Harmonic decomposition of three-particle azimuthal correlations at RHIC

TL;DR

This paper reports measurements of three-particle azimuthal correlations in gold-gold collisions at RHIC across various energies, revealing insights into the initial collision geometry, medium expansion, and transport properties of the quark-gluon plasma.

Contribution

It provides detailed three-particle correlation data across multiple energies, offering new constraints on models of heavy-ion collision dynamics and medium properties.

Findings

Strong $ riangle m exteta$ dependence indicating breaking of boost invariance

Energy-dependent changes in two-particle correlations relative to the event-plane

Data constrains theoretical models of collision dynamics and medium properties

Abstract

We present measurements of three-particle correlations for various harmonics in Au+Au collisions at energies ranging from $\sqrt{s_{{\rm NN}}}=7.7$ to 200 GeV using the STAR detector. The quantity $\langle\cos(m\phi_1+n\phi_2-(m+n)\phi_3)\rangle$ is evaluated as a function of $\sqrt{s_{{\rm NN}}}$, collision centrality, transverse momentum, $p_T$, pseudo-rapidity difference, $\Delta\eta$, and harmonics ($m$ and $n$). These data provide detailed information on global event properties like the three-dimensional structure of the initial overlap region, the expansion dynamics of the matter produced in the collisions, and the transport properties of the medium. A strong dependence on $\Delta\eta$ is observed for most harmonic combinations consistent with breaking of longitudinal boost invariance. Data reveal changes with energy in the two-particle correlation functions relative to the second-harmonic event-plane and provide ways to constrain models of heavy-ion collisions over a wide range of collision energies.

Tables

Table 1. Table 1: Values for C m , n , m + n subscript 𝐶 𝑚 𝑛 𝑚 𝑛 C_{m,n,m+n} / v 2 subscript 𝑣 2 v_{2} for specific cases of ϕ 1 ′ superscript subscript italic-ϕ 1 ′ \phi_{1}^{\prime} and ϕ 2 ′ superscript subscript italic-ϕ 2 ′ \phi_{2}^{\prime} where ϕ ′ = ϕ − Ψ RP superscript italic-ϕ ′ italic-ϕ subscript Ψ RP \phi^{\prime}=\phi-\Psi_{\mathrm{RP}} (see Eq.  3 ). The first column ( ϕ 1 ′ = ϕ 2 ′ = 0 superscript subscript italic-ϕ 1 ′ superscript subscript italic-ϕ 2 ′ 0 \phi_{1}^{\prime}=\phi_{2}^{\prime}=0 ) corresponds to a particle pair with Δ ​ ϕ = 0 Δ italic-ϕ 0 \Delta\phi=0 emitted in the direction of the reaction plane (in-plane). The second column corresponds to back-to-back ( Δ ​ ϕ = π Δ italic-ϕ 𝜋 \Delta\phi=\pi ) particles emitted in-plane. The third and fourth columns correspond to pairs of particles emitted perpendicular to the reaction plane (out-of-plane) with either Δ ​ ϕ = 0 Δ italic-ϕ 0 \Delta\phi=0 or Δ ​ ϕ = π Δ italic-ϕ 𝜋 \Delta\phi=\pi respectively. The right-most column is a scenario consistent with the correlations observed in mid-central collisions at s NN = 200 subscript 𝑠 NN 200 \sqrt{s_{{\rm NN}}}=200  GeV.

	(${\varphi }_1^{\prime },{\varphi }_2^{\prime }$) [rad]
	(0, 0)	(0, $\pi$)	$\pm (\frac{\pi }{2},\frac{\pi }{2}$)	($\frac{\pi }{2},-\frac{\pi }{2}$)	$\pm (\frac{\pi }{3},\frac{2\pi }{3})$
$C_{1,1,2}$/$v_2$	+1	-1	-1	+1	-1
$C_{1,2,3}$/$v_2$	+1	-1	-1	+1	+$\frac{1}{2}$
$C_{2,2,4}$/$v_2$	+1	+1	-1	-1	+1
$C_{2,3,5}$/$v_2$	+1	-1	-1	+1	+$\frac{1}{2}$

Equations

$$\begin{gathered} C_{m,n,m+n}=⟨\cos (m{\varphi }_1+n{\varphi }_2-(m+n){\varphi }_3)⟩= \\ ⟨\left(\frac{{\sum }_{i,j,k}{w}_i{w}_j{w}_k\cos (m{\varphi }_i+n{\varphi }_j+(m+n){\varphi }_k)}{{\sum }_{i,j,k}{w}_i{w}_j{w}_k}\right)⟩ \end{gathered}$$

$$\begin{gathered} ⟨\cos (m{\varphi }_1(p_T)+n{\varphi }_2-(m+n){\varphi }_3)⟩\approx \\ \frac{v_m(p_T)}{{\epsilon }_m}\frac{v_n}{{\epsilon }_n}\frac{v_{m+n}}{{\epsilon }_{m+n}}\times \\ ⟨{\epsilon }_m{\epsilon }_n{\epsilon }_{m+n}\cos (m{\Psi }_m+n{\Psi }_n-(m+n){\Psi }_{m+n})⟩, \end{gathered}$$

$$⟨\cos (1{\varphi }_1+1{\varphi }_3-2{\varphi }_2)⟩/v_2$$

$$⟨\cos (1{\varphi }_1+2{\varphi }_3-3{\varphi }_2)⟩/v_2$$

$$⟨\cos (2{\varphi }_1+2{\varphi }_3-4{\varphi }_2)⟩/v_2$$

$$⟨\cos (2{\varphi }_3+3{\varphi }_1-5{\varphi }_2)⟩/v_2$$

$$⟨\cos (m{\varphi }_1$$

Full text

I Introduction

Heavy nuclei are collided at facilities like the Relativistic Heavy Ion Collider (RHIC) and the Large Hadron Collider (LHC) in order to study the emergent properties of matter with quarks and gluons as the dominant degrees-of-freedom: a quark-gluon plasma (QGP) Collins:1974ky ; Chin:1978gj ; Kapusta:1979fh ; Anishetty:1980zp . The QGP is a form of matter that existed in the early universe when its ambient temperature was more than 155 MeV or 200 thousand times hotter than the center of the sun Borsanyi:2010bp ; Bhattacharya:2014ara . As temperatures drop, quarks and gluons no longer possess the energy necessary to overcome the confining forces of QCD and they become confined into color neutral hadrons and the QGP transitions smoothly and continuously into a gas of hadrons Aoki:2006we . This transition occurred in the early universe at about one microsecond after the big bang. Heavy-ion collisions provide the only known method to recreate and study that phase transition in a laboratory setting.

To provide the clearest possible picture of this phase transition, a beam energy scan was carried out at RHIC with collision energies ranging from $\sqrt{s_{{\rm NN}}}$=200 GeV down to 7.7 GeV. Lowering the beam energy naturally reduces the initial temperature of the matter created in the collisions providing information on how the transport properties and equilibrium of the matter vary with temperature Aggarwal:2010cw . These heavy-ion collisions however create systems that are both very small and short-lived. The characteristic size of the collision region is the size of a nucleus or approximately $10^{-14}$ meters across. This system expands in the longitudinal direction and eventually in the transverse direction so that the energy density drops quickly. Any quark gluon plasma that exists will only survive for on the order $5\times 10^{-23}$ seconds. Given the smallness of the system and its very brief lifetime, it is challenging to determine the nature of the matter left behind after the initial collisions. Physicists rely on indirect observations based on particles streaming from the collision region which are observed long after any QGP has ceased to exist. Correlations between these produced particles have provided insight into the early phases of the expansion as well as the characteristics of the matter undergoing the expansion reviews . The dependence of the correlations on the azimuthal angle between particles $\Delta\phi=\phi_{1}-\phi_{2}$ has proven to be particularly informative. Data have revealed that even when particle pairs are separated by large angles in the longitudinal direction (large $\Delta\eta$), they remain strongly correlated in the azimuthal direction. This correlation manifests itself as a prominent ridge-like structure in two-particle, $\Delta\eta$, $\Delta\phi$, correlation functions ridgedata . The origin of this ridge has been traced to the initial geometry of the collision region where flux tubes are localized in the transverse direction but stretch over a long distance in the longitudinal direction radflow ; Mishra:2007tw ; Sorensen:2008dm ; Takahashi:2009na . How well these structures from the initial geometry are translated into correlations between particles emitted from the collision region reveals information about the medium’s viscosity: the larger the viscosity, the more washed out the correlations will become Sorensen:2011hm . To study these effects, it is convenient to examine the coefficients of a Fourier transform of the $\Delta\phi$ dependence of the two-particle correlation functions earlyv3 . These coefficients have been variously labeled as $V_{n}$, $a_{n}$, or $v_{n}^{2}\{2\}$ where $n$ is the harmonic. Although the latter is perhaps more cumbersome, we have maintained its usage owing to its connection to the original terminology used for two-particle cumulants which has been in use for more than a decade Adler:2002pu . While $v_{n}^{2}\{2\}=\langle\cos n(\Delta\phi)\rangle$ has been studied as a function of $\sqrt{s_{{\rm NN}}}$, centrality, harmonic $n$, $p_{T}$, and $\Delta\eta$ besv3 , in this paper we extend this analysis from two-particle correlations to three-particle mixed harmonic correlations of the form $\langle\cos(m\phi_{1}+n\phi_{2}-(m+n)\phi_{3})\rangle$ Bhalerao:2013ina where $m$ and $n$ are positive integers.

Extending the analysis of azimuthal correlations from two to three particles provides several benefits. First, the three particle correlations provide greater sensitivity to the three-dimensional structure of the initial state by for example revealing information about the two-particle $\Delta\eta-\Delta\phi$ correlations with respect to the reaction plane. Many models of heavy-ion collisions make the simplifying assumption that the initial geometry of the collision overlap does not vary with rapidity and that a boost invariant central rapidity plateau may be considered Bjorken:1982qr . It is likely however that this assumption is broken by the asymmetric nature of the initial state and that precision comparisons between models and data will require a better understanding of the initial state fluctuations in all three dimensions Denicol:2015nhu . Second, the new measurements can constrain models Teaney:2010vd ; Qiu:2012uy ; Niemi:2015qia ; Teaney:2013dta . When signals seen in two-particle correlations may be mocked up by multiple effects, three-particle correlations can break those ambiguities. This is important as models become more sophisticated by including for example bulk viscosity, shear viscosity, and their temperature dependence Ryu:2015vwa . Also, three-particle correlations can reveal information about how two-particle correlations change as a function of their angle with respect to the reaction plane. When one of the harmonics $m$, $n$, or $m+n$ is equal to two, that harmonic will be dominated by the preference of particles to flow in the direction of the reaction plane. This feature has been exploited to study charge separation relative to the reaction plane through measurements of the charge dependence of $\langle\cos(\phi_{1}+\phi_{2}-2\phi_{3})\rangle$ Abelev:2009ac ; Abelev:2009ad . The motivation for those measurements was to search for evidence of the chiral magnetic effect (CME) in heavy-ion collisions Kharzeev:1998kz ; Kharzeev:2004ey ; Voloshin:2004vk . By extending the measurements to other harmonics we can ascertain more information about the nature of the correlations interpreted as evidence for CME. Finally, three-particle correlations reveal information about how various harmonics are correlated with each other. For example, Teaney and Yan Teaney:2010vd originally proposed the measurement of $\langle\cos(\phi_{1}+2\phi_{2}-3\phi_{3})\rangle$ because initial state models predict a strong correlation between the first, second and third harmonics of the spatial density distribution. That correlation can be traced to collision geometries where a nucleon from one nucleus fluctuates toward the edge of that nucleus and impinges on the oncoming nucleus. This leads to something similar to a $p+A$ collision and a high density near the edge of the main collision region. That configuration increases the predicted $v_{3}$ by a factor of 2-3 in noncentral collisions so that $v_{3}$ deviates from the 1/$\sqrt{N_{\mathrm{part}}}$ one would expect from random fluctuations in the positions of the nucleons participating in the collision Sorensen:2011hm ; earlyv3 ; besv3 . That configuration should also be asymmetric in the forward and backward rapidity directions, again pointing to the importance of understanding the three dimensional structure of the initial state. If the evidence proposed by Teaney and Yan is not confirmed, then one may question the validity of any model that predicts the centrality dependence of $v_{n}$ based on those initial condition models. In this paper we present measurements of $\langle\cos(m\phi_{1}+n\phi_{2}-(m+n)\phi_{3})\rangle$ as a function of energy, centrality, $\Delta\eta$, $p_{T}$, and harmonics $m$ and $n$. Data confirm the predicted correlation between the first, second and third harmonics but the $\Delta\eta$ dependence points to the potential importance of including the three-dimensional structure of the initial state in the model calculations.

In the following, we first describe the experiment and the analysis (Sec. II). We then present the results in Sec. III including the $\Delta\eta$ dependence (Sec. III.1), the centrality dependence (Sec. III.2), the $p_{T}$ dependence (Sec. III.3), and the beam energy dependence (Sec. III.4). Conclusions are presented in Sec. IV. We include measurements of $v_{n}^{2}\{2\}$ for n=1,2,4, and 5 in the appendix.

II Experiment and Analysis

Our measurements make use of data collected from Au+Au collisions with the STAR detector at RHIC in the years 2004, 2010, 2011, 2012, and 2014. The charged particles used in this analysis are detected through their ionization energy loss in the STAR Time Projection Chamber STAR . The transverse momentum $p_{T}$, $\eta$, and charge are determined from the trajectory of the track in STAR’s solenoidal magnetic field. With the 0.5 Tesla field used during data taking, particles can be reliably tracked for $p_{T}>0.2$ GeV/$c$. The efficiency for finding particles drops quickly as $p_{T}$ decreases below this value Abelev:2008ab . Weights have been used to correct the three-particle correlation functions for the $p_{T}$-dependent efficiency and for imperfections in the detector acceptance. The quantity analyzed and reported is

[TABLE]

where $\langle\rangle$ represents an average over events and $\sum_{i,j,k}$ is a sum over unique particle triplets within an event. Each event is weighted by the number of unique triplets in that event. The weights $w_{i,j,k}$ are determined from the inverse of the $\phi$ distributions after they have been averaged over many events (which for a perfect detector should be flat) and by the $p_{T}$ dependent efficiency. The $w_{i,j,k}$ depend on the particles’ $p_{T}$, $\eta$, and charge and the collisions’ centrality and z-vertex location. The correction procedure is verified by checking that the $\phi$ distributions are flat after the correction so that $\langle\cos n(\phi)\rangle$ and $\langle\sin n(\phi)\rangle$ are near zero. With these corrections, the data represent the $C_{m,n,m+n}$ that would be seen by a detector with perfect acceptance for particles with $p_{T}>0.2$ GeV/$c$ and $|\eta|<1$. In practice, calculating all possible combinations of three particles individually would be computationally too costly to be practical, particularly for the larger data sets at 200 GeV. In that case we use algebra based on Q-vectors ($Q_{n}=\Sigma\exp(in\phi)$) to reduce the computational challenge Bilandzic:2010jr . Differential measurements like the $\Delta\eta$ dependence of the correlations, however, require explicit calculations for at least two of the particles. Studying the $\Delta\eta$ dependence of the correlations also allows us to correct for the effect of track-merging on the correlations. Track-merging leads to a large anti-correlation between particle pairs that are close to each other in the detector. The effect becomes large in central collisions where the detector occupancy is largest. After weight corrections have been applied to correct for single particle acceptance effects, the effect of track-merging is the largest remaining correction. Data have been divided into standard centrality classes (0-5%, 5-10%, 10-20%,… 70-80%) based on the number of charged hadrons within $|\eta|<0.5$ observed for a given event. In some figures, we will report the centrality in terms of the number of participating nucleons ($N_{\mathrm{part}}$) estimated from a Monte Carlo Glauber calculations Abelev:2008ab ; glauber .

The three-particle correlations presented in this paper are related to the low-resolution limit of the event-plane measurements that have been explored at the LHC Aad:2014fla . Practically this would be carried out by dividing $C_{m,n,m+n}$ by $\langle v_{m}v_{n}v_{m+n}\rangle$. Typically, however, $v_{n}$ is measured from a two-particle correlation function such as the two-particle cumulants $v_{n}=\sqrt{v_{n}^{2}\{2\}}$ or a similar measurement and the $v_{n}^{2}\{2\}$ are not positive-definite quantities. As such, $\sqrt{v_{n}^{2}\{2\}}$ can, and often does, become imaginary. This is particularly true for the first harmonic and also at lower collision energies. For this reason we report the pure three-particle correlations which, in any case, do not suffer from the ambiguities related to the low- and high-resolution limits associated with reaction plane analyses Bhalerao:2013ina ; Luzum:2012da and are therefore easier to interpret theoretically.

III Results

In the following, we present the $\Delta\eta$ dependence of the three-particle correlations for several harmonic combinations corrected for track-merging. After removing the effects of track merging and Hanbury Brown and Twiss (HBT) correlations Lisa:2005dd , we integrate over the $\Delta\eta$ dependence of the correlations and present the resulting integrated correlations as a function of centrality for the energies $\sqrt{s_{{\rm NN}}}$=200, 62.4, 39, 27, 19.6, 14.5, 11.5, and 7.7 GeV. We also investigate the $p_{T}$ dependence of the correlations by plotting them as a function of the $p_{T}$ of either the first or second particle used in the correlation. Finally, we study how the data depends on the beam energy.

III.1 $\Delta\eta$ Dependence

Figure 1 shows the $\Delta\eta$ dependence of $C_{1,1,2}$ scaled by $N_{\mathrm{part}}^{2}$ for charged hadrons with $p_{T}>0.2$ GeV/$c$ and $|\eta|<1$. The scaling accounts for the natural dilution of correlations expected if the more central collisions can be treated as a linear superposition of nucleon-nucleon collisions. Results for nine different centrality intervals from 200 GeV Au+Au collisions are shown. We do not include the uncertainty on $N_{\mathrm{part}}$ in the uncertainties in our figures. The left panels show the correlations as a function of the difference in $\eta$ between the first and second particle. Note that the subscripts in $C_{m,n,m+n}$ refer to the harmonic number while the subscripts for the $\eta$ refers to the particle number. The right panels show the same but as a function of the difference between particles 1 and 3. The $C_{1,1,2}$ correlation is similar to the correlation used in the search for the chiral magnetic effect except that we do not separate out the cases when particles 1 and 2 have like-sign charges vs unlike-sign charges as is done when looking for charge separation with respect to the reaction plane. These measurements can be approximately related to the reaction-plane based measurements by scaling the three-particle correlations by 1/$v_{2}$. We note that the difference in $C_{1,1,2}$ for different charge combinations is as large as the signal with $C_{1,1,2}$ being nearly zero for unlike-sign combitions of particle 1 and 2. This correlation may also be influenced by momentum conservation effects as well. It’s not clear however how those effects would be distributed with respect to $\Delta\eta$.

In the left panels of Fig. 1, we see a strong dependence for $C_{1,1,2}$ on $|\eta_{1}-\eta_{2}|$. In central collisions, the data starts out negative at the smallest values of $|\eta_{1}-\eta_{2}|$ but then begins to increase and becomes close to zero or even positive near $|\eta_{1}-\eta_{2}|=1.5$. At small $|\eta_{1}-\eta_{2}|$, a narrow peak is seen in the correlation that is related to HBT. As we progress from central to peripheral collisions, the trends change with $C_{1,1,2}$ in peripheral collisions exhibiting a positive value at small $|\eta_{1}-\eta_{2}|$, perhaps signaling the dominance of jets in the correlation function in these peripheral collisions.

The left panels share the same scales as the right panels making it clear that the dependence of $C_{1,1,2}$ on $|\eta_{1}-\eta_{3}|$ is much weaker than the dependence on $|\eta_{1}-\eta_{2}|$. This is expected since the $e^{-2i\phi_{3}}$ term in $C_{1,1,2}$=$\langle e^{i\phi_{1}}e^{i\phi_{2}}e^{-2i\phi_{3}}\rangle$ will be dominated by the global preference of particles to be emitted in the direction of the reaction plane. For all but the most central collisions, the almond shaped geometry of the collision overlap region is approximately invariant with rapidity. This is not likely the case for other harmonics.

Figure 2 shows $C_{1,2,3}$ scaled by $N_{\mathrm{part}}^{2}$ as a function of $|\eta_{1}-\eta_{2}|$ (left panels) and $|\eta_{1}-\eta_{3}|$ (right panels). In this case, $C_{1,2,3}$ exhibits a stronger dependence on $|\eta_{1}-\eta_{3}|$ than on $|\eta_{1}-\eta_{2}|$. The variation with $|\eta_{2}-\eta_{3}|$ is very similar to the variation with $|\eta_{1}-\eta_{2}|$ and is omitted from the figures to improve legibility. Again, the $e^{i2\phi_{2}}$ component of $C_{1,2,3}$ is dominated by the reaction plane which is largely invariant within the $\eta$ range covered by these measurements so that $C_{1,2,3}$ depends very little on the $\eta_{2}$, $|\eta_{1}-\eta_{2}|$, or $|\eta_{2}-\eta_{3}|$. However, $C_{1,2,3}$ depends very strongly on $|\eta_{1}-\eta_{3}|$. This dependence may arise from the longitudinal asymmetry inherent in the fluctuations that lead to predictions for large values of $C_{1,2,3}$ Teaney:2013dta . In models for the initial geometry, the correlations are induced between the first, second, and third harmonics of the eccentricity by cases where a nucleon fluctuates towards the edge of the nucleus Shou:2014eya . If that occurs in the reaction plane direction and towards the other nucleus in the collision, then that nucleon can collide with many nucleons from the other nucleus. This geometry will cause the first and third harmonics to become correlated with the second harmonic. Since the collision of one nucleon from one nucleus with many nucleons in the other nucleus is asymmetric along the rapidity axis, we argue that we can expect a strong dependence on $|\eta_{1}-\eta_{3}|$. Models that assume the initial energy density is symmetric with rapidity (boost invariant) will likely fail to describe this behavior. One may also speculate that the variation with $|\eta_{1}-\eta_{3}|$ could arise from sources like jets or resonances particularly if they interact with the medium so that they become correlated with the reaction plane. Making use of the full suite of measurements provided here will help delineate between these two scenarios.

In Fig. 3 we present the $|\eta_{1}-\eta_{2}|$ and $|\eta_{1}-\eta_{3}|$ dependence of $C_{2,2,4}$. This correlation is more strongly influenced by the reaction plane correlations and exhibits much larger values than either $C_{1,1,2}$ or $C_{1,2,3}$. The dependence on $|\eta_{1}-\eta_{2}|$ and $|\eta_{1}-\eta_{3}|$ are also weaker with $C_{2,2,4}$ in central and mid-central collisions showing little variation over the $|\eta_{1}-\eta_{2}|$ range, consistent with a mostly $\eta$-independent reaction plane within the measured range. A larger variation is observed with $|\eta_{1}-\eta_{3}|$ which in mid-central collisions amounts to an approximately 20% variation. We also note that in mid-central collisions, the change in value of $C_{2,2,4}$ over the range $0<|\eta_{1}-\eta_{3}|<2$ is similar in magnitude to the change of $C_{1,1,2}$ over $0<|\eta_{1}-\eta_{2}|<2$ and $C_{1,2,3}$ over $0<|\eta_{1}-\eta_{3}|<2$.

In Fig. 4 we present the $|\eta_{1}-\eta_{2}|$ and $|\eta_{2}-\eta_{3}|$ dependence of $C_{2,3,5}$. Again, $C_{2,3,5}$ only exhibits a weak dependence on $|\eta_{1}-\eta_{2}|$ but a stronger dependence on $|\eta_{2}-\eta_{3}|$. In central and mid-central collisions, a strong short-range correlation at $|\eta_{2}-\eta_{3}|<0.4$ is apparent consistent with HBT and Coulomb correlations that vary with respect to the reaction plane. In addition to that peak, $C_{2,3,5}$ decreases as $|\eta_{2}-\eta_{3}|$ increases. Although the relative variation of ${C_{2,3,5}}$ is similar to $C_{2,2,4}$, the absolute change is much smaller than for $C_{1,1,2}$, $C_{1,2,3}$, or $C_{2,2,4}$.

The combination of the various $C_{m,n,m+n}$ can help elucidate the nature of the three-particle correlations. If the $|\eta_{1}-\eta_{3}|$ dependence of $C_{1,2,3}$ arises from correlations between particles from jets correlated with the reaction plane, we would expect the particles at small $\Delta\eta$ to predominantly come from the near-side jet (at $\Delta\phi\approx 0$) and particles at larger $\Delta\eta$ to come from the away-side jet (at $\Delta\phi\approx\pi$ radians). In that case, at small $\Delta\eta$, $C_{m,n,m+n}$ for all harmonics will have a positive contribution from the jets. The same is not true however for large $\Delta\eta$ where we would expect the correlations to be dominated by the away-side jet separated by $\pi$ radians. For this case at large $\Delta\eta$, $C_{1,1,2}$ and $C_{1,2,3}$ would receive negative contributions from the away side jet while $C_{2,2,4}$ and $C_{2,3,5}$ would both receive positive contributions. The trends observed across the variety of $C_{m,n,m+n}$ measurements are inconsistent with this simple picture with $C_{2,2,4}$ decreasing by nearly the same amount as $C_{1,2,3}$ as $\Delta\eta$ is increased. A more complicated picture of the effect of jets would therefore be required to account for the observed data but it appears difficult to construct a non-flow scenario that can account for the long-range variation of $C_{m,n,m+n}$. Breaking of boost-invariance in the initial density distributions may provide an explanation for the observed variations but we do not know of any specific model that has been shown to describe our data.

III.2 Centrality Dependence

In Figs. 5 and 6 we show $C_{m,n,m+n}$ correlations scaled by $N_{\mathrm{part}}^{2}$ with $(m,n)$ = $(1,1)$, $(1,2)$, $(1,3)$, $(2,2)$, $(2,3)$, $(2,4)$, $(3,3)$, and $(3,4)$ for $\sqrt{s_{{}_{\rm NN}}}$=200, 62.4, 39, 27, 19.6, 14.5, 11.5, and 7.7 GeV Au+Au collisions as a function of $N_{\mathrm{part}}$. Data are for charged particles with $|\eta|<1$ and $p_{T}>0.2$ GeV/$c$. The correlation $C_{2,2,4}$, by far the largest of the measured correlations, has been scaled by a factor of 1/5. Otherwise, the scales on each of the three panels are kept the same for each energy to make it easier to compare the magnitudes of the different harmonic combinations.

At 200 GeV, $C_{1,1,2}$ is negative for all centralities except for the most peripheral where it is slightly positive but consistent with zero. $C_{1,2,3}$ is consistent with zero in peripheral collisions, positive in mid-central collisions but then becomes negative in central collisions. If the second and third harmonic event planes are uncorrelated, then $C_{1,2,3}$ should be zero. The $C_{1,2,3}$ correlation is non-zero deviating from that expectation. The magnitude is however much smaller than originally anticipated based on a linear hydrodynamic response to initial state geometry fluctuations Teaney:2010vd . Non-linear coupling between harmonics, where the fifth harmonic for example is dominated by a combination of the second and third harmonic, has been shown to be very important Qiu:2012uy ; Teaney:2012ke . In the case of $C_{1,2,3}$, the non-linear contribution has an opposite sign to the linear contribution and similar magnitude canceling out most of the expected strength of $C_{1,2,3}$. This suggests that $C_{1,2,3}$ is very sensitive to the nonlinear nature of the hydrodynamic model. $C_{1,3,4}$ is close to zero for all centralities indicating little or no correlation between the first, third, and fourth harmonics. The other $C_{m,n,m+n}$ correlations are positive for all centralities. When considering the comparison of this data to hydrodynamic models, it is important to also consider the strong $\Delta\eta$ dependence of the correlations as shown in the previous section.

The correlations involving a second harmonic are largest with $C_{2,2,4}$ being approximately 5 times larger in magnitude than the next largest correlator $C_{2,3,5}$. The correlations decrease quickly as harmonics are increased beyond n=2. The higher harmonic correlations $C_{3,3,6}$ and $C_{3,4,7}$ are both small but non-zero. The correlations $C_{1,1,2}$, $C_{1,2,3}$, $C_{2,2,4}$, $C_{2,3,5}$, and $C_{3,3,6}$ scaled by $N_{\mathrm{part}}^{2}$ all exhibit extrema in mid central collisions where the initial overlap geometry is predominantly elliptical. We note that the centrality at which N_{\mathrm{part}}^{2}$$C_{2,2,4} reaches a maximum is different than the centrality at which N_{\mathrm{part}}^{2}$$C_{2,3,5} reaches a maximum.

As the collision energy is reduced, although the magnitude of the correlations becomes smaller, the centrality dependence and ordering of the different harmonics seems to remain mostly the same. The $C_{1,2,3}$ correlation however is an exception. While at 200 GeV, $C_{1,2,3}$ is mostly positive, at 62.4 GeV it is consistent with zero or slightly negative and at lower energies it becomes more and more negative. We speculate that this behavior may be related to the increasing importance of momentum conservation as the number of particles produced in the collision decreases. No theoretical guidance exists however for the energy dependence of these correlations at energies below 200 GeV. This data should provide useful constraints for the models being developed to describe lower energy collisions associated with the energy scan program at RHIC.

Figure 6 shows the same correlations as Fig. 5 except for lower energy data sets: $\sqrt{s_{{}_{{\rm NN}}}}=$ 19.6, 14.5, 11.5, and 7.7 GeV. Trends similar to those seen in Fig. 5 are for the most part also exhibited in this figure. Although the statistical precision is poor for the lowest energy points, it appears that $C_{1,1,2}$ at 7.7 GeV is smaller in magnitude than at higher energies, becoming consistent with zero. This was also observed in the charge dependent measurements of $C_{1,1,2}$ Adamczyk:2014mzf . A second phase of the RHIC beam energy scan planned for 2019 and 2020 will significantly increase the number of events available for analysis at these lower energies while expanding the $\eta$ acceptance from $|\eta|<1$ to $|\eta|<1.5$ itpc so that this intriguing observation can be further investigated. The increased acceptance will increase the number of three-particle combinations by approximately a factor of three and will make it possible to measure the $\Delta\eta$ dependence of the $C_{m,n,m+n}$ correlations to $|\Delta\eta|\approx 3$.

III.3 $p_{T}$ Dependence

If the three-particle correlations presented here are dominated by correlations between event planes, then one might expect that the $p_{T}$ dependence of the three-particle correlations will simply track the $p_{T}$ dependence of the relevant $v_{n}$ Teaney:2010vd :

[TABLE]

where $\varepsilon_{m}$ is the $m^{{\mathrm{th}}}$ harmonic eccentricity and $\Psi_{m}$ is the $m^{{\mathrm{th}}}$ harmonic participant plane angle. For the purpose of simplicity in this publication, we have scaled the correlations by $N^{2}_{\mathrm{part}}/p_{T}$ to account for the general increase of $v_{n}(p_{T})$ with $p_{T}$ v2papers . That simple scaling is only valid at lower $p_{T}$ and for $n\neq 1$. It does, however, aid in visualizing trends in the data which would otherwise be visually dominated by the larger $p_{T}$ range. Our primary reason for introducing Eq. 2 is to provide a context for understanding the $p_{T}$ dependence of $C_{m,n,m+n}$. The relationship between $C_{m,n,m+n}$ and harmonic planes in Eq. 2 is not guaranteed to hold and is particularly likely to be broken for correlations involving the first harmonic where momentum conservation effects will likely play an important role or where a strong charge sign dependence has been observed Abelev:2009ac ; Abelev:2009ad .

In Fig. 7 we show N_{\mathrm{part}}^{2}$$C_{1,1,2}/$p_{T}$ as a function of the $p_{T}$ of particle one. The top panel shows the more central collisions while the bottom panel shows more peripheral collisions. In this and in the following figures related to the $p_{T}$ dependence, we sometimes exclude centrality bins and slightly shift the positions of the points along the $p_{T}$ axis to make the figures more readable. For more central collisions, $C_{1,1,2}$/$p_{T,1}$ is negative and slowly decreases in magnitude as $p_{T,1}$ increases. This indicates that $C_{1,1,2}$ is generally increasing with the $p_{T}$ of particle one but that for central collisions at high $p_{T}$, $C_{1,1,2}$ starts to saturate. For the more peripheral 30-40% and 40-50% collision however, $C_{1,1,2}$ appears to be linear in $p_{T}$ without an indication of saturation even up to $p_{T}\approx 10$ GeV/$c$. For the much more peripheral 60-70% and 70-80% centrality intervals, $C_{1,1,2}$ starts out at or above zero then becomes more and more negative as $p_{T}$ is increased. The trends in the most peripheral centrality intervals, particularly at high $p_{T}$, are consistent with being dominated by momentum conservation and jets. A pair of back-to-back particles aligned with the reaction plane will lead to a negative value for ${C_{1,1,2}}$. Although the data exhibit a smooth transition from the trends in more central collisions to the trends in more peripheral collisions, the trends are quite distinct and indicative of very different correlations in those different regions. In peripheral collisions, the correlations get stronger as $p_{T}$ is increased. In central collisions, the opposite is observed.

For the case of $C_{1,2,3}$ in Fig. 8, we show the $p_{T}$ dependence of both particle one (left panels) and particle two (right panels). The dependence of C_{1,2,3}$$/p_{T,2} on $p_{T,2}$ is quite weak indicating that where $C_{1,2,3}$ is non-zero, it increases roughly linearly with $p_{T,2}$. The dependence of $C_{1,2,3}$/$p_{T,1}$ on $p_{T,1}$, however, exhibits several notable trends. First we note that for the 20-30% centrality interval, C_{1,2,3}$$/p_{T,1} changes sign up to three times. In hydrodynamic models, the value of $C_{1,2,3}$ is very sensitive to the interplay between linear and non-linear effects and to viscous effects. The sign oscillations exhibited in the data may be a consequence of subtle changes in the relevant sizes of those effects. If this is the case, then this confirms that $C_{1,2,3}$ is a powerful measurement to help tune those models. At intermediate $p_{T,1}$ (2-5 GeV/$c$), $C_{1,2,3}$ is positive for central collisions but negative for peripheral collisions. At $p_{T}>7$ GeV/$c$, $C_{1,2,3}$ is strongly negative, perhaps again, indicative of the contribution of back-to-back jets to the correlations. Such strong negative correlation seems to be absent in central collisions where $C_{1,2,3}$ appears to remain positive, although with large error bars. This is consistent with a scenario where di-jets have been quenched in central collisions. As with $C_{1,1,2}$, the $p_{T}$ trends for $C_{1,2,3}$ are very different in the most peripheral and most central collisions.

The $C_{2,2,4}$ correlation is the largest of the $C_{m,n,m+n}$ correlations since it is strongly affected by the tendency of particles to preferentially line up with the reaction plane. In Fig. 9 we show N_{\mathrm{part}}^{2}$$C_{2,2,4}$$/p_{T,1} as a function of $p_{T,1}$. At low $p_{T,1}$, the centrality dependence of the correlations is as expected from Fig. 5 (top panels) where we saw that the integrated value of N_{\mathrm{part}}^{2}$$C_{2,2,4} is largest for mid-central collisions. This is a natural consequence of the fact that the initial second harmonic eccentricity decreases as collisions become more central while the efficiency of converting that eccentricity into momentum-space correlations increases (with multiplicity). The competition of these two trends leads to a maximum for second harmonic correlations in mid-central collisions. This well-known v2papers and generic trend does not persist to higher values of $p_{T,1}$. We see a clear change in trends at $p_{T,1}>5$ GeV/$c$ with the most peripheral collisions having the largest correlation strength while N_{\mathrm{part}}^{2}$$C_{2,2,4}$$/p_{T,1} drops significantly as a function of $p_{T,1}$ for the mid-central collisions. We note that past measurements of $p_{T}$ spectra and $v_{2}(p_{T})$ for identified particles have indicated that the effects of flow may persist up to 5 or 6 GeV/$c$ v2papers . This observation is consistent with model calculations that show in a parton cascade even up to $p_{T}\approx 5$ GeV/$c$ there are a significant number of partons whose final momentum has been increased by interactions with the medium Molnar:2005hb . The $p_{T,1}$ dependence of $C_{2,2,4}$/$p_{T,1}$ supports that picture as well.

In Fig. 10, we show the $p_{T}$ dependence of N_{\mathrm{part}}^{2}$$C_{2,3,5}/$p_{T}$ where $p_{T}$ is either the $p_{T}$ of particle one (left panels) or particle two (right panels). Again, the top panels show more central collisions and the bottom panels more peripheral. For $p_{T}<5$, C_{2,3,5}$$/p_{T} is mostly flat as a function of the $p_{T}$ of either particle one or particle two. Above that, the correlations seem to become smaller but with large statistical errors. One can discern a slight difference between the trends in the left and right panels: $C_{2,3,5}$/$p_{T,1}$ seems to decrease slightly as a function of $p_{T,1}$, while $C_{2,3,5}$/$p_{T,2}$ as a function of $p_{T,2}$ seems to increase slightly. This is likely related to the different $p_{T}$ dependences of $v_{2}$ and $v_{3}$ where $v_{2}$ has been found to saturate at lower $p_{T}$ while $v_{3}$ is still growing. In central collisions, it is even found that $v_{3}$ becomes larger than $v_{2}$ at intermediate $p_{T}$ earlyv3 .

We have tried to point out interesting features in the $p_{T}$ dependence of the correlations. In particular, we note that the $p_{T}$ trends are very different when comparing central collisions to peripheral collisions. We expect that when these data are compared to model calculations, they will provide even greater insights into the interplay between the effects of hard scattering, shear viscosity, bulk viscosity, the collision life-time and non-linear couplings between harmonics.

III.4 Energy Dependence

While Figs. 5 and 6 show the centrality dependence of 8 different $C_{m,n,m+n}$ correlations for 8 beam energies, in this section we will investigate the energy dependence in greater detail by first showing the centrality dependence of individual $C_{m,n,m+n}$ correlations for a variety of energies in single panels for easier comparison. We will then show correlations at specific centrality intervals as a function of $\sqrt{s_{{}_{\rm NN}}}$ scaled by $v_{2}$. Finally we will discuss implications of the energy dependence of the correlations.

Figure 11 shows the centrality dependence of N_{\mathrm{part}}^{2}$$C_{1,1,2} (left) and N_{\mathrm{part}}^{2}$$C_{1,2,3} (right) for 200, 62.4, 27, 14.5, and 7.7 or 11.5 GeV collisions. Some energies are omitted for clarity. For N_{\mathrm{part}}^{2}$$C_{1,1,2}, the general centrality trend appears to remain the same at all energies except 7.7 GeV, even though the magnitude slightly decreases. For mid-central collisions, $C_{1,1,2}$ is negative for all the energies shown. The 7.7 GeV data may deviate from the trend observed for the other energeis as will be discussed later. For N_{\mathrm{part}}^{2}$$C_{1,2,3}, the energy dependence is quite different. The only positive values for $C_{1,2,3}$ are for 200 GeV collisions. At 62.4 GeV, N_{\mathrm{part}}^{2}$$C_{1,2,3} has a slightly negative value that is within errors, independent of centrality. As the energy decreases, $C_{1,2,3}$ becomes more negative so that the centrality dependence of $C_{1,2,3}$ at 14.5 GeV is nearly the mirror reflection of the 200 GeV data. As will be discussed below, the change in sign of $C_{1,2,3}$ has interesting implications for how two-particle correlations relative to the reaction plane change as a function of beam energy.

Figure 12 shows the centrality dependence of N_{\mathrm{part}}^{2}$$C_{2,2,4} and N_{\mathrm{part}}^{2}$$C_{2,3,5} for a selection of collision energies. Both $C_{2,2,4}$ and $C_{2,3,5}$ remain positive for the centralities and energies shown with no apparent changes in the centrality trends. We note that although $C_{2,2,4}$ drops significantly from 200 down to 19.6 GeV, we observe little change with energy below 19.6 GeV. A similar lack of energy dependence between 7.7 and 19.6 GeV was also observed in recent measurements of $v_{3}^{2}\{2\}$ besv3 . This is notable since one would naively expect either of these correlation measurements to continuously increase as the density of the collision region increases.

To better view the energy trends, in Fig. 13 we show N_{\mathrm{part}}$$C_{m,n,m+n}$$/v_{2} as a function of $\sqrt{s_{{}_{\rm NN}}}$ for three centrality intervals: 10-20%, 20-30%, and 30-40%. The $v_{2}$ values are based on a two-particle cumulant analysis as discussed in Appendix A. The scaling will be further discussed in the next paragraph. For all centrality intervals shown, C_{1,1,2}$$/v_{2} is negative at the highest energy but the magnitude of the correlation decreases as the energy decreases and becomes consistent with zero, although with large errors, at 7.7 GeV. This behavior was also observed in the charge dependence of this correlator which has been studied to search for the charge separation predicted to be a consequence of the chiral magnetic effect Adamczyk:2014mzf . As noted above, both $C_{2,2,4}$ and $C_{2,3,5}$ are positive for all energies. The energy dependence of C_{1,2,3}$$/v_{2} is unique in that it is positive at 200 GeV but then drops below zero near 62.4 GeV and continues to become more negative at lower energies. In the following paragraph, we discuss the implications that this trend has for how two-particle correlations with respect to the reaction plane change with energy.

The correlations $C_{1,1,2}$, $C_{1,2,3}$, $C_{2,2,4}$, and $C_{2,3,5}$ presented in Fig. 13 have either $m=2$, $n=2$, or $m+n=2$. When $v_{2}$ is large, as it is for the 10-20%, 20-30% and 30-40% centrality intervals, then $\langle\cos(1\phi_{1}+1\phi_{2}-2\phi_{3})\rangle/v_{2}\approx\langle\cos(1\phi_{1}+1\phi_{2}-2\Psi_{\mathrm{RP}})\rangle$ and $\langle\cos(2\phi_{1}+m\phi_{2}-(m+2)\phi_{3})\rangle/v_{2}\approx\langle\cos(2\Psi_{\mathrm{RP}}+m\phi_{2}-(m+2)\phi_{3})\rangle$ where $\Psi_{\mathrm{RP}}$ is the reaction plane angle. Correlations including a second harmonic should then provide information about two-particle correlations with respect to the second harmonic reaction plane:

[TABLE]

where $\phi^{\prime}=\phi-\Psi_{\mathrm{RP}}$. Since we are integrating over all particles in these correlations, the subscript label for the particles is arbitrary so we have reassigned them so that particle 3 is always associated with the second harmonic. For illustration, Table 1 shows values for C_{m,n,m+n}$$/v_{2} for specific values of $\phi_{1}^{\prime}$ and $\phi_{2}^{\prime}$. At 200 GeV, all measured correlations are positive except $\langle\cos(\phi_{1}^{\prime}+\phi_{2}^{\prime})\rangle$. This points to an enhanced probability for a pair of particles in one of two possible configurations: either $\phi_{1}^{\prime}\approx\pi/3$ and $\phi_{2}^{\prime}\approx 2\pi/3$ or $\phi_{1}^{\prime}\approx-\pi/3$ and $\phi_{2}^{\prime}\approx-2\pi/3$ (these correspond to the right-most column of Table 1). This result is surprising since it implies a preference for both of the correlated particles to either be in the upper hemisphere, or both in the lower hemisphere. We note however, that hydrodynamic models with fluctuating initial conditions correctly predict this trend shortpaper which could arise from increased density fluctuations at either the top or the bottom of the almond shaped overlap region. A high density fluctuation in the lower half of the almond zone naturally leads to particles moving upward and away from that density fluctuation so that they both end up in the upper hemisphere. This response was described in Ref. Teaney:2010vd and was illustrated as “Position B” in Fig. IV of that reference. For energies below 200 GeV, $C_{1,2,3}$ changes sign so that $\langle\cos(\phi_{1}^{\prime}+\phi_{2}^{\prime})\rangle$ and $\langle\cos(1\phi_{1}^{\prime}-3\phi_{2}^{\prime})\rangle$ are both negative while $\langle\cos(2\phi_{1}^{\prime}-4\phi_{2}^{\prime})\rangle$ and $\langle\cos(3\phi_{1}^{\prime}-5\phi_{2}^{\prime})\rangle$ are both positive. This condition does not match any of the scenarios in the table but it could indicate an increased preference for particle pairs with $\phi_{1}^{\prime}\approx 0$ and $\phi_{2}^{\prime}\approx\pi$. A preference for back-to-back particle pairs aligned with the reaction plane would be consistent with an increased importance for momentum conservation at lower energies. Momentum conservation naturally leads to a tendency for particles to be emitted with back-to-back azimuth angles Adamczyk:2013hsi . As the beam energy is decreased, the multiplicity decreases and we should expect the effects of momentum conservation to become more prominent (in the case that only two particles are emitted, they must be back-to-back). The implications of this change in the configuration of two-particle correlations with respect to the reaction plane deserves further theoretical investigation.

The discussion in the above paragraph illustrates how measurements of $C_{m,n,m+n}$ reveal information about two-particle correlations with respect to the reaction plane and we pointed out two specific conclusions based on the $p_{T}$- and $\Delta\eta$-integrated measurements. The value of $C_{1,2,3}$ changes sign as a function of centrality, $\Delta\eta$ and $p_{T}$ suggesting that further specific configurations may arise when triggering on a particular $p_{T}$ or investigating particles separated by an $\eta$-gap. We have not examined the charge dependence of $C_{m,n,m+n}$ but future work placing a like-sign or unlike-sign requirement on $\phi_{1}^{\prime}$ and $\phi_{2}^{\prime}$ may be useful for interpreting charge separation measurements and determining whether they should be taken as evidence for the chiral magnetic effect.

IV Conclusions

We presented measurements of the energy, centrality, $p_{T}$, and $\Delta\eta$ dependence of three-particle azimuthal correlations $C_{m,n,m+n}$ for a variety of combinations of $m$ and $n$. We find a strong dependence of $C_{1,1,2}$ on $|\eta_{1}-\eta_{2}|$ and a strong dependence of $C_{1,2,3}$ on $|\eta_{1}-\eta_{3}|$. Meanwhile, $C_{2,2,4}$ and $C_{2,3,5}$ exhibit a smaller but still appreciable dependence on $|\eta_{1}-\eta_{3}|$. This may indicate either the presence of short-range non-flow correlations or a rapidity dependence to the initial energy density signaling a breaking of longitudinal invariance. Simple pictures of non-flow however, appear to be inconsistent with the overall trends observed in the data. The integrated correlations with $m=1$ are generally negative or consistent with zero except for $C_{1,2,3}$ which, at 200 GeV, is positive for mid-central collisions while it is negative for all centralities at all of the lower energies. Nonzero values for $C_{1,2,3}$ imply correlations between the second and third harmonic event plane that are predicted from models of the initial overlap geometry. The $p_{T}$ dependence of the correlations exhibits trends suggesting significant differences between the correlations in peripheral collisions and more central collisions as well as differences for $p_{T}>5$ GeV/$c$ and $p_{T}<5$ GeV/$c$. The quantity $C_{1,2,3}$ as a function of $p_{T,1}$ changes sign as many as three times. While $C_{1,1,2}$ is negative for higher energies, it becomes positive or consistent with zero at 7.7 GeV. By examining the energy dependence of $C_{1,1,2}$, $C_{1,2,3}$, $C_{2,2,4}$, and $C_{2,3,5}$ divided by $v_{2}$ we are able to infer that in mid-central collisions at 200 GeV, there is a preference for particle pairs to be emitted with angles relative to the reaction plane of either $\phi_{1}\approx\pi/3$ and $\phi_{2}\approx 2\pi/3$ or $\phi_{1}\approx-\pi/3$ and $\phi_{2}\approx-2\pi/3$. At 62.4 GeV and below, this appears to change due to a possible preference for back-to-back pairs ($\phi_{1}\approx 0$ and $\phi_{2}\approx\pi$) aligned with the reaction plane. These data will be useful for constraining hydrodynamic models shortpaper . In order to facilitate such future data-model comparisons we also include the measurements of $v_{n}^{2}\{2\},n=1,2,4,5$, over a wide range of energy, in the appendix of this paper. Measurements of the charge dependence of the correlations presented here, by revealing information about the preferred directions of correlated particles with respect to the reaction plane, should provide valuable insights into whether or not the charge separation observed in heavy-ion collisions is related to the chiral magnetic effect.

V Summary

The very first measurement of charge inclusive three-particle azimuthal correlations from the RHIC beam energy scan program, presented in this paper, can provide several new insights into the initial state and transport in heavy ion collisions. These observables go beyond conventional flow harmonics and provide the most efficient way of studying the correlation between harmonic amplitudes and their phases over a wide range of multiplicities. These observables are well defined and of general interests even when the azimuthal correlations are not dominated by hydrodynamic flow. The major finding of this analysis is the strong relative pseudorapidity ($\Delta\eta$) dependence between the particles associated with different harmonics, observed up to about two units ($\Delta\eta\sim$ 2) of separation. Non-flow based expectations such as fragmentation ($\Delta\eta\sim$ 1) or momentum conservation (flat in $\Delta\eta$) can not provide a simple explanation to such observations. If the observed correlations are dominated by flow, the current results strongly hint at a breaking of longitudinal invariance of the initial state geometry at RHIC. The comprehensive study of momentum and centrality dependence of three-particle correlations over a wide range of energy (7.7-200 GeV), presented here, will help reduce the large uncertainties in the transport parameters involved in hydrodynamic modeling of heavy ion collisions over a wide range of temperature and net-baryon densities. In addition, the charge inclusive three-particle correlations will provide baselines for the measurements of the chiral magnetic effect.

Acknowledgments

We thank the RHIC Operations Group and RCF at BNL, the NERSC Center at LBNL, and the Open Science Grid consortium for providing resources and support. This work was supported in part by the Office of Nuclear Physics within the U.S. DOE Office of Science, the U.S. National Science Foundation, the Ministry of Education and Science of the Russian Federation, National Natural Science Foundation of China, Chinese Academy of Science, the Ministry of Science and Technology of China and the Chinese Ministry of Education, the National Research Foundation of Korea, GA and MSMT of the Czech Republic, Department of Atomic Energy and Department of Science and Technology of the Government of India; the National Science Centre of Poland, National Research Foundation, the Ministry of Science, Education and Sports of the Republic of Croatia, RosAtom of Russia and German Bundesministerium fur Bildung, Wissenschaft, Forschung and Technologie (BMBF) and the Helmholtz Association.

Appendix A Two-particle Cumulants $v_{n}^{2}\{2\}$

In this appendix we present the measurements of $v_{n}^{2}\{2\}$ for n=1, 2, 4 and 5. The second harmonic $v_{2}^{2}\{2\}$ was used to scale $C_{m,n,m+n}$ in Fig. 13. Under the assumption that

[TABLE]

where $\Psi_{m}$ is the participant plane angle for harmonic $m$, one can convert the $C_{m,n,m+n}$ correlations into reaction plane correlations in the low-resolution limit by dividing by $\sqrt{v_{m}^{2}\{2\}v_{n}^{2}\{2\}v_{m+n}^{2}\{2\}}$. The relationship of the $C_{m,n,m+n}$ to $v_{m}$ and $\Psi_{m}$ assumes that non-flow correlations are minimal. Similar assumptions must also be made when using event-plane angles in the analysis. The analysis of $v_{n}^{2}\{2\}$ was performed in a similar manner to that of $v_{3}^{2}\{2\}$ presented in Ref. besv3 . The $\Delta\eta$ dependence of $\langle\cos 2(\phi_{1}-\phi_{2})\rangle$ is analyzed for $p_{T}>0.2$ GeV/$c$ and $|\eta|<1$. Short-range correlations are parameterized with a narrow Gaussian peak centered at $\Delta\eta=0$ and the remaining longer-range correlations are integrated (weighting by the number of pairs at each $\Delta\eta$) to obtain the $\Delta\eta$-integrated $v_{n}^{2}\{2\}$ results. The quantity labeled $v_{2}$ in Fig. 13 is $\sqrt{v_{2}^{2}\{2\}}$.

Figure 14 shows the results for $v_{1}^{2}\{2\}$ (left) and $v_{2}^{2}\{2\}$ (right) as a function of centrality for 200, 62.4, 39, 27, 19.6, 14.5, 11.5, and 7.7 GeV Au+Au collisions. The data are scaled by $N_{\mathrm{part}}$ and plotted verses $N_{\mathrm{part}}$ for convenience. At 200 GeV, $v_{1}^{2}\{2\}$ is positive for central collisions but becomes negative for N_{\mathrm{part}}$$<150. The negative values are expected from momentum conservation and present a conceptual challenge for dividing $C_{m,n,m+n}$ by $\sqrt{v_{1}^{2}\{2\}}$. The values of $v_{1}^{2}\{2\}$ become more negative at lower energies. This is consistent again with momentum conservation effects which are expected to become stronger as multiplicity decreases. In the limit of a collision that produces only two particles, momentum conservation would require that $v_{1}^{2}\{2\}=-1$. The $v_{1}^{2}\{2\}$ results follow a monotonic energy trend except for peripheral collisions at 19.6 GeV which appear to be elevated with respect to the trends.

The right panel of Fig. 14 shows the results for N_{\mathrm{part}}$$v_{2}^{2}\{2\} which remain positive for all energies and collision centralities. While it is unusual to scale $v_{2}^{2}\{2\}$ by $N_{\mathrm{part}}$, we keep this format for consistency. The scaled results exhibit a strong peak for mid-central collisions due to the elliptic geometry of those collisions.

Figure 15 shows the data for N_{\mathrm{part}}$$v_{4}^{2}\{2\} (left) and N_{\mathrm{part}}$$v_{5}^{2}\{2\} (right) for a more limited energy range. Results for N_{\mathrm{part}}$$v_{3}^{2}\{2\} are available in Ref. besv3 . At the lower energies the relative uncertainties on these data become too large to be of use. This presents another challenge to recasting $C_{m,n,m+n}$ in terms of reaction plane correlations because scaling by $\sqrt{v_{4}^{2}\{2\}}$ or $\sqrt{v_{5}^{2}\{2\}}$ leads to a large uncertainty on the resulting ratios.