Two-particle correlations on transverse momentum and momentum dissipation in Au-Au collisions at $\sqrt{s_{NN}}$ = 130 GeV

0.6 MB·English

by  STAR Collaboration

Two-particle correlations on transverse momentum and momentum dissipation in
Au-Au collisions at √sNN = 130 GeV
J. Adams,2 M.M. Aggarwal,29 Z. Ahammed,44 J. Amonett,19 B.D. Anderson,19 M. Anderson,6 D. Arkhipkin,12
G.S. Averichev,11 Y. Bai,27 J. Balewski,16 O. Barannikova,2 L.S. Barnby,2 J. Baudot,17 S. Bekele,28 V.V. Belaga,11
A. Bellingeri-Laurikainen,39 R. Bellwied,47 B.I. Bezverkhny,49 S. Bhardwaj,34 A. Bhasin,18 A.K. Bhati,29
H. Bichsel,46 J. Bielcik,49 J. Bielcikova,49 L.C. Bland,3 C.O. Blyth,2 S-L. Blyth,21 B.E. Bonner,35 M. Botje,27
J. Bouchet,39 A.V. Brandin,25 A. Bravar,3 M. Bystersky,10 R.V. Cadman,1 X.Z. Cai,38 H. Caines,49
M. Caldero´n de la Barca Sa´nchez,6 J. Castillo,27 O. Catu,49 D. Cebra,6 Z. Chajecki,28 P. Chaloupka,10
S. Chattopadhyay,44 H.F. Chen,37 J.H. Chen,38 Y. Chen,7 J. Cheng,42 M. Cherney,9 A. Chikanian,49 H.A. Choi,33
W. Christie,3 J.P. Coffin,17 T.M. Cormier,47 M.R. Cosentino,36 J.G. Cramer,46 H.J. Crawford,5 D. Das,44 S. Das,44
M. Daugherity,41 M.M. de Moura,36 T.G. Dedovich,11 M. DePhillips,3 A.A. Derevschikov,31 L. Didenko,3
T. Dietel,13 P. Djawotho,16 S.M. Dogra,18 W.J. Dong,7 X. Dong,37 J.E. Draper,6 F. Du,49 V.B. Dunin,11
7
0 J.C. Dunlop,3 M.R. Dutta Mazumdar,44 V. Eckardt,23 W.R. Edwards,21 L.G. Efimov,11 V. Emelianov,25
0 J. Engelage,5 G. Eppley,35 B. Erazmus,39 M. Estienne,17 P. Fachini,3 R. Fatemi,22 J. Fedorisin,11 K. Filimonov,21
2 P. Filip,12 E. Finch,49 V. Fine,3 Y. Fisyak,3 J. Fu,48 C.A. Gagliardi,40 L. Gaillard,2 J. Gans,49 M.S. Ganti,44

n V. Ghazikhanian,7 P. Ghosh,44 J.E. Gonzalez,7 Y.G. Gorbunov,9 H. Gos,45 O. Grebenyuk,27 D. Grosnick,43
a S.M. Guertin,7 K.S.F.F. Guimaraes,36 Y. Guo,47 N. Gupta,18 T.D. Gutierrez,6 B. Haag,6 T.J. Hallman,3
J
A. Hamed,47 J.W. Harris,49 W. He,16 M. Heinz,49 T.W. Henry,40 S. Hepplemann,30 B. Hippolyte,17 A. Hirsch,32
2
E. Hjort,21 G.W. Hoffmann,41 M.J. Horner,21 H.Z. Huang,7 S.L. Huang,37 E.W. Hughes,4 T.J. Humanic,28 G. Igo,7
1
A. Ishihara,41 P. Jacobs,21 W.W. Jacobs,16 P. Jakl,10 F. Jia,20 H. Jiang,7 P.G. Jones,2 E.G. Judd,5 S. Kabana,39

5 K. Kang,42 J. Kapitan,10 M. Kaplan,8 D. Keane,19 A. Kechechyan,11 V.Yu. Khodyrev,31 B.C. Kim,33 J. Kiryluk,22
v A. Kisiel,45 E.M. Kislov,11 S.R. Klein,21 D.D. Koetke,43 T. Kollegger,13 M. Kopytine,19 L. Kotchenda,25
2
V. Kouchpil,10 K.L. Kowalik,21 M. Kramer,26 P. Kravtsov,25 V.I. Kravtsov,31 K. Krueger,1 C. Kuhn,17
1
0 A.I. Kulikov,11 A. Kumar,29 A.A. Kuznetsov,11 M.A.C. Lamont,49 J.M. Landgraf,3 S. Lange,13 S. LaPointe,47
8 F. Laue,3 J. Lauret,3 A. Lebedev,3 R. Lednicky,12 C-H. Lee,33 S. Lehocka,11 M.J. LeVine,3 C. Li,37 Q. Li,47 Y. Li,42
0 G. Lin,49 S.J. Lindenbaum,26 M.A. Lisa,28 F. Liu,48 H. Liu,37 J. Liu,35 L. Liu,48 Z. Liu,48 T. Ljubicic,3
4
W.J. Llope,35 H. Long,7 R.S. Longacre,3 M. Lopez-Noriega,28 W.A. Love,3 Y. Lu,48 T. Ludlam,3 D. Lynn,3
0
/ G.L. Ma,38 J.G. Ma,7 Y.G. Ma,38 D. Magestro,28 D.P. Mahapatra,14 R. Majka,49 L.K. Mangotra,18 R. Manweiler,43
x
S. Margetis,19 C. Markert,19 L. Martin,39 H.S. Matis,21 Yu.A. Matulenko,31 C.J. McClain,1 T.S. McShane,9
e
- Yu. Melnick,31 A. Meschanin,31 M.L. Miller,22 N.G. Minaev,31 S. Mioduszewski,40 C. Mironov,19 A. Mischke,27
l
c D.K. Mishra,14 J. Mitchell,35 B. Mohanty,44 L. Molnar,32 C.F. Moore,41 D.A. Morozov,31 M.G. Munhoz,36
u B.K. Nandi,15 C. Nattrass,49 T.K. Nayak,44 J.M. Nelson,2 P.K. Netrakanti,44 V.A. Nikitin,12 L.V. Nogach,31
n
S.B. Nurushev,31 G. Odyniec,21 A. Ogawa,3 V. Okorokov,25 M. Oldenburg,21 D. Olson,21 M. Pachr,10 S.K. Pal,44
:
v Y. Panebratsev,11 S.Y. Panitkin,3 A.I. Pavlinov,47 T. Pawlak,45 T. Peitzmann,27 V. Perevoztchikov,3 C. Perkins,5
i
X W. Peryt,45 V.A. Petrov,47 S.C. Phatak,14 R. Picha,6 M. Planinic,50 J.Pluta,45 N. Poljak,50 N. Porile,32 J. Porter,46
A.M. Poskanzer,21 M. Potekhin,3 E. Potrebenikova,11 B.V.K.S. Potukuchi,18 D. Prindle,46 C. Pruneau,47
r
a J. Putschke,21 G. Rakness,30 R. Raniwala,34 S. Raniwala,34 R.L. Ray,41 S.V. Razin,11 J.G. Reid,46 J. Reinnarth,39
D. Relyea,4 F. Retiere,21 A. Ridiger,25 H.G. Ritter,21 J.B. Roberts,35 O.V. Rogachevskiy,11 J.L. Romero,6
A. Rose,21 C. Roy,39 L. Ruan,21 M.J. Russcher,27 R. Sahoo,14 I. Sakrejda,21 S. Salur,49 J. Sandweiss,49
M. Sarsour,40 P.S. Sazhin,11 J. Schambach,41 R.P. Scharenberg,32 N. Schmitz,23 K. Schweda,21 J. Seger,9
I. Selyuzhenkov,47 P. Seyboth,23 A. Shabetai,21 E. Shahaliev,11 M. Shao,37 M. Sharma,29 W.Q. Shen,38
S.S. Shimanskiy,11 E Sichtermann,21 F. Simon,22 R.N. Singaraju,44 N. Smirnov,49 R. Snellings,27 G. Sood,43
P. Sorensen,3 J. Sowinski,16 J. Speltz,17 H.M. Spinka,1 B. Srivastava,32 A. Stadnik,11 T.D.S. Stanislaus,43
R. Stock,13 A. Stolpovsky,47 M. Strikhanov,25 B. Stringfellow,32 A.A.P. Suaide,36 E. Sugarbaker,28 M. Sumbera,10
Z. Sun,20 B. Surrow,22 M. Swanger,9 T.J.M. Symons,21 A. Szanto de Toledo,36 A. Tai,7 J. Takahashi,36
A.H. Tang,3 T. Tarnowsky,32 D. Thein,7 J.H. Thomas,21 A.R. Timmins,2 S. Timoshenko,25 M. Tokarev,11
T.A. Trainor,46 S. Trentalange,7 R.E. Tribble,40 O.D. Tsai,7 J. Ulery,32 T. Ullrich,3 D.G. Underwood,1 G. Van
Buren,3 N. van der Kolk,27 M. van Leeuwen,21 A.M. Vander Molen,24 R. Varma,15 I.M. Vasilevski,12
A.N. Vasiliev,31 R. Vernet,17 S.E. Vigdor,16 Y.P. Viyogi,44 S. Vokal,11 S.A. Voloshin,47 W.T. Waggoner,9
F. Wang,32 G. Wang,7 J.S. Wang,20 X.L. Wang,37 Y. Wang,42 J.W. Watson,19 J.C. Webb,43 G.D. Westfall,24
A. Wetzler,21 C. Whitten Jr.,7 H. Wieman,21 S.W. Wissink,16 R. Witt,49 J. Wood,7 J. Wu,37 N. Xu,21 Q.H. Xu,21
Z. Xu,3 P. Yepes,35 I-K. Yoo,33 V.I. Yurevich,11 W. Zhan,20 H. Zhang,3 W.M. Zhang,19 Y. Zhang,37
2
Z.P. Zhang,37 Y. Zhao,37 C. Zhong,38 R. Zoulkarneev,12 Y. Zoulkarneeva,12 A.N. Zubarev,11 and J.X. Zuo38
(STAR Collaboration)
1Argonne National Laboratory, Argonne, Illinois 60439
2University of Birmingham, Birmingham, United Kingdom
3Brookhaven National Laboratory, Upton, New York 11973
4California Institute of Technology, Pasadena, California 91125
5University of California, Berkeley, California 94720
6University of California, Davis, California 95616
7University of California, Los Angeles, California 90095
8Carnegie Mellon University, Pittsburgh, Pennsylvania 15213
9Creighton University, Omaha, Nebraska 68178
10Nuclear Physics Institute AS CR, 250 68 Rˇeˇz/Prague, Czech Republic
11Laboratory for High Energy (JINR), Dubna, Russia
12Particle Physics Laboratory (JINR), Dubna, Russia
13University of Frankfurt, Frankfurt, Germany
14Institute of Physics, Bhubaneswar 751005, India
15Indian Institute of Technology, Mumbai, India
16Indiana University, Bloomington, Indiana 47408
17Institut de Recherches Subatomiques, Strasbourg, France
18University of Jammu, Jammu 180001, India
19Kent State University, Kent, Ohio 44242
20Institute of Modern Physics, Lanzhou, China
21Lawrence Berkeley National Laboratory, Berkeley, California 94720
22Massachusetts Institute of Technology, Cambridge, MA 02139-4307
23Max-Planck-Institut fu¨r Physik, Munich, Germany
24Michigan State University, East Lansing, Michigan 48824
25Moscow Engineering Physics Institute, Moscow Russia
26City College of New York, New York City, New York 10031
27NIKHEF and Utrecht University, Amsterdam, The Netherlands
28Ohio State University, Columbus, Ohio 43210
29Panjab University, Chandigarh 160014, India
30Pennsylvania State University, University Park, Pennsylvania 16802
31Institute of High Energy Physics, Protvino, Russia
32Purdue University, West Lafayette, Indiana 47907
33Pusan National University, Pusan, Republic of Korea
34University of Rajasthan, Jaipur 302004, India
35Rice University, Houston, Texas 77251
36Universidade de Sao Paulo, Sao Paulo, Brazil
37University of Science & Technology of China, Hefei 230026, China
38Shanghai Institute of Applied Physics, Shanghai 201800, China
39SUBATECH, Nantes, France
40Texas A&M University, College Station, Texas 77843
41University of Texas, Austin, Texas 78712
42Tsinghua University, Beijing 100084, China
43Valparaiso University, Valparaiso, Indiana 46383
44Variable Energy Cyclotron Centre, Kolkata 700064, India
45Warsaw University of Technology, Warsaw, Poland
46University of Washington, Seattle, Washington 98195
47Wayne State University, Detroit, Michigan 48201
48Institute of Particle Physics, CCNU (HZNU), Wuhan 430079, China
49Yale University, New Haven, Connecticut 06520
50University of Zagreb, Zagreb, HR-10002, Croatia
(Dated: February 5, 2008)
Measurements of two-particle correlations on transverse momentum p for Au-Au collisions at
t
√sNN = 130 GeV are presented. Significant large-momentum-scale correlations are observed for
chargedprimaryhadronswith0.15 p 2GeV/candpseudorapidity η 1.3. Suchcorrelations
t
≤ ≤ | |≤
were not observed in a similar study at lower energy and are not predicted by theoretical collision
models. Their direct relation to mean-p fluctuations measured in the same angular acceptance is
t
demonstrated. Positive correlations are observed for pairs of particles which have large p values
t
whilenegativecorrelationsoccurforpairsinwhichoneparticlehaslargep andtheotherhasmuch
t
lower p . The correlation amplitudes per final state particle increase with collision centrality. The
t
3
observed correlations are consistent with a scenario in which the transverse momentum of hadrons
associated with initial-stage semi-hard parton scattering is dissipated by themedium to lower p .
t
PACSnumbers: 25.75.-q,25.75.Gz
I. INTRODUCTION (p ,p ) for all charged particles with 0.15 p 2
t1 t2 t
≤ ≤
GeV/cand η 1.3(pseudorapidity)usingthe√sNN =
| |≤
130GeVAu-AucollisionsobservedwiththeSTARdetec-
Studying two-particlecorrelationsandevent-wisefluc-
tor [17]. This analysis is intended to reveal the response
tuations can provide essential information about the
ofthebulkmediumtostrongmomentumdissipationand
medium produced in ultrarelativistic heavy ion colli-
probe the dynamical origins of p fluctuations. The
sions [1, 2, 3]. At the collision energies available at the t
h i
data used in this analysis are described in Sec. II and
Relativistic Heavy Ion Collider (RHIC) energetic parton
theanalysismethod,correctionsanderrorsarediscussed
scattering occurs at sufficient rate to enable quantita-
in Sec. III. Models and fits to the data are presented in
tive studies ofin-mediummodificationofpartonscatter-
Secs. IV and V, respectively. A discussion and summary
ing and the distribution of correlated charged hadrons
are presented in the last two sections VI and VII.
associated with those energetic partons. Modification
of those correlation structures is expected as the bulk
mediumproducedinultrarelativisticheavyioncollisions
II. DATA
increases in spatial extent and energy density with in-
creasing collision centrality. Analyses of the central-
ity dependence in Au-Au collisions of high-p back-to- DataforthisanalysiswereobtainedwiththeSTARde-
t
back jet angular correlations based on a leading-particle tector [17] using a 0.25T uniformmagnetic fieldparallel
technique (e.g., leading-particle p > 4 GeV/c, associ- to the beam axis. A minimum-bias event sample (123k
t
ated particle p < 4 GeV/c) reveal strong suppression triggeredevents)requiredcoincidenceoftwoZero-Degree
t
for central collisions [4, 5], suggesting the development Calorimeters(ZDC); a0-15%oftotalcrosssectionevent
of a medium which dramatically dissipates momentum. sample (217k triggered events) was defined by a thresh-
Complementary studies of the lower-p bulk medium, its old on the Central Trigger Barrel (CTB) scintillators,
t
correlationstructure on transverse momentum, and how with ZDC coincidence. Event triggering and charged-
those correlations evolve with collision centrality pro- particle measurements with the Time Projection Cham-
videameasureofthe momentumtransportfromthefew ber (TPC) are described in [17]. Approximately 300k
GeV/crangetolowerp oforderafewtenthsofaGeV/c events were selected for use in this analysis. A primary
t
wherethebulkhadronicproductionoccurs. Suchstudies eventvertexwithin 75cmofthe axialcenter ofthe TPC
are an essential part of understanding the nature of the was required. Valid TPC tracks fell within the detector
medium produced in heavy ion collisions at RHIC. acceptance used here,defined by 0.15<pt <2.0 GeV/c,
In addition to jet angular correlations at high-p sub- η <1.3and2πinazimuth. Primarytracksweredefined
t | |
stantial nonstatistical fluctuations in event-wise mean as having a distance of closest approach less than 3 cm
transverse momentum p of charged particles from from the reconstructed primary vertex which included a
t
Au-Au collisions were rhepiorted by the STAR [6] and largefractionoftrueprimaryhadronsplusapproximately
PHENIX [7] experiments at RHIC. p fluctuations at 7%backgroundcontamination[18]fromweakdecaysand
t
RHIC aremuchlargerthanthoserephortiedatthe CERN interactions with the detector material. In addition ac-
SuperProtonSynchrotron(SPS)withone-tenththeCM cepted particle tracks were required to include a mini-
energy [8], and were not predicted by theoretical mod- mum of 10 fitted points (the TPC contains 45 pad rows
els [6, 9, 10, 11]. p fluctuations could result from in each sector) and, to eliminate split tracks (i.e., one
t
several sources incluhdinig collective flow (e.g., elliptic particle trajectory reconstructedas two or more tracks),
flow [12] when azimuthal acceptance is incomplete), lo- the fraction of space points used in a track fit relative
cal temperature fluctuations, quantum interference [13], to the maximum number expected was required to be
final-state interactions, resonance decays, longitudinal >52%. Particle identification was not implemented but
fragmentation [14], and initial-state multiple scatter- charge sign was determined. Further details associated
ing [15] including hard parton scattering [9] with sub- with track definitions, efficiencies and quality cuts are
sequentin-mediumdissipation[16]. p fluctuationscan described in [18, 19].
t
h i
bedirectlyrelatedtointegralsoftwo-particlecorrelations
over the p acceptance. Correlations on p , by provid-
t t
III. DATA ANALYSIS
ing differential information, better reveal the underlying
dynamics for the observed nonstatistical fluctuations in
p . A. Analysis Method
t
h i
InthispaperwereportthefirstmeasurementsatRHIC
of two-particle correlations (based on number of pairs) Our eventual goal is to determine the complete struc-
on two-dimensional (2D) transverse momentum space ture of the six-dimensional two-particle correlation for
4
all hadron pair charge combinations. Toward this goal tities, resulting in the correlation structures common to
thetwo-particlemomentumspacewasprojectedonto2D all charge-sign combinations. Hence we refer to these fi-
subspace(p ,p )byintegratingthepseudorapidityand nalresults as charge-independent (CI = LS + US) corre-
t1 t2
azimuth coordinates (η ,η ,φ ,φ ) over the detector ac- lationseventhoughthey areconstructedfromquantities
1 2 1 2
ceptance for this analysis, η 1.3 and full 2π azimuth. which depend on the charge signs of the hadron pairs.
| |≤
Projectiononto2Dsubspace (p ,p )is achievedbyfill- The correlation measure reported here is therefore the
t1 t2
ing2Dbinnedhistogramsofthenumberofpairsofparti- CI combination for rˆ[X(p ),X(p )] 1.
t1 t2
−
cles for allvalues ofη,φ within the acceptance. Comple- Deviations of event-wise p fluctuations from a
t
h i
mentarycorrelationstructuresonrelativepseudorapidity central-limit-theorem reference [25, 26] are measured by
andazimuthcoordinateswithintegrationovertransverse scale-dependent (i.e., η,φ bin sizes) variance difference
momentum acceptance are reported in [20, 21]. ∆σ2 introduced in [6], where it was evaluated at the
pt:n
The quantities obtained here are ratios of normalized STAR(η,φ)detectoracceptancescale. ∆σ2 canbeex-
histogramsofsiblingpairs(particlesfromthesameevent) pressedasaweighted integralon(p ,p )opft:pnair-density
t1 t2
to mixed-event pairs (each particle of the pair is from a difference ρ ρ , where two-particle densities ρ
sib mix sib
−
different, but similar event) in an arbitrary 2D bin with and ρ are approximated by the event-averaged num-
mix
indices a,b representing the values of pt1 and pt2 (see ber of sibling and mixed-event pairs per 2D bin, re-
discussionbelow). Thenormalizedpair-numberratiorˆab spectively. Both densities are normalized to the event-
introduced in [22] is here defined by averagedtotal number of pairs. ∆σ2 can be rewritten
pt:n
exactly as a discrete sum over p products [25] [first line
t
rˆ nˆ /nˆ , (1)
ab ≡ ab,sib ab,mix in Eq. (2) below], and the summations approximated in
turn by the weighted integral of the pair density differ-
where nˆ = n / n (sum over all 2D
ab,sib ab,sib ab ab,sib ence [second line in Eq. (2)] according to,
bins), nˆ = n / n , and n and
ab,mix ab,mixP ab ab,mix ab,sib
n are the inclusive number of sibling and mixed-
evaeb,nmtixpairs, respectively, inP2D bin a,b. Histograms and 1 1 ǫ Nj
ratios rˆab were constructed for each charge-sign combi- ∆σp2t:n ≡ N¯ ǫ ptjiptji′ −pˆ2t (2)
nation: (+,+), ( , ), (+, ) and ( ,+). Ratio rˆab is Xj=1i6=Xi′=1(cid:0) (cid:1)
− − − − 1
approximately1,while difference (rˆ 1) measurescor-
ab dp dp p p (ρ ρ )
relation amplitudes and is the quantit−y reported here. ≈ N¯ t1 t2 t1 t2 sib− mix
Z Z
The exponential decrease in particle yield with in- pˆ2N¯ r(p ,p ) 1 ,
creasing p degrades the statistical accuracy of rˆ at ≡ t h t1 t2 − i
t ab
larger transverse momentum, thus obscuring the statis- where weighted average r(p ,p ) 1 is defined in the
t1 t2
h − i
tically significant correlation structures there. In or- lastline with weightp p ρ (p ,p )and the integral
t1 t2 mix t1 t2
dertoachieveapproximatelyuniformstatisticalaccuracy of ρ , dp dp p p ρ is N¯2pˆ2 [27]. In Eq. (2)
mix t1 t2 t1 t2 mix t
across the full p domain considered here, nonuniform N is the event-wise number of accepted particles, N¯ is
t j
RR
bin sizes on p were used. This was done by noting that the mean of N in the centrality bin, ǫ is the number of
t j
thechargedhadronp distribution,dN/p dp ,forAu-Au events,j theeventindex,pˆ isthemeanoftheensemble-
t t t t
collisions at √s = 130 GeV is approximately expo- average p distribution (all accepted particles from all
NN t
nential for 0.15 p 2 GeV/c [18] and by dividing events in a centrality bin), and i,i′ are particle indices.
t
≤ ≤
therunningintegralofthatexponentialdistributioninto Eq. (2) relates nonstatistical p fluctuations at the ac-
t
h i
equal bin sizes. This procedure provides a convenient ceptancescaletotheweightedintegralofρ ρ ,the
sib mix
−
mapping from p to function X(p ) 1 exp (m latter difference being related to the two-particle num-
t t t
≡ − {− −
m )/0.4 GeV where 0 X 1, m = p2+m2, and ber correlation density. In the present analysis we mea-
0 } ≤ ≤ t t 0
m0 (here assumed to be the pion mass mπ) is a map- surenormalizedpair-ratiodistributionsrˆ[X(pt1),X(pt2)]
p
ping parameter from coordinate pt to X [23]. Equal exhibiting two-particle number correlations on pt which
bin sizes in X therefore have approximately the same correspond to excess pt fluctuations.
h i
number of sibling pairs. For this analysis 25 equal
width bins on X from X(p = 0.15 GeV/c) = 0.15 to
t
X(p =2.0 GeV/c)=0.99 were used [24]. B. Corrections and centrality
t
Normalized pair-number ratios were formed from sub-
sets of events with similar centrality (multiplicities dif- Corrections were applied to ratio rˆ for two-particle
fer by 100, except 50 for the most-central event reconstruction inefficiencies due to overlapping space
≤ ≤
class) and primary-vertex location (within 7.5 cm along points in the TPC (two trajectories merged into one
the beam axis) and combined as weighted (by sibling reconstructed track) and intersecting trajectories which
pair number) averages within each centrality class. The cross paths within the TPC and are reconstructed as
normalized pair-number ratios for each charge-signwere more than two tracks. These corrections were imple-
combined to form like-sign (LS: ++, ) and unlike- mentedusingtwo-trackproximitycuts[28]atvariousra-
−−
sign (US: + , +) quantities. The final correlations re- dialpositionsintheTPCinboththe longitudinal(drift)
− −
portedherewereaveragedoverallfourcharge-signquan- and transverse directions (approximately along the pad
5
rows). The track pair cuts were applied to both ρ
sib 1
sacnadleρmcoixrrealsatiinonHBstTrucatnuarleyssedsu[e13t].o Sqmuaanltl-ummominetnetrufemr-- r ˆ - 10.002 r ˆ - 00..000012
0.001
ence, Coulomb and strong final-state interactions [13]
0
were suppressed by eliminating sibling and mixed-event 0
track pairs (∼3% of total pairs) with |η1 − η2| < 0.3, -0.001 -0.001
φ φ < π/6 (azimuth), p p < 0.15 GeV/c,
1 2 t1 t2
| − | | − |
if pt < 0.8 GeV/c for either particle. The small- 0.8 0.8
smpproaomcmeeinnaetluonnmtg-isntchatelheepct1olorw=reelrap-ttpi2tonddioa(mSgSoaCnina)losatfnrudthctewuer2reDes sa(hrpoetw1m,npott2so)t X0(p.t260).40.20.01.52 (0a.4) 00p..56t (G0.e18V.0/c2.)0 X0(p.t260).40.2 0.2 (0b.4) 0.6 X0(.p8t1)
be similar in amplitude and location to simulations [29]
1 1
which accountfor quantum interference correlationsand r ˆ - 0.004 r ˆ - 0.015
Coulombfinal-stateinteractioneffectsusingpairweights 0.01
0.002
determinedbyHBTanalysesforthesedata[13]. Thepre-
0.005
ceding cuts were optimized [19] to eliminate SSC struc-
0
ture without affecting the large-momentum-scale corre- 0
lation (LSC) structure which is of primary interest here.
The track-pair cuts generally have small effects on the 0.8 0.8
LcuStCs;aurnecderistcauinstsieedswinhiScehc.reIsIuIlCtfraonmd aapreplniceagtliiogniboleftchoemse- 0.60.40.2 0.2 0.4 0.6 X0(.p8t1) 0.60.40.2 0.2 0.4 0.6 X0(.p8t1)
pared to the large momentum scale structures studied X(p (c) X(p (d)
here. t2) t2)
Four centrality classes labeled (a) - (d) for central to FIG. 1: Symmetrized pair-density net ratios
peripheralweredefinedbycutsonTPCtrackmultiplicity rˆ[X(pt1),X(pt2)] 1 for all nonidentified charged pri-
−
N within the acceptance by (d) 0.03<N/N 0.21,(c) mary particles for (a) most-central, (b) mid-central, (c)
0
0.21 < N/N 0.56, (b) 0.56 < N/N 0.7≤9 and (a) mid-peripheral, and (d) peripheral Au-Au collision events at
0 0
N/N >0.79,c≤orrespondingrespectively≤toapproximate √sNN = 130 GeV/c. Note the scale change for panels (c)
0
and (d) and auxiliary p scale in units GeV/c in panel (a).
fraction of total cross section ranges 40-70%, 17-40%, 5- t
SSC were removed using track pair cuts (see text). Errors
17%and0-5%. N isthe end-point[30]oftheminimum-
0 are discussed in Sec. IIIC.
bias multiplicity distribution.
Thecentralitydependenceofquantityrˆ 1isshownin
−
Fig. 1 as perspective views for the four centrality classes
used here. This correlationmeasure represents the num-
berofcorrelatedparticlepairsperfinal-statepairineach
2D bin, andtherefore contains a dilution factor 1/N¯ rel-
An upper limit estimate for resonance contributions
ative to the LSC measure presented in [21], N¯(rˆ 1)
was obtained using Monte Carlo simulations [29] assum-
−
whose amplitudes are of order one. The structures in
ing 70% of the primary charged particle production is
Fig. 1 are therefore numerically a few permil for cen-
from resonance decays. The correlations were simulated
tral Au-Au collisions but are highly significant statisti- bypopulatingtheeventswithasufficientnumberofρ0,ω
cally as seen by comparing to the statistical errors. The
two-bodydecaystoaccountfor70%oftheobservedmul-
dominant features in Fig. 1 are 1) a large-momentum-
tiplicity. These two-body decay processes produced a
scale correlation ‘saddle’ structure with positive curva-
small saddle-shape correlation with curvature opposite
ture along the X(p ) X(p ) + X(p ) sum direction
t Σ t1 t2 to the data and amplitude at the corners approximately
≡
from [X(pt1),X(pt2)] (0,0) to (1,1) and a correspond- 0.0002 for the most-central data, increasing as 1/N¯ for
ing negative curvature along the X(p ) X(p )
t ∆ t1 the remaining centrality bins. The saddle-shape struc-
≡ −
X(p )difference directionfrom[X(p ),X(p )] (0,1)to
t2 t1 t2 tures in Fig. 1 cannot be explained with resonance de-
(1,0), and 2) a narrow peak structure at large X(p )
t cays.
(p > 0.6 GeV/c). With increasing centrality the neg-
t
ativecurvature ofthe LSC saddleshape alongthe differ-
ence variable increases in magnitude, the positive curva- ThesameanalysisappliedtoPb-Pbcollisionsin1.1<
turealongthesumvariabledecreases,andthemagnitude y < 2.6 at the CERN SPS did not reveal any sta-
cm
of the peak at large X(p ) also decreases. Without the tistically significant CI correlations [31] when SSC (see
t
SSC cuts a relatively small peaked structure with am- Sec. IIIB) were removed with pair cuts. The analysis in
plitude of order 0.004 (peripheral) to 0.0005 (central) [32] of proton + proton and various nucleus + nucleus
is present for X(p ) < 0.3 (p < 0.25 GeV/c) which collision data from the CERN SPS for 1.1 < y < 2.6
t t cm
weakensin amplitude but visibly persists to X(p )<0.6 withoutthosepaircuts revealedSSC peaksatlowX(p )
t t
(p <0.5 GeV/c). along the X(p ) direction.
t t Σ
6
C. Error analysis p acceptance cut and also by the pair cuts described in
t
Sec. IIIB and also makes negligible contribution to the
systematic error.
Per-bin statistical errors for rˆ 1 in Fig. 1 range
−
from 6-9% of the maximum correlation amplitude for
∼
each centrality [typically 0.00015, 0.00011, 0.00035 and
0.001 for centralities (a)-(d) respectively] and are ap- IV. MODELING ONE- AND TWO-PARTICLE
proximately uniform, by design, over the 2D domain on DISTRIBUTIONS ON pt
X(p ). Statistical errors for N¯(rˆ 1) ( 0.1 - 0.15) are
t
− ∼
less dependent on centrality. Two features dominate the data in Fig. 1: 1) a large-
Systematic errors were estimated as in [6, 21] and momentum scale saddle shape and 2) a peak at large
are dominated by the 7% non-primary background con- X(pt1) and X(pt2). In this section results from Monte
tamination [18] whose correlation with primary parti- Carlo collision models are analyzed in order to gain in-
cles is unknown. The upper limit on the systematic sightintothedynamicalorigin(s)ofthesetwocorrelation
error from this source was estimated by assuming the structuresinthe data. Basedonthisstudy ananalytical
number of correlated pairs associated with background- function is obtained which accurately describes the sad-
primary pairs of particles could range from zero up to dle shape and in Sec. V this function is used to fit the
the amount which would occur among 7% of the pri- 2D correlation data.
mary particles and the remaining primaries. This con- The high energy nuclear collision model hijing [9],
servative assumption produces an overall 7% uncer- which includes longitudinal color-string fragmentation
±
tainty relative to the correlation amplitudes in Fig. 1 and perturbative quantum chromodynamics (pQCD)
throughoutthedomainforX(p )>0.4. Thiserrorin- based jet production and jet quenching, exhibits signifi-
t1,2
creasesto 16%atlowerX(p )wherethecontamination cantcorrelationstructureathigherp [X(p )+X(p )>
t t t1 t2
±
fraction is larger and is about 12% in the off-diagonal 1.6] as shown in the left-hand panel of Fig. 2 for cen-
±
corners of the [X(p ),X(p )] domain. Multiplicative tral Au-Au collisions. The predictions, which include jet
t1 t2
factors for quantity rˆ 1 which correct for the non- production with jet quenching (default parameters) are
−
primary background contamination range from 1.0, as- qualitatively different from the data in Fig. 1, failing to
suming background-primaryparticle pairs arecorrelated produce anysaddle-shape,but suggestthe type ofcorre-
and increase both nˆ and nˆ by 2 7% = 14%, lation structure produced by jets. The general structure
ab,sib ab,mix
to 1.14if background-primaryparticle pairs×areuncorre- of the hijing predictions suggests that the peaks in the
lated but the non-primary background contributes 14% data at higher X(p ) are at least partly due to initial-
t
to nˆ . Multiplication of the data in Fig. 1 by aver- state partonic scattering and fragmentation. Other the-
ab,mix
age factor 1.07 provides an estimate of the background oretical models which combine initial-state parton scat-
corrected correlation amplitudes. tering,energyloss,dissipation,rescatteringandrecombi-
Additional sources of systematic error were evalu- nation [11, 33] may eventually explain these correlation
ated. Uncertainty in the two-track inefficency correc- data, but relevant predictions are not available at this
tions have modest effects along the X(p ) = X(p ) di- time.
t1 t2
agonal (< 2%) and are negligible elsewhere. Tracking
anomaliescausedwhen particletrajectoriesintersectthe 1 1
TPC high-voltage central membrane significantly affect r ˆ - 0.004 r ˆ - 0.002
theX(p )<0.2domaincorrespondingtothesinglebin 0.001
t1,2 0.002
atlowestX(p ),andthe diagonalbins forX(p )<0.4
t t1,2 0
by 20%. Final multiplicative correction estimates (not
0
applied in Fig. 1) and total systematic errors for rˆ 1 -0.001
−
variedrespectivelyfrom1.07and 7%forX(p )>0.4
t1,2
±
up to 1.16 and 16-20%for X(p )<0.4 and 1.12 and 0.8 0.8
±(11,02)%]. in the o±ff-diagonal cornetr1s,2[i.e., near (0,1) and 0.60.40.2 0.2 0.4 0.6 X0(.p8t1) 0.60.40.2 0.2 0.4 0.6 X0(.p8t1)
X( X(
Other potential sources of systematic error were stud- pt2) pt2)
ied and determined to have negligible effects including
primaryvertexpositionuncertaintyperpendiculartothe FIG. 2: Symmetrized pair-density ratio rˆ[X(pt1),X(pt2)]
−
beamdirection,variationoftrackingacceptanceandeffi- 1 for unidentified charged particles and for central Au-Au
ciency withprimaryvertexlocationalongthe axisofthe collisions. Left panel: Default Hijing [9] with jet quenching,
Rightpanel: aMonteCarlomodel[29]whichsimulatesevent-
TPC, TPC drift speed and/or timing-offset fluctuation,
wise global temperaturefluctuations (see text).
sporadicoutagesofTPCread-outelectroniccomponents,
angular resolution, multiplicity and primary vertex po-
sition bin sizes used for producing mixed events, and The saddle-shape correlationspans the entire momen-
charge sign dependence of the tracking efficiency. Con- tumscalestudiedhere,suggestingevent-wisefluctuations
versionelectroncontaminationissuppressedbythelower of global event characteristics (e.g. temperature and/or
7
collective velocity of the bulk medium) as a possible distribution exp[ m /T(η ,ϕ)] = exp[ β(η ,ϕ)m ] as
t z z t
− −
source. If heavy ion collisions at RHIC thermalize then illustrated in the diagram in Fig. 3(a). In general the
an ensemble of collision events would be characterized histogram of sampled T(η ,ϕ) or β(η ,ϕ) values for all
z z
by a distribution of event-wise equilibrium temperatures particlesinalleventsinthe eventensemble,g (β), could
1
reflecting event-to-eventfluctuations in the initial condi- be like the generic peaked distribution in Fig. 3(b) with
tions and time evolution of each colliding system. Based mean β and standard deviation σ . The inclusive m
0 β t
onthisideathetransversemomentumcorrelationsforan distributionisthenobtainedbyconvolutingthermaldis-
ensemble of such events can be predicted using a Monte tribution exp[ β(m m )] with g (β) given by
t 0 1
− −
Carlomodelinwhichchargedparticleproductionisgen-
erated by sampling the inclusive single-particle (pt,η,φ) dN = A ∞dβg (β)e−β(mt−m0) (3)
1
distributionobtainedfromthedata. Atmid-rapiditythe m dm
t t Z0
inclusive distribution on p for 0.15 p 2 GeV/c
t t
≤ ≤ where A is a normalization constant. The global tem-
is well approximated by exp( m /T) exp( βm ) [18]
t t
− ≡ − perature fluctuation model is recoveredwhen T(η ,ϕ) is
where T is an effective temperature [34] or inverse slope z
independent of source coordinate but varies from event-
parameter and β = 1/T. Events were generated by
to-event.
samplingexp( m /T)whereT fluctuatesrandomlyfrom
t
− Inthe MonteCarlomodelevent-wiseT =1/β wasob-
event-to-eventaccordingtoagaussiandistributionabout
tainedbysamplingagaussiandistribution. Itistherefore
mean value T = 1/β ; T was determined by the mea-
0 0 0
reasonable to represent g (β) by a peaked distribution
sured p spectrum. 1
t
which is here assumed to be a gamma distribution [36]
The result of this Monte Carlo model for central Au-
in order to obtain an analytic solution of the integral in
Aucollisionsat130GeVisshownintheright-handpanel
Eq.(3) given by
of Fig. 2 where the mean and standard deviation (gaus-
sian sigma) of the event-wise temperature distribution
dN
are T0 = 200 MeV and σT/T0 = 1.5%. The predicted m dm = A/[1+β0(mt−m0)/nfluct]nfluct, (4)
correlations are not sensitive to T but the overall cor- t t
0
relation amplitude is directly sensitive to σT/T0 which a L´evy distribution [37], where 1/n = σ2/β2 is the
fluct β 0
was adjusted to approximate the overall amplitude of
relative variance of g (β). The finite width of g (β) pro-
1 1
the data in Fig. 1(a). The global temperature fluctu-
duces a net increase in the yield at higher m as illus-
t
ation model accurately describes the saddle-shape. An
trated in Fig. 3(c). We emphasize that any finite-width
analytical function based on this approach is derived in
peakedfunctiong (β)resultsinanm distributionwhich
1 t
the remainderofthis sectionandis usedin the following
decreases less rapidly with increasing m than thermal
t
section (Sec. V) to fit the data. spectrum e−β0mt. The assumption of a gamma distri-
We seek an analytical representation of the LSC bution for g (β) is therefore not essential but is used for
1
saddle-shape structure of the data in Fig. 1 that is mathematicalconvenienceandisjustifiedbythecapabil-
both mathematically compact and physically motivated ityofthem distributioninEq.(4)todescribetheinclu-
t
in order to conveniently characterize the centrality de- sive data. We note however that deviations of the mea-
pendence and to infer thermodynamic properties of the sured m distribution from a thermal spectrum, quanti-
t
medium. The above Monte Carlo results indicate that fiedbyexponentninthe power-lawm distribution[18],
t
a successful representation should involve an averaging can result from transverse expansion [34] in addition to
over the inverse slope parameter. In general the inverse localandevent-to-eventfluctuationsinβ(η ,ϕ)assumed
z
temperature β can vary from event-to-event as well as in deriving Eqs. (3) and (4). Consequently, fitting the
internally within each event, reflecting the possibility of 1/m dN/dm spectra to obtain the power-law exponent
t t
relative “hot spots” and “cold spots” in the final-state n cannot by itself determine the relative variance of the
particle distributions. The number, location in source effective temperature distribution, 1/n , which is re-
fluct
coordinates (e.g., ηz space-time rapidity [34] and ϕ lated to the degree of equilibration.
−
azimuth), amplitude, and angularextent of these per- Similarlythe two-particledistributionon(m ,m )is
− t1 t2
turbations in β may vary for each event. In addition, obtained by convoluting the two-particle thermal distri-
forrealisticcollisionsystemsboththermalandcollective butionexp[ β (m m )]exp[ β (m m )]withthe
1 t1 0 2 t2 0
motions are involved such that parameter β becomes an 2D distribu−tion of pa−irs of inver−se effecti−ve temperature
inverseeffectivetemperature[34]wherefluctuationsinβ parameters (β ,β ), where particles 1 and 2 sample lo-
1 2
could result from fluctuations in the local temperature cal thermal distributions determined by β(η ,ϕ ) and
z1 1
of the flowing medium, the collective flow velocity itself, β(η ,ϕ ), respectively [see Fig. 3(a)]. The distribution
z2 2
or a combination of both effects [35]. Event-wise effec- of (β ,β ) for all pairs of particles used in all events in
1 2
tive temperature is therefore representedby distribution theensembledefinesa2Dhistogramand2Ddistribution,
T(ηz,ϕ) on source coordinates ηz and ϕ, and similarly g2(β1,β2), illustrated in Fig. 3, panels (d)-(f) for three
for β(ηz,ϕ). hypotheticalcases. If the eventensemble distributionon
The momentum ofa particle at η ,ϕ in the finalstage β hasfinitewidth(σ >0),butarepoint-to-pointuncor-
z β
of the collision system is obtained by sampling thermal relatedwithineachevent,theng (β ,β )issymmetricon
2 1 2
8
(a) y (b) g (b) (global temperature fluctuation model) g2(β1,β2) limits
1 toadiagonallinedistributionillustratedinFig.3(e)and
p units) givenby g2(β1,β2)∝g1′(β1)δ(β1−β2), where δ(β1−β2)
2 b ary is a Dirac delta-function. In this case g2 has maximum
1 bitr sb covariance and represents the conventional picture of an
b2 p1 (ar ensemble of equilibrated events with event-wise fluctu-
ations in the global temperature. In general g (β ,β )
2 1 2
0 b b may have an intermediate covariance as illustrated in
z x 0 Fig. 3(f). In this case if g (β ,β ) is expressed as a
2 1 2
product of a gamma distribution on the sum direction,
b
2 (d) βΣ = β1+β2 multiplied by a gaussian on β∆ = β1 β2
g (b ,b ) −
2 1 2 (formathematicalconvenience),thenananalyticexpres-
units) (c) s2iDonLf´eovrytdhiesttrwibou-ptiaornticle distribution results, given by a
(arbitrary sb> 0 b0 F 1+ β0mtΣ −2nΣ 1 β0mt∆ 2 −(n5∆)
dmt s sib ∝ (cid:18) 2nΣ (cid:19) " −(cid:18)2n∆+β0mtΣ(cid:19) #
m t b= 0
dN/ 0 mt- m0 0 b0 b1 aonndsummt∆and≡dmiffte1re−ncmetv2a.riaIbnlveesrmsetΣex≡pomnet1n+tsm1/t2n−Σ 2amnd0
1/n are the relative variances of g (β ,β ) along sum
b b ∆ 2 1 2
2 (e) 2 (f) and difference variables βΣ and β∆ respectively, and
g2 (b1 ,b2 ) g2 (b1 ,b2 ) ∆(1/n)tot ≡ 1/nΣ − 1/n∆ is the relative covariance
of g [38], measuring velocity/temperature correlations.
2
For the examples in panels (d), (e) and (f) of Fig. 3,
b0 b0 1/nΣ = 1/n∆, 1/nΣ > 0 and 1/n∆ = 0, and 1/nΣ >
1/n > 0, respectively. Mixed-event pair distribution
∆
F (p ,p ), a product of one-dimensional L´evy dis-
mix t1 t2
tributions [Eq. (4)], has the form of Eq. (5) but with
n =n =n .
Σ ∆ fluct
0 b b 0 b b Ratio
0 1 0 1
r F /F , (6)
model sib mix
≡
FIG. 3: Diagrams illustrating the temperature fluctuation
referred to as a 2D L´evy saddle, predicts a saddle-shape
model. Panel (a): Source coordinates with two final state
when g (β ,β ) has non-zero covariance and is the ana-
particles sampling local inversetemperatures β1 =β(ηz1,ϕ1) 2 1 2
and β2 = β(ηz2,ϕ2). Panel (b): Distribution g1(β) of sam- lyticalquantitytobe comparedto data. Itcanbe tested
pled β values for all particles in all events of a centrality by comparison to the data in Fig. 1 via chi-square fits.
bin with mean β0 and standard deviation σβ. Panel (c): We emphasize for this 2D case that any peaked function
Thermal model inclusive charged particle yield dN/mtdmt g2(β1,β2)withnonzero covariance resultsina2Dsaddle
at mid-rapidity versus mt −m0 with no temperature fluc- shape distribution for rmodel. The gamma distribution
tuations (σβ = 0, solid line) and with temperature fluctu- timesgaussian2Dmodelforg waschosenformathemat-
2
ations (σ > 0, dashed line). Panel (d): 2D distribution,
β icalconveniencebutitisreasonablegiventheformofthe
g2(β1,β2), of sampled pairs β(ηz1,ϕ1) and β(ηz2,ϕ2) when measured event-wise p distribution. The variance of
there are no point-to-point temperature correlations within h ti
g alongthedifferencedirectionβ measurestheaverage
each source but large temperature variations within each 2 ∆
degreeofequilibrationoftheeventsintheensemble. Rel-
event (non-equilibrium sources). Panel (e): Same as (d) ex-
cept for global temperature fluctuations where each event is ative variance differences ∆(1/n)Σ (1/nΣ 1/nfluct)
≡ −
equilibrated buttheequilibrium temperaturefluctuatesfrom and ∆(1/n)∆ (1/n∆ 1/nfluct) measure the saddle
≡ −
event-to-event. Panel (f): Same as (d) except point-to-point curvaturesofrmodel (andhence the data)alongsumand
temperaturecorrelationsoccurwithineacheventasevidenced difference directions at the origin,and are the quantities
by thepositive covariance of distribution g2(β1,β2). best determined by these fits. Sensitivity to the magni-
tudesoftherelativevariances1/n and1/n isdiscussed
Σ ∆
in the next section.
β vs β (zerocovariance)asshowninFig.3(d). Forun-
1 2
correlatedβ fluctuationsorformixed-eventpairs,g fac-
2
V. ANALYTICAL MODEL FITS
torizesasg (β ,β )=g (β )g (β ),implyingzerocovari-
2 1 2 1 1 1 2
ance. Ontheotherhand,ifeveryeventisthermallyequi-
librated, then each pair of particles from a given event Data in Fig. 1 (excluding peak region X(pt)Σ > 1.6)
samplesthesamevalueofβ whereβ =β . Forthiscase were fitted with r 1+C˜ by varying parameters
1 2 model
−
9
x 10-2 x 10-2
TABLE I: Parameters and fitting errors (only) for 2D veloc-
r ˆ - 1 0.01.52 r ˆ - 1 0.01.52 ity/temperaturefluctuationmodelforeachcentralitybin,(a)
- (d) (central - peripheral as in Fig. 1). Errors (last column)
0.1 0.1
0.05 0.05 represent fitting uncertainties. Systematic errors are 7-12%
0 0 [39]. Mean multiplicities of used particles in the acceptance,
-0.05 -0.05 N¯, are listed for each centrality bin. Quantities (last row)
-0.1 -0.1 are correction factors for contamination and traScking ineffi-
ciency [6].
0X.80(p.t260).40.2 0.2 0.4 0.6 X0(.p8t1) 0X.80(p.t260).40.2 0.2 (0b.4) 0.6 X0(.p8t1) ∆(c1Ce/˜nn×tN)r¯Σa1l0it4y104 1-31(1.d155.).465 -4002.(.6c841)2.190 700(9..71b088).732 9008..(71a351).080 err66o--r12a44(%)
FIG. 4: Left: pair-density net ratio rmodel[X(pt1),X(pt2)] ∆(1/n)∆×104 -8.61 -3.33 -2.53 -2.04 6-3
1 for model fit to mid-central (b) Au-Au collisions. Righ−t: ∆(1/n) × 104 12.2 3.95 2.71 2.16
tot
residuals (data model) for mid-centralcollisions. χ2/Do×F 348 313 475 402
− 286 286 286 286
1.19 1.22 1.25 1.27 8b
S
aRangeoffittingerrorsinpercentfromperipheraltocentral.
nΣ, n∆ and C˜ (offset). Parameters β0 = 5 GeV−1 and bSystematicerror.
m = m were fixed by the (pion dominated) inclusive
0 π
single-particlep spectrumforp <1GeV/c. Thefitsare
t t
insensitivetotheabsolutevalueof1/n ;itsvaluewas temperature fluctuation model adequately describes the
fluct
fixed as follows. Parameter 1/n when fitted to the sin- saddle structure. Residuals from the fit for mid-central
gle particle m spectrum [18], using an analog of Eq. (4) events(b)areshowninFig.5(left panel)projectedonto
t
with nfluct replacedby n, accounts for the deviation be- sum variable X(pt)Σ. Errors are included in the data
tween the measured distribution and e−βmt. In general, symbolsandaresmallerthanthoseinFig.4(rightpanel)
both collective radial expansion velocity [34] and effec- due to bin averaging. Residuals for other centralities are
tive temperature fluctuations contribute to the curva- similar,butdifferinamplitude. Wehypothesizethatthis
ture (decreasing slope) of the m spectrum relative to residualstructureisduetocorrelatedfinal-statehadrons
t
Boltzmann reference e−βmt at increasing mt shown by associated with initial-state semi-hard parton scatter-
the dashedcurve in Fig.3(c). Bothcontributions arein- ing [40].
cludedinparameterninEq.(4),whenfittedtothesingle Centrality dependences of efficiency-corrected model
particle distribution, resulting in an apparent variance, parameters N¯∆(1/n) [41], which determine saddle-
S
1/n,givenbyanincoherentsumofcontributionsfromra- shape correlation amplitudes in Fig. 1, are shown in
dial flow, 1/n , and effective temperature fluctations, Fig. 5 (right panel). The linear trends suggested by
flow
1/n , where 1/n = 1/n +1/n . However, for the solid lines are notable. Multiplication by factor
fluct flow fluct
the effective temperature fluctuationmodel developedin N¯ estimates correlation amplitudes per final state pri-
S
the preceding section only component 1/n is rele- mary particle as discussed below. Centrality measure
fluct
vant to the 2D L´evy saddle fit but it is not accessible ν estimates the mean participant path length as the
because fits to correlation data (rˆ 1) poorly constrain average number of encountered nucleons per partici-
−
absolutequantities1/n and1/n . However,differences pant nucleon in the incident nucleus. For this anal-
Σ ∆
∆(1/n)Σ,∆ arewelldeterminedbythesaddlecurvatures, ysis ν ≡ 5.5(N/N0)1/3 ≃ 5.5(Npart/Npart,max)1/3 ≃
nearly independently of the assumed value of 1/nfluct 2Nbin/Npart,basedonGlauber-modelsimulationswhere
in rmodel. The maximum value for 1/nfluct corresponds Npart (Nbin) is the number of participant nucleons (bi-
to 1/n = 1/13 in the no-flow limit, 1/n = 0, where nary collisions).
flow
n = 13 is obtained from the L´evy distribution fit to the The reasons for multiplying the parameters in Ta-
single particle m spectrum [18]. The minimum value of ble I by and N¯ are the following. Multiplication of
0.0009correspontdsto thatnecessitatedbythe fittedval- (rˆ 1) bSy N¯ yields the density of correlated pairs per
−
ues of ∆(1/n) in Table I in the limit 1/n 0. The final-state particle [21], typically O(1) forallcentralities.
∆ ∆
fitswereinsensitivetovariationsof1/n int→hisrange, N¯(rˆ 1) would be independent of centrality if Au-Au
fluct −
intermediate value 1/n =0.03nearthe center ofthe collisionswerelinearsuperpositionsofp-pcollisions(par-
fluct
allowed range provided stable ∆(1/n) and ∆(1/n) fit ticipantscaling)becausetheamplitudeofthe numerator
Σ ∆
values. Best-fit parameters and χ2/DoF for the saddle of (rˆ 1), which is proportional to the density of corre-
−
fits are listed in Table I. The model function and resid- lated pairs, would scale with participant number, or in
uals for the fit to centrality (b) are shown in Fig. 4. thismodelwithN¯,whilethedenominatorisproportional
Two-dimensionalsaddle-fitresiduals,asinFig.4(right to N¯2. Therefore variation of N¯(rˆ 1) with centrality
−
panel), are approximately constant along directions par- directly displays the effects of those aspects of Au-Au
alleltotheX(p ) =X(p ) X(p )axisforeachvalue collisions which do not follow na¨ıve p-p superposition.
t ∆ t1 t2
−
of X(p ) and are small for X(p ) < 1.5. The L´evy Factor is defined as the ratio of true, primary particle
t Σ t Σ
S
10
model 0.006 (1/n) 00.3.35 - SSNNDD ((11//nn))tDot −2.∆9%(1g/lno)b∆a,lerveesnutl-ttion-gevienntσtβe/mβp0er∼=atuσreT//vTe0loc=ity1fl.4u%ctutao-
-= data 0.004 DSN 00.2.25 SND (1/n)S ttihoinscfraosme1c/ennt∼=ra1l/tnofploewripahnedragllocboalllistieomnsp,erreastpuercet/ivveelloy.ciItny
als 0.002 0.15 fluctuations contribute negligibly to the upward curva-
residu 0 00.0.15 ftourrethoef tvhaeridaNnc/ems tcdomrrtessppoencdtrutmo .1/Tnhfelowma=xim0,umresvualtluinegs
in 1/n = 1/n and σ /β = σ /T = 1/n = 30%,
fluct β 0 ∼ T 0
0.5 1 1.5 XS 01 2 3 4 5 n6 awthuerree/vσeβlo∆ci∼ty σflβu,ctcuoartrieosnpsonwditinhginteoac3h0%evpeloncta,latesmigpneifir--
cantlynon-equilibratedsystem. Thus, localtemperature
FIG.5: Left: Residualsfrom2DL´evysaddlefittomid-central
variation could range between 0 and 30%. One can ask
(b) data in Fig. 1 projected onto sum variable X(pt)Σ = what is the source of the fluctuating effective tempera-
sXad(pdtl1e)-c+urvXa(tputr2e).meaRsiugrhets:[4E1]ffiocniecnecnyt-rcaolritryecνte:d pN¯er∆-p(a1r/tnic)lΣe ture,andislocalsourcevelocityratherthantemperature
(dots), N¯∆(1/n)∆ (triangles) and N¯∆(1S/n)tot (open a more appropriate quantity?
−S S
circles). Data symbols include fitting errors only [39]. Solid Given the correlation peaks at higher pt it is reason-
lines are linear fits. able to offer the hypothesis that the saddle-shape corre-
lation structure in Fig. 1 results from in-medium modi-
fication, specifically momentum dissipation on (p ,p )
t1 t2
yield (i.e., 100% tracking efficiency and no background of a two-particle distribution from fragmenting, semi-
contamination) estimated for these data in Ref. [18] di- hard scattered partons in the initial-stage of the colli-
videdbytheactualmultiplicity usedinthisanalysiscor- sion. Since no selection was made on leading particle or
rected for the 7% background contamination. is high-p “trigger” particle for these data we refer to the
∼ S t
essentiallythe reciprocalofthe charged-particletracking hadrons associated with a semi-hard, initial-state scat-
efficiency, specific for the present analysis. Multiplica- tered parton as a minijet [9, 42]. Minijet production in
tion by factor N¯ of the parameters in Table I there- Au-Au collisions should increase approximately linearly
S
fore estimates the correlation amplitudes per final-state withN [43,44]whilethesubsequentmomentumdissi-
bin
particle for 100% tracking efficiency and no background pation should monotonically increase with greater mini-
contamination, assuming the measured correlations in- jet production. Correlation amplitudes per final state
clude background-primary particle correlations half-way particle(the latterapproximatelyproportionalto N )
part
between the limits described in Sec. IIIC. The uncer- should therefore increase monotonically with mean par-
tainty inextrapolatingto the true primary particleyield ticipant path length ν =N /(N /2), thus providing
∼ bin part
is estimated to be 8%, most of which is due to the 7% a basis for experimental tests of this hypothesis.
systematic uncertainty in the measured charged hadron The linear trends in Fig.5 (rightpanel)thereforesup-
yield [18]. The combined systematic uncertainty for the port,butdo notrequire, aminijet momentumdissipa-
efficiency correctedamplitudes is from 11 14%across tion mechanism for the observed c−orrelations on p . In
± − t
the X(pt1) vs X(pt2) space. Fig. 5 we also observe 1) reduced curvature along the
sum direction and 2) increased curvature along the dif-
ferencedirectionwhichmayrepresentrespectivelytrans-
VI. DISCUSSION portofsemi-hardpartonstructuretolowerptandamore
correlated bulk medium. Given a minijet interpretation
of N¯∆(1/n) , the combined trends 1) and 2) repre-
Correlationsonpthavetwomaincomponents,asaddle senSt strong evtiodtence for increased parton dissipation in
shape and a peak at higher p . By measuring the saddle
t the more central Au-Au collisions. The present results
curvatures we infer the relative covariance of two-point
complement the observed suppression of high-p spectra
distribution g (β ,β ) and hence the average two-point t
2 1 2 (R ) [43, 44] and suppression of large angle trigger-
correlationamplitude of the temperature/velocity struc- AA
particle–associated-particleconditionaldistributionson
ture of the composite particle source. We now consider
∆φ [4, 5] in central Au-Au collisions at RHIC. It is very
possible dynamical origins of that structure.
likely that the lower-p fluctuations and correlations re-
t
Theanalysisofthesaddle-shapeproducesaccuratere-
ported here are, at least in large part, a consequence of
sults for relative variance differences ∆(1/n) = (σ2
Σ βΣ − the processes which lead to the above suppressions at
σβ2)/β02, ∆(1/n)∆ =(σβ2∆ −σβ2)/β02, and the correspond- higher-pt.
ing ∆(1/n)tot = (σβ2Σ −σβ2∆)/β02 for effective tempera- Itisimportanttonotethatthesecorrelationsontrans-
ture fluctuations. The measurements do not constrain verse momentum observed at relatively low p reveal
t
the absolute magnitudes of the individual variances,σβ2Σ nominally ‘soft’ structure in relativistic heavy ion col-
and σ2 . The minimum possible values, consistent lisions which scales with the number of binary collisions
β∆
with the saddle-shape conditions and the single-particle N , whereas a low-p inclusive quantity such as mul-
bin t
m spectra, correspond to σ2 = 0 and 1/n = tiplicity scales with participant number N . Binary-
t β∆ fluct part