Measurements of dihadron correlations relative to the event plane in Au+Au collisions at ${\sqrt {{{ s}_{{\bf{NN}}}}}{\bf{ = 200}}}$ GeV

Collisions at the Relativistic Heavy Ion Collider (RHIC) at Brookhaven National Laboratory have created a medium with properties that resemble a nearly perfect liquid of strongly interacting quarks and gluons [1-4]. This conclusion is based upon two pillars of evidence [5]: (i) the strong elliptic flow and (ii) jet-quenching– suppression of high transverse momentum ($ p_{T} $) single hadron yields and dihadron correlations in heavy-ion collisions relative to elementary p+p interactions. While suppression of high $ p_{T} $ single hadron yields has limited sensitivity to the medium core, dihadron correlation measurements provide richer and more valuable information about the properties of the created medium [6-8]. There are several key observations that can be made from dihadron correlations with a high $ p_{T} $ trigger particle. (i) The correlated hadron yield at high $ p_{T} $, while not much changed on the near side of the trigger particle (where azimuth difference between correlated and trigger particles $ | \Delta\phi|<\pi/2 $), is strongly suppressed on the away side (where $ | \Delta\phi|>\pi/2 $) [9]. This lends strong support to the partonic energy loss picture [10-12]. (ii) The correlated hadron yields at low $ p_{T} $ are strongly enhanced on both the near and away side [13]. In particular, the near-side enhancement is tied to long-range correlations in pseudo-rapidity – the ridge [13-15]. (iii) The away-side correlation broadens from peripheral to central collisions, and exhibits double peaks for select trigger and associated particle $ p_{T} $ ranges [13, 16, 17]. The double-peak structure is peculiar and may provide an opportunity to study the underlying physics mechanisms for partonic energy loss, such as gluon radiation [18, 19], Mach-cone shock-wave excitation [20-29], or simply the bulk medium response [30]. Three-particle jet-like correlation studies indicate that the double-peak emission pattern of correlated hadrons is characteristic of medium triangular flow [30] and/or Mach-cone shock-waves [31]. (iv) The away-side associated particles are partially equilibrated with the bulk medium in mid-central to central collisions, and a higher degree of equilibration is observed for particles which are more aligned back-to-back with the trigger particles [13, 17]. This observation may underscore the connection between the medium's path-length and partonic energy loss.

We study the path-length dependence of partonic energy loss in detail in non-central collisions where the overlap region between the two colliding nuclei is anisotropic: the size in the reaction-plane direction is shorter than that perpendicular to it. The reaction plane (RP) is defined by the beam direction and the line connecting the centers of two colliding nuclei. It can be estimated in non-central collisions by determining the azimuthal angle with the highest particle emission probability, using the fact that the particles have an elliptic emission pattern [32]. The estimated angle is called the event plane (EP), to emphasize that it is an experimental estimate of the reaction plane with finite resolution. By selecting the trigger particle direction with respect to the event plane, $ \phi_{s} = | \phi_{t}- \psi_{{\rm{EP}}}| $ (where $ \phi_{t} $ is the trigger particle azimuth and $ \psi_{{\rm{EP}}} $ is the event-plane azimuth), we aim to select different path-lengths through the medium that the away-side parton traverses, providing differential information unavailable to inclusive jet-correlation measurements.

Previously, the Solenoidal Tracker at the RHIC (STAR) experiment has performed an exploratory measurement of azimuthal correlations at high $ p_{T} $ with trigger particles in-plane ($ \phi_{s}<\pi/4 $) and out-of-plane ($ \phi_{s}>\pi/4 $) using non-central 20%-60% Au+Au collisions [33]. The results hinted that the away-side correlation with out-of-plane trigger particles is more strongly suppressed than that with in-plane trigger particles. In this paper, we extend those measurements to finer bins in $ \phi_{s} $ and to lower associated and trigger $ p_{T} $ ranges [34]. We also present inclusive jet-like correlation results from minimum bias d+Au collisions as a reference to the Au+Au data. We further study the ridge as a function of $ \phi_{s} $, and investigate the systematics of the ridge in an attempt to further identify the underlying physics mechanism for the formation of the ridge. The highlights of the results have been published in Ref. [35]. This paper provides extensive results and analysis details. Similar results on the event-plane dependent dihadron correlations have recently been reported by the PHENIX collaboration [36].

The paper is organized as follows. In Sec. II we describe in detail our data analysis of dihadron correlations relative to the event plane. In Sec. III we discuss our extensive studies of the systematic uncertainties of our results. In Sec. IV we report and discuss our results of dihadron correlations relative to the event plane. We finally conclude in Sec. V. We present all raw and background-subtracted dihadron correlation functions relative to the event plane in Appendix C.

The data used in this analysis were taken by the STAR experiment [37] at RHIC at the nucleon-nucleon center of mass energy of $ \sqrt{s_{_{{\rm{NN}}}}} = 200 $ GeV. The minimum-bias Au+Au data were from Run IV in 2004 at RHIC. The reference minimum-bias d+Au data used for comparison were from Run III in 2003. The minimum-bias triggers for Au+Au and d+Au collisions were provided by the Central Trigger Barrel [38] and the Zero Degree Calorimeters [39].

The details of the STAR experiment can be found in Ref. [37]. The main detector used for this analysis is the Time Projection Chamber (TPC) [40, 41]. The TPC is surrounded by a solenoidal magnet providing a nearly uniform magnetic field of 0.5 Tesla along the beam direction. Particle tracks are reconstructed in the TPC. The primary event vertex was fit using reconstructed tracks which pass certain quality cuts. Events with a primary vertex within $ \pm 30 $ cm of the geometric center of the TPC along the beam axis are used in the analysis. With this range of primary vertex position, the TPC has good acceptance within the pseudo-rapidity region of $ |\eta|\leq 1.1 $. The Au+Au collision centrality is defined according to the measured charged hadron multiplicity in the TPC within $ |\eta|<0.5 $ (reference multiplicity) [42]. We choose the 20%-60% centrality data, where good event-plane resolution is achieved, for our analysis (see later).

In our analysis, only tracks that extrapolate to within 2 cm of the primary vertex are used. Tracks are required to be reconstructed with at least 20 out of a maximum of 45 hits in the TPC. The ratio of the number of hits used in track reconstruction to the number of possible hits is required to be greater than 0.51, to eliminate multiple track segments being reconstructed from a single particle trajectory. The same event and track cuts are applied to particle tracks used for event-plane reconstruction and for the subsequent correlation analysis. Particle tracks within $ |\eta|<1 $ are used in the correlation analysis.

High $ p_{T} $ particles are selected as triggers off-line to perform the correlation analyses. We select high $ p_{T} $ trigger particles within the $ p_{T}^{(t)} $ ranges of 3-4 GeV/c and 4-6 GeV/c. A total of 4.4 million Au+Au events with centrality ranging from 20%-60% are used in this analysis. From the event sample we find 2.1 million trigger particles with $ p_{T} $ values ranging between 3-4 GeV/c, and 0.36 million trigger particles with $ p_{T} $ values between 4-6 GeV/c. Associated particles, i.e. all particles in the event including those correlated with the trigger particles, are grouped into the $ p_{T}^{(a)} $ ranges of 0.15-0.5, 0.5-1.0, 1.0-1.5, 1.5-2.0, 2.0-3.0 (or 2.0-4.0) GeV/c. The low $ p_{T}^{(a)} $ cut-off of 0.15 GeV/c is imposed by the magnetic field strength and the TPC aperture. The azimuthal correlation functions in $ \Delta\phi $ (azimuthal angle difference between associated particle and trigger particle) are analyzed separately for trigger particles at different azimuthal angles ($ \phi_{s} $) relative to the event plane.

The associated particle yields are corrected for single-particle track reconstruction efficiency, which is obtained from embedding simulated tracks into real events [43]. It depends on both centrality and $ p_{T} $. The efficiency is found to be insensitive to $ \eta $ and is therefore averaged over $ \eta $. The $ \phi $-dependent part of the acceptance and track reconstruction efficiency are corrected for both the trigger and associated particle yields. This $ \phi $-dependent correction is obtained from the inverse of the single-particle $ \phi $ distribution, whose average is normalized to unity. Correction for the $ \phi $-dependent efficiencies for both trigger and associated particles removes the majority of the non-uniformity caused by the TPC sector boundaries. The remaining non-uniformity in $ \Delta\phi $ is corrected by using an event-mixing technique, where the trigger particle from one event is paired with associated particles from another event within the same centrality bin [13]. The two-particle acceptance in $ \Delta\eta $ (pseudo-rapidity difference between associated particle and trigger particle), which is approximately triangle-shaped in $ \Delta\eta $, is not corrected to be consistent with earlier publications [13]. The correlation function is normalized by the corrected number of trigger particles in its corresponding $ \phi_{s} $ bin. The centrality and $ p_{T} $ dependent aspects of the trigger particle efficiency cancel out in the normalization.

Tracks that are spatially near each other can be combined into a single reconstructed track due to merged space points of ionization in the STAR TPC. This track merging results in a pair inefficiency at $ \Delta\eta\sim0 $ and a small but finite $ \Delta\phi $ whose value depends on the magnetic field polarity, charge combination and the $ p_{T} $'s of the trigger and associated particles [15]. The track merging effect is most significant in central collisions where the TPC hit occupancy is high. The track merging effect in our centrality range of 20%-60% is negligible.

We use the second Fourier harmonic in azimuthal angle to determine the event-plane angle $ \psi_{{\rm{EP}}} $ [32], which is not identical to the real reaction-plane angle ($ \psi_{{\rm{RP}}} $). The event plane is an estimate, with finite resolution, of the second harmonic participant plane (the plane defined by the beam direction and the minor axis of the overlap geometry of participant nucleons) [44]. The participant plane angle, $ \psi_2 $, fluctuates about the reaction plane direction. The particles used to determine the event plane are below $ p_{T} = 2 $ GeV/c. To avoid self-correlations, particles from the $ p_{T} $ bin that is used in the correlation analysis are excluded from event-plane reconstruction. For example, for the associated particle $ p_{T} $ bin of $ 1.0< p_{T}^{(a)}<1.5 $ GeV/c, the particles used to calculate the event plane are from $ p_{T} $ ranges of $ 0.15< p_{T}<1.0 $ GeV/c plus $ 1.5< p_{T}<2.0 $ GeV/c. We use the $ p_{T} $-weight method [32], which gives better event-plane resolution due to the stronger anisotropy at larger $ p_{T} $. The slight non-uniform efficiency and acceptance in azimuthal angle were corrected as mentioned previously in the event-plane reconstruction. Figure 1 shows examples of the constructed event plane azimuthal angle distributions. As seen from the figure, the constructed event plane $ \psi_{{\rm{EP}}} $ distribution is approximately uniform. We weight the events by the inverse of the event-plane angle distributions in Fig. 1 in our correlation analysis. However, we find negligible difference in our results with and without this event-plane weighting.

Figure 1.  (color online) Constructed event plane azimuthal angle ($ \psi_{{\rm{EP}}} $) distributions by the modified reaction-plane (MRP) method (points) and the traditional reaction-plane method (histogram). The particles used for constructing the event plane shown in this figure are from $ 0.15< p_{T}<1 $ GeV/c or $ 1.5< p_{T}<2 $ GeV/c, to be used for correlation analysis for the associated particle $ p_{T}^{(a)} $ bin of $ 1< p_{T}^{(a)}<1.5 $ GeV/c.

Nonflow correlations, such as di-jets, can influence the determination of the event plane. To reduce this effect, we exclude from EP reconstruction particles within pseudorapidity difference of $ | \Delta\eta| = |\eta-\eta_{\rm{trig}}|<0.5 $ from the trigger particle. (In other words, we reconstruct an EP for each trigger particle; if there are multiple trigger particles in an event, their event planes are different even though they belong to the same event.) This method is called the modified reaction-plane (MRP) method [45]. The traditional reaction-plane method, on the other hand, does not exclude from EP reconstruction those particles in the $ \eta $ vicinity of the trigger particle. Remaining possible biases due to correlations between trigger particles and EP particles may be assessed by comparing our results relative to the EP reconstructed from these two different methods with their respective EP resolutions. The results are found to be qualitatively similar, suggesting that any biases may be small. See Appendix A for details.

To extract the near-side jet-like component, we use the difference in azimuthal correlations between those analyzed at small and large $ | \Delta\eta| $. The MRP method, which excludes particles within $ | \Delta\eta|<0.5 $ of the trigger particle in the event, would have different systematic biases in the $ \Delta\phi $ correlations at small and large $ | \Delta\eta| $. Thus, we use the traditional reaction-plane method for the jet-like component. Figure 1 shows the $ \psi_{{\rm{EP}}} $ distributions from the modified reaction-plane method (data points) and the traditional reaction-plane method (histogram). We have checked the correlation between the event plane angles constructed from the traditional method and the MRP method, and found they are correlated as expected; the spread between their numerical values is consistent with that caused by EP resolutions.

We divide our data into six equal-size slices of trigger particle azimuthal angle relative to the event plane, $ \phi_{s} $, and analyze azimuthal correlations separately in each slice. Figure 2 shows a schematic view, with the slices labeled numerically 1 to 6, corresponding to $ \phi_{s} = | \phi_{t}- \psi_{{\rm{EP}}}| = 0 $-$ \pi/12 $, $ \pi/12 $-$ \pi/6 $, $ \pi/6 $-$ \pi/4 $, $ \pi/4 $-$ \pi/3 $, $ \pi/3 $-$ 5\pi/12 $, and $ 5\pi/12 $-$ \pi/2 $. We form azimuthal correlations with trigger particles in each slice separately. Figure 3 shows, as examples, the raw azimuthal correlations in 20%-60% Au+Au collisions for six slices in $ \phi_{s} $ for trigger and associated particle $ p_{T} $ ranges of $ 3< p_{T}^{(t)}<4 $ GeV/c and $ 1< p_{T}^{(a)}<2 $ GeV/c (upper panel), and $ 4< p_{T}^{(t)}<6 $ GeV/c and $ 2< p_{T}^{(a)}<4 $ GeV/c (lower panel), respectively. All raw correlation functions are presented in Figs. C1, C2, C3, and C4 in Appendix C as a function of trigger $ p_{T}^{(t)} $, associated $ p_{T}^{(a)} $, and $ \phi_{s} $.

Figure 2.  Sketch of six slices in trigger particle azimuthal angle relative to the event plane, $ \phi_{s} = | \phi_{t}- \psi_{{\rm{EP}}}| $.

Figure 3.  (color online) Raw dihadron $ \Delta\phi $ correlations with trigger particles in six slices of azimuthal angle relative to the event plane, $ \phi_{s} = | \phi_{t}- \psi_{{\rm{EP}}}| $. The data are from minimum-bias 20%-60% Au+Au collisions. The trigger and associated particle $ p_{T} $ ranges are $ 3< p_{T}^{(t)}<4 $ GeV/c and $ 1< p_{T}^{(a)}<2 $ GeV/c (upper panel), and $ 4< p_{T}^{(t)}<6 $ GeV/c and $ 2< p_{T}^{(a)}<4 $ GeV/c (lower panel), respectively. Note the lower panels correspond to the kinematic range used in Ref. [9]. Both the trigger and associated particles are restricted to within $ |\eta|<1 $. The triangle two-particle $ \Delta\eta $ acceptance is not corrected. Statistical errors are smaller than the symbol size. The raw data in the upper panels have been published in Ref. [35]. The curves are flow modulated ZYAM background including $ v_2 $ and $ v_4\{\psi_2\} $ from Eq. (1). The $ v_2 $ values used are given in Table 1 from four-particle $ v_2\{{4}\} $ and two-particle $ v_{{2}}\{2, \eta_{{\rm{gap}}}={0.7}\} $ (dashed curves) and the average $ v_2 $ from the two methods (solid curve). The $ v_4\{\psi_2\} $ is taken from the parameterization in Eq. (15).

Figure C1.  (color online) Raw dihadron $ \Delta\phi $ correlations with trigger particles in six slices of azimuthal angle relative to the event plane, $ \phi_{s} = | \phi_{t}- \psi_{{\rm{EP}}}| $. The data are from minimum-bias 20%-60% Au+Au collisions. The trigger $ p_{T} $ range is $ 3< p_{T}^{(t)}<4 $ GeV/c. Five associated particle $ p_{T}^{(a)} $ bins are shown. Both the trigger and associated particles are restricted within $ |\eta|<1 $. The triangle two-particle $ \Delta\eta $ acceptance is not corrected. Statistical errors are smaller than the symbol size. The curves are flow modulated ZYAM background including $ v_2 $ and $ v_4\{\psi_2\} $ by Eq. (1). The used $ v_2 $ values are given in Table 1 from four-particle $ v_2\{{4}\} $ and two-particle $ v_{{2}}\{2, \eta_{{\rm{gap}}}={0.7}\} $ (dashed curves) and the average $ v_2 $ from the two methods (solid curve). The $ v_4\{\psi_2\} $ is taken from the parameterization in Eq. (15).

Figure C2.  (color online) Same as in Fig. C1 but for trigger particle $ 4< p_{T}^{(t)}<6 $ GeV/c and six bins in associated particle $ p_{T}^{(a)} $.

Figure C3.  (color online) Same as in Fig. C1 but for $ | \Delta\eta|>0.7 $.

Figure C4.  (color online) Same as in Fig. C2 but for $ | \Delta\eta|>0.7 $.

The correlation structure sits atop a large combinatorial background. The background has a flow modulation induced by the anisotropies of the trigger particle and the background particles with respect to the participant plane [46]. In this analysis we use anisotropic flow parameters measured by two- and multi-particle cumulants [32] for the combinatorial background. An alternative approach that has been used to describe dihadron correlation data treats the anisotropic flow modulations as free parameters in a multi-parameter model fit to the dihadron correlation functions in 2-dimensional $ \Delta\eta $-$ \Delta\phi $ space [47, 48]. Results from this alternative approach to the inclusive dihadron correlation data (without a high-$ p_{T} $ trigger or cutting on $ \phi_{s} $) can be found in Refs. [47, 48]. The multi-parameter fit approach to our $ \phi_{s} $-dependent high-$ p_{T} $ triggered dihadron correlations is considered in Sec. III E, but a detailed discussion of the differences in assumptions and conclusions of the two approaches is beyond the scope of this paper.

In this analysis, first, only the $ v_2 $ and $ v_4 $ harmonic flow backgrounds are considered. The triangular harmonic flow background and other high-order effects are described in the next subsection, Sec. II C. Considering only the $ v_2 $ and $ v_4 $ harmonics, the flow correlated background is given by [49]

where B is the background normalization. In Eq. (1), $ v_2^{(a)} $ and $ v_4^{(a)}\{\psi_2\} $ are the associated particle's second and fourth harmonics with respect to the second harmonic event plane, $ \psi_2 $, and $ v_2^{(t,R)} $ and $ v_4^{(t,R)}\{\psi_2\} $ are the average harmonics of the trigger particles, $ v_2^{(t,R)} = \left\langle\cos\left[2\left( \phi_{t}- \psi_2\right)\right]\right\rangle^{(R)} $ and $ v_4^{(t,R)}\{\psi_2\} = \left\langle\cos\left[4\left( \phi_{t}- \psi_2\right)\right]\right\rangle^{(R)} $, respectively. The superscript ‘$ (R) $’ indicates that the averages are taken within the $ \phi_{t} $ region of a slice of width $ 2c $ at $ \phi_{s} $: $ \phi_{s}-c< \left| \phi_{t}- \psi_{{\rm{EP}}}\right|< \phi_{s}+c $ (where $ c = \pi/24 $ in our analysis). Note that we have used $ \phi_{s} $ here and in Eq. (3) to be the center value of a $ | \phi_{t}- \psi_{{\rm{EP}}}| $ bin, while elsewhere we simply use $ \phi_{s} = | \phi_{t}- \psi_{{\rm{EP}}}| $ to indicate a narrow bin in $ | \phi_{t}- \psi_{{\rm{EP}}}| $. For the $ n^{\rm{th}} $ harmonic anisotropy we have [49]

Here $ T_k $ is a short-hand notation for

$ \langle {\cos(k \Delta\psi)}\rangle\equiv\langle {\cos k( \psi_{{\rm{EP}}}- \psi_2)}\rangle $ is the event-plane resolution with respect to the $ k^{\rm{th}} $ harmonic, and $ \delta $ is Kronecker's delta. Since the correlation signal we are studying is of the order of a few percent of the background, we need to keep the flow correction in Eq. (2) up to the order $ v_2 v_4\sim0.1 $%. Keeping terms for $ v_2^{(t,R)} $ up to $ v_4 $ and for $ v_4^{(t,R)}\{\psi_2\} $ up to $ v_2 $, we have

and

Note the $ v_4^{(t)}\{\psi_2\} $ above is with respect to the second harmonic plane $ \psi_2 $. The final flow correction is given by Eqs. (1), (4), and (5).

The event-plane resolutions, $ \langle {\cos(k \Delta\psi)}\rangle $ ($ k = 2,4,6 $), are obtained from the sub-event method [32]. The event is randomly divided into two sub-events a and b with equal multiplicities. The sub-events, excluding the associated particle $ p_{T} $ region, are analyzed to yield event-plane angles which, ideally, should be identical. The difference between the obtained event-plane angles, $ \psi_{a}-\psi_{b} $, gives the uncertainty in the event-plane determination of the sub-events [32]

The event-plane resolution of the full event can be approximated by [32]

in the limit of small event-plane resolution. The factor $ \sqrt{2} $ comes in because the multiplicities of the sub-events are smaller than the full event multiplicity by a factor of 2. We use the approximate form of Eq. (7) to assess systematic uncertainties in the event-plane resolutions from different ways of dividing the event into sub-events (see Sec. III B).

The precise form of the event-plane resolution of the full event is given by [32]

where

depends on the harmonic anisotropy magnitude $ v_{k} $ and the number of particles N used in event-plane reconstruction. The $ p_{T} $ enters into Eq. (9) because we weighted each particle by its $ p_{T} $ in constructing the event plane. In data analysis we solve for the sub-event $ \chi_{k}(N/2) $ by Eq. (8) and the known event-plane resolution of the sub-events from Eq. (6), employing an iterative procedure [32]. From Eq. (9) we obtain the full event $ \chi_{k}(N) = \sqrt{2}\chi_{k}(N/2) $. We then use Eq. (8) to determine the event-plane resolution of the full event [32]. The event-plane resolutions are listed in Table 1. The resolutions depend on the $ p_{T} $ bin because particles in a given $ p_{T} $ bin (to be used for correlation analysis) are excluded from the event-plane reconstruction to avoid self-correlations as aforementioned.

$p_T$ /(GeV/c)	$v_2$	$\left\langle {\cos (2\Delta \psi)} \right\rangle $	$\left\langle {\cos (4\Delta \psi)} \right\rangle$	$\left\langle {\cos (6\Delta \psi)} \right\rangle $
0.15 - 0.5	0.038±0.003	0.673	0.324	0.127
0.5 - 1	0.082±0.006	0.596	0.247	0.082
1 - 1.5	0.128±0.010	0.637	0.286	0.104
1.5 - 2	0.164±0.011	0.676	0.328	0.129
2 - 3	0.189±0.012	0.704	0.360	0.150
3 - 4	0.194±0.013
4 - 6	0.163±0.020

Table 1.  Elliptic flow and event-plane resolutions as a function of $ p_{T} $ in 20%-60% minimum-bias Au+Au collisions. The resolutions depend on the $ p_{T} $ bin because particles in a given $ p_{T} $ bin are excluded from the event-plane reconstruction to avoid self-correlations. The errors on $ v_2 $ are systematic uncertainties given by two-particle $ v_{{2}}\{2, \eta_{{\rm{gap}}}={0.7}\} $ (with a reference particle $ 0.15< p_{T}<2 $ GeV/c) and four-particle $ v_{{2}}\{{4}\} $ (with three reference particles). Systematic uncertainties on the resolutions are negligible.

One would naively expect that the event-plane resolution should be different for different trigger particle orientations from the event plane because the influence of di-jets on the event-plane determination should vary: a di-jet aligned with the reaction plane enhances the event-plane reconstruction, resulting in a better resolution, whereas a di-jet perpendicular to the reaction plane reduces the accuracy of the constructed event plane, resulting in a poorer resolution. However, this is a post effect due to the selection based on the relative angle between the trigger particle and the corresponding event plane. The resolutions used in Eq. (1), on the other hand, are those of all triggered events before any selection of the trigger particle orientation is made. We have also verified this with Monte Carlo toy model simulations.

Since only triggered events enter into our correlation measurements, the event-plane resolutions are measured using only these events. The event-plane resolutions from inclusive events (minimum-bias events within the given centrality bin) are found to be within a couple of percent of that from the triggered events (see systematic uncertainty discussion in Sec. III B).

We analyzed the elliptic flow in each of the $ p_{T} $ bins used in our correlation analysis. The obtained elliptic flow parameters are tabulated in Table 1 together with their systematic uncertainties. The analysis of the elliptic flow and the assessment of its systematic uncertainty are both described in Sec. III A. We used these $ v_2 $ parameters for background subtraction. The calculated magnitudes of the elliptic flow modulation, $ 2 v_2^{(a)} v_2^{(t,R)} $, are listed in Table 2 together with their systematic uncertainties. The calculated background curves are superimposed in Fig. 3. As seen from Fig. 3 and Table 2, the flow modulation is relatively small for $ \pi/4< \phi_{s}<\pi/3 $; this is because $ v_2^{(t,R)} $ is the average within a given $ \phi_{s} $ bin, as aforementioned, which is close to zero for $ \phi_{s}\sim 45^\circ $. It is more so for the $ \pi/4< \phi_{s}<\pi/3 $ bin than for its “symmetric” $ \pi/6< \phi_{s}<\pi/4 $ bin because other harmonics also contribute to the average only within the limited $ \phi_{s} $ bin (cf. Eq. (2)).

$p_T^{\left( a \right)}$ /(GeV/c)	$0 - \pi/12$	$\pi/12 - \pi/6$	$\pi/6 - \pi/4$	$\pi/4- \pi/3$	$\pi/3 - 5\pi/12$	$5\pi/12 - \pi/2$
	$3 < p_T^{\left( t \right)} < 4$ GeV/c
0.15 - 0.5	$0.0544\pm0.0046$	$0.0433\pm0.0039$	$0.0229\pm0.0025$	$-0.0028\pm0.0006$	$-0.0270\pm0.0015$	$-0.0416\pm0.0028$
0.5 - 1	$0.1098\pm0.0096$	$0.0884\pm0.0082$	$0.0490\pm0.0055$	$-0.0004\pm0.0018$	$-0.0466\pm0.0022$	$-0.0745\pm0.0045$
1 - 1.5	$0.1793\pm0.0149$	$0.1435\pm0.0128$	$0.0776\pm0.0085$	$-0.0054\pm0.0024$	$-0.0831\pm0.0042$	$-0.1301\pm0.0081$
1.5 - 2	$0.2376\pm0.0178$	$0.1892\pm0.0152$	$0.0999\pm0.0100$	$-0.0128\pm0.0025$	$-0.1186\pm0.0057$	$-0.1825\pm0.0105$
2 - 3	$0.2814\pm0.0194$	$0.2233\pm0.0166$	$0.1159\pm0.0108$	$-0.0199\pm0.0024$	$-0.1473\pm0.0067$	$-0.2243\pm0.0121$
	$4 < p_T^{\left( t \right)} < 6$ GeV/c
0.15 - 0.5	$0.0535\pm0.0047$	$0.0421\pm0.0041$	$0.0213\pm0.0028$	$-0.0045\pm0.0008$	$-0.0284\pm0.0013$	$-0.0427\pm0.0026$
0.5 - 1	$0.1073\pm0.0101$	$0.0853\pm0.0088$	$0.0451\pm0.0062$	$-0.0045\pm0.0025$	$-0.0502\pm0.0017$	$-0.0777\pm0.0041$
1 - 1.5	$0.1758\pm0.0156$	$0.1390\pm0.0136$	$0.0717\pm0.0095$	$-0.0115\pm0.0034$	$-0.0883\pm0.0035$	$-0.1344\pm0.0074$
1.5 - 2	$0.2337\pm0.0186$	$0.1838\pm0.0162$	$0.0928\pm0.0113$	$-0.0201\pm0.0038$	$-0.1246\pm0.0048$	$-0.1872\pm0.0097$
2 - 3	$0.2773\pm0.0202$	$0.2174\pm0.0177$	$0.1080\pm0.0123$	$-0.0280\pm0.0039$	$-0.1537\pm0.0057$	$-0.2291\pm0.0113$

Table 2.  The elliptic flow modulation in the correlation background, $ 2 v_2^{(a)} v_2^{(t,R)} $, calculated using measurements in Table 1, as a function of $ p_{T}^{(a)} $ (in rows) and $ \phi_{s} = | \phi_{t}- \psi_{{\rm{EP}}}| $ (in columns) in minimum-bias 20%-60% Au+Au collisions. Both trigger particle $ p_{T} $ ranges of $ 3< p_{T}^{(t)}<4 $ GeV/c and $ 4< p_{T}^{(t)}<6 $ GeV/c are listed. Quoted errors are systematic uncertainties. Note the significantly smaller systematic uncertainties out-of-plane than in-plane.

As mentioned previously, our trigger particle $ p_{T} $ ranges are $ 3< p_{T}^{(t)}<4 $ GeV/c and $ 4< p_{T}^{(t)}<6 $ GeV/c. In elementary p+p and d+Au collisions, the particles in these $ p_{T} $ ranges originate mainly from hard-scatterings and jets. In relativistic heavy ion colllisions, however, a large baryon to meson ratio has been observed in the $ p_{T} $ region around 3 GeV/c [50, 51]. The reason for the large ratio and the sources of those high $ p_{T} $ particles are still under debate. The coalescence and recombination models [52-54] can elegantly explain the large baryon to meson ratio from a thermal bath of constituent quarks. On the other hand, the jet-like correlations at small angles relative to trigger particles of $ p_{T}^{(t)}>3 $ GeV/c, with the long range ridge correlation removed, are measured to be invariant from p+p, d+Au, peripheral to central Au+Au collisions [55], and independent of the reaction plane direction in Au+Au collisions, as will be shown in this work. This experimental evidence strongly suggests that those $ p_{T}^{(t)}>3 $ GeV/c particles are mostly of jet origin in Au+Au collisions, just as in p+p and d+Au collisions. It is possible that recombination may still be at work in our trigger particle $ p_{T} $ ranges, but in such a fashion that the parton(s) prior to recombination have already imprinted angular correlations related to the hard-scatterings [56].

Different sources, such as the recombination [52-54] and jet fragmentation discussed above, will likely give different anisotropies to those high $ p_{T} $ particles. However, the anisotropy of the trigger particles to be used in the background subtraction in Eqs. (1), (2), (4), and (5) should be the experimentally measured net anisotropy [57], as we have done in this work, irrespective of the different origins.

In Eq. (1) we have neglected the odd harmonic terms, such as $ 2v_1^{(a)}v_1^{(t,R)}\cos( \Delta\phi) $ and $ 2 v_3^{(a)} v_3^{(t,R)}\cos(3 \Delta\phi) $. Due to symmetry at mid-rapidity, the averages of the odd harmonic coefficients $ v_1 $, $ v_3 $, and etc. vanish. However, their fluctuations would yield non-vanishing averages of the products of $ v_1^{(a)}v_1^{(t)} $ and $ v_3^{(a)} v_3^{(t)} $, thereby contributing to the background in the dihadron correlations. If one assumes that the amplitude of the $ v_1 $ (directed flow) fluctuations is of the same order of magnitude as the maximum $ v_1 $ in our pseudorapidity range (which was measured to be small [58]), then the $ v_1 $ fluctuation contribution can be neglected [59, 60]. In the present work we neglect any direct flow fluctuation effect in our background subtraction. More recent developments [61, 62] in the understanding of initial geometry fluctuations, however, suggest that $ v_1 $ fluctuation effects (sometimes called rapidity-even $ v_1 $) may not be small as originally thought [59, 60]. We remark in Sec. IV F on the magnitude of the possible $ v_1 $ fluctuation effects using recent measurements.

Note that the possible effect of statistical global momentum conservation can generate a negative dipole [63] which has the same shape as the $ v_1 $ fluctuation effect. However, the statistical momentum conservation effect is not from $ v_1 $ fluctuations, but part of the correlation signal, the same as momentum conservation by any other mechanisms, such as dijet production.

It has been shown that the initial fluctuations in the overlap geometry (spatial distribution of participating nucleons) give rise to $ v_3 $ (triangular flow) fluctuations [30, 64, 65]. It was found from the Monte Carlo Glauber model [66] that the triangularity due to geometry fluctuations can be comparable to the magnitude of the eccentricity, which is connected to the elliptic flow [30]. It is thus possible that large triangular flow fluctuations can arise which would give triangular peaks in the flow background [30, 64, 65]. This appears to be the case in the AMPT (A Multi-Phase Transport) model and the UrQMD (Ultrarelativistic Quantum Molecular Dynamics) model studied in Refs. [30, 67, 68] and [65], respectively. Hydrodynamic calculations with event-by-event geometry fluctuations confirm that sizeable $ v_3 $ can be generated from initial geometry fluctuations [69-73]. The $ v_3 $ magnitude is smaller than that of $ v_2 $ despite the similar initial triangular and elliptic eccentricities of $ \epsilon_3 $ and $ \epsilon_2 $, respectively. This is likely due to the larger damping power of shear viscosity on $ v_3 $ than on $ v_2 $ [69-72, 74].

Since the orientation of the triangular overlap shape due to fluctuations is random relative to the event-plane direction [75, 76], determined by the elliptic anisotropy, the effect of any triangular flow is independent of the event plane. In other words, the triangular flow background would be proportional to $ 2 v_3^{(a)} v_3^{(t,R)}\cos(3 \Delta\phi) = 2 v_3^{(a)} v_3^{(t)}\cos(3 \Delta\phi) $ independent of $ \phi_{s} $. With triangular flow, the flow background of Eq. (1) becomes

We may estimate the effect of triangular flow fluctuations in our correlation measurements. The AMPT and UrQMD models indicate that in the 20%-60% centrality range the triangular flow fluctuation effect is about 10% of the elliptic flow for our trigger and associated $ p_{T} $ bins, $ v_{3}^{2}/v_{2}^{2}\approx0.1 $ [30, 65, 77]. Event-by-event hydrodynamic calculations yield a similar magnitude of $ v_{3}^{2}/v_{2}^{2} $ [78]. Experimental data on inclusive two-particle correlations at $ p_{T}>2 $ GeV/c indicate a ratio of the harmonic coefficients also of magnitude $ v_{3}^{2}/v_{2}^{2}\approx0.1 $ within 20%-60% centrality [79]. More recent measurements on triangular anisotropy are consistent with these estimates [80, 81]. This suggests that the measured third harmonic term in the inclusive two-particle correlations at low $ p_{T} $ may be dominated by triangular flow fluctuations, just as the second harmonic term is dominated by elliptic flow. As we will show in Sec. IV C, the effect of a triangular flow of this magnitude is sizeable in our dihadron correlation measurements with high $ p_{T} $ trigger particles as well. In the main work of our study of high-$ p_{T} $ dihadron correlations relative to the EP, we do not include the possible contributions from $ v_3 $ anisotropy in the flow background subtraction. In Sec. IV F we discuss the effect of the presently measured $ v_3 $ on our dihadron correlation results.

So far only the $ v_4 $ contribution correlated with the second harmonic plane $ \psi_2 $ has been considered as in Eq. (1). This part of $ v_4 $ is referred to as $ v_4\{\psi_2\} $. The other part of $ v_4 $ arises from fluctuations and is uncorrelated to $ \psi_{{\rm{EP}}} $. We refer to this part as $ V_4\{{{\rm{uc}}}\} $. The flow background is then given by

Section IV F discusses how $ V_4\{{{\rm{uc}}}\} $ is obtained in the present analysis.

Glauber model [66] calculations also show that the quadrangularity, pentagonality, and hexagonality due to geometry fluctuations are equal to the triangularity, all large and comparable to the eccentricity [77, 82]. However, it has been suggested that those higher order eccentricities are inefficient in generating sizeable high-order harmonic flow in final state momentum space [77]. Experimental data also indicate that the magnitudes and fluctuations of $ v_4 $ and $ v_6 $ are small relative to the magnitude of $ v_2 $ [83]. Although we include $ V_4\{{{\rm{uc}}}\} $ in our flow background of Eq. (11), the effect of $ V_4\{{{\rm{uc}}}\} $ is small, as will be discussed in Sec. IV F. It is safe to neglect $ v_5^2 $ and the higher order anisotropic fluctuation terms in the flow background of Eq. (1).

The flow correlated backgrounds given by Eq. (1), as an example, are shown in Fig. 3 as solid curves. The background curves have been normalized assuming that the background-subtracted signal has Zero Yield At Minimum (ZYAM) [13, 84]. To obtain the ZYAM normalization factor, we fold the raw correlation function to within the range of $ 0< \Delta\phi<\pi $ because of the symmetry of the correlation function. We take the ratio of the folded raw correlation to the background curve of Eq. (1), where B is set to unity before taking the ratio. We obtain a continuous range of the size of $ \pi/6 $ where the average ratio is the smallest. This smallest average ratio is the normalization factor B to be used in the flow background of Eq. (1), which is then subtracted from the raw correlation function to obtain the final correlation signal.

The background levels can be different for the different $ \phi_{s} $ slices because of the net effect of the variations in jet-quenching with $ \phi_{s} $ and the centrality cuts in total charged particle multiplicity in the TPC within $ |\eta|<0.5 $. Thus, in our correlation analysis, the background level B is treated independently in individual $ \phi_{s} $ slices. In the recent proposal [85] to fit the $ \phi_{s} $-dependent near-side correlations at large $ \Delta\eta $ (i.e. the ridge region) by Fourier coefficients and treat them as backgrounds, the background level B is required to be the same in all $ \phi_{s} $ slices.

Table 3 lists the obtained background level B as a function of $ \phi_{s} $ and $ p_{T}^{(a)} $ in 20%-60% Au+Au collisions. Results from both trigger particle $ p_{T} $ ranges of $ 3< p_{T}^{(t)}<4 $ GeV/c and $ 4< p_{T}^{(t)}<6 $ GeV/c are listed. The background levels listed are not only for the correlation functions with the $ | \Delta\eta|<2 $ region within our acceptance, but also for those in the large $ \Delta\eta $ region of $ | \Delta\eta|>0.7 $. The latter is used for the ridge studies (see Sec. IV C). The background level for the lower trigger particle $ p_{T}^{(t)} $ range is slightly larger. This is due to the fact that relatively more events contain multiple jets with the lower trigger particle $ p_{T}^{(t)} $ and those events are used multiple times in our di-hadron correlation analysis [17].

$p_T^{\left( a \right)}$ /(GeV/c)	$0 - \pi/12$	$\pi/12 - \pi/6$	$\pi/6 - \pi/4$	$\pi/4- \pi/3$	$\pi/3 - 5\pi/12$	$5\pi/12 - \pi/2$
	$3 < p_T^{\left( t \right)} < 4$ GeV/c
0.15 - 0.5	$47.41\pm 0.01^{+ 0.06}_{- 0.07}$	$47.37\pm 0.01^{+ 0.07}_{- 0.09}$	$47.28\pm 0.02^{+ 0.02}_{- 0.17}$	$47.22\pm 0.02^{+ 0.06}_{- 0.08}$	$47.16\pm 0.02^{+ 0.02}_{- 0.10}$	$47.04\pm 0.02^{+ 0.04}_{- 0.06}$
0.5 - 1	$22.47\pm 0.01^{+ 0.01}_{- 0.10}$	$22.67\pm 0.01^{+ 0.03}_{- 0.15}$	$22.92\pm 0.01^{+ 0.03}_{- 0.20}$	$23.31\pm 0.01^{+ 0.06}_{- 0.18}$	$23.40\pm 0.01^{+ 0.06}_{- 0.11}$	$23.59\pm 0.01^{+ 0.03}_{- 0.09}$
1 - 1.5	$6.023\pm0.005^{+0.008}_{-0.015}$	$6.072\pm0.005^{+0.020}_{-0.089}$	$6.128\pm0.005^{+0.016}_{-0.106}$	$6.177\pm0.005^{+0.033}_{-0.081}$	$6.128\pm0.005^{+0.022}_{-0.042}$	$6.199\pm0.006^{+0.014}_{-0.028}$
1.5 - 2	$1.683\pm0.002^{+0.005}_{-0.007}$	$1.691\pm0.003^{+0.002}_{-0.034}$	$1.698\pm0.002^{+0.003}_{-0.046}$	$1.700\pm0.003^{+0.010}_{-0.034}$	$1.694\pm0.003^{+0.006}_{-0.036}$	$1.694\pm0.003^{+0.001}_{-0.013}$
2 - 3	$0.655\pm0.002^{+0.004}_{-0.002}$	$0.662\pm0.002^{+0.003}_{-0.017}$	$0.663\pm0.002^{+0.003}_{-0.028}$	$0.660\pm0.002^{+0.002}_{-0.026}$	$0.654\pm0.002^{+0.001}_{-0.014}$	$0.659\pm0.002^{+0.008}_{-0.011}$
	$4 < p_T^{\left( t \right)} < 6$ GeV/c
0.15 - 0.5	$46.63\pm 0.04^{+ 0.02}_{- 0.12}$	$46.56\pm 0.04^{+ 0.08}_{- 0.16}$	$46.72\pm 0.04^{+ 0.08}_{- 0.24}$	$46.77\pm 0.04^{+ 0.08}_{- 0.27}$	$46.67\pm 0.05^{+ 0.12}_{- 0.07}$	$46.76\pm 0.05^{+ 0.12}_{- 0.15}$
0.5 - 1	$22.16\pm 0.02^{+ 0.01}_{- 0.07}$	$22.30\pm 0.02^{+ 0.09}_{- 0.22}$	$22.42\pm 0.02^{+ 0.00}_{- 0.32}$	$23.11\pm 0.03^{+ 0.06}_{- 0.19}$	$23.07\pm 0.03^{+ 0.11}_{- 0.09}$	$23.42\pm 0.03^{+ 0.07}_{- 0.17}$
1 - 1.5	$5.947\pm0.012^{+0.003}_{-0.049}$	$5.989\pm0.012^{+0.001}_{-0.084}$	$5.985\pm0.012^{+0.006}_{-0.109}$	$6.113\pm0.013^{+0.040}_{-0.101}$	$6.076\pm0.014^{+0.021}_{-0.061}$	$6.174\pm0.014^{+0.037}_{-0.057}$
1.5 - 2	$1.659\pm0.006^{+0.003}_{-0.041}$	$1.664\pm0.006^{+0.003}_{-0.035}$	$1.673\pm0.006^{+0.017}_{-0.050}$	$1.671\pm0.007^{+0.024}_{-0.051}$	$1.674\pm0.007^{+0.013}_{-0.022}$	$1.712\pm0.007^{+0.014}_{-0.038}$
2 - 3	$0.611\pm0.004^{+0.001}_{-0.006}$	$0.618\pm0.004^{+0.003}_{-0.017}$	$0.613\pm0.004^{+0.001}_{-0.024}$	$0.621\pm0.004^{+0.010}_{-0.028}$	$0.615\pm0.004^{+0.008}_{-0.012}$	$0.615\pm0.005^{+0.004}_{-0.014}$
3 - 4	$0.058\pm0.001^{+0.001}_{-0.002}$	$0.060\pm0.001^{+0.001}_{-0.009}$	$0.061\pm0.001^{+0.002}_{-0.014}$	$0.063\pm0.001^{+0.000}_{-0.016}$	$0.061\pm0.001^{+0.001}_{-0.020}$	$0.062\pm0.001^{+0.001}_{-0.004}$
	$3 < p_T^{\left( t \right)} < 4$ GeV/c, $|\Delta |>0.7$
0.15 - 0.5	$19.37\pm 0.01^{+ 0.02}_{- 0.05}$	$19.35\pm 0.01^{+ 0.02}_{- 0.06}$	$19.29\pm 0.01^{+ 0.00}_{- 0.12}$	$19.25\pm 0.01^{+ 0.00}_{- 0.06}$	$19.28\pm 0.01^{+ 0.02}_{- 0.05}$	$19.22\pm 0.01^{+ 0.00}_{- 0.03}$
0.5 - 1	$9.187\pm0.006^{+0.001}_{-0.043}$	$9.268\pm0.006^{+0.022}_{-0.100}$	$9.356\pm0.006^{+0.022}_{-0.100}$	$9.507\pm0.007^{+0.022}_{-0.102}$	$9.548\pm0.007^{+0.009}_{-0.024}$	$9.603\pm0.007^{+0.041}_{-0.028}$
1 - 1.5	$2.452\pm0.003^{+0.004}_{-0.006}$	$2.475\pm0.003^{+0.006}_{-0.050}$	$2.497\pm0.003^{+0.012}_{-0.061}$	$2.512\pm0.003^{+0.014}_{-0.053}$	$2.493\pm0.003^{+0.012}_{-0.025}$	$2.517\pm0.004^{+0.017}_{-0.015}$
1.5 - 2	$0.683\pm0.002^{+0.003}_{-0.005}$	$0.688\pm0.002^{+0.003}_{-0.020}$	$0.691\pm0.002^{+0.003}_{-0.026}$	$0.689\pm0.002^{+0.005}_{-0.017}$	$0.686\pm0.002^{+0.007}_{-0.016}$	$0.685\pm0.002^{+0.005}_{-0.004}$
2 - 3	$0.264\pm0.001^{+0.001}_{-0.004}$	$0.269\pm0.001^{+0.002}_{-0.008}$	$0.269\pm0.001^{+0.002}_{-0.014}$	$0.267\pm0.001^{+0.002}_{-0.015}$	$0.265\pm0.001^{+0.003}_{-0.010}$	$0.264\pm0.001^{+0.004}_{-0.002}$
	$4 < p_T^{\left( t \right)} < 6$ GeV/c, $|\Delta |>0.7$
0.15 - 0.5	$18.98\pm 0.02^{+ 0.00}_{- 0.05}$	$18.99\pm 0.02^{+ 0.01}_{- 0.10}$	$19.01\pm 0.03^{+ 0.00}_{- 0.11}$	$18.98\pm 0.03^{+ 0.02}_{- 0.16}$	$18.98\pm 0.03^{+ 0.01}_{- 0.01}$	$19.00\pm 0.03^{+ 0.02}_{- 0.07}$
0.5 - 1	$9.014\pm0.015^{+0.003}_{-0.045}$	$9.065\pm0.015^{+0.014}_{-0.116}$	$9.110\pm0.015^{+0.017}_{-0.122}$	$9.374\pm0.017^{+0.035}_{-0.108}$	$9.391\pm0.018^{+0.025}_{-0.077}$	$9.518\pm0.018^{+0.004}_{-0.120}$
1 - 1.5	$2.421\pm0.008^{+0.008}_{-0.012}$	$2.434\pm0.008^{+0.003}_{-0.036}$	$2.435\pm0.008^{+0.009}_{-0.078}$	$2.456\pm0.008^{+0.023}_{-0.045}$	$2.457\pm0.009^{+0.011}_{-0.034}$	$2.501\pm0.009^{+0.008}_{-0.019}$
1.5 - 2	$0.673\pm0.004^{+0.001}_{-0.011}$	$0.669\pm0.004^{+0.003}_{-0.017}$	$0.677\pm0.004^{+0.009}_{-0.032}$	$0.681\pm0.004^{+0.010}_{-0.039}$	$0.677\pm0.005^{+0.007}_{-0.010}$	$0.691\pm0.005^{+0.004}_{-0.018}$
2 - 3	$0.241\pm0.003^{+0.002}_{-0.004}$	$0.250\pm0.003^{+0.004}_{-0.010}$	$0.248\pm0.003^{+0.001}_{-0.014}$	$0.253\pm0.003^{+0.002}_{-0.014}$	$0.245\pm0.003^{+0.004}_{-0.007}$	$0.247\pm0.003^{+0.003}_{-0.004}$
3 - 4	$0.023\pm0.001^{+0.001}_{-0.001}$	$0.024\pm0.001^{+0.000}_{-0.003}$	$0.025\pm0.001^{+0.000}_{-0.004}$	$0.025\pm0.001^{+0.001}_{-0.008}$	$0.023\pm0.001^{+0.001}_{-0.007}$	$0.024\pm0.001^{+0.000}_{-0.002}$

Table 3.  Background level B in flow subtraction by Eq. (1) as a function of $ p_{T}^{(a)} $ (in rows) and $ \phi_{s} = | \phi_{t}- \psi_{{\rm{EP}}}| $ (in columns) in minimum-bias 20%-60% Au+Au collisions. Both trigger particle $ p_{T} $ ranges of $ 3< p_{T}^{(t)}<4 $ GeV/c and $ 4< p_{T}^{(t)}<6 $ GeV/c are listed. The trigger and associated particles are within $ |\eta|<1 $. Backgrounds are tabulated for the entire $ | \Delta\eta|<2 $ range of our acceptance as well as for the large $ \Delta\eta $ cut of $ | \Delta\eta|>0.7 $. The first error is statistical. The second error is the quadratic sum of the ZYAM systematic uncertainty and the one-sided systematic uncertainty due to background deviation from ZYAM. The former is assessed by varying the $ \Delta\phi $ normalization range. The latter is assessed by comparing our ZYAM background to those obtained from asymmetric correlations of the separate positive and negative $ \phi_{t}- \psi_{{\rm{EP}}} $ regions.

It is worth emphasizing here that our quantitative results depend on the assumption of the ZYAM background normalization, and the effects of variations in the ZYAM normalization within a reasonable range are assessed by systematic uncertainties. However, as will be discussed in Sec. III E, our qualitative conclusions are not affected by the ZYAM normalization.

Background subtraction is the major source of systematic uncertainty in our results. We first study the dihadron correlations with even harmonic flow subtraction. The even harmonic flow background, as given by Eq. (1), has three important ingredients: the anisotropic flow measurements $ v_2 $ and $ v_4 $, the event-plane resolutions, and the background magnitude B. We discuss these systematic uncertainties in Sections III A-III E, respectively. They have effects on the dihadron correlation functions presented in Sec. IV A and the away-side correlation widths and magnitudes presented in Sec. IV B.

We also report results on near-side jet-like and ridge correlations in Sec. IV C. Uncertainties in $ v_2 $ and the ZYAM background normalization contribute to the uncertainties in the ridge correlation results. They do not affect the jet-like correlation results, in which they largely cancel because $ v_2 $ is approximately independent of pseudo-rapidity within our acceptance. Additional systematic uncertainties arise from the assumption of a uniform ridge in $ \Delta\eta $, which affects both the ridge and jet-like results. These additional systematic uncertainties are discussed in Sec. III F.

We also study the dihadron correlations with background subtraction, including odd harmonic flow. Various systematics are discussed together with the correlation results in Sec. IV F.

The anisotropic flow (mainly elliptic flow) background which is to be subtracted from the dihadron correlation is the anisotropy caused by particle correlations to the participant plane [86, 87]. There are several measurements of elliptic flow; many of them are affected to various degrees by nonflow contributions that are caused by particle correlations unrelated to the reaction plane (or participant plane), such as resonance decays and jet-correlations. One technique, called the event-plane method, is to construct the event plane from all charged particles except those of interest and then calculate $ v_2 ${EP}$ = \langle {\cos2(\phi- \psi_{{\rm{EP}}})}\rangle/\langle {\cos2 \Delta\psi}\rangle $ for the particles of interest, where $ \langle {\cos2 \Delta\psi}\rangle $ is the event-plane resolution [32]. This method is affected by nonflow contributions in both sets of particles, those of interest and those used to construct the event plane. The $ v_2 ${EP} already contains flow fluctuation effects which should be included in the jet-correlation background.

Another method, called the two-particle method, is to calculate $ v_2\{{2}\} = \sqrt{\langle {\cos2 \Delta\phi}\rangle} $ using all particle pairs of interest [32]. This method is affected by nonflow only in the particles of interest used for correlation studies. This flow parameter also contains flow fluctuation effects. The two-particle cumulant method can also be applied between the particle of interest and a reference particle. The anisotropy of the particle of interest is then the ratio of the two-particle cumulant to the anisotropy of the reference particles, which can be in turn obtained from the two-particle cumulant between reference particle pairs. (More details are given in Sec. IV F.) This method of mixed pair cumulant is intrinsically similar to the event-plane method.

The third method, called the four-particle method, is to obtain $ v_2\{{4}\} $ from the four-particle cumulant [88]. This method is less affected by nonflow from particle clustering because the nonflow arising from two particle correlations is eliminated, and the nonflow from three particle correlations does not contribute. This method is subject to nonflow from higher orders (four-particle correlation and above) but those contributions are suppressed by high orders of multiplicity [88]. The flow fluctuation will give a negative contribution to $ v_2\{{4}\} $ [88].

The fourth method is to decompose the low $ p_{T} $ two-particle correlation (the so-called untriggered correlation, without the requirement of a trigger particle) into a near-angle Gaussian, a dipole, and a quadrupole, and infer $ v_2 ${2D} from the fitted quadrupole [89]. The method attempts to geometrically separate the reaction-plane correlated $ v_2 $ from other (i.e. nonflow) correlations (small-angle correlations and large-angle dipole). However, the method assumes a particular functional form for those nonflow correlations, whereas the goal of this paper is to study the magnitude and shape of those nonflow (jet) correlations, defined to be the data minus harmonic (flow) backgrounds.

The measured $ v_2\{{2}\} $ and $ v_2 ${MRP} are similar and they both significantly overestimate elliptic flow due to large contributions from nonflow and fluctuations. While the flow fluctuation effect should be included in our background subtraction, nonflow should be excluded. The major component of nonflow is the measured small-angle two-particle correlation [47, 48, 90]. To suppress nonflow, a pseudo-rapidity $ \eta $-gap ($ \eta_{{\rm{gap}}} $) is often applied between the particle pair in the $ v_{{n}}\{{2}\} $ measurement, and in the $ v_2 ${EP} measurement, between the particle of interest and the particles used in EP reconstruction. In this analysis, we apply $ \eta_{{\rm{gap}}} = 0.7 $ to obtain the two-particle cumulant elliptic flow, $ v_{{2}}\{2, \eta_{{\rm{gap}}}={0.7}\} $. However, the away-side two-particle correlations, presumably due to jet-like correlations, cannot be eliminated [8, 91, 92]. This is because the inter-jet correlation in $ \eta $ is broad (nearly uniform in the STAR TPC acceptance) due to the unconstrained underlying parton kinematics in the longitudinal direction.

We use $ v_{{2}}\{{2}\} $ as our upper systematic bound for $ v_2 $. The $ v_{{2}}\{{2}\} $ is measured in 10%-size centrality bins. Two-particle cumulants between the particle of interest and a reference particle, $ V_{{n}}\{{ p_{T}{\rm{-ref}}}\} $, and between two reference particles, $ V_{{n}}\{{\rm{ref-ref}}\} $, are calculated. The particle of interest is from a particular $ p_{T} $ bin, while the reference particle is from $ 0.15< p_{T}<2 $ GeV/c. To reduce nonflow one particle is taken from $ \eta<-0.35 $ and the other from $ \eta>0.35 $, with an $ \eta_{{\rm{gap}}} = 0.7 $ in-between. The $ v_n $ are referred to as $ v_{{n}}\{2, \eta_{{\rm{gap}}}={0.7}\} $ or simply as $ v_{{n}}\{{2}\} $. The cumulants are calculated by the Q-cumulant method and divided by the corresponding number of pairs in each event. The cumulants are averaged over the event sample with a unit weight (not weighted by the number of pairs). The anisotropy of the particle of interest is simply given by

The $ v_{{n}}\{{2}\} $ of the four individual centralities are averaged by weighting each centrality by the number of particles of interest.

The measured $ v_2\{{4}\} $ likely underestimates elliptic flow because the flow fluctuation effect in $ v_2\{{4}\} $ is negative [88]. We note that $ v_2\{{4}\} $ may still contain some nonflow effects. However, the agreement between $ v_2\{{4}\} $ and the elliptic flow measurement using the Lee-Yang-Zero method suggests that such nonflow effects are small [93]. We therefore use $ v_2\{{4}\} $ as our lower bound of $ v_2 $ systematic uncertainty, the same as in Refs. [13, 17]. The $ v_{{2}}\{{4}\} $ is obtained as follows. Two four-particle cumulants are calculated. One is for quadralets of one particle of interest and three reference particles, referred to as $ V_{{2}}\{{ p_{T}{\rm{-ref}}^3}\} $. The other is for quadralets of four reference particles, referred to as $ V_{{2}}\{{{\rm{ref}}^4}\} $. Since nonflow is negligible in $ v_{{n}}\{{4}\} $, no $ \eta_{{\rm{gap}}} $ is applied; all four particles are from the entire region of $ |\eta|<1 $. Similar to $ v_{{2}}\{{2}\} $, the Q-cumulant method is used to calculate $ v_{{2}}\{{4}\} $. Self-correlations are properly removed. The four-particle anisotropy of the particle of interest is given by

Again the $ v_{{2}}\{{4}\}( p_{T}) $ of the four individual centralities are averaged by weighting each centrality by the number of particles of interest.

As the default $ v_2 $, we use the average,

We use the range bracketed by $ v_{{2}}\{{2}\} $ and $ v_{{2}}\{{4}\} $ as our systematic uncertainty on $ v_2 $. Table 1 lists the default $ v_2 $ values together with systematic uncertainties for different $ p_{T} $ bins in 20%-60% Au+Au collisions.

We parameterized the $ v_4 $ measurement [45] as

and used this parameterization for both trigger and associated particles in our flow correction [31]. The uncertainties in $ v_2 $ are propagated to $ v_4 $. Note that the $ v_4 $ fluctuation effects related to the second harmonic event plane, which should be included in our flow background, are already included in the $ v_4 $ measurement which was carried out with respect to the second harmonic event plane [45, 83]. Fluctuations in $ v_4 $ related to the fourth harmonic event plane could be potentially not small [83] and are not included in the available measurement of $ v_4 $. However, these fluctuation effects come into our two-particle correlation background as $ v_4^{2} $ (not through the cross-term of $ v_2 v_4 $) and are therefore negligible for our centrality range. Nevertheless, in Sec. IV F, we also include this fluctuation effect in flow subtraction.

The flow backgrounds are shown by the solid curves in Fig. 3. The systematic uncertainties due to anisotropic flow parameters are shown by the dashed curves. The normalization of each background curve is adjusted by ZYAM to match the raw correlation function such that the background-subtracted correlation is zero at the minimum (see Sec. II D). As seen from the figures, the dashed curves are not symmetric about the solid curve. This is mainly due to the ZYAM normalization, as the normalization region is around $ \Delta\phi\approx\pm1 $, not at $ \pm\pi/2 $.

The coefficient $ v_2^{(a)} v_2^{(t,R)} $ in Eq. (1) determines the size of the modulation in the flow background. These coefficients are tabulated in Table 2. For in-plane trigger particles, $ v_2^{(t,R)} $ is positive as given by Eq. (2) or (4). The correlated elliptic flow uncertainties in $ v_2^{(a)} $ and $ v_2^{(t,R)} $ gives a large uncertainty in $ v_2^{(a)} v_2^{(t,R)} $. For out-of-plane trigger particles, however, $ v_2^{(t,R)} $ is negative. The correlated uncertainties in $ v_2^{(a)} $ and $ v_2^{(t,R)} $ tend to cancel each other, resulting in a small uncertainty in $ v_2^{(a)} v_2^{(t,R)} $. This is apparent in the systematic uncertainties listed in Table 2. It is shown in the systematic uncertainty background curves in Fig. 3, where the uncertainty for in-plane correlations is large, while for out-of-plane correlations it is small.

The event-plane resolutions enter into the flow background modulation together with the anisotropic flow parameters, via $ v_{n}\langle {\cos(k \Delta\psi)}\rangle $. Terms with $ k = n $ are not affected by uncertainties in the event-plane resolutions, because $ v_{n}\langle {\cos(n \Delta\psi)}\rangle $ are the measured anisotropic flow parameters. The event-plane resolutions $ \langle {\cos(k \Delta\psi)}\rangle $ of different k's are likely correlated, hence the uncertainty in $ v_{n}\langle {\cos(k \Delta\psi)}\rangle $ for $ k\neq n $ due to uncertainties in the resolutions may be greatly reduced. To be conservative, we assume the uncertainties in the event-plane resolutions to be uncorrelated in our estimation of their effects on our correlation results.

The systematic uncertainty of the event-plane resolution was determined by repeating the sub-event method, but splitting the particles by charge instead of by random determination, as done in the default case. It was also assessed by comparing the event-plane resolution from triggered events only (default) to inclusive events, and by applying a weighting of the number of trigger particles (default) and not applying this weighting. In addition, differences in event-plane resolutions were assessed with (default) and without event-plane flattening by weighting of the inverse of $ \phi $-dependent efficiencies. The event-plane resolution uncertainties thus estimated are typically less than 1% for $ \langle {\cos(2 \Delta\psi)}\rangle $, and less than 2%-3% for $ \langle {\cos(4 \Delta\psi)}\rangle $ and $ \langle {\cos(6 \Delta\psi)}\rangle $.

The effects of the estimated event-plane resolution uncertainties on the final background-subtracted correlation functions are significantly smaller than those caused by the uncertainties on anisotropic flow, and are therefore neglected.

For the data reported in this paper, the entire 20%-60% Au+Au centrality range is treated as a single centrality bin in which the event-plane resolutions and elliptic flow are obtained and the azimuthal correlation is analyzed. Alternatively, the analysis was repeated in each of the four 10%-size centrality bins using the corresponding event-plane resolutions and the elliptic flow measurements. Those correlation results were added together, weighted by the number of trigger particles in each centrality bin. The recombined results are consistent with using a single 20%-60% centrality bin, well within the systematic uncertainties due to those in flow subtraction and ZYAM normalization. This is because the measured elliptic flow $ v_2 $ is fairly constant over the entire 20%-60% centrality range, so that $ \langle { v_2^{(t)} v_2^{(a)}}\rangle\approx\langle { v_2^{(t)}}\rangle\langle { v_2^{(a)}}\rangle $. The event-plane resolutions vary with centrality mainly due to the multiplicity change. However, the event-plane resolutions enter into the flow background of Eq. (1) linearly, and because the high $ p_{T} $ trigger particle multiplicity scales almost linearly with the total multiplicity, the effect of the centrality-varying event-plane resolution is minimal in the flow correction calculated from the single 20%-60% centrality bin or summed from multiple narrower centrality bins.

Naively one would expect the background level B in Eq. (1) to be the same for all $ \phi_{s} $ slices because the underlying background should not depend on the signal (or orientation of the trigger particle). However, there could be biases in the event samples with trigger particles at different $ \phi_{s} $ such that they contain slightly different underlying background multiplicities due to the possible difference in jet-like correlated multiplicities at different $ \phi_{s} $ and the overall constraints caused by centrality cuts on the reference multiplicity. In our analysis we use different B values for different $ \phi_{s} $ slices, each independently obtained using ZYAM on the correlation function of the corresponding slice.

One source of systematic uncertainty on B is due to the limited range in $ \Delta\phi $ where the background-subtracted correlations appear to have a minimum ‘plateau’. This part of the systematic uncertainty is assessed by varying the size of the normalization range in $ \Delta\phi $ between $ \pi/12 $ and $ \pi/4 $ (the default range is $ \pi/6 $), similar to Ref. [13].

The ZYAM assumption likely gives an upper limit to the underlying background level. One could make an improved assessment of the background level with more stringent requirements, such as using three-particle correlation ZYAM [31]. However, the analysis of three-particle correlation within a limited $ \phi_{s} $ range of the trigger particle is difficult.

In this paper, we assess this part of the systematic uncertainty on B by comparing to the ZYAM backgrounds obtained separately from correlation functions at positive $ \phi_{t}- \psi_{{\rm{EP}}} $ and negative $ \phi_{t}- \psi_{{\rm{EP}}} $. Those ZYAM backgrounds are always lower than our default B from ZYAM of the combined correlation function of positive and negative $ \phi_{t}- \psi_{{\rm{EP}}} $. This is because the separately analyzed correlation functions are asymmetric about $ \Delta\phi = 0 $ and $ \Delta\phi = \pi $, and the ZYAM is determined by only one side of the correlation function [94, 95], whereas in our combined correlation functions reported here, the two sides of the separately analyzed asymmetric correlation functions are averaged. We treat the difference between the ZYAM background from this paper and that obtained from the asymmetric correlation functions as an additional, one-sided systematic uncertainty on B.

We may also study the background level by fitting the ZYAM-background-subtracted correlation functions with a combination of Gaussians and a free parameter for an offset from zero. Specifically, we fit the correlation data to three Gaussians (a near-side Gaussian at $ \Delta\phi = 0 $ and two away-side Gaussians symmetric about $ \Delta\phi = \pi $), and four Gaussians (adding a fourth Gaussian at $ \Delta\phi = \pi $ with the same width as the near-side Gaussian). Some of the fits yielded unphysical offsets because of the limited constraint of the correlation data on the fit model. For the other fits, the fitted offsets are comparable to the systematic uncertainty obtained from the comparisons to the asymmetric correlation functions discussed above. The Gaussian fits to the correlation functions without the offset will be discussed in Sec. IV E.

The different sources of systematic uncertainties on B are added in quadrature. The total systematic uncertainty is listed in Table 3 together with the statistical uncertainty. We take the quadratic sum of the statistical and systematic uncertainties as the total uncertainty for B on our correlation results.

As will be shown in Sec. IV A, the $ v_2 $ and $ v_4 $ background-subtracted correlation functions on the away side are single-peaked at $ \Delta\phi = \pi $ for triggered particles in-plane, but double-peaked for trigger particles out-of-plane beyond the flow systematic uncertainties. Since the subtracted background is flow-modulated, the natural question is whether the away-side double-peak structure is due to an unrealistic systematic uncertainty. To address this question, it is worth noting that the flow background modulation changes phase when the trigger particle moves from in-plane to out-of-plane, as shown in Fig. 3. A smaller elliptic flow would make the in-plane correlation more peaked at $ \Delta\phi = 0 $ and $ \pi $ and the out-of-plane correlation more dipped at $ \pi $ (hence more double peaked on the away side). On the other hand, a larger elliptic flow would make the out-of-plane away-side correlation less double-peaked. One would need a $ \sim $15% larger $ v_2^{(a)} v_2^{(t,R)} $ than in Table 2, significantly beyond the systematic uncertainty from the anisotropy measurements, to eliminate the away-side double-peak for the out-of-plane $ \phi_{s} $ slice. However, this large $ v_2^{(a)} v_2^{(t,R)} $ would result in double-peaked away-side correlations for some of the other $ \phi_{s} $ slices.