Harmonic decomposition of three-particle azimuthal correlations at RHIC

0.55 MB·

by  STAR Collaboration

Harmonic decomposition of three-particle azimuthal correlations at RHIC
L. Adamczyk,1 J. K. Adkins,19 G. Agakishiev,17 M. M. Aggarwal,31 Z. Ahammed,50 N. N. Ajitanand,40
I. Alekseev,15,26 D. M. Anderson,42 R. Aoyama,46 A. Aparin,17 D. Arkhipkin,3 E. C. Aschenauer,3 M. U. Ashraf,45
A. Attri,31 G. S. Averichev,17 X. Bai,7 V. Bairathi,27 A. Behera,40 R. Bellwied,44 A. Bhasin,16 A. K. Bhati,31
P. Bhattarai,43 J. Bielcik,10 J. Bielcikova,11 L. C. Bland,3 I. G. Bordyuzhin,15 J. Bouchet,18 J. D. Brandenburg,36
A. V. Brandin,26 D. Brown,23 I. Bunzarov,17 J. Butterworth,36 H. Caines,54 M. Caldero´n de la Barca Sa´nchez,5
J. M. Campbell,29 D. Cebra,5 I. Chakaberia,3 P. Chaloupka,10 Z. Chang,42 N. Chankova-Bunzarova,17
A. Chatterjee,50 S. Chattopadhyay,50 X. Chen,37 J. H. Chen,39 X. Chen,21 J. Cheng,45 M. Cherney,9 W. Christie,3
G. Contin,22 H. J. Crawford,4 S. Das,7 L. C. De Silva,9 R. R. Debbe,3 T. G. Dedovich,17 J. Deng,38
A. A. Derevschikov,33 L. Didenko,3 C. Dilks,32 X. Dong,22 J. L. Drachenberg,20 J. E. Draper,5 L. E. Dunkelberger,6
7 J. C. Dunlop,3 L. G. Efimov,17 N. Elsey,52 J. Engelage,4 G. Eppley,36 R. Esha,6 S. Esumi,46 O. Evdokimov,8
1 J. Ewigleben,23 O. Eyser,3 R. Fatemi,19 S. Fazio,3 P. Federic,11 P. Federicova,10 J. Fedorisin,17 Z. Feng,7 P. Filip,17
0 E. Finch,47 Y. Fisyak,3 C. E. Flores,5 L. Fulek,1 C. A. Gagliardi,42 D. Garand,34 F. Geurts,36 A. Gibson,49
2
M. Girard,51 D. Grosnick,49 D. S. Gunarathne,41 Y. Guo,18 A. Gupta,16 S. Gupta,16 W. Guryn,3 A. I. Hamad,18

n A. Hamed,42 A. Harlenderova,10 J. W. Harris,54 L. He,34 S. Heppelmann,32 S. Heppelmann,5 A. Hirsch,34
a
G. W. Hoffmann,43 S. Horvat,54 T. Huang,28 B. Huang,8 X. Huang,45 H. Z. Huang,6 T. J. Humanic,29
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P. Huo,40 G. Igo,6 W. W. Jacobs,14 A. Jentsch,43 J. Jia,3,40 K. Jiang,37 S. Jowzaee,52 E. G. Judd,4
3
2 S. Kabana,18 D. Kalinkin,14 K. Kang,45 K. Kauder,52 H. W. Ke,3 D. Keane,18 A. Kechechyan,17 Z. Khan,8
D. P. Kikol a,51 I. Kisel,12 A. Kisiel,51 L. Kochenda,26 M. Kocmanek,11 T. Kollegger,12 L. K. Kosarzewski,51

] A. F. Kraishan,41 P. Kravtsov,26 K. Krueger,2 N. Kulathunga,44 L. Kumar,31 J. Kvapil,10 J. H. Kwasizur,14
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e R. Lacey,40 J. M. Landgraf,3 K. D. Landry,6 J. Lauret,3 A. Lebedev,3 R. Lednicky,17 J. H. Lee,3 X. Li,37 C. Li,37
- W. Li,39 Y. Li,45 J. Lidrych,10 T. Lin,14 M. A. Lisa,29 H. Liu,14 P. Liu,40 Y. Liu,42 F. Liu,7 T. Ljubicic,3
l
c W. J. Llope,52 M. Lomnitz,22 R. S. Longacre,3 S. Luo,8 X. Luo,7 G. L. Ma,39 L. Ma,39 Y. G. Ma,39 R. Ma,3
u
N. Magdy,40 R. Majka,54 D. Mallick,27 S. Margetis,18 C. Markert,43 H. S. Matis,22 K. Meehan,5 J. C. Mei,38
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[ Z. W. Miller,8 N. G. Minaev,33 S. Mioduszewski,42 D. Mishra,27 S. Mizuno,22 B. Mohanty,27 M. M. Mondal,13
D. A. Morozov,33 M. K. Mustafa,22 Md. Nasim,6 T. K. Nayak,50 J. M. Nelson,4 M. Nie,39 G. Nigmatkulov,26
1
T. Niida,52 L. V. Nogach,33 T. Nonaka,46 S. B. Nurushev,33 G. Odyniec,22 A. Ogawa,3 K. Oh,35 V. A. Okorokov,26
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6 D. Olvitt Jr.,41 B. S. Page,3 R. Pak,3 Y. Pandit,8 Y. Panebratsev,17 B. Pawlik,30 H. Pei,7 C. Perkins,4 P. Pile,3
9 J. Pluta,51 K. Poniatowska,51 J. Porter,22 M. Posik,41 A. M. Poskanzer,22 N. K. Pruthi,31 M. Przybycien,1
4 J.Putschke,52 H. Qiu,34 A. Quintero,41 S.Ramachandran,19 R. L.Ray,43 R.Reed,23 M. J.Rehbein,9 H. G. Ritter,22
6
J. B. Roberts,36 O. V. Rogachevskiy,17 J. L. Romero,5 J. D. Roth,9 L. Ruan,3 J. Rusnak,11 O. Rusnakova,10
0
. N. R. Sahoo,42 P. K. Sahu,13 S. Salur,22 J. Sandweiss,54 M. Saur,11 J. Schambach,43 A. M. Schmah,22
1
W. B. Schmidke,3 N. Schmitz,24 B. R. Schweid,40 J. Seger,9 M. Sergeeva,6 P. Seyboth,24 N. Shah,39 E. Shahaliev,17
0
7 P. V. Shanmuganathan,23 M. Shao,37 A. Sharma,16 M. K. Sharma,16 W. Q. Shen,39 Z. Shi,22 S. S. Shi,7
1 Q. Y. Shou,39 E. P. Sichtermann,22 R. Sikora,1 M. Simko,11 S. Singha,18 M. J. Skoby,14 N. Smirnov,54 D. Smirnov,3
v: W. Solyst,14 L. Song,44 P. Sorensen,3 H. M. Spinka,2 B. Srivastava,34 T. D. S. Stanislaus,49 M. Strikhanov,26
i B. Stringfellow,34 T. Sugiura,46 M. Sumbera,11 B. Summa,32 Y. Sun,37 X. M. Sun,7 X. Sun,7 B. Surrow,41
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D. N. Svirida,15 A. H. Tang,3 Z. Tang,37 A. Taranenko,26 T. Tarnowsky,25 A. Tawfik,53 J. Tha¨der,22
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a J. H. Thomas,22 A. R. Timmins,44 D. Tlusty,36 T. Todoroki,3 M. Tokarev,17 S. Trentalange,6 R. E. Tribble,42
P. Tribedy,3 S. K. Tripathy,13 B. A. Trzeciak,10 O. D. Tsai,6 T. Ullrich,3 D. G. Underwood,2 I. Upsal,29
G. Van Buren,3 G. van Nieuwenhuizen,3 A. N. Vasiliev,33 F. Videbæk,3 S. Vokal,17 S. A. Voloshin,52 A. Vossen,14
G. Wang,6 Y. Wang,7 F. Wang,34 Y. Wang,45 J. C. Webb,3 G. Webb,3 L. Wen,6 G. D. Westfall,25 H. Wieman,22
S. W. Wissink,14 R. Witt,48 Y. Wu,18 Z. G. Xiao,45 W. Xie,34 G. Xie,37 J. Xu,7 N. Xu,22 Q. H. Xu,38 Y. F. Xu,39
Z. Xu,3 Y. Yang,28 Q. Yang,37 C. Yang,38 S. Yang,3 Z. Ye,8 Z. Ye,8 L. Yi,54 K. Yip,3 I. -K. Yoo,35 N. Yu,7
H. Zbroszczyk,51 W. Zha,37 Z. Zhang,39 X. P. Zhang,45 J. B. Zhang,7 S. Zhang,37 J. Zhang,21 Y. Zhang,37
J. Zhang,22 S. Zhang,39 J. Zhao,34 C. Zhong,39 L. Zhou,37 C. Zhou,39 X. Zhu,45 Z. Zhu,38 and M. Zyzak12
(STAR Collaboration)
1AGH University of Science and Technology, FPACS, Cracow 30-059, Poland
2Argonne National Laboratory, Argonne, Illinois 60439
3Brookhaven National Laboratory, Upton, New York 11973
4University of California, Berkeley, California 94720
5University of California, Davis, California 95616
6University of California, Los Angeles, California 90095
2
7Central China Normal University, Wuhan, Hubei 430079
8University of Illinois at Chicago, Chicago, Illinois 60607
9Creighton University, Omaha, Nebraska 68178
10Czech Technical University in Prague, FNSPE, Prague, 115 19, Czech Republic
11Nuclear Physics Institute AS CR, 250 68 Prague, Czech Republic
12Frankfurt Institute for Advanced Studies FIAS, Frankfurt 60438, Germany
13Institute of Physics, Bhubaneswar 751005, India
14Indiana University, Bloomington, Indiana 47408
15Alikhanov Institute for Theoretical and Experimental Physics, Moscow 117218, Russia
16University of Jammu, Jammu 180001, India
17Joint Institute for Nuclear Research, Dubna, 141 980, Russia
18Kent State University, Kent, Ohio 44242
19University of Kentucky, Lexington, Kentucky, 40506-0055
20Lamar University, Physics Department, Beaumont, Texas 77710
21Institute of Modern Physics, Chinese Academy of Sciences, Lanzhou, Gansu 730000
22Lawrence Berkeley National Laboratory, Berkeley, California 94720
23Lehigh University, Bethlehem, PA, 18015
24Max-Planck-Institut fur Physik, Munich 80805, Germany
25Michigan State University, East Lansing, Michigan 48824
26National Research Nuclear University MEPhI, Moscow 115409, Russia
27National Institute of Science Education and Research, Bhubaneswar 751005, India
28National Cheng Kung University, Tainan 70101
29Ohio State University, Columbus, Ohio 43210
30Institute of Nuclear Physics PAN, Cracow 31-342, Poland
31Panjab University, Chandigarh 160014, India
32Pennsylvania State University, University Park, Pennsylvania 16802
33Institute of High Energy Physics, Protvino 142281, Russia
34Purdue University, West Lafayette, Indiana 47907
35Pusan National University, Pusan 46241, Korea
36Rice University, Houston, Texas 77251
37University of Science and Technology of China, Hefei, Anhui 230026
38Shandong University, Jinan, Shandong 250100
39Shanghai Institute of Applied Physics, Chinese Academy of Sciences, Shanghai 201800
40State University Of New York, Stony Brook, NY 11794
41Temple University, Philadelphia, Pennsylvania 19122
42Texas A&M University, College Station, Texas 77843
43University of Texas, Austin, Texas 78712
44University of Houston, Houston, Texas 77204
45Tsinghua University, Beijing 100084
46University of Tsukuba, Tsukuba, Ibaraki, Japan,
47Southern Connecticut State University, New Haven, CT, 06515
48United States Naval Academy, Annapolis, Maryland, 21402
49Valparaiso University, Valparaiso, Indiana 46383
50Variable Energy Cyclotron Centre, Kolkata 700064, India
51Warsaw University of Technology, Warsaw 00-661, Poland
52Wayne State University, Detroit, Michigan 48201
53World Laboratory for Cosmology and Particle Physics (WLCAPP), Cairo 11571, Egypt
54Yale University, New Haven, Connecticut 06520
Wepresentmeasurementsofthree-particlecorrelationsforvariousharmonicsinAu+Aucollisions
atenergiesrangingfrom√sNN =7.7to200GeVusingtheSTARdetector. Thequantity cos(mφ1+
h
nφ2 (m+n)φ3) isevaluatedasafunctionof√sNN,collisioncentrality,transversemomentum,pT,
− i
pseudo-rapiditydifference,∆η,andharmonics(mandn). Thesedataprovidedetailed 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 ∆η 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-ioncollisions over a wide range of collision energies.
I. INTRODUCTION Collider (LHC) in order to study the emergent proper-
ties of matter with quarks and gluons as the dominant
HeavynucleiarecollidedatfacilitiesliketheRelativis- degrees-of-freedom: a quark-gluon plasma (QGP) [1–4].
tic Heavy Ion Collider (RHIC) and the Large Hadron
3
TheQGPisaformofmatterthatexistedintheearlyuni- toitsconnectiontotheoriginalterminologyusedfortwo-
verse when its ambient temperature was more than 155 particlecumulantswhichhasbeeninuseformorethana
MeVor200thousandtimeshotterthanthecenterofthe decade[17]. Whilev2 2 = cosn(∆φ) hasbeenstudied
n{ } h i
sun [5, 6]. As temperatures drop, quarks and gluons no as a function of √sNN, centrality, harmonic n, pT, and
longerpossesstheenergynecessarytoovercomethecon- ∆η [18], in this paper we extend this analysis from two-
finingforcesofQCDandtheybecomeconfinedintocolor particle correlations to three-particle mixed harmonic
neutral hadrons and the QGP transitions smoothly and correlationsoftheform cos(mφ +nφ (m+n)φ ) [19]
1 2 3
h − i
continuously into a gas of hadrons [7]. This transition where m and n are positive integers.
occurred in the early universe at about one microsecond
Extending the analysis of azimuthal correlations from
after the big bang. Heavy-ioncollisions provide the only
two to three particles provides several benefits. First,
known method to recreate and study that phase transi-
the three particle correlationsprovide greater sensitivity
tion in a laboratory setting.
to the three-dimensional structure of the initial state by
To provide the clearest possible picture of this phase forexamplerevealinginformationaboutthe two-particle
transition, a beam energy scan was carried out at RHIC ∆η ∆φ correlationswith respect to the reactionplane.
−
with collision energies ranging from √sNN=200 GeV Many models of heavy-ion collisions make the simplify-
down to 7.7 GeV. Lowering the beam energy naturally ing assumption that the initial geometry of the collision
reduces the initial temperature of the matter created in overlap doesn’t vary with rapidity and that a boost in-
thecollisionsprovidinginformationonhowthetransport variant central rapidity plateau may be considered [20].
properties and equilibrium of the matter vary with tem- It is likely however that this assumption is broken by
perature [8]. These heavy-ion collisions however create the asymmetric nature of the initial state and that pre-
systems that are both very small and short-lived. The cisioncomparisonsbetweenmodelsanddatawillrequire
characteristic size of the collision region is the size of a fuller understanding of the initial state fluctuations in
a nucleus or approximately 10−14 meters across. This allthreedimensions[21]. Second,thenewmeasurements
systemexpandsinthelongitudinaldirectionandeventu- can constrainmodels [22–25]. When signals seen in two-
allyinthetransversedirectionsothattheenergydensity particle correlations may be mocked up by multiple ef-
drops quickly. Any quark gluon plasma that exists will fects, three-particle correlations can break those ambi-
only survive for on the order 5 10−23 seconds. Given guities. This is important as models become more so-
×
the smallness of the system and its very brief lifetime, phisticatedbyincludingforexamplebulkviscosity,shear
it is challenging to determine the nature of the matter viscosity, and their temperature dependence [26]. Also,
left behind after the initial collisions. Physicists rely on three-particle correlations can reveal information about
indirect observations based on particles streaming from howtwo-particlecorrelationschangeasafunctionoftheir
the collision region which are observed long after any anglewithrespecttothereactionplane. Whenoneofthe
QGP has ceased to exist. Correlations between these harmonicsm, n, orm+n is equalto two,thatharmonic
produced particles have provided insight into the early will be dominated by the preference of particles to flow
phases of the expansion as well as the characteristics of in the direction of the reaction plane. This feature has
the matter undergoing the expansion [9]. The depen- been exploitedto study chargeseparationrelativeto the
denceofthe correlationsontheazimuthalanglebetween reaction plane through measurements of the charge de-
particles ∆φ=φ φ has provento be particularly in- pendenceof cos(φ +φ 2φ ) [27,28]. Themotivation
1 2 1 2 3
− h − i
formative. Data have revealed that even when particle forthose measurementswasto searchforevidence ofthe
pairsareseparatedbylargeanglesinthelongitudinaldi- chiralmagnetic effect(CME) inheavy-ioncollisions[29–
rection(large∆η),theyremainstronglycorrelatedinthe 31]. By extending the measurements to other harmonics
azimuthal direction. This correlation manifests itself as we can ascertain more information about the nature of
aprominentridge-likestructureintwo-particle,∆η,∆φ, thecorrelationsinterpretedasevidenceforCME.Finally,
correlation functions [10]. The origin of this ridge has three-particle correlations reveal information about how
beentracedtotheinitialgeometryofthe collisionregion variousharmonicsarecorrelatedwitheachother. Forex-
where flux tubes are localizedin the transversedirection ample,TeaneyandYan[22]originallyproposedthemea-
butstretchoveralongdistanceinthelongitudinaldirec- surement of cos(φ +2φ 3φ ) because initial state
1 2 3
h − i
tion [11–14]. How well these structures from the initial modelspredictastrongcorrelationbetweenthefirst,sec-
geometry are translated into correlations between parti- ond and third harmonics of the spatial density distribu-
clesemitted fromthe collisionregionrevealsinformation tion. That correlation can be traced to collision geome-
about the medium’s viscosity: the larger the viscosity, trieswhereanucleonfromonenucleusfluctuates toward
the more washed out the correlations will become [15]. the edge of that nucleus and impinges on the oncom-
To study these effects, it is convenient to examine the ing nucleus. This leads to something similar to a p+A
coefficients of a Fourier transformof the ∆φ dependence collisionandahighdensityneartheedgeofthemaincol-
of the two-particle correlation functions [16]. These co- lision region. That configuration increases the predicted
efficients have been variouslylabeled as V , a , orv2 2 v by a factor of 2-3 in noncentral collisions so that v
n n n{ } 3 3
where n is the harmonic. Although the latter is perhaps deviates from the 1/ N one would expect from ran-
part
more cumbersome, we have maintained its usage owing dom fluctuations in the positions of the nucleons partic-
p
4
ipating in the collision [15, 16, 18]. That configuration is verified by checking that the φ distributions are flat
should also be asymmetric in the forward and backward after the correction so that cosn(φ) and sinn(φ) are
h i h i
rapidity directions, again pointing to the importance of nearzero. Withthesecorrections,thedatarepresentthe
understandingthethreedimensionalstructureoftheini- C that would be seen by a detector with per-
m,n,m+n
tial state. If the evidence proposed by Teaney and Yan fect acceptance for particles with p > 0.2 GeV/c and
T
is not confirmed, then one may question the validity of η < 1. In practice, calculating all possible combina-
| |
any model that predicts the centrality dependence of v tions of three particles individually would be computa-
n
basedonthoseinitialconditionmodels. Inthispaperwe tionally too costly to be practical, particularly for the
presentmeasurementsof cos(mφ +nφ (m+n)φ ) as larger data sets at 200 GeV. In that case we use alge-
1 2 3
h − i
a function of energy, centrality, ∆η, p , and harmonics bra basedonQ-vectors(Q =Σexp(inφ)) to reduce the
T n
m and n. Data confirm the predicted correlation be- computational challenge [33]. Differential measurements
tween the first, second and third harmonics but the ∆η like the ∆η dependence of the correlations, however, re-
dependencepointstothepotentialimportanceofinclud- quire explicit calculations for at least two of the parti-
ing the three-dimensionalstructure ofthe initial state in cles. Studying the ∆η dependence of the correlations
the model calculations. also allows us to correct for the effect of track-merging
In the following, we first describe the experiment on the correlations. Track-merging leads to a large anti-
and the analysis (Sec. II). We then present the results correlation between particle pairs that are close to each
in Sec. III including the ∆η dependence (Sec. IIIA), otherinthedetector. Theeffectbecomeslargeincentral
the centrality dependence (Sec. IIIB), the p depen- collisions where the detector occupancy is largest. After
T
dence (Sec. IIIC), and the beam energy dependence weightcorrectionshavebeenappliedtocorrectforsingle
(Sec. IIID). Conclusions are presented in Sec. IV. We particle acceptance effects, the effect of track-merging is
include measurements ofv2 2 for n=1,2,4,and 5 in the thelargestremainingcorrection. Datahavebeendivided
n{ }
appendix. into standard centrality classes (0-5%, 5-10%, 10-20%,...
70-80%)basedonthe number ofchargedhadronswithin
η < 0.5 observed for a given event. In some figures,
| |
II. EXPERIMENT AND ANALYSIS we will report the centrality in terms of the number of
participating nucleons (N ) estimated from a Monte
part
Carlo Glauber calculations [34, 35].
Our measurements make use of data collected from
The three-particle correlations presented in this pa-
Au+AucollisionswiththeSTARdetectoratRHICinthe
per are related to the low-resolution limit of the event-
years2004,2010,2011,2012,and2014. Thechargedpar-
plane measurements that have been explored at the
ticlesusedinthisanalysisaredetectedthroughtheirion-
LHC[36]. Practicallythiswouldbecarriedoutbydivid-
ization energy loss in the STAR Time ProjectionCham-
ing C by v v v . Typically, however, v is
ber[32]. Thetransversemomentump ,η,andchargeare m,n,m+n m n m+n n
T h i
measured from a two-particle correlation function such
determined from the trajectory of the track in STAR’s
as the two-particle cumulants v = v2 2 or a simi-
solenoidal magnetic field. With the 0.5 Tesla field used lar measurement and the v2 2 nare not pno{si}tive-definite
during data taking, particles can be reliably tracked n{ } p
for pT > 0.2 GeV/c. The efficiency for finding parti- quantities. Assuch, vn2{2}can,andoftendoes,become
cles drops quickly as p decreases below this value [34]. imaginary. Thisisparticularlytrueforthefirstharmonic
T p
Weightshavebeenusedtocorrectthe three-particlecor- and also at lower collision energies. For this reason we
relationfunctionsforthep -dependentefficiencyandfor report the pure three-particle correlations which, in any
T
imperfections in the detector acceptance. The quantity case, do not suffer from the ambiguities related to the
analyzed and reported is low- and high-resolution limits associated with reaction
plane analyses [19, 37] and are therefore easier to inter-
pret theoretically.
C = cos(mφ +nφ (m+n)φ ) =
m,n,m+n 1 2 3
h − i
w w w cos(mφ +nφ +(m+n)φ )
i,j,k i j k i j k
w w w III. RESULTS
* P i,j,k i j k !+
(1)
P
In the following, we presentthe ∆η dependence of the
where represents an average over events and is three-particlecorrelationsfor severalharmoniccombina-
hi i,j,k
asumoveruniqueparticletripletswithinanevent. Each tions corrected for track-merging. After removing the
P
eventisweightedbythenumberofuniquetripletsinthat effects of track merging and Hanbury Brown and Twiss
event. Theweightsw aredeterminedfromtheinverse (HBT) correlations [38], we integrate over the ∆η de-
i,j,k
ofthe φdistributions after they havebeen averagedover pendence of the correlations and present the resulting
manyevents(whichforaperfectdetectorshouldbe flat) integrated correlationsas a function of centrality for the
andbythepT dependentefficiency. Thewi,j,k dependon energies √sNN=200, 62.4, 39, 27, 19.6, 14.5, 11.5, and
the particles’ p , η, and charge and the collisions’ cen- 7.7 GeV. We also investigate the p dependence of the
T T
trality and z-vertex location. The correction procedure correlations by plotting them as a function of the p of
T
5
either the firstor secondparticle usedin the correlation. refers to the particle number. The right panels show the
Finally, we study how the data depends on the beam samebutasafunctionofthedifferencebetweenparticles
energy. 1 and 3. The C correlation is similar to the correla-
1,1,2
tion used in the search for the chiral magnetic effect ex-
ceptthatwedonotseparateoutthecaseswhenparticles
A. ∆η Dependence 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 ap-
proximatelyrelatedtothereaction-planebasedmeasure-
200 GeV Au+Au ments by scaling the three-particle correlations by 1/v .
2
0.15 N2part × C112 We note that the difference in C1,1,2 for different charge
0.1 combinations is as large as the signal with C being
1,1,2
0.05
nearlyzeroforunlike-signcombitionsofparticle1and2.
0
−0.05 cen This correlation may also be influenced by momentum
−0.1 tra conservation effects as well. It’s not clear however how
−0.15 0.5% l those effects would be distributed with respect to ∆η.
−0.2 5-10%
−0.25 10-20% IntheleftpanelsofFig.1,weseeastrongdependence
forC on η η . Incentralcollisions,thedatastarts
1,1,2 1 2
0.15 | − |
out negative at the smallest values of η η but then
0.1 | 1− 2|
beginstoincreaseandbecomesclosetozeroorevenpos-
0.05
m
−0.050 id c ipteivaekniseasree|ηn1i−n ηth2|e=co1rr.5el.atAiotnsmthaaltl |iηs1r−elaηt2e|d, atonaHrBroTw.
e
−0−.01.51 20-30% ntra As we progress from central to peripheral collisions, the
−0.2 30-40% l trendschangewithC1,1,2 inperipheralcollisionsexhibit-
−0.25 40-50% ing a positive value at small |η1−η2|, perhaps signaling
the dominanceofjets inthe correlationfunctioninthese
0.15 50-60% peripheral collisions.
0.1 60-70%
0.05 70-80% Theleftpanelssharethesamescalesastherightpanels
0 pe makingitclearthatthe dependence ofC1,1,2 on η1 η3
−0.05 rip ismuchweakerthanthedependence on η η .| Th−isis|
−0−.01.51 hera expectedsincethee−2iφ3 terminC1,1,2=|he1i−φ1e2iφ|2e−2iφ3i
−0.2 l willbedominatedbytheglobalpreferenceofparticlesto
−0.25 be emitted in the direction of the reaction plane. For
0.5 |η-η1 | 1.5 0.5 |η-η1 | 1.5 all but the most central collisions, the almond shaped
1 2 1 3 geometryofthe collisionoverlapregionis approximately
invariant with rapidity. This is not likely the case for
FIG. 1. (color online) The ∆η dependence of C1,1,2 scaled other harmonics.
byNp2art for9centralityintervalswiththethreemostcentral Figure 2 shows C scaled by N2 as a function of
classesshowninthetoppanelsandthethreemostperipheral 1,2,3 part
η η (left panels)and η η (rightpanels). Inthis
in the bottom. The Npart values used for the corresponding | 1− 2| | 1− 3|
case, C exhibits a stronger dependence on η η
centralities are 350.6, 298.6, 234.3, 167.6, 117.1, 78.3, 49.3, 1,2,3 | 1 − 3|
28.2 and 15.7. Inthepanelson theleft, ∆η is takenbetween thanon η1 η2 . Thevariationwith η2 η3 isverysim-
| − | | − |
particles 1 and 2 while on the right it is between particles ilar to the variation with η1 η2 and is omitted from
1 and 3 (which is identical to 2 and 3). Data are from 200 the figuresto improvelegib|ilit−y. Ag|ain,the ei2φ2 compo-
GeVAu+AucollisionsandforchargedhadronswithpT >0.2 nent of C1,2,3 is dominated by the reaction plane which
GeV/c, η <1. is largely invariant within the η range covered by these
| |
measurements so that C depends very little on the
1,2,3
Figure 1 shows the ∆η dependence of C scaled by η , η η , or η η . However, C depends very
1,1,2 2 1 2 2 3 1,2,3
N2 for charged hadrons with p > 0.2 GeV/c and stro|ngly−on| η | η−. T|his dependence may arise from
part T | 1 − 3|
η < 1. The scaling accounts for the natural dilution the longitudinal asymmetry inherent in the fluctuations
| |
ofcorrelationsexpectedifthe morecentralcollisionscan that lead to predictions for large values of C [24]. In
1,2,3
be treated as a linear superposition of nucleon-nucleon models for the initial geometry, the correlations are in-
collisions. Results for nine different centrality intervals duced between the first, second, and third harmonics of
from 200 GeV Au+Au collisions are shown. We do not the eccentricity by cases where a nucleon fluctuates to-
include the uncertainty on N in the uncertainties in wards the edge of the nucleus [39]. If that occurs in the
part
our figures. The left panels show the correlations as a reactionplanedirectionandtowardstheothernucleusin
function of the difference in η between the first and sec- thecollision,thenthatnucleoncancollidewithmanynu-
ond particle. Note that the subscripts in C refer cleons from the other nucleus. This geometry will cause
m,n,m+n
to the harmonic number while the subscripts for the η the first and third harmonics to become correlated with
6
220000 GGeeVV Au+Au 200 GeV Au+Au
0.2 N2part × C123 1 N2part × C224 05--51%0%
0.15
0.8 10-20%
0.1
0.05 c 0.6 c
e e
−0.005 0.5% ntral 0.4 ntral
0.2
−0.1 5-10%
−0.15 10-20% 0
0.2
1
0.15
0.8
0.1 m m
0.05 id 0.6 id
c c
0 en 0.4 en
−0−.00.51 2300--3400%% tral 0.2 2300--3400%% tral
−0.15 40-50% 0 40-50%
0.2 50-60% 0.18 50-60%
0.15 6700--7800%% 00..1146 6700--7800%%
0.1 0.12
0.05 pe 0.1 pe
0 riph 00..0068 riph
−0.05 era 0.04 era
−0.1 l 0.02 l
0
−0.15 −0.02
0.5 1 1.5 0.5 1 1.5 0.5 1 1.5 0.5 1 1.5
|η-η| |η-η| |η-η| |η-η|
1 2 1 3 1 2 1 3
FIG. 2. (color online). The ∆η dependence of C1,2,3 scaled FIG. 3. (color online) The ∆η dependence of C2,2,4 scaled
byN2 for9centralityintervalswiththethreemostcentral byN2 for9centralityintervalswiththethreemostcentral
part part
classesshowninthetoppanelsandthethreemostperipheral classesshowninthetoppanelsandthethreemostperipheral
inthebottom. Inthepanelsontheleft,∆η istakenbetween inthebottom. Inthepanelsontheleft,∆η istakenbetween
particles 1 and 2 while on the right it is between particles particles 1 and 2 while on the right it is between particles 1
1 and 3. Data are from 200 GeV Au+Au collisions and for and3(identicalto2and3). Dataarefrom 200GeVAu+Au
charged hadrons with pT >0.2 GeV/c, η <1. collisionsandforchargedhadronswithpT >0.2GeV/c, η <
| | | |
1.
the second harmonic. Since the collision of one nucleon η η which in mid-central collisions amounts to an
1 3
fromonenucleuswithmanynucleonsintheothernucleus a|pp−roxim| ately 20% variation. We also note that in mid-
is asymmetric along the rapidity axis, we argue that we central collisions, the change in value of C over the
2,2,4
cthaanteaxspsuecmteathsetrionnitgiadleepneenrgdyendceensointy|ηis1s−ymηm3|.etrMicowdietlhs rcahnagnege0o<f C|η1 −oηv3|er<02<isηsimilηar i<n m2aagnndituCde tootvheer
1,1,2 1 2 1,2,3
rapidity (boost invariant) will likely fail to describe this 0< η η <2. | − |
1 3
behavior. Onemayalsospeculatethatthevariationwith In|Fi−g. 4|we present the η η and η η de-
1 2 2 3
|η1−η3| could arise from sources like jets or resonances pendence of C2,3,5. Again, C| 2,3−,5 on|ly exh|ibit−s a w| eak
particularlyiftheyinteractwiththemediumsothatthey dependence on η η but a stronger dependence on
1 2
become correlated with the reaction plane. Making use η η . In cen|tra−l and| mid-central collisions, a strong
2 3
of the full suite of measurements provided here will help s|hor−t-ra|ngecorrelationat η η <0.4isapparentcon-
2 3
delineate between these two scenarios. sistent with HBT and Co|ulo−mb|correlations that vary
In Fig. 3 we present the η η and η η de- with respect to the reaction plane. In addition to that
1 2 1 3
| − | | − |
pendence of C2,2,4. This correlation is more strongly in- peak, C2,3,5 decreases as |η2 −η3| increases. Although
fluenced by the reaction plane correlations and exhibits therelativevariationofC2,3,5 issimilartoC2,2,4,theab-
much larger values than either C1,1,2 or C1,2,3. The solute change is much smaller than for C1,1,2, C1,2,3, or
dependence on η1 η2 and η1 η3 are also weaker C2,2,4.
| − | | − |
with C in central and mid-central collisions show- ThecombinationofthevariousC canhelpelu-
2,2,4 m,n,m+n
ing little variation over the η η range, consistent cidate the nature of the three-particle correlations. If
1 2
| − |
with a mostly η-independent reaction plane within the the η η dependenceofC arisesfromcorrelations
1 3 1,2,3
| − |
measured range. A larger variation is observed with between particles from jets correlated with the reaction
7
specific modelthathas beenshownto describeour data.
200 GeV Au+Au
0.2 N2 × C
part 235
B. Centrality Dependence
0.1 c
en InFigs.5 and6 we showCm,n,m+n correlationsscaled
tra by N2 with (m,n) = (1,1), (1,2), (1,3), (2,2), (2,3),
l part
0.5%
0 (2,4), (3,3), and (3,4) for √s =200, 62.4, 39, 27, 19.6,
5-10% NN
10-20% 14.5, 11.5, and 7.7 GeV Au+Au collisions as a function
0.2 20-30% of Npart. Data are for charged particles with |η| < 1
30-40% and pT > 0.2 GeV/c. The correlation C2,2,4, by far the
40-50% largestofthemeasuredcorrelations,hasbeenscaledbya
m
0.1 id factor of 1/5. Otherwise, the scales on each of the three
ce panels are kept the same for each energy to make it eas-
n
tra ier to compare the magnitudes of the different harmonic
0 l combinations.
At200GeV,C isnegativeforallcentralitiesexcept
1,1,2
for the most peripheral where it is slightly positive but
50-60%
0.04 60-70% consistentwith zero. C1,2,3 is consistentwith zero in pe-
70-80% ripheral collisions, positive in mid-central collisions but
0.02 perip tahnedntbheircdomheasrmneognaictiveeveinntcepnlatrnaelscaorlleisuionncso.rrIefltahteeds,ectohnend
h
0 era C1,2,3 should be zero. The C1,2,3 correlation is non-zero
l deviating fromthat expectation. The magnitude is how-
ever much smaller than originally anticipated based on
−0.02
0.5 |η-η1 | 1.5 0.5 |η-1η| 1.5 a linearhydrodynamicresponseto initial stategeometry
1 2 2 3 fluctuations [22]. Non-linear coupling between harmon-
ics, where the fifth harmonic for example is dominated
FIG. 4. (color online) The ∆η dependence of C2,3,5 scaled by a combination of the second and third harmonic, has
byNp2art for9centralityintervalswiththethreemostcentral been shown to be very important [23, 40]. In the case of
classesshowninthetoppanelsandthethreemostperipheral
C ,thenon-linearcontributionhasanoppositesignto
1,2,3
inthebottom. Inthepanelsontheleft,∆η istakenbetween
the linear contribution and similar magnitude canceling
particles 1 and 2 while on the right it is between particles 1
out most of the expected strength of C . This sug-
and3(identicalto2and3). Dataarefrom200GeVAu+Au 1,2,3
gests that C is very sensitive to the nonlinear nature
collisionsandforchargedhadronswithpT >0.2GeV/c, η < 1,2,3
1. | | ofthe hydrodynamic model. C1,3,4 is close to zeroforall
centralitiesindicatinglittleornocorrelationbetweenthe
first, third, and fourth harmonics. The other C
m,n,m+n
plane, we would expect the particles at small ∆η to pre- correlations are positive for all centralities. When con-
dominantly come fromthe near-sidejet (at∆φ 0) and sidering the comparison of this data to hydrodynamic
particlesatlarger∆η to comefromthe away-sid≈ejet (at models, it is important to also consider the strong ∆η
∆φ π radians). In that case, at small ∆η, C dependence of the correlations as shown in the previous
m,n,m+n
for a≈ll harmonics will have a positive contribution from section.
thejets. Thesameisnottruehoweverforlarge∆ηwhere The correlations involving a second harmonic are
wewouldexpectthecorrelationstobedominatedbythe largestwith C2,2,4 being approximately5 times largerin
away-side jet separated by π radians. For this case at magnitude than the next largest correlator C2,3,5. The
large ∆η, C and C would receive negative con- correlations decrease quickly as harmonics are increased
1,1,2 1,2,3
tributions from the away side jet while C2,2,4 and C2,3,5 beyond n=2. The higher harmonic correlations C3,3,6
would both receive positive contributions. The trends andC3,4,7 are both small but non-zero. The correlations
observed across the variety of Cm,n,m+n measurements C1,1,2, C1,2,3, C2,2,4, C2,3,5,andC3,3,6 scaledbyNp2art all
are inconsistent with this simple picture with C de- exhibitextremainmidcentralcollisionswheretheinitial
2,2,4
creasing by nearly the same amount as C as ∆η is overlap geometry is predominantly elliptical. We note
1,2,3
increased. A more complicated picture of the effect of that the centrality at which Np2artC2,2,4 reaches a maxi-
jets would therefore be required to account for the ob- mumis differentthanthe centralityatwhichNp2artC2,3,5
served data but it appears difficult to construct a non- reaches a maximum.
flow scenario that can account for the long-range vari- As the collision energy is reduced, although the mag-
ation of C . Breaking of boost-invariance in the nitude of the correlations becomes smaller, the central-
m,n,m+n
initial density distributions may provide an explanation ity dependence and ordering of the different harmonics
for the observed variations but we do not know of any seems to remainmostly the same. The C correlation
1,2,3
8
0.2
N2 × C N2 × C
part 112 part 336
0.15 N2 × C N2 × C
part 123 part 347
0.1 N2part × C134
2
0.05 0
0
0 G
e
−0.05 N2 × C / 5 V
part 224
−0.1 N2part × C235
N2 × C
−0.15 part 246
0.1
0.05
6
2
0 .4
G
−0.05 e
V
−0.1
−0.15
0.1
0.05
3
9
0 G
e
V
−0.05
−0.1
0.06
0.04
0.02
2
0 7
−0.02 G
e
−0.04 V
−0.06
−0.08
−0.1
0 100 200 300 0 100 200 300 0 100 200 300
N
part
FIG.5. (coloronline)ThecentralitydependenceoftheCm,n,m+ncorrelationsscaledbyNp2artforchargedhadronswithpT >0.2
GeV/c and η < 1 from 200, 62.4, 39, and 27 GeV Au+Au collisions for (m,n) = (1,1),(1,2),(1,3) (left) (2,2),(2,3),(2,4)
| |
(center) and (3,3),(3,4) (right). Systematic errors are shown as bands. All panels in the same row share the same scale but
C2,2,4 hasbeendividedbyafactorof5tofitonthepanel. Thelabelsinthetoppanelsapplytoallthepanelsinsamecolumn.
however is an exception. While at 200 GeV, C is energiesbelow 200GeV.This data shouldprovideuseful
1,2,3
mostly positive, at 62.4GeV it is consistentwith zeroor constraints for the models being developed to describe
slightly negative and at lower energies it becomes more lower energy collisions associated with the energy scan
andmorenegative. Wespeculate thatthisbehaviormay programat RHIC.
be related to the increasing importance of momentum
conservation as the number of particles produced in the Figure 6 shows the same correlations as Fig. 5 except
collision decreases. No theoretical guidance exists how- for lower energy data sets: √s = 19.6, 14.5, 11.5, and
NN
ever for the energy dependence of these correlations at 7.7 GeV. Trends similar to those seen in Fig. 5 are for
the mostpartalso exhibited in this figure. Although the
9
0.06 N2 × C N2 × C
part 112 part 336
0.04 N2part × C123 N2part × C347
N2 × C
0.02 part 134
1
0 9.6
−0.02 G
e
−0.04 N2part × C224 / 5 V
N2 × C
−0.06 part 235
N2 × C
−0.08 part 246
0.06
0.04
0.02
0 1
4
−0.02 .5
−0.04 G
e
−0.06 V
−0.08
−0.1
0.1
0.05
1
1
0 .5
G
−0.05 e
V
−0.1
0.1
0.05
7
0 .7
G
−0.05 e
V
−0.1
0 100 200 300 0 100 200 300 0 100 200 300
N
part
FIG.6. (color online) Thesame quantitiesas Fig. 5butfor thelower energy Au+Aucollisions 19.6, 14.5, 11.5, and 7.7 GeV.
statistical precision is poor for the lowest energy points, tionsby approximatelyafactor ofthree andwillmakeit
itappearsthatC at7.7GeVissmallerinmagnitude possible to measurethe ∆η dependence ofthe C
1,1,2 m,n,m+n
than at higher energies, becoming consistent with zero. correlations to ∆η 3.
| |≈
Thiswasalsoobservedinthechargedependentmeasure-
ments of C [41]. A second phase of the RHIC beam
1,1,2
energy scan planned for 2019 and 2020 will significantly
C. p Dependence
T
increase the number of events available for analysis at
these lower energies while expanding the η acceptance
If the three-particle correlations presented here are
from η <1to η <1.5[42]sothatthisintriguingobser-
| | | | dominated by correlations between event planes, then
vation can be further investigated. The increased accep-
one might expect that the p dependence of the three-
tancewillincreasethenumberofthree-particlecombina- T
particlecorrelationswillsimplytrackthep dependence
T
10
relationship between C and harmonic planes in
m,n,m+n
Eq. 2 is not guaranteedto hold and is particularly likely
N2 × C /p tobe brokenforcorrelationsinvolvingthe firstharmonic
0.1 part 112 T,1 where momentum conservationeffects will likely play an
importantroleorwhereastrongchargesigndependence
0 has been observed [27, 28].
In Fig. 7 we show N2 C /p as a function of the
−0.1 part 1,1,2 T
p ofparticleone. Thetoppanelshowsthemorecentral
T
collisions while the bottom panel shows more peripheral
−0.2 collisions. In this and in the following figures related to
thep dependence,wesometimesexcludecentralitybins
T
−0.3 0-5% andslightlyshiftthepositionsofthe pointsalongthepT
5-10% axistomakethe figuresmorereadable. Formorecentral
−0.4 10-20% collisions, C1,1,2/pT,1 is negative and slowly decreases in
20-30% magnitude as pT,1 increases. This indicates that C1,1,2
is generally increasing with the p of particle one but
T
0.1 N2part × C112 /pT,1 that for centralcollisionsat highpT, C1,1,2 startsto sat-
urate. For the more peripheral 30-40% and 40-50% col-
lision however, C appears to be linear in p without
0 1,1,2 T
an indication of saturation even up to p 10 GeV/c.
T
≈
For the much more peripheral 60-70% and 70-80% cen-
−0.1
trality intervals, C starts out at or above zero then
1,1,2
becomesmoreandmorenegativeasp isincreased. The
T
−0.2
trends inthe mostperipheralcentralityintervals,partic-
ularly at high p , are consistent with being dominated
T
−0.3 30-40% by momentum conservation and jets. A pair of back-to-
40-50% back particles aligned with the reaction plane will lead
−0.4 60-70% to a negative value for C1,1,2. Although the data exhibit
70-80% a smooth transition from the trends in more central col-
lisions to the trends in more peripheral collisions, the
1 10
p (GeV) trends are quite distinct and indicative of very different
T,1 correlations in those different regions. In peripheral col-
lisions, the correlations get stronger as p is increased.
FIG. 7. (color online) Three-particle azimuthal correlations T
In central collisions, the opposite is observed.
C1,1,2 scaled by Np2art/pT,1 as a function of the first particles
pT for 200 GeV Au+Au collisions for charged hadrons with For the case ofC1,2,3 inFig. 8, we show the pT depen-
pT > 0.2 GeV/c and η < 1. The top and bottom panels dence of both particle one (left panels) and particle two
| |
show the same quantity but for a different set of centrality (right panels). The dependence of C /p on p
1,2,3 T,2 T,2
intervals. Systematicerrorsareshownassolidlinesenclosing is quite weak indicating that where C is non-zero,
1,2,3
therespective data points.
it increases roughly linearly with p . The dependence
T,2
of C /p on p , however, exhibits several notable
1,2,3 T,1 T,1
trends. First we note that for the 20-30% centrality in-
of the relevant v [22]:
n terval, C /p changes sign up to three times. In
1,2,3 T,1
hydrodynamic models, the value of C is very sensi-
cos(mφ (p )+nφ (m+n)φ ) 1,2,3
1 T 2 3
h − i≈ tivetotheinterplaybetweenlinearandnon-lineareffects
vm(pT)vnvm+n and to viscous effects. The sign oscillations exhibited in
εm εn εm+n× the data may be a consequence of subtle changes in the
ε ε ε cos(mΨ +nΨ (m+n)Ψ ) , (2) relevantsizesofthoseeffects. Ifthisisthecase,thenthis
m n m+n m n m+n
h − i confirms that C is a powerful measurement to help
1,2,3
where ε is the mth harmonic eccentricity and Ψ is tune those models. At intermediate p (2-5 GeV/c),
m m T,1
the mth harmonic participant plane angle. For the pur- C ispositiveforcentralcollisionsbutnegativeforpe-
1,2,3
pose of simplicity in this publication, we have scaled the ripheral collisions. At p > 7 GeV/c, C is strongly
T 1,2,3
correlations by N2 /p to account for the general in- negative,perhapsagain,indicativeofthe contributionof
part T
creaseofv (p )withp [43]. Thatsimplescalingisonly back-to-back jets to the correlations. Such strong nega-
n T T
valid at lower p and for n = 1. It does, however, aid tive correlation seems to be absent in central collisions
T
6
in visualizing trends in the data which would otherwise where C appears to remain positive, although with
1,2,3
be visually dominated by the larger p range. Our pri- largeerrorbars. This isconsistentwithascenariowhere
T
maryreasonforintroducingEq.2istoprovideacontext di-jets have been quenched in central collisions. As with
for understanding the p dependence of C . The C , the p trends for C are very different in the
T m,n,m+n 1,1,2 T 1,2,3