Flow Harmonics Research Articles

Participation of nucleons among different harmonics of anisotropic flow

We present the comprehensive interpretation of the momentum distribution of nucleons participating in different kinds of nuclear flow, within the framework of isospin-dependent quantum molecular dynamics (IQMD) model. Our study reveals that the number of nucleons decreases with increasing order of different harmonics ⟨v1⟩, ⟨v2⟩, ⟨v3⟩, and ⟨v4⟩. The magnitude of flow harmonics depends strongly on transverse momentum.

Scaling behavior of anisotropic flow harmonics in the far from equilibrium regime

We consider the development of anisotropic flow in an expanding system of particles undergoing very few rescatterings, using a kinetic-theoretical description with a nonlinear collision term. We derive the scaling behaviors of the harmonic coefficients v_n with the initial-state eccentricities and the mean number of rescatterings, and argue that hexagonal flow v_6 should follow a nontrivial behavior, different from that of the lower harmonics. Our findings should be observable in experimental data for small systems.

Standardized cumulants of flow harmonic fluctuations

The distribution of flow harmonics in heavy ion experiment can be characterized by standardized cumulants. We first model the ellipticity and power parameters of the elliptic-power distribution by employing MC-Glauber model. Then we use the elliptic-power distribution together with the hydrodynamic linear response approximation to study the two dimensional standardized cumulants of elliptic and triangular flow ($v_2$ and $v_3$) distribution. For the second harmonic, it turns out that finding two dimensional cumulants in terms of $2q$-particle correlation functions $c_2\{2q\}$ is limited to the skewness. We also show that $c_3\{2\}$, $c_3\{4\}$, and $c_3\{6\}$, are related to the second, fourth, and sixth standardized cumulants of the $v_3$ distribution, respectively. The cumulant $c_{n}\{2q\}$ can be also written in terms of $v_n\{2q\}$. Specifically, $-(v_3\{4\}/v_3\{2\})^4$ turns out to be the kurtosis of the $v_3$ event-by-event fluctuation distribution. We introduce a new parametrization for the distribution $p(v_3)$ with $v_3\{2\}$, kurtosis and sixth-order standardized cumulant being its free parameters. Compared to the Gaussian distribution, it indicates a more accurate fit with experimental results. Finally, we compare the kurtosis obtained from simulation with that of extracted from experimental data for the $v_3$ distribution.

Calculation of power spectrum of the little bangs

The power spectrum of the momentum distributions of particles have been estimated with Optical Glauber and Monte Carlo Glauber initial conditions for relativistic heavy ion collisions. The evaluation procedure adopted in this work is analogous to the one used in the calculation of power spectrum in Cosmic Microwave Background Radiation (CMBR). The power spectrum due to perturbations in the phase space distribution of the particles has also been evaluated. The perturbation in phase space has been evolved through the Boltzmann transport equation in an expanding quark–gluon plasma (QGP) background. The expansion of the QGP has been treated within the purview of (3[Formula: see text]+[Formula: see text]1)-dimensional relativistic hydrodynamics. We observe that the nonequilibrium effects introduced as perturbations in the phase space distributions can be traced through its variation with temperature which is distinctly different from the case of vanishing perturbation. A relation has been derived between the power spectrum and the flow harmonics.

Effects of equation of state on hydrodynamic expansion, spectra, flow harmonics and two-pion interferometry

We perform an extensive study of the role played by the equation of state (EoS) in the hydrodynamic evolution of the matter produced in relativistic heavy ion collisions. By using the same initial conditions and freeze-out scenario, the effects of different equations of state are compared by calculating their respective hydrodynamical evolution, particle spectra, harmonic flow coefficients [Formula: see text], [Formula: see text] and [Formula: see text] and two-pion interferometry radius parameters. The equations of state investigated contain distinct features, such as the nature of the phase transition, as well as strangeness and baryon density contents, which are expected to lead to different hydrodynamic responses. The results of our calculations are compared to the data recorded at two RHIC energies, 130[Formula: see text]GeV and 200[Formula: see text]GeV. The three equations of state used in the calculations are found to describe the data reasonably well. Differences can be observed among the studied observables, but they are quite small. In particular, the collective flow parameters are found not to be sensitive to the choice of the EOS, whose implications are discussed.

Energy dependence and fluctuations of anisotropic flow in Pb-Pb collisions at sqrt{s_{mathrm{NN}}}=5.02 and 2.76 TeV

Measurements of anisotropic flow coefficients with two- and multi-particle cumulants for inclusive charged particles in Pb-Pb collisions at sqrt{s_{mathrm{NN}}}=5.02 and 2.76 TeV are reported in the pseudorapidity range |η| < 0.8 and transverse momentum 0.2 < pT < 50 GeV/c. The full data sample collected by the ALICE detector in 2015 (2010), corresponding to an integrated luminosity of 12.7 (2.0) μb−1 in the centrality range 0-80%, is analysed. Flow coefficients up to the sixth flow harmonic (v6) are reported and a detailed comparison among results at the two energies is carried out. The pT dependence of anisotropic flow coefficients and its evolution with respect to centrality and harmonic number n are investigated. An approximate power-law scaling of the form vn(pT) ∼ pTn/3 is observed for all flow harmonics at low pT (0.2 < pT < 3 GeV/c). At the same time, the ratios vn/vmn/ m are observed to be essentially independent of pT for most centralities up to about pT = 10 GeV/c. Analysing the differences among higher-order cumulants of elliptic flow (v2), which have different sensitivities to flow fluctuations, a measurement of the standardised skewness of the event-by-event v2 distribution P(v2) is reported and constraints on its higher moments are provided. The Elliptic Power distribution is used to parametrise P(v2), extracting its parameters from fits to cumulants. The measurements are compared to different model predictions in order to discriminate among initial-state models and to constrain the temperature dependence of the shear viscosity to entropy-density ratio.

Directed, elliptic and triangular flow of protons in Au+Au reactions at 1.23 A GeV: a theoretical analysis of the recent HADES data

Recently, the HADES experiment at GSI has provided preliminary data on the directed flow, v1 elliptic flow, v2 and triangular flow, v3 of protons in Au+Au reactions at a beam energy of 1.23 A GeV. Here we present a theoretical discussion of these flow harmonics within the UrQMD transport approach. We show that all flow harmonics, including the triangular flow, provide a consistent picture of the expansion of the system, if potential interactions are taken into account. Investigating the dependence of the flow harmonics on the nuclear interaction potentials it is shown that especially v3 can serve as a sensitive probe for the nuclear equation of state at such low energies. The triangular flow and its excitation function with respect to the reaction-plane were calculated for the first time and indicate a complex interplay of the time-evolution of the system and the initial conditions at low beam-energies. Our study also indicates a significant softening of the equation of state at beam energies above Elab > 7 A GeV which can be explored by at the future FAIR facility.

Correlation measurements between flow harmonics in Au+Au collisions at RHIC

Flow harmonics (vn) in the Fourier expansion of the azimuthal distribution of particles are widely used to quantify the anisotropy in particle emission in high-energy heavy-ion collisions. The symmetric cumulants, SC(m,n), are used to measure the correlations between different orders of flow harmonics. These correlations are used to constrain the initial conditions and the transport properties of the medium in theoretical models. In this Letter, we present the first measurements of the four-particle symmetric cumulants in Au+Au collisions at sNN=39 and 200 GeV from data collected by the STAR experiment at RHIC. We observe that v2 and v3 are anti-correlated in all centrality intervals with similar correlation strengths from 39 GeV Au+Au to 2.76 TeV Pb+Pb (measured by the ALICE experiment). The v2–v4 correlation seems to be stronger at 39 GeV than at higher collision energies. The initial-stage anti-correlations between second and third order eccentricities are sufficient to describe the measured correlations between v2 and v3. The best description of v2–v4 correlations at sNN=200GeV is obtained with inclusion of the system's nonlinear response to initial eccentricities accompanied by the viscous effect with η/s>0.08. Theoretical calculations using different initial conditions, equations of state and viscous coefficients need to be further explored to extract η/s of the medium created at RHIC.

Nonlinear coupling of flow harmonics: Hexagonal flow and beyond

Higher Fourier harmonics of anisotropic flow ($v_4$ and beyond) get large contributions induced by elliptic and triangular flow through nonlinear response. We present a general framework of nonlinear hydrodynamic response which encompasses the existing one, and allows to take into account the mutual correlation between the nonlinear couplings affecting Fourier harmonics of any order. Using Large Hadron Collider data on Pb+Pb collisions at ~$\sqrt[]{s}=2.76$ TeV, we perform an application of our formalism to hexagonal flow, $v_6$, a coefficient affected by several nonlinear contributions which are of the same order of magnitude. We obtain the first experimental measure of the coefficient $\chi_{624}$, which couples $v_6$ to $v_2$ and $v_4$. This is achieved by putting together the information from several analyses: event-plane correlations, symmetric cumulants, as well as new higher-order moments recently analyzed by the ALICE collaboration. The value of $\chi_{624}$ extracted from data is in fair agreement with hydrodynamic calculations, although with large error bars, which would be dramatically reduced by a dedicated analysis. We argue that within our formalism the nonlinear structure of a given higher harmonic can be determined more accurately than the harmonic itself, and we emphasize potential applications to future measurements of $v_7$ and $v_8$.

Predictions for multiplicities and flow harmonics in 5.44 TeV Xe+Xe collisions at the CERN Large Hadron Collider

We present the next-to-leading-order event-by-event EKRT model predictions for the centrality dependence of the charged hadron multiplicity in the pseudorapidity interval $|\eta|\le 0.5$, and for the centrality dependence of the charged hadron flow harmonics $v_n\{2\}$ obtained from 2-particle cumulants, in $\sqrt{s_{NN}}=5.44$ TeV Xe+Xe collisions at the CERN Large Hadron Collider. Our prediction for the 0-5 \% central charged multiplicity is $dN_{\rm ch}/d\eta =1218\pm 46$. We also predict $v_n\{2\}$ in Xe+Xe collisions to increase more slowly from central towards peripheral collisions than those in a Pb+Pb system. We find that at $10 \dots 50$\% centralities $v_2\{2\}$ is smaller and $v_3\{2\}$ is larger than in the Pb+Pb system while $v_4\{2\}$ is of the same magnitude in both systems. We also find that the ratio of flow harmonics in Xe+Xe collisions and in Pb+Pb collisions shows a slight sensitivity to the temperature dependence of the shear-viscosity-to-entropy ratio. As we discuss here, the new nuclear mass-number systematics especially in the flow harmonics serves as a welcome further constraint for describing the space-time evolution of a heavy-ion system and for determining the shear viscosity and other transport properties of strongly interacting matter.

Systematic studies of correlations between different order flow harmonics in Pb-Pb collisions at sNN=2.76 TeV

The correlations between event-by-event fluctuations of anisotropic flow harmonic amplitudes have been measured in Pb-Pb collisions at sNN=2.76 TeV with the ALICE detector at the Large Hadron Collider. The results are reported in terms of multiparticle correlation observables dubbed symmetric cumulants. These observables are robust against biases originating from nonflow effects. The centrality dependence of correlations between the higher order harmonics (the quadrangular v4 and pentagonal v5 flow) and the lower order harmonics (the elliptic v2 and triangular v3 flow) is presented. The transverse momentum dependences of correlations between v3 and v2 and between v4 and v2 are also reported. The results are compared to calculations from viscous hydrodynamics and a multiphase transport (AMPT) model calculations. The comparisons to viscous hydrodynamic models demonstrate that the different order harmonic correlations respond differently to the initial conditions and the temperature dependence of the ratio of shear viscosity to entropy density (η/s). A small average value of η/s is favored independent of the specific choice of initial conditions in the models. The calculations with the AMPT initial conditions yield results closest to the measurements. Correlations among the magnitudes of v2, v3, and v4 show moderate pT dependence in midcentral collisions. This might be an indication of possible viscous corrections to the equilibrium distribution at hadronic freeze-out, which might help to understand the possible contribution of bulk viscosity in the hadronic phase of the system. Together with existing measurements of individual flow harmonics, the presented results provide further constraints on the initial conditions and the transport properties of the system produced in heavy-ion collisions.6 MoreReceived 1 October 2017DOI:https://doi.org/10.1103/PhysRevC.97.024906Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.©2018 CERN, for the ALICE CollaborationPhysics Subject Headings (PhySH)Research AreasCollective flowNuclear reactionsParticle correlations & fluctuationsRelativistic heavy-ion collisionsNuclear Physics

Development of heavy-flavour flow-harmonics in high-energy nuclear collisions

We employ the POWLANG transport setup, developed over the last few years, to provide new predictions for several heavy-flavour observables in relativistic heavy-ion collisions from RHIC to LHC center-of-mass energies. In particular, we focus on the development of the flow-harmonics v2 and v3 arising from the initial geometric asymmetry in the initial conditions and its associated event-by-event fluctuations. Within the same transport framework, for the sake of consistency, we also compare the nuclear modification factor of the pT spectra of charm and beauty quarks, heavy hadrons and their decay electrons. We compare our findings to the most recent data from the experimental collaborations. We also study in detail the contribution to the flow harmonics from the quarks decoupling from the fireball during the various stages of its evolution: although not directly accessible to the experiments, this information can shed light on the major sources of the final measured effect.

Coupled dynamics of heavy and light flavor flow harmonics from EPOSHQ

We pursue the study of event by event correlations between low-mass particles and heavy mesons flow harmonics in ultrarelativistic heavy ion collisions and clarify some ambiguities found in one of our previous work.

Measurements of charm hadron production and anisotropic flow in Au+Au collisions at 200 GeV with the STAR experiment at RHIC

Heavy flavor quarks, owing to their large masses, are predominantly produced through initial hard parton scatterings in heavy-ion collisions, and thus are excellent probes to study properties of the strongly coupled Quark Gluon Plasma (sQGP) medium produced in these collisions. Measurements of anisotropic flow harmonics of heavy flavor hadrons can provide information on the properties of the medium, including the heavy flavor transport coefficient. Charm quark hadronization mechanism in the sQGP medium can be studied through measurements of yields of different charm hadrons. In these proceedings we report on the measurements of elliptic and triangular flow harmonics of D0 mesons as well as the yield ratios of D±s/D0 and Λ±c/D0 in Au+Au collisions at [see formula in PDF] = 200 GeV at RHIC with the STAR detector. These measurements use the STAR Heavy Flavor Tracker (HFT) to reconstruct charm hadrons via their hadronic decay channels. Results are compared to model calculations and the implications on the understanding of charm quark dynamics in the medium are discussed.

Comparisons of Rare-Earth and Rare-Earth-Free External Rotor Permanent Magnet Assisted Synchronous Reluctance Motors

This paper presents the comparisons of five-phase external rotor permanent magnet assisted synchronous reluctance motors (PMa-SynRMs) with neodymium (Nd) and ferrite (Fe) based PMs. Five-phase PMa-SynRMs, in general, have been proposed as an alternative to conventional three-phase motors utilized in automobile applications. However, with increased attention towards higher torque density and lower torque pulsations in electric motor designs, five-phase external rotor PMa-SynRMs can be highly attractive. In addition to increased fault tolerance and reliability of five-phase PMa-SynRMs, an adaptation of external rotor design can significantly increase the average torque developed with reduced torque pulsations and back-electromotive force (EMF) harmonics. However, with the design complexity involved in such high performance five-phase PMa-SynRM designs, the shape of flux barrier and choice of PMs have been one of the main challenges. In this study, novel five-phase external rotor PMa-SynRM designs with optimal flux barrier number and shape with Nd and Fe PMs are presented. Finite element simulations are conducted on the 2-D designs to compare their performance characteristics such as torque developed, torque pulsations, structural stability, thermal heat flow, demagnetization effects, and back-EMF harmonics. Optimal 3.8 kW Nd-based and 3.7 kW Fe-based five-phase external rotor PMa-SynRM designs are fabricated, and experimental tests are conducted on the prototypes to compare the results with a benchmark internal rotor PMa-SynRM.

General formulae for dipole Wilson line correlators with the Color Glass Condensate

We present general formulae to compute Wilson line correlators with the Color Glass Condensate described by the McLerran-Venugopalan model. We explicitly construct a complete and non-orthogonal set of color-singlet bases and write matrix elements down, so that the exponential of the matrix leads to the Wilson line correlators. We further develop a systematic perturbative expansion of dipole Wilson line correlators in terms of 1/Nc where Nc is the color number. As a phenomenological application we calculate the flow harmonics vn{m} in the dipole model and discuss the Nc scaling.

Dynamical vs. geometric anisotropy in relativistic heavy-ion collisions: Which one prevails?

We study the influence of geometric and dynamical anisotropies on the development of flow harmonics and, simultaneously, on the second- and third-order oscillations of femtoscopy radii. The analysis is done within the Monte Carlo event generator HYDJET++, which was extended to dynamical triangular deformations. It is shown that the merely geometric anisotropy provides the results which anticorrelate with the experimental observations of either $v_2$ (or $v_3$) or second-order (or third-order) oscillations of the femtoscopy radii. Decays of resonances significantly increase the emitting areas but do not change the phases of the radii oscillations. In contrast to the spatial deformations, the dynamical anisotropy alone provides the correct qualitative description of the flow and the femtoscopy observables simultaneously. However, one needs both types of the anisotropy to match quantitatively the experimental data.

Flow harmonics from self-consistent particlization of a viscous fluid

The quantitative extraction of quark-gluon plasma (QGP) properties from heavy-ion data, such as its specific shear viscosity $\ensuremath{\eta}/s$, typically requires comparison to viscous hydrodynamic or ``hybrid'' hydrodynamics + transport simulations. In either case, one has to convert the fluid to hadrons, yet without additional theory input the conversion is ambiguous for dissipative fluids. Here, shear viscous phase-space corrections calculated using linearized transport theory are applied in Cooper-Frye freeze-out to quantify the effects on anisotropic flow coefficients ${v}_{n}({p}_{T})$ at the energies available at both the BNL Relativistic Heavy Ion Collider and the CERN Large Hadron Collider. Expanding upon our previous flow harmonics studies [D. Molnar and Z. Wolff, Phys. Rev. C 95, 024903 (2017); Z. Wolff and D. Molnar, J. Phys.: Conf. Ser. 535, 012020 (2014)], we calculate pion and proton ${v}_{2}({p}_{T})$, ${v}_{4}({p}_{T})$, and ${v}_{6}({p}_{T})$, but here we incorporate a hadron gas that is chemically frozen below a temperature of 175 MeV and use hypersurfaces from realistic viscous hydrodynamic simulations. For additive quark model cross sections and relative phase-space corrections with ${p}^{3/2}$ momentum dependence rather than the quadratic Grad form, we find at moderately high transverse momentum noticeably higher ${v}_{4}({p}_{T})$ and ${v}_{6}({p}_{T})$ for protons than for pions. In addition, the value of $\ensuremath{\eta}/s$ deduced from elliptic flow data differs by nearly 50% from the value extracted using the naive ``democratic Grad'' form of freeze-out distributions. To facilitate the use of the self-consistent viscous corrections calculated here in hydrodynamic and hybrid calculations, we also present convenient parametrizations of the corrections for the various hadron species.

Measurement of the Width and Skewness of Elliptic Flow Fluctuations in PbPb Collisions at 5.02 TeV with CMS

Flow harmonic fluctuations are studied for PbPb collisions at sNN=5.02 TeV using the CMS detector at the LHC. Flow harmonic probability distributions p(v2) are obtained by unfolding smearing effects from observed azimuthal anisotropy distributions using particles of 0.3 <pT < 3.0 GeV/c and |η| < 1.0. Cumulant flow harmonics are determined from the moments of p(v2) and used to estimate the standardized elliptic flow skewness. Hydrodynamic models predict this skewness to be negative with respect to the reaction plane. A statistically significant negative skewness is observed for all centrality bins as evidenced by a splitting between v2 {4}, v2 {6}, and v2 {8} cumulants. Elliptic power law distribution fits are made to p(v2) distributions to infer information on the nature of initial-state eccentricity distributions and found to provide a more accurate description of the fluctuations than a Bessel-Gaussian parametrization.

Mixed Harmonic Correlations: Hydrodynamic Predictions at RHIC using Experimental Analysis Techniques

Correlations of different azimuthal flow harmonics vn and symmetry planes Ψn can add constraints to theoretical models, and probe aspects of the system that are independent of the traditional single-harmonic measurements. Using NeXSPheRIO, a hydrodynamical model which has accurately reproduced a large set of single-harmonic correlations at RHIC, we make predictions of these new observables for 200 A GeV Au+Au Collisions, providing an important baseline for comparison to correlations of flow harmonics, which contain non-trivial information about the initial state. We also point out how to properly compare theoretical calculations to measurements using wide centrality bins and non-trivial event weighting.