Systematic Studies of Elliptic Flow Measurements
in Au+Au Collisions at = 200 GeV
Abstract
We present inclusive charged hadron elliptic flow () measured over the pseudorapidity range 0.35 in Au+Au collisions at = 200 GeV. Results for are presented over a broad range of transverse momentum ( = 0.2–8.0 GeV/) and centrality (0–60%). In order to study nonflow effects that are not correlated with the reaction plane, as well as the fluctuations of , we compare two different analysis methods: (1) event plane method from two independent subdetectors at forward ( = 3.1–3.9) and beam () pseudorapidities and (2) twoparticle cumulant method extracted using correlations between particles detected at midrapidity. The two eventplane results are consistent within systematic uncertainties over the measured and in centrality 0–40%. There is at most 20% difference of the between the two event plane methods in peripheral (40–60%) collisions. The comparisons between the twoparticle cumulant results and the standard event plane measurements are discussed.
pacs:
25.75.LdPHENIX Collaboration
I Introduction
Collisions of Au+Au nuclei at the Relativistic Heavy Ion Collider (RHIC) produce matter at very high energy density Arsene et al. (2005); Adcox et al. (2005); Back et al. (2005); Adams et al. (2005a). The dynamical evolution of this hot and dense medium reflects its state and the degrees of freedom that govern the different stages it undergoes Gyulassy and McLerran (2005); Muller (); Shuryak (2005). Azimuthal anisotropy measurements serve as a probe of the degree of thermalization, transport coefficients and the equation of state (EOS) Ollitrault (1992); Kolb et al. (2001); Hirano and Nara (2004) of the produced medium.
Azimuthal correlation measurements in Au+Au collisions at RHIC have been shown to consist of a mixture of jet and harmonic contributions Ajitanand (2003); Chiu (2003); Adler et al. (2003a, 2005). Jet contributions are found to be relatively small for 2.0 GeV/, with awayside jet yields strongly suppressed Adler et al. (2003a). Significant modifications to the awayside jet topology have also been reported Adams et al. (2005b); Adler et al. (2006a); Adare et al. (2008). The harmonic contributions are typically characterized by the Fourier coefficients,
(1) 
where represents the azimuthal emission angle of a charged hadron and is the azimuth of the reaction plane defined as containing both the direction of the impact parameter vector and the beam axis. The brackets denote statistical averaging over particles and events. The first two harmonics and are referred to as directed and elliptic flow, respectively.
It has been found that at low ( GeV/) the magnitude and trends of are underpredicted by hadronic cascade models supplemented with string dynamics Bleicher and Stoecker (2002), but are well reproduced by models which either incorporate hydrodynamic flow Shuryak (2005); Kolb et al. (2001) with a first order phase transition and rapid thermalization, fm/c Adler et al. (2003b), or use a quasiparticle ansatz but include more than just 2to2 interactions Xu et al. (2008).
The mass dependence of as a function of has been studied using identified baryons and mesons Adler et al. (2003b); Adams et al. (2004a) and empirical scaling of elliptic flow per constituent quark was observed when the signal and the of the hadron were divided by the number of constituent quarks ( = 2 for mesons, 3 for baryons). This scaling is most clearly observed by plotting the data as a function of transverse kinetic energy Adare et al. (2007), where and denote the transverse mass and mass of the particle, respectively. A recent study Huang (2008) finds that the constituent quark scaling holds up to GeV. This indicates partonic, rather than hadronic flow, and suggests that the bulk matter collectivity develops before hadronization takes place Molnár and Voloshin (2003); Fries et al. (2003); Greco et al. (2003). Results for the of the meson further validate the observation of partonic collectivity. The is not expected to be affected by hadronic interactions in the late stages of the medium evolution, due to its small interaction cross section with nonstrange hadrons Shor (1985).
All of the measurements referenced above were performed using the event plane method Poskanzer and Voloshin (1998). In PHENIX the event plane was determined at forward and backward pseudorapidities ( = 3.1–3.9) with the assumption that correlations induced by elliptic flow dominate over all other nonflow correlations Adler et al. (2003b). Nonflow correlations are those which are not correlated with the reaction plane. Common sources of nonflow correlations include jets, the nearside ridge, quantum correlations and resonance decays. Simulation studies Adler et al. (2003b); Jia (2007) have shown that the correlations from jets and dijets become negligible when the rapidity separation between the particles and the event plane is greater than three units. Thus we expect that the event plane at forward pseudorapidities = 3.1–3.9 in the PHENIX experiment would not have significant jetcorrelation with particles measured within the PHENIX central arm spectrometer covering the pseudorapidity window . PHOBOS has observed that in central Au+Au collisions there is a ridge of particles Alver et al. () that are correlated in azimuthal angle with a high particle and that this ridge of particles extends to (for midrapidity triggers). The ridge could produce a nonflow correlation that we can provide information by using our measurements that are made with different techniques and at different rapidities.
Eventbyevent flow fluctuations can also affect the magnitude of the extracted flow signal Sorensen (2008). This occurs because the event plane at forward pseudorapidities is reconstructed using particles from participant nucleons whose positions fluctuate eventbyevent. Assuming that fluctuates according to a Gaussian distribution, the fluctuation is proportional to the fluctuation of the initial geometry. This effect scales as , where denotes the number of participant nucleons. The difference between values obtained from different methods can be quantitatively understood in terms of nonflow and fluctuation effects Ollitrault et al. ().
Hence in this paper we will compare the results from the event plane determined at two different pseudorapidities with the goal to investigate the sensitivity of to nonflow and fluctuation effects. Additionally, we extract the elliptic flow with the twoparticle cumulant method, which is expected to have higher sensitivity to nonflow contributions to .
In this paper, we describe the PHENIX measurements of elliptic flow () at midrapidity () in Au+Au collisions at = 200 GeV obtained from a cumulant analysis of twoparticle azimuthal correlations and the event plane method over a broad range of ( = 0.2–8 GeV/) and centrality (0–60%). The paper is organized as follows: Section II describes the PHENIX apparatus, with an emphasis on the detectors relevant to the presented results, as well as the track selections used in the analysis. Section III gives details of the eventplane and the cumulant methods as applied in PHENIX. Section IV discusses the systematic uncertainties of the eventplane and cumulant methods. The results from the two methods are reported in Section V. Section VI presents a comparison of results across different experiments and discussion. The values obtained from the different methods are tabulated in the Appendix.
Ii Experimental Analysis
ii.1 The PHENIX detector
The PHENIX detector consists of two central spectrometer arms at midrapidity that are designated East and West for their location relative to the interaction region, and two muon spectrometers at forward rapidity, similarly called North and South. A detailed description of the PHENIX detector can be found in Ref. Adcox et al. (2003). The layout of the PHENIX detector during data taking in 2004 is shown in Fig. 1. Each central spectrometer arm covers a pseudorapidity range of subtending degrees in azimuth and is designed to detect electrons, photons and charged hadrons. Charged particles are tracked by drift chambers (DC) positioned between 2.0 m and 2.4 m radially outward from the beam axis and layers of multiwire proportional chambers with pad readout (two in the east arm and three in the west arm) PC1, PC2 and PC3 located at a radial distance of 2.4 m, 4.2 m and 5 m, respectively. Particle identification is provided by Ring Imaging erenkov counters (RICH), a timeofflight scintillator wall (TOF), and two types of electromagnetic calorimeters (EMCAL), the lead scintillator (PbSc) and lead glass (PbGl).
The detectors used to characterize each event are the beambeam counters (BBCs) Allen et al. (2003) and the zerodegree calorimeters (ZDCs) Adler et al. (2003c). These detectors are used to determine the time of the collision, the position of the collision vertex along the beam axis and the collision centrality and also provide the event trigger. In this analysis the BBCs are also used to determine the event plane. Each BBC is composed of 64 elements and a single BBC element consists of a oneinch diameter mesh dynode photomultiplier tube (PMT) mounted on a 3 cm long quartz radiator. The BBCs are installed on the north and south sides of the collision point along the beam axis at a distance of 144 cm from the center of the interaction region and surround the beam pipe. The BBC acceptance covers the pseudorapidity range and the full range of azimuthal angles.
The ZDCs are hadronic calorimeters located on both sides of the PHENIX detector. Each ZDC is mechanically subdivided into 3 identical modules of two interaction lengths. They cover a pseudorapidity range of and measure the energy of the spectator neutrons with a 20 GeV energy resolution Adler et al. (2003c). The shower maximum detectors (ZDCSMDs) are scintillator strip hodoscopes between the first and second ZDC modules. This location approximately corresponds to the maximum of the hadronic shower. The horizontal coordinate is sampled by 7 scintillator strips of 15 mm width, while the vertical coordinate is sampled by 8 strips of 20 mm width. The active area of a ZDCSMD is 105 mm 110 mm (horizontal vertical dimension). Scintillation light is delivered to a multichannel PMT M16 by wavelengthshifter fibers. The ZDCSMD position resolution depends on the energy deposited in the scintillator. It varies from 3 mm when the number of particles exceeds 100, to 10 mm for a smaller number of particles.
ii.2 Event selection
For the analyses presented here we used approximately 850 10 minimumbias triggered events. The minimumbias trigger was defined by a coincidence between North and South BBC signals and an energy threshold of one neutron in the ZDCs. The events are selected offline to be within a vertex of less than 30 cm from the nominal center of the PHENIX spectrometer. This selection corresponds to % of the 6.9 barn Au+Au inelastic cross section at = 200 GeV Miller et al. (2007). The event centrality was determined by correlating the charge detected in the BBCs with the energy measured in the ZDCs, as shown in Fig. 2.
A Glauber model MonteCarlo simulation Glauber and Matthiae (1970); Adcox et al. (2001); Adler et al. (2004) that includes the responses of BBC and ZDC gives an estimate of the average number of participating nucleons for each centrality class. The simulation did not include fluctuations in the positions of the nucleons which give rise to eccentricity fluctuations. Table 1 lists the calculated values of for each centrality class.
Centrality  

0–10%  
10–20%  
20–30%  
30–40%  
40–50%  
50–60% 
ii.3 Track selection
Charged particle tracks are measured using information from the DC, PC1 and PC3 detectors and the vertex from the BBC. The DC has 12 wire planes which are spaced at 0.6 cm intervals along the radial direction from the beam axis. Each wire provides a track position measurement, with better than 150 m spatial resolution in the azimuthal () direction. The PC1 provides a space point in the and beam directions, albeit with lower resolution. This space point and the vertex position help determine the threedimensional momentum vector by providing the polar angle for charged tracks at the exit of the DC. Trajectories are confirmed by requiring matching hits at PC3 to reduce secondary background. Tracks are then projected back to the collision vertex through the magnetic field to determine the momentum Mitchell et al. (2002). The momentum resolution is (GeV/). The momentum scale is known to 0.7%, as determined from the reconstructed proton mass using the TOF detector. Further details on track reconstruction and momentum determination can be found in Refs. Mitchell et al. (2002); Adler et al. (2004).
The tracks reconstructed by the DC which do not originate from the event vertex have been investigated as potential background to the charged particle measurement. The main background sources include secondary particles from decays and pairs from the conversion of photons in the material between the vertex and the DC Adler et al. (2004). Tracks are required to have a hit in the PC3, as well as in the EMCAL, within at most 2 of the expected hit location in both the azimuthal and beam directions. This cut reduces the background not originating in the direction of the vertex. In order to reduce the conversion background we further require tracks to have , where denotes the energy deposited in the EMCAL and is the transverse momentum of particles measured in the DC. Since most of the electrons from photon conversion are genuine low particles that were reconstructed as high particles, requiring a large deposit of energy in the EMCAL suppresses the electron background Adler et al. (2006b). We also require that there are no associated hits in the RICH. The RICH is filled with CO gas at atmospheric pressure and has a charged particle threshold to emit erenkov photons.
Iii Methods of azimuthal anisotropy measurement
In this section we introduce the methods for azimuthal anisotropy measurements as used in the PHENIX experiment. Section III.A describes the event plane method using the BBCs and ZDCSMDs detectors and Sec. III.B describes the twoparticle cumulant method.
iii.1 Event plane method
The event plane method Poskanzer and Voloshin (1998) uses the azimuthal anisotropy signal to estimate the angle of the reaction plane. The estimated reaction plane is called the “event plane” and is determined for each harmonic of the Fourier expansion of the azimuthal distribution. The event flow vector and azimuth of the event plane for th harmonic of the azimuthal anisotropy can be expressed as
(2)  
(3)  
(4) 
where denotes the number of particles used to determine the event plane, is the azimuthal angle of each particle, and is the weight chosen to optimize the event plane resolution. Once the event plane is determined, the elliptic flow can be extracted by correlating the azimuthal angle of emitted particles with the event plane
(5) 
where is the azimuthal angle of tracks in the laboratory frame, is the th order event plane and the brackets denote an average over all charged tracks and events. The denominator Res{} is the event plane resolution that corrects for the difference between the estimated event plane and true reaction plane .
In this paper the secondharmonic event planes were independently determined with two BBCs located at forward (BBC South, referred to as BBCS) and backward (BBC North, referred to as BBCN) pseudorapidities = 3.1–3.9 Adler et al. (2003b). The difference between the two independent event planes was used to estimate the event plane resolution. The planes were also combined to determine the event plane for the full event. A large pseudorapidity gap between the charged particles detected in the central arms and the event plane at the BBCs reduces the effect of possible nonflow contributions, especially those from dijets Jia (2007). The measured of hadrons in the central arms with respect to the combined secondharmonic BBC event plane will be denoted throughout this paper as {BBC}.
Two firstharmonic event planes were also determined using spectator neutrons at the two shower maximum detectors (ZDCSMDs) that are sandwiched between the first and second modules of the ZDCs. Forward (ZDCS) and backward (ZDCN) SMDs which cover pseudorapidity 6.5 were used. The measured of hadrons in the central arms determined with respect to the firstharmonic ZDCSMD event plane will be denoted as {ZDCSMD}.
The pseudorapidity gap between the hadrons measured in the central arms and the ZDCSMDs is larger than that for the BBCs which could cause a further reduction of nonflow contributions on {ZDCSMD}. Since the ZDCSMD measures spectator neutrons, the ZDCSMD event plane should be insensitive to fluctuations in the participant event plane. Hence fluctuations in {ZDCSMD} should be suppressed up to fluctuations in the spectator positions.
For completeness, two further event planes are defined 1) a combined event plane defined by the weighted average of event planes at the forward and backward pseudorapidities for both BBCs and ZDCSMDs, and 2) an event plane found using tracks in the central arm. The event plane at the central arms (CNT) is only used to estimate the resolution of BBC and ZDCSMD event planes by using three subevents combination of the ZDCSMD, BBC and CNT.
iii.1.1 Event plane determination
To determine an event plane the contribution at each azimuthal angle needs to be appropriately weighted depending on the detector used. For the BBC we chose the weights to be the number of particles detected in each phototube, while for the ZDCSMD the weights were based on the energy deposited in each of the SMD strips. For the CNT event plane the weight was taken to be proportional to up to 2 GeV/ and constant for 2 GeV/. For the CNT event plane we also adopted a unit weight () and found that the resulting CNT event plane resolution extracted by comparing the CNT event plane with the BBC and ZDCSMD planes was nearly identical when using the dependent or unit weights.
Corrections were performed to remove possible biases from the finite acceptance of the BBC and ZDCSMD. In this analysis, we applied two corrections called the recentering and shift methods. In the recentering method, event flow vectors are shifted and normalized to a Gaussian distribution by using the mean and width of flow vectors;
(6) 
This correction reduces the dependence of the event plane resolution on the laboratory angle. Most acceptance effects were removed by the application of the recentering method. However, remaining small corrections were applied after recentering using the shift method Poskanzer and Voloshin (1998), in which the reaction plane is shifted by defined by
(7)  
where = 8 in this analysis. The shift ensures that is isotropic. When was reduced to , the difference in the extracted was negligible and thus we include no systematic uncertainty due to the choice of in our results.
Independent corrections were applied to each centrality selection in 5% increments and in 20 cm steps in vertex in order to optimize the event plane resolution. The corrections were also done for each experimental run (the duration of a run is typically 13 hours) to minimize the possible timedependent response of detectors.
Figure 3 shows event plane distributions for a subsample of the entire data set. After all corrections are applied the event plane distributions are isotropic.
iii.1.2 Event plane resolution
The event plane resolution for was evaluated by both the twosubevents and threesubevents methods. In the twosubevents method the event plane resolution Poskanzer and Voloshin (1998) is expressed as
(8) 
where , is the number of particles used to determine the event plane , is the modified Bessel function of the first kind and = 1 for the second harmonic BBC event plane. For the ZDCSMD event plane the resolution is estimated with both = 1 or 2 in Eq. (8). We will discuss the difference between these estimates in Sec. IV.1.
To determine the event plane resolution we need to determine . Since the North and South BBCs have approximately the same coverage, the event plane resolution of each subdetector is expected to be the same. The same is true for the North and South ZDCSMDs. Thus, the subevent resolution for South and North event planes can be expressed as
(9) 
where denotes the event plane determined by the South (North) BBC or ZDCSMD. Once the subevent resolution is obtained from Eq. (9), one can calculate using Eq. (8). The for the full event can then be estimated by . This is then substituted into Eq. (8) to give the full event resolution. Since the multiplicity of the full event is twice as large as that of the subevent, is proportional to .
In the threesubevents method the resolution of each subevent is calculated by adding a reference event plane in Eq. (9):
(10) 
where are the harmonics of the event plane for subevent A, B and C, respectively. The multiplicity of each subevent is not necessarily the same in Eq. (10).
The resolution of each subdetector for the BBC and ZDCSMD can be evaluated with the threesubevents method. For the BBC event plane the reference event plane is chosen to be the ZDCSMD event plane and vice versa. We found that the agreement of the event plane resolutions for BBCS and BBCN is much better than 1%, while the ZDCS and ZDCN resolutions are comparable with each other within 2%.
Figure 4 shows the fullevent resolution as a function of centrality. The resolution of ZDCSMD is much smaller than that of BBC because the resolution of the firstharmonic event plane is proportional to . The dashed lines are the resolutions obtained from the threesubevents method with the CNT event plane as the reference plane. For example, the BBC event plane resolution is estimated by substituting , , and in Eq. (10). By including the CNT event plane, the BBC resolution increases by about 3% compared to that of the twosubevents method. For the ZDCSMD we observe the opposite effect, namely the resolution decreases by about 8%. In Sec. VI the resulting {BBC} and {ZDCSMD}, corrected by the resolution obtained using the ZDCBBCCNT combination, will be compared to those with the resolution determined from SouthNorth subevents. Table 2 summarizes the event plane resolutions.
Res{}  

Centrality  SN  ZDCBBCCNT 
0–10%  0.2637 0.0003  0.272 0.003 
10–20%  0.3809 0.0002  0.394 0.001 
20–30%  0.3990 0.0002  0.4106 0.0008 
30–40%  0.3634 0.0002  0.3759 0.0007 
40–50%  0.2943 0.0003  0.3067 0.0007 
50–60%  0.2106 0.0004  0.2240 0.0009 
Res{}  
Centrality  SN  ZDCBBCCNT 
0–10%  0.02 0.01  0.0223 0.0003 
10–20%  0.059 0.003  0.0574 0.0002 
20–30%  0.087 0.002  0.0818 0.0002 
30–40%  0.100 0.002  0.0928 0.0002 
40–50%  0.102 0.002  0.0920 0.0002 
50–60%  0.100 0.002  0.0798 0.0003 
iii.1.3 Correlation of event planes
Figure 5 shows the correlation of two different event planes as a function of centrality. The first harmonic event plane correlation for SouthNorth detector combinations is negative both for the ZDCSMDs and the BBCs over all centrality bins, as shown in Fig. 5(a). This is due to the fact that is an odd function of . The magnitude of the ZDCSMDs correlation is about a factor of two larger than that of the BBCs for midcentral collisions. This indicates that the magnitude of and/or the subevent multiplicity at higher pseudorapidities are larger compared to that at the BBC location, since the magnitude of the correlation is proportional to . Fig. 5(b) shows the correlation of the first harmonic event planes between BBC and ZDCSMD. The sameside correlation is negative while the oppositeside correlation is positive, which shows that the particles detected at the BBCs (dominantly charged pions emitted from participant nucleons) have the opposite sign of compared to the spectator neutrons detected at the ZDCsSMDs.
The correlation of the mixed harmonic event planes provides the sign of since the correlation is given by the expression Poskanzer and Voloshin (1998)
(11) 
Three assumptions were made to obtain Eq. (11): (1) the BBC and ZDCSMD are statistically independent, (2) the weak flow limit is applicable, and (3) the subevent multiplicity is equal in the NorthSouth direction for the same detector type. Thus the sign of the correlation of the mixed harmonic event planes in Eq. (11) is determined by the term Res, which in turn determines the sign of measured at the BBC.
Figure 6 shows the mixed harmonic correlation of the ZDCSMD and BBC event planes as a function of centrality. The approximations in Eq. (11) provide a good description of the magnitude of the measured correlation as shown by the dashed line. The correlation is positive over all centrality bins. This result indicates that the sign of at the BBC is positive.
iii.2 Cumulant method
In this section, we present the application of the cumulant method for azimuthal anisotropy measurements in PHENIX. This method uses cumulants of multiparticle correlations Borghini et al. (2001a, b) to extract the azimuthal anisotropy. The cumulant method has been successfully applied in several heavyion experiments utilizing detectors with full azimuthal coverage (NA49, STAR) Alt et al. (2003); Adler et al. (2002). Here, we describe the first application of the method for a detector with only partial azimuthal coverage. The cumulant method does not require the measurement of the reaction plane, instead the cumulants of multiparticle azimuthal correlations are related to the flow harmonics , where is the harmonic being evaluated. The cumulants can be constructed in increasing order according to the number of particles that are correlated with each other. Since PHENIX has partial azimuthal coverage, reliable extraction of azimuthal anisotropy requires the choice of a fixed number of particles from each event in order to avoid additional numerical errors Borghini et al. (2001a).
Particles in an event are selected over a fixed range where there is sufficient multiplicity. These particles (called “integral particles” hereafter) are used to determine integrated flow, that is flow measured over a large (,) bin. For differential flow measurement, we select particles (called “differential” particles) over small (,) bins, from which the integral particles are excluded so as to avoid autocorrelations. For each event a fixed number of particles, chosen at random among the integral particles in the event, are used to reconstruct the integrated flow through the generating function defined by:
(12) 
where is the weight, chosen to be equal to 1 in our analysis, is the azimuth of the detected particles, and is the multiplicity chosen for the integrated flow reconstruction. is a function of the complex variable . The average of over events is then expanded in a power series to generate multiparticle azimuthal correlations. The generating function of the cumulants, defined by
(13) 
generates cumulants of azimuthal correlations to all orders, the lowest being the second order, as detailed in Section II.B of Ref. Borghini et al. (2001a). The formulas used to compute the cumulants from which the is computed are given in Appendix B of Ref. Borghini et al. (2001a). In the case of a perfect acceptance the relations between the anisotropy parameter and the lowest order cumulants are
(14)  
(15) 
for the integrated anisotropy. Here and are the second and fourth order , respectively; whereas, and are the second and fourth order cumulants. Because the typical multiplicity of charged hadrons in PHENIX did not allow a reliable calculation of {4}, we report here only the {2} results.
The remaining differential particles in the same event are selected in different (, ) bins and the differential cumulants are calculated from the generating function
(16) 
where denotes an average over all events, and is the azimuth of each differential particle. denotes the second order differential cumulant computed with respect to the second order integral cumulant.
The differential , the second order differential with respect to the second order integrated , is calculated from the relation
(17) 
where is the second order differential cumulant. These relations have to be modified through acceptance corrections which are detailed below.
iii.2.1 Acceptance/efficiency corrections
The central arms detectors in PHENIX have only partial azimuthal coverage and the implementation of the cumulant method requires an additional acceptance correction. In order to correct for the influence of the detector acceptance on the raw anisotropy values, we apply a correction factor using the prescription described in Ref. Borghini et al. (2001a). The acceptance and efficiency of the detector is characterized by a function which is expressed in terms of the Fourier series
(18) 
The Fourier coefficients for the detector acceptance were extracted from the fit of the respective azimuthal distributions of integral and differential particles. The coefficients resulting from such fits were then used to calculate the correction factor for the raw values of the following the procedure detailed in Appendix C of Ref. Borghini et al. (2001a).
Figure 7 shows a typical azimuthal angular distribution of differential particles detected in the PHENIX central arms and the corresponding Fourier fit used to correct for acceptance inhomogeneities. The Fourier fit reproduces well the overall features of the acceptance profile. This produces typical correction factors that are in the range 1.1–1.2 for the differential flow and depend very little on centrality and , as shown in Fig. 8.
iii.2.2 Simulations
While Fig. 7 shows that the uneven detector acceptance is reproduced by the Fourier fit, a better test of the cumulant method is to use MonteCarlo simulations, as was done in Ref. Borghini et al. (2001a). For these tests events were generated with particles having a distribution of the form , with known integrated and differential azimuthal anisotropies. The anisotropy was introduced into the events by way of a Fourier weighted selection of the azimuthal angles followed by a random event rotation designed to simulate the random orientation of the reaction plane. The multiplicity of these events was chosen to reflect the typical multiplicity measured with the PHENIX detector and the angles were chosen from a filter that is representative of the PHENIX acceptance. We extracted Fourier components from these simulated results and applied these to extract corrected elliptic flow values.
Figure 9 shows selected results from these simulations. Corrected differential anisotropy values are compared for various input differential values, with the integral kept fixed. The dotted line shows the trend expected if the extracted is identical to the input value used to generate the events. The good agreement between the input and extracted attests to the reliability of the analysis method within the acceptance of the PHENIX central arms.
Iv Systematic Uncertainties
In this section, we present the systematic uncertainties on the from the event plane method (Sec. IV.1) and the twoparticle cumulant method (Sec. IV.2). Table 3 lists the different sources of systematic errors for each method. The errors in Tab. 3 are categorized by type:

pointtopoint error uncorrelated between bins,

correlated, all points move in the same direction but not by the same factor,

an overall normalization error in which all points move by the same factor independent of .
Error source  Percentage error  Type  

{BBC}  {ZDCSMD}  
Background contribution  5% in 4 GeV/  B  
5–30% in 4 GeV/  B  
Event plane calibration  1–5%  C  
Event plane determination  1–4%  1–16%  C 
Acceptance effect  1%  1–25%  C 
on event planes  
{2}  
Fixed multiplicity  5%  B  
Integrated range  3–8%  B  
Background correction  6–10%  B 
iv.1 Event plane method
iv.1.1 Background contributions
In order to study the influence of background on our results, we varied one of the track selections while keeping other cuts fixed and investigated the effect on in the following two cases: (i) the PC3 and EMCAL matching cuts, 1.5 and 2.5 matching cuts and (ii) and . For both conditions, we found that the difference of the is 1–2% for 4 GeV/, and 5–20% for 4 GeV/ depending on and centrality.
The effect of the RICH veto cut has also been studied. Since the contribution of charged increases without the RICH veto cut, the ratio decreases at high . Thus, the for charged hadrons could be modified due to the difference of between protons and in the range 4 8 GeV/. We found that is 10–20% different without the RICH veto cut for 4–5 GeV/, where the charged starts firing the RICH.
One of the remaining sources of background contribution comes from the random tracks that are accidentally associated with the tracks in PC3. These random tracks have been estimated by swapping the zcoordinate of the PC3 hits and then by associating those hits with the real tracks. Figure 10 shows the comparison of the radial PC3 matching distribution between the real and random tracks for 6 8 GeV/. The signal to background ratio is evaluated in the window, and is 52 for 6 8 GeV/ in centrality 0–60%.
The ratio of real and random tracks with and without the cut is shown as a function of for centrality 0–60% in Fig. 11. The cut reduces the random tracks and improves the ratio by a factor of 10–24 for 4 GeV/. Since random tracks are not expected to be correlated with the event plane, we assume that their and evaluate the systematic uncertainty on to be less than 2% for 4 GeV/, increasing to 5% for 0.5 GeV/.
There is a finite residual background contribution even after the has been applied, as observed in Fig. 10. The residual backgrounds have been estimated by fitting the with a double Gaussian while requiring that the signal and residual background distribution have the same mean. For the highest bin, we found that the signal to background ratio is 5 for . The systematic error on is evaluated by comparing the measured with that of signal
(19) 
where , and are respectively of signal, background estimated for , and measured within the 2 matching window. The systematic uncertainties are less than 5% for 4 GeV/, and 5–10% for higher . All the above systematic errors are added in quadrature and the overall systematic error from the background contribution is estimated to vary from 5% for 4 GeV/ to 30% for higher .
iv.1.2 Event plane calibrations
The procedures used in the determination and calibration of event planes are the dominant sources of systematic errors on and are discussed in the following sections.
Different calibration procedures of the BBC event plane were extensively studied for previous Au + Au data sets Adler et al. (2003b). We followed the same procedure to study the systematic errors on the BBC and ZDCSMD event planes. Systematic uncertainties from the shift methods on {BBC} are 15% depending on the centrality. The systematic errors on the {ZDCSMD} are 12% larger than those on {BBC} for centrality 10–30% and 50–60%, although those are still less than 5%.
iv.1.3 Event plane determination
Figure 12 shows the comparison of for different subdetectors with respect to the BBC and ZDCSMD event planes as a function of centrality. Systematic errors are estimated by taking the maximum difference of the from the South and North event planes to that from the combined SouthNorth event plane scaled by for each centrality. Systematic errors range from 14% for the BBC, and 116% for the ZDCSMD event planes depending on the centrality bins.
iv.1.4 Effect of nonuniform acceptance on
In this subsection we discuss the effect of nonuniform acceptance on the measured . In practice, the imperfect azimuthal acceptance of the BBC or ZDCSMD or the central arms could induce an azimuthaldependent event plane resolution and/or smear the magnitude of . In order to study the possible effect of nonuniform acceptance, the measured is decomposed into X and Y components Selyuzhenkov and Voloshin (2008):
(20) 
where denotes the azimuthal angle of hadrons measured in the central arms and are the acceptance correction factors of the measured in the central arms. The coefficient should be unity in the case of perfect azimuthal acceptance. Res and Res denote the event plane resolution for and respectively and are expressed as
(21)  
where are the harmonics of event planes for subevents A, B, and C, respectively. Another acceptance effect from the difference between Res and Res is discussed below.
Figure 13 shows the acceptance correction factor as a function of in the central arms for centrality 0–60%. The dependence is parameterized by
(22) 
where ( = 0,1,…,5) are free parameters. From the fit, we get , , , , and . There is no centrality dependence of the acceptance corrections in the measured centrality range and these same correction factors are applied for all centrality bins.
Figure 14 shows the raw {BBC} as a function of in 20–60% centrality bin. is systematically higher than for 1 GeV/ as shown in Fig. 14(a). Figure 14(b) shows that and agree with each other after dividing by , the remaining difference between them being accounted for as a systematic error. For the ZDCSMD event plane we observed a similar trend for and .
A possible nonuniform acceptance of the BBC and ZDCSMD could lead to the difference between Res and Res. If the azimuthal coverage of both detectors is perfect, Res and Res should be identical. Therefore, the effect of the acceptance of the detector on the event plane resolution can be assessed by comparing Res and Res.
Figure 15 shows Res and Res of the BBC and ZDCSMD as a function of centrality. The resolutions are calculated by using Eq. (21) with the ZDCSMD, BBC and CNT event planes. Res was comparable with Res for both the BBC and ZDCSMD event planes. They also agreed, within statistical errors, with the expected resolution, namely the full event resolution scaled by 1/. We also evaluated Res and Res of BBC and ZDCSMD for the twosubevents method. Res was consistent with Res. However, for the ZDCSMD event plane, Res (Res) was systematically higher (lower) by about 30% than the expected resolution when the resolutions were calculated with = 1 in Eq. (8). The difference between Res and Res for the twosubevents method is attributed to the nonuniform acceptance between horizontal (x) and vertical (y) directions of the ZDCSMD. Those resolutions of the ZDCSMD were consistent with each other using = 2. For = 2, the nonuniform acceptance in the azimuthal directions cancels out since Res contain both and terms. Thus, Res should be the same and consistent with that from the expected resolution.
The comparison of and with with respect to the BBC and ZDCSMD event planes is shown in Fig. 16. The maximum difference of and relative to {BBC} is about 2% for the centrality range 20–60% and is independent of centrality. Systematic uncertainties are evaluated by scaling the maximum difference by . The same comparison is also made for {ZDCSMD} as shown in the bottom panel in Fig. 16. The systematic errors range from 1–25% and in this case strongly depend on the centrality, as well as on the corrections by the different event plane resolutions. and are 10–25% different from {ZDCSMD} in the 0–20% centrality bin due to the very low resolution. This systematic uncertainty is denoted as “Acceptance effect on event planes” in Table 3.
iv.2 Cumulant method
The potential sources of systematic errors on the cumulant measurements are detailed below.
iv.2.1 Fixed multiplicity cut
Following Ref. Borghini et al. (2001a) a fixed multiplicity is used to reconstruct the integrated flow to avoid introducing additional errors arising from a fluctuating multiplicity. In our analysis the systematic errors were estimated by varying the fixed multiplicity cut used for the reconstruction of the integrated flow and studying its effect on the differential flow values.
iv.2.2 range for integrated flow
In order to assess the influence of the range used to estimate the integrated flow on the differential flow, we chose different ranges over which the integral particles were selected. Differential results were obtained for three ranges: 0.25  2.0 GeV/, 0.25  1.5 GeV/ and 0.3  1.5 GeV/. The systematic error from this source is estimated to be 38% depending on centrality and .
iv.2.3 Background contribution
The procedures followed for studying the background contribution to {2} were the same as for the event plane method. After background subtraction the systematic error is calculated by determining the difference between the obtained from using 2 and 3 association cuts. We determined that the overall systematic error due to these differences is 6–10% depending on and centrality.
V Results
v.1 dependence of
The dependence of has been instrumental in revealing the hydrodynamic properties of the matter formed at RHIC Adler et al. (2003b); Adams et al. (2004a). In this context, it is important to compare the dependence of from different methods to establish the robustness of our measurements. This comparison is displayed in Fig. 18 which shows the differential charged hadron as a function of from the event plane and cumulant methods for different centrality bins in the range 0–60% in Au+Au at = 200 GeV. {2} increases up to 3 GeV/ and saturates at 0.1–0.25, depending on centrality, for higher . On the other hand, {BBC} and {ZDCSMD} reach their maximum value at 3 GeV/, and decrease for higher .
The differences between {BBC} and {ZDCSMD} are independent of within systematic errors in the measured centrality range. {ZDCSMD} is consistent with {BBC} within systematic errors in the 0–40% centrality range, but is 10–20% smaller than {BBC} in the 40–60% centrality range. These results could indicate that the influence of nonflow effects on {BBC} is small and within the systematic errors, because nonflow effects are not expected to influence {ZDCSMD}. The difference between {BBC} and {ZDCSMD} in peripheral collisions could be attributed to nonflow contributions that might be proportionally larger in more peripheral collisions.
The cumulant and event plane agree well within systematic uncertainties in the centrality range 0–40%. In more peripheral collisions, there may be some differences developing above 4 GeV/. Correlations between particles from jets affect the cumulant results, but have less influence on {BBC}, as explained in Ref. Jia (2007), where it was shown that the smaller the rapidity gap between the leading particle from a jet and the event plane, the greater the of the leading particle of the jet.
In order to illustrate more clearly the differences between the different methods, Fig. 19 shows the ratio of {ZDCSMD} and {2} to {BBC}. The results from the three methods are comparable in magnitude within systematic errors, except for the central and peripheral bins where the largest deviations occur. In addition, {2} and {ZDCSMD} show different behaviors at 3 GeV/, with {2} being larger, and {ZDCSMD}, smaller than {BBC}.
v.2 Centrality dependence of
Figure 20 shows the dependence of from different methods for charged hadrons in the range 1.0 1.2 GeV/. is observed to increase with decreasing and then decrease slightly for 75. Note that values obtained with the different methods agree well within systematic errors for all centralities. This is dependent, as shown in Fig. 18.
v.3 Pseudorapidity dependence of v
Figure 21 compares the pseudorapidity dependence of the of charged hadrons within the range ( 0.35) of the PHENIX central arms for different selections. It can be observed that is constant over the coverage of the PHENIX detector and the constancy does not depend on . This is not the case when the is measured far from midrapidity where the PHOBOS and STAR collaborations observe a drop in for Adams et al. (2005c); Alver et al. (2007).
Vi Discussion
vi.1 Effect of CNT event plane resolution
Figure 22 shows the comparison of {ZDCSMD} and {BBC} as a function of corrected either by the resolution from SouthNorth correlations from the same detectors or by the resolution from ZDCSMDCNT correlations in the 20–60% centrality bin. Figures 22(a) and (b) compare the obtained by using two different corrections from the SouthNorth and ZDCBBCCNT subevents for the BBC (a) and ZDCSMD event planes (b). The from the SouthNorth subevent is consistent with that from the ZDCBBCCNT subevent, within systematic uncertainties. The small difference between SouthNorth and ZDCBBCCNT subevents is attributed to the difference between the event plane resolution, as shown in Fig. 4. Figures 22(c) and (d) compare {ZDCSMD} with {BBC} for the SouthNorth (c) and ZDCBBCCNT subevent (d). The data points in Fig. 22(c) and (d) are the same as in Fig. 22(a) and (b). Figure 22(c) shows that {ZDCSMD} is about 10% smaller than {BBC} for the SouthNorth subevent. The ratio of {ZDCSMD} to {BBC} is found to be independent of except for GeV/. If jets are the dominant source of nonflow, one expects its contribution to to become larger at higher . The constant ratio suggests that the nonflow contribution from jets is small and fluctuations may affect {BBC} below GeV/ since the effect of fluctuations is expected to be independent of . {ZDCSMD} agrees with {BBC} within systematic uncertainties for the ZDCBBCCNT subevent as shown in Fig. 22(d). The event plane resolution from the ZDCBBCCNT subevents includes the effect of nonflow contributions and fluctuations since the CNT and BBC event planes are sensitive to both effects, though nonflow effects especially from jets could be negligible in the BBC event plane, as discussed earlier. The consistency between from the ZDCSMD and BBC event planes may suggest that {ZDCSMD} becomes sensitive to fluctuations by the inclusion of the BBC and CNT event planes to estimate the resolution.
vi.2 Comparison with other experiments
It is instructive to compare measurements made by different experiments at RHIC. Figure 23 shows a comparison of the dependence of charged hadron in the 20–60% centrality range between PHENIX and STAR experiments Adams et al. (2004b). The relative systematic errors on the STAR {2} and {4} measurements range up to 10% for 1 GeV/, with the lowest bin having the largest error 10%, while they are of the order of 1% above 1 GeV/ Adams et al. (2004b). The {2} from PHENIX is lower than that from STAR, but they are comparable within systematic uncertainties, as shown in Fig. 23(a). Figure 23(b) compares {BBC} and {ZDCSMD} with {4}, obtained from four particle cumulants, as measured in STAR. For 2 GeV/, the STAR {4} is systematically smaller than the PHENIX event plane , while {ZDCSMD} is lower than {BBC}. However, the three set of measurements are consistent within systematic errors. The order of , {BBC} {ZDCSMD} {4} could be explained by the effect of flow fluctuations Bhalerao and Ollitrault (2006); Miller and Snellings () if other nonflow contributions are small.
Figure 24 shows the comparison of our charged hadron from the BBC and ZDCSMD event planes to from a modified event plane method Adams et al. (2005c), labelled {EP}, from the STAR experiment for three centrality bins in the range 10–40%. Particles within around the highest particle were excluded for the determination of the modified event plane in order to reduce some of the nonflow effects at high . We find that