Measurement of two-particle correlations with respect to second- and third-order event planes in Au+Au collisions at \sqrt{s_{{}_{NN}}}=200 GeV

Measurement of two-particle correlations with respect to second- and third-order event planes in AuAu collisions at GeV


We present measurements of azimuthal correlations of charged hadron pairs in GeV AuAu collisions after subtracting an underlying event using a model that includes higher-order azimuthal anisotropy , , and . After subtraction, the away-side ( of the highest transverse-momentum trigger ( GeV/) correlations is suppressed compared to that of correlations measured in collisions. At the lowest associated particle , the away-side shape and yield are modified. These observations are consistent with the scenario of radiative-jet energy loss. For the lowest- trigger correlations, an away-side yield exists and we explore the dependence of the shape of the away-side within the context of an underlying-event model. Correlations are also studied differentially versus event-plane angle . The angular correlations show an asymmetry when selecting the sign of the trigger-particle azimuthal angle with respect to the event plane. This asymmetry and the measured suppression of the pair yield out of plane is consistent with a path-length-dependent energy loss. No dependence can be resolved within experimental uncertainties.

PHENIX Collaboration

I Introduction

Energy loss of hard-scattered partons (jet quenching Wang and Gyulassy (1992)) resulting from the interaction of a colored parton in the quark gluon plasma (QGP) formed in relativistic heavy ion collisions Arsene et al. (2005); Back et al. (2005); Adams et al. (2005a); Adcox et al. (2005) has been observed in several different ways. Suppression of single-particle and single-jet invariant yields in central A+A collisions Adler et al. (2003a, 2004); Adams et al. (2003); Chatrchyan et al. (2012a); Aad et al. (2015a, 2013) provide a baseline measurement of jet quenching. Measurements of correlations between two particles and/or jets give more detailed spatial information of the jet quenching process inside the medium Renk (2013); Qin and Wang (2015). The first jet suppression effect observed in these correlations was an attenuation of the away-side yields in high-transverse momenta () correlations in the most central AuAu collisions at  200 GeV Adler et al. (2003b). The centrality dependence of high- -hadron correlations Adare et al. (2011a) shows a monotonic attenuation of the away-side yields with increasing propagation length of partons through the medium. In addition to away-side yield suppression, direct photon-hadron correlations Abelev et al. (2010a); Chatrchyan et al. (2013); Adamczyk et al. (2016), two-particle correlations Adare et al. (2013); Agakishiev et al. (2012); Nattrass et al. (2016), and jet-hadron correlations Adamczyk et al. (2014a); Aad et al. (2014a); Chatrchyan et al. (2014a); Khachatryan et al. () show that low-momentum particles correlated with high- jets are enhanced in yield, especially at large angles with respect to the jet axis. This may be attributable to the radiation from the parent parton or other lost energy recovered by the nearby medium. Thus, two-particle angular correlations have provided much of the experimental information we have about jet energy loss Aamodt et al. (2012); Adare et al. (2010a, 2011a); Adams et al. (2006); Abelev et al. (2010b); Agakishiev et al. (2011).

It is important to understand the interactions of partons with the QGP at all scales from the hard-scattering scale to the thermal scale. Below 10 GeV, full jet reconstruction is much more difficult due the underlying event subtraction. Two-particle correlations are important because they can probe lower energies. However, observations of many of the energy loss effects mentioned above, especially for lower jet and particle momenta in dijet two-particle correlations, have been obscured due to the much larger contribution from the underlying event at these momenta.

The underlying event modulations are attributed to hydrodynamic collective flow patterns where the importance of higher-order terms was established more recently Alver and Roland (2010); Xu and Ko (2011); Aad et al. (2012); Adare et al. (2011b); Aamodt et al. (2011); Chatrchyan et al. (2014b). These patterns are thought to result from the hydrodynamic response of the QGP to fluctuating initial geometrical shapes of the interacting portions of the colliding nuclei. Many hydrodynamic models have been developed which capture these effects Shen et al. (2016); Lin et al. (2005); Romatschke (2015), but to date, important details of these models are still under development, and their full implementation requires involved calculations. This motivates the use of a simpler data-driven model, which will be explored in this work.

The shape of the collective flow in the transverse plane is parameterized Voloshin and Zhang (1996); Ollitrault (1992); Poskanzer and Voloshin (1998); Adare et al. (2010b) by a Fourier expansion with


where is the -order component of single-particle correlation due to flow, is the azimuthal angle of emitted particles, and is the event plane angle defined by the -harmonic number. For the first decade of the Relativistic Heavy Ion Collider (RHIC), only the even harmonics and frequently only the term, were considered. The shapes of two-particle correlations after subtraction of the -only background motivated the introduction of the other harmonics, most importantly Sorensen (2010); Alver and Roland (2010); Adare et al. (2011b); Aad et al. (2012); Aamodt et al. (2011). Under the two-source (flow + jet) model assumption Adler et al. (2006a), this underlying event is directly subtracted to obtain the jet contributions. In our previous measurements and most RHIC results, the subtracted flow modulations of the underlying event were limited to contributions of and the fourth-order harmonic component with respect to the second-order event plane  Adler et al. (2003b); Abelev et al. (2009a); Adare et al. (2008a); Adler et al. (2006a); Adare et al. (2008b); Adamczyk et al. (2014a); Adare et al. (2011a, 2010a, 2007, 2013). Only the recent STAR measurement Agakishiev et al. (2014) took into account contributions from and the fourth-order harmonic component uncorrelated to the second-order event-plane .

At low to intermediate in two-particle correlations, intricate features appear such as the near-side long-range rapidity correlations called the “ridge” Abelev et al. (2009a); Alver et al. (2010) and the away-side “double-humped” structures Adler et al. (2006a); Adare et al. (2007, 2008b); Abelev et al. (2009b); Adare et al. (2008a); Aggarwal et al. (2010); Adare et al. (2010a); Agakishiev et al. (2014). Across the large rapidity ranges available at the Large Hadron Collider, the rapidity-independence and hence the likely geometrical origin of most of these structures have been established. Experiments have shown that the ridge and the double-hump structures in the two-particle azimuthal correlations for for ALICE and for ATLAS and CMS measured in , Pb, and PbPb collisions at and  TeV Aamodt et al. (2011); Aad et al. (2012); Chatrchyan et al. (2012b) are the same in shape and size at much larger rapidity differences. Both the ridge and hump are successfully explained by the higher-order harmonics. Despite this conclusion, how the jet correlations combine with the flow correlations, especially at small , to yield the total two-particle correlation has not been quantified. In particular, the correlations left after subtracting a flow-based model at small have not been analyzed in detail.

In this work, we assume a two-source model where the total pair yield is a sum of a jet-like component and an underlying-event component. The underlying-event components is modeled using the flow harmonics , event plane resolutions, and the most important event plane correlations between and . We assume that the measured through the event plane method are the the same as those in the correlation functions. Event-by-event fluctuations Aad et al. (), correlations between different order Aad et al. (2015b), and rapidity dependent event plane decorrelations Khachatryan et al. (2015) are not considered in this background model. Measurements of using the event plane method Adams et al. (2005b); Back et al. (2006); Abelev et al. (2008); Adamczyk et al. (2014b) generally yield at integrated over all . Finite values of differential measurements of  Abelev et al. (2008) include momentum conservation and jet (mini-jet) effects which are considered signal in two-particle correlations. We therefore excluded contributions from and event plane correlations involving from the background model. For the inclusive trigger correlations, we estimated a potential impact of modulation using an empirical relation found in ATLAS measurements Aad et al. (2012).

After subtracting the underlying event with the model, we study the structures observed at high where the flow backgrounds are negligible. Because the jet signal-to-flow background is significantly reduced in the low to intermediate region, studying the correlations there provides a more stringent test of such a background model. Any features left in the residuals can be used to reveal jet energy-loss effects at low and intermediate . However, because of our simple model, only substantially significant correlations can be attributable to the medium effect on jets (i.e. broadening or suppression) or the medium response (i.e. yields at large angles from the jet).

Figure 1: Schematic picture of a trigger particle selection with respect to event-planes and pairing a trigger particle with an associate particle.

We also study two-particle correlations measured differentially with respect to and event planes as depicted in Fig. 1. Such differential correlations probe the path-length and geometrical dependence of energy loss with more event-by-event sensitivity and also extend similar studies of high- correlations Adare et al. (2011a) down to lower-. We also use a new method of distinguishing “left/right” asymmetry in the correlations, which can be employed to provide more information on the background dominated low and intermediate regions.

In this article: Section II describes the detector set-up of the PHENIX Experiment. Sections III.1III.2, and III.3 describe the analysis methodology of particle selections, higher-order flow harmonics, and two-particle correlations, respectively. Section IV presents analysis results and discusses their interpretations. This section first starts with the highest trigger selections, GeV/, and makes connections to known energy-loss effects. Next, lower trigger correlations down to 1 GeV/ are presented. Finally the event-plane dependence of the intermediate selections are investigated. Section V summarizes this article.

Ii Phenix Detector

Figure 2: The PHENIX detector configuration in the 2007 experimental run period. The top panel shows the central arm detectors viewed from the beam direction. The bottom panel shows the global detectors and muon arm viewed from the side perpendicular to the beam direction.

The PHENIX detector Adcox et al. (2003) was designed to measure charged hadrons, leptons, and photons to study the nature of the QGP formed in ultra-relativistic heavy ion collisions. Figure 2 shows the beam view and side view of the PHENIX detector including all subsystems for this data taking period.

The global detectors, which include the beam-beam counters (BBC), the zero-degree calorimeters (ZDC), and the reaction-plane detector (RXN), were used to determine event characterizing parameters such as the collision vertex, collision centrality, and event-plane directions. They are located on both the south and north side of the PHENIX detectors. The BBC is located at 144 cm () from the beam interaction point and surrounds the beam pipe with full azimuthal acceptance. Each BBC module comprises 64 quartz Čerenkov radiators equipped with a photomultiplier tube (PMT) and measures the total charge (which is proportional to the number of particles) deposited in its acceptance. The BBC determines the beam collision time, beam collision position along the beam axis direction, and collision centrality. The ZDCs Adler et al. (2001), located at 18 m away from the nominal interaction point, detect the energy deposited by spectator neutrons of the two colliding nuclei. The PHENIX minimum-bias trigger is provided by the combination of hit information in the ZDC and BBC, which requires at least one hit in both the ZDC modules and two hits in the BBC modules.

The orientation of higher-order event planes is determined by the BBC and the RXN Richardson et al. (2011), which have different acceptance. The RXNs are located at 38 cm from the beam interaction point and have two rings in each module; RXN-inner and RXN-outer are installed to cover and , respectively. Each ring has 12 scintillators in its azimuthal angle acceptance .

Charged hadron tracks are reconstructed in the PHENIX central arm spectrometer (CNT), which is comprised of two separate arms, east and west. Each arm covers and .

The PHENIX tracking system is composed of the drift chamber (DC) in addition to two layers of pad chambers (PC1 and PC3) in the east arm and three layers of pad chambers (PC1, PC2, and PC3) in the west arm. Momentum is determined by measuring the track curvature through the magnetic field by means of a Hough transform with hit information from the DC and PC1 with a momentum resolution of  Adare et al. (2012). Additional track position information is provided by the outer layers of the pad chambers and the electromagnetic calorimeter (EMCal).

The ring imaging Čerenkov counter (RICH) and the EMCal identify and exclude electron tracks from the analysis. The RICH produces a light yield for electrons with 30 MeV and for pions with 5 GeV, meaning that a signal in the RICH can be used to separate electrons and pions below 5 GeV. Above 5 GeV where this is no longer possible, the energy deposited in the EMCal can be used for this separation. Electrons will deposit much more of their total energy than pions will, so that the ratio of deposited energy to track momentum is significantly higher for electrons than for pions.

Iii Analysis Methodology

The results presented are based on an analysis of 4.38 billion minimum-bias events for AuAu collisions at  200 GeV recorded by the PHENIX detector at RHIC in 2007.

iii.1 Particle Selection

Charged hadrons are selected from candidate tracks using cuts similar to previous correlation analyses Adare et al. (2013). One important cut to reject fake tracks, especially decays in the central magnetic field before the drift chamber, is an association cut to outer CNT detectors. The track trajectories are projected onto outer CNT detectors. The nearest hits in the PC3 and the EMCal from the projections are identified as hits for the track. The distributions of the distance in the azimuthal () and beam () directions between the hits in the PC3 and the EMCal and the extrapolated line are fitted with a double-Gaussian. One Gaussian arises from background and the other from the signal. Hadron tracks are required to be within of the signal Gaussian mean in both the and directions in both the PC3 and the EMCal. To veto conversion electrons, tracks with 5 GeV/ having one or more Čerenkov photons in the RICH are excluded from this analysis. For 5 GeV/, we require c GeV Adler et al. (2006b); Adare et al. (2008b), where is the cluster energy associated with the track.

iii.2 Higher-Order Flow Harmonics

Event-plane and Resolution

Each event plane is determined event-by-event for different harmonic numbers using the RXN and BBC detectors. The RXN detectors are used to measure the nominal values of while the BBC detectors provide systematic checks to the extracted values. The observed event-plane is reconstructed as


Here and are the flow vectors


where is the azimuthal angle of the -th segment in the event-plane detector and is the weight proportional to multiplicity in the -th segment.

The  th-order resolution of  th-order event plane is defined as and can be expressed as Poskanzer and Voloshin (1998)


where , is the multiplicity used to determine the event-plane , and is the modified Bessel function of the first kind.

Because the north (N) and south (S) modules of a given event-plane detector have the same pseudorapidity coverage and see the same multiplicity and energy for symmetric nucleus-nucleus collisions, the north and south modules should have identical resolution. We obtain the event-plane resolution of an event-plane detector using the two subevent method Poskanzer and Voloshin (1998).


The north+south combined event-plane resolution is determined from Eq. (5) with = . The factor of accounts for twice the multiplicity in north+south compared to north or south. Figure 3 shows the the north+south combined event-plane resolution for both RXN and BBC.

Figure 3: Event-plane resolutions , , , and obtained by the combination of the north and south modules of RXN and BBC.


Higher-order flow harmonics  Voloshin and Zhang (1996); Poskanzer and Voloshin (1998); Alver and Roland (2010) are measured by the event-plane method Poskanzer and Voloshin (1998). Charged hadron tracks with azimuthal angle are measured with respect to the event plane angle . The flow coefficients are measured as an event-average and track-average and correcting by the event plane resolution.


Four different event planes are studied: , , , and . The flow harmonics are measured by the nine possible combinations of RXN modules: south-inner, south-outer, south-inner+outer, north-inner, north-outer, north-inner+outer, south+north-inner, south+north-outer, and south+north-inner+outer. The reported is an average over the nine different possible RXN combinations, , where is the flow harmonic in one of the nine RXN module combinations.

Systematic uncertainties and results

The systematic uncertainties in measurements are from the following sources:

  • systematic differences among RXN modules,

  • matching cut width for CNT hadron tracks,

  • rapidity-separation dependence between event-planes and CNT tracks.

The systematic uncertainties in the RXN detector are defined by the standard deviation of


As an example, in 20%–30% central collisions measured by different RXN event-planes are shown in Fig. 4 (a, d, g, j). The (blue) band indicates .

To evaluate the systematic uncertainty due to track matching, the matching cut was varied by from the nominal window. We calculated the uncertainty as the average deviation between the with the nominal cut and the varied cut.


The variation due to the track matching cut is illustrated in Fig. 4 (b, e, h, k) by showing the in 20%–30% central collisions measured with tracks having a matching cut of 1.5, 2 and 2.5. The differences between the nominal and both and are also shown and scatter around zero indicating the size of .

The systematic uncertainties associated with the rapidity gap between particles and the event-plane are defined by the absolute difference between determined by the RXN average and determined by the BBC.


The measured with the RXN, the BBC, and their difference are shown in Fig. 4 (c, f, i, l). Except in the case of , this systematic uncertainty is much less than the uncertainty due to the RXN module variation. The small uncertainty in the rapidity gap supports the claim that, even though the rapidity gap between the central arm and the RXN is less than one unit of pseudorapidity, the contamination from nonflow correlations does not dominate the uncertainty on the extraction of .

The total systematic uncertainties are the quadrature sum of these individual systematic uncertainties.


These total systematic uncertainties are applied symmetrically for a conservative estimate of the systematic uncertainties in the correlation measurements. In nearly all and centrality selections, the RXN systematic uncertainty dominates the total uncertainty.

The results are shown in Fig. 5 and compared with previous PHENIX measurements Ref. Adare et al. (2011b). They are consistent within uncertainties where they overlap. For the two-particle correlations, we use larger bins (four bin selections) as indicated in Table 1, which lists the results. A test of scaling between higher-order flow harmonics and the second-order flow harmonics is in Table 2 similar to that done in Ref. Aad et al. (2012). These ratios are constant as a function of within systematic uncertainties, which are dominant over the statistical uncertainties in 0%–50% central collisions. A possible interpretation of this relation is the acoustic scaling described in Ref. Lacey et al. (2011).

Figure 4: Higher-order flow harmonics for charged hadrons at midrapidity in AuAu collisions at and their systematics: (a-c), (d-f), (g-i), and (j-l). The source of systematic uncertainties are difference among RXN event-planes (a, d, g, j), matching cut width for CNT hadron tracks (b, e, h, k), and difference between measured with RXN and BBC event planes (c, f, i, l). Systematic uncertainties are shown as a shaded band in (a, d, g, j) and as an open marker in (b, e, h, k) and (c, f, i, l).
Centrality GeV/
0%–10% 0.5–1.0
10%–20% 0.5–1.0
20%–30% 0.5–1.0
30%–40% 0.5–1.0
40%–50% 0.5–1.0
Table 1: Data table for , , and (multiplied by 100) in AuAu collisions at . The first uncertainties are statistical while the second uncertainties are total systematic. In all instances the statistical error is not identically zero but it is much smaller than the systematic uncertainty.
Centrality GeV/
0%–10% 0.5–1.0
10%–20% 0.5–1.0
20%–30% 0.5–1.0
30%–40% 0.5–1.0
40%–50% 0.5–1.0
Table 2: Data table for and in AuAu collisions at  GeV. The uncertainties are systematic uncertainties, which are dominant over the statistical uncertainties.
Figure 5: Higher-order flow harmonics for charged hadrons at midrapidity in AuAu collisions at  GeV. Coefficients are determined using the event-plane method for ((a)-(e)), ((f)-(j)), ((k)-(o)), and ((p)-(t)). The columns represent centrality bins 0%–10% (a,f,k,p), 10%–20% (b,g,l,q), 20%–30% (c,h,m,r), 30%–40% (d,i,n,s), and 40%–50% (e,j,o,t). Coefficients obtained in this analysis are shown by blue points and those measured in Ref. Adare et al. (2011b) are shown by magenta points. Shaded bands and magenta lines indicate systematic uncertainties of those measurements.

iii.3 Two-Particle Correlations

Pair Selections

Tracks that pass the single-particle cuts are paired for correlations. In real events two tracks cannot come arbitrarily close together. The tracking algorithm would split or merge the tracks. Therefore, there is an acceptance difference in real and mixed events. These effects are estimated by calculating the distances (rad) and (cm) between hits in the PC1 and the PC3, where (rad) is the relative azimuthal angle and (cm) is the relative length between two track hits in both real and mixed events. The ratios of these distributions are shown in Fig. 6. The ratio is normalized to one where the detectors work well to reconstruct hadrons, and inefficient and over-efficient regions deviate from unity. The dashed lines indicate the cuts used to remove the inefficient regions:

Figure 6: The ratio of the real-event to mixed-event distributions of distances - between hits in a pair of tracks in (a) PC1 and (b) PC3 after the PC1 cut. The region encircled by dashed (magenta) curves are excluded from this analysis.

Inclusive Trigger Correlations

Two-particle correlations are calculated as


where is the relative azimuthal angle between trigger and associated hadrons and and are pair distributions in the real and mixed events, respectively. reflects the physical correlation among trigger and associated hadrons from jets and from the underlying event as well as the dihadron detector acceptance effects. is obtained by pairing trigger and associated hadrons from randomly selected pairs of events that have similar collision vertices and centralities so that it reflects only the dihadron acceptance effects. Taking the ratio between the real and mixed distributions corrects for the nonuniform azimuthal acceptance for dihadrons so that contains only physical effects. The correlation function is normalized so that the underlying event modulates approximately around unity.

Within the two-source model Adler et al. (2006a), the correlation function is composed of a jet-like term and an underlying-event term that includes modulations from flow . We use the following model for the underlying event Poskanzer and Voloshin (1998)


The jet-like correlation is then obtained by subtracting from as


The scaling factor is determined with the zero yield at minimum (ZYAM) method Ajitanand et al. (2005); Adams et al. (2005c); Adler et al. (2006a). In the ZYAM assumption, is scaled such that has a minimum of exactly zero. This therefore gives the lower boundary of possible jet-like correlations. The ZYAM scaling factor is determined by fitting the correlation function with Fourier series for and identifying the single point where this fit and have the minimum contact point. The statistical uncertainty of the bin containing the ZYAM point is used to scale to estimate the systematic uncertainty due to ZYAM


An example ZYAM identification is shown in Fig. 7.

Figure 7: Example of ZYAM extraction where the correlation function (open circle) is fitted (dashed line). The normalization of the underlying event model (red solid line) is adjusted to match the minimum value of the fit. The (blue) band indicates the uncertainty on the ZYAM extraction determined by the statistical uncertainty of near the minimum.

The jet-like correlations are scaled to the per-trigger yield


where is the number of trigger hadrons, is the number pairs, and is the single-hadron tracking efficiency in the associated hadron range. The efficiency is estimated via detector simulations for acceptance and occupancy effects as discussed in Ref. Adler et al. (2004); Adare et al. (2008b, 2010a, 2011a). The tracking efficiency of a trigger particle is canceled by the ratio of .

Event Plane-Dependent Correlations

Event plane-dependent two-particle correlations are defined as


where and and are the event plane-dependent pair distributions in real and mixed events, respectively.

Similar to inclusive correlations, event plane-dependent jet-like correlations are obtained by subtracting the event plane-dependent flow background term from with a ZYAM scale factor as


We use the same as determined from the inclusive correlations from the same trigger, associated and centrality selection. An analytical formula for including the event plane dependence exists Bielcikova et al. (2004), however, it is not easily applied with finite correlations between the and event planes. For this reason, a Monte Carlo simulation is employed to estimate . This is described in Sec. III.3.4 below.

The event plane-dependent jet-like correlations are converted into event plane-dependent per-trigger yield as


where is the number of trigger hadrons and is the number of pairs in the trigger event-plane bin.

Flow Background Model Including Event Plane Dependence

With the assumption that the measured from the event plane method are purely from collective dynamics of the medium, flow-like azimuthal distributions of single hadrons can be generated by performing a Monte Carlo simulation inputting the experimentally measured , the resolution of the event planes, and the strength of correlation among different order event planes. The single-hadron azimuthal distributions due to collective flow can be described by a superposition of as


where is the azimuthal angle of the emitted hadrons and is a true th-order event plane defined over . Separate distributions using for each ranges of trigger and associated particles are used in the simulation. The trigger and associated distributions in real events share a common , while those in mixed events do not.

The experimental event plane resolution is introduced through a dispersion term where . We calculate as


where and is the error function Ollitrault (); Poskanzer and Voloshin (1998). This equation can be solved for by using the experimentally-determined from the measured event plane resolutions using Eq. (5).

Because a weak correlation between and exists Adare et al. (2011b), the directions of and are generated independently. The direction of is generated assuming a correlation with , . We estimate assuming the correlation between the two event planes follow similar functional forms as the dispersion of event planes due to the resolution. That is, we assume,


where . The parameter is assumed to be similar to Eq. (5)


where  Yan and Ollitrault (2015). The functional shape of Eq. (23) is verified by event plane correlation studies using the BBCs and the RXNs following the method described in Ref. Aad et al. (2014b). The correlation strength between and , , is measured to be consistent with zero within large statistical uncertainties. Potential impacts of to the event plane dependent correlations are estimated using the value of reported in Ref. Aad et al. (2014b) by the ATLAS Experiment. The impact of is within the systematic uncertainties described later.

We use the averaged value between  GeV/ and  GeV/ for event plane-dependent correlations of ( (), ( (), and ( () GeV/ because would contain auto-correlations from jets at high .

Figure 8: Event plane-dependent (black circles) and event plane-dependent model flow background (blue lines)of ( () GeV/. Trigger particles are selected in (a) out-of-plane of , (b) in-plane of , (c) out-of-plane of , (d) in-plane of . Schematic pictures in each panel also depict these ranges of the trigger particle selections with respect to event plane .
Figure 9: Event plane-dependent (black circles) and event plane-dependent model flow background (blue lines) of ( () GeV/. Trigger particles are selected in (a) out-of-plane of , (b) in-plane of , (c) out-of-plane of , (d) in-plane of . Schematic pictures in each panel also depict these ranges of the trigger particle selections with respect to event-plane .

The event plane-dependent background shapes are determined by generated particles in this simulation using Eq. (18). Figure 9 shows event plane-dependent correlations and backgrounds with a selection of the absolute trigger azimuthal angle relative to the event-planes . The backgrounds agree with the experimental correlations except at where contributions from jets are expected. Figure 9 shows event plane-dependent correlations and backgrounds with a selection of trigger azimuthal angle relative to event planes . Agreement between the experimental correlations and the background except at is also observed here. Other event-plane dependent correlations and backgrounds with a selection of trigger azimuthal angle relative to event-planes for different collision centralities and selections are shown in the Appendix.

iii.4 Unfolding of Event Plane-Dependent Correlations

In this analysis, is divided into 8 bins. The width of the bins is and when correlating with and , respectively. The event plane-dependent per-trigger yields are smeared across neighboring event plane bins due to limited experimental resolution of the event planes. We unfold the smearing to obtain the true event-plane dependence of the correlations. Two different methods are used to check the unfolding procedure: (I) iterative Bayesian unfolding, , and (II) correcting the event plane-dependence of the per-trigger yield based on a Fourier analysis, .

Iterative Bayesian Unfolding

The Iterative Bayesian Unfolding Method presented in Ref. D’Agostini (1995); Adye (2011) is applied to this analysis with the following formulation


where is the unfolded distribution, is the experimentally observed distribution, is the prior distribution, is the conditional probability matrix where is measured to be , and is the efficiency. In the iterative calculation, also serves as the prior distribution of the next loop. We perform this unfolding separately for every bin.

We define the experimentally observed distribution as using the measured event plane-dependent per-trigger yield. The offset is to prevent a divergence in the iteration due to small yields near the ZYAM point. In the initial loop of the iteration, we define the prior distribution as