transport properties of the quark–gluon plasma (QGP) and a diagnostic of the interplay between early-time dynamics, fluctuating initial conditions, and the medium's response to pressure gradients. The paper of $v_2$ leverages both experimental data (including cumulant and event-plane methods) and a diverse set of theoretical and simulation-based frameworks, ranging from ideal and viscous hydrodynamics to machine learning–based inference, covering system sizes from nucleus–nucleus to proton–proton and small collision systems.

1. Definition and Physical Significance of the Elliptic Flow Coefficient

The elliptic flow coefficient $v_2$ is defined as the amplitude of the second Fourier harmonic in the decomposition of the single-particle azimuthal distribution,

$$\frac{dN}{d\phi} = \frac{N}{2\pi} \left( 1 + 2v_2 \cos[2(\phi - \Psi_2)] + \cdots \right)$$

where $\phi$ is the azimuthal angle, and $\Psi_2$ is the event symmetry plane (reaction plane, participant plane, or event plane depending on the analysis technique) (0806.1116, Collaboration, 2017).

In ideal hydrodynamics, non-central heavy-ion collisions produce an almond-shaped overlap zone, leading to spatial eccentricity $\varepsilon_2$. Subsequent anisotropic pressure gradients convert this spatial asymmetry into a momentum anisotropy in the final particle spectra, measured by $v_2$. For a fluid with negligible viscosity, the mapping between initial eccentricity and final $v_2$ is approximately linear: $v_2 \approx \kappa_2 \varepsilon_2$ where $\kappa_2$ is a response coefficient dependent on the equation of state and transport properties of the QGP (Alqahtani et al., 27 Jul 2025).

The $p_T$-dependent $v_2(p_T)$ provides insight into the interplay of collective expansion (radial and elliptic flow), hadronization, and possible dissipative effects. Integrated $v_2$ values are sensitive to the lifetime of the QGP phase, system size, and initial geometry (0806.1116, Collaboration, 2015).

2. Hydrodynamic Modeling, Initial Conditions, and Centrality Dependence

2.1 Hydrodynamics and Evolution Equations

Relativistic (viscous or ideal) hydrodynamics forms the theoretical baseline for modeling $v_2$ in nucleus–nucleus collisions. The key equations are local conservation of energy–momentum and baryon number: $$\partial_\mu T^{\mu\nu} = 0,\qquad \partial_\mu (n_B u^\mu) = 0$$ with $$T^{\mu\nu} = (\epsilon + P) u^\mu u^\nu - P g^{\mu\nu}$$ in the ideal fluid limit, where $\epsilon$ is energy density, $P$ is pressure, and $u^\mu$ is the four-velocity (0806.1116).

Boost-invariant (2+1)D treatments are common for ultrarelativistic energies. Pressure gradients—encoded in the hydrodynamic equations—drive anisotropic expansion, leading to $v_2>0$ for noncentral collisions (Schenke et al., 2010, Holopainen et al., 2011).

2.2 Initial-State Modeling: Geometric Eccentricity and Fluctuations

For central (head-on) collisions, energy density initialization can be based on the EKRT (pQCD+saturation) model, calibrated to match minijet production with a saturation scale $p_{\text{sat}}$ (0806.1116). For noncentral collisions, the optical Glauber model defines spatial distributions of energy and baryon densities via binary collision (BC) scaling or wounded nucleon (WN) scaling: $$\varepsilon_\mathrm{BC}(\mathbf{r};b)\propto T_A(\mathbf{r}+\mathbf{b}/2)T_A(\mathbf{r}-\mathbf{b}/2)$$ with centrality classes mapped to impact parameters via the total cross section (0806.1116).

Event-by-event models based on Monte Carlo Glauber or hot-spot-based initializations are essential for capturing fluctuations. These fluctuations generate local pressure inhomogeneities and influence not only the mean $v_2$ but also its distribution and correlation with other harmonics (Holopainen et al., 2011, Chaudhuri, 2011).

2.3 Centrality Dependence and Scaling Observables

The centrality dependence of $v_2$—maximal at mid-centrality—traces back to the competition between geometric eccentricity and dynamic equilibration. Scaling observables such as $v_2/\varepsilon$ serve as proxies for the degree of local thermalization (MeiJuan et al., 2012):

• For near-central collisions (small impact parameter, large number of participants), $v_2/\varepsilon$ saturates, indicating local equilibrium.
• For peripheral events, $v_2/\varepsilon$ increases with more participants, suggesting incomplete relaxation.

Comparison of $v_2$ between systems (Pb+Pb vs Xe+Xe) at matched initial entropy density reveals deviations (on the order of $7\%$) in $v_2/\varepsilon_2$, ascribed to viscous attenuation and system-size effects (Collaboration, 2022).

3. Fluctuations, Non-Gaussianity, and Multi-Particle Cumulants

3.1 Event-by-Event $v_2$ Probability Distributions

Fluctuations in the initial geometry (participant nucleon positions) induce non-Gaussian $v_2$ distributions. The measured $v_2$ is related to the true distribution $p(v_2)$ via experimental response functions and unfolding techniques (Collaboration, 2017). The higher moments of $p(v_2)$ are extracted using multi-particle cumulant analyses: $$\begin{aligned} v_2\{2\}^2 &= \big\langle v_2^2 \big\rangle, \ v_2\{4\}^4 &= -\big\langle v_2^4 \big\rangle + 2 \big\langle v_2^2 \big\rangle^2, \ \cdots \end{aligned}$$ The ordering $v_2\{2\} > v_2\{4\} > v_2\{6\} > v_2\{8\}$, with finite (negative) standardized skewness, signals non-Gaussian fluctuations (Collaboration, 2017).

3.2 Nonflow Contributions, Momentum Conservation Effects

In small systems or at low multiplicity, $v_2$ extracted from multi-particle cumulants may receive substantial contributions from nonflow effects, such as momentum conservation (Bzdak et al., 2017). Analytic calculations show that transverse momentum conservation (TMC) generates positive cumulant coefficients $c_2\{k\}$ for $k=2,4,6,8$, implying that even in the absence of hydrodynamic collectivity, measurable $v_2$-like signals can emerge.

3.3 Spectator Plane vs Participant Plane

Ratios of $v_2$ measured with respect to different symmetry planes, such as $v_2\{\Psi_\mathrm{SP}\}$ (spectator plane) and $v_2\{4\}$ (four-particle cumulant), deviate by up to 20% from unity, indicating decorrelation between the spectator plane and participant plane. This decorrelation likely arises from nuclear fragmentation and other unmodeled initial-state features, challenging the completeness of current fluctuating initial-state models (Collaboration, 2022).

4. Flow Correlations: Elliptic and Triangular Flow, and Mixed Skewness

Correlations between $v_2$ and higher-order flow harmonics, notably triangular flow ($v_3$), are essential observables for characterizing initial-state non-Gaussianity and disentangling the origins of collective behavior (Alqahtani et al., 27 Jul 2025). The normalized symmetric cumulant,

$$\mathrm{nsc}_{2,3}\{4\} = \frac{ \langle v_2^2 v_3^2 \rangle - \langle v_2^2 \rangle \langle v_3^2 \rangle }{ \langle v_2^2 \rangle \langle v_3^2 \rangle }$$

changes sign and varies non-monotonically with centrality. This behavior is attributed to two key effects:

• Mixed skewness in initial-state fluctuations, quantified by an intensive parameter: $$\Gamma_S = \frac{ \langle \varepsilon_2^* \varepsilon_3 \varepsilon_3^* \rangle_c }{ \bar{\varepsilon}_2 \sigma_{\varepsilon_2} (\sigma_{\varepsilon_3})^3 }$$ with $\Gamma_S$ of order unity and insensitive to centrality or system size.
• Fluctuations in the impact parameter relative to experimental centrality classifiers (e.g., $N_{ch}$ or $E_T$), which further modulate the observed correlation.

Monte Carlo initial-state models demonstrate that $\Gamma_S$ (and thus the observed $v_2$–$v_3$ correlation) is sensitive to the nucleon width parameter in the spatial entropy deposition profile (Alqahtani et al., 27 Jul 2025). Fitting ATLAS data yields intensive mixed skewness values consistent with theoretical expectations.

5. Machine Learning Approaches for $v_2$ Estimation

Deep learning frameworks have been introduced to estimate $v_2$, leveraging the high-dimensional, image-like nature of particle kinematic information in the $\eta$–$\phi$ plane (Mallick et al., 2022, Murali et al., 17 Nov 2024). Two paradigms have emerged:

• Feed-forward deep neural networks (DNNs) that encode kinematic and event-level observables into fixed-length feature arrays (e.g., $32\times32\times3$) and regress $v_2$ with high precision. These models preserve the centrality, energy, and $p_T$ dependencies of $v_2$ even under simulated detector noise (Mallick et al., 2022).
• Convolutional neural networks (CNNs) that take as input multi-layered $(\eta,\phi)$ images (weighted by $p_T$, mass, $m_T$, etc.), jointly predicting $v_2$ and the impact parameter $b$. CNN architectures with regularization and group normalization preserve the physics correlations identified in traditional analyses and offer direct mapping of event geometry to collective observables. Visual interpretability techniques (e.g., Grad-CAM) reveal that crucial event features are captured by network attention (Murali et al., 17 Nov 2024).

Both approaches report sub-6% mean absolute errors with respect to simulation or experiment. The ability of these networks to simultaneously infer $v_2$ and $b$ has opened new directions for real-time event characterization and may provide additional constraints when integrated with experimental workflows.

6. System Dependence, Small Systems, and Non-hydrodynamic Sources of $v_2$

While $v_2$ is traditionally associated with hydrodynamic collective flow in large systems, its extraction and interpretation in small systems require careful treatment. In proton–proton and proton–nucleus collisions, $v_2$ signals extracted from angular correlation measurements can reach values in the range $0.04-0.10$ (Bozek, 2010), with evidence for long-range pseudorapidity correlations. These signals may arise from a combination of final-state collective effects, initial-state parton correlations (such as elliptic gluon Wigner distributions) (Hagiwara et al., 2017), or kinematic constraints like TMC (Bzdak et al., 2017).

The elliptic gluon Wigner distribution predicts nontrivial, $v_2$-like azimuthal modulations purely from QCD initial-state correlations, demonstrating that a cos $2\phi$ modulation in two-particle production can be realized without final-state collective flow. This mechanism is particularly relevant at high energy and opens connections to the gluon tomography of hadrons.

In the context of polarized light nuclei colliding with a heavy target, the intrinsic quadrupole deformation (manifest in the polarization state) is converted to measurable $v_2$, offering a direct handle on the relationship between initial geometry and final-state collectivity in small systems (Bozek et al., 2018, Broniowski et al., 2019).

7. Electromagnetic Field Effects and Photonic $v_2$

The influence of time-dependent electromagnetic fields on $v_2$ has been quantified in both hadronic and photonic observables. Magnetic field–induced photon production via gluon fusion and splitting provides an excess photon yield and an anisotropic emission pattern, leading to additional, centrality-dependent $v_2$ contributions at low $p_T$ (Ayala et al., 2019). Hydrodynamic simulations incorporating electromagnetic forces (Coulomb, Lorentz, Faraday, plasma-based) find that $v_2$ can be enhanced at lower $p_T$ and higher collision energies, and that the early-time "kick" is more significant for heavier particles than for pions or anti-particles (Gezhagn et al., 2021). Accurate modeling of field evolution and conductivity is necessary for quantitative agreement.

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In summary, the elliptic flow coefficient $v_2$ is an essential, multifaceted observable that encodes the mapping from initial spatial asymmetry—determined by geometry, fluctuations, and intrinsic nuclear structure—to final-state momentum-space anisotropy. Interpretation of $v_2$ requires integrated understanding of hydrodynamic response, initial-state fluctuation statistics (including non-Gaussianity and mixed skewness), system size and initial entropy dependence, and both final-state and initial-state nonflow sources. Advanced statistical, machine learning, and experimental techniques continue to refine the extraction and physical interpretation of $v_2$ across the spectrum of collision systems and energies.