Experimental studies of heavy-ion collisions at the Relativistic Heavy Ion Collider (RHIC) indicate that a state of matter predicted by Quantum Chromodynamics (QCD), called Quark-Gluon Plasma (QGP), is formed in these collisions. Many of the ongoing studies are aimed at characterizing the transport properties (particularly, the specific shear viscosity: the ratio of shear viscosity to entropy density η/s) of the QGP. The azimuthal anisotropy of particle production relative to the collision symmetry planes, known as anisotropic flow, is a key observable in many such studies because it displays the viscous hydrodynamic response to the initial spatial distribution created in the early stages of the collision [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16].
The anisotropic flow can be characterized by the Fourier expansion [2,17] of the particle azimuthal angle (φ) distributions,
where n is the n-th order flow symmetry plane. The n-th complex anisotropic flow vector with v n magnitude and n direction is defined as V n = v n e in n . The flow coefficient v 1 is commonly termed as directed flow, v 2 is the elliptic flow, and v 3 is the triangular flow. Anisotropic flow studies of higher-order flow harmonics v n>3 [12,[18][19][20][21][22][23][24], correlation between different flow harmonics [22,[25][26][27][28][29][30][31] and flow fluctuations [20,[32][33][34] have led to a deeper understanding of the initial conditions [35] and the properties of the matter created in heavy-ion collisions.
In the hydrodynamic models, anisotropic flow arises from the evolution of the medium in the presence of initial-state energy density anisotropies, characterized by the complex eccentricity vectors [29,[36][37][38][39]]:
where ρ e (r, ϕ) is the initial anisotropic energy density profile, ε n = |E n | 2 1/2 represents the eccentricity vectors magnitude and n denotes the azimuthal direction of the eccentricity vector [39][40][41].
The elliptic and triangular flow harmonics are, to a reasonable approximation, linearly proportional to the initial-state anisotropies, ε 2 and ε 3 , respectively [9,29,[42][43][44][45][46][47][48]:
where k n is the proportionality factor that encodes the medium response, and is expected to be sensitive to η/s and the system lifetime [49]. Therefore, the ratio v n /ε n (for n = 2, 3) could be used as a tool to probe η/s of the QGP [19]. In contrast, the higher-order flow harmonics are expected to arise from a modecoupled (nonlinear) response to the lower-order eccentricities, ε 2 and/or ε 3 [14,40,41] in addition to linear response to the sameorder initial-state anisotropies [50]:
where V L n and V mc n represents the linear and the mode-coupled contributions to the flow vector V n respectively. The χ 4,22 and χ 5,23 are the mode-coupled response coefficients which define the magnitude of the V mc n>3 measured with respect to the lower-order symmetry plane angle(s). Also, the mode-coupled contribution of V n is expected to reflect the correlation between different order flow symmetry planes, n , which could shed light on the initial stage dynamics [25,27,28,40,[51][52][53][54][55][56].
The v 2 and v 3 harmonics are sensitive to the respective influence of the initial-state eccentricity and the final-state viscous attenuation, which have proven difficult to disentangle. The modecoupled coefficients show characteristically different dependencies on the viscous attenuation and the initial-state eccentricity [48]. Therefore, they can be used in conjunction with measurements for the v 2 and v 3 harmonics to leverage additional unique constraints for initial-state models, as well as reliable extraction of transport coefficients.
In this paper we report new differential and integral measurements of v 4 and v 5 and their mode-coupled response coefficients, obtained with the two-and multiparticle cumulant methods described in Section 2. Measurements of these quantities as functions of collision centrality and charged particle transverse momentum, p T , in Au+Au collisions at
s N N = 200 GeV, are reported in Section 3. The presented results and conclusions are summarized in Section 4.
The data reported in this analysis were collected with the STAR detector at RHIC using a minimum-bias trigger [57] in 2011.
Charged particle tracks, measured in pseudorapidity range |η| < 1.0 and covering all azimuthal angles of the Time Projection Chamber (TPC) [58], are used to reconstruct the collision vertices. Collision centrality is determined from the measured event-by-event multiplicity with the assistance of the Monte Carlo Glauber simulation [59,60]. Tracks included in the analysis are required to have a distance of closest approach to the primary vertex of less than 3 cm, and to have at least 15 TPC space points used in their reconstruction. In order to remove track splitting, the ratio of the number of fit points to the maximum possible number of TPC fit points was required to be larger than 0.52. The transverse momentum (p T ) threshold of these tracks is ∼0.2 GeV/c; the cut p T ≤ 4 GeV/c was employed to reduce a possible nonflow influence from jets. Events are chosen with vertex positions within ±30 cm from the TPC center (along the beam direction), and within ±2 cm in the radial direction relative to the center of the beam intersection. Also, the absolute difference between the two z-vertex positions defined by the TPC and Vertex Position Detector is required to be less than 3 cm to decrease beam-induced background and pileup.
The systematic uncertainties associated with the measurements presented in this work are estimated by changing different parameters of the analysis and comparing the results with their baseline values. The systematic uncertainty associated with the event selection is estimated by using more restrictive requirements for the vertex positions determined by the TPC along the beam direction (-30 to 0 cm or 0 to 30 cm instead of the nominal value of ±30 cm). The systematic uncertainty arising from track selection is evaluated by employing more strict requirements: (i) Distance of Closest Approach (DCA) is changed to be less than 2 cm instead of the standard value of 3 cm, and (ii) number of TPC space points from more than 15 points to more than 20 points. The systematic uncertainty associated with the nonflow effects, due to Bose-Einstein correlations, resonance decays and the fragments of individual jets, is estimated by investigating the impact of a pseudorapidity gap, η = η 1η 2 , for the track pairs used in the measurements. Studies were performed for η values of 0.6, 0.7, and 1.0. Table 1 shows the systematic uncertainties evaluated for this work. The overall systematic uncertainty was calculated by combining uncertainties from different sources in quadrature. In the ensuing figures, the overall systematic uncertainties (which do not include those from η variation) are shown as open boxes; statistical uncertainties are shown as vertical lines.
The two-and multiparticle cumulant techniques are used in this work. The framework for the cumulant method is described in Refs. [51,61], which was extended to the case of subevents in Refs. [62,63]. In this work, the two-and multiparticle correlations were constructed using the two-subevents cumulant method with particle weights [as outlined in Ref. [63]], for η > 0.7 between the subevents A and B (i.e., η A > 0.35 and η B < -0.35). The use of the two-subevents method helps to suppress the nonflow correlations. The two-and multiparticle correlations can be written as:
where indicates the average over all particles in a single event and then the average over all events, k = n + m, n = 2, m = 2 or 3, and ϕ i is the azimuthal angle of the i-th particle.
Using Eqs. ( 6)-( 8), the mode-coupled contribution in higherorder anisotropic flow harmonics, v 4 and v 5 , can be approximated as [41,64]:
and the linear contribution to v 4 and v 5 can be given as:
Equation (11) assumes that the linear and mode-coupled contributions in v 4 and v 5 are independent [41,65]. The ratios of the mode-coupled contribution to the inclusive v 4 and v 5 are expected to measure the correlations between different order flow symmetry planes [66] and are expressed as ρ 4,22 and ρ 5,23 , respectively. The ρ 4,22 and ρ 5,23 can be approximated as: The mode-coupled response coefficients, χ 4,22 and χ 5,23 , which quantify the contributions of the mode-coupling to the higherorder anisotropic flow harmonics, are defined as
In Eq. ( 15) for the differential χ 5,23 , this work further makes the [40]. These dimensionless ratios that represent the mode-coupled coefficients in Eq. ( 4) are expected to be weakly sensitive to viscous effects [48].
In A+A collisions, short-range nonflow correlations contribute to the measured three-particle correlators C 4,22 and C 5,23 [65]. However, such correlations can be reduced by using subevents cumulant methods [63]. Fig. 1 compares the C 4,22 and C 5,23 values obtained from the standard (i.e., the three particles are selected using the entire detector acceptance) and the two-subevents cumulant methods as a function of centrality in the range 0.2 < p T < 4.0 GeV/c for Au+Au collisions at
s N N = 200 GeV. For mid-central to peripheral collisions, the magnitudes of the measured C 4,22 and C 5,23 from the standard cumulant method are larger than those from the subevents cumulant method, compatible with the expectation that the subevents cumulant method can further reduce the nonflow correlations. The shaded bands in Fig. 1 indicate viscous hydrodynamic model predictions [67,68], as summarized in Table 2. Note that these model predictions include an influence from changes in the initial-and final-state assumptions incorporated in model calculations. The model predictions, which were generated with the standard cumulant method, show good qualitative agreement with both C 4,22 and C 5,23 . However, Hydro-2 b with no hadronic cascade gives a better description of the data for C 4,22 and C 5,23 obtained with the two-subevents cumulant method.
The centrality dependence of the inclusive, linear and modecoupled v 4 and v 5 in the p T range from 0.2 to 4.0 GeV/c for Au+Au collisions at
s N N = 200 GeV are shown in Fig. 2. They indicate that the linear mode of v 4 and v 5 depends weakly on the collision centrality and constitutes the dominant contribution to the inclusive v 4 and v 5 in central collisions. These results are compared to similar LHC measurements in the p T range from 0.2 to 5.0 GeV/c and pseudorapidity range |η| < 0.8 for Pb+Pb collisions at √ s N N = 2.76 TeV [66]. The comparison indicates strikingly similar patterns for the RHIC and LHC measurements, albeit with a difference in the magnitude of the measurements. This observed difference could result from a sizable difference in the p T for the p T -integrated v 4 and v 5 measurements at RHIC and the LHC, respectively. Here, it is noteworthy that even though the p T range for both measurements is similar, the inverse slopes of the hadron p T spectra are larger at the LHC than at RHIC. Subtleties related to a difference in the viscous properties of the medium created at RHIC and LHC energies could also contribute to the observed difference in the magnitude of the measurements [67].
The centrality dependence of the mode-coupled response coefficients, χ 4,22 and χ 5,23 , for Au+Au collisions, is presented in Fig. 3(a) and (b) for the range 0.2 < p T < 4.0 GeV/c. They show a weak centrality dependence, akin to the patterns observed for similar measurements at the LHC for Pb+Pb collisions at
76 TeV [66] (closed symbols). These patterns suggest that (i) the centrality dependence observed for the mode-coupled v 4 and v 5 (cf., Figs.
) stems from the lower-order flow harmonics and (ii) the mode-coupled response coefficients are dominated by initial-state eccentricity couplings which have a weak dependence on beam energy. The shaded bands in Figs. 3(a) and (b) show that the predictions from the viscous hydrodynamic models [67,68] summarized in Table 2, give a good qualitatively description of the χ 4,22 and χ 5,23 data. However, the predictions from Hydro-1 and Hydro-2 b (cf. Table 2), give the overall closest description to χ 4,22 and χ 5,23 .
Figs.
In summary, we have presented new differential measurements of the charge-inclusive, linear and mode-coupled contributions to the higher-order anisotropic flow coefficients v 4 and v 5 , modecoupled response coefficients χ 4,22 and χ 5,23 and the correlations to the data indicate good qualitatively agreement. However, none of the models provide a simultaneous description of the threeparticle correlations, the mode-coupled response coefficients, and the correlations of event plane angles. These higher-order flow measurements could provide additional stringent constraints to discern between initial state models and aid precision extraction of the transport properties of the medium produced in the collisions.
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.