Elliptic FlowEdit
Elliptic flow is a key observable in the study of hot, dense matter created in high-energy heavy-ion collisions. It refers to the second harmonic modulation of the angular distribution of produced particles around the beam axis, relative to the reaction plane. In practical terms, experiments quantify elliptic flow with the coefficient v2, which measures how preferentially particles are emitted along the short axis of the almond-shaped overlap region formed when two nuclei collide off-center. This observable is intimately connected to the properties of the quark–gluon plasma and the collective dynamics that emerge in such collisions. The concept sits at the intersection of experimental measurement, relativistic hydrodynamics, and the physics of strong interactions described by Quantum chromodynamics.
Elliptic flow emerges most clearly in non-central collisions, where the overlap geometry is anisotropic. The initial spatial anisotropy translates, through pressure gradients, into anisotropic momentum distributions of the emitted hadrons. The formalism is built around the azimuthal angle φ of each particle relative to a reference direction called the reaction plane, often denoted by ΨR P. The distribution can be expanded in a Fourier series in φ − ΨR P, with v2 as the second-order coefficient: dN/dφ ∝ 1 + 2v1 cos(φ − ΨR P) + 2v2 cos[2(φ − ΨR P)] + … . Thus, v2 encapsulates the collective flow pattern that develops as the system expands and cools. See Azimuthal anisotropy for broader context, and Reaction plane for the geometric reference used in experimental analyses.
Observables and methods
Measuring elliptic flow involves several approaches, each with its own strengths and sources of systematic uncertainty. The reaction-plane method uses an estimate of the event-by-event reaction plane to extract v2, while the event-plane and two-particle correlation methods exploit correlations among emitted particles without needing a perfect reaction-plane determination. More sophisticated techniques employ multi-particle cumulants to suppress nonflow correlations from jets or resonance decays. These methods are discussed in detail in literature on Elliptic flow and Flow harmonics.
Experiments across different facilities have established a robust pattern for v2 as a function of centrality, transverse momentum pT, and particle species. Notable data come from the Relativistic Heavy Ion Collider (RHIC), including measurements by the STAR (experiment) and PHENIX collaborations, and from the Large Hadron Collider (LHC) with results from ALICE, CMS (experiment), and ATLAS (experiment). The complementary information from these detectors helps constrain models of the initial state and the subsequent evolution of the system.
Physical interpretation and modeling
A central interpretation is that elliptic flow signals the development of a collectively expanding medium that behaves, at least for a significant portion of its evolution, like a relativistic fluid. In hydrodynamic descriptions, the early pressure gradients convert the initial spatial eccentricity ε2 into a measurable momentum anisotropy v2. The degree to which this conversion occurs is sensitive to the medium’s transport properties, most notably its shear viscosity to entropy density ratio, η/s. A lower η/s generally leads to stronger collective flow and a larger v2, at least within the validity range of the hydrodynamic description.
The linkage between v2 and the medium’s properties relies on the concept of eccentricity, ε2, which characterizes the shape of the overlap region. The relationship between v2 and ε2 is sharpened in models that assume near-thermalized matter and strong collectivity, though departures from ideal hydrodynamics are expected due to finite viscosity, non-equilibrium effects, and hadronic rescattering in the late stages. See Quark–Gluon Plasma for the broader state of matter created in these collisions and Relativistic hydrodynamics for the framework used to describe its evolution.
Initial-state fluctuations play a crucial role. Even in events with roughly the same centrality, fluctuations in the positions of nucleons lead to a nonuniform geometry that seeds higher-order flow harmonics such as v3 (triangular flow) and v4. The study of these harmonics has become a standard way to test the degree of collectivity and the fluid-like behavior of the system. The interplay between initial geometry and final-state dynamics is a central focus of modern heavy-ion theory, with connections to :glasma ideas about the early-time state and to various initial-condition models like IP-Glasma.
Experimental systematics and particle dependence
The pT dependence of v2 exhibits characteristic trends. At low pT, lighter particles tend to show larger v2 than heavier ones at the same momentum per nucleon, a pattern known as mass ordering that emerges naturally in hydrodynamic treatment and is contrasted with non-flow effects. At intermediate pT, a scaling behavior attributed to the number of constituent quarks (NCQ scaling) has been observed, suggesting that flow develops at the partonic level before hadronization. These features have been reported across RHIC and LHC energies and provide important cross-checks for the underlying dynamics. See Mass ordering (nuclear physics) and Number of constituent quarks for related concepts.
The field has also explored how v2 behaves in smaller collision systems, such as proton–nucleus or deuteron–nucleus collisions. Signals resembling elliptic flow in these systems have sparked debate about whether hydrodynamics applies in small volumes or whether alternative, principally initial-state or momentum-correlation mechanisms can mimic flow-like patterns. See discussions under Color-glass condensate and Small systems (heavy-ion physics) for more on this topic.
Controversies and debates
Hydrodynamic applicability and η/s: A long-running debate centers on how fluid-like the quark–gluon plasma really is, and how small the η/s value must be to reproduce the observed v2 across energies and centralities. Critics point to sensitivity to initial conditions and to the regime of validity of hydrodynamics, while supporters emphasize the success of viscous hydrodynamics in describing a broad range of data.
Initial-state versus final-state origins: While many features of elliptic flow arise from the collective expansion of a medium, some researchers argue that part of the observed anisotropy, especially in smaller systems or at very early times, can originate from initial-state correlations present before substantial interactions occur. The balance between these contributions remains a focus of ongoing work, with implications for how we interpret v2 in different collision systems.
NCQ scaling and hadronization: The apparent approximate scaling of v2 with the number of constituent quarks at intermediate pT has been cited as evidence for partonic collectivity prior to hadronization. Yet not all datasets cleanly follow NCQ scaling, and questions persist about the universality of this behavior and its interpretation within competing hadronization scenarios.
Role of fluctuations and higher harmonics: The discovery and systematic study of v3, v4, and beyond have sharpened the picture of initial fluctuations and their propagation through the medium. Critics caution that extracting precise transport properties from higher harmonics demands careful modeling of the initial state, nonflow effects, and the interplay with non-hydrodynamic processes.
Small-system collectivity: The observation of flow-like signals in p+Pb and d+Au collisions challenges a clean hydrodynamic interpretation in tiny systems. Proponents of hydrodynamics argue that sufficient particle interactions can generate collective behavior, while skeptics point to alternative explanations based on initial-state dynamics or momentum correlations that do not require a strongly interacting medium.
Broader context and connections
Elliptic flow is part of a broader program to characterize the quark–gluon plasma and the behavior of QCD matter at extreme temperatures and densities. It connects to measurements of jet quenching and energy loss, which probe how high-energy partons traverse and interact with the medium, and to studies of the phase structure of QCD, including the onset of deconfinement and chiral symmetry restoration. The field relies on a synergy between state-of-the-art detectors, such as those described in the pages for STAR (experiment) and ALICE, and advances in theoretical frameworks like Relativistic hydrodynamics and transport models such as AMPT.
For readers seeking a broader primer or related topics, consider exploring Quark–Gluon Plasma for the nature of the matter created, Initial-state fluctuations for the origins of geometric irregularities, and Flow harmonics for the full spectrum of anisotropic flow observables. Experimental context is enriched by discussions of key facilities and collaborations, including RHIC and the LHC, as well as the experiments that have contributed the bulk of elliptic-flow measurements: PHENIX and STAR (experiment) at RHIC, and ALICE, CMS (experiment), and ATLAS (experiment) at the LHC.