Measurements of 1st, 2nd and 3rd azimuthal√
anisotropy in sNN = 200 GeV Cu+Au collisions at
RHIC-PHENIX

Hiroshi NAKAGOMI

December 2016

Measurementsof 1st, 2nd and 3rd azimuthal√
anisotropy in sN N = 200 GeV Cu+Au collisions at
RHIC-PHENIX

Hiroshi NAKAGOMI
Doctoral Program in Physics

Submitted to the Graduate School ofPure and Applied Sciencesin Partial Fulfillment of the Requirements
for the Degree of Doctor of Philosophy in
Scienceat theUniversity of Tsukuba

Abstract
Relativistic high energy heavy ion collision is a unique tool in the laboratory to create a newstate of nuclear matter composed of de-confined quarks and gluons, ”Quark Gluon Plasma (QGP )”, which is predicted to exist under extreme high temperature and/or high density bylattice Quantum Chromodynamics. Azimuthal anisotropies of particle production, defined as
Fourier coeﬃcients vn , in relativistic heavy ion collisions have proven the importance of the initialcondition and the bulk property of the QGP, because the azimuthal anisotropies originate fromthe initial spatial geometry εn and develops during the entire evolution of the expanding system.
In spite of many experimental results and theoretical calculations, there are still uncertainties of
the initial geometry and the viscosity of QGP.
In 2012, Cu+Au collisions, which are the first asymmetric heavy ion collisions at collider
energy, were delivered at the Relativistic Heavy Ion Collider (RHIC) to study the influence ofasymmetric initial condition on azimuthal anisotropy of produced particles. In Cu+Au collisions,the large directed azimuthal anisotropy of produced particles at mid-rapidity η ∼ 0 is observed intransverse plane, where η describes the angle of a particle with respect to the beam axis. Largeabsolute values of η correspond to the angles close to the beam axis. The forward/backward
asymmetry of produced particle multiplicity and azimuthal anisotropy, which have not been
observed in the symmetric collisions, are observed. These observations could be due to thesideward asymmetric shape at the mid-rapidity and the forward/backward asymmetry in numberof participants and eccentricity εn . In this dissertation, the directed, elliptic and triangular flow√
in Cu+Au collisions at sN N = 200GeV have been measured at mid-rapidity (|η| < 0.35) for
charged hadrons, π ± ,K ± , p and p and forward/backward-rapidity (3 < |η| < 3.9) for charged
hadrons via event plane technique by using the PHENIX detectors at the RHIC.
At mid-rapidity region, the elliptic flow v2 and the triangular flow v3 in asymmetric Cu+Au
collisions show similar transverse momentum pT dependence and collision centrality dependence
as seen in symmetric Au+Au and Cu+Cu collisions. The pT integrated v2 in Cu+Au collisions isalways between those in Au+Au and Cu+Cu collisions. On the other hand, the values of Cu+Au
v3 is almost same as the Au+Au results. The system size dependence of v2 is expected from theinitial spatial ellipticity ε2 calculated in Monte Carlo Glauber model which is commonly used inheavy ion field as the initial geometry model. On the other hand, the system size dependenceof v3 is not ordered according to the initial spatial triangularity ε3 in Monte Carlo Glaubermodel. The large directed flow v1 is observed at mid-rapidity and indicates that large numberof high pT particles are emitted toward the hemisphere of Au spectator. By studying Monte
Carlo Glauber simulation and Blast wave model, the largest pressure gradient in Au hemisphereside pushes more particles to higher pT , which leads the larger v1 for high pT particles at midrapidity. For identified hadron vn , mass ordering is observed for all three harmonics at low pTregion and baryon/meson splitting is observed for the v2 and v3 measurements but not for thev1 measurements. At forward/backward rapidity region, the Au-going (−3.9 < η < −3) v2 andv3 in Cu+Au collisions have larger values than the Cu-going (3 < η < 3.9) results. Like the midrapidity v2 measurements, we found that the forward/backward Cu+Au v2 is always between
the Au+Au and Cu+Cu results. The Au-going v3 in Cu+Au collisions shows larger values thanthose in Au+Au collisions.
Since the vn originates from the εn , we studied the influence of the initial condition on vn

by scaling the vn with εn in Monte Carlo Glauber model. For the mid-rapidity vn measurements, the scaled v2 with ε2 in Cu+Au collisions are good agreement with those in Au+Au andCu+Cu collisions, while the diﬀerence of the scaled v3 between Au+Au and Cu+Au collisions isobserved. These results suggest that the Glauber model is not favored for describing the initialgeometry because ε2 and ε3 have to be described simultaneously by one model. Although one
might expect in model calculations that the forward/backward asymmetry of vn is caused by theforward/backward asymmetry of εn , we conclude that the forward/backward asymmetry of vn
arises from the forward/backward asymmetry of initial energy density by testing various diﬀerent
eccentricity assumptions as well as diﬀerent energy density assumptions.
The charged hadron vn is compared to event by event hydrodynamical and the combined
parton cascade and hadron casede model (AMPT) calculations. The hydrodynamical calculations
reproduce vn at mid-rapidity reasonably. Although the AMPT model calculations reproduce midrapidity v2 and v3 well, the sign of mid-rapidity v1 is found to be opposite with respect to theexperimental data. The forward/backward vn measurements are also compared to the predictions
from the hydrodynamical and the AMPT model calculations. Although the hydrodynamical
calculation predicts the larger values of vn than those of the measured vn , the hydrodynamical
calculations reproduce the forward/backward asymmetry of v2 except for peripheral collisions
but underpredict the forward/backward asymmetry of v3 . The AMPT model reproduces themagnitude of measured vn reasonably. However, the AMPT model calculations overpredict theforward/backward asymmetry of v2 and v3 . The longitudinal fluctuation has been naturally
embedded in AMPT model, which is based on the Hijing (pythia pp superpositions), whilesmooth longitudinal profile is assumed in the hydrodynamical model. Therefore this comparisonhas shed light on the importance of longitudinal initial condition and longitudinal dynamics.

Contents
1 Introduction
1.1 Quark Gluon Plasma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2 Relativistic Heavy Ion Collisions . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2.1 Participant Spectator Picture . . . . . . . . . . . . . . . . . . . . . . . . .
1.2.2 Nuclear stopping power . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2.3 Space-Time Evolution of Collisions . . . . . . . . . . . . . . . . . . . . . .
1.3 Major Features of Experimental Observables . . . . . . . . . . . . . . . . . . . .
1.3.1 Energy Density . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.3.2 Particle Production . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.3.3 Azimuthal anisotropy . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Event by event fluctuation . . . . . . . . . . . . . . . . . . . . . . . . . . .
Identified hadron vn (pT ) . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Pseudorapidity dependence of vn . . . . . . . . . . . . . . . . . . . . . . .
1.4 Cu+Au collision and thesis motivation . . . . . . . . . . . . . . . . . . . . . . . .

1
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2 Experimental Apparatus
2.1 Relativist Heavy Ion Collider(RHIC) . . . . . . . . . . . . . . . . . . . . . . . . .
2.2 PHENIX Detectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2.1 Magnet systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
PHENIX Magnet system . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2.2 Global detector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Beam-Beam Counter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Zero Degree Calorimeter(ZDC) . . . . . . . . . . . . . . . . . . . . . . . .
Shower Max Detector(SMD) . . . . . . . . . . . . . . . . . . . . . . . . .
2.2.3 Central arm detector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Drift Chamber . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Pad chamber . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Time of flight . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Time of flight East(TOF.E) . . . . . . . . . . . . . . . . . . . . . . . . . .
Time of flight West . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Electro Magnetic Calorimeter(EMCal) . . . . . . . . . . . . . . . . . . . .
Pb scintillator sampling calorimeter . . . . . . . . . . . . . . . . . . . . .
Pb glass Cherenkov calorimeter . . . . . . . . . . . . . . . . . . . . . . . .
Reaction Plane Detector(RXNP) . . . . . . . . . . . . . . . . . . . . . . .

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I

II

CONTENTS
2.2.4

Data acquisition(DAQ) . . . . . . . . . . . . . . . . . . . . . . . . . . . .

39

3 Analysis
42
3.1 Event selection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
Centrality Determination . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
3.2 Track selection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
3.2.1 Track reconstruction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
3.2.2 Momentum determination . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
3.2.3 Track Quality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
Track Matching . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
3.2.4 Particle Identification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
3.3 Event Plane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
3.3.1 Azimuthal anisotoropy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
3.3.2 Event Plane Determination . . . . . . . . . . . . . . . . . . . . . . . . . . 51
3.3.3 Event Plane Calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
3.3.4 Re-centering Calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
3.3.5 Flattening Calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
3.3.6 Event Plane QA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
3.3.7 Event Plane Resolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
3.4 Measurements of charged hadron vn at mid-rapidity . . . . . . . . . . . . . . . . 58
3.5 Measurements of particle identified vn at mid-rapidity . . . . . . . . . . . . . . . 58
3.6 Measurements of hadron vn at forward/backward-rapidity . . . . . . . . . . . . . 59
3.6.1 Single particle simulation for forward/Backward-rapidity vn measurements 62
3.7 Measurements of charged hadron dNch /dη at forward/backward-rapidity . . . . . 63
3.7.1 Single particle simulation for forward/backward-rapidity charged particle
multiplicity dNch /dη measurements . . . . . . . . . . . . . . . . . . . . . . 63
3.8 Initial spatial anisotropy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
3.8.1 Glauber model Monte Carlo simulation . . . . . . . . . . . . . . . . . . . 65
3.8.2 Initial spatial anisotropy . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
3.9 Systematic Uncertainty . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
3.9.1 Systematic Uncertainties at mid-rapidity vn . . . . . . . . . . . . . . . . . 66
Uncertainty from Event plane determination . . . . . . . . . . . . . . . . 66
Uncertainty from East and West . . . . . . . . . . . . . . . . . . . . . . . 68
Uncertainty from hadron misidentification . . . . . . . . . . . . . . . . . . 69
3.9.2 Systematic Uncertainties of forward/backward vn measurements . . . . . 71
Uncertainty from Event plane determination . . . . . . . . . . . . . . . . 71
Uncertainty from Input pT spectra in Single particle simulation . . . . . . 71
Uncertainty from input η probability distribution in Single particle simulation 72
Uncertainty from input vn (η) shape in Single particle simulation . . . . . 73
Uncertainty from magnitude of vn (pT ) . . . . . . . . . . . . . . . . . . . . 74
Uncertainty from the shape of vn (pT ) . . . . . . . . . . . . . . . . . . . . 75
3.9.3 Systematic Uncertainties of forward/backward dNch /dη measurements . . 78
3.9.4 Summary of systematic uncertainties . . . . . . . . . . . . . . . . . . . . . 78

CONTENTS

III

4 Results
4.1 Charged hadron vn at mid-rapidity . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2 π ± , K ± , p and p vn results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.3 Charged hadron vn at forward/backward rapidity . . . . . . . . . . . . . . . . . .
4.4 Charged hadron dN/dη . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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5 Discussions
5.1 Interpretation of sign of v1 (pT ) . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.1.1 Approach with Monte Carlo Glauber model . . . . . . . . . . . . . . . . .
5.1.2 Approach with Blast wave model . . . . . . . . . . . . . . . . . . . . . . .
5.2 Interpretation of forward/backward asymmetry of dNch /dη with Monte Carlo
Glauber model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.3 Eccentricity scaling of vn . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.3.1 Mid-rapidity vn /εn . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.3.2 Forward/Backward-rapidity vn /εn . . . . . . . . . . . . . . . . . . . . . .
5.4 Theory comparison . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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6 Conclusion

125

A intrinsic triangularity
127
A.1 v3 (Ψ1 ) at mid-rapidity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
B Initial geometry model131
B.1 Initial geometry model dependence . . . . . . . . . . . . . . . . . . . . . . . . . . 131
C Eccentricity and Npart scaling
136
C.1 Weighted eccentricity and Npart scaling of forward/backward vn . . . . . . . . . . 136

List of Figures
1.1
1.2

Summary of measurement of αs as a function of momentum transfer Q [4] . . . .
2(2 + 1) flavor lattice QCD prediction of the potential energy between a pair ofquark and anti-quark as a function of the distance[5] . . . . . . . . . . . . . . . .
2
1.3 (2 + 1) flavor lattice QCD prediction on pressure, energy density and entropy
density normalized by T14 as a function of temperature. The dark tree lines are theresults of hadron resonance gas model. The ideal gas limit for the energy density
is shown as a horizontal line at 95π 2 /60[6] . . . . . . . . . . . . . . . . . . . . . .
3
1.4 Schematic view of the colliding nuclei before(left) and after(right) collision. . . .
6
1.5 Net proton dN/dη distribution at 5(AGS), 17(SPS) and 200(RHIC) GeV. . . . .
7
1.6 A sketch for the space time history of a relativistic heavy ion collision in the beam
direction z and the time t . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9
1.7 Geometry for the initial state of a central collision in nucleus-nucleus collisions[8]. 10
1.8 Bjorken energy density as a function of numeber of participants at mid-rapidity
in Au+Au collisions observed by PHENIX[9] . . . . . . . . . . . . . . . . . . . . 10
1.9 Charged hadron pT spectrum for diﬀerent multiplicity class in Au+Au collisions√
at sN N = 200GeV [16] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
1.10 π ± , K ± , p and p transverse mass spectra at mid-rapidity for diﬀerent multiplicity√
class in Au+Au collisions at sN N = 200GeV [18] . . . . . . . . . . . . . . . . . 12
1.11 Inverse slope parameters for π, K and p mT distributions in Pb+Pb 17.2 GeV,
S+S 19.4 GeV and p+p 23 GeV collisions[17] . . . . . . . . . . . . . . . . . . . . 13
1.12 Inverse slope parameters for diﬀerent multiplicity class in Au+Au 200 GeV collisions[18] 13√
1.13 Charged particle dN/dη in central Au+Au collisions at sN N = 200, 130, 62.4
and 19.6 GeV[13] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
1.14 Overlap region in beam direction and transverse plane . . . . . . . . . . . . . . . 14
1.15 Overlap region in transverse plane . . . . . . . . . . . . . . . . . . . . . . . . . . 15
1.16 The proper time dependence of spatial eccentricity ϵx and momentum eecentricity
ϵp in Au+Au collisions for two diﬀerent sets of EOS [19] . . . . . . . . . . . . . . 16
1.17 parton v2 as a function of proper time calculated by parton transport model for√several parton cross sections in Au+Au collisions at sN N = 200GeV [20] . . . . 16
1.18 Sketches of heavy ion collisions for directed, ellitpic and triangular flow . . . . . 17
1.19 Transverse momentum dependence of charged hadron vn at mid-rapidity for different multiplicity class in Au+Au 200GeV measured by the PHENIX collaboration 18
1.20 Multiplicity dependence of charged hadron vn at mid-rapidity for two transverse
momentum intervals in Au+Au 200GeV measured by the PHENIX collaboration
19
IV

LIST OF FIGURES
1.21 Charge combined pion, kaon and proton vn at mid-rapidity in Au+Au collisions√
at sN N = 200GeV . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
√1.22 Pseudo-rapidity dependence of charge hadron v2 in Au+Au collisions at sN N =200GeV in comparison to the hydrodynamic calculations. Left:Glauber Monte
Carlo model(nucleon base) initial condition. Right:KLN model(gluon base) initialcondition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
√1.23 Pseudo-rapidity dependence of charge hadron vn in Pb+Pb collisions at sN N =200GeV . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
√1.24 Pseudo-rapidity dependence of charge hadron vn in Au+Au collisions at sN N =200GeV . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.25 Cu+Au collisions in longitudinal direction(a) and transverse direction(b) . . . . .
2.1
2.2
2.3

RHIC accelerator complex . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Schematic of procedure of Au ion injection. . . . . . . . . . . . . . . . . . . . . .
Schematic view of the PHENIX subsystems. Top panel:Beam View, Bottom
panel:Side View . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.4 Schematic view of Central and Muon magnets. . . . . . . . . . . . . . . . . . . .
2.5 Left:One BBC counter composed of 64 elements. Right:Single BBC element composed of PMT equipped with quartz . . . . . . . . . . . . . . . . . . . . . . . . .
2.6 Schematic view of the colliding nuclei before(left) and after(right) collision. . . .
2.7 Schematic view of the colliding nuclei before(left) and after(right) collision. . . .
2.8 Schematic view of the colliding nuclei before(left) and after(right) collision. . . .
2.9 Schematic view of the colliding nuclei before(left) and after(right) collision. . . .
2.10 A plane cutting through by a Pad chamber . . . . . . . . . . . . . . . . . . . . .
2.11 Left:The pad and pixel geometry. Right:A cell made of three pixels is at center ofthis picture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.12 TOF.E installed at PHENIX Central Arm East Arm . . . . . . . . . . . . . . . .
2.13 TOF.E each panel composed of scintilator slats and PMTs . . . . . . . . . . . . .
2.14 Schematic view of MRPC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.15 Schematic view of each Pb-Schintilator tower . . . . . . . . . . . . . . . . . . . .
2.16 Schematic view of each Lead Glass tower . . . . . . . . . . . . . . . . . . . . . .
2.17 Schematic view of Reaction plane detector . . . . . . . . . . . . . . . . . . . . . .
2.18 Schematic design of PHENIX DAQ system . . . . . . . . . . . . . . . . . . . . .
3.1

3.2
3.3
3.4

A schematic of the correlation between measured charged particles and Npart andb calculated by Glauber Monte Carlo. The plotted distribution is not actual
measurements.[7] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
√The total bbc charge distribution in Cu+Au collisioins at sN N = 200GeV fromcombined South and North Bbc detectors . . . . . . . . . . . . . . . . . . . . . .
A cartoon of the ϕ and α for the DC track reconstruction. The dashed lines
represent DC West arm. The small circles are DC hits along the track . . . . . .
A cartoon of the track in r and z plane. The red line corresponds to the reference
radius of the DC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

V

21

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VI

LIST OF FIGURES
3.5

Left figure:Simulated hit informations in a central Au+Au collisions with HIJING
for a small region of the DC. Right figure:The Hough transform feature space(ϕ-αplane) for this small DC region. The peaks corresponds to tracks. . . . . . . . . .
3.6 The squared mass distributions as functions of pT . In each plot, the lowest horizontal band corresponds to pions, the second bad are kaons and the third andforth bands are protons and deuterons. Top Left: Positive and TOF.E tracks.
Bottom Left:Negative and TOF.E tracks. Top Right:Positive and TOF.W tracks.
Bottom Right:Negative and TOF.W tracks. . . . . . . . . . . . . . . . . . . . . .
3.7 Squared mass distribution for medium pT bin and positive particles in TOF.W.
The fitted functions are for pions, kaons and protons. . . . . . . . . . . . . . . . .
3.8 The mean values of squared mass distributions for pT bins and positive particles
in TOF.W. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.9 The width of squared mass distributions for pT bins and positive particles in
TOF.W. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.10 The squared mass distributions as functions of pT after the PID cuts(2σ from thepeaks and 2σ veto cuts from their adjacent species peaks). In each plot, the lowest
horizontal band corresponds to pions, the second bad are kaons and the third andforth bands are protons and deuterons. Top Left: Positive and TOF.E tracks.
Bottom Left:Negative and TOF.E tracks. Top Right:Positive and TOF.W tracks.
Bottom Right:Negative and TOF.W tracks. . . . . . . . . . . . . . . . . . . . . .
3.11 The Qvector distribution determined the combined South and North Bbc detectors
before and after the Re-centring correction. Left:Raw Qvector. Right:Corrected
Qvector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.12 The 2nd harmonic event plane determined by the combined South and North
Bbc detectors for the no correction, only the Re-centering correction and the Re√
centering and the Flattening corrections in Cu+Au collisions at sN N = 200GeV.
3.13 The 2nd order event plane resolutions of South and North side of Bbc and Cnt asfunctions of centrality in CuCu collisions at 200GeV. The resolutions are determined by BbcS-Cnt-BbcN 3-subevent combination. . . . . . . . . . . . . . . . .
3.14 The 2nd and 3rd order event plane resolutions of the South and North side of Bbc,
the South and North side of Rxn and the Cnt as functions of centrality in Au+Au
collisions at 200GeV. The event plane resolution of the Rxn is determined from
2-subevent method. The others are determined from BbcS-Cnt-BbcN 3-subevent
combination. Left panel: 2nd order event plane. Righ panel: 3rd order event plane.
3.15 The 1st , 2nd and 3rd order event plane resolutions as functions of centrality. Leftpanel:The 1s event plane for the South side of Smd determined from 3-subevent
combinations, SmdS-BbcS-SmdN and SmdS-BbcN-SmdN. Middle panel:The 2ndorder event plane for the South, North and combined South and North Bbc and
the Cnt. Right panel:3rd order event plane for the Bbc and Cnt. The 2nd and3rd order event plane resolutions for the Bbc and the Cnt are determined from
BbcS-Cnt-BbcN 3-subevent combination. . . . . . . . . . . . . . . . . . . . . . .
3.16 Charged hadron v1 as a function of pT in Cu+Au collisions for diﬀerent centrality
bins . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

44

47
48
48
48

49

53

54

55

56

57
58

LIST OF FIGURES
3.17 Charged hadron v2 as a function of pT in Cu+Au collisions for diﬀerent centrality
bins . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.18 Charged hadron v3 as a function of pT in Cu+Au collisions for diﬀerent centrality
bins . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.19 PID v2 as a function of pT in Cu+Au collisions . . . . . . . . . . . . . . . . . . .
3.20 PID v3 as a function of pT in Cu+Au collisions for diﬀerent centrality bins . . .
3.21 PID v1 as a function of pT in Cu+Au collisions . . . . . . . . . . . . . . . . . . .
3.22 Charged hadron v2 as a function of η in Cu+Au collisions for diﬀerent centrality
bins . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.23 Charged hadron v3 as a function of η in Cu+Au collisions for diﬀerent centrality
bins . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.24 Bbc charge sum as a function of centrality in Au+Au, Cu+Au and Cu+Cu collisions. In Cu+Au collisions, BbcS and BbcN indicate Au-going side and Cu-goingside respectively . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.25 Left:Impact parameter distribution in Monte Carlo Glauber model. Right:Npart
distribution estimated from Monte Carlo Glauber model. . . . . . . . . . . . . .
3.26 2nd and 3rd order eccentricity ε2 in Monte Carlo Glauber model as a function ofcentrality for Au+Au, Cu+Au and Cu+Cu collisions . . . . . . . . . . . . . . . .
3.27 v1 as a function of pT with respect to SMD S for diﬀerent centrality bins . . . . .
3.28 v2 as a function of pT with respect to BBC S, BBC N and BBC S and N combined
event plane for diﬀerent centrality bins . . . . . . . . . . . . . . . . . . . . . . . .
3.29 v3 as a function of pT with respect to BBC S, BBC N and BBC S and N combined
event plane for diﬀerent centrality bins . . . . . . . . . . . . . . . . . . . . . . . .
3.30 v1 as a function of pT with respect to SMD S for diﬀerent centrality bins . . . . .
3.31 v2 as a function of pT with respect to BBC S and N combined event plane fordiﬀerent track matching cuts and centrality bins . . . . . . . . . . . . . . . . . .
3.32 v3 as a function of pT with respect to BBC S and N combined event plane fordiﬀerent track matching cuts and centrality bins . . . . . . . . . . . . . . . . . .
3.33 v1 as a function of pT with East, West and East and West combined central arms
for diﬀerent centrality bins . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.34 v2 as a function of pT with East, West and East and West combined central arms
for diﬀerent centrality bins . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.35 v3 as a function of pT with East, West and East and West combined central arms
for diﬀerent centrality bins . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.36 Integrated pT v2 as a function of η for diﬀerent event plane selections in Au+Au√
collisions at sN N = 200GeV . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.37 Integrated pT v3 as a function of η for diﬀerent event plane selections in Au+Au√
collisions at sN N = 200GeV . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.38 Input pT spectra for three inverse parameters . . . . . . . . . . . . . . . . . . . .
3.39 Input η probability distribution. The blue data points represent central Au+Au
collisions case and the red data points corresponds to peripheral Cu+Cu collisions
case. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.40 Input vn as a function of η for four type slopes . . . . . . . . . . . . . . . . . . .
3.41 Input vn (η) probability disribution . . . . . . . . . . . . . . . . . . . . . . . . . .

VII

59
59
60
60
60
61
61

64
66
67
67
68
68
69
69
69
70
70
70
71
72
72

73
74
75

VIII

LIST OF FIGURES

3.42 Input vn (η) probability disribution . . . . . . . . . . . . . . . . . . . . . . . . . .
3.43 Input vn (η) probability disribution . . . . . . . . . . . . . . . . . . . . . . . . . .
3.44 Input vn (η) probability disribution . . . . . . . . . . . . . . . . . . . . . . . . . .
4.1

76
76
77

Charged hadron v1 at mid-rapidity (|η| < 0.35) as a function of pT in Cu+Au
collisions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
4.2 Charged hadron v2 at mid-rapidity (|η| < 0.35) as a function of pT in Cu+Au
collisions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
4.3 Charged hadron v3 at mid-rapidity (|η| < 0.35) as a function of pT in Cu+Au
collisions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
4.4 Charged hadron v2 at mid-rapidity (|η| < 0.35) as a function of pT in Au+Au,
Cu+Au and Cu+Cu collisions for diﬀerent centrality classes . . . . . . . . . . . . 84
4.5 Charged hadron v3 at mid-rapidity (|η| < 0.35) as a function of pT in Au+Au andCu+Au collisions for diﬀerent centrality classes . . . . . . . . . . . . . . . . . . . 85
4.6 Charged hadron v2 with polynomial fitting functions at mid rapidity in Au+Au,
Cu+Au and Cu+Cu collisions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
4.7 Charged hadron v3 with polynomial fitting functions at mid rapidity in Au+Au
and Cu+Au collisions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
4.8 Charged hadron pT spectra at mid rapidity in Au+Au collisions [16] . . . . . . . 87nofor charged hadron
4.9 Parameters A, p0 and n of fitting function f (pT ) = A pop+pT
pT spectra at mid rapidity in Au+Au collisions [16] . . . . . . . . . . . . . . . . . 88
4.10 pT integrated v2 (0 < pT < 3GeV/c) at mid rapidity Au+Au, Cu+Cu and Cu+Au
collisions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
4.11 PID v2 as a function of pT in Cu+Au collisions . . . . . . . . . . . . . . . . . . . 89
4.12 PID v3 as a function of pT in Cu+Au collisions for diﬀerent centrality bins . . . 90
4.13 PID v1 as a function of pT in Cu+Au collisions . . . . . . . . . . . . . . . . . . . 90
4.14 The v2 measurements for 0 ∼ 40 % centrality class in Au+Au and Cu+Cu compared to the PHOBOS results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
4.15 The v2 measurements for diﬀerent centrality classes in Au+Au and Cu+Cu compared to the PHOBOS results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
4.16 v2 at BBC rapidity in the three collision systems for diﬀerent centrality classes . 92
4.17 Left:pT integrated v2 as a function of Npart at forward/backward rapidity in√
Au+Au, Cu+Au and Cu+Cu collisions at sN N = 200GeV. Right:ε2 in Monte
Carlo Glauber model as a function of Npart in Au+Au, Cu+Au and Cu+Cu collisions 93
4.18 v3 at BBC rapidity in the two collision systems for diﬀerent centrality classes . . 93
4.19 Left:pT integrated v3 as a function of Npart at forward/backward rapidity in√
Au+Au and Cu+Au collisions at sN N = 200GeV. Right:ε3 in Monte Carlo√
Glauber model as a function of Npart in Au+Au and Cu+Au collisions at sN N =200GeV . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
4.20 The measurements of charged particle multiplicity dN/dη at 3 < |η| < 3.9 in√
Au+Au and Cu+Cu collisions at sN N = 200GeV in comparison to PHOBOS
results[12],[53] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
4.21 Charged particle multiplicity dN/dη as a function of η in Au+Au, Cu+Au and√Cu+Cu collisions at sN N = 200GeV . . . . . . . . . . . . . . . . . . . . . . . . 95

LIST OF FIGURES
5.1
5.2

IX

Participant nucleon distribution in Cu+Au collisions for 20-30 % centrality class97
Number of participant distribution and density gradient distribution in x direction in Cu+Au collisions for 20-30% centrality class. Left:Number of participant
distribution. Right:Density gradient distribution . . . . . . . . . . . . . . . . . . 97
5.3 Azimuthal correlation between 1st order participant plane Ψ1,PP and impact parameter Ψimp . Negative sign indicates the highest density gradient is Au nucleus
side. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
5.4 Identified charged hadron v1 at mid-rapidity fitted with the Blast wave model for
10-50 % centrality class in Cu+Au collisions . . . . . . . . . . . . . . . . . . . . . 100
5.5 The Blast wave model parameters for identified charged hadron v1 at mid-rapidity
for 10-50 % centrality class in Cu+Au collisions. Top left: Freeze-out temperature
Tf , Top right: Averaged radial velocity < ρ >, Bottom left:Directed anisotropyof radial velocity ρ1 , Bottom right:Directed anisotropy of spatial density s1 . The
parameters Tf and < ρ > are obtained from the PHENIX Au+Au publication[24]. The anisotropy parameters ρ1 and s1 are extrapolated from the fitting. . . . 101
5.6 Total charged particle multiplicity per participant nucleon in Au+Au collisions at√
sN N = 200, 130, 62.4 and 19.6 GeV measured by PHOBOS [48] . . . . . . . . . 102
5.7 The comparison of the charged particle multiplicity distributions dNch /dη among
Au+Au, Cu+Au(Cu-going) and Cu+Cu collisions collisions . . . . . . . . . . . . 104
5.8 The comparison of the charged particle multiplicity distributions dNch /dη among
Au+Au, Cu+Au(Au-going) and Cu+Cu collisions collisions . . . . . . . . . . . . 104
5.9 The χ2 /N DF distributions for the comparison of dNch /dη among Au+Au, Cu+Au
and Cu+Cu collisions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
5.10 Left:pT integrated v2 as a function of Npart at mid-rapidity in Au+Au, Cu+Au√
and Cu+Cu collisions at sN N = 200GeV. The integrated pT range is from 0 to
3 GeV. Right:ε2 in Monte Carlo Glauber model as a function of Npart in Au+Au,
Cu+Au and Cu+Cu collisions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
5.11 v2 /ε2 at mid-rapidity as a function of Npart in Au+Au, Cu+Au and Cu+Cu collisions107
5.12 Left:pT integrated v3 as a function of Npart at mid-rapidity in Au+Au and Cu+Au√
collisions at sN N = 200GeV. The integrated pT range is from 0 to 3 GeV.
Right:ε3 in Monte Carlo Glauber model as a function of Npart in Au+Au andCu+Au collisions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
5.13 v3 /ε3 as a function of Npart in Au+Au and Cu+Au collisions . . . . . . . . . . . 108
5.14 Forward/backward-rapidity v2 for charged hadrons as a function of dNch /dη measured at forward/backward rapidity(3 < |η| < 3.9) in Au+Au, Cu+Au andCu+Cu collisions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111√
5.15 ε2,A(B) , ε2,A+B and ε2,B(A) as a function of forward/backward dNch /dη at sN N =200GeV in Au+Au, Cu+Au and Cu+Cu collisions. . . . . . . . . . . . . . . . . . 111
5.16 v2 /ε2,A(B) (Left), v2 /ε2,A+B (Middle) v2 /ε2,B(A) (Right) as a function of dN/dη atforward/backward rapidity(3 < |η| < 3.9) in Au+Au, Cu+Au and Cu+Cu colli√
sions at sN N = 200GeV. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
5.17 v2 /ε2 as a function of dNch /dη at measured rapidity region (3 < |dNch /dη| < 3.9)
in Au+Au, Cu+Au(Cu-going, Au-going) and Cu+Cu collisions. ε2 for Cu+Au
collisions is defined as ε2,Au(Cu)−going = wAu(Cu)−going ε2,Au + (1 − wAu(Cu)−going )ε2,Cu 113

X

LIST OF FIGURES
5.18 χ2 /N DF contour distribution for wCu−going vs wCu−going . χ2 /N DF is obtained
from the diﬀerence between the Cu-going and the Au-going side v2 . . . . . . . . . 114
5.19 χ2 /N DF as a function of weight for the Au-going and the Cu-going side. χ2 /N DF
is obtained from the diﬀerencee between Cu+Au and Au+Au collisions . . . . . 114
5.20 Forward/backward-rapidity v3 for charged hadrons as a function of dNch /dη measured at forward/backward rapidity(3 < |η| < 3.9) in Au+Au and Cu+Au collisions.115√
5.21 ε3,A(B) , ε3,A+B and ε3,B(A) as a function of forward/backward dNch /dη at sN N =200GeV in Au+Au and Cu+Au collisions. . . . . . . . . . . . . . . . . . . . . . . 115
5.22 v3 /ε3,A(B) (Left), v3 /ε3,A+B (Middle) v3 /ε3,B(A) (Right) as a function of dN/dη atforward/backward rapidity(3 < |η| < 3.9) in Au+Au, Cu+Au and Cu+Cu colli√
sions at sN N = 200GeV. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
5.23 v3 /ε3 as a function of dNch /dη at measured rapidity region (3 < |dNch /dη| < 3.9)
in Au+Au and Cu+Au(Cu-going, Au-going) collisions. ε3 for Cu+Au collisions isdefined as ε3,Au(Cu)−going = wAu(Cu)−going ε3,Au + (1 − wAu(Cu)−going )ε3,Cu . . . . . 117
5.24 χ2 /N DF contour distribution for wCu−going vs wAu−going . χ2 /N DF is obtained
from the consisitency between the Cu-going and the Au-going side v3 . . . . . . . 118
5.25 χ2 /N DF as a function of weight for the Au-going and the Cu-going side. χ2 /N DF
is obtained from the consistensy between Cu+Au and Au+Au collisions . . . . . 118
5.26 Structure of AMPT model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119√
5.27 Charged hadron v1 (pT ) measured at mid-rapidity in Cu+Au collisions at sN N =200GeV in comparison to theory calculations for 20-30% centrality class. The
theory calculations shown in this figure are viscous hydrodynamic calculation with
η/s = 0.08(Bozek et al[51]), ideal hydrodynamic calculation(Hirano et al[52]) and
AMPT model with σ = 3mb parton cross section[50] . . . . . . . . . . . . . . . . 120√
5.28 Charged hadron v2 (pT ) measured at mid-rapidity in Cu+Au collisions at sN N =200GeV in comparison to theory calculations for diﬀerent % centrality classes.
The theory calculations shown in this figure are the ideal hydrodynamic calculation(Hirano et al[52]) and the AMPT model with σ = 3mb parton cross section[50].121√
5.29 Charged hadron v3 (pT ) measured at mid-rapidity in Cu+Au collisions at sN N =200GeV in comparison to theory calculations for diﬀerent % centrality classes.
The theory calculations shown in this figure are the ideal hydrodynamic calculation(Hirano et al[52]) and the AMPT model with σ = 3mb parton cross section[50].121√
5.30 Charged hadron v2 (pT ) measured at mid-rapidity in Cu+Au collisions at sN N =200GeV in comparison to theory calculation for 0-5% and 20-30%centrality classes.
The theory calculation shown in this figure is the viscous hydrodynamic calculation(Bozek et al[51]) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122√
5.31 Charged hadron v3 (pT ) measured at mid-rapidity in Cu+Au collisions at sN N =200GeV in comparison to theory calculation for 0-5% and 20-30%centrality classes.
The theory calculation shown in this figure is the viscous hydrodynamic calculation(Bozek et al[51]) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122√
5.32 Charged hadron v2 (η) measured in Cu+Au collisions at sN N = 200GeV incomparison to theory calculations for diﬀerent % centrality classes. The theory
calculations shown in this figure are ideal hydrodynamic calculation(Hirano etal[52]) and AMPT model with σ = 3mb parton cross section[50]. . . . . . . . . . 123

LIST OF FIGURES

XI√
5.33 Charged hadron v3 (η) measured in Cu+Au collisions at sN N = 200GeV incomparison to theory calculations for diﬀerent % centrality classes. The theory
calculations shown in this figure are ideal hydrodynamic calculation(Hirano etal[52]) and AMPT model with σ = 3mb parton cross section[50]. . . . . . . . . . 123
5.34 The ratio of forward/backward v2 values as a function of centrality in Cu+Au√
collisions at sN N = 200GeV in comparison to the theory calculations. The
theory calculations shown in this figure are ideal hydrodynamic calculation(Hirano
et al[52]) and AMPT model with σ = 3mb parton cross section[50]. . . . . . . . . 124
5.35 The ratio of forward/backward v3 values as a function of centrality in Cu+Au√
collisions at sN N = 200GeV in comparison to the theory calculations. The
theory calculations shown in this figure are ideal hydrodynamic calculation(Hirano
et al[52]) and AMPT model with σ = 3mb parton cross section[50]. . . . . . . . . 124
A.1 Participant nucleon distributions for the impact parameter range 4 < b < 5 in√
Au+Au and Cu+Au collisions at sN N = 200GeV simulated by Monte Carlo
Glauber model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128
A.2 Participant plane correlations between Ψ3,PP and the impact parameter Ψ1,imp in
Au+Au and Cu+Au collisions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
A.3 v3 (Ψ1,Imp ) at mid-rapidity as a function of pT in Au+Au and Cu+Au collisions√
at sNN = 200GeV calculated from the ideal hydrodynamics and the AMPT. . . 129
A.4 v3 (Ψ1,SMDS ) as a function of pT measured without the resolution correction at√
mid-rapidity (|η| < 0.35) in Au+Au and Cu+Au collisions at sNN = 200GeV . 130
B.1 Initial energy density in transverse plane for nulceon base Monte Carlo Glaubermodel and IPGlasma model. Left:nucleon base Monte Carlo Glauber model.
Right:IPGlasma model [55] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132
B.2 ϵ2 for Au+Au, Cu+Au and Cu+Cu collisions at 200GeV in nucleon base GlauberMonte Carlo, quark base Glauber Monte Carlo and IPGlasma models . . . . . . 133
B.3 ϵ3 for Au+Au and Cu+Au collisions at 200GeV in nucleon base Glauber Monte
Carlo, quark base Glauber Monte Carlo and IPGlasma models . . . . . . . . . . 133
B.4 Scaled v2 for charged hadrons at mid-rapidity with ϵ2 estimated in the three initial√condition models in Au+Au, Cu+Au and Cu+Cu collisions at sN N = 200GeV.
Left:Nucleon base Monte Carlo Glauber model, Middle:Quark base Monte Carlo
Glauber model and Right:IPGlasma model . . . . . . . . . . . . . . . . . . . . . 134
B.5 Scaled v3 for charged hadrons at mid-rapidity with ϵ3 estimated in the three√
initial condition models in Au+Au and Cu+Au collisions at sN N = 200GeV.
Left:Nucleon base Monte Carlo Glauber model, Middle:Quark base Monte Carlo
Glauber model and Right:IPGlasma mode . . . . . . . . . . . . . . . . . . . . . . 134
B.6 Scaled v2 for charged hadrons at forward/backward-rapidity with ϵ2 estimated in
the three initial condition models in Au+Au, Cu+Au and Cu+Cu collisions at√
sN N = 200GeV. Left:Nucleon base Monte Carlo Glauber model, Middle:Quarkbase Monte Carlo Glauber model and Right:IPGlasma mode . . . . . . . . . . . 135

XII

LIST OF FIGURES
B.7 Scaled v3 for charged hadrons at forward/backward-rapidity with ϵ3 estimated in√
the three initial condition models in Au+Au and Cu+Au collisions at sN N =200GeV. Left:Nucleon base Monte Carlo Glauber model, Middle:Quark baseMonte Carlo Glauber model and Right:IPGlasma mode . . . . . . . . . . . . . . 135
C.1 Weighted eccentricity scaling of v2 as a function of weighted Npart for Au-goingside in Cu+Au collisions compared to the Au+Au and Cu+Cu results. . . . . . .
C.2 Weighted eccentricity scaling of v2 as a function of weighted Npart for Cu-goingside in Cu+Au collisions compared to the Au+Au and Cu+Cu results. . . . . . .
C.3 Weighted eccentricity scaling of v3 as a function of weighted Npart for Au-goingside in Cu+Au collisions compared to the Au+Au result. . . . . . . . . . . . . .
C.4 Weighted eccentricity scaling of v3 as a function of weighted Npart for Cu-goingside in Cu+Au collisions compared to the Au+Au result. . . . . . . . . . . . . .

137
138
139
140

List of Tables
1.1

Summary of heavy ion program. Accelerators, Locations of the accelerators, Collision species, Energy and operation years are written. . . . . . . . . . . . . . . .

3.1
3.2
3.3
3.4
3.5
3.6
3.7

Default setting for single particle simulation . . . . . . . . . . . . . . . . . . . .
Nuclear density parameters used in Glauber model . . . . . . . . . . . . . . . . .
Systematic uncertainties in the v1 measurements at mid rapidity . . . . . . . . .
Systematic uncertainties given in percent on the v2 measurements at mid rapidity
Systematic uncertainties given in percent on the v3 measurements at mid rapidity
Systematic uncertainties in the measured v1 for identified particles at mid rapidity
Systematic uncertainties in percent on the measured v2 and v3 for identified particles at mid rapidity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.8 Event Plane systematic uncertainties given in percent on the v2 measurements at
forward/backward rapidity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.9 Event Plane systematic uncertainties given in percent on the v3 measurements at
forward/backward rapidity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.10 Correction factors and Systematic uncertainties given in percent on the simulated
vn (n=2,3) at forward/backward rapidity . . . . . . . . . . . . . . . . . . . . . . .
3.11 Correction factors and Systematic uncertainties given in percent on the simulateddNch /dη at forward/backward rapidity . . . . . . . . . . . . . . . . . . . . . . . .

XIII

4
62
65
78
79
79
79
79
80
81
81
81

Acknowledgements
Fist of all, I would like to express my gratitude to Prof. Y. Miake for giving me an opportunity
to study heavy ion physics and have great experience in this field. He gave me essential andimportant suggestions and advises. His crucial comments led improvement of my research andthis dissertation. I also would like to acknowledge Prof. S. Esumi. My dissertation would neverbeen finished without him. He alway gave me a great deal of comments and advices, discussedwith me and kindly answered my questions. When I talked at PHENIX physics working groupmeeting, he alway attended at the meeting till night. I learned many things from him, theknowledge on the heavy ion physics, experimental technique and analysis of experimental data.
I am thankful to the staﬀ at our laboratory. Prof. T. Chujo, Prof. M. Inaba, Prof. S. Sakai and
Mr. S. Kato for giving me comments, advices and suggestions and supporting me to develop the
comfortable research environment.
I would like to acknowledge my colleagues at our laboratory. Especially, I would like to thankDr. S. Mizuno, Mr. N. Tanaka, Mr. S. Horiuchi, Dr. D. Watanabe, Ms. K. Gunji and Ms. T.
Nakajima. I enjoyed my student life with them. I would like to thank Mr. H. Yokoyama, Mr. R.
Hosokawa, Mr. N. Toshihiro, Mr. R. Aoyama, Mr. L. Joonil, Mr. T. Sugiura, Mr. Y. Fukuda,
Mr. K. Ito, Mr. B. Kim, Mr. R. Koyama, Ms. S. Kudo, Mr. K. Matsunaga, Mr. K. Sato, Mr.
Y. Rebaza, Mr. T. Ichisawa, Mr. H. Kato, Mr. D. Taichi, Mr. Y. Kawamura, Mr. T. Suzuki,
Mr. K. Nakagawa, Mr. T. Nishimatsu and Mr. H. Jeong for their friendship and discussions.
I would like to acknowledge RIKEN Radiation Laboratory members, Dr. H. En’yo, Dr. Y.
Akiba, Dr. I. Nakagawa, Dr. T. Hachiya, Dr. Y. Yamaguchi, Ms. K. Suzuki, Dr. C. Kim, Dr. I.
Yoon, Mr. T. Moon, Ms. S. Han, Ms. M. Kim and the other members for their support, many
useful comments and discussions.
I am very grateful to Prof. M. Shimomura, Dr. M. Kurosawa, Dr. Y. Ikeda, Dr. T. Niida
and Dr. T. Todoroki for their useful comments, discussions and friendship.
I would like to acknowledge the students of Hiroshima University, Mr. T. Hoshino, Mr. K.
Nagashima and Mr. Y. Ueda for their friendship at BNL.
I would express my many thanks to PHENIX collaborator, Prof. J. Nagle, Dr. D. Morrison,
Prof. J. Velkovska, Dr. T. Sakaguchi, Dr. P. Stankus, Prof. R. Seto, Prof. V. Riavov, Dr. J.
Haggaty, Dr. Mickey Chiu, Dr. S. Huang, Dr. B. Schaefer and Dr. Y. Haiwang for many advises
and comments.
Finally, I would like to express my deepest appreciation to my family, Keiichi Nakagomi,
Toshie Nakagomi and Miku Nakagomi for their understanding, continuous support and encouragements.

XIV

Chapter 1

Introduction
1.1

Quark Gluon Plasma

An atom consists of electrons and a nucleus that is composed of protons and neutrons. The
electrons are considered to be point like particles and the protons and neutrons are composedof three fundamental particles, so called quarks. The quarks have the flavor degrees of freedom(up, down, strange, charm, bottom and top) and color degrees of freedom (red, blue and green).
Experimentally, quarks have been never observed in isolation, because they are always combined
to form composite particles ”hadrons” that are white or neutral in terms of color charges. Hadrons
are categorized into two familiar groups, baryons and mesons. Baryons are fermions such as
protons and neutrons and compose of three quarks. On the other hand mesons are bosons, suchas pions and kaons, and are formed by a pair of quark and anti-quark.
The interaction(strong interaction) acting among quarks are called the strong interaction anddescribed by quantum chromodynamics (QCD). QCD is the gauge field theory based on color
charge in analogous to quantum electrodynamics (QED). In QCD (QED) theory, the gluons(photons) as gauge bosons that carry the strong force (electromagnetic force) between quarks(charged particles). Although QCD is similar to QED, QCD has SU(3) algebra and QED has
U(1) algebra. This is because gluons are not neutral in color charge. Gluons interact with eachother as well as with quarks. On the other hand, photons do not interact with each other due
to no electric charge.
QCD has two important characteristics. One is ”Asymptotic freedom” [1, 2] and the otheris ”Color confinement” [3]. In QCD, the asymptotic freedom describes the interaction between
quarks and gluons to become asymptotically weaker at large energy (short distance) and larger
at small energy (large distance). Fig. 1.1[4] shows the (running) coupling constant of QCD as afunction of momentum transfer Q. At the small Q region, the coupling constant becomes larger
and is described in non-pertubative way. On the other hand, the coupling constant become
smaller at the high Q region and is described pertubatively.
As described at the beginning of this section, the quarks that compose a hadron can not be
separated from the hadron. This phenomena is called Color confinement. Due to asymptotic
freedom, the interaction becomes stronger at long distance. Although there is no analytic proofof the confinement, the potential energy between quark and anti-quark pair is calculated in
1

2

CHAPTER 1. INTRODUCTION

lattice QCD. Figure 1.2[5] shows the potential energy as a function of the distance of a quark
and anti-quark pair r. The potential energy V (r) is fitted with
V (r) = V0 −α+ σ0 rr(1.1)

where V0 , α and σ are unknown parameters (free parameters). As a pair of constituent quark andanti-quark are separated from each other, the potential energy increases linearly. If the distance
between them is beyond some critical distance, energetically it is favorable to create a new quark
and anti-quark pair. Then, the original pair is divided into two pair. Thus quarks are always
bound together and are inside hadrons.

Figure 1.1: Summary of measurement of αs as afunction of momentum transfer Q [4]Figure 1.2: (2 + 1) flavor lattice QCD predictionof the potential energy between a pair of quark
and anti-quark as a function of the distance[5]Quark-Gluon-Plasma(QGP) is a new state of nuclear matter that is formed by asymptotically free quarks and gluons. Due to asymptotic freedom, a phase transition from normal nuclear
matter to the QGP should occur at high temperature(high energy). Indeed, lattice QCD theory
predicts the QCD phase transition at high temperature and zero-baryon density by computing
from first principles[6]. Fig. 1.3 shows the temperature dependences of pressure, energy densityand entropy density normalized by T14 calculated by (2 + 1) flavor lattice QCD. These thermodynamic variables are smoothly changed when increasing the temperature, which means thephase transition from hadronic matter to the QGP state is a crossover. The critical temperature
Tc = 154 ± 9 is shown as a yellow vertical band in Figure 1.3. At the critical temperature, thecorresponding energy density is ϵ = 0.18 − 0.5GeV.

1.2. RELATIVISTIC HEAVY ION COLLISIONS

3

Figure 1.3: (2 + 1) flavor lattice QCD prediction on pressure, energy density and entropy density
normalized by T14 as a function of temperature. The dark tree lines are the results of hadronresonance gas model. The ideal gas limit for the energy density is shown as a horizontal line at
95π 2 /60[6]1.2

Relativistic Heavy Ion Collisions

As described in previous chapter, Lattice QCD calculation predicts the existence of new stateof nuclear matter called Quark-Gluon-Plasma(QGP) which is composed of de-confined quarks
and gluons under extremely high and/or dense. Experimentally to create such a hot and/or dense
matter, relativistic heavy ion collisions is considered a unique tool on the earth. By heavy ion
collisions, longitudinal kinetic energies of incoming nuclei are deposited into the region between
the receding nuclei. If the released energy is highly enough, the QGP state can be formed.
Historically, some high energy heavy ion accelerators have been built to create QGP state.
From the middle of 1970’s to the end of 1980’s, BEVALAC accelerator at Lawrence Berkeley
Laboratory in United States of America was operated. In the middle of 1980’s, Alternating Gradient Synchrotron(AGS) at Brookhaven National Laboratory(BNL) in USA and Super Proton
Synchrotron(SPS) at European Organization for Nuclear Research (CERN) in Switzerland wereoperated. In 2000, Relativistic Heavy Ion Collider(RHIC) at BNL began operation. And Large
Hadron Collider(LHC) at CERN started its first run in 2010. Table 1.1 is the summary of theaccelerators, heavy ion collision species, energies and operation years.

4

CHAPTER 1. INTRODUCTION

Table 1.1: Summary of heavy ion program. Accelerators, Locations of the accelerators, Collision
species, Energy and operation years are written.

Accelerators

Location

Species

Energies (GeV)

Year

AGS

BNL

16 O, 28 Si

5.4
4.8
19.4
17.4
130
200
200
200,62.4
200,62.4,22.4
200
200,62.4
200,62.4,39,11.5, 7.7
200,19.6, 27
193
200
200,14.6
200
200
200
200
200,62.4,19.6,39
2760
2760
5020
5500

1986
1992
1986
1994
2000
2001
2003
2004
2005
2007
2008
2010
2011
2012
2012
2014
2014
2015
2015
2016
2016
2010
2011
2013
2015

197 AuSPS

CERN

16 O, 32 S
208 PbRHIC

BNL

197 Au197 AuLHC

CERN

d+197 Au
197 Au63 Cu
197 Aud+197 Au
197 Au197 Au238 U
63 Cu+197 Au
197 AuHe+197 Au
p+197 Au
p+27 Al197 Aud+197 Au
208 Pb208 Pbp+208 Pb
208 Pb

1.2. RELATIVISTIC HEAVY ION COLLISIONS

1.2.1

5

Participant Spectator Picture

In a heavy ion collision, the initial collision geometry play a important role to understand thecollision dynamics. In relativistic heavy ion collisions, nuclei collide at nearly the speed of light.
Because they move at relativistic speed, they are Lorentz-contracted. The longitudinal thicknessof nuclei becomes 2R/γ where R is radius of nuclei and γ is Lorentz factor. γ is expressed as
γ = E/M using colliding energy E and nuclei mass M .
In high energy nucleus-nucleus collisions, ”participant-spectator model” is established andexperimentally observed features are described well using this model. In relativistic heavy ion
collisions, the speed of colliding nuclei is much faster than fermi motion of nucleons. Therefore
nucleons do not approximately move in transverse direction. Fig. 1.4 is a schematic picture of
the colliding nuclei before and after a collision. Then the overlap region can be described using
impact parameter b. Impact parameter b is defined as the distance between the centers of twocolliding nuclei. If b is less than 2R where R is a radius of the nucleus, inelastic collisions occur.
As impact parameter decreases, overlap region increases. If b ≃ 0, almost all particles participate
in a collision. Such a collision is called a central collision. If b ≃ 2R, the overlap region of two
nuclei is small. Such collision is called a peripheral collision. In nucleus-nucleus collisions, thecollision systems consist of two components. One is ”participant”, which is the overlapped region
shown by elliptical region in Fig. 1.4 and the other is ”spectator”, which is the region other than”participant” shown in Fig. 1.4. Glauber model described below can successfully evaluate thenumber of nucleons which participate collisions[7].
In order to describe high energy nuclear reactions, the Glauber model has been employed.
By using the Glauber model, the total reaction cross-sections and the number of nucleons which
participate in collisions, the number of nucleon-nucleon binary collisions can be evaluated. In the
Glauber model, the nucleus-nucleus collision are treated as the multiple nucleon-nucleon collisions
and nucleons are assumed to go through in straight lines without any deflection. In Au+Au√
collisions at sN N = 200GeV, this approximation works well because the nucleon can move 0.12
fm at maximum in transverse plane during the beam crossing time 0.12 fm/c. Compared to theradius of Au nuclear 6.38 fm, this moving distance is relativiely small. In this model, secondary
particle production and possible excitation of nucleons are not included. For the simplest typeof the Glauber model, a nucleon-nucleon collision occurs when the distance of the two nucleonsd is less than,√d ≦ σN N /π
(1.2)
where σN N is the total inelastic nucleon-nucleon cross section. The nucleus A sickness function
TAB (b) with impact parameter b is given by,∫
TA (s) = dzρA (z, s)(1.3)
where ρA (s, z) is the probability per unit volume for the nucleon at (s, z). The product ofTA (s)TB (s − b)d2 s provides the probability per unit area ds2 of target nuclear A and projectile
nuclear B. Then the overlap thickness function of A and B nuclei defined by integrating the
product over all s is given by,∫
TAB (b) = d2 sTA (s)TB (s − b)
(1.4)

6

CHAPTER 1. INTRODUCTION

Spectatorb

Par%cipant

Spectator

Figure 1.4: Schematic view of the colliding nuclei before(left) and after(right) collision.
The number of participant nucleons Npart and the number of binary collisions Ncoll are expressed
as,
∫
∫
2
−σN N TB (s)Npart (b) =ds TA (s)(1 − e) + d2 s(TB (s − b)(1 − e−σN N TA (s) )(1.5)∫
Ncoll (b) =d2 sσN N TA (s)TB (s − b)
(1.6)

1.2.2

Nuclear stopping power

In heavy ion collisions, the energy deposition of colliding two nuclei is a fundamental quantitythat is converted into particle production. For the QGP formation, the colliding nuclei have toloss suﬃcient kinetic energy. Since the baryon number is conserved, the energy deposition could
be estimated by the measured net-baryon rapidity distribution( the baryon rapidity distribution
is subtracted by the anti-baryon rapidity disritibution). The rapidity is defined asy=

1 E + pzln
2 E − pz

(1.7)

where E and px are energy and momentum in beam direction respectively. The energy deposition

1.2. RELATIVISTIC HEAVY ION COLLISIONS

7

Figure 1.5: Net proton dN/dη distribution at 5(AGS), 17(SPS) and 200(RHIC) GeV.

is quantified by the average net-baryon rapidity loss. The average rapidity < y > after thecollision is expressed as
∫ yp
∫ yp
dN
dN
< y >=
y
dy/
dy
(1.8)
dy
dy
0
0
where yp is rapidity of the incoming projectile particles(incident particles) and dN
dy is the numberof net-baryons per rapidity unit. The average rapidity loss is < δy >= yp − < y >. If the incident
particles loss all their energy, δ < y >= 0.
Fig. 1.5 shows the net-proton rapidity distribution (the p rapidity distribution is subtracted√
by the p rapidity distribution) for central Pb+Pb and Au+Au collisions at AGS( sN N = 5GeV),√√SPS( sN N = 17GeV) and RHIC sN N = 200GeV. The distributions have strong beam energy
dependence. At AGS energy, the central peak is seen at mid-rapidity. But at SPS and RHIC
energies, the dip and plateau structures are seen at mid-rapidity and the central peak as seen
at AGS energy is changed into a double hump structure. This energy dependence implies thatthe nuclear collision dynamics is changed from ”stopping” to ”transparent”. Nuclear stoping
power is considered to saturate in these energies, namely the incident particles don’t lose all
their kinetic energy but pass though each other.
The BRAHMS collaboration estimated the rapidity loss in central Au+Au collisions at√
sN N =200 GeV. The rapidity loss is < δy >= 2.0 ± 0.4. The corresponding energy loss isaround 70 GeV per nucleon. The total kinetic energy deposition in central Au+Au collision at√sN N =200 GeV is about 28 TeV (70×197×2).

8

1.2.3

CHAPTER 1. INTRODUCTIONSpace-Time Evolution of Collisions

The time history of the hot and dense matter formed in relativistic heavy ion collisions iscategorized into following stages. Fig. 1.6 is a sketch for the space time history of a relativistic
heavy ion collision in the beam direction z and the time t.
−Parton cascade stage(pre-equilibrium stage) : 0 < τ < τ0
After the collisions, a large amount of energy is deposited in the overlap region. The energy
density is expected to be high. Many free partons are produced because of huge energy densityand frequently parton-parton scattering occurs.
−QGP phase(thermal equilibrium)
After multi parton scattering occurs, local thermal equilibrium is achieved. Once the local thermal equilibrium is achieved, relativistic hydrodynamics could describe the evolution of
the interaction region. In the relativistic hydrodynamics, the equation of motion are given by
conservation laws for the energy-momentum tensor T µν and the i-th conserved charge currents
jiµ (in heavy ion collisions, there are some conserved charges, for example baryon number andstrangeness.)∂µ T µν = 0(1.9)∂jiµ = 0(1.10)

When the fluid achieve the local thermal equilibrium, in the perfect fluid approximation, themomentum-energy tensor and the conserved charge current are given by
T µν
jiµ

= (ϵ + P )uµ uν − P g νµ
= ni uν(1.11)(1.12)

where ϵ and P are the energy and the thermodynamic pressure in the local rest frame of thefluid, g νµ is the Minkowski metric tensor, ni is the charge density and uν is the fluid flow fourvelocity. The perfect fluid approximation reduces the number of unknown variables to 5+n. The
unknown 5+n variables are the energy density, pressure, the charge density for n types and threecomponent of four-velocity vx , vy , vz . Eq. 1.10 contain 4+n equation. If the equation of state
P = P (ϵ, {ni }) is provided, the total unknown variables and equations will be same, then one
can solve these equations with boundary conditions (initial conditions).
−Mixed state between QGP and hadrons
As the medium expands, the temperature becomes low. If the temperature reaches thecritical temperature Tc , the medium starts to hadronize. The medium consists of free partonsand hadrons. If the phase transition is first order transition, this state would exist.
−Chemical Freeze-out and thermal Freeze-out
While the medium hadronize, inelastic scattering among hadrons is kept until the temperature
become below Tch , inelastic scattering is terminated and hadron yield is determined. We callthis temperature Tch ”Chemical Freeze-out”. Finally hadron elastic scattering is also terminatedand hadrons stream out at temperature Tth . This temperature is called ”Thermal freeze-out”.

1.3. MAJOR FEATURES OF EXPERIMENTAL OBSERVABLES

9

Figure 1.6: A sketch for the space time history of a relativistic heavy ion collision in the beam
direction z and the time t

1.3

Major Features of Experimental Observables

1.3.1

Energy Density

Bjorken gave a prescription for the initial energy density estimation in [8]. In Fig. 1.7,
the original diagram in his paper is shown. Once the two pancakes recede from the collision
points, particles are produced and occupy the region between two pancakes. At time t after thecollision, the volume that contains all of the produced particles will be longitudinal thickness dz
with the nuclear-nuclear overlap region A. The number of produced particles in this volume canbe written
dN =

dz dN
τ dβ

(1.13)√
If the produced particles have an average total energy < mT >(mT = p2T + m2 no longitudinalvelocity), the energy density is calculated by the total volume at time τ from the collision and

10

CHAPTER 1. INTRODUCTION

the total energy< ϵ(τ ) > ===dN < mT >dzAdN < mT >dy cτ A1 dET
τ A dy(1.14)

2∈Bjτ [GeV/fm /c]dN
Twhere dE
dy =< mT > dy . This equation is referred to as Bjorken energy density ϵBj . The
PHENIX experiment measured the ϵBj in Au+Au collisions at three collision energies as shownin Fig. 1.8 [9]. In the estimation of ϵBj , the time τ is used 1 fm/c at the thermalization timefrom the collision and the overlap zone A is calculated by the Monte Carlo Glauber model
simulation. As described in Section 1.1, the energy density for the phase transition is expected
to be approximately ϵ = 0.18 − 0.5GeV/fm3 by the Lattice QCD. Except peripheral collisions,the ϵBj are larger than the predicted energy density.

4

2
200 GeV
130 GeV19.6 GeV
0

100

200

300

Np

Figure 1.8: Bjorken energy density as a function of numeber of participants at mid-rapidity
in Au+Au collisions observed by PHENIX[9]Figure 1.7: Geometry for the initial state of acentral collision in nucleus-nucleus collisions[8].

1.3. MAJOR FEATURES OF EXPERIMENTAL OBSERVABLES

1.3.2

11

Particle Production

10

2

Au+Au 200 GeV-2

1/(2 πp TN evt) dN /dp Td η [ (GeV/c) ]√Measurement of single particle transverse momentum pT = p2x + p2y distribution is one of theuseful tools in studying collision dynamics because the transverse motion of produced particles
is generated during the collisions and hence the produced particles carries informations of thecollision dynamics. The transverse momentum spectra have been measured in several collisionsystems and at several energies. The charged hadron transverse spectra at mid-rapidity for√diﬀerent centrality in Au+Au collisions at sN N = 200 GeV are shown in Fig. 1.9 [16]. It is
known that the pT spectra are described by an exponential function at pT ≦ 2 and power law
function at higher pT range. This trend implies the transition of particle production mechanism
from soft production at low pT to hard particle production hight pT . It is also known that the

20-30%
1

10

2min. bias × 5
10-20%
30-40%

0-10%

10

10
10
10
10
10

40-50%-1

50-60%
70-80%

60-70%
80-92%

-2

-3

+

h +h
2

-4

-

-5

-6

0

1

2

3

4

5

6

7

8

9

0

1

2

3

4

p T (GeV/c)

5

6

7

8

9

pT (GeV/c)

Figure 1.9: Charged hadron pT spectrum for diﬀerent multiplicity class in Au+Au collisions at√
sN N = 200GeV [16]√single particle spectra of transverse mass mT =as an exponential function at low pT region.
E

p2T + m20 (m0 is particle mass) are expressedd3 σ 1d2 σ∝ exp(−mT /T )dp3 2πmT dmT dy(1.15)

where the inverse slope parameter T . The inverse slope parameter is called eﬀective temperature
which increases with increasing collision energy and depend on system size. In high energy√
nucleon-nucleon collisions above sN N ≧ 5 GeV, the inverse slope parameters are common forparticle species. This scaling is called mT scaling. In Fig.1.10, the π ± , K ± , p and p transverse√
mass spectra at mid-rapidity for diﬀerent multiplicity class in Au+Au collisions at sN N =200 GeV[18]. The overlaid solid lines are the exponential fitting functions. The fitting ranges
are 0.2 − 1.0GeV for π and 0.1 − 1.0 GeV for K and p. The inverse slope parameters for π, K
and p mT distributions at mid-rapidity for diﬀerent energies and collision species are shown in
Fig. 1.11 and 1.12. In Fig.1.11, the parameters are measured by NA44 collaboration at SPS√√
accellerator and compared among Pb+Pb sN N = 17.2 GeV, S+S sN N = 19.4 GeV and p√+ p sN N = 23 GeV. The parameters in p+p collisions case are almost same for the particles
species as referred to above. Whereas the parameters for the nucleus - nucleus collisions increase

12

CHAPTER 1. INTRODUCTION

π+
+
K
p

2

10
10
1
10
10
10

-1
-2
-3

Positive

Negative(0 - 5% central)(0 - 5% central)-5

Positive

Negative-6(40 - 50% mid-central)(40 - 50% mid-central)Positive

Negative(60 - 92% peripheral)(60 - 92% peripheral)

-4
2

10
10

2

4

(1/2π mT) d N / dmTdy [c / GeV2]

10

πK

p

1
10
10
10
10
10
10

-1
-2
-3
-4

10
1
10
10
10
10
10
10
10

-1
-2
-3
-4
-5
-6
-7

0

0.5

1

1.5

2

2.5 3 3.5 0
2mT - m0 [GeV/c ]

0.5

1

1.5

2

2.5 3 3.5
2mT - m0 [GeV/c ]Figure 1.10: π ± , K ± , p and p transverse mass spectra at mid-rapidity for diﬀerent multiplicity√
class in Au+Au collisions at sN N = 200GeV [18]with increasing particle mass and the size of colliding system. In Fig. 1.12, the parameters
extracted from the fitting functions in Fig.1.10[18] are shown. The parameters at RHIC energy
have particle mass dependence as seen at SPS energy in Fig.1.11. This particle mass dependence
has been considered as the influence of a common radial velocity field directed outward. This
radial flow is caused by the strong interaction among the produced particles and described byhydrodynamics. In the low pT , the inverse slope parameter is written as
1
Teﬀ ≃ Tf + m0 < β 2 >, pT ≤ 2GeV
2

(1.16)

where Tf is the temperature at thermal freeze out, m0 is a particle mass and < β > is theaverage radical velocity. Eq. 1.16 shows the eﬀective temperature becomes higher for the heavier
particles and the larger eﬀective temperature for the higher collision energy and the larger systemsize imply the stronger radial flow.
Bjorken proposed the collision reaction picture based on the parton model of hadrons for
relativistic high energy collision. By the deep inelastic lepton-hadron scattering experiments,
the nucleon is considered to consist of three valence quark(proton:uud,neutron:udd) and thewee partons (gluons and sea quarks). Compared to the valence quarks, most of wee partons
have much smaller momentum fraction of the colliding nucleon. Thus the two incoming nuclei

Inverse Slope T [GeV/c]

1.3. MAJOR FEATURES OF EXPERIMENTAL OBSERVABLES

13

0.6
0.5

Central (0-5%)
Mid-central (40-50%)
Peripheral (60-92%)

0.4
0.3
0.2
0.1
0

π+
0.2

K
0.4

+

0.6

π-

p
0.8

1 0
2

Mass [GeV/c ]

0.2

K
0.4



-

0.6

p
0.8

1
2

Mass [GeV/c ]Figure 1.11: Inverse slope parameters for π, K
and p mT distributions in Pb+Pb 17.2 GeV, Figure 1.12: Inverse slope parameters for difS+S 19.4 GeV and p+p 23 GeV collisions[17] ferent multiplicity class in Au+Au 200 GeV
collisions[18]Figure 1.13: Charged particle dN/dη in central Au+Au collisions at
19.6 GeV[13]√sN N = 200, 130, 62.4 andbefore the collision wear the ”fur coat of wee partons”. When the two nuclei collide, the valencequarks punch through each other because On the other hand, the wee partons scattering takeplace. After head-on collisions, the region between the two nuclei is supposed to achieve thelocal thermal equilibrium within the time scale 1 fm/c though the multiple scattering of thewee patrons. Fig. 1.13 shows the pseudo-rapidity distributions of charged particles for diﬀerent√
multiplicity classes for -5.4< η <5.4 in Au+Au collisions at sN N = 200, 130, 62.4 and 19.6GeV
measured by the PHOBOS collaboration. Since the PHOBOS detectors have large acceptancein η direction, dNdη is measured over the almost all η region. The number of produced particle at
central plateau (mid-rapidity) is about 650. The distributions become larger density and wider
at central plateau with increasing collision energy.

14

CHAPTER 1. INTRODUCTION

Figure 1.14: Overlap region in beam direction and transverse plane

1.3.3

Azimuthal anisotropy

In relativistic heavy ion collisions, measurements of azimuthal anisotropies of particle emission
is a strong tool for investigating initial spatial condition and bulk property of the QGP. The originof this phenomena is considered to be the initial spatial participant geometry. Fig.1.14 shows anucleus - nucleus collision in beam direction and transverse plane. In such a collision, the overlap
region has a rugby ball shape. If the overlap region rapidly reaches local thermal equilibrium
state while keeping the initial geometry, the final particle distribution could be proportional tothe initial spatial geometry as described below. Strengths of these anisotropies are evaluated by
the coeﬃcients of the Fourie expansion of the produced particle distribution in azimuthal planewith respect to event plane Ψn .
∑
dN
∝1+2
vn cos(n[ϕ − Ψn ])dϕ(1.17)

n=1

where n is the order of the harmonics, ϕ denotes the azimuthal angle of produced particle andΨn is the event plane for the nth order. The event plane is the azimuthal angle of nth order
event plane where the particles emission become largest and determined for each order of theharmonics on event by event basis. Mainly measurements of the second harmonic coeﬃcientwhich corresponds to the magnitude of the second harmonic flow so-called ”Elliptic flow” havebeen studied. Elliptic flow is considered to arise from initial elliptical participant shape in nucleus- nucleus overlap zone. If the mean free path of produced particles l is much larger than systemsize R, l ≫ R, the produced particles are emitted radially without any interactions. In this case,the azimuthal distribution of produced particles does not have azimuthal angle dependence. In
the other case, when the mean free path is much smaller than the system size l ≪ R, multi
particle interaction occurs and leads to the thermalization of the medium. After the medium
achieve the thermalization, the medium expands by the pressure gradient. The pressure gradient
toward the minor axis is larger than the major axis due to the elliptic geometry. The azimuthal
dependence of the pressure gradient leads to that the particle production toward the minor axis
is enhanced rather than toward the major axis. Thus the azimuthal distribution of produced
particles has azimuthal angle dependence and its oscillation amplitude corresponds to the elliptic
flow v2 .

1.3. MAJOR FEATURES OF EXPERIMENTAL OBSERVABLES

15

Figure 1.15: Overlap region in transverse plane

The hydrodynamic/transport simulation demonstrate the important property of elliptic flow.
Fig.1.16 shows the spatial anisotropy of the participant geometry εx and the participant momentum εp as a function of the proper time from hydrodynamic simulations for diﬀerent sets of theequation of state in non central Au+Au collisions. The spacial eccentricity decrease and themomentum eccentricity increase with expanding the medium. Therefore the spacial anisotropy
is converted into the anisotropy in momentum space though the hydrodynamic expansion. Since
the momentum anisotropy is created at the beginning and saturate at the first few fm/c, the v2
is a sensitive observable to the early stage of the hydrodynamic evolution.
In Fig.1.17, a parton transport theory based on the Boltzmann equation for undergoingelastic scattering process of gluons only shows parton v2 as a function of proper time in Au+Au√
collisions at sN N = 200 GeV. At the beginning, the multi gluon medium has an rugby ballspacial geometry in transverse plane and thermal parton distribution in momentum space. The
medium with several gluon-gluon scattering cross section expands as the proper time goes by. For
free streaming case(without any parton scattering) v2 are consistent with zero, so that secondary
collisions are necessary to generate azimuthal anisotropy. The v2 for the all three parton crosssections is generated at the early stage of collisions and saturate at around 2 fm/c. The v2 forlarger parton cross section is larger than that for smaller parton cross section. The parton crosssection is proportional to11σ∝ ∝(1.18)
λη
where λ is the mean free path and η is the shear viscosity calculated in the kinetic theory ofgases. The v2 values with the finite parton cross section are smaller than the ideal hydrodynamic
limit (η ∼ 0).

16

CHAPTER 1. INTRODUCTION

Figure 1.16: The proper time dependence of spatial eccentricity ϵx and momentum eecentricity
ϵp in Au+Au collisions for two diﬀerent sets of EOS [19]Figure 1.17: parton v2 as a function of proper time calculated by parton transport model for√several parton cross sections in Au+Au collisions at sN N = 200GeV [20]

1.3. MAJOR FEATURES OF EXPERIMENTAL OBSERVABLES

17

Figure 1.18: Sketches of heavy ion collisions for directed, ellitpic and triangular flow

Event by event fluctuation
Lately significant attention from both of experimentalists and theorists have been attracted forstudying the initial geometry fluctuation. If the colliding nuclei have smooth density distributions, the participant region in the symmetric nuclear-nuclear collisions is always almond shape.
Such a participant geometry does not make odd harmonics at mid-rapidity. However, in actual
fact, the event by event fluctuations of participant nucleons lead to odd harmonic anisotropies
at mid-rapidity. Figure 1.18 shows the cartoons of the initial geometries for directed flow, elliptic flow and triangular flow. The violet small circles denote the participant nucleons and theother small circles are the spectator nucleons. The first order eccentricity implies the centerof mass, the second order eccentricity is an almond shape of the overlapping region with theparticipants fluctuations and the third order eccentricity is the triangular shape caused by theparticipant fluctuations alone. Indeed, sizeable odd harmonics have been observed at RHIC and
LHC [21, 22, 23]. Figure 1.19 shows the pT dependence of charged hadron vn at mid-rapidity
for diﬀerent multiplicity classes in Au+Au 200GeV collisions measured by the PHENIX collaboration. The values of vn increase with transverse momentum for all event classes. As observed
before, the values of the v2 increase from central collision event(left side panel) to peripheral
collision event (right side panel). On the other hand, the v3 has weak multiplicity dependence.
These observed trend in v2 and v3 measurements are expected by the eccentricity εn [14]. In Fig.
1.20, the PHENIX collaboration compared the multiplicity dependence of vn for two transverse
momentum range to several theoretical predictions. The black data points denote the resultsfrom the PHENIX collaboration and the other symbols are the theoretical predictions. The
purpose of this comparison with theoretical models is to constrain the initial conditions and theviscosity (shear viscosity over entropy density ratio η/s of QGP) because these model presented
in this Fig.1.20 employed the diﬀerent initial conditions and the values of the viscosity. The
initial conditions are introduced from Glauber Monte Calro simulation and MC-KLN model. In
the Glauber Monte Carlo simulation, the participant geometry in transverse plane is determinedby the nucleon positions. On the other hand, the MC-KLN model describes the participant geometry by the transverse gluon positions based on the Color Glass Condensate(CGC). Since ε2 in
MC-Glauber is smaller than that in MC-KLN, MC-Glauber is combined with the value of viscosity 4πη/s = 1 and MC-KLN is paired with the value 4πη/s = 2 to reproduce the v2 . The valuesof ε3 from the two models are similar for reproduction of the v2 and therefore the larger viscous

18

CHAPTER 1. INTRODUCTION

Figure 1.19: Transverse momentum dependence of charged hadron vn at mid-rapidity for diﬀerent
multiplicity class in Au+Au 200GeV measured by the PHENIX collaboration

correction makes smaller v3 in MC-KLN model. Additionally, the calculation from the transportmodel ”UrQMD+4π/s = 0” is shown in Fig.1.20. That model employ MC-Glauber with the
UrQMD transport model for reproducing the pre-equilibrium combined as the initial condition
and the ideal hydrodynamics (4πη/s = 0). Because the ideal hydrodynamics reproduces thehighest value of v2 and v3 due to no viscous correction, the ideal hydrodynamics provides theupper limit of v2 and v3 . All theoretical predictions describe the v2 in both of the pT range
except for the low multiplicity bins. However in the lower panel for the v3 comparison, ”KLN
+4πη/s = 2” shows the smaller value of the v3 compared to the PHENIX experimental results.
Although MC-KLN model with 4πη/s = 2 combination is disfavored by the v3 measurements,
this comparison does not imply the initial condition is not CGC. Thus the vn measurements
could constrain the initial condition and the value of viscosity of the QGP.

Identified hadron vn (pT )Measurements of azimuthal anisotropies for pions, kaons and protons provide further insight into
the property of the QGP. Figure 1.21 shows the charge combined pion, kaon and proton vn at√
mid-rapidity in Au+Au collisions at sN N = 200 GeV collisions [24]. All vn coeﬃcients seen in
Fig.1.21 have two common trends that are referred to as ”mass ordering” and ”baryon and meson
splitting”. First, the feature of the mass ordering is seen in the low pT region. The anisotropiesof the lighter hadrons are larger than those of heavier hadrons. This trend reflects the massdependence from the radial flow eﬀect as described in the previous section. Second, baryon andmeson splitting appears at the higher pT region. The anisotropy for baryons becomes lager thanthat for mesons. This behaviour is explained by quark coalescence hadronization mechanism.
Quark coalescence model starts from the idea that the invariant spectra for emitted particles
is proportional to the product of the invariant spectra of constituent quarks. In the quark

1.3. MAJOR FEATURES OF EXPERIMENTAL OBSERVABLES

19

Figure 1.20: Multiplicity dependence of charged hadron vn at mid-rapidity for two transverse
momentum intervals in Au+Au 200GeV measured by the PHENIX collaboration

20

CHAPTER 1. INTRODUCTION

coalescence model, the meson and baryon spectra are given by those of constituent quarks
dNq pT
dNM(pT ) = CM (pT )( 2 ( ))22d pT
d pT 2dNq pTdNB(pT ) = CB (pT )( 2 ( ))3d2 pT
d pT 3(1.19)(1.20)

where NM , NB and Nq are the number of mesons, baryons and quarks. The CM (pT ) and CB (pT )are the probability for qq → meson and qqq → baryon respectively. These relations are valid atintermediate pT region, which is moderate constituent phase space density. In the intermediatepT region, anisotropies of produced hadrons are given by parton anisotropies,
pT
)
2
pT
vn,B ≈ 3vn,q ( )3vn,M ≈ 2vn,q ((1.21)(1.22)(1.23)

where vn,M , vn,B and vq,n are meson, baryon and parton vn respectively.
Pseudorapidity dependence of vn
At RHIC, pseud-rapidity dependence of vn in Au+Au collisions is measured by the PHOBOS
Collaboration and STAR Collaboration[25] and [26]. Figure 1.22 shows rapidity dependence of
charged hadron v2 for three diﬀerent multiplicity classes measured by the PHOBOS Collaboration. The black squares represent the experimental results. The v2 for entire rapidity region
increases from the high multiplicity class (top panel) to the low multiplicity class (bottom panel).
For all multiplicity classes, the v2 at mid-rapidity is largest and decreases with increasing absolute value of rapidity. In Fig.1.22, several hydrodynamics calculations for two diﬀerent initialconditions are compared to the experimental results. In left panel, the Glauber type initialcondition is employed and the KLN initial condition is used in right panel. For both of thepanels, the pure ideal hydrodynamic simulation with freeze out(kinetic decopling) temperature
Tdec = 100 MeV shows weak rapidity dependence of v2 and overpredicts the experimental data at
entire rapidity region except for most central collisions for the Glauber initial condition case. On
the other hand, the pure hydrodynamic simulation with freeze out temperature Tdec = 169 MeV
underpredicts the data at entire rapidity region for the Glauber initial condition case and overpredicts the peripheral collision data at |η| < 2 for the KLN initial condition case. In these purehydrodynamic calculations, the phase transition temperature from the QGP to hadron phase is
Tc = 170 MeV. Thus the hydrodynamics calculations with Tdec = 100 MeV and Tdec = 169MeV
correspond to ”QGP+hadron fluid” and ”only QGP” state, respectively. The red circles shown
between the two theory curves represent ”QGP + hadron cascade”. Compared to ”QGP+hadron
fluid”, hadron cascade phase makes smaller v2 value. Especially this dissipative eﬀect is seen atforward and backward rapidity region. The hydrodynamics with the KLN initial condition isalways higher than that with the Glauber initial condition for all three multiplicity classes. The
PHOBOS results prefer the calculation from the ”QGP+hadronic cascade” with the KLN initialcondition for highest multiplicity class. In the lowest multiplicity class, the ”QGP+hadronic
cascade” with the Glauber initial condtion model agrees with the PHOBOS data.

1.3. MAJOR FEATURES OF EXPERIMENTAL OBSERVABLES

21

Figure 1.21: Charge combined pion, kaon and proton vn at mid-rapidity in Au+Au collisions at√
sN N = 200GeV22

CHAPTER 1. INTRODUCTION

Figure 1.22: Pseudo-rapidity dependence of charge hadron v2 in Au+Au collisions at√
sN N = 200GeV in comparison to the hydrodynamic calculations. Left:Glauber Monte Carlomodel(nucleon base) initial condition. Right:KLN model(gluon base) initial condition

Although the event averaged longitudinal multiplicity distribution in A+A collisions is symmetric and boost invariance at |η| < 2(plateau region), the event by event distribution is notnecessarily a symmetric shape. The particle production at the forward(backward) rapidity is
influenced by the participant nucleons moving toward the forward(backward) rapidity. ThusFB ,the diﬀerent number of forward going and backward going participant nucleons,Npart
̸= Npart
would lead to the forward/backward asymmetric rapidity distribution. Indeed, the experimental results from the PHOBOS and STAR collaboration reveal a forward/backward asymmetricparticle production in rapidity[27],[28].
Jia and Huo[29] predict forward/backward asymmetry of azimuthal anisotropies of particle√
production in Pb+Pb collisions at sN N = 2.76TeV using a multiphase transport model(AMPT)[50], which is frequently used for the study of azimuthal anisotropies in relativistic heavy ion collisions. In order to study forward/backward asymmetry in azimuthal anisotropies, initial participant eccentricities of forward going and backward going participant nucleons,ϵFn , ϵBn , are defined
separately on event by event basis in their study. In each event, they calculate the diﬀerence of
forward and backward eccentricity, ∆ϵn = ϵFn −ϵB
n . Fig. 1.23 shows the simulated rapidity dependence of v2 and v3 for selecting similar forward and backward eccentricities event,|δϵn | < 0.02.
The flow harmonics are calculated with respect to the three event planes Ψn defined by thethree rapidity region, 4 < η < 6,−6 < η < −4 and −1 < η < 1 and the participant plane ,Φ∗n ,that is the reference axis for the ϵn . The participant plane is calculated by all the participantnucleons at the thermalization time before the collective expansion. The deviations among thethree event planes and the participant plane selections are observed in Fig. 1.23 and are mainly
contributed from non-flow eﬀect In Fig. 1.24, the simulated rapidity dependence of v2 and v3 for1.3. MAJOR FEATURES OF EXPERIMENTAL OBSERVABLES

Figure 1.23: Pseudo-rapidity dependence of charge hadron vn in Pb+Pb collisions at
200GeV

Figure 1.24: Pseudo-rapidity dependence of charge hadron vn in Au+Au collisions at
200GeV

23√sN N =√sN N =|δϵn | > 0.125 event are shown. Significantly the values of v2 and v3 are larger at forward rapidity(positive rapidity) than at backward rapidity(negative rapidity). These asymmetric rapidity
dependences are attributed to the event selection of ϵFn > ϵB
n . Thus this rapidity asymmetry ofvn implies the independent fluctuations of the participant nucleons in the two colliding nuclei
survive even after the collective expansion and the initial geometry would not be boost invariant
but have a strong rapidity dependence.

24

1.4

CHAPTER 1. INTRODUCTION

Cu+Au collision and thesis motivation

As it is introduced in the previous sections, the azimuthal anisotropies are strong tool tounderstand the initial condition and the bulk property of the QGP. Until now, many experimentalists and theorists have concentrated on studying the azimuthal anisotropies in high energy
heavy ion collisions and concluded the QGP formed at RHIC behaves nearly perfect fluid. However, in spite of the many experimental observables and theoretical predictions, the uncertainties
of the initial condition and the viscosity of the QGP still remain. The understanding of theinitial condition in heavy ion collisions is crucial for the subsequent hydrodynamic expansion.
Thus the determination of the initial condition help us to quantify the viscosity of the QGP.
√In 2012, Cu+Au collisions at sN N = 200GeV were operated at RHIC for controlling theinitial geometry. Until now, symmetric collision systems such as Au+Au collisions, have been
operated at high energy collisions so that this is the first test to collide diﬀerent heavy nuclei.
Since the size of Au nucleus (A=197, the radius of Au is 7fm) is larger by factor about 3
than Cu nucleus (A=63 and the radius of Cu is 4.5fm), the participant geometry would beasymmetric in transverse and longitudinal directions. The asymmetric initial geometry provides
diﬀerent geometry situations from symmetric collision systems such as Au+Au collisions. Thus
the asymmetric collisions provide opportunities to investigate the influence of the initial geometryon the collective dynamics and particle production.
In this dissertation, we present the measurements of the azimuthal anisotropies of charged√
particles at mid- and forward/backward-rapidities in Cu+Au collisions at sN N = 200 GeV
using PHENIX detectors. As it is noted in the above paragraph, the initial geometry of Cu+Au
collisions has two features, sideward asymmetry and longitudinal asymmetry. In non-central
collisions, the participant geometry has sideward asymmetry in transverse plane as shown in
Figure1.25 (b). The sideward asymmetry of initial geometry would lead directed flow at midrapidity. Therefore we examine the v1 measurements at mid-rapidity with model calculations
and discuss about the influence of the initial sideward asymmetry on the directed flow.
Until now, the longitudinal structure is considered to be boost invariant. However, lately
some experimental results which indicate breaking boost invariant longitudinal structure areobserved and there are some theoretical calculations which predict longitudinal dependence ofinitial condition. In the asymmetric collisions, longitudinally, the forward (Cu-going side) andbackward (Au-going side) diﬀerent collective dynamics and particle production are expected,
because the participant geometry and number of participants in Au nucleus and those in Cu
nucleus influence Au-going side and Cu-going side, respectively. Therefore the measurements ofvn and particle multiplicity in Cu+Au collisions will provide us further insight into the initialcondition. By introducing weighted εn and Npart scalings, we discuss about longitudinal initialgeometry and density in Cu+Au collisions.

1.4. CU+AU COLLISION AND THESIS MOTIVATIONa) Reaction Planex

Cu Spectatorb) Transverse Plane

Autarget(η<0)

25Cu

projectile(η>0)
zxparticipant zone
Au Spectator

Figure 1.25: Cu+Au collisions in longitudinal direction(a) and transverse direction(b)Chapter 2

Experimental Apparatus
2.1

Relativist Heavy Ion Collider(RHIC)

Relativistic Heavy Ion Collider(RHIC) was built as the first heavy ion collider at Brookhaven
National Laboratory in Upton New York the United State of America and is the second highest
energy heavy ion collider in the world at present. RHIC is composed of two 3.834km long rings
which circulate in opposite directions. One ring is called “Blue ring” which is clock-wise and theother is “Yellow ring” which is anti-clock-wise. The length of circumference of RHIC is 3.8km.
At RHIC, a wide variety of particle species can be operated from A = 1 (proton) to A = 238(uranium) at present and nuclei are collided at several energies. In heavy ion experiment, heavy
ions are accelerated in several stages to achieve relativistic speed. Thus several accelerators areneeded. At BNL, five accelerators are used before the injection to RHIC, Electron Beam Ion
Source (EBIS), Radio Frequency Quadrupole (RFQ) accelerator, Linear accelerator (LINAC),
the Booster Synchrotron,the Alternating Gradient Synchrotron(AGS). The accelerator complex
is shown in Fig. 2.1. In Fig 2.2, the procedure for accelerating gold ions is shown. At thefirst step, heavy nuclei are created and accelerated by EBIS. In the gold ions case, the ions areaccelerated to 2A MeV through EBIS, RFQ and LINAC before the Booster accelerator. At thesecond stage, the Booster Synchrotron accelerate ions to 70A MeV and the ions are grouped
into three bunches by a RF electric field. At the exit of the Booster, all of atomic electrons
are stripped oﬀ by the foil. These positive gold ions are injected into the AGS. At the AGS,
the three bunches of gold ions are further accelerated to 9A GeV and then injected to RHIC
ring. The two RHIC rings are rounded hexangular concentric superconducting magnet rings andhave six intersection locations where collisions occur. At the present, two experiments records
collisions data at two intersection points. To guide and focus ion beam, each beam pipe uses 396
superconducting dipole magnets and 492 superconducting quadrupole magnets. In total, there
are 1740 magnets including other correction magnets. The magnets are cooled less than 4.6 K bysupercritical helium. In the two RHIC rings, RF cavities are implemented. The injected bunches
are accelerated by electric filed in RF cavities. As increasing collision energy, the magnitude ofmagnetic filed are increased accordingly up to 3.5 T which corresponds to Au-Au collisions at
200 GeV case and p+p collisions at 500 GeV case. The ions are stored in the rings for a periodof 6 to 12 hours and collided at six intersection points. For the description of the performance of26

2.1. RELATIVIST HEAVY ION COLLIDER(RHIC)

27

Figure 2.1: RHIC accelerator complex

Figure 2.2: Schematic of procedure of Au ion injection.

the accelerator, the parameter of collisions rate is important. The luminosity L is the parameterswhich is used for the description of the collisions rate. The number of events N are expressed asfollows using L and a cross section σ
N = Lσ

(2.1)

28

2.2

CHAPTER 2. EXPERIMENTAL APPARATUS

PHENIX Detectors

PHENIX (the Pioneering High Energy Nuclear Interaction eXperiment) is one of the two
experiments at BNL. PHENIX is located at 8 o’clock, if the injection point of the beam pipe
from AGS is 6 o’clock. Fig.2.3 is the schematic view of the PHENIX subsystems. PHENIX is designed to measure hadrons, leptons and photons in high rate events and high multiplicity events.
PHENIX consists of 4 spectrometers. The two spectrometers at mid-rapidity are instrumented tomeasure electrons, photons and hadrons and the two spectrometers at forward/backward rapidity
are instrumented to measure muons.

2012

PHENIX Detector
PC3
PC2

PbSc

PbSc

DC

PbSc

PbSc

DC
BBC

RICH

RICH

PbSc

PbGl

BB
MPC (F)VTX
PC1

PbSc

PC1

7.9 m = 26 ft

TOF-W

PC3
TEC

CentralMagnetPbGl

AerogelTOF-E

Beam View

West
RPC3

So

MuID

10.9 m = 36 ft

ZDC South

uth

M
uo

East
et

n
ag

Central Magnet
nM

ag

ne

rt

No

t

RPC3

nM
uo
M
h

MPC
BBC
ZDC NorthMuID(F)VTX
MuTr

RPC1

South

Side View

North

18.5 m = 60 ft

Figure 2.3: Schematic view of the PHENIX subsystems.
panel:Side View

Top panel:Beam View, Bottom

2.2. PHENIX DETECTORS

2.2.1

29

Magnet systems

PHENIX Magnet system

At PHENIX, the magnet systems is composed of three spectrometer magnets which are madeof warn iron yokes and water-cooled copper coils. One is the Central Magnet (CM) and the others
are the North and South Muon Magnets. The CM use concentric coil and provide magnetic fieldaround the collision vertex position which is parallel to the beam axis for measurements ofcharged particle tracks. The Muon Magnets use solenoid coils produce radial magnetic filed formuon analysis. Each of three magnets produce about 0.8 Tesla-meters. The design of CM forthe physics-driven requirements- To minimize multiple scattering and interactions of particles from primary vertex, there isno mass in the apertures of the central spectrometer arms.
- To create a ”zero-filed” region near R=0, the radial magnetic field has to be controlled.
- The magnitude of magnetic field for the region R>200 cm is required to be minimum.
As described, CM uses two sets of circular coils as shown in Fig.2.4. These “inner” and“outer” coils produce the “++” configuration and the “+-” configuration.

Figure 2.4: Schematic view of Central and Muon magnets.

2.2.2

Global detector

For the event categorization, Beam - Beam counters(BBC) installed at forward/backward
rapidity is employed. A pair of BBCs provides time of flight of particles emitted to forward/backward rapidity for the determination of the collision time and position. The BBCs

30

CHAPTER 2. EXPERIMENTAL APPARATUS

provides multiplicity at forward/backward rapidity to determine centrality. At this subsection
these detectors are explained.
Beam-Beam Counter

The main role of Beam-Beam counter (BBC) is to provide the vertex position along withbeam axis and the multiplicity of produced particles at forward/backward rapidity region for the
event categorization, to provide the collisions time for the time of flight of particles for particle
identification using TOF and EMCAL detectors and to trigger the PHENIX LVL1 trigger. The
BBC is composed of two identical counters installed on South side and North side of collision
point along with beam pipe. One installed on the South side and the other installed on the
North side are named BBC S and BBC N respectively. The both of counter are placed at the
distance of 144cm from the center of interaction point and surround the beam pipe. The BBC
S and BBC N cover full azimuthal angle and pseudo-rapidity range from 3 to 3.9. The BBC isrequired to satisfy the following conditions:(i) The BBC has a capability to function the dynamic range from 1MIP to 30 MIPs. Because
the number of particles generated in central Au+Au collision at 200GeV is about few
thousands at BBC rapidity and BBC has to operate p+p collisions case.
(ii) BBC has to be radiation hard. The place where BBC is installed is around beam pipe near
the collision point. At the place, a very high level radiation is expected.
(iii) Since BBC is placed behind the PHENIX central magnet, BBC has to work under high ahigh magnetic filed environment.
Each detector consist of 64 Cherenkov radiators. Each radiator is composed of one-inch diameter mesh-dynode photomultiplier tubes(Hamamatsu R6178) attached with 3cm quartz on the
top of PMT. In Fig.2.5, each BBC counter is shown on left side picture and each individual
element is shown on right side picture. The outer diameter is 30cm and the inner diameter is
10cm. In central AuAu collisions at 200GeV, 15 particles per each element are expected for thisconfiguration.
Zero Degree Calorimeter(ZDC)

Zero Degree Calorimeter(ZDC) is installed to detect spectator neutrons and to measure thetotal energy of them. The purpose of ZDC is to detect spectator neutrons and measurementof the total energy of them. ZDC is also used as a event trigger and a luminosity monitor.
Because of this reason, identical detectors are installed in all RHIC experiments. ZDC systems
are composed of two identical calorimeters. ZDC systems are placed at a distance of 18m frominteraction point along the beam line for South and North side separately and the coverage ofpseudo-rapidity range is |η| > 6.5. Since ZDC systems are placed at DX magnets, the charged
particles are bended and swept away and only neutral particles are detected at ZDC. The someof charged particles may hit beam pipe and induce showers. The induced shower may hit the2.2. PHENIX DETECTORS

31

Figure 2.5: Left:One BBC counter composed of 64 elements. Right:Single BBC element composedof PMT equipped with quartz

ZDC. A scintillation counter is placed in front of ZDC for charged particle veto. ZDC is a
hadronic sampling calorimeter. Each ZDC system consists of three ZDC modules. Each module
is composed of 27 Tungsten absorber layer and polymethylmethacrylate fiber layers(PMMA).
In Fig 2.7, the schematic view of fiber layer and the schematic view of single ZDC module areshown. The fiber ribbons were impregnated with a low viscosity silicone rubber glue. The active
region of the fibers are covered and the surface of fibers in the fiber and absorber sandwich region
are protected by the glue. The one side of fibers are connected to PMT and the other side offibers were left untreated.

Figure 2.6: Schematic view of the colliding nuclei before(left) and after(right) collision.

Shower Max Detector(SMD)
Shower Max Detector is installed to measure the center of neutron induced neutron shower.

32

CHAPTER 2. EXPERIMENTAL APPARATUS

Figure 2.7: Schematic view of the colliding nuclei before(left) and after(right) collision.

The SMD is composed of 7 scintillator strips for the determination of x-coordinate of center ofneutron induced shower and 8 scintillator strips for the determination of y-coordinate of centerof neutoron induced shower. The SMD is placed between first and second ZDC modules where
the maximum energy of neutron induced shower is reached. The position resolution is about
1mm for neutrons at 100GeV energy.

2.2.3

Central arm detector

Central magnet provide magnetic field parallel to the beam axis and the magnetic field bend
the tracks of emitted charged particles. Central arm is composed of two hybrid tracking detectors
for charged particles and photons. They are installed in East and West side respectively. On
the central arm, electro magnetic calorimeters are installed in both side for electron and photon
measurement. For charged particles tracking, Pad and Drift chamber are employed. In thissection, each subsystem at Central Arm are explained.
Drift Chamber

2.2. PHENIX DETECTORS

33

Drift Chamber(DC) systems provide, measurement of charged particle trajectories and determine transverse momentum of charged particles. DC is composed of two identical detectors
located East and West side of Central arm spectrometers respectively. DC are inner most detector of Central arm subsystems and placed at the region from 2 to 2.4 m from the beam pipe.
The magnetic filed is applied in the region where DC is placed to bend charged particle tracks
for the determination the transverse momentum. In heavy ion simulation study, the requirementof the transverse momentum resolution and double track spatial resolution are specified for themeasurement of ϕ mass with a resolution better than 4.4 MeV and good tracking eﬃciency forthe highest multiplicities. The drift chamber is imposed the following requirements,- Resolution of single wire is better than 150 µm in r-ϕ plane- Two track separation of single wire is better than 1.5 mm.
- Eﬃciency of single wire is better than 99%.
- Spatial resolution in the z direction is better than 2 mm
Each detectors are divided 20 sectors which cover 4.5 in ϕ◦. In each sector, there are six diﬀerent
types of wire modules. The six modules are called X1,U1,V1,X2,U2 and V2. Each module has
4 cathode planes and 4 anode planes. The X1 and X2 are placed in the direction parallel to thebeam to measure trajectories of charged particles precisely in transverse plane. The U and V are
used for the track pattern recognition and determine the z coordinate of the track. The U1(2)and V1(2) are placed at 6◦ relative to X1(2) plane.
Pad chamber
Pad chamber(PC) provides three dimensional coordinate of charged particle track, especiallyprecise measurement of z-coordinate. Since PC systems are located outside of magnetic filed, thehit points are along the straight line particle trajectories. The charged particle track coordinates
at PC are used for tagging charged particle and track matching to reduce background fromdecays and γ conversions. The PC is Multi Wire Proportional Chambers which consists of 3
separated layers(PC1, PC2, PC3). PC1 is most inner layer of 3 PC layers. PC1 is located atradial distance of 2.5m and is present behind DC for East and West both side. PC2 is the second
inner layer. The radial distance of PC2 from beam pipe is 4.2m. PC2 is present behind Ring
Image CHerenkov counter(RICH) in West arm only. PC3 is installed between RICH and Electro
Magnetic CaLolimeter(EMCAL) in East and West arms and the radial distance from beam pipe
is 4.9m. Each layer consists of a signal plate of anode and field wires inside a gas bounded by
two cathode plates as shown in Fig. 2.10. Each cathode plate is segmented into an array of
pixels as shown in Fig.2.11. The inside gas was composed of 50% Argon and 50% of Ethan underatmospheric pressure.

34

CHAPTER 2. EXPERIMENTAL APPARATUS

Figure 2.8: Schematic view of the colliding nuclei before(left) and after(right) collision.

Figure 2.9: Schematic view of the colliding nuclei before(left) and after(right) collision.

2.2. PHENIX DETECTORS

35

Figure 2.10: A plane cutting through by a Pad chamber

Figure 2.11: Left:The pad and pixel geometry. Right:A cell made of three pixels is at center ofthis picture

36

CHAPTER 2. EXPERIMENTAL APPARATUS

Time of flight

Time of flight detectors provides charged particle identification. For particle identification,
Time of flight detectors measure charged particle time of flight from collision vertex. Timeof flight detectors are composed of two diﬀerent detectors which are installed in Central arm
detectors for East and West side separately.
Time of flight East(TOF.E)

Time of flight East(TOF.E) detector is a scintillator detector and placed at a distance of
5.1m from beam pipe, in between PC3 and EMCal in Central arm East side. TOF.E covers
70◦ ≤θ≤110◦ in azimuth. The TOF.E consists of 960 scintillator slats and 1920 PMTs. Scintillator
slats are oriented along r-ϕ and parallel to z-axis. Each scintillator slat is equipped with two
PMTs which are attached at the both side edge of a scintillator slat. The designed timing
resolution is about 100ps. π and K could be separated up to 2.4 GeV/c and K and proton
separation could be up to 4 GeV/c.

Figure 2.12: TOF.E installed at PHENIX
Central Arm East Arm

Time of flight West

Figure 2.13: TOF.E each panel composedof scintilator slats and PMTs

2.2. PHENIX DETECTORS

37

Time of flight West(TOF.W) detector is a Multi-Gap Resistive Plate Chamber(MRPC) which
is a gas chamber with multi-layers and multi-gas gaps. TOF.W is composed of two identicalcounters which are placed at a radial distance of 4.81 m from collision point in Central Arm
West side separately. The TOF.W covers pseudo-rapidity acceptance of |η| < 0.35 and azimuth
acceptance of 22◦ in two separated counters. Each counter is composed of 6 0.23mm gas gaps
separated by 5 0.55mm glass layers in between anode and cathode plates. If charged particles
pass though the MRPC, the charged particles ionize the gas between glass layers. The image
charge is collected at both top and bottom side of chamber on copper readout strips. Since thereadout strips are placed at top and bottom side of chamber, the hit position is determined fromthe time diﬀerence between top and bottom readout chips. The timing resolution is achieved 90
ps for the transverse momentum range of 1.3≤ pT ≤1.7.

Figure 2.14: Schematic view of MRPCElectro Magnetic Calorimeter(EMCal)

Electromagnetic Calorimeter(EMCal) [32]system provides the hit position of particles andenergy of photons and electrons. EMCal systems are placed at most outer layer of Central arm
and covers pseudo-rapidity range of |η| < 0.35 and azimuthal range of 70◦ ≤ θ ≤ 110◦ . EMCal
systems is composed of two walls which are installed in Central arm East side and West side
separately. West side wall comprised four Pb scintillator sampling calorimeters. East side wallcomprises two Pb scintillator sampling calorimeters and two Pb glass Cherenkov calorimeters. Pb
scintillator and Pb glass have diﬀerent properties. From following the properties are explained.
Pb scintillator sampling calorimeter

38

CHAPTER 2. EXPERIMENTAL APPARATUS

The PHENIX Pb scintillator calorimeter consists of 15552 individual towers which are madeof a pile of alternating tiles of Pb and scintillator. This type of calorimeter is referred to shashlik
type sampling calorimeter. Each tower contains 66 cells made of alternating tiles of Pb andscintillator. At the edge of tiles, Al tiles are placed. For light collection, 36 longitudinallypenetrating wavelength shifting fibers are inserted into each tower and at the back of the towers,
light is read out by 30mm phototubes. Mechanically four tower form together one module which isa single structural entity. Thirty six modules are grouped into super-module and eighteen super√⊕
module form one sector. The Pb scintillator has a energy resolution of 8.1%/ E(GeV ) 2.1%
and a timing resolution is better than 200 ps.

Figure 2.15: Schematic view of each Pb-Schintilator tower

Pb glass Cherenkov calorimeter

The Pb glass Cherenkov calorimeter is most outer layer of Central arm and placed at lower
sectors of East side. Each sector contains 192 super-modules arranged widthwise 16 supermodules and heightwise 12 super-modules as shown in Fig. Each super-module contains twenty
four modules arranged widthwise 6 modules and hightwise 4 modules. The size of each module
is widthwise 40mm and heightwise 40 mm and depthwise 400mm. Each module are wrappedwith aluminized mylar and shrink tube. To form super-module, twenty four modules are glued
with carbon fiber and epoxy resin. For Cherenkov radiation collection, FEU-84 photomultiplieris used at the back of module. At the front of super-module, LED light is attached for gain
monitoring and a polystyrene reflective dome is also attached to enclose the LED system. The√
Pb glass calorimeter has energy resolution of 6%/ (E)(GeV ) . The timing resolution is better
than 300ps.

2.2. PHENIX DETECTORS

39

Figure 2.16: Schematic view of each Lead Glass tower

Reaction Plane Detector(RXNP)

Reaction Plane Detector(RXNP) system is composed of two identical detectors which areinstalled at a distance of 39 cm from the nominal interaction point along with beam pipe inSouth and North side of PHENIX respectively. Each detector is composed of concentric inner andouter rings and contains 24 scintillators in these rings. Each ring has twelve equally sized segment
scintillator arranged perpendicular to beam pipe and azimuthal acceptance of 2π, pseudo-rapidity
acceptance of 1 < η <2.8. Each scintillator is trapezoidal shape with 2cm thickness and wrappedwith aluminized mylar inner layer and black plastic outer layer. A schematic view of RXNP isshown in Fig.2.17. The inner segments cover from 5cm to 18 cm from the beam pipe and outer
segments cover from 18cm to 33cm. These radial acceptance corresponds to 1.5< |η| <2.8 and1< |η| <1.5. The inner segment has 2cm inner and 9cm outer edges respectively. The outer
segment has 17cm outer edge.

2.2.4

Data acquisition(DAQ)The PHENIX data acquisition(DAQ) system is designed to collect the event data in a wide
range of colliding systems and the interaction rate at design luminosity. The number of produced
particles within PHENIX acceptance is from a few tracks in p+p and several hundred tracks incentral Au+Au collisions. The collision rate at RHIC varies from approximately 500kHz forminimum bias p+p collisions to a few kHz for Au+Au collisions at design beam luminosity. The
PHENIX DAQ system is able to seamlessly accommodate the variety of event size and event rate
through the dead timeless and pipelined features of the detector front ends and the ability to40

CHAPTER 2. EXPERIMENTAL APPARATUS

Figure 2.17: Schematic view of Reaction plane detector

accommodate higher level trigger. Fig 2.18 represent the general schematic for PHENIX DAQ
flow. At PHENIX, the Master Timing Module(MTM), the Global Level-1 Trigger System(GL1)

Figure 2.18: Schematic design of PHENIX DAQ systemand the Granule Timing Module(GTM) control overall data acquisition. The MTM receive RHIC
beam clock and distribute it to GL1 and GTM. The GL1 receive and combine the data to provide
the first Level-1(LVL1) trigger decision. In addition, the GL1 also manages busy signals. The
GTM deliver the RHIC clock, trigger information and event accepts to the Front End Modules
which convert detector analog signal into digitized signal. While the GL1 is making the decision,
where the event data is recorded or not, temporally the event data is stored in AMU. After FEM’s

2.2. PHENIX DETECTORS

41

receive the accept decision, the digitization of the analog signals are started. The digitized data
are collected by Data Collection Modules(DCM) which communicate with rest the rest of the
DAQ system. via G-LINK. The DCM is able to receive 100 Gbytes/sec of uncompressed event
data and perform zero suppression, error checking and data reformation. From DCM’s, many
parallel informations are sent to Event Builder(EvB) which is the final stage of data collection.
The EvB also provides an environment for the LVL2 trigger to operate. In order to record
interesting events, it is necessary to discard the number of events. The LVL2 trigger provides asecond filter for uninteresting events to assemble interesting events. Then EvB sends the accepted
event data to the PHENIX Online Control System(ONCS) to log and monitor the data. The
Common Object Request Broker Architecture(CORBA) system is the technology used to handle
the many components. Throughout the network, the CORBA access to the object on the remotecomputers. The Run Control(RC) is the main control process. The role of the RC is access andcommunication with remote object that handle each of the hardware. The configuration of DAQ
is determined by RC system for example the trigger, the subsystems and run type and so on.

Chapter 3

Analysis
In this thesis, the data sets used are Minimum-bias triggered Cu+Au, Au+Au and Cu+Cu√
collisions at sN N = 200 GeV recorded by PHENIX experiment in 2012(Run12), 2007(Run7)
and 2005(Run5). In this chapter, we describe the data reduction and the analysis method
performed in this study.

3.1

Event selection

As described above, minimum bias data set is used in this study. The minimum bias event
is defined as more than two PMTs fired in each BBC counter. In Cu+Au collision case, the MB
sample covers 93±3% of total inelastic collision cross-section simulated by a comparison of BBC
multiplicity distribution to Monte Carlo Glauber model. Z vertex position is determined by theaverage hit time diﬀerence between BBCS and BBCN. Moreover Z vertex region within +-30 cm
is required.
Centrality Determination

In order to classify event geometry, “centrality” is introduced. The centrality is the degreeof overlap region in two nuclei, which is related to impact parameter b. The centrality from 0 %to 100 % corresponds to impact parameter from 0 fm to R + R ′ fm(R, R ′ are radius of diﬀerent
nuclei). The centrality 0 % and 100 % corresponds to the most central and the most peripheral
collisions respectively. Under the assumption that impact parameter b is related to emitted
particle multiplicity, experimentally centrality class is determined by comparing the measured
particle distribution and the simulated particle distribution by Monte Carlo Glauber [7]. Figure
3.1 shows the total emitted charged particles as a function of Npart or b calculated by GlauberMonte Carlo. In the PHENIX experiment, centrality is defined as a percentile of the total charge
distribution on combined BBC S and N side, which corresponds to charged particle multiplicities
at forward/backward rapidity. Fig.3.2 shows BBCS and N combined charge sum distribution
with the boundary lines which represent centrality classes in Cu+Au collisions.
42

3.2. TRACK SELECTION

43

103
102

10

1
0

Figure 3.1: A schematic of the correlation
between measured charged particles andNpart and b calculated by Glauber Monte
Carlo. The plotted distribution is not actual measurements.[7]3.2

Track selection

3.2.1

Track reconstruction

200

400

600

800
1000 1200
Bbc Charge Sum

Figure 3.2:
The total bbc charge
distribution in Cu+Au collisioins at√
sN N = 200GeV from combined Southand North Bbc detectors

In each event, a lot of number of particles are emitted and pass through PHENIX detectors.
At the Central Arm spectrometer, charged particle tracks can be reconstructed on event by evenbasis. For data analysis, the track reconstruction is important for momentum determination,
identification of particle and background estimation, and so on. In order to reconstruct the
trajectories of charged particles, the hit informations on the Central Arm subsystems have to
be combined. Mainly the DC and the PC1-3 reconstruct track from collision vertex. The otherdetectors, such as TOF.E, TOF.W and EMCAL are employed for particle identification andadditional background removal.
The track reconstruction is performed in the r − ϕ plane and the r − z plane of the DC andthe PC1 respectively. Fig. 3.3 and 3.4 show the typical tracks in these planes. The variables
illustrated in Fig. 3.3 and 3.4 are used for the track reconstruction with the DC and the PC1,of which definitions are summarized below• ϕ : The azimuthal angle of the intersection of the track with the reference radius at midpoint of the DC• α: The azimuthal inclination angle of the track at the intersection point.
• zed : z coordinate of the intersection point

44

CHAPTER 3. ANALYSIS• β : The inclination angle of the intersection point in the r − z plane.

δβ
PC1 radius 245cm
DC reference radius 220 cm

zed

r

z

θ
collision vertex

Figure 3.3: A cartoon of the ϕ and α for theDC track reconstruction. The dashed lines
represent DC West arm. The small circles
are DC hits along the track

Figure 3.4: A cartoon of the track in r andz plane. The red line corresponds to thereference radius of the DC

The reconstruction of tracks within the DC is based on combinatorial Hough transform technique[31].
Fig. 3.5 shows HIJING simulation results in central Au+Au collisions at RHIC energy for thesmall region of the DC x-y plane and the associated feature space. After a track is reconstructedin the magnetic field, two parameters ϕ and α specify the direction of the track. In the non-bendplane, track reconstruction is first attempted by information of PC1 hits have z coordinate information. The non-bend vector is determined from z coordinate from PC1 and Z vertex position
information from the BBC.

Figure 3.5: Left figure:Simulated hit informations in a central Au+Au collisions with HIJING
for a small region of the DC. Right figure:The Hough transform feature space(ϕ-α plane) for thissmall DC region. The peaks corresponds to tracks.

3.2. TRACK SELECTION

3.2.2

45

Momentum determination

In the magnetic field which is parallel to the beam axis, the transverse momentum of charged
particles can be determined from the curvature of track trajectory. In the PHENIX experiment,
the transverse momentum of charged particles are obtained from the inclination of the track at
the reference radius of the DC, α, as described in the previous section. The relation between pT
and α is expressed as
K
α≃
,
(3.1)
pT
where K=101 mrad GeV/c is the eﬀective field integral. One can derive the relation between
the momentum resolution and the angular resolution using Eq. (3.1),
δp
p

=
=

δα
α
√(
)
1
σms 2
+ (σα p)2
K
β

(3.2)

where δα is the measured angular spread, σms is the contribution from multiple scattering andσα is the contribution from angular resolution of the DC respectively.

3.2.3

Track Quality

The tracks reconstructed by the DC and the PC1 have quality parameter which represents
track hit pattern recognition of the DC wires and the PC1 and are expressed as 6 bit number
from 17 to 63. In this analysis, we select the quality values of 31 and 63 as the good tracks.
Good track, 31 and 63 , require a hit in both X1 and X2 wires, the unique hit in the UV layers
and the (unique) hit in the PC1. For the quality value of 31 case, there are multiple hits in PC1
and for the quality value of 63 case, the unique hit in PC1 is required.
Track Matching

Track model projects a charged track trajectory reconstructed by the DC and the PC1 tothe outer layer subsystems, such as PC3 and EMCAL and find candidates of real hit withina fixed ϕ − z window around real hit in the each outer layer subsystem. The hit associatedwith the track is identified by the closest distance to the projection point. Since the rack model
assume the track is from primary vertex, the background track from non-primary vertex could
be removed by the distance between the projection point and the real hit. The track matching
residual distribution which is the distribution of the diﬀerences between the projection points
and real hits is approximately Gaussian distribution with a width of√σmatch =(2
σdetector
+

σms
pβ

)2
,

(3.3)

46

CHAPTER 3. ANALYSIS

where σdetector is the position resolution of the detector and σms is the contribution from multiple
scattering. The contribution of multiple scattering becomes large at low pT and small andnegligible compared to the position resolution at high pT . If the mean of the residual distribution
is non-zero value, this is considered to be imperfect detector alignment calibration or the magnetic
filed parameters in the track model. For the analysis, the track matching distributions arenormalized by the width of the distribution and the mean of the distribution is shifted towards
zero such that the standard deviation is one and the mean is zero in azimuthal(r − ϕ) andlongitudinal(z) directions respectively.

3.2.4

Particle Identification

Particle identification for π, K, p is performed via Time of flight method using Time of flight
East and West detectors. In the time of flight method, particles are identified by squared mass
distribution. The squared mass and the time of flight relation is expressed as,
(

m

2

)
1
= p
β2 − 1
(( )
)
ct 2
2
= p
−1 ,
L
2

(3.4)

where p is the particle momentum and β is particle velocity expressed as a ratio relative tothe speed of light c, the time of flight of particle t and flight path length L. The particle
momentum is measured by the DC and the time of flight of particle is measured by TOF.E and
W. The particle flight path length corresponds to the distance between the interaction point
and projection point at TOF.E and W. Fig. 3.6 shows the squared mass distribution in Cu+Au
collisions for positive and negative, TOF.E and TOF.W separately. In each plot, the lowest
horizontal band corresponds to pions, the second band is kaons and the third and forth bands
are protons and deuterons
The particle identification cuts are determined based on the distance from the peaks of particle
species in the squared mass distribution. The values of peaks and σ for pions, kaons and protons
are extracted from fits. For the pions and kaons identification, one fitting function that has two“Crystal Ball” functions is used and for the protons case, a single Crystal Ball function is used.
The Crystal Ball function has five parameters, three parameters from a Gaussian function(height,
mean and σ) and two parameters from a power function which reproduce tail. The fitting are
perfomed for every 100 MeV/cpT bin, charge and TOF.E and TOF.W separately. Fig. 3.7
represents squared mass distribution for the positive pions, kaons and protons at inter-mediatepT bin in the TOF.W fitted with the Crystal Ball functions and three single Gaussian functionof which the parameters are fixed by the Crystal Ball functions(height, mean and σ). Once themeans and σ are determined, the values of them are plotted as functions of pT , then they are fit
with polynomials. The polynomials used in our analysis are the following formula,
a4,i
a5,i
⟨m2i ⟩(pT ) = a0,i + a1,i pT + a2,i p2T + a3,i p3T + √ +
pT
pT
b2,i
b5,i
σi (pT ) = b0,i + b1,i pT ++ b3,i p2T + b4,i p3T + √
pT
pT

(3.5)
(3.6)

3.2. TRACK SELECTION
m2 vs pT

547
m2 vs p

tofe_pos
m2tof_pos_east

T

5

107

tofw_pos
m2tof_pos_west

Entries 7.693577e+09
0.7153

RMS x
RMS y

0.424

Entries 8.216503e+09

10

0.10770.270810m2 (GeV2/c 2)

3

6

5

Mean xMean y0.7327

RMS x
RMS y

0.404

0.1711
0.35

104
3

1

5

104
10

3

102
0

10

0.5

1

1.5

2

2.5

3

p (GeV/c)

3.5

4

4.5

5

1

-1

10

0.5

1

1.5

2

T

m2 vs p

T

5

2.5

3

p (GeV/c)

3.5

4

4.5

5

m2 vs p

tofe_neg

T

5

107

m2tof_neg_east
Mean xMean ytofw_neg
m2tof_neg_west
Entries 7.345445e+09

0.7012
0.0955

RMS x
RMS y

10

6

10

5

Mean xMean y40.40510.2534

3

104
2
10

0.7405

3

1

RMS x
RMS y

10

1.5

2

2.5

3

p (GeV/c)
T

3.5

4

4.5

5

1

10

5

104
2
10
1

1

6

0.41710.2528

0

-1

3

102

102
0

10

0.10693m2 (GeV2/c 2)

4

0.5

1

T

Entries 6.677533e+09

m2 (GeV2/c 2)

10

1

0

-1

6

2

102

-1

10

3

2
10

4

m2 (GeV2/c 2)

4

Mean xMean y

10

0.5

1

1.5

2

2.5

3

p (GeV/c)

3.5

4

4.5

5

1

T

Figure 3.6: The squared mass distributions as functions of pT . In each plot, the lowest horizontal
band corresponds to pions, the second bad are kaons and the third and forth bands are protons
and deuterons. Top Left: Positive and TOF.E tracks. Bottom Left:Negative and TOF.E tracks.
Top Right:Positive and TOF.W tracks. Bottom Right:Negative and TOF.W tracks.
where i corresponds to pions,kaons and protons.
In Fig. 3.8 and 3.9, the mean and width values for positive and TOF.W tracks as functions of
pT with the fitting functions are shown.
In our analysis, the 2 σ from the mean values cuts are applied to identify particle species.
Additionally to maintain good signal to background ratio, we used a veto cut which require 2
σ away from the adjacent species peak. The squared mass distribution after the PID cuts areshown in Fig. 3.10. Although deuterons are identified using the PID functions, we do not use
them in any measurements.

48

CHAPTER 3. ANALYSIS

west pos m2tof, p : 1.6 to 1.7 GeV/c
T

Entriesχ2 / ndfalpha1n1
mu1
sigma1N1alpha2n2
mu2
sigma2N2

105

pion_kaon_14
3.847549e+07
4981 / 134
0.7694 ± 0.0024
16.03 ± 0.41
0.01776 ± 0.00005-0.0332 ± 0.0000
1.094e+05 ± 1.032e+02
0.716 ± 0.003
132.5 ± 7.9
0.2582 ± 0.0001-0.03616 ± 0.00010
2.611e+04 ± 4.924e+01

proton_fit_14
Entries
3.847549e+07
χ2 / ndf
581.6 / 93
alpha0.9795 ± 0.0089
n28.58 ± 7.92
mu
0.9447 ± 0.0001
sigma-0.071 ± 0.000
2.839e+04 ± 3.509e+01
N

N-tracks

104

103

102
-0.2

0

0.2

0.4

0.6

0.8
2

1

1.2

1.4

2

m2 (GeV /c )Figure 3.7: Squared mass distribution for medium pT bin and positive particles in TOF.W. The
fitted functions are for pions, kaons and protons.

pid_pos_width_full

pid_pos_mean_full
0.4
1

0.35

proton0.3

0.8

m2 (GeV 2/c 2)

m2 (GeV 2/c 2)

0.25
0.6

0.4

0.2

kaon

0.2
0.15
0.1

proton
0.05
0

0

-0.2

pion
0.5

kaon
pion

-0.05
1

1.5

2

p (GeV/c)

2.5

3

3.5

4

T

Figure 3.8: The mean values of squared
mass distributions for pT bins and positive
particles in TOF.W.

-0.1

0.5

1

1.5

2

p (GeV/c)

2.5

3

3.5

4

T

Figure 3.9: The width of squared mass distributions for pT bins and positive particles
in TOF.W.

3.2. TRACK SELECTION

m2 vs pT

549

m2 vs p

tofe_pos

T

5

107

m2tof_pos_east_sigma

tofw_pos
m2tof_pos_west_sigma

Entries 8.766321e+08
0.7348

RMS x
RMS y

0.4105

Entries 7.982225e+08

100.10730.253810m2 (GeV2/c 2)

3

6

4
5

104
2
10

Mean xMean y0.7448

RMS x
RMS y

0.39510.164
0.3303

3

1

5

104
10

3

102
0

10

0.5

1

1.5

2

2.5

3

p (GeV/c)

3.5

4

4.5

5

1

-1

10

0.5

1

1.5

2

T

m2 vs p

T

5

2.5

3

p (GeV/c)

3.5

4

4.5

5

1

T

m2 vs p

tofe_neg

T

5

107

m2tof_neg_east_sigma

tofw_neg
m2tof_neg_west_sigma

Entries 8.684021e+08
Mean xMean y4

Entries 5.838471e+08

0.7171
0.09799

RMS x
RMS y

10

6

10

5

4

0.38780.2409Mean xMean y0.7524

RMS x
RMS y

0.4077

10

6

10

5

0.1114
0.253

3
104

2
10

3

1

m2 (GeV2/c 2)3m2 (GeV2/c 2)

10

1

0

104
2
10
1

0

10

0.5

1

1.5

2

2.5

3

p (GeV/c)
T

3.5

4

4.5

5

1

0

-1

3

102

102

-1

6

2

102

-1

10

3
m2 (GeV2/c 2)

4

Mean xMean y

10

0.5

1

1.5

2

2.5

3

p (GeV/c)

3.5

4

4.5

5

1

T

Figure 3.10: The squared mass distributions as functions of pT after the PID cuts(2σ from thepeaks and 2σ veto cuts from their adjacent species peaks). In each plot, the lowest horizontal
band corresponds to pions, the second bad are kaons and the third and forth bands are protons
and deuterons. Top Left: Positive and TOF.E tracks. Bottom Left:Negative and TOF.E tracks.
Top Right:Positive and TOF.W tracks. Bottom Right:Negative and TOF.W tracks.

50

CHAPTER 3. ANALYSIS

3.3

Event Plane

3.3.1

Azimuthal anisotoropy

Azimuthal distribution of emitted particle dN/dϕ is written in the form of Fourier expansion
with 2π period,dN
x0
1∑[xn cos(nϕ) + yn sin(nϕ)],
(3.7)
=
+
dϕ
2π π
n=1

The coeﬃcients xn and yn are obtained by integrating cos(nϕ) and sin(nϕ) with the azimuthal
probability distribution r(ϕ) or by counting cosine and sine of each particle with normalized
weight r
∫ π
xn =
=

r(ϕ) cos(nϕ)dϕ(3.8)

rν cos(nϕν )(3.9)r(ϕ) sin(nϕ)dϕ(3.10)rν sin(nϕν )(3.11)

−π

∑
ν

∫ π
yn =
=

−π

∑
ν

∫
∑
where r(ϕ) = 1 andν rν = 1, ν represents particle number and runs all particles, ϕν isthe azimuthal angle of ν-th particle. If there are no azimuthal anisotropic flow eﬀect, r(ϕ) isconstant, r(ϕ) = 1/(2π). The Fourier coeﬃcients xn and yn give the corresponding harmonic√component, defined as vn = x2n + yn2 /x0 , and the n-th harmonic order event plane angle Ψn(−π/n ≤ Ψn ≤ π/n).
xn= vn cos(nΨn )x0yn= vn sin(nΨn )x0
Ψn = tan−1 (yn /xn )/n(3.12)(3.13)(3.14)

Eq. 3.7 can be re-written using vn and Ψn ,
dN
dϕ

=
=
=

x0
1∑
+
[xn cos(nϕ) + yn sin(nϕ)]
2π π
n=1
x0
1∑
+
[vn cos(nΨn ) cos(nϕ) + vn sin(nΨn ) sin(nϕ)]
2π π
n=1
x0
1∑
+
[vn cos(n[ϕ − Ψn ])]2π π(3.15)(3.16)(3.17)

n=1

In experimentally, the real azimuthal anisotropy vntrue is obtained from following relation between

3.3. EVENT PLANE

51

the observed anisotropy and the real anisotropy in enough number of events,
vnobs = ⟨cos(n[ϕ − Ψobs
n ]⟩=(3.18)⟨cos(n[ϕ − Ψn + Ψn − Ψobs
n ]⟩(3.19)

obs= ⟨cos(n[ϕ − Ψn ])⟩⟨cos(n[Ψn − Ψobs
n ])⟩ + ⟨sin(n[ϕ − Ψn ])⟩⟨sin(n[Ψn − Ψn ])⟩(3.20)

vntrue= ⟨cos(n[ϕ − Ψn ])⟩⟨cos(n[Ψn − Ψobs
n ])⟩(3.21)= vntrue Res{Ψobs
n }
obs
vn
=
Res{Ψobs
n }(3.22)(3.23)

obs
where Res{Ψobsn } = ⟨cos(n[Ψn − Ψn ])⟩ is the resolution of the observed event plane determination. This equation is given under the assumption that sine terms of ϕ−Ψn and Ψobsn −Ψn vanish
because the distributions of ϕ and Ψobs
withrespecttoΨ
becomesymmetric
distribution innnenough number of events.

3.3.2

Event Plane Determination

The event plane is defined as the average angle of emitted particles and reconstructed on
event by event basis. Since the azimuthal angle of the impact parameter can not be controlled
by the beam, the azimuthal angle of the impact parameter is distributed randomly. Thus inenough number of events, the azimuthal distribution in the event plane angle should be flat. The
event plane is reconstructed using flow vector(Q-vector),∑Qxn =wı cos(nϕı )(3.24)
ıQyn =∑wı sin(nϕı )(3.25)

ı

where i is the index for the reconstructed particles, ϕi and wi correspond to azimuthal angel
and weight of the i-th particle respectively. If the Q-vector is reconstructed using a detector
which can not reconstruct charged particle tracks, such as BBC, i-th particle is changed to i-th
segment in the detector. In this analysis, BBC, RXN, SMD and CNT detectors are employedfor Q-vector i.e. event plane determination. The BBC is located at 3 < |η| < 4 and has 64
PMTs for South and North side respectively. The charge information in each PMT is used asweight and the azimuthal angle of each PMT is used as ϕi for Q-vector calculation. The RXN
sits on 1 < |η| < 2.8 and has 24 PMTs in each side of North and South. The weigh for RXN
Q-vector calculation is ADC value in each PMT. The SMD provide the centroid of spectator
neutrons, which corresponds to Q-vector. The CNT is located at −0.35 < |η| < 0.35 and canreconstruct charged tracks. The weight of CNT is pT of each charged track and pT < 2 GeV/c
is used due to reduction of the high pt particle contribution which has non flow correlates with
event plane angle, such as jet. Ideally, the azimuthal angle of event plane is flat distribution.
However normally the reconstructed event plane distribution is not alway flat because imperfectazimuthal acceptance of detector, the shifted beam line and the dead area eﬀect and so on. In
order to correct these eﬀects, we perform two step calibrations. One is “Re-centering collection”
and the other is “Fourier Flattening collection”.

52

CHAPTER 3. ANALYSIS

3.3.3

Event Plane Calibration

In this section, we describe two steps for event plane calibration mentioned in previous section.

3.3.4

Re-centering Calibration

At the first step, we calibrate Qvector distribution of X component and Y component respectively. We perform Re-centering procedure to correct beam shift and non-uniform detector
acceptance. This correction method is used to obtain isotropic distribution of Qvector in laboratory frame. The event by event Qvector X and Y componentsQx , Qy are subtracted by Qvector
X and Y components ⟨Qnx ⟩, ⟨Qny ⟩ averaged over all event and divided by the standard deviations of Qvector X and Y distributions, σnx , σny . With this correction, the corrected Qvector
components become,
Qcorr
=
nx
Qcorr
=
ny

Qnx − ⟨Qnx ⟩σnx
Qny − ⟨Qny ⟩σny(3.26)(3.27)

In practice, this correction is applied for each centrality and Z vertex class. After applying theRe-centering correction, the event plane angle is given fromΨRe−centered= tan−1 (QRe−centered, QRe−centered
)/n
n
ny
nx

(3.28)

Fig.3.11 shows the Qvector distributions determined the combined South and North Bbc detectors before and after the Re-centring correction. In both plots, the horizontal and vertical axes
are the x and y components of the Qvector. In the raw Qvector plot, the mean are slightly shiftedfrom the center due to the shifted beam position and non-uniform azimuthal acceptance. Afterthe Re-centering correction, the corrected Qvector distribution is centered and has RMS = 1. If
the event plane corrected by Re-centering is non-flat distribution, additional calibration step isneeded. In our analysis, we apply “Flattening” calibration to the corrected event plane.

3.3.5

Flattening Calibration

For the second additional event plane correction, we perform Flattening procedure. Basically,
the Re-centering procedure remove almost all beam shift bias and detector acceptance bias.
However there are the residual non-flat components which can not be removed by the Re-centering
procedure. The Flattening calibration is defined as,nΨn = nΨcorr+ n∆ΨRe−centerednnn∆ΨRe−centered= Σk=1 [Ak cos(knΨRe−centered) + Bk sin(knΨRe−centered
)].
n
n
n

(3.29)

Since the performance of EP detectors and the beam condition are changed run by run, we
generate the Re-centering and the Flattening calibration parameters for each centrality class and3.3. EVENT PLANE

53
BbcS+N Re-centered Qx2 vs Qy2

Bbc_Raw_QxQy_K2_H2

Entries 9270044

1

Mean x -0.03013
5

10

Mean y -0.007672

0.5

Bbc_Re_QxQy_K2_H2

Entries

Qy2

Qy2

BbcS+N Raw Qx2 vs Qy2

4

Mean y -5.058e-07

RMS x

0.05974

RMS x

RMS y

0.06171
4

RMS y

10

9270044

Mean x -7.999e-06
104

1

103

2

1

103

0

0

102

102

-2

-0.5

10
10

-4
-1
-1

-0.5

0

0.5

1
Qx2

1

-4

-2

0

2

4
Qx2

1

Figure 3.11: The Qvector distribution determined the combined South and North Bbc detectors
before and after the Re-centring correction. Left:Raw Qvector. Right:Corrected Qvector
Z vertex class in each run. Figure 3.12 shows the 2nd harmonic event plane determined the
combined South and North Bbc detectors for the no correction, only the Re-centering correction
and the Re-centering and the Flattening corrections. The red line is no correction case, theblue line is the application of Re-centering and the green line is the application of Re-centeringand Flattening. Although the Re-centring correction could be enough for the Bbc detector, theadditional Flattening correction make flatter event plane distribution as shown in the insert box
in Fig 3.12.

3.3.6

Event Plane QA

After applying the two step calibration, the event plane distribution for diﬀerent centrality
class, Zvertex class and detectors should be flat in each run. We check the flatness of the event
plane distribution in each run. In order to check the flatness of the event plane distribution, we
fit the distribution with a constant function and the extract χ2 /N DF . The χ2 /N DF value isselected below 3.

3.3.7

Event Plane Resolution

Since the finite number of emitted particles make the azimuthal angle resolution of the event
plane Res{Ψn }, the observed Fourie coeﬃcients vnobs with respect to the event plane have to becorrected by the event plane resolution. If the number of emitted particles that determine theevent plane is large enough, the correction is done by dividing the observed vnobs by the event
plane resolution vn = vnobs /Res{Ψn }. This correction is analytically introduced in Eq.(3.18).
The event plane resolution is obtained from the correlation of the two event planes ⟨cos[n(Ψn,A −54

CHAPTER 3. ANALYSIS

×10

3

94000
180
160
93000
140
92000
120
-1
0
1
Ψ
100
80
60
BbcS+N40
Raw
After re-centering
20
After re-centering+flattening
0-1.5 -1 -0.5
0
0.5
1
2

1.5
Ψ2

Figure 3.12: The 2nd harmonic event plane determined by the combined South and North Bbc
detectors for the no correction, only the Re-centering correction and the Re-centering and the√Flattening corrections in Cu+Au collisions at sN N = 200GeV.

Ψn,B )]⟩ determined by independent rapidity windows A, B. When the distance of the two rapidity
windows A and B is large, the event plane correlation of the two subevents could be expressedas the product of the event plan resolutions of the two subevents. If the distance of the two
subevents is close, other corrections from non-flow eﬀect such as multi-particle correlations fromresonance decay and jets, make the stronger event plane correlation.
⟨cos[n(Ψn,A − Ψn,B )]⟩ = ⟨cos[n(Ψn,A − Ψn − (Ψn,B − Ψn ))]⟩= ⟨cos[n(Ψn,A − Ψn )]⟩⟨cos[n(Ψn,B − Ψn )]⟩+ ⟨sin[n(Ψn,A − Ψn )]⟩⟨sin[n(Ψn,B − Ψn )]⟩= ⟨cos[n(Ψn,A − Ψn )]⟩⟨cos[n(Ψn,B − Ψn )]⟩= Res{Ψn,A }Res{Ψn,B }(3.30)

where Ψn is the true event plane and sin term (odd function) should vanish because the observed
event plane distributions with respect to the true event plane are symmetric. If the two subevents

3.3. EVENT PLANE

55

are symmetric in rapidity and have same multiplicity, the resolution of each of them becomes,√
Res{Ψn,A } = Res{Ψn,B } = ⟨cos[n(Ψn,A − Ψn,B )]⟩.
(3.31)
This method is so-called “2-subevent method” and can be performed in symmetric collision cases.
For asymmetric collision cases or asymmetric rapidity window cases, “3-subevent method” isgenerally used. In the 3-subevent method, the 3 event plane correlations from 3 independent
subevents A,B and C are used to determine the event plane resolution of each of them. The
relation of the resolution and the correlations are√⟨cos[n(Ψn,A − Ψn,B )]⟩⟨cos[n(Ψn,A − Ψn,C )]⟩.
(3.32)
Res{Ψn,A } =⟨cos[n(Ψn,B − Ψn,C )]⟩Res{Ψ2}

In Cu+Cu collisions, the 2nd order event plane is measured by the BbcS, the BbcN and theCnt. Each of the event plane resolutions are determined by the 3-subevent combination BbcSCnt-BbcN. Fig. 3.13 shows the 2nd order event plane resolutions as functions of centrality for
the BbcS, the BbcN and the Cnt.

0.3
CuCu 200GeV
Cnt
BbcS
BbcN

0.25
0.2
0.15
0.1
0.05
0
0

10

20

30

40

50
60
centraliy[%]

Figure 3.13: The 2nd order event plane resolutions of South and North side of Bbc and Cnt asfunctions of centrality in CuCu collisions at 200GeV. The resolutions are determined by BbcSCnt-BbcN 3-subevent combination.

In Au+Au collisions, the 2nd order and 3rd order event planes are measured by the RxnS, theRxnN, the BbcS, the BbcN and the Cnt. The RxnS and the RxnN event plane resolutions are
determined by the 2-subevent combination RxnS-RnxN. Therefore the resolutions for the RxnS
and the RxnN are identical. The other event plane resolutions are determined by the 3-subevent
combination BbcS-Cnt-BbcN. Fig. 3.14 shows the 2nd order and 3rd order event plane resolutions
as functions of centrality for the RxnS, the RxnN, the BbcS, the BbcN and the Cnt.
In Cu+Au collisions, the 1st order coeﬃcient is measured with respect to the Au spectator
neutrons measured by the SmdS on the Au-going side. The event plane resolution for SmdS is

CHAPTER 3. ANALYSIS

0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0

0.5
0.45
0.4
0.35
0.3
0.25
0.2
0.15
0.1
0.05
0
0

Res{Ψ3}

Res{Ψ2}

56

10

20

30

40
50
60
centraliy[%]

AuAu 200GeV
RxnS(N)
Cnt
BbcS
BbcN

10

20

30

40
50
60
centraliy[%]

Figure 3.14: The 2nd and 3rd order event plane resolutions of the South and North side of Bbc,
the South and North side of Rxn and the Cnt as functions of centrality in Au+Au collisions
at 200GeV. The event plane resolution of the Rxn is determined from 2-subevent method. The
others are determined from BbcS-Cnt-BbcN 3-subevent combination. Left panel: 2nd order event
plane. Righ panel: 3rd order event plane.

estimated by the 3-subevent combinations, the Cu spectator neutrons measured by the SmdNon the Cu-going side and the 1st order participant event plane determined the South side of Bbc,
SmdS-BbcS-SmdN. Event by event fluctuations in the participants and spectators will makediﬀerence between the 1st order participant event plane and the 1st order spectator plane. To
cover this uncertainty, the resolution of SmdS is also estimated by the 3-subevent combinations
including the participant event plane observed by the BbcN, SmdS-BbcN-SmdN.
For the 2nd and 3rd harmonics, the event plane resolutions for the BbcS, the BbcN and the Cnt
are estimated by the 3-subevent combinations of them, BbcS-Cnt-BbcN. The combined Southand North side of Bbc event plane resolution is calculated by dividing the correlation of the
BbcSN Ψn,BbcSN and the Cnt Ψn,Cnt with the Cnt resolution Res{Ψn,Cnt }, Res{Ψn,BbcSN } =⟨cos(n[Ψn,BbcSN − Ψn,Cnt ])⟩/Res{Ψn,Cnt }. In Fig. 3.15, the 1st , 2nd and 3rd order event plane
resolutions as functions of centrality are shown.

3.3. EVENT PLANE

57

SMD S: BBC S-SMD S-SMD N

0.3

0.3
Res{Ψn}Res{Ψn}SMD S: BBC N-SMD S-SMD N

0.2

0.2

10

20

30
40
Centrality(%)

0
50 0Res{Ψ3}

0.2

0.1

0.1

0

CNT
BBC S+N
BBC S
0.3
BBC N
Res{Ψn}Res{Ψ2}

Res{Ψ1}0.1

20
40Centrality(%)

0
60 0

10
20
Centrality(%)

30

Figure 3.15: The 1st , 2nd and 3rd order event plane resolutions as functions of centrality. Leftpanel:The 1s event plane for the South side of Smd determined from 3-subevent combinations,
SmdS-BbcS-SmdN and SmdS-BbcN-SmdN. Middle panel:The 2nd order event plane for theSouth, North and combined South and North Bbc and the Cnt. Right panel:3rd order event
plane for the Bbc and Cnt. The 2nd and 3rd order event plane resolutions for the Bbc and the
Cnt are determined from BbcS-Cnt-BbcN 3-subevent combination.

58

CHAPTER 3. ANALYSIS

3.4

Measurements of charged hadron vn at mid-rapidity

The charged hadron vn is measured via the Event Plane method as described at the previous
section. Experimentally, the magnitude of vn are obtained from following formula as shown at
previous section,
vntrue ==vnobs
Res{Ψn }⟨cos(n[ϕ − Ψobs
n ])⟩
obs
⟨cos(n[Ψn − Ψtruen ])⟩(3.33)(3.34)

where ϕ is the azimuthal angle of the track and Ψtrue
are true Event plane azimuthal angle andnobs
Ψn are the observed Event plane azimuthal angle. At the mid-rapidity, inclusive charged hadronand π, K, p vn (n=1 3) are measured for diﬀerent centrality classes. The azimuthal angle ϕ aredetermined by Central Arm detector and the Event plane is estimated by BBC detector. In thisobs
analysis, ⟨cos(n[ϕ − Ψobs
n ])⟩ are the average of the observed event by event cos(n[ϕ − Ψn ]) andthe resolution of Ψobs
n are estimated from 2-subevent or 3-subevent method. Inclusive charged
hadron vn (n=1 3) are measured for each centrality bin and π, K and proton vn are measured.
π, K and p v2 are measured for each 10% centrality class and v1 and v3 are measured for one
wider centrality class.

0.01

E+W

0

0.01

PC3 2.5σ+EMCAL 2.5σ

v1

0.02

v1

0.02

v1

0.02

v1

0.02

0.01 Res{Smd}:SmdS-BbcS-SmdN

0.01

0

0

0

-0.01

-0.01

-0.01

-0.01

-0.02

-0.02

-0.02

-0.02

-0.03

-0.03

-0.03

-0.03

-0.04

SmdS

-0.04

10~20%-0.04

20~30%-0.04

30~40%

40~50%-0.05
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5-0.05
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5-0.05
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5-0.05
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

pT

pT

pT

pT

Figure 3.16: Charged hadron v1 as a function of pT in Cu+Au collisions for diﬀerent centrality
bins

3.5

Measurements of particle identified vn at mid-rapidity

The identified particle vn (PID vn ) at the Cnt region are also measured via the event plane
technique. The used event planes are observed by the SmdS for the 1st order and the BbcSN for
the 2nd and 3rd order flow harmonics. For the PID vn measurements, the azimuthal angles ϕ inthe observed PID vnobs = ⟨cos(n[ϕ − Ψobs
n ])⟩ are used for the each particle species. The particle
species are determined on track by track basis with the PID functions. Fig. 3.19, 3.20 and 3.21
show the PID v2 , v3 and v1 with PID 2 sigma cut in Cu+Au collisions separately.

v2

v2

v2

v2

v2

v2

v2

v2

3.6. MEASUREMENTS OF HADRON VN AT FORWARD/BACKWARD-RAPIDITY

0.25

0.25

0.25

0.25

0.25

0.25

0.25

0.25

0.2

0.2

PC3 2.5σ+EMCAL 2.5σ

0.2

E+W

0.2

0.2

0.2

0.2

0.2

59

BbcSN
0.15

0.15

0.15

0.15

0.15

0.15

0.15

0.15

0.1

0.1

0.1

0.1

0.1

0.1

0.1

0.1

0.05

0.05

0.05

0~5%
0

5~10%

0.05

0.05

0~10%

0.05

10~20%

20~30%

0.05

0.05

30~40%

40~50%

50~60%

0

0
0
0
0
0
0
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5

pT

pT

pT

pT

pT

pT

pT

pT

0.2

0.15

BbcSN

0.1

0.05

0.2

0.2

0.15

PC3 2.5σ+EMCAL 2.5σ 0.15

0.1

0.1

0.05

0~5%
0

0.05

5~10%
0

E+W

v3

v3

v3

v3

v3

Figure 3.17: Charged hadron v2 as a function of pT in Cu+Au collisions for diﬀerent centrality
bins

0.2

0.2

0.15

0.15

0.1

0.1

0.05

0~10%
0

0.05

10~20%
0

20~30%
00 0.5 1 1.5 2 2.5 3 3.5 4 4.5 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5

p

T

pT

pT

pT

pT

Figure 3.18: Charged hadron v3 as a function of pT in Cu+Au collisions for diﬀerent centrality
bins

3.6

Measurements of hadron vn at forward/backward-rapidity

At the forward/backward rapidity, v2 and v3 are measured using the BBC PMTs for each
10% centrality class. Since the BBC can not reconstruct charged hadron tracks, the statisticaluncertainty of vn can not be calculated single particle basis. Thus we extract vn coeﬃcients byfitting the azimuthal PMT angle distribution with respect to the Event Plane, ∆ϕP M T (n) =ϕP M T − Ψn with a Fourier expansion serious formula. The ∆ϕP M T (n) distribution is measured
for each Fourier harmonic. In the ∆ϕP M T (n) distribution, the charge information from single
PMT is used as weight for the ∆ϕP M T (n) bin which corresponds to the azimuthal angle of that
PMT, because each PMT charge information is proportional to the number of particles detected
at the PMT. Since each BBC counter has 64 PMTs, the azimuthal coverage of the BBC is not
uniform. Although the calibrated event plane can reduce such detector acceptance eﬀects, weapply an additional acceptance correction that is event mixing method. In the event mixing
method, the ∆ϕP M T (n) is corrected by being divided by the product of one PMT and the event

v2

v2

v2

v2

0.2

v2

CHAPTER 3. ANALYSIS

v2

60

0.2

0.2

0.2

0.2

0.2

0.15

0.15

0.15

0.15

0.15

0.1

0.1

0.1

0.1

0.1

0.05

0.05

0.05

0.05

0.05

π++π0.15

+

K +K
0.1

-

p+p

0.05
0-10%

10-20%

0

0

0

1

2

3

20-30%
0

0

1

2

3

p

0

1

2

3

p

40-50%
0

0

1

2

3

p

T

T

30-40%
0

0

1

2

p

T

50-60%
0

3

0

1

2

3

p

T

p

T

T

0.15

π++πK++K
p+p

v1

v3

Figure 3.19: PID v2 as a function of pT in Cu+Au collisions

0.02

0

0.1

-0.02
0.05
-0.04

0-30%
0

1

2

3

pT

-0.06
0

π++πK++Kp+p1

10-50%
2

3

p (GeV/c)T

Figure 3.20: PID v3 as a function of pT
in Cu+Au collisions for diﬀerent centrality
bins

Figure 3.21: PID v1 as a function of pT in
Cu+Au collisions

plane probability distributions.

C(∆ϕ)n =P (ϕP M T , Ψn )P (ϕP M T )P (Ψn )(3.35)

Since PMTs fired by emitted charged particles and the Event Plane preserve physical correlationin real (same) event, the measurement of the azimuthal angle of fired PMTs with respect to the
Event Plane become the pair of one fired PMT and the Event Plane probability distribution.
If those pair is measured in mixed (diﬀerent) events, the pair distribution corresponds to the
product of one fired PMT and the Event Plane probability distributions because there is no
physical correlation between fired PMTs and the Event plane in diﬀerent event. Because thelimited acceptance eﬀect is included in real event and mixed event, the limited acceptance eﬀect
is canceled by taking ration of real event and mixed event. As described before, vn coeﬃcients3.6. MEASUREMENTS OF HADRON VN AT FORWARD/BACKWARD-RAPIDITY

61

are extracted by fitting the above correlation function with following Fourier function,r(∆ϕ)n = 1 + 2vnobs,pmt cos(n[ϕpmt − Ψobs
n ])(3.36)⟨cos(n[ϕpmt − Ψobs
n ])⟩obs,pmtvn(3.37)

vnpmt=

vncorr,pmt =(3.38)

Res{Ψn }

v2

v2

v2

Run12CuAu 0~5%
0.03

v2

0.03

v2

v2

v2

where ϕpmt is the azimuthal angle of PMT and vnobs,pmt is free parameter in r(∆ϕ)n function.
The vn coeﬃcient extracted from fitting function is divided by the Event plane resolution tocorrect the uncertainty of the estimated Event Plane. The vn coeﬃcients measured with BBC
PMTs have to be converted to track based vn , since BBC can not reconstruct charged particle
track. In order to obtain track based vn at BBC rapidity, we run Single Monte Carlo simulation to
produce correction parameters which translate PMT based vn to track based vn . To remove nonflow eﬀect, we employ the Event Plane estimated from the detector placed at opposite rapidity
side of vn measurement rapidity region.

5~10%
0.03

10~20%
0.03

20~30%
0.03

30~40%
0.03

40~50%
0.03

50~60%

Cnt

0.02

0.02

0.02

0.02

0.02

0.02

0.02

0.01

0.01

0.01

0.01

0.01

0.01

0.01

0

0

-4

-2

0

2

4

0
-4

-2

0

2

η

4

0
-4

-2

0

2

η

4

0
-4

-2

0

2

η

4

0
-4

-2

0

2

η

0

4

-4

-2

0

2

η

4

-4

-2

0

2

4

η

η

Run12CuAu 0~5%
0.006

v3

0.006

v3

v3

v3

Figure 3.22: Charged hadron v2 as a function of η in Cu+Au collisions for diﬀerent centrality
bins

5~10%
0.006

10~20%
0.006

20~30%

Cnt

0.004

0.004

0.004

0.004

0.002

0.002

0.002

0.002

0

0

-4

-2

0

2

4

0
-4

η

-2

0

2

4

0
-4

η

-2

0

2

4

-4
η

-2

0

2

4
η

Figure 3.23: Charged hadron v3 as a function of η in Cu+Au collisions for diﬀerent centrality
bins

62

CHAPTER 3. ANALYSIS

3.6.1

Single particle simulation for forward/Backward-rapidity vn measurements

Since the BBC detector can not reconstruct charged hadron tracks, the measurements of the
PMT based vn include the background eﬀect such as γ conversion. In order to estimate thebackground contribution, we run single particle Monte Carlo simulation with PISA which is thePHENIX simulation software based on the GEANT libraries. The PMT based vn including
background contribution is corrected by following steps
1. Single particle Monte Carlo simulation (input)
2. Single particle Monte Carlo simulation with the PISA to evaluate material interactions andsecondery particle generation
3. Reconstruct Bbc information from the PISA hits(output)
4. Measure vn using the reconstructed Bbc information
5. Estimate the correction factor
The correction factor Rn is estimated by comparing the input vn and the output vn from the
PISA simulation.
vninput
Rn = output(3.39)
vn
where vninput and vnoutput are the vn estimated by the single particle simulation without and with
the PISA respectively. We obtain the corrected vn as
vn = Rn vnpmt
The default set-up of single particle simulation is summarized in Table 3.1.
Table 3.1: Default setting for single particle simulation

Parameter
Event typeSpeciespT distribution
pT rangeη disribution
η rangeazimuth range

Value
Single particle
π ± and π0T
pT exp( −pT )0 < pT < 5 GeV/c
Trapezoidal−5 < η < 5−π/2 < ϕ < π/2(3.40)

3.7. MEASUREMENTS OF CHARGED HADRON DNCH /Dη AT FORWARD/BACKWARD-RAPIDITY

3.7

Measurements of charged hadron dNch /dη at forward/backwardrapidity

Measurements of dN/dη at forward/backward-rapidity (3 < |η| < 3.9) is also performed in√
Au+Au, Cu+Au and Cu+Cu collisions at sN N = 200 GeV using Bbc detectors. As notedin the previous section about the vn measurements at forward/backward-rapidity, Bbc can notreconstruct charged particle tracks. Thus the Bbc can not remove the background contribution
and count the number of charged particles directly. Figure. 3.24 shows the Bbc charge sum whichis ADC information corresponding the charged particle multiplicity as a function of centrality
in Au+Au, Cu+Au and Cu+Cu collisions. In Cu+Au collisions, BbcS and BbcN indicate Augoing side and Cu-going side respectively. The Bbc charge sum become the largest in the central
collisions and the smallest in the peripheral collisions, because the charged particle multiplicity
and the background particles increase from the peripheral to the central collisions. In the next
subsection, we will explain how to remove the background contribution using simulation study.

3.7.1

Single particle simulation for forward/backward-rapidity charged particle multiplicity dNch /dη measurements

In order to measure the number of charged particles correctly, the ADC in Bbc for charged
particles is converted into the number of charged particles with the correction factor R. For
estimating the correction factor R, we perform the single particle simulation which is used forthe correction in the forward/backward vn measurements described in the previous section. The
simulation set up is completely same as the vn simulation listed in Table 3.1. Both of correction
factors R for the multiplicity measurements and Rn for azimuthal measurements are obtained
simultaneously in the same simulation run. The correction factor R is defined as following,
R=

1
2
,
sim
3 ADCoutput(3.41)

where 23 is the fraction of charged π and 1 is the number of emitted particle in each event.
sim
ADCoutput
is the ADC for 1 particle passing through Bbc. Once the correction factor R is
estimated, the charge particle multiplicity is given by following formula,dNch= R × ADC obs ,dη(3.42)

where ADC obs is the ADC for charged particles and backgrounds measured at Bbc.

3.8

Initial spatial anisotropy

In this study, the initial spatial anisotropy εn are estimated from the Glauber model Monte
Carlo simulation[7]. The Glauber model Monte Carlo simulation is the simplest approach todescribe the initial collision geometry as described. This simulation model is widely used for
determination of centrality and the initial conditions in hydrodynamical models and event generators.

CHAPTER 3. ANALYSIS

Bbc charge sum

64

1000

AuAu BbcS
AuAu BbcN
CuAu BbcS
CuAu BbcN
CuCu BbcS
CuCu BbcN

500

0
0

20

40

60
80
centrality(%)

Figure 3.24: Bbc charge sum as a function of centrality in Au+Au, Cu+Au and Cu+Cu collisions.
In Cu+Au collisions, BbcS and BbcN indicate Au-going side and Cu-going side respectively

3.8. INITIAL SPATIAL ANISOTROPY

65

Table 3.2: Nuclear density parameters used in Glauber modelSpecies
Au
Cu

3.8.1

R[fm]
6.38
4.20

a[fm]
0.535
0.5960σN N [mb]42
42Glauber model Monte Carlo simulation

In the Glauber model Monte Carlo simulation, the nucleons of nucleus A and the nucleons of
nucleus B are distributed inside A and B respectively according to the Woods-Saxon distribution.
ρ(r) =

1
1 + exp( r−Ra )(3.43)

where r is the distance of each nucleon from the center of nucleus, R is the nuclear radius and a isthe skin depth. The Glauber model assumes that the nucleons travel on straight line trajectories
and the inelastic nucleon-nucleon cross section is independent of the number of collisions. Under
these assumptions, an A-B collision is treated as independent binary nucleon-nucleons collisions.
The impact parameter b is randomly sampled. A nucleon-nucleon collision occur if the distanceof two nucleons d in the transverse plane(x-y plane) becomes less than√d ≤ σN N /π
(3.44)
where σN N is the total inelastic nucleon-nucleon cross section. The value of R, a and σN N
are listed in Table 3.2. The nucleons participating in collisions provide the number of binary
collisions ⟨Ncoll ⟩, the number of participant nucleons ⟨Npart and the initial spatial anisotropy εn .
The impact parameter and ⟨Npart ⟩ distribution in Cu+Au collisions are shown in Fig.3.25. The
centrality class for this simulation is determined by ⟨Npart ⟩ distribution.

3.8.2

Initial spatial anisotropy

The origin of the azimuthal anisotropy of produced particles with respect to the event plane
is considered to be the initial spatial anisotropy. The initial spatial anisotropy is converted tothe anisotropy in the momentum space through the pressure gradient. Thus it is important tocharacterize the initial geometry. The initial spatial anisotropy is defined as the following formula[34],√⟨rn cos(nϕ)⟩2 + ⟨rn sin(nϕ)⟩2εn =(3.45)⟨rn ⟩⟨rn cos(n[ϕ − Ψn,PP ])⟩= −,(3.46)⟨rn ⟩where r and ϕ are the distance and azimuthal angle of participant nucleon in the polar coordinate
where the center of participant nucleons is the origin and Ψn,PP is the average angle of over all

66

CHAPTER 3. ANALYSIS

4000
104
3500
3000

103

2500
2000

102

1500
1000

10

500
0
0

2

4

6

8 10 12 14 16 18 20
Impact parameter(fm)

1
0

50

100

150

200
250
<Npart>Figure 3.25: Left:Impact parameter distribution in Monte Carlo Glauber model. Right:Npart
distribution estimated from Monte Carlo Glauber model.
participant nucleons so called the participant plane. The participant plane is determined foreach harmonictan−1 [⟨rn sin(nϕ)⟩, ⟨rn cos(nϕ)⟩] π+ .
(3.47)
Ψn,PP =
nn
Figure 3.26 shows 2nd and 3rd order eccentricities in Monte Carlo Glauber model as a function√of centrality for Au+Au, Cu+Au and Cu+Cu collisions at sN N = 200GeV.

3.9

Systematic Uncertainty

3.9.1

Systematic Uncertainties at mid-rapidity vn

Contributions to systematic uncertainties of mid-rapidity vn are from the following sources• Event plane• Background contribution to vn measurement• East and West arm acceptance diﬀerence• Particle identification purity
Uncertainty from Event plane determination

3.9. SYSTEMATIC UNCERTAINTY

0.8

0.8

ε2

ε3

0.6

MC-Glauber
Cu+Cu
Cu+Au
Au+Au

εn

εn

0.6
0.4

0.4

0.2
0

67

0.2
0

20
40
60
centrality(%)

0
80 0

20
40
60
centrality(%)

80

Figure 3.26: 2nd and 3rd order eccentricity ε2 in Monte Carlo Glauber model as a function ofcentrality for Au+Au, Cu+Au and Cu+Cu collisions

The systematic uncertainties from the event plane measurements using the diﬀerent detectors
are considered not to depend on pT . For the v1 measurement, the uncertainties are estimatedby comparing the v1 with respect to SMD S with alternately BBC N or BBC S used for theestimation of SMDS event plane resolution. For the v2 and v3 measurements, the uncertainties
are obtained by comparing the v2 and v3 measured with respect to BBC S and BBC N. Figure
3.28 and 3.29 show the v2 and v3 measured with respect to BBC S and BBC N separately as afunction of pT for diﬀerent centrality bins.

0.01

E+W

0.01

0

PC3 2.5σ+EMCAL 2.5σ

0

0.02

v1

v1

0.02

v1

0.02

v1

0.02

Res{SmdS} w BbcN

0.01

0.01

Res{SmdS} w BbcS

0

0

-0.01

-0.01

-0.01

-0.01

-0.02

-0.02

-0.02

-0.02

-0.03

-0.03

-0.03

-0.03

-0.04

-0.04

10~20%-0.04

20~30%-0.04

30~40%

40~50%-0.05
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5-0.05
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5-0.05
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5-0.05
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

pT

pT

pT

pT

Figure 3.27: v1 as a function of pT with respect to SMD S for diﬀerent centrality bins
The tracking selection can not remove all of background tracks which may distort themagnitude of vn . The background is caused by γ conversion, particle decay and track misreconstruction. The systematic uncertainties from the background contribution is obtained byvarying the diﬀerent PC3 and EMCAL matching width cut. The following are the diﬀerent PC3

v2

v2

v2

v2

v2

v2

v2

CHAPTER 3. ANALYSIS

v2

68

0.25

0.25

0.25

0.25

0.25

0.25

0.25

0.25

BbcS
BbcN
BbcSN

0.2

0.15

0.1

0.05

0.2

PC3 2.5σ+EMCAL 2.5σ

E+W

0.2

0.2

0.2

0.2

0.2

0.15

0.15

0.15

0.15

0.15

0.15

0.15

0.1

0.1

0.1

0.1

0.1

0.1

0.1

0.05

0.05

0~5%
0

0.2

5~10%

0.05

0.05

0~10%

0.05

10~20%

20~30%

0.05

0.05

30~40%

40~50%

50~60%

0

0
0
0
0
0
0
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5

pT

pT

pT

pT

pT

pT

pT

pT

0.2

BbcS
BbcN
BbcSN

0.15

0.1

0.05

0.2

0.2

0.15

PC3 2.5σ+EMCAL 2.5σ 0.15

0.1

0.1

0.05

0~5%
0

E+W

0.05

5~10%

v3

v3

v3

v3

v3

Figure 3.28: v2 as a function of pT with respect to BBC S, BBC N and BBC S and N combined
event plane for diﬀerent centrality bins

0.2

0.2

0.15

0.15

0.1

0.1

0.05

0~10%

0

0

0.05

10~20%
0

20~30%
00 0.5 1 1.5 2 2.5 3 3.5 4 4.5 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5

p

T

pT

pT

pT

pT

Figure 3.29: v3 as a function of pT with respect to BBC S, BBC N and BBC S and N combined
event plane for diﬀerent centrality bins
and EMCAL matching width cut.
• |pc3| < 2.5 σ, |emcal| < 2 σ• |pc3| < 2.5 σ, |emcal| < 3 σ• |pc3| < 2 σ, |emcal| < 2.5 σ• |pc3| < 3 σ, |emcal| < 2.5 σ
Figure 3.30, 3.31 and 3.32 show the v1 , v2 and v3 measured with the diﬀerent PC3 and EMCAL
matching width cut as a function of pT for diﬀerent centrality bins.
Uncertainty from East and West

Systematic uncertainties from acceptance were obtained from the measurements using Eastand West separately. The diﬀerences in the vn measurements using East and West arm arecaused by diﬀerent alignment and performance. Figure 3.33, 3.34 and 3.35 show the v1 , v2 andv3 measured with East and West separately as a function of pT for diﬀerent centrality bins.

3.9. SYSTEMATIC UNCERTAINTY

SmdS

0.01

v1

pc3 3σ+emcal 2.5σ

0.01 Res{Smd}:SmdS-BbcS-SmdN

pc3 2.5σ+emcal 2.5σ

0.01

0.02

v1

0.02

v1

0.02

v1

0.02

69

0.01

pc3 2σ+emcal 2.5σ

0

0

-0.01

-0.01

-0.02
-0.03
-0.04

10~20%

0

0

-0.01

-0.01

-0.02

-0.02

-0.02

-0.03

-0.03

-0.03

-0.04

-0.04

pc3 2.5σ+emcal 3σ
pc3 2.5σ+emcal 2σ

20~30%-0.04

30~40%

40~50%-0.05
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5-0.05
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5-0.05
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5-0.05
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

pT

pT

pT

pT

0.15

0.15

0.1

0.1

0.1

0.2

0.05

0.05

v2

v2

v2

0.25

0.25

0.25

0.25

E+W

0.05

0~5%
0

0.25

0.25

pc3 3σ +emcal 2.5σ
pc3 2.5 σ +emcal 2.5σ 0.2
pc3 2σ +emcal 2.5σ
pc3 2.5 σ +emcal 3σ
pc3 2.5 σ +emcal 2σ 0.15

0.2

v2

0.25

BbcSN

v2

0.25

v2

v2

v2

Figure 3.30: v1 as a function of pT with respect to SMD S for diﬀerent centrality bins

5~10%

0.2

0.2

0.2

0.2

0.2

0.15

0.15

0.15

0.15

0.15

0.1

0.1

0.1

0.1

0.1

0.05

0.05

0~10%

0.05

10~20%

20~30%

0.05

0.05

30~40%

40~50%

50~60%

0

0
0
0
0
0
0
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5

pT

pT

pT

pT

pT

pT

pT

pT

0.2

0.2

BbcSN

0.15

0.15

0.1

pc3 3σ+emcal 2.5σ
pc3 2.5σ+emcal 2.5σ0.15
pc3 2σ+emcal 2.5σ
pc3 2.5σ+emcal 3σ
pc3 2.5σ+emcal 2σ

E+W

0.1

0.05

0~5%
0

0.2

0.1

0.05

0.05

5~10%
0

v3

v3

v3

v3

v3

Figure 3.31: v2 as a function of pT with respect to BBC S and N combined event plane fordiﬀerent track matching cuts and centrality bins

0.2

0.2

0.15

0.15

0.1

0.1

0.05

0~10%
0

0.05

10~20%
0

20~30%
00 0.5 1 1.5 2 2.5 3 3.5 4 4.5 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5

p

T

pT

pT

pT

pT

Figure 3.32: v3 as a function of pT with respect to BBC S and N combined event plane fordiﬀerent track matching cuts and centrality bins
Uncertainty from hadron misidentification

For the PID vn measurements, an additional systematic uncertainty from the purity of particle
species needs to be considered. The purity of particle species is aﬀected by the particle masssquare distribution width cut. This uncertainty is estimated from the diﬀerence among PID vn
with the diﬀerent PID cuts. Veto cut is also applied to establish boundaries between particle
species. The diﬀerent PID cuts with the veto cut are following

70

CHAPTER 3. ANALYSIS

SmdS

0.01

0.01

East
West
E+W

0

PC3 2.5σ+EMCAL 2.5σ

v1

0.02

v1

0.02

v1

0.02

v1

0.02

0.01 Res{Smd}:SmdS-BbcS-SmdN

0.01

0

0

0

-0.01

-0.01

-0.01

-0.02

-0.02

-0.02

-0.02

-0.03

-0.03

-0.03

-0.03

-0.04

-0.04

-0.04

-0.01

10~20%

20~30%-0.04

30~40%

40~50%-0.05
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5-0.05
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5-0.05
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5-0.05
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

pT

pT

pT

pT

v2

v2

v2

v2

0.25

0.25

0.25

0.25

0.25

0.25

0.25

BbcSN

0.15

East
0.2
West
East+West
0.15

0.1

0.1

0.2

0.05

PC3 2.5σ+EMCAL 2.5σ

0.05

0.2

0.2

0.2

0.2

0.2

0.2

0.15

0.15

0.15

0.15

0.15

0.15

0.1

0.1

0.1

0.1

0.1

0.1

0.05

0~5%
0

v2

0.25

v2

v2

v2

Figure 3.33: v1 as a function of pT with East, West and East and West combined central arms
for diﬀerent centrality bins

5~10%

0.05

0.05

0~10%

0.05

10~20%

20~30%

0.05

0.05

30~40%

40~50%

50~60%

0

0
0
0
0
0
0
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5

pT

pT

pT

pT

pT

pT

pT

pT

0.2

BbcSN

0.15

v3

0.2

0.2

0.2

0.2

East
0.15
West
East+WestPC3 2.5σ+EMCAL 2.5σ 0.15

0.15

0.15

0.1

0.1

0.1

0.1

0.1

0.05

0.05

0~5%
0

v3

v3

v3

v3

Figure 3.34: v2 as a function of pT with East, West and East and West combined central arms
for diﬀerent centrality bins

0.05

5~10%
0

0.05

0~10%
0

0.05

10~20%
0

20~30%
00 0.5 1 1.5 2 2.5 3 3.5 4 4.5 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5

p

T

pT

pT

pT

pT

Figure 3.35: v3 as a function of pT with East, West and East and West combined central arms
for diﬀerent centrality bins

• 1.5 σ with 2σ veto
• 2 σ with 2σ veto• 2.5 σ with 2σ veto

3.9. SYSTEMATIC UNCERTAINTY

3.9.2

71

Systematic Uncertainties of forward/backward vn measurements

Sources of the systematic uncertainties of the forward/backward-rapidity vn are following• Event plane determination (experimental data)• pT spectra(simulation data)• dN/dη distribution (simulation data)• pT dependence of vn (simulation data)• η dependence of vn (simulation data)
Uncertainty from Event plane determination

v2

v2

Bbc

v2

Run7AuAu 0~5%
0.03

v2

0.03

v2

v2

v2

For the η dependence of v2 and v3 measurements, the uncertainties from event plane determination are included in systematic uncertainties. The systematic uncertainties from event
plane measurements are obtained by comparing the v2 and v3 measured with respect to BbcSand BbcN in Cu+Cu and Cu+Au collisions. In Au+Au collisions, the measurements using RxnS
and RxnN are also included since Rxn EP is available. Fig. 3.36 and 3.37 show the v2 and v3
measured with respect to Bbc, Cnt and Rxn in Au+Au collisions separately as a function of pT
for diﬀerent centrality bins.

5~10%0.03

10~20%
0.03

20~30%
0.03

30~40%
0.03

40~50%
0.03

0.02

0.02

0.02

0.02

0.02

0.01

0.01

0.01

0.01

0.01

50~60%

Rxn

0.02

0.02

0.01

0.01

0

Cnt

0

-4

-2

0
η

2

4

0
-4

-2

0
η

2

4

0
-4

-2

0
η

2

4

0
-4

-2

0
η

2

4

0
-4

-2

0

2

4

0
-4

-2

η

0
η

2

4

-4

-2

0

2

4

η

Figure 3.36: Integrated pT v2 as a function of η for diﬀerent event plane selections in Au+Au√
collisions at sN N = 200GeV

Uncertainty from Input pT spectra in Single particle simulation
The systematic uncertainties for pT spectra are estimated by comparing three inverse slope
parameters T of pT spectra. The tested three inverse slope parameters T are 235, 300 and 150
MeV. The pT spectra for π ± and π0 in Au+Au 200 GeV are measured by BRAHMS experimentand paramerterized using the pT exp(−pT /T ) [36]. In the Ref. [36], the inverse slope parameters
are extracted from fit to the spectra and are between 230 to 240 MeV.

Bbc

v3

Run7AuAu 0~5%
0.006

v3

v3

0.006

v3

CHAPTER 3. ANALYSIS

v3

72

5~10%
0.006

10~20%
0.006

20~30%
0.006

0.004

0.004

0.004

0.002

0.002

0.002

30~40%

Rxn

0.004

0.004

0.002

0.002

0

Cnt

0

-4

-2

0

2

4

0
-4

η

-2

0

2

4

0
-4

-2

η

0

2

4

0
-4

-2

η

0

2

4

-4

-2

η

0

2

4

η

T

P(p )

Figure 3.37: Integrated pT v3 as a function of η for diﬀerent event plane selections in Au+Au√
collisions at sN N = 200GeV

1
p exp(-p /235 ×10 )
-3

-1

T

10

T

p exp(-p /300 ×10-3)TT

p exp(-p /150 ×10-3)

10-2

T

T

10-3
10-4
10-5
0

0.5

1 1.5

2

2.5

3

3.5

4

4.5 5
p (GeV)T

Figure 3.38: Input pT spectra for three inverse parameters
Uncertainty from input η probability distribution in Single particle simulation

The systematic uncertainties from single particle pseudorapidity distributions are obtainedby comparing two diﬀerent probability distributions P (η) shown in Fig. 3.39. The two η probability distributions are selected from most central Au+Au collision and most peripheral Cu+Cu
collision [37]. The probability distributions in central collision case and peripheral collision case

3.9. SYSTEMATIC UNCERTAINTY73

are described as follows,
0.3η + 1.6
P (η) =

(η < −2)1(−2 ≦ η ≦ 2)
−0.3η + 1.6 (η > 2)
0.2η + 1.4

P(η)

P (η) =

(3.48)(η < −2)1(−2 ≦ η ≦ 2)
−0.2η + 1.4 (η > 2)

(3.49)

1
0.8
0.6
0.4
central0.2
0-5

peripheral
-4

-3

-2

-1

0

1

2

3

4

5
η

Figure 3.39: Input η probability distribution. The blue data points represent central Au+Au
collisions case and the red data points corresponds to peripheral Cu+Cu collisions case.

Uncertainty from input vn (η) shape in Single particle simulation
The simulated shape of η dependence of vn might influence the measured vn . In out simulation
study, four type of slope of vn (η) at |η| > 2 are tested. Since the PHOBOS experiment measuredη dependence of v2 in 200GeV Au+Au collisions[25], [39], the slope for central Au+Au collision(315%) case is used as the default setting and the slopes for and peripheral collision case(25-50%)

74

CHAPTER 3. ANALYSIS

is included in the systematic uncertainties in our simulation. For the v3 simulation, the same
slope parameters as used in the v2 study are used. The two input vn are expressed using theslope parameters respectively.
71(η < −1)6η + 6P (vn (η)) =1

− 1 η + 7
6

6


1
9


8η + 8P (vn (η)) = 1

− 1 η + 9
8

8(−1 ≦ η ≦ 1)

(3.50)(η < −1)(η < −1)(−1 ≦ η ≦ 1)

(3.51)(η < −1)v2v3

Additionally, the asymmetric η dependence of vn are tested and included in the systematic
uncertainties because the η dependence of vn are asymmetric distribution in heavy ion generator
study. Fig. 3.40 shows the input η dependence of vn for the four slopes. In the Fig.3.40, thetype1 represents the slope for

0.04

0.04
default slope

0.03

0.03

type1 slope
type2 slope

0.02

0.02

0.01

0.01

0
-5 -4 -3 -2 -1 0 1 2 3 4 5η0-5 -4 -3 -2 -1 0 1 2 3 4 5
ηFigure 3.40: Input vn as a function of η for four type slopes

Uncertainty from magnitude of vn (pT )3.9. SYSTEMATIC UNCERTAINTY

75

v3

v3

v2

v2

Since the magnitude of vn widely vary with centrality class, the systematic uncertainties ofthe magnitude of vn are obtained by comparing the central, mid-central and peripheral collisioncases. Fig. 3.41 shows the input v2 and v3 as a function of pT at mid-rapidity and forwardrapidity. In out simulation study, the pT dependence of vn at mid-rapidity are referred from thePHENIX vn results [21] as the input. At forward rapidity, the maximum pT dependence of vn
is determined by the slope of default η dependence of vn . In the Fig. 3.41, the default setting
represent mid-central collision case, the type1 corresponds to central collision case and the type2
is peripheral collision case. Fig. 3.42 shows the input η dependence of v2 and v3 for the default,

0.2

0.2

0.2

0.2

0.15 0<η<10.15 3<η<40.15 0<η<10.15 3<η<4default
type1
type2

0.1

0.1

0.1

0.1

0.05

0.05

0.05

0.05

0
0

1

2

0
0

3

1

2

0
0

3

p

p

T

T

1

2

0
0

3
p

T

1

2

3
p

T

Figure 3.41: Input vn (η) probability disribution
type1 and type2.
Uncertainty from the shape of vn (pT )Additionally, we studied the influence of the shape of pT dependence of vn . The systematic
uncertainties are estimated by varying the pT bin for the maximum value of pT dependence ofvn . In the default setting, the pT bin for the maximum value of vn is 2 GeV. For the systematic
study, we test 1.5 GeV and 2.5 GeV case as shown in Fig.3.43. In the Fig.3.43, default means
the maximum value of vn is at 2 GeV, type1 is 2.5 GeV and type2 is 1.5 GeV case. In Fig.3.44,
the input η dependence of vn for the pT bin cases are shown.

v3

CHAPTER 3. ANALYSIS

v2

76

default
0.04

type1

0.04

type2

0.02

0.02

0
-5 -4 -3 -2 -1 0 1 2 3 4 5η0-5 -4 -3 -2 -1 0 1 2 3 4 5
η

v3

v3

v2

v2

Figure 3.42: Input vn (η) probability disribution

0.2

0.2

0.2

0.2

0.15 0<η<10.15 3<η<40.15 0<η<10.15 3<η<4default
type1
type2

0.1

0.1

0.1

0.1

0.05

0.05

0.05

0.05

0
0

1

2

0
0

3
p

T

1

2

0
0

3
p

T

1

2

0
0

3
p

T

Figure 3.43: Input vn (η) probability disribution

1

2

3
p

T

77

v3

v2

3.9. SYSTEMATIC UNCERTAINTY

default
0.04

0.04

type1
type2

0.02

0.02

0
-5 -4 -3 -2 -1 0 1 2 3 4 5η0-5 -4 -3 -2 -1 0 1 2 3 4 5
ηFigure 3.44: Input vn (η) probability disribution

78

CHAPTER 3. ANALYSIS

3.9.3

Systematic Uncertainties of forward/backward dNch /dη measurements

Since the correction factor for charge particle multiplicity dN/dη is obtained from the same
simulation run in the vn measurements, the systematic sources are same as the vn measurements
except for the pT and η dependences of vn distributions. Sources of the systematic uncertainties
of the forward/backward dNch /dη measurements are following,• pT spectra (simulation data)• dN/dη distribution (simulation data)
For the systematic uncertainty from the diﬀerent pT spectra, the input pT spectra is shown in
3.38. As described in the previous section, the probability function for the pT distribution is
given by the pT ∗ exp(−pT /T ). The single particle is generated according to that function withthe three inverse slope parameters T 235, 300 and 150 MeV.
The systematic uncertainty for dN/dη distribution is also same as the vn measurements. The
input dN/dη is shown in Fig. 3.39.

3.9.4

Summary of systematic uncertainties

Here are the tables for systematic uncertainties of the mid-rapidity and the forward/backwardrapidity vn and the forward/backward-rapidity dN/dη.
Table 3.3: Systematic uncertainties in the v1 measurements at mid rapidity
Centrality class
10%–20%20%–30%30%–40%40%–50%

Event-plane
24%
12%
10%
13%

Background(absolute value)
4×10−55×10−5
6×10−5
6×10−5

Acceptance (absolute value)
3×10−3
2×10−3
2×10−3
2×10−3

3.9. SYSTEMATIC UNCERTAINTY

79

Table 3.4: Systematic uncertainties given in percent on the v2 measurements at mid rapidity
Centrality class0%–5%
5%–10%
0%–10%
10%–20%20%–30%30%–40%40%–50%50%–60%

Event-plane
3%
2%
2%
1%
1%
2%
2%
4%

Background
3×10−1 %2×10−1 %2×10−1 %1×10−1 %1×10−1 %2×10−1 %2×10−1 %4×10−1 %Acceptance
2%
2%
2%
2%
2%
3%
3%
3%

Table 3.5: Systematic uncertainties given in percent on the v3 measurements at mid rapidity
Centrality class0%–5%
5%–10%
0%–10%
10%–20%20%–30%

Event-plane
1%
3%
2%
4%
7%

Background
1%
1%
1%
1%
1%

Acceptance
3%
3%
3%
8%
22%

Table 3.6: Systematic uncertainties in the measured v1 for identified particles at mid rapidity
speciespion (absolute value)kaon (absolute value)
proton (absolute value)pT ≤ 2 GeV/c
1×10−3
1×10−3
1×10−3pT ≥ 2 GeV/c
2×10−3
3×10−3
3×10−3

Table 3.7: Systematic uncertainties in percent on the measured v2 and v3 for identified particles
at mid rapidity
speciespion
kaon
protonpT ≤ 2 GeV/c
3%
3%
3%

pT ≥ 2 GeV/c
5%
10%
5%

80

CHAPTER 3. ANALYSIS

Table 3.8: Event Plane systematic uncertainties given in percent on the v2 measurements at
forward/backward rapidity.
South/North
0%–5%
South
North
5%–10%
SouthNorth
10%–20%SouthNorth
20%–30%SouthNorth
30%–40%SouthNorth
40%–50%SouthNorth
50%–60%
South
North

Au+Au

Cu+Au

Cu+Cu

1%
2%

4×10−1 %4×10−1 %

1%
3%

1%
1%

2%
1%

3×10−1 %2%

2×10−1 %1%

2%
1%

1×10−1 %2%

3×10−1 %
1%

2%
1%

1%
5×10−1 %4×10−1 %2%

2%
1%

1×10−1 %1×10−1 %

1%
2%

2%
2%

2×10−1 %2×10−1 %

1%
3%

1%
3%

2%
4×10−1 %3.9. SYSTEMATIC UNCERTAINTY

81

Table 3.9: Event Plane systematic uncertainties given in percent on the v3 measurements at
forward/backward rapidity.
South/North
0%–5%
South
North
5%–10%
SouthNorth
10%–20%SouthNorth
20%–30%
South
North

Au+Au

Cu+Au

3%
5%

5×10−1 %1%

3%
5%

1×10−1 %
1%

7%
6%

1%
1×10−1 %

8%
7%

4%
3%

Table 3.10: Correction factors and Systematic uncertainties given in percent on the simulated
vn (n=2,3) at forward/backward rapidity
South/North
v2
South
North
v3
South
North

Correction factor

Systematic uncertainty

74%
74%

4%
3%

67%
66%

4%
4%

Table 3.11: Correction factors and Systematic uncertainties given in percent on the simulateddNch /dη at forward/backward rapidity
South/North
dNch /dη
South
North

Correction factor

Systematic uncertainty

49%
49%

7.4%
7.6%

Chapter 4

Results
4.1

Charged hadron vn at mid-rapidity√
The final result of charged hadron vn at mid-rapidity in Cu+Au sN N = 200 GeV with
event plane technique is obtained. The charged hadron vn as a function of pT in diﬀerent
centrality classes are shown in Fig. 4.1, 4.2 and 4.3. The red solid symbols represent the vn
data and the shaded boxes are the systematic uncertainties. In Fig. 4.1, the v1 measurements
are shown. The v1 measurements are performed with respect to the Ψ1 event plane determinedby the Au spectator neutrons. Since spectator neutrons going positive rapidity region have beenused as reference axis for v1 measurements in previous conventions. To align with the previous
conventions, the sign of v1 is flipped so that eﬀectively the v1 is measured with respect to thefirst order event plane determined by spectator neutrons from the Cu nucleus. In all centrality
classes, the absolute values of v1 are consistent with zero within the systematic uncertainties
below pT = 1GeV/c and increases with pT . The negative values of v1 at high pT region indicate
the high pT particles at mid-rapidity are emitted towards Cu hemisphere side. Although in low
pT regions, the more particles are expected to move towards the opposite direction of high pTparticles emission due to the momentum conservation eﬀect, the values of v1 at low pT region
can not be distinguished whether the signals are positive values or negative values beyond thelarge systematic uncertainties. From the central to the peripheral collisions, the absolute value
decrease since the participant zone become the symmetric shape in peripheral collisions.
For the v2 and v3 measurements shown in Fig. 4.2 and 4.3, the v2 and v3 values are positiveand increase with pT for all centrality classes. The v2 show strong centrality dependence wherethe values increase from the most central 0-10 % and the mid-central 30-40 % centrality classes.
This trend is considered to be originated from the initial spatial ellipticity of participant nucleons
ε2 because the initial spatial ellipticity shown in Fig. 3.26 increases in the peripheral collisions.
Unlike the v2 measurements, the v3 components show weak centrality dependence as seen in
those in symmetric collisions. In the symmetric collisions, the triangular flow is driven by theevent by event initial spatial fluctuation of participant nucleons. In Cu+Au collisions, the eventby event spatial fluctuation are also the origin of the triangular flow.
In all three Fourier harmonics, the values of these coeﬃcients increase with pT up to about
3 GeV/c. Except for the v1 measurements, the signal decrease with pT above around 3 GeV/c.
82

4.1. CHARGED HADRON VN AT MID-RAPIDITY

83

This trend might indicate a change of particle production mechanism for example high pT particle
emission with fewer interactions. It is interesting to look system size dependence of the charged

0.02

0.02

0.02

0.02

0

0

0

Cu+Au 200GeV
h±

v1

v1

v1

v1

0
-0.02

-0.02

-0.02

-0.02

-0.04

-0.04

10~20%02

p (GeV/c)

4

-0.04

20~30%02

4

0p (GeV/c)

T

-0.04

30~40%
2

4

40~50%0p (GeV/c)T24p (GeV/c)T

T

Figure 4.1: Charged hadron v1 at mid-rapidity (|η| < 0.35) as a function of pT in Cu+Au
collisions

0.15

Cu+Au 200GeV
0.15
h±

0.15

0.15

0.1

0.1

0.1

v2

0.2

v2

0.2

v2

0.2

v2

0.2

0.05

0.05

0.05

0.05

0~5%

5~10%

2

p (GeV/c)

0~10%

0

0

0.2 0

0.1

10~20%

04 0.2 0

T

2

4

0

0.2 0

p (GeV/c)

2

4

0.2 0

p (GeV/c)

0.15

0.15

0.15

0.15

0.1

0.1

0.1

v2

v2

0.05

0.05

0.05

0.05

20~30%002
p (GeV/c)T

430~40%002
p (GeV/c)
T

4

2

4

p (GeV/c)
T

v2

T

v2

T

0.1

40~50%002
p (GeV/c)T

450~60%002

4p (GeV/c)T

Figure 4.2: Charged hadron v2 at mid-rapidity (|η| < 0.35) as a function of pT in Cu+Au
collisions
hadron vn at the same center of mass energy measured by the PHENIX Collaboration. In early
study, the PHENIX experiment has measured azimuthal anisotropic flow in Au+Au and Cu+Cu√
collisions at sN N = 200GeV. By comparing vn in diﬀerent collision systems and centralities,
one can study the influence of the initial spatial anisotropy on the measured azimuthal anisotropy.
Fig. 4.4 shows the v2 values as a function of pT in Au+Au, Cu+Au and Cu+Cu collisions fordiﬀerent centrality classes. In each centrality class, we observe the v2 coeﬃcients in Cu+Au are

84

CHAPTER 4. RESULTS

0.15

0.15

0.15

0.15

0.15

0.1

0.1

0.1

0.1

Cu+Au 200GeV
h±

v3

v3

v3

v3

v3

0.1

0.05

0.05

0.05

0.05

0.05

0~5%

5~10%

0

0

0

2

4

0~10%

10~20%002

p (GeV/c)

402pT (GeV/c)

T

20~30%

0
4

0
0

2

pT (GeV/c)

4

0

2

pT (GeV/c)

4pT (GeV/c)

Figure 4.3: Charged hadron v3 at mid-rapidity (|η| < 0.35) as a function of pT in Cu+Au
collisions
between those in Cu+Cu and Au+Au collisions. Except for central 0 to 10% class, the observed
v2 values are not ordered according to the magnitude of initial ellipticity in diﬀerent systems
obtained in Monte Carlo Glauber model shown in the left side of Fig. 3.26. In Fig. 4.5, thecharged hadron v3 coeﬃcients as a function of pT in Au+Au and Cu+Au collisions for three
diﬀerent centrality classes are compared. Since the measurements of v3 in Cu+Cu collisions havenot been performed, the data points of v3 measurements in Cu+Cu collisions are not shown.
Unlike the v2 coeﬃcients, the v3 in Cu+Au collisions shows larger values than Au+Au collisions.
This ordering of the v3 values are same as the initial triangularities shown in the right side ofFig. 3.26. Although the vn (pT ) values in the diﬀerent collision systems are compared at same
centrality bins, it is diﬃcult to conclude that the diﬀerence of the vn (pT ) values among thecollision systems is caused by the initial geometry. Because at each centrality class, the numberof participants Npart or the number of produced particles dN/dη which are related to vn values
depend on collision systems.

0.25

0.25

Cu+Au 200 GeV

Au+Au 200 GeV
PRC92, 034913
Cu+Cu 200 GeV
PRC92, 034913

0.25

0.25

0.25

0.25

0.2

0.2

0.2

0.2

0.2

0.15

0.15

0.15

0.15

0.15

0.15

0.1

0.1

0.1

0.1

0.1

0.1

v2

0.2

0.05

0.05

0.05

0~10%
0

0

0

1

2

3

4

0.05

10~20%

5 0

0
1

2

3

4

0.05

20~30%

5 0

0
1

2

3

4

0.05

30~40%

5 0

40~50%
0

1

2

3

4

5 0

50~60%
0

1

2

3

4

5 0

1

2

3

4

5

pT (GeV/c)

Figure 4.4: Charged hadron v2 at mid-rapidity (|η| < 0.35) as a function of pT in Au+Au,
Cu+Au and Cu+Cu collisions for diﬀerent centrality classes

4.1. CHARGED HADRON VN AT MID-RAPIDITY

0.14

0.12

0.12
Au+Au 200GeV PRL 107. 252301

0.12

0.1

0.1

0.1

0.08

0.08

0.08

v3

Cu+Au 200GeV

v3

0.14

0.14

v3

85

0.06

0.06

0.06

0.04

0.04

0.04

0.02
0

0-10%
0

1

2
3
pT (GeV/c)

4

0.02
0
5 0

10-20%
1

2
3
pT (GeV/c)

4

0.02
0
5 0

20-30%
1

2
3
pT (GeV/c)

45Figure 4.5: Charged hadron v3 at mid-rapidity (|η| < 0.35) as a function of pT in Au+Au andCu+Au collisions for diﬀerent centrality classes

86

CHAPTER 4. RESULTS

For studying the eﬀect of initial geometry on vn coeﬃcients by comparing diﬀerent collisionsystems, the measurements of pT integrated vn as a function of Npart or dN/dη are commonlyused because the diﬀerence of < vn > values among the collision systems at a given Npart ordN/dη bin is considered to be caused by the initial geometry. pT integrated vn is calculated∫∑
dpT dN/dpT v2 (pT )dNi /dpT vn,i (pT )∫< vn >=
= i∑
,
(4.1)
dpT dN/dpT
i dNi /dpT
where dN/dpT is the transverse momentum distribution, and vn (pT ) is the vn as a function of
pT . They are obtained by fitting functions to the experimental data. At low pT region where
there no experimental data points, dN/dpT and vn are estimated by extrapolating the fitting
results. In our < vn > measurements, we integrate pT from 0 to 3GeV/c.
Fig. 4.6 and 4.7 show vn with 3rd polynomial functions as a function of pT in Au+Au, Cu+Au
and Cu+Cu collisions. The vn in the three collision systems are parameterized by the fitting
functions.

v2

0.3

0.3

0.3

0.3

0.3

0.3

0.25

Cu+Au 200 GeV0.25
Polynomial Fit

Au+Au 200 GeV
PRC92, 034913 0.25
Polynomial Fit

Cu+Cu 200 GeV
PRC92, 034913 0.25
Polynomial Fit

0.25

0.25

0.2

0.2

0.2

0.2

0.2

0.2

0.15

0.15

0.15

0.15

0.15

0.15

0.1

0.1

0.1

0.1

0.1

0.1

0.05

0.05

0.05

0~10%

0
0

1

2

3

4

0.05

10~20%

0
5 0

1

2

3

4

0.05

20~30%

0
5 0

1

2

3

4

0.05

30~40%

0
5 0

1

2

3

4

40~50%

0
5 0

1

2

3

4

50~60%

0
5 0

1

2

3

4

5

pT(GeV/c)

Figure 4.6: Charged hadron v2 with polynomial fitting functions at mid rapidity in Au+Au,
Cu+Au and Cu+Cu collisions

0.12

0.12

v3

0.1

0.12

0.12

0.12

0.12

0.1

0.1

0.1

0.1

Au+Au 200 GeV

Cu+Au 200 GeV

PRL 107. 252301

0.1

Polynomial Fit

Polynomial Fit

0.08

0.08

0.08

0.08

0.08

0.08

0.06

0.06

0.06

0.06

0.06

0.06

0.04

0.04

0.04

0.04

0.04

0.04

0.02

0.02

0.02

0.02

0.02

0~10%

10~20%

20~30%

30~40%

40~50%

0

0

0

0

0

0.02

0
0

1

2

3

4

0

1

2

3

4

0

1

2

3

4

0

1

2

3

4

0

1

2

3

4

50~60%
0

1

2

3

4

pT(GeV/c)

Figure 4.7: Charged hadron v3 with polynomial fitting functions at mid rapidity in Au+Au andCu+Au collisions
In Fig. 4.8, the transverse distribution at mid rapidity in Au+Au collisions measured by4.1. CHARGED HADRON VN AT MID-RAPIDITY

87

PHENIX[16] is shown. To parameterize these pT spectra, the following function is used,po nf (pT ) = A
po + pT(4.2)

where A, p0 and n are free parameters. These parameters are obtained by fitting procedure.
Since PHENIX have not measured pT spectra in Cu+Au and Cu+Cu collisions, we estimate pT102
dN2 [(GeV/c)-2]12π p Nevtdp dη

Au+Au 200GeV

10
1

1

-2

10

10-3
10-4 010-1

0~10%Power Fit
20~30%
Power Fit
40~50%
Power Fit
60~70%
Power Fit

1

T

T

10-1

1

10

T

T

dN2 [(GeV/c)-2]2π p Nevtdp dη

102

-2

10

10-3

2

10-4

3
0
p (GeV/c)
T

10~20%
Power Fit
30~40%
Power Fit
50~60%
Power Fit
70~80%
Power Fit

12

3
pT(GeV/c)

Figure 4.8: Charged hadron pT spectra at mid rapidity in Au+Au collisions [16]distributions for Cu+Cu and Cu+Au collisions using the fitting results of Au+Au pT spectra
based on the fact that dN/dη and < pT > do not depend on collision system but Npart [47],[18].
Fig. 4.9 shows the free parameters as a function of Npart for Au+Au pT spectra obtained by thefitting results in Fig. 4.8. In Fig. 4.9, we fit A, p0 and n with polynomial functions to extract
their values at corresponding Npart for Cu+Au and Cu+Cu collisions. Then we make the pT
distributions for Cu+Cu and Cu+Au.
po (Npart,Cu+Cu ))n(Npart,Cu+Cu )po (Npart,Cu+Cu ) + pT
po (Npart,Au+Au )f (pT , Au + Au) = A(Npart,Au+Au )()n(Npart,Au+Au )po (Npart,Au+Au ) + pT
f (pT , Cu + Cu) = A(Npart,Cu+Cu )(

(4.3)

(4.4)
Figure 4.10 shows the pT integrated v2 (0< pT <3GeV ) as a function of Npart in Au+Au, Cu+Au
and Cu+Cu collisions. Since the data points of v2 in Au+Au and Cu+Cu measured by the
PHOBOS (|η| <1,pT > 0GeV) are available, the our data points of v2 are compared to PHOBOS’s
results. In all collision systems, the pT integrated v2 increase from the central to the peripheral
collisions. Although pT and η ranges are diﬀerent between PHENIX and PHOBOS, the pT
integrated v2 from the two experiments are good agreement within the systematic uncertainties.

88

CHAPTER 4. RESULTS

600

20

4

n

A

p0

400

10

2

AuAu 200GeV

200

Polynomial Fit

Polynomial Fit

Linear Fit

0

0

100

200

300

0

400
Npart

0

100

200

300

400
Npart

0

0

100

200

300

400
Npart

n
o
Figure 4.9: Parameters A, p0 and n of fitting function f (pT ) = A pop+p
for charged hadron pTT
spectra at mid rapidity in Au+Au collisions [16]

0.1

0.1

Au+Au 200GeV

0.1

Cu+Cu 200GeV

Cu+Au 200GeV

0.06
0.04
0.02

0.08

0.08

0.06

0.06

v2

Au+Au 200GeV PHOBOS

v2

v2

0.08

0.04

0.04

0.02

0.02

Cu+Cu 200GeV PHOBOS

0<pT<3(GeV/c)00

100 200 300 400
Npart

0
0

100 200 300 400
Npart

0
0

100 200 300 400
Npart

Figure 4.10: pT integrated v2 (0 < pT < 3GeV/c) at mid rapidity Au+Au, Cu+Cu and Cu+Au
collisions

4.2. π ± , K ± , P AND P VN RESULTS

89

π ± , K ± , p and p vn results

4.2

Figure 4.11, 4.12 and 4.13 show the final results of particle species dependence of vn at√
mid-rapidity in Cu+Au sN N =200 GeV collisions. The results presented in the Fig.s are charge
combined π ± , K ± , p and p. The red solid circles represent π ± , the open squares represent K ± andthe open cross symbols represent p and p. The shaded boxes are the systematic uncertainties fromthe hadron misidentification based on the PID purity. In the systematic uncertainties, the othersystematic uncertainties such as the event plane determination and the acceptance diﬀerence are
not shown since these uncertainties are common for π ± , K ± , p and p± . To improve the statisticaluncertainties for the v1 and v3 , we have merged centrality classes into a single centrality class.
For the v2 and v3 measurements, there are two trend. In low pT region, the vn for the lightest
hadron is the largest and those for the heaviest hadron is the smallest. Hydrodynamical theory
predict same mass dependence in low pT caused by radial flow eﬀect that makes all particles
move in the same velocity. Above mid-pT region, this particle mass trend is reversed, such thatthe anisotropy for baryons is larger than that for mesons. This baryon and meson dependence at
the high pT , namely valence quark number dependence, is associated with the partonic flow at
the QGP phase and subsequent hadronization by quark coalescence. These two trends for PID
vn measurements have been seen in the symmetric collision systems at RHIC and LHC. However
the PID v1 measurements show mass ordering at the mid pT region and do not show baryon andmeson splitting at the high pT . Although the values of PID v1 are positive at the low pT region,
if the full systematic uncertainty is taken into account, the sign of the bulk directed flow can not
be concluded.

π++π-

v2

-

(b)
0.2

0.15

0.15

v2

+

K +K
p+p

0.15

(a)
0.2

0.1

0.1

0.05

0.1

0.05

0.05

0-10%

10-20%

020-30%

0
0

1

0.2

2
pT(GeV/c)

3

0
0

1

(d)
0.2

2
pT(GeV/c)

3

0

0.05

0.05
40-50%

50-60%

0
1

2
pT(GeV/c)

3

(f)

0.1

30-40%
0

3

v2
0.1

0.05

2
pT(GeV/c)

0.15

v2
0.1

1

(e)
0.2

0.15

v2

0.15

0

(c)

v2

PHENIX Cu+Au 200 GeV

0.2

0
0

1

2
pT(GeV/c)

3

0

1

2
pT(GeV/c)

Figure 4.11: PID v2 as a function of pT in Cu+Au collisions3

90

CHAPTER 4. RESULTS

0.15

PHENIX
Cu+Au 200 GeV
π+++π- K +K
p+p

0.02
0

0.1

v1

v3

PHENIX Cu+Au 200 GeV
v1 w.r.t. Cu spectator neutrons

-0.02

0.05
0-30%
0

1

2
pT (GeV/c)

3

-0.04
-0.06
0

π++π+
K +K
p+p

10-50%

1

2

3

p (GeV/c)T

Figure 4.12: PID v3 as a function of pT
in Cu+Au collisions for diﬀerent centrality
bins

4.3

Figure 4.13: PID v1 as a function of pT in
Cu+Au collisions

Charged hadron vn at forward/backward rapidity

The measurements of the η dependence of v2 in Au+Au and Cu+Cu collisions are comparedto the PHOBOS Au+Au and Cu+Cu collisions [25, 38, 39] in Fig. 4.14 and 4.15. The v2 shownin Fig. 4.14 and 4.15 are corrected with the correction factors Rn estimated by the full Geantbased Monte Carlo simulation described in the previous analysis section. The v2 measurements
in Au+Au and Cu+Cu collisions show the highest values at mid-rapidity and the lower values atforward/backward-rapidity. The PHENIX results for Au+Au and Cu+Cu collisions corresponds
to the open cross and the open triangle. and the PHOBOS results are the open circles. Goodagreements between the PHENIX results and the PHOBOS results are seen at the three rapidity
regions for Au+Au and Cu+Cu collisions, which indicate the correction factor R is estimatedcorrectly.
√
The final results of v2 and v3 at forward/backward-rapidity in Cu+Au collisions at sN N
= 200 GeV are obtained. The results presented in Fig.4.16 and 4.18 are the v2 and v3 as a√function of η in Au+Au, Cu+Au and Cu+Cu collisions at sN N = 200 GeV. Both of the v2
and v3 measurements are corrected for background eﬀect. Since the v3 measurements in Cu+Cu
collisions have not been performed, the data points in Cu+Cu collisions are not shown. The
solid symbols represent the data points and the shaded boxes are the systematic uncertainties.
The dashed straight lines are drawn to look the slope between the data points. In all collisionsystems, the values of v2 at forward/backward-rapidity are smaller than those at mid-rapidity.
In symmetric Au+Au and Cu+Cu collisions, the η dependence of vn is η symmetric distribution.
On the other hand, a diﬀerence of the v2 values in Au-going side (negative η side) and Cugoing side (positive η side) is seen in mid-central collision classes. The v2 measurements at
forward/backward-rapidity in all collision systems have the centrality dependence as seen at
mid-rapidity. From central to peripheral, the value of v2 increases in all collision systems. The
forward/backward asymmetry of v2 in Cu+Au collisions becomes small in the most central andthe peripheral collision class.
In the left side of Fig. 4.17, the v2 at forward/backward rapidity as a function of Npart in4.3. CHARGED HADRON VN AT FORWARD/BACKWARD RAPIDITY

91

0.06

0.08
0.06

v2

v2

0.04
0.04

0.02
0.02

0~40% Cu+Cu
PHENIXPHOBOS

0~40% Au+Au
PHENIXPHOBOS0-5 -4 -3

-2

-1

0

1

2

3

4

0
-5 -4 -3

5

-2

-1

0

1

2

3

4

5

η

η

0.08

0.08

0.06

0.06

0.06

0.04
0.02

v2

0.08

v2

v2

Figure 4.14: The v2 measurements for 0 ∼ 40 % centrality class in Au+Au and Cu+Cu comparedto the PHOBOS results

0.04

0.04

0.02

0.02

PHOBOS 3~15%
PHENIX 0~20%

PHOBOS 15~25%
PHENIX 10~30%

PHOBOS 25~50%
PHENIX 30~50%

0-5 -4 -3 -2 -1 0 1 2 3 4 5

0-5 -4 -3 -2 -1 0 1 2 3 4 5

0-5 -4 -3 -2 -1 0 1 2 3 4 5

η

η

η

Figure 4.15: The v2 measurements for diﬀerent centrality classes in Au+Au and Cu+Cu compared to the PHOBOS results

Au+Au, Cu+Au and Cu+Cu collisions are shown. In Au+Au and Cu+Cu collisions, the data
points shown in Fig. 4.17 are averages of v2 values at forward and backward rapidities because
rapidity dependence of v2 in symmetric collision systems is symmetric. In Cu+Au collisions, thev2 values at Au-going rapidity (−3.9 < η < −3) and these at Cu-going rapidity (3 < η < 3.9)
are shown with blue and red circles, respectively. Like the v2 measurements at mid-rapidity,
the system size dependence of v2 holds at forward/backward rapidity. In Cu+Au collisions,the v2 for Au-going side shows larger values than that for Cu-going side except for the mostcentral and peripheral collisions. This Npart dependence is predicted by the ε2 shown in theright side of Fig. 4.17. In Fig. 4.17, we add Cu+Au ε2,A(B) which are ε2,Au estimated from
Au participant nucleons and ε2,Cu estimated from Cu participant nucleons. The red open circles
represent ε2,Cu+Au given by all participant nucleons in Cu and Au nuclei. In both ε, the common

92

CHAPTER 4. RESULTS

0.08

0.08

0.08

0.08

0.08

0.08

0~10%
0.06

10~20%
0.06

20~30%
0.06

30~40%
0.06

40~50%
0.06

0.04

0.04

0.04

0.04

0.04

0.04

0.02

0.02

0.02

0.02

0.02

0.02

Au+Au 200GeV

0.06

Cu+Au 200GeV

50~60%

v2

Cu+Cu 200GeV

0

0

-4

-2

0

2

4

0
-4

-2

0

2

4

0
-4

-2

0

2

4

η

0
-4

-2

0

2

4

0
-4

-2

0

2

4

-4

-2

0

2

4

Figure 4.16: v2 at BBC rapidity in the three collision systems for diﬀerent centrality classes
participant plane estimated from all participant nucleons in Au and Cu nuclei are used.
εn,Au =
εn,Cu =

< rn cos(n[ϕAu − Ψn,PP,Cu+Au ]) >< rn >n< r cos(n[ϕCu − Ψn,PP,Cu+Au ]) >< rn >

(4.5)
(4.6)

where n is harmonics order and ϕAu and ϕCu are azimuthal angle of participant Au and Cu
nucleon positions respectively. Ψn,PP,Cu+Au is the participant plane for n harmonics given by
combined Au and Cu participant nucleons. The ε2,Au values are generally higher than the ε2,Cuvalues. However the diﬀerence of ε2,Au and ε2,Cu at the most central and the peripheral bins
become small.
For the v3 measurements shown in Fig. 4.18, the v3 in Au+Au collisions is symmetric
distribution and the v3 in Cu+Au collisions is asymmetric distribution. Unlike the mid-rapidity
v3 measurements, the centrality dependence of v3 in Cu+Au collisions decrease from central toperipheral collision class. The values of v3 in Au-going side are always higher than those in
Cu-going side.
For the v3 measurements at the forward/backward rapidity, the forward/backward asymmetryof v3 in Cu+Au collisions is seen in Fig. 4.18 and the left figure of Fig. 4.19. From the peripheral
to the central collisions, the v3 values in Au+Au and Cu+Au collisions increase. In Fig. 4.19, thev3 values for Au going side are always larger than those for Cu going side in Cu+Au collisions.
Surprisingly the v3 values at Au-going side in Cu+Au collisions have the larger values than thatin Au+Au collisions. This trend is not seen in the initial spatial triangularity shown in the rightside figure of Fig. 4.19.

4.4

Charged hadron dN/dη

Figure 4.20 shows the dN/dη at 3 < |η| < 3.9 for charged hadrons in Au+Au and Cu+Cu√
collisions at sN N = 200GeV in comparison to the results in PHOBOS [12, 53]. Our measurements presented in Fig. 4.20 are corrected from Bbc charge sum to number of charged particles

4.4. CHARGED HADRON DN/Dη

0.06

93

0.8

Au+Au 200 GeV
Cu+Au 200 GeV Au-going
Cu+Au 200 GeV Cu-going
Cu+Cu 200 GeV
Hadron v2 3<|η|<3.9

0.6
ε2

v2

0.04

MC-Glauber
Au+Au
Cu+Au Au
Cu+Au
Cu+Au Cu
Cu+Cu

0.02

0.4
0.2

0
0

0
0

100 200 300 400
Npart

100 200 300 400
Npart

Figure 4.17: Left:pT integrated v2 as a function of Npart at forward/backward rapidity in Au+Au,√
Cu+Au and Cu+Cu collisions at sN N = 200GeV. Right:ε2 in Monte Carlo Glauber model asa function of Npart in Au+Au, Cu+Au and Cu+Cu collisions

0.03

0.03
Au+Au 200GeV

0.03

0~10%

0.03

10~20%

20~30%

30~40%

Cu+Au 200GeV

0.02

0.02

0.02

0.01

0.01

0.01

v3

0.02

0.01

0

-4

-2

0

2

4

0

-4

-2

0

2

4

0
η

-4

-2

0

2

4

0

-4

-2

0

2

4

Figure 4.18: v3 at BBC rapidity in the two collision systems for diﬀerent centrality classes
by the correction factors estimated from the full Geant based Monte Carlo simulation. A good
agreement between our measurements and PHOBOS results in both of Au+Au and Cu+Cu collisions. Figure 4.21 shows the measurements of rapidity dependence of dN/dη in Au+Au, Cu+Au
and Cu+Cu collisions for diﬀerent centrality bins. The mid-rapidity data points are obtained
from the previous PHENIX publication [47]. Like vn measurements, dN/dη at forward/backward
rapidity is smaller than that at mid-rapidity in all collision systems. Au+Au collisions show thelargest values of dN/dη at mid, forward/backward-rapidities for all centrality bins, while Cu+Cu
collisions show the smallest values in the three collision systems. The measurements of dN/dηin Cu+Au collisions are always between those in Au+Au and Cu+Cu collisions. This systemdependence of dN/dη is supposed to arise from the system dependence of number of participants.
In Cu+Au collisions, Au-going side have the larger values of dN/dη than Cu-going side except

94

CHAPTER 4. RESULTS

0.8

Au+Au 200 GeV
Cu+Au 200 GeV Au-going

0.01

ε3

v3

MC-Glauber
Au+Au
Cu+Au Au
Cu+Au
Cu+Au Cu

0.6

Cu+Au 200 GeV Cu-going
Hadron v3 3<|η|<4

0.4

0.005
0.2
0
0

0
0

100 200 300 400
Npart

100 200 300 400
Npart

Figure 4.19: Left:pT integrated v3 as a function of Npart at forward/backward rapidity in Au+Au√
and Cu+Au collisions at sN N = 200GeV. Right:ε3 in Monte Carlo Glauber model as a function√of Npart in Au+Au and Cu+Au collisions at sN N = 200GeVPHOBOS PRL91 052303|η|=3.45

Au+Au 200GeV

400

dN/dη

dN/dη

|η|=3.45

for peripheral collisions. In peripheral collisions, the diﬀerence of dN/dη in Au-going side andCu-going side becomes small, because the number of participants in Au nucleus and Cu nucleus
would become similar.

200
Cu+Cu 200GeV
150

PHOBOS PRL102 142301

BbcS

BbcS

BbcN

BbcN

100

200
50
0
0

20

40

60
80
centrality(%)

0
0

20

40

60
80
centrality(%)

Figure 4.20: The measurements of charged particle multiplicity dN/dη at 3 < |η| < 3.9 in Au+Au√
and Cu+Cu collisions at sN N = 200GeV in comparison to PHOBOS results[12],[53]4.4. CHARGED HADRON DN/Dη

95

0~10%

10~20%

20~30%

600

600

600

600

30~40%

40~50%

600

600

400

400

200

200

Au+Au 200GeV

50~60%dN/dη

Cu+Au 200GeV

400

400

400

400

200

200

200

200

0

0

-4

-2

0

2

4

0
-4

-2

0

2

4

Cu+Cu 200GeV

0
-4

-2

0

2

4

η

0
-4

-2

0

2

4

0
-4

-2

0

2

4

-4

-2

0

2

4

Figure 4.21: Charged particle multiplicity dN/dη as a function of η in Au+Au, Cu+Au and√Cu+Cu collisions at sN N = 200GeV

Chapter 5

Discussions
5.1

Interpretation of sign of v1 (pT )5.1.1

Approach with Monte Carlo Glauber model

In asymmetric collisions, the sign of mid-rapidity v1 value reveals the direction of azimuthal
angle where the density gradient becomes largest. Since particles are pushed to high pT by radialflow, the high pT particles move along the direction of the largest density gradient, while the low
pT particles move the opposite direction due to momentum conservation eﬀect Ref. [34, 40].
Figure 5.1 shows the nucleon participant distribution in Cu+Au collisions for 20-30 % centrality class. In this figure, positive x direction is toward the Cu nucleus. This participant
distribution is re-plotted with respect to the x-y position of center of mass. The red area denotes
the larger number of participants, and the blue area denotes the lower number of participants.
Although it looks symmetric distribution, the width of green and yellow bands is narrower innegative x direction ( Au nucleus side ) than that in positive x direction ( Cu nucleus side ), whichindicates the larger density gradient in Au nucleus side. For the further study, the number ofparticipant and density gradient in x direction are shown in Fig. 5.2. The number of participant
distribution in x direction looks almost symmetric shape. In the density gradient distribution,
the value of the density gradient is lower at the center and the surface of the participant zone
and become higher at the middle of the participant zone. The Au side show the higher valueof maximum density gradient than the Cu side. Figure 5.3 show the correlation of the participant plane for 1st harmonic Ψ1,PP , which indicates the highest number density direction with r3
weighting, with respect to the impact parameter Ψimp , < cos(Ψ1,PP − Ψimp ) >, as a function ofcentrality. In our calculation, we treat Ψimp as Cu nucleus side. Therefore positive and negative
sign indicates the highest density gradient is Cu and Au side, respectively. In Fig. 5.3, thecorrelation shows the negative values, and the absolute value of the correlation increase from 0
to 30% centrality and decrease from 30 to 80 % centrality. In our measurements of directed flow
(Fig. 4.1), the maximum absolute value decreases from 10-20 % to 40-50 % centrality. Thus
the centrality dependence of directed flow measurements is consistent with the participant plane
correlation from Monte Carlo Glauber model.

965.1. INTERPRETATION OF SIGN OF V1 (PT )

97

×10

3

10
Cu+Au 200GeV
20~30%

8

300

6
4

250

2

200

0
150

-2
-4

100

-6
50

-8
-10
-10 -8 -6

-4

-2

0

2

4

6

8

10

0

Cu+Au 200GeV

4

20~30%

3

2d(x-<x>)5

dNpart(x-<x>)

Npart(x-<x>)

Figure 5.1: Participant nucleon distribution in Cu+Au collisions for 20-30 % centrality classAu sideCu side

1.5
1

2
0.5

1
0
-6

-4

-2

0

2

4

6

x-<x>

0
-6

-4

-2

0

2

4

6

x-<x>

Figure 5.2: Number of participant distribution and density gradient distribution in x direction in Cu+Au collisions for 20-30% centrality class. Left:Number of participant distribution.
Right:Density gradient distribution

CHAPTER 5. DISCUSSIONS<cos(Ψ1 - Ψimp)>

98

0

-0.1

-0.2
0

20

40

60

80
100
centrality(%)

Figure 5.3: Azimuthal correlation between 1st order participant plane Ψ1,PP and impact parameter Ψimp . Negative sign indicates the highest density gradient is Au nucleus side.

5.1. INTERPRETATION OF SIGN OF V1 (PT )

5.1.2

99

Approach with Blast wave model

In order to study the direction of larger pressure gradient in Cu+Au collisions, we apply
the Blast wave model to the directed flow results at mid-rapidity. The Blast wave model iscommonly used in the study of the property of the medium at the freeze-out stage with its
freeze-out temperature Tf and its integrated ϕ maximal flow velocity in radial direction ρ0 . In
our study, the extended Blast wave model which incorporate the azimuthal anisotropy of radial
velocity ρn and spatial density sn for nth harmonics defined as the coeﬃcients in the Fourier
expansion of the empirically introduced quantities ρ(n, m, ϕ, r) = ρ0 [1 + 2ρn cos(nϕ)]r/Rmax
and S(n, m, ϕ) = 1 + 2sn cos(nϕ) [24]. The charged hadron transverse momentum spectra andazimuthal anisotropy at the freeze-out of the medium are predicted by the following formulas,dN
pT dpT∝vn =∫ Rmax∫rdr dϕmT I0 (αt )K1 (βt ),∫ Rmax∫rdr dϕ cos(nϕ)In (αt )K1 (βt )S(n, m, ϕ),∫ Rmax∫rdr dϕI0 (αt )K1 (βt )S(n, m, ϕ)

(5.1)
(5.2)

where In and K1 are modified Bessel functions of the first and second kinds, αt and βt are(pT /Tf ) sinh(ρ(n, m, ϕ, r)) and (mT /Tf ) cosh(ρ(n, m, ϕ, r)), respectively. In above formulas, theBlast wave model parameters Tf , ρ0 , ρn and sn are determined simultaneously by fitting single
particle transverse momentum pT spectra and azimuthal anisotropy vn (pT ) for π ± , K ± and p + p
with minimizing χ2 /N DF . Tf and ρ0 are mainly determined by the single particle spectra,
whereas anisotropy parameters ρn and sn are primarily determined by vn .
√Since the π ± , K ± and p + p single particle spectra in Cu+Au collisions at sN N = 200 GeV
have not been measured at PHENIX, we utilize the Tf and ρ0 parameters which are estimated in
Au+Au collision at same energy by PHENIX [24]. The Tf and ρ0 for lowest Npart,AuAu ∼ 94 bin
are selected to be similar Npart,CuAu ∼ 84 bin corresponding to 10 − 50% centrality class. The
fitting ranges of directed flow for π pm , K ± and p + p are 0.5 < pT < 1.1 GeV/c, 0.4 < pT < 1.3
GeV/c and 0.6 < pT < 1.7 GeV/c,respectively. These fitting ranges are same as it is done forthe determination of the Tf and ρ0 in Au+Au collisions [24].
In Fig. 5.4, the identified π ± , K ± and p + p v1 as a function of pT for 10 50% centrality classfitted with the Blast wave model is shown. The experimental data points are shown as markers
and the smooth curves are the Blast wave fitting functions. The fitting functions for the fitting
ranges are shown as bold line. The Blast wave model well reproduce the experimental data at
the each fitting range. The Blast wave model parameters of freeze-out temperature Tf , radially∫∫integrated flow velocity < ρ >= ρ0 [r/Rmax ]rdr/ rdr, directed anisotropy of radial velocity
ρ1 and directed anisotropy of spatial density s1 for 10 - 50 % centrality class are shown in Fig.
5.5. The Tf and < ρ > are same values in the PHENIX publication. The anisotropy parameterρ1 and s1 are extrapolated from the fitting results. The ρ1 shows the negative value, while thes1 has the positive value. Since the v1 measurements performed with respect to the Cu nucleus
side, the negative ρ1 indicates stronger radial flow in Au nucleus side, which is predicted by theprevious Glauber Monte Carlo model approach. On the other hand, the positive s1 indicatesmore particles are emitted toward Cu nucleus side.

CHAPTER 5. DISCUSSIONS

v1

100

0.01
10~50%

0-0.01

π++πK++K
p+p

-0.02
-0.03
-0.04

0

1

2

3
4
p (GeV/c)T

Figure 5.4: Identified charged hadron v1 at mid-rapidity fitted with the Blast wave model for
10-50 % centrality class in Cu+Au collisions

5.1. INTERPRETATION OF SIGN OF V1 (PT )

101

0.13

0.7

Tf (GeV)

0.65

<ρ>

0.12
0.11

0.6

0.55
0.1
0

20

40

60

0

centrality(%)

40

60

centrality(%)

0.02

0.02

0.01

0.01

ρ

s1

1

20

0

0

-0.01

-0.01

0

20

40

60

centrality(%)

0

20

40

60

centrality(%)

Figure 5.5: The Blast wave model parameters for identified charged hadron v1 at mid-rapidity
for 10-50 % centrality class in Cu+Au collisions. Top left: Freeze-out temperature Tf , Top right:
Averaged radial velocity < ρ >, Bottom left:Directed anisotropy of radial velocity ρ1 , Bottom
right:Directed anisotropy of spatial density s1 . The parameters Tf and < ρ > are obtained fromthe PHENIX Au+Au publication [24]. The anisotropy parameters ρ1 and s1 are extrapolated
from the fitting.

102

CHAPTER 5. DISCUSSIONS

5.2

Interpretation of forward/backward asymmetry of dNch /dηwith Monte Carlo Glauber model

Wounded nucleon model
In the collision dynamics, the number of produced particles is related to the amount of thereleased the degree of freedom. In the 1970s, experimentally it was found that the numberof produced charged particles in nucleon-nucleus collisions was proportional to the number ofparticipants [46],Nch ∝ Npart,pA = Npart,A + 1,
(5.3)
where Npart,pA is the total number of participants, Npart,A is the number of participants in A
nucleus and 1 is the projectile nucleon. Surprisingly this experimental feature also helds innucleus-nucleus collisions at RHIC energy. Figure 5.6 shows the charged particle multiplicity
per participant nucleon in Au+Au collisions at several collision energies measured by PHOBOS
collaboration [48]. From the central to the peripheral collisions, the normalized charged particle
multiplicities at the three energies are almost constants, which indicates the charged particle
production is determined by the number of participants, Nch ∝< Npart /2 > or one nucleon
produces same charged particles. Since participant nucleon is also called ”wounded nucleon”,
this experimental fact is called ”wounded nucleon model”.

Figure 5.6: Total charged particle multiplicity per participant nucleon in Au+Au collisions at√
sN N = 200, 130, 62.4 and 19.6 GeV measured by PHOBOS [48]Weighted Npart scaling based on wounded nucleon model
As it is seen in Fig. 4.21, the number of charged particles dNch /dη in Cu+Au collisions
shows the forward/backward asymmetry. The forward/backward asymmetry of charged particle
production is considered to arise from the diﬀerent number of participants in Au nucleus andCu nucleus. In order to investigate the contributions of the Au participants, Npart,Au , and theCu participants, Npart,Cu , to the particle production at forward/backward rapidity, we introduce
the weighted Npart with Npart,Au and Npart,Cu based on wounded nucleon model. The weighted

5.2. INTERPRETATION OF FORWARD/BACKWARD ASYMMETRY OF DNCH /Dη WITH MONTE C
Npart for Au-going and Cu-going side in Cu+Au collisions is defined as the following formulas,
Npart,Au−going = wAu−going Npart,Au + (2 − wAu−going )Npart,Cu(5.4)

Npart,Cu−going = wCu−going Npart,Au + (2 − wCu−going )Npart,Cu ,(5.5)

where wAu−going and wCu−going are weights for the Au-going and the Cu-going sides, respectively.
Since the weight is defined to change from 0 to 1, the weighed Npart ranges from 2Npart,Cu to
2Npart,Au ,2Npart,Cu ≤ Npart,Cu−going , Npart,Au−going ≤ 2Npart,Au .
Weighted εn scaling based on wounded nucleon model
Like the dNch /dη measurements at forward/backward-rapidity, the forward/backward asymmetry of charged particle vn in Cu+Au collisions is seen in Fig. 4.16 and 4.18. Based on
wounded nucleon model, the weighted initial spatial anisotropy εn with the Au nucleus εn,Au
and Cu nucleus εn,Cu is introduced for studying the influence of Au and Cu geometries on theforward/backward vn .
εn = wεn,Au + (1 − w)εn,Cu(5.6)

where w is the weights changing from 0 to 1 (εn,Cu < εn < εn,Au ). This model is used in thenext section for studying the initial geometry at the forward/backward-rapidity.
The charged particle multiplicity distributions dNch /dη as a function of weighted Npart at
Cu-going and Au-going side in Cu+Au collisions are compared to those in Au+Au and Cu+Cu
collisions shown in Fig. 5.7 and 5.8. The values of dNch /dη in all collision systems increase
from the peripheral to the central collisions. The almost linear relation between dNch /dη andNpart is seen, whose feature is described by the wounded nucleon model. In all panels, thevalues of dNch /dη in the Au+Au and the Cu+Cu collisions are common. From the left to theright panel, both of the Au-going and the Cu-going side Npart range from 2Npart,Cu to 2Npart,Au,2Npart,Cu ≤ Npart,Cu−going , Npart,Au−going ≤ 2Npart,Au . In the left side panels of Fig. 5.7, a good
agreement among Au+Au, Cu+Au and Cu+Cu collisions is seen. However, the discrepancy
between the symmetric Au+Au and Cu+Cu collisions and the Cu+Au collisions become larger
with increasing the fraction of the Npart,Au as in the right panels in Fig. 5.7. In Fig. 5.8,
the Au-going side dNch /dη in Cu+Au collisions become slightly close to the symmetric Au+Au
and Cu+Cu results from the left to the right panel. For 1.5Npart,Au + 0.5Npart,Cu in Cu+Au
collisions, the Cu+Au result is closest to the symmetric collision systems. Compared to theCu-going results, dNch /dη at Au-going side needs larger opposite beam participant (Npart,Cu )contribution. In Fig. 5.9, the χ2 /N DF as a function of the weight for the diﬀerence of dNch /dη
between the symmetric Au+Au and Cu+Cu collisions and the Cu+Au collisions is shown. In
the left side panel, χ2 /N DF for Au-going side is shown. For the Au-going side, the χ2 /N DF
decreases from the lower to larger weight. The lowest χ2 /N DF is expected to be between
w = 1 ∼ 1.5, which indicates the dNch /dη at Au-going side is originated from not only Npart,Au
but also the Npart,Cu . On the other hand, the χ2 /N DF for Cu-going side becomes smaller at thelarger weight of Npart,Cu . Thus the dNch /dη at Cu-going side is described by the almost pure
Npart,Cu contribution.

104

CHAPTER 5. DISCUSSIONS

dNch/dηAu+Au
Cu+Au Cu-going

400

400

400

400

400

Cu+Cu

200

200

200

χ /NDF=2.2
0.0Npart,Au+2.0Npart,Cu

000
100 200 300 0

200χ /NDF=6.1
0.5Npart,Au+1.5Npart,Cu

2

100

Npart

200

300

200

χ /NDF=16.7
1.0Npart,Au+1.0Npart,Cu

2

χ /NDF=34.0
1.5Npart,Au+0.5Npart,Cu

2

0

0

100

Npart

200

300

χ2/NDF=58.0
2.0Npart,Au+0.0Npart,Cu

2

0

0

100

Npart

200

300

0

0

100

Npart

200

300

Npart

Figure 5.7: The comparison of the charged particle multiplicity distributions dNch /dη among
Au+Au, Cu+Au(Cu-going) and Cu+Cu collisions collisions

dNch/dηAu+Au
Cu+Au Au-going

400

400

400

400

400

200

200

200

200

Cu+Cu

200

χ2/NDF=11.0
0.0Npart,Au+2.0Npart,Cu

000
100 200 300 0
Npart

χ2/NDF=4.6
0.5Npart,Au+1.5Npart,Cu

100

200
Npart

300

χ2/NDF=1.3
1.0Npart,Au+1.0Npart,Cu

0

0

100

200
Npart

300

χ2/NDF=1.2
1.5Npart,Au+0.5Npart,Cu

0

0

100

200
Npart

300

χ2/NDF=4.3
2.0Npart,Au+0.0Npart,Cu

0

0

100

200

300

Npart

Figure 5.8: The comparison of the charged particle multiplicity distributions dNch /dη among
Au+Au, Cu+Au(Au-going) and Cu+Cu collisions collisions

60

χ2/NDF

χ2/NDF5.2. INTERPRETATION OF FORWARD/BACKWARD ASYMMETRY OF DNCH /Dη WITH MONTE C

Au-going

60
Cu-going

40 Npart=wNpart,Au+(2-w)Npart,Cu

40

20

20

0

0

0.5

1

1.5

2
w

0

0

0.5

1

1.5

2
w

Figure 5.9: The χ2 /N DF distributions for the comparison of dNch /dη among Au+Au, Cu+Au
and Cu+Cu collisions

106

CHAPTER 5. DISCUSSIONS

5.3

Eccentricity scaling of vn

5.3.1

Mid-rapidity vn /εnIn order to study the eﬀect of initial geometry on azimuthal anisotropies, eccentricity scaling
vn /εn is commonly used, because vn coeﬃcients are mostly driven originally from εn . In thissection, various eccentricity scalings of vn mid, forward and backward rapidities are discussed to
study the initial geometry eﬀect. Especially the forward/backward asymmetry of vn in Cu+Au
collisions is discussed.
In Fig. 5.10, the average < v2 > coeﬃcients (0 < pT < 3GeV/c) as a function of number ofparticipants, Npart for Au+Au, Cu+Au and Cu+Cu collisions are shown. At all Npart bins, thev2 coeﬃcients in Au+Au collisions are the largest and those in Cu+Cu collisions are smallest
in the three collision systems. The v2 values increase from the large Npart (central collision) to
the small Npart (peripheral collision). These system size dependence and Npart dependence are
expected from the initial geometry in Monte Carlo Glauber model for the three collision systems
shown in Fig. 5.10. From the central to peripheral collisions, the nucelei overlap region become
more elliptical shape. In the earlier studies [38], [41] and [42], PHOBOS, PHENIX and STAR
experiments compared the scaled v2 values in Au+Au and Cu+Cu collisions at 200 GeV with
their 2nd participant eccentricity ε2 to study the initial geometry and the collective response. In
this comparison, the scaled v2 /ε2 values in the two symmetric collision systems show a universal
behavior. This trend indicates that the relation between v2 and ε2 is not linear and the collective
responses depend on the Npart of the symmetric heavy ion collisions. We add the results fromthe asymmetric Cu+Au data to this comparison in Fig. 5.11. The Cu+Au collisions are in good
agreement with the symmetric collision systems. All scaled v2 increase from small Npart to large
Npart . Namely, the strength of elliptic flow is mainly determined from initial geometry shapeand system size.
For the < v3 > measurements, the < v3 > values in Au+Au and Cu+Au collisions as afunction of Npart are shown in Fig. 5.12. Unlike the < v2 > measurements, the Cu+Au v3
is consistent with the Au+Au v3 within the systematic uncertainties. This is not predicted by
the simple Glauber model as shown in Fig.5.12. Although both of v3 and ε3 do not have same
system size trend, we scale v3 in Au+Au and Cu+Au with their ε3 shown in 5.13. Except formost central bin, the scaled v3 values in Au+Au and Cu+Au collisions are consistent withinerror. Compared to the scaled v2 values, the scaled v3 values are smaller, which means that theconversion from ε3 to v3 through collective expansion is smaller than that from ε2 to v2 .

5.3. ECCENTRICITY SCALING OF VN

0.8

Au+Au 200GeV
Cu+Au 200GeV
Cu+Cu 200GeV
Charged Hadron v2 |η|<0.35

v2

0<p <3(GeV/c)
T

0.05

MC-Glauber
Au+Au
Cu+Au
Cu+Cu

0.6
ε2

0.1

107

0.4
0.2

0
0

0
0

100 200 300 400
Npart

100 200 300 400
Npart

Figure 5.10: Left:pT integrated v2 as a function of Npart at mid-rapidity in Au+Au, Cu+Au and√Cu+Cu collisions at sN N = 200GeV. The integrated pT range is from 0 to 3 GeV. Right:ε2 inMonte Carlo Glauber model as a function of Npart in Au+Au, Cu+Au and Cu+Cu collisions

0.4
Au+Au 200GeV

v2/ε2

0.3

Cu+Au 200GeV
Cu+Cu 200GeV

0.2
0.1
MC-Glauber

00100 200 300 400
Npart

Figure 5.11: v2 /ε2 at mid-rapidity as a function of Npart in Au+Au, Cu+Au and Cu+Cu collisions

108

CHAPTER 5. DISCUSSIONS

0.06

0.8
Au+Au 200GeV
Cu+Au 200GeV
Charged Hadron v |η|<0.35

ε3

30<pT <3(GeV/c)

v3

0.04
0.02
0
0

MC-Glauber
Au+Au
Cu+Au

0.6
0.4
0.2
0
0

100 200 300 400
Npart

100 200 300 400
Npart

Figure 5.12: Left:pT integrated v3 as a function of Npart at mid-rapidity in Au+Au and Cu+Au√
collisions at sN N = 200GeV. The integrated pT range is from 0 to 3 GeV. Right:ε3 in Monte
Carlo Glauber model as a function of Npart in Au+Au and Cu+Au collisions

0.3

Au+Au 200GeV

v3/ε3

Cu+Au 200GeV

0.2
0.1
MC-Glauber

00100 200 300 400
Npart

Figure 5.13: v3 /ε3 as a function of Npart in Au+Au and Cu+Au collisions

5.3. ECCENTRICITY SCALING OF VN

5.3.2

109

Forward/Backward-rapidity vn /εnWe also perform the eccentricity scaling of vn at forward/backward-rapidity. As it is seen in
the above study of the eccentricity scaling of mid-rapidity vn , the magnitude of vn is describedby not only initial geometry but also Npart , which is related to the initial energy density andthe volume of formed matter. In Cu+Au collisions, the forward/backward asymmetry of dN/dη
is observed in Fig. 4.21. Since the dN/dη is proportional to Npart , this rapidity asymmetry ofdN/dη in Cu+Au collisions also could lead to the rapidity asymmetry of vn . By taking thisasymmetry of dN/dη into account, we measure the vn and the scaled vn as a function of dN/dηmeasured at the forward/backward-rapidity.
Figure 5.14 shows the v2 measurements as a function of the forward/backward dN/dη in
Au+Au, Cu+Au and Cu+Cu collisions at the forward/backward-rapidity. In all collision systems, the v2 increase from the central to the peripheral collisions. The v2 in Au+Au collisions
shows the largest values and the Cu+Cu v2 is lowest value at entire measured dN/dη region. Thev2 values in Cu+Au are always between those in Au+Au and Cu+Cu collisions. In Cu+Au collisions, the v2 at Au-going side is higher than that at Cu-going side. This system size dependence
is expected from the ε2,A+B shown in Fig.5.15. In Fig. 5.15, the ε2,A(B) , ε2,A+B and ε2,B(A) in
Au+Au, Cu+Au and Cu+Cu collisions as a function of the forward/backward dN/dη are shown.
For the system size dependence of ε2,A(B) , the ε2,Cu in Cu+Au collisions is same as the ε2,Cu+Cu .
This consistency is not seen in the v2 measurements and indicates the initial geometry for the
Cu-going side v2 is not described by the ε2,Cu alone. Figure 5.16 shows the eccentricity scalingof v2 with the ε2 . In the three panels, each of the ε2,Au+Au and ε2,Cu+Cu uses common values.
Therefore the scaled v2 in Au+Au and Cu+Cu collisions is not changed in the three panels. For
the Cu+Au collisions, the values of ε2 are diﬀerent in the three panels. In the left side panel,the ε2,Au and ε2,Cu are used to Au-going and Cu-going side v2 , respectively. In the middle panel,
the common ε2,Cu+Au is used to the both of side. In the right side panel, the ε2,Cu and ε2,Au areused to Au-going side and Cu-going side, respectively, which is the opposite combination of v2
and ε2 compared to the combination in the left side panel. Like the mid-rapidity scaled v2 , thescaled v2 at forward/backward-rapidity increase from the peripheral to the central in the threepanels. From the left side to the right side panels, the scaled Au-going side v2 increase, while thescaled Cu-going side of v2 decrease. In the right and left side panels, the discrepancies between
the Au going side and Cu going side are observed. The eccentricity scaling with independent
eccentricity definitions fails in the v2 measurements. In the middle of panel, a good agreement
between Au going side and Cu going side is observed, and a universal behavior is seen except
for the central Cu+Au collisions, which indicates that the forward/backward asymmetry of vn iscaused by the forward/backward asymmetry of dN/dη with a common eccentricity ε2,A+B . To
confirm the results that the forward/backward asymmetry of v2 in Cu+Au collisions is canceledwith the common eccentricity, the scaled v2 with the weighted eccentricity which is already introduced in the description about wounded nucleon model Sec. 5.2. We defined the eccentricity
as following formula,
ε2,Au−going = wAu−going ε2,Au + (1 − wAu−going )ε2,Cu(5.7)

ε2,Cu−going = wCu−going ε2,Au + (1 − wCu−going )ε2,Cu

(5.8)

110

CHAPTER 5. DISCUSSIONS

where wAu−going and wCu−going are the weights changing from 0 to 1 (ε2,Cu < wAu−going , wCu−going <ε2,Au ), which means the weighted eccentricity range from the the ε2,Cu to the ε2,Au . In Fig. 5.17,
the row from the bottom to the top corresponds to the weight for Au-going side wAu−going from0 to 1 and the column from the left to the right corresponds to the weight for Cu-going side
wCu−going from 0 to 1. Namely, the ε2,Au (ε2,Cu ) contribution in ε2,Au−going increases (decreases)from the bottom to the top row, and the ε2,Au (ε2,Cu ) contribution in ε2,Cu−going increases (decreases) from the left to the right column. Thus the scaled Au-going side v2 decreases from thebottom to the top row and the scaled Cu-going side v2 decreases from the left to the right column.
On the top left side and the bottom right side panels, there are the deviations between Au-going
and Cu-going sides are seen. Around middle of row and column, a good agreement between thescaled Au-going and Cu-going v2 is observed. An agreement between the Cu+Au and Au+Au
collisions is also observed. Fig. 5.18 shows the χ2 /N DF contour distributions for wAu−going vs
wCu−going . In Fig. 5.18, the χ2 /N DF is estimated from the diﬀerence between the scaled Augoing side and Cu-going side v2 in Cu+Au collisions. The χ2 /N DF increase at the top left andthe bottom right. The lower χ2 /N DF distribution is seen at around wAu ∼ wCu band. In Fig.
5.19, the χ2 /N DF map is shown as a function of the weight w for ε , where χ2 /N DF is definedby the diﬀerence between each of the scaled Au-going and Cu-going side v2 in Cu+Au collisions
and that in Au+Au collisions. For both of Au-going and Cu-going side, the χ2 /N DF becomelarge at w = 1 and w = 0, and the lowest χ2 /N DF is found to be between 0.25 and 0.5. For theforward/backward v2 measurements in Cu+Au collisions, the almost same ε2 is expected to be
the initial geometry at forward/backward rapidity and the forward/backward asymmetry of thev2 arise from the forward/backward asymmetry of the dN/dη which is supposed to be originatedfrom the initial energy density that i later driving the strength of the collective expansion.
To confirm this indication, the v3 as a function of dN/dη is scaled as well. In Fig.5.20 ,the forward/backward v3 as a function of the forward/backward dN/dη in Au+Au and Cu+Au
collisions is shown. Figure 5.21, show the ε3,A(B) , ε3,A+B and ε3,B(A) as a function of dN/dη. Allε3 cases, does not describe the system size diﬀerence of v3 . However we perform the eccentricity
scaling of v3 in the same way as done in the v2 . In Fig. 5.22, the scaled v3 values in Au+Au andCu+Au collisions are compared. Like the mid-rapidity v3 , the scaled v3 at forward/backwardrapidity in both of Au+Au and Cu+Au collisions increase from the peripheral to the central
collisions. The scaled Au+Au v3 is common in all panels. The v3 in Cu+Au collisions is scaledby the diﬀerent ε3 . In the left panel, the Cu+Au v3 at Au-going and Cu-going side are scaledwith the ε3,Au and the ε3,Cu respectively. The Au-going side v3 in Cu+Au collisions is consistent
with that in Au+Au collisions within the error, while the Cu-going side v3 in Cu+Au collisions islarger than that in Au+Au collisions. The discrepancy between the Au-going and the Cu-goingside is obserbed. In the middle panel, the common ε3,Cu+Au which is the average of the ε3,Auand the ε3,Cu is used for both of the Au-going and the Cu-going v3 in Cu+Au collisions. The
consistency between the Au-going and Cu-going side in Cu+Au collisions is seen as seen in theforward/backward v2 measurements. The both of Au-going and Cu-going side is not consistent
with Au+Au collisions. In the right side panel, the ε3,Cu and the ε3,Au in Cu+Au collisions areused for the Au-going and the Cu-going side v3 respectively. The discrepancy among the Au-going
and the Cu-going and Au+Au collisions is seen. From the left to the right panel, the Au-goingside v3 increase and the Cu-going side v3 decreases. This is because the ε3 for the Au-going anddecreases and the ε3 for the Cu-going side increases from the left to the right side panel. Like the5.3. ECCENTRICITY SCALING OF VN

0.06

111

Au+Au 200 GeV
Cu+Au 200 GeV Au-going
Cu+Au 200 GeV Cu-going
Cu+Cu 200 GeV
Hadron v2 3<|η|<3.9

v2

0.04
0.02

0
0 100 200 300 400 500
dNch/dηFigure 5.14: Forward/backward-rapidity v2 for charged hadrons as a function of dNch /dη measured at forward/backward rapidity(3 < |η| < 3.9) in Au+Au, Cu+Au and Cu+Cu collisions.

MC-Glauber nucleon
Au+Au
Cu+Au Au
Cu+Au Cu
Cu+Cu

0.4
0.2
0
0 100 200 300 400 500
dNch/dη0.6
ε2,A+B

ε2,A(B)

0.6

0.8

0.8
MC-Glauber nucleon
Au+Au
Cu+Au Au
Cu+Au Cu
Cu+Cu

0.4
0.2
0
0 100 200 300 400 500
dNch/dη0.6
ε2,B(A)

0.8

MC-Glauber nucleon
Au+Au
Cu+Au Au
Cu+Au Cu
Cu+Cu

0.4
0.2
0
0 100 200 300 400 500
dNch/dηFigure 5.15: ε2,A(B) , ε2,A+B and ε2,B(A) as a function of forward/backward dNch /dη at
200GeV in Au+Au, Cu+Au and Cu+Cu collisions.

√sN N =

112

CHAPTER 5. DISCUSSIONS

0.15 MC-Glauber nucleon
0.1

0.05
0
0 100 200 300 400 500
dNch/dη

0.2

0.25
Au+Au 200 GeV
Cu+Au 200 GeV Au-going
Cu+Au 200 GeV Cu-going
Cu+Cu 200 GeV

0.15 MC-Glauber nucleon

v2/ε2

v2/ε2,A(B)

0.2

0.25
Au+Au 200 GeV
Cu+Au 200 GeV Au-going
Cu+Au 200 GeV Cu-going
Cu+Cu 200 GeV

0.1

0.05
0
0 100 200 300 400 500
dNch/dη0.2
v2/ε2,B(A)

0.25

Au+Au 200 GeV
Cu+Au 200 GeV Au-going
Cu+Au 200 GeV Cu-going

0.15 MC-Glauber nucleon
0.1

0.05
0
0 100 200 300 400 500
dNch/dηFigure 5.16: v2 /ε2,A(B) (Left), v2 /ε2,A+B (Middle) v2 /ε2,B(A) (Right) as a function of dN/dη at√forward/backward rapidity(3 < |η| < 3.9) in Au+Au, Cu+Au and Cu+Cu collisions at sN N =200GeV.
v2 measurements, the forward/backward asymmetry of v3 in Cu+Au collisions is scaled with thecommon eccentricity. We also test the scaling of the forward/backward v3 with the eccentricity
ranging from ε3,Cu to ε3,Au , (ε3,Cu ≤ ε3,Au−going , ε3,Cu−going ≤ ε3,Au ). In Fig. 5.23, the scaled v3
values in Au+Au and Cu+Au collisions are shown. In all panel, the scaled v3 in Au+Au andCu+Au collisions increase from the peripheral to the central collisions. The values of the scaled
v3 in Au+Au collisions are common in all panels. For the Cu+Au collisions, the ε3,Au−goingvaries from the ε3,Cu to the ε3,Au from the bottom to the top row and the ε3,Cu−going varies fromthe ε3,Cu to the ε3,Au from the left to the right column. Thus the scaled v3,Au−going decreases
from the bottom to the top row and the v3,Cu−going decreases from the left to the right column.
From the bottom right to the top left, the ordering of the scaled v3 magnitudes at Au-going andCu-going sides is reversed. In the middle of panels, the good agreements between the Au-going
and Cu-going sides are seen as seen the v2 measurements. However, the discrepancy between
Au+Au cand Cu+Au collisions is seen except for the top right panel where the eccentricities for
Au-going and Cu-going are ε3,Au . In Fig. 5.24, the χ2 /N DF contour distributions for wAu−going
vs wCu−going are shown. The χ2 /N DF is estimated for evaluating the diﬀerence between the
Au-going and the Cu-going side v3 /ε3 . The chi2 /N DF increases at the top left and bottom right.
The lower values of the χ2 /N DF are around wAu ∼ wCu range as seen in the v2 measurements.
Fig. 5.25 shows the χ2 /N DF contour distributions given by the diﬀerence between Au+Au andCu+Au collisions. The χ2 /N DF for the Au-going and Cu-going sides are shown in the left sidepanel and the right side panel, respectively. In both panel, the values of χ2 /N DF decrease fromthe low to the large weight. Unlike the χ2 /N DF for the v2 measurements, a minimum point is
not seen in both of panels, which might indicates the Glauber Monte Carlo model is not favoredfor the third order eccentricity.

5.3. ECCENTRICITY SCALING OF VN

ε2,Au-going=1.00ε2,Au+0.00ε2,Cu

ε2,Au-going=1.00ε2,Au+0.00ε2,Cuχ2/ndf=12.35

χ2/ndf=6.98

113

ε2,Au-going=1.00ε2,Au+0.00ε2,Cu

ε2,Au-going=1.00ε2,Au+0.00ε2,Cu

ε2,Au-going=1.00ε2,Au+0.00ε2,Cu

0.2 ε2,Cu-going=0.00ε2,Au+1.00ε2,Cu 0.2 ε2,Cu-going=0.25ε2,Au+0.75ε2,Cu 0.2 ε2,Cu-going=0.50ε2,Au+0.50ε2,Cu 0.2 ε2,Cu-going=0.75ε2,Au+0.25ε2,Cu 0.2 ε2,Cu-going=1.00ε2,Au+0.00ε2,Cu

0.2

100 200 300 400

0

0

0.15

v2/ε2

χ2/ndf=1.65

0.05

0.05

0.1

0
0
0
100 200 300 400 0 100 200 300 400 0 100 200 300 400 0 100 200 300 400
ε2,Au-going=0.75ε2,Au+0.25ε2,Cu
ε2,Au-going=0.75ε2,Au+0.25ε2,Cu
ε2,Au-going=0.75ε2,Au+0.25ε2,Cu
dN
/d
η
dN
/d
η
dNε ch/dη
0.2 ε2,Cu-going=0.50ε2,Au
ch +0.50ε
0.2 ε2,Cu-going=0.75ε2,Au
ch +0.25ε
0.2 ε2,Cu-going=1.00
2,Cu
2,Cu
2,Au+0.00ε2,Cu

ε2,Au-going=0.75ε2,Au+0.25ε2,Cu
ε=0.75ε2,Au+0.25ε2,Cu
dN /d+1.00
η ε 0.2 ε2,Au-going=0.25
dNε ch/dηε2,Cu-going=0.00εch
2,Au
2,Cu
2,Cu-going
2,Au+0.75ε2,Cuχ2/ndf=1.61

0.15

χ2/ndf=2.79

0.15

v2/ε2

0.15

v2/ε2

χ2/ndf=3.84

v2/ε2

0.15

v2/ε2

χ2/ndf=8.30

0.05

0.05

0.05

0.05

0.05

0.1

0
0.2

0.1

0

100 200 300 400

0

0.1

0

0.1

0
0
0
100 200 300 400 0 100 200 300 400 0 100 200 300 400 0 100 200 300 400
ε2,Au-going=0.50ε2,Au+0.50ε2,Cu
ε2,Au-going=0.50ε2,Au+0.50ε2,Cu
ε2,Au-going=0.50ε2,Au+0.50ε2,Cu
dN
/d
η
dN
/d
η
dNε ch/dη
0.2 ε2,Cu-going=0.50ε2,Au
ch +0.50ε
0.2 ε2,Cu-going=0.75εch0.2 ε2,Cu-going=1.00
2,Cu
2,Au+0.25ε2,Cu
2,Au+0.00ε2,Cu

ε2,Au-going=0.50ε2,Au+0.50ε2,Cu
ε=0.50ε2,Au+0.50ε2,Cu
dN /d+1.00
η ε 0.2 ε2,Au-going=0.25
dNε ch/dηε2,Cu-going=0.00εch
2,Au
2,Cu
2,Cu-going
2,Au+0.75ε2,Cuχ2/ndf=2.13

0.15

χ2/ndf=6.98

0.15

v2/ε2

0.15

v2/ε2

χ2/ndf=1.89

v2/ε2

0.15

v2/ε2

χ2/ndf=4.87

0.05

0.05

0.05

0.05

0.05

0.1

0
0.2

0.1

0

100 200 300 400

0

0.1

0

0.1

χ2/ndf=17.74

0.1

0
0
0
100 200 300 400 0 100 200 300 400 0 100 200 300 400 0 100 200 300 400
ε2,Au-going=0.25ε2,Au+0.75ε2,Cu
ε2,Au-going=0.25ε2,Au+0.75ε2,Cu
ε2,Au-going=0.25ε2,Au+0.75ε2,Cu
dN
/d
η
dN
/d
η
dNε ch/dη
0.2 ε2,Cu-going=0.50ε2,Au
ch +0.50ε
0.2 ε2,Cu-going=0.75εch0.2 ε2,Cu-going=1.00
2,Cu
2,Au+0.25ε2,Cu
2,Au+0.00ε2,Cu

ε2,Au-going=0.25ε2,Au+0.75ε2,Cu
ε=0.25ε2,Au+0.75ε2,Cu
dN /d+1.00
η ε 0.2 ε2,Au-going=0.25
dNε ch/dηε2,Cu-going=0.00εch
2,Au
2,Cu
2,Cu-going
2,Au+0.75ε2,Cuχ2/ndf=5.72

0.15

χ2/ndf=15.76

0.15

v2/ε2

0.15

v2/ε2

χ2/ndf=1.83

v2/ε2

0.15

v2/ε2

χ2/ndf=2.50

v2/ε2

0.15

χ2/ndf=8.56

0.1

v2/ε2

0.15

χ2/ndf=3.78

0.1

v2/ε2

0.15

0

0.15

0.05

0.1

Au+Au
Cu+Au Au-going
0.05
Cu+Au Cu-going
Cu+Cu

0.05

χ2/ndf=3.06

v2/ε2

0.1

0.15

v2/ε2

0.1

0

0.15

v2/ε2

v2/ε2

0.15

0.05

0.05

0.05

0.05

0.05

0.1

0.1

0.1

0.1

χ2/ndf=33.48

0.1

0

0
0
0
0
0ε 100 =0.00
200 ε 300
400 0 100 200 300 400 0ε 100=0.00
200ε 300
0ε 100=0.00
200ε 300
0ε 100=0.00
200ε 300+1.00400
ε 2,Cu+1.00400
ε 2,Cu+1.00400
ε 2,Cu2,Au-going
2,Au+1.00ε2,Cu ε 2,Au-going=0.00ε 2,Au+1.00ε 2,Cu
2,Au-going
2,Au
2,Au-going
2,Au
2,Au-going
dN
/d
η
dN
/d
η
dN
/d
η
dN
/d
η
dNεch2,Au/dη
0.2 ε2,Cu-going=0.00
ε 2,Au+0.25ε 2,Cu0.2 ε 2,Cu-going=1.00+0.00ε 2,Cu
chε2,Au+1.00ε0.2
ch2,Au+0.75ε 2,Cu0.2 ε 2,Cu-going=0.50ε
ch2,Au+0.50ε 2,Cu0.2 ε 2,Cu-going=0.75ch
2,Cu ε 2,Cu-going=0.25ε2,Auχ2/ndf=14.17

0.15

χ2/ndf=31.73

0.15

0.1

0.1

0.1

v2/ε2

0.15

v2/ε2

χ2/ndf=4.88

v2/ε2

0.15

v2/ε2

χ2/ndf=1.96

v2/ε2

0.15

0.05

0.05

0.05

0.05

0.05

0.1

0

0 100 200 300 400
dNch/dη00100 200 300 400
dNch/dη00100 200 300 400
dNch/dη0

χ2/ndf=59.34

0.1

0100 200 300 400
dNch/dη00100 200 300 400
dNch/dηFigure 5.17: v2 /ε2 as a function of dNch /dη at measured rapidity region (3 < |dNch /dη| < 3.9) in
Au+Au, Cu+Au(Cu-going, Au-going) and Cu+Cu collisions. ε2 for Cu+Au collisions is definedas ε2,Au(Cu)−going = wAu(Cu)−going ε2,Au + (1 − wAu(Cu)−going )ε2,Cu

CHAPTER 5. DISCUSSIONS

wAu-going

114

1
50
40

0.5

30
20
10

0
0

0.5

1
wCu-going

20
18
16
14

χ2/NDF

χ2/NDF

Figure 5.18: χ2 /N DF contour distribution for wCu−going vs wCu−going . χ2 /N DF is obtained
from the diﬀerence between the Cu-going and the Au-going side v2 .

v2/ε2 Au-going
ε2=wε2,Au+(1-w)ε2,Cu

20
18
16
14

12

12

10

10

8

8

6

6

4

4

2

2

0

0

0.2 0.4 0.6 0.8

0

1

w

v2/ε2 Cu-going0

0.2 0.4 0.6 0.8

1w

Figure 5.19: χ2 /N DF as a function of weight for the Au-going and the Cu-going side. χ2 /N DF
is obtained from the diﬀerencee between Cu+Au and Au+Au collisions

5.3. ECCENTRICITY SCALING OF VN

115

Au+Au 200 GeV
Cu+Au 200 GeV Au-going
Cu+Au 200 GeV Cu-going
Hadron v3 3<|η|<4

v3

0.01

0.005

0
0 100 200 300 400 500
dNch/dηFigure 5.20: Forward/backward-rapidity v3 for charged hadrons as a function of dNch /dη measured at forward/backward rapidity(3 < |η| < 3.9) in Au+Au and Cu+Au collisions.

MC-Glauber nucleon

Cu+Au Au

0.4

0.6

Au+Au

Cu+Au Cu

0.2
00 100 200 300 400 500
dNch/dηε3,A+B

ε3,A(B)

0.6

0.8

0.8
MC-Glauber nucleon

Cu+Au Au

0.4

0.6

Au+Au

Cu+Au Cu

0.2
00 100 200 300 400 500
dNch/dηε3,B(A)

0.8

MC-Glauber nucleon
Au+Au
Cu+Au Au

0.4

Cu+Au Cu

0.2
00 100 200 300 400 500
dNch/dηFigure 5.21: ε3,A(B) , ε3,A+B and ε3,B(A) as a function of forward/backward dNch /dη at
200GeV in Au+Au and Cu+Au collisions.

√sN N =

116

CHAPTER 5. DISCUSSIONSMC-Glauber nucleon

0.05

00 100 200 300 400 500
dNch/dη

0.1

Au+Au 200 GeV
Cu+Au 200 GeV Au-going
Cu+Au 200 GeV Cu-going

MC-Glauber nucleon

0.05

00 100 200 300 400 500
dNch/dη0.1
v3/ε3,B(A)

Au+Au 200 GeV
Cu+Au 200 GeV Au-going
Cu+Au 200 GeV Cu-goingv3/ε3,A+B

v3/ε3,A(B)

0.1

Au+Au 200 GeV
Cu+Au 200 GeV Au-going
Cu+Au 200 GeV Cu-going

MC-Glauber nucleon

0.05

00 100 200 300 400 500
dNch/dηFigure 5.22: v3 /ε3,A(B) (Left), v3 /ε3,A+B (Middle) v3 /ε3,B(A) (Right) as a function of dN/dη at√forward/backward rapidity(3 < |η| < 3.9) in Au+Au, Cu+Au and Cu+Cu collisions at sN N =200GeV.

5.3. ECCENTRICITY SCALING OF VN

117

ε3,Cu+Au,Au-going=1.00ε3,Au+0.00ε3,Cu ε3,Cu+Au,Au-going=1.00ε3,Au+0.00ε3,Cu

ε3,Cu+Au,Au-going=1.00ε3,Au+0.00ε3,Cu

ε3,Cu+Au,Au-going=1.00ε3,Au+0.00ε3,Cu

v3/ε3

v3/ε3

v3/ε3

v3/ε3

v3/ε3

0.05

ε3,Cu+Au,Au-going=1.00ε3,Au+0.00ε3,Cu

ε3,Cu+Au,Cu-going=0.00ε3,Au+1.00ε3,Cu ε3,Cu+Au,Cu-going=0.25ε3,Au+0.75ε3,Cu ε3,Cu+Au,Cu-going=0.50ε3,Au+0.50ε3,Cu ε3,Cu+Au,Cu-going=0.75ε3,Au+0.25ε3,Cu ε3,Cu+Au,Cu-going=1.00ε3,Au+0.00ε3,Cu2
2
22
χ2/ndf=14.40
0.05 χ /ndf=9.52
0.05 χ /ndf=6.42
0.05 χ /ndf=5.36
0.05 χ /ndf=6.52

Au+Au
Cu+Au Au-going
Cu+Au Cu-going00

100 200 300 400
dNch/dη000100 200 300 400
dNch/dηε3,Cu+Au,Au-going=0.75ε3,Au+0.25ε3,Cu ε3,Cu+Au,Au-going=0.75ε3,Au+0.25ε3,Cu

00100 200 300 400
dNch/dηε3,Cu+Au,Au-going=0.75ε3,Au+0.25ε3,Cu

00100 200 300 400
dNch/dηε3,Cu+Au,Au-going=0.75ε3,Au+0.25ε3,Cu

0100 200 300 400
dNch/dηε3,Cu+Au,Au-going=0.75ε3,Au+0.25ε3,Cu

ε3,Cu+Au,Cu-going=0.00ε3,Au+1.00ε3,Cu ε3,Cu+Au,Cu-going=0.25ε3,Au+0.75ε3,Cu ε3,Cu+Au,Cu-going=0.50ε3,Au+0.50ε3,Cu ε3,Cu+Au,Cu-going=0.75ε3,Au+0.25ε3,Cu ε3,Cu+Au,Cu-going=1.00ε3,Au+0.00ε3,Cu
2
2
2
2
2
0.05 χ /ndf=6.69
0.05 χ /ndf=4.73
0.05 χ /ndf=5.09
0.05 χ /ndf=7.96

00

100 200 300 400
dNch/dη000100 200 300 400
dNch/dηε3,Cu+Au,Au-going=0.50ε3,Au+0.50ε3,Cu ε3,Cu+Au,Au-going=0.50ε3,Au+0.50ε3,Cu

0

v3/ε3

v3/ε3

v3/ε3

v3/ε3

v3/ε3

0.05 χ /ndf=10.71

0100 200 300 400
dNch/dηε3,Cu+Au,Au-going=0.50ε3,Au+0.50ε3,Cu

00100 200 300 400
dNch/dηε3,Cu+Au,Au-going=0.50ε3,Au+0.50ε3,Cu

0100 200 300 400
dNch/dηε3,Cu+Au,Au-going=0.50ε3,Au+0.50ε3,Cu

ε3,Cu+Au,Cu-going=0.00ε3,Au+1.00ε3,Cu ε3,Cu+Au,Cu-going=0.25ε3,Au+0.75ε3,Cu ε3,Cu+Au,Cu-going=0.50ε3,Au+0.50ε3,Cu ε3,Cu+Au,Cu-going=0.75ε3,Au+0.25ε3,Cu ε3,Cu+Au,Cu-going=1.00ε3,Au+0.00ε3,Cu
2
2
2
2
2
0.05 χ /ndf=4.52
0.05 χ /ndf=4.00
0.05 χ /ndf=6.13
0.05 χ /ndf=11.09

00

100 200 300 400
dNch/dη000100 200 300 400
dNch/dηε3,Cu+Au,Au-going=0.25ε3,Au+0.75ε3,Cu ε3,Cu+Au,Au-going=0.25ε3,Au+0.75ε3,Cu

0

v3/ε3

v3/ε3

v3/ε3

v3/ε3

v3/ε3

0.05 χ /ndf=7.41

0

100 200 300 400
dNch/dηε3,Cu+Au,Au-going=0.25ε3,Au+0.75ε3,Cu

00100 200 300 400
dNch/dηε3,Cu+Au,Au-going=0.25ε3,Au+0.75ε3,Cu

0100 200 300 400
dNch/dηε3,Cu+Au,Au-going=0.25ε3,Au+0.75ε3,Cu

ε3,Cu+Au,Cu-going=0.00ε3,Au+1.00ε3,Cu ε3,Cu+Au,Cu-going=0.25ε3,Au+0.75ε3,Cu ε3,Cu+Au,Cu-going=0.50ε3,Au+0.50ε3,Cu ε3,Cu+Au,Cu-going=0.75ε3,Au+0.25ε3,Cu ε3,Cu+Au,Cu-going=1.00ε3,Au+0.00ε3,Cu
2
2
2
2
2
0.05 χ /ndf=3.31
0.05 χ /ndf=4.63
0.05 χ /ndf=8.98
0.05 χ /ndf=16.54

χ2/ndf=3.52

0 100 200 300 400
dNch/dη

3,Au

3,Cu

0.05

0

0

100 200 300 400
dNch/dη3,Cu+Au,Cu-goingχ2/ndf=7.21

3,Au

v3/ε3

3,Cu

0.05

0

0

100 200 300 400
dNch/dη3,Cu+Au,Cu-goingχ2/ndf=14.37

3,Au

3,Cu 3,Cu+Au,Cu-goingχ2/ndf=25.20

3,Au

0.05
v3/ε3

3,Cu
3,Cu+Au,Cu-going

0.05
v3/ε3

v3/ε3

3,Au

v3/ε3

3,Cu+Au,Cu-going2

0.05 χ /ndf=3.05

0

v3/ε3

0
0
0
0
0ε 100 200=0.00
300ε 400
0ε 100 200
300 400 0 100 200 300 400 0 100 200 300 400 0 100 200 300 400=0.00ε 3,Au+1.00ε 3,Cu ε 3,Cu+Au,Au-going=0.00ε 3,Au+1.00ε 3,Cu ε 3,Cu+Au,Au-going=0.00ε 3,Au+1.00ε 3,Cu ε 3,Cu+Au,Au-going=0.00ε 3,Au+1.00ε 3,Cu
3,Cu+Au,Au-going
3,Au+1.00ε3,Cu
3,Cu+Au,Au-goingdNch=0.00/dη ε +1.00ε ε
dN=0.25dN=0.50dN=0.75dN=1.00εch/dηε
+0.75εεch/dηε
+0.50εεch/dηε
+0.25εεch/dηε
+0.00ε

v3/ε3

0

v3/ε3

v3/ε3

v3/ε3

0.05 χ /ndf=4.75

00100 200 300 400
dNch/dη00100 200 300 400
dNch/dηFigure 5.23: v3 /ε3 as a function of dNch /dη at measured rapidity region (3 < |dNch /dη| < 3.9)
in Au+Au and Cu+Au(Cu-going, Au-going) collisions. ε3 for Cu+Au collisions is defined as
ε3,Au(Cu)−going = wAu(Cu)−going ε3,Au + (1 − wAu(Cu)−going )ε3,Cu

3,CuCHAPTER 5. DISCUSSIONS

wAu-going

118

25

1
20

15

0.5

10

0

5

0

0.5

1
wCu-going50

χ2/NDF

χ2/NDF

Figure 5.24: χ2 /N DF contour distribution for wCu−going vs wAu−going . χ2 /N DF is obtained
from the consisitency between the Cu-going and the Au-going side v3

45
40
35

50
45
40
35

30

30

25

25

20

20

15
10
5
0

15

v3/ε3 Au-going
ε3=wε3,Au+(1-w)ε3,Cu0

0.2 0.4 0.6 0.8

10v3/ε3 Cu-going

5
0

1

w

0

0.2 0.4 0.6 0.8

1w

Figure 5.25: χ2 /N DF as a function of weight for the Au-going and the Cu-going side. χ2 /N DF
is obtained from the consistensy between Cu+Au and Au+Au collisions

5.4. THEORY COMPARISON

5.4

119

Theory comparison−A Multiphase TransportModel(AMPT)For the study of azimuthal anisotropy in relativistic heavy ion collisions, the A-MultiphaseTransport Model(AMPT) generator[50] is a useuful simulation tool. In the AMPT model, initial
partonic interaction and final hadronic interaction are included. For the initial condition, the
AMPT model uses the heavy ion jet interaction generator(HIJING) to produce the minijet(hardprosess) and excited strings(soft prosess). The excited strings are melted into partons. Afterthe partons are generated in the nucleus-nucleus collisions, Zhang’s parton cascade(ZPC) to
describe multiple-interaction of partons, which is followed by a quark coalescence model in which
hadrons are formed from quarks that are closest in phase space for a hadronization process. In the
AMPT models, the parton cascade stage corresponds to the hydrodynamic stage in hydrodynamic
model. After hadronization from partons, a relativistic transport(ART) is employed as describing
scattering among the formed hadrons. In Fig. 5.26, the interaction process and the models usedin AMPT is summarized.

Figure 5.26: Structure of AMPT model−Event by event hydrodynamic calculation√
For vn measurements in Cu+Au collisions at sN N = 200GeV, event by event hydrodynamic
calculations are available[51],[52]. In [51], on event by event Monte Carlo Glauber model generates the initial energy density and then hydrodynamic expansion with shear viscosity is followed
until a freeze-out temperature. This model does not use hadronic cascade model which cause
hadronic re-scattering after the hydrodynamic expansion. In [52], ideal hydrodynamics approachis employed in the QGP phase after initial geometry is generated from Monte Carlo Glaubermodel in each event. As an afterburner, hadronic cascade model, JAM is used for hadronic
re-scattering.
Theory comparison for mid − rapidity vn (pT )At low pT region, the two hydrodynamic models predict directed flow is in Cu hemisphere side,

120

CHAPTER 5. DISCUSSIONS

while at high pT the two models predict directed flow is in Au hemisphere side in Fig. 5.27. In thetheory comparison, the viscous hydrodynamic calculation with η/s = 0.08 [51] predicts stronger
directed flow compared to the prediction from the ideal hydrodynamic calculation [52]. But
basically, ideal hydrodynamics predict larger magnitude of azimuthal anisotropies than viscoushydrodynamics. However, the diﬀerence of the two theory calculations are not only viscouscorrection but also model composition such as whether hadronic cascade is turned on or not.
Thus we can not conclude that the viscous correction make stronger directed flow signal. On theother hand, the AMPT shows the opposite pT dependence of v1 predicted by the hydrodynamic
calculations. The AMPT predicts directed flow is in Au hemisphere side at low pT and in Cu
hemisphere side at high pT . The v1 (pT ) values are well reproduced by the ideal hydrodynamic
calculation. However the v1 measurement shows the large systematic uncertainties and smallvalues at low pT . Therefore we can not conclude whether many particle move to Cu hemisphereside or Au hemisphere side.
Theory comparison for vn (pT ) at mid − rapidity

0.02

v1

0
-0.02
-0.04
0

Cu+Au 200GeVPHENIX
Bozek η/s=0.08
Hirano η/s=0.0
AMPT 3mb

1

2

3

p (GeV/c)T√
Figure 5.27: Charged hadron v1 (pT ) measured at mid-rapidity in Cu+Au collisions at sN N =200GeV in comparison to theory calculations for 20-30% centrality class. The theory calculations
shown in this figure are viscous hydrodynamic calculation with η/s = 0.08(Bozek et al[51]),
ideal hydrodynamic calculation(Hirano et al[52]) and AMPT model with σ = 3mb parton crosssection[50]

In Fig.5.28 and 5.29, the ideal hydrodynamic calculation [52] and the AMPT model withthe values of parton cross section σ = 3.0mb are compared to the experimental results. The
ideal hydrodynamic calculation reproduce the experimental results for pT < 1GeV/c . The vn
values from the ideal hydrodynamic increase with pT due to no viscous correction. The AMPT

5.4. THEORY COMPARISON

121

model well reproduce the v2 values for pT < 1.5GeV/c except for 50-60% and the v3 values for
pT < 2GeV/C.

0.3

0.3

0.3

0.3

0.3

0.3

0.3

0.2

0.2

0.2

0.2

0.2

0.2

0.1

0.1

0.1

0.1

0.1

0.1

Cu+Au 200 GeV
PHENIX
Hydro η/s=0
0.2

v2

AMPT 3mb

0.1

0~5%

5~10%

0

10~20%

0
0

1

2

3

4 0

0
1

2

3

4

20~30%
0

0

1

2

3

4

30~40%
0

0

1

2

3

4

40~50%
0

0

1

2

3

4

50~60%
0

0

1

2

3

4

0

1

2

3

4

p (GeV/c)T√
Figure 5.28: Charged hadron v2 (pT ) measured at mid-rapidity in Cu+Au collisions at sN N =200GeV in comparison to theory calculations for diﬀerent % centrality classes. The theory
calculations shown in this figure are the ideal hydrodynamic calculation(Hirano et al[52]) andthe AMPT model with σ = 3mb parton cross section[50].

0.2

0.2

0.2

0.15

Cu+Au 200 GeV
PHENIX
Hydro η/s=0
0.15
AMPT 3mb

0.15

0.15

0.1

0.1

0.1

0.1

0.05

0.05

0.05

0.05

v3

0.2

0~5%
0
0

5~10%

10~20%

0
1

2

3

4 0

0
1

2

3

4

20~30%
0

0

1

2

3

4

0

1

2

3

4

p (GeV/c)T√
Figure 5.29: Charged hadron v3 (pT ) measured at mid-rapidity in Cu+Au collisions at sN N =200GeV in comparison to theory calculations for diﬀerent % centrality classes. The theory
calculations shown in this figure are the ideal hydrodynamic calculation(Hirano et al[52]) andthe AMPT model with σ = 3mb parton cross section[50].

The v2 (pT ) and v3 (pT ) values are well predicted by the viscous hydrodynamic calculation[51]shown in Fig. 5.30 and Fig. 5.31. The viscous hydrodynamic calculation with the two diﬀerent
values of shear viscosity η/s = 0.08, 0.16 are shown. In 20-30% centrality class, viscous hydrodynamics with both of values reproduce vn well. In the 0-5% centrality class, hydrodynamics with

122

CHAPTER 5. DISCUSSIONS

η/s = 0.08 is closer to the experimental data.
Theory comparison for vn (η)

0.15

0.15
PHENIX Cu+Au 200 GeVBozek(η/s=0.08)
Bozek(η/s=0.16)
v2

0.2

v2

0.2

0.1

0.05

0.1

0.05

0-5%

20-30%00

02p (GeV/c)

4

0

2

4

pT (GeV/c)

T√
Figure 5.30: Charged hadron v2 (pT ) measured at mid-rapidity in Cu+Au collisions at sN N =200GeV in comparison to theory calculation for 0-5% and 20-30%centrality classes. The theory
calculation shown in this figure is the viscous hydrodynamic calculation(Bozek et al[51])

0.15

0.15

PHENIX Cu+Au 200 GeVBozek(η/s=0.08)
Bozek(η/s=0.16)

0.1
v3

v3

0.1

0.05

0.05

0-5%

20-30%00

02p (GeV/c)
T

4

0

2

4

pT (GeV/c)√
Figure 5.31: Charged hadron v3 (pT ) measured at mid-rapidity in Cu+Au collisions at sN N =200GeV in comparison to theory calculation for 0-5% and 20-30%centrality classes. The theory
calculation shown in this figure is the viscous hydrodynamic calculation(Bozek et al[51])In Fig.5.32 and 5.33, the v2 (η) and v3 (η) results are compared to the predictions from theideal hydrodynamic calculation and the AMPT model with σ = 3mb. The ideal hydrodynamic
calculation show the larger values of vn due to no viscous correction. The AMPT model reproduce
well in terms of forward/backward asymmetry in vn . In Fig. 5.34 and 5.35, the ratios of Au-goingvn and Cu-going vn as a function of centrality are shown and compared to the ideal hydrodynamic
and the AMPT model calculations are shown. For the v2 measurement, the maximum value ofthe ratio is around 20% at mid-central collisions, and the ratio becomes small at central and5.4. THEORY COMPARISON

0.08

0.08

0.08

0~5%
0.06

123

5~10%

0.08

0.08

0.08

0.08

10~20%

20~30%

30~40%

40~50%

0.06

0.06

0.06

0.06

0.06

0.04

0.04

0.04

0.04

0.04

0.04

0.02

0.02

0.02

0.02

0.02

0.02

0.06
Cu+Au 200 GeV

50~60%PHENIX

v2

Hydro η/s=0
AMPT 3mb

0.04

0.02

0

0
-4

-2

0

2

4

0
-4

-2

0

2

4

0
-4

-2

0

2

4

0
-4

-2

0

2

4

0
-4

-2

0

2

4

0
-4

-2

0

2

4

-4

-2

0

2

4

η

√
Figure 5.32: Charged hadron v2 (η) measured in Cu+Au collisions at sN N = 200GeV in comparison to theory calculations for diﬀerent % centrality classes. The theory calculations shown inthis figure are ideal hydrodynamic calculation(Hirano et al[52]) and AMPT model with σ = 3mb
parton cross section[50].

0.05

0.05

0.05

0.05

0.04

0~5%

5~10%

10~20%

0.03

Cu+Au 200 GeV 0.04
PHENIX
Hydro η/s=0
0.03
AMPT 3mb

0.03

0.03

0.02

0.02

0.02

0.02

0.01

0.01

0.01

0.01

20~30%

0.04

v3

0.04

0

0
-4

-2

0

2

4

0
-4

-2

0

2

0

4

-4

-2

0

2

4

-4

-2

0

2

4

η

√
Figure 5.33: Charged hadron v3 (η) measured in Cu+Au collisions at sN N = 200GeV in comparison to theory calculations for diﬀerent % centrality classes. The theory calculations shown inthis figure are ideal hydrodynamic calculation(Hirano et al[52]) and AMPT model with σ = 3mb
parton cross section[50].

peripheral collisions. The ideal hydrodynamic calculation show similar values as seen in theexperimental data. However the forward and backward asymmetry is still remained at peripheral
collisions with flatter centrality dependence. The AMPT model predict larger values of forward
and backward asymmetry except for central collisions. Like the ideal hydrodynamic calculation,

124

CHAPTER 5. DISCUSSIONS

2

v3(Au-going)/v3(Cu-going)v2(Au-going)/v (Cu-going)

the large forward and backward asymmetry is still remained at peripheral collisions in the AMPT
calculation. For the v3 measurements, the value of the ratio of forward and backward v3 slightly
increase from central to mid-central collisions. The ideal hydrodynamic calculation underpredicts
the values of v3 ratio. On the other hand, the AMPT model calculation overpredicts the valuesof the ratio.

Cu+Au 200 GeV
PHENIX
Hydro η/s=0

3

Cu+Au 200 GeV
PHENIX
Hydro η/s=0

2.5

AMPT 3mb

2

1.5

AMPT 3mb

2

1.5

1
0

20

40

60

centrality(%)
Figure 5.34: The ratio of forward/backward v2
values as a function of centrality in Cu+Au col√lisions at sN N = 200GeV in comparison to
the theory calculations. The theory calculations shown in this figure are ideal hydrodynamic
calculation(Hirano et al[52]) and AMPT modelwith σ = 3mb parton cross section[50].

1
0

10

20

30

centrality(%)
Figure 5.35: The ratio of forward/backward v3
values as a function of centrality in Cu+Au col√lisions at sN N = 200GeV in comparison to
the theory calculations. The theory calculations shown in this figure are ideal hydrodynamic
calculation(Hirano et al[52]) and AMPT modelwith σ = 3mb parton cross section[50].

Chapter 6

Conclusion
We have measured the directed flow v1 , elliptic flow v2 , and triangular flow v3 for charged√
hadrons, pions, kaons and protons at mid-rapidity (|η| < 0.35) in Cu+Au collisions at sN N =200GeV and the v2 and v3 for charged hadrons at forward and backward-rapidity (3 < |η| < 3.9)√
in Au+Au, Cu+Au and Cu+Cu collisions at sN N = 200GeV by the PHENIX experiment at
RHIC using event plane techniques.
At mid-rapidity, the measurements of charged hadron v1 , v2 and v3 in Cu+Au collisions were
performed as a function of transverse momentum pT over a wide range of collision centralities.
pT integrated v2 and v3 were also measured as a function of the number of participants Npart .
We found the following features of charged hadron vn in Cu+Au collisions,• The negative v1 is observed at high pT , which means the more number of high pT particles
are emitted toward the Au spectator side.
• The v2 and v3 have similar pT dependence, centrality and Npart dependence as seen in
Au+Au and Cu+Cu collisions• The v2 is always between Au+Au and Cu+Cu results• The v3 is similar to the Au+Au results
We also measured particle identified, π, K and p, vn (n = 1 ∼ 3) at mid-rapidity in Cu+Au
collisions. In the identified particle measurements, particle mass ordering was observed for all
three harmonics at low pT as predicted by hydrodynamics. Baryon/mass splitting was also
observed for 2nd and 3rd harmonics, but not observed for first harmonic. By studying with
Monte Carlo Glauber model and Blast wave model for interpreting the negative v1 at high pT ,we found the larger pressure gradient is Au nucleus side, which push more particles to high pT
region. We performed the eccentricity scaling of vn with Monte Carlo Glauber model, vn /εn ,to study the initial spatial geometry. At mid-rapidity, the consistency of the scaled v2 among
Au+Au, Cu+Au and Cu+Cu collisions is seen. On the other hand, the eccentricity scaling of v3
shows the discrepancy between Au+Au and Cu+Au collisions.
At forward/backward-rapidity, the measurements of the charged hadron v2 in Au+Au, Cu+Au
and Cu+Cu collisions and v3 in Au+Au and Cu+Au collisions were performed. Charged particle multiplicity dN/dη at in Au+Au, Cu+Au and Cu+Cu collisions were also measured. The
125

126

CHAPTER 6. CONCLUSION

charged hadron v2 and v3 were plotted as a function of pseudo-rapidity η over a wide range ofcollision centralities and as a function of the dNch /dη. In Cu+Au collisions, the main featuresof vn and dNch /dη are following,• The v2 at Au-going side is larger than that at Cu-going for mid-central collisions, v2,Cu−going <v2,Au−going• The v2 at Au-going side becomes similar to that at Cu-going for most-central and peripheral
collisions, v2,Cu−going ∼ v2,Au−going• Both of Au-going and Cu-going side v2 are between those in Au+Au and Cu+Cu collisions,v2,Cu+Cu < v2,Cu−going , v2,Au−going < v2,Au+Au• The v3 at Au-going side is larger than that at Cu-going, v3,Cu−going < v3,Au−going• Compared to the Au+Au collision results, the Au-going side v3 is larger and the Cu-goingside v3 is smaller, v3,Cu−going < v3,Au+Au < v3,Au−going• The dNch /dη at Au-going side is larger than that at Cu-going side dN/dη(Cu − going) <dN/dη(Au − going)
For studying of the forward/backward asymmetry of vn and dN/dη, we introduced weighted
Npart scaling for dNch /dη and weighted εn scaling for vn are tested based on the Glauber Monte
Carlo, where the weighted average of Npart and εn are calculated with Cu-participants and Auparticipants with varying the relative contributions between Cu and Au nuclei. By performing
the weighted Npart scaling, we found that the dNch /dη at Au-going side is described by both ofNpart,Au and Npart,Cu and the dNch /dη at Cu-going side is mainly determined Npart,Cu . From
the weighted εn scaling, we found that the forward/backward asymmetry of vn in Cu+Au collisions mainly arises from the forward/backward asymmetry of dN/dη which is supposed to be
proportional to the initial energy density, and the forward/backward rapidity vn is originatedfrom the common initial spatial anisotropy ϵn .
The measurements of inclusive charged hadron vn were compared to theoretical predictions.
In the measurements of v1 at mid-rapidity, the negative v1 values at high pT are observedwith Cu beam spectator defining the sign of v1 . The negative values of v1 indicate that moreparticles are emitted in transverse plane preferentially toward the spectators from Au nucleus.
The event by event ideal and viscous hydrodynamic calculations predict the positive values ofv1 at low pT and the negative values of v1 at high pT as observed in the experimental data.
Although the AMPT transport model reproduces the similar magnitude of the measured v1
signals, the sign of v1 is opposite compared with the experimental data. At low pT , the AMPT
models with parton cross section σ = 3mb reproduces the elliptic flow and triangular flowat mid-rapidity in Cu+Au collisions. The event by event viscous hydrodynamic calculations
with shear viscosity η/s = 0.08 − 0.16 also reproduce the measured v2 and v3 values at midrapidity in Cu+Au collisions. In case of the vn measurements at forward/backward-rapidity for
Cu+Au collisions, the event by event ideal hydrodynamic calculation reproduce the magnitude
of forward/backward-rapidity v2 and the forward/backward ratio of v2 , but fails to explain themagnitude of the v3 . Compared to the hydrodynamics, the magnitudes of v2 and v3 are better
described by the AMPT.

Appendix A

intrinsic triangularity
A.1

v3 (Ψ1 ) at mid-rapidity

In Cu+Au collisions, the intrinsic triangularity which does not only come from the geometricalfluctuations but also from the shape of the overlap region is expected to provide the non zero
value of v3 (Ψ1 ) at mid-rapidity. Figure A.1 shows the nucleon participant distributions in Au+Au
and Cu+Au collisions for the impact parameter range 4 < b < 5 fm. In Fig. A.1, the densityof participant nucleons decrease from the center to the surface, the red area represent higher
density and the blue area is lowest density. The black dashed circles represent the radiuses of Au
nucleus (RAu = 6.38) and Cu nucleus (RCu = 4.2). In the Au+Au collisions case, the overlap
area is the symmetric elliptic shape. On the other hand, the overlap area in the Cu+Au collisions
is the sideward asymmetric triangular shape. Thus in Cu+Au collisions, the initial triangular
overlap shape will be converted into the triangular flow.
In order to measure the triangular flow driven by the triangular overlap region in Cu+Au
collisions, the measurement of v3 with respect to 1st order event plane determined from spectator
neutrons is useful. If the v3 (Ψ1 ) shows non-zero values, the triangular overlap region leads to thetriangualr flow. Figure A.2 shows the azimuthal angle correlation between 3rd order participant
plane Ψ3,PP and the angle of impact parameter Ψ1,imp which points toward Cu side as a function√of centrality in Au+Au and Cu+Au collisions at sN N = 200GeV simulated by Monte Carlo
Glauber model. The values of the participant event plane correlaion in Au+Au collision are
zero due to the sideward symmetric overlap region. In Cu+Au collisions, the correlation shows
positive values and have centrality dependence. The values of the azimuthal correlation in
Cu+Au collisions increase from the central to the mid-central collisions and then decrese withincreasing with centrality. In Fig. A.3, the v3 (Ψ1,Imp ) at mid-rapidity as a function of pT fordiﬀerent centrality classes in Cu+Au collisions calculated from the ideal hydrodynamics [52]and the combined parton cascade and hadron cascade model (AMPT) [50] are shown. In bothof calculations, Glauber Monte carlro simulation are usede as the initial condition. Like theparticipant correlation shown in Fig. A.2, the ideal hydrodynamic calculation shows the positive
values. The value of v3 (Ψ1,Imp ) from the ideal hydrodynamics increase with pT , which is seen in
the v2 and v3 pT dependence. However AMPT does not show finite values. In Fig. A.4, v3 (Ψ1,Imp )as a function of pT measured without the resolution correction at mid-rapidity (|η| < 0.35) in
127

APPENDIX A. INTRINSIC TRIANGULARITY

10 Au+Au 200 GeV 5<b<6 fm

y[cm]

y[cm]

128

10 Cu+Au 200 GeV

0

0

-10

-10
-10

0

10
x[cm]

-10

0

10
x[cm]

Figure A.1: Participant nucleon distributions for the impact parameter range 4 < b < 5 in√
Au+Au and Cu+Au collisions at sN N = 200GeV simulated by Monte Carlo Glauber model√
Cu+Au collisions at sNN = 200GeV. The values of the measured v3 (Ψ1,Imp ) are consistent
with zero within the error.

A.1. V3 (Ψ1 ) AT MID-RAPIDITY

129<cos(3[Ψ3 - Ψ1,imp])>0.06

Au+Au 200 GeV
Cu+Au 200 GeV

0.04
0.02
0
0

20

40

60

80
100
centrality(%)

Figure A.2: Participant plane correlations between Ψ3,PP and the impact parameter Ψ1,imp in
Au+Au and Cu+Au collisionsv3(Ψ1,imp)

0.02

0.02

0.02

0.02

Hydro η/s=0
Ampt σ=3mb
0.01

10~20%

0.01

020~30%

0.01

0

0

1
2
p (GeV/c)T

330~40%

0.01

0

0

1

2
pT (GeV/c)

340~50%

00

1

2
pT (GeV/c)

3

0

1

2
pT (GeV/c)

3

Figure A.3: v3 (Ψ1,Imp ) at mid-rapidity as a function of pT in Au+Au and Cu+Au collisions at√
sNN = 200GeV calculated from the ideal hydrodynamics and the AMPT.

v3obs(Ψ1)

130

APPENDIX A. INTRINSIC TRIANGULARITY

×10-3

×10-3

×10-3

×10-3

1

1

1

1

Cu+Au 200GeV

10~20%

h±0.5

20~30%

30~40%

40~50%

0.5

0.5

0.5

0

0

0

0

-0.5

-0.5

-0.5

-0.5

0

2
p (GeV/c)
T

4

0

2
p (GeV/c)
T

4

0

2
p (GeV/c)
T

4

0

2

4

p (GeV/c)T

Figure A.4: v3 (Ψ1,SMDS ) as a function of pT measured without the resolution correction at mid√
rapidity (|η| < 0.35) in Au+Au and Cu+Au collisions at sNN = 200GeVAppendix B

Initial geometry model
B.1

Initial geometry model dependence

One of the uncertainties in theory model calculation of relativistic heavy ion collision reactionis the initial geometry condition. Since azimuthal anisotropies are strongly aﬀected by initial
spatial geometry of nucleus-nucleus collisions, eccentricity scaling of vn with diﬀerent initial geometry models can provide further insights and hints for understanding initial geometry models.
Historically, Monte Carlo Glauber model which describes initial spatial geometry with nucleon
positions has been used as initial geometry. In our study, we compare nucleon base Monte Carlo
Glauber model, quark base Monte Carlo Glauber model and IPGlasma model.
−Monte Carlo Glauber model : quark base
In dNch /dη measurements, PHOBOS dNch /dη data at mid-rapidity in Au+Au collisions at√
sN N = 130 and 200GeV are better described charged particle multiplicity at mid-rapidity in
Au+Au collisions by scaling with number of participant constituent quarks, Nqp than numberof participant nucleons,Npart [43]. Additionally, PHOBOS data are extended down to 62.4 and
19.6GeV and concluded the scaling with Nqp works well in the lower energies[45]. In PHENIX
experiment, dNch /dη at midrapidity is measured in various collision systems and energies[44],[47].
In Ref. [44] and [47], PHENIX verified the scaling with Nqp holds for Au+Au collisioins from
200 to 7.7GeV, for Cu+Au 200GeV and for Cu+Cu collisions at 200 and 62.4 GeV.
In our study, the number of participant constituent quarks is estimated using a modified
Monte Carlo Glauber model thats treat multiple quark-quark collisions instead of multiple
nucleon-nucleon collisions. First, the nuclei are assembled by the nucleons distributed according to a WoodsSaxon distribution. Second, the three constituent quarks are distributed aroundthe center of each nucleon. The constituent quarks are distributed radially by an empiricalfunction[47],f (r) = r2 exp −4.27r(1.21466 − 1.888r + 2.03r2 )(1 + 1/r − 0.03/r2 )(1 + 0.15r)(B.1)

where r is the radial distance of constituent quark in fm. Randomly the transverse positionsof three constituent quarks in each nucleon are assigned to become a spherically symmetric
distribution. Lastly, the center of mass of three constituent quarks in each nucleon is shiftedto be the center of the nucleon position. The above empirical quark distribution is choosen
131

132

APPENDIX B. INITIAL GEOMETRY MODEL

Figure B.1: Initial energy density in transverse plane for nulceon base Monte Carlo Glaubermodel and IPGlasma model. Left:nucleon base Monte Carlo Glauber model. Right:IPGlasma
model [55]to reproduce the measured the Fourier transform of the proton form factor in electron-proton
elastic√scattering[49]. If the distance between two centers of quarks in transverse plane is lessthaninel /π quark-quark collisions will occur.
σqq

√
d<

inel
σqq
π

(B.2)

inel = 8.17mb is the inelastic quark-quark cross section. The σ inel value is chosen towhere σqq
√ qq
reproduce the inelastic nucleon nucleon cross section σnn = 42mb at sN N = 200GeV.
−IPGlasma model : gluon baseThe values of eccentricity in Au+Au and Cu+Au collisions in IPGlasma model which is agluon base model are available[54]. At relativistic high energy, the density of gluons inside nucleus increase greatly. Both of nucleon base and quark base Monte Carlo Glauber models do nottake into account this phenomena. Although quark base Monte Carlo Glauber model include
fluctuations in nucleon and quark distributions, there are fluctuations in nucleon, auark gluon
distributions in IPGlasma model. In Fig.B.1, initial energy density distributions in transverse
plane for nulceon base Monte Carlo Glauber model and IPGlasma model are shown. The IPGlasma model have finner stuructures relative to the nucleon base Monte Carlo Glauber model.
The quark base Monte Carlo Glauber model is considered to be in between the two models.
In Fig. B.2 and Fig. B.3, ϵn as a function of Npart for Au+Au, Cu+Au and Cu+Cu
collisions at 200GeV in nucleon base Glauber Monte Carlo, quark base Glauber Monte Carloand IPGlasma models are compared. In Au+Au collisions, ϵ2 is largest in IPGlasma modeland smallest in nucleon base Monte Carlo Glauber. Quark base Monte Carlo Glauber is always
between them. In Cu+Au collisions, there is no significant diﬀerence between the three models.
Because IPGlasma data points for Cu+Cu collisions are not available, we compare only the two
Glauber Monte Carlo model. Like Cu+Au collisions, no model dependence is seen. For ϵ3 , themodel dependence is reversed. In Au+Au and Cu+Au collisions, nucleon base Monte Carlo
Glauber shows the highest values of ϵ3 , and IPGlasma shows the lowest values of ϵ3 except for
central collisions.
Fig. B.4 shows the scaled v2 with ϵ2 in nucleon base Monte Carlo Glauber, quark base Monte
Carlo Glauber and IPGlasma models. In both of nucleon and quark base Glauber Monte CarloB.1. INITIAL GEOMETRY MODEL DEPENDENCE

0.6
ε2

ε2

0.6
0.4
0.2
0
0

0.8

Au+Au
MC-Glauber nucleon base
MC-Glauber quark base
IPGlasma

0.4
0.2

100 200 300 400
Npart

0
0

0.8

Cu+Au
MC-Glauber nucleon base
MC-Glauber quark base
IPGlasma

0.6
ε2

0.8

133

Cu+Cu
MC-Glauber nucleon base
MC-Glauber quark base

0.4
0.2

100 200 300 400
Npart

0
0

100 200 300 400
Npart

Figure B.2: ϵ2 for Au+Au, Cu+Au and Cu+Cu collisions at 200GeV in nucleon base GlauberMonte Carlo, quark base Glauber Monte Carlo and IPGlasma models

ε3

0.6
0.4
0.2
0
0

0.8

Au+Au
MC-Glauber nucleon base
MC-Glauber quark base
IPGlasma

0.6
ε3

0.8

Cu+Au
MC-Glauber nucleon base
MC-Glauber quark base
IPGlasma

0.4
0.2

100 200 300 400
Npart

0
0

100 200 300 400
Npart

Figure B.3: ϵ3 for Au+Au and Cu+Au collisions at 200GeV in nucleon base Glauber Monte
Carlo, quark base Glauber Monte Carlo and IPGlasma models

134

APPENDIX B. INITIAL GEOMETRY MODEL

0.3

0.2
0.1
0
0

0.4

Au+Au 200GeV
Cu+Au 200GeV
Cu+Cu 200GeV
Hadron v2 |η|<0.35
MC-Glauber quark

0.3
v2/ε2

v2/ε2

0.3

0.4

Au+Au 200GeV
Cu+Au 200GeV
Cu+Cu 200GeV
Hadron v2 |η|<0.35
MC-Glauber nucleon

v2/ε2

0.4

0.2
0.1

200 400
dNch/dη00600

Au+Au 200GeV
Cu+Au 200GeV
Hadron v2 |η|<0.35
IPGlasma

0.2
0.1

200 400
dN/dη

0
0

600

200 400dN/dη

600

Figure B.4: Scaled v2 for charged hadrons at mid-rapidity with ϵ2 estimated in the three√
initial condition models in Au+Au, Cu+Au and Cu+Cu collisions at sN N = 200GeV.
Left:Nucleon base Monte Carlo Glauber model, Middle:Quark base Monte Carlo Glauber modeland Right:IPGlasma model

0.25

Cu+Au 200GeV
Hadron v3 |η|<0.35

0.2 MC-Glauber nucleon

v3/ε3

v3/ε3

0.25

0.3

Au+Au 200GeV

0.15

0.3

Au+Au 200GeV

0.25

Cu+Au 200GeV
Hadron v3 |η|<0.35

0.2 MC-Glauber quark

v3/ε3

0.3

0.15

0.2 IPGlasma

0.15

0.1

0.1

0.1

0.05

0.05

0.05

0
0

200 400
dNch/dη600

00

Au+Au 200GeV
Cu+Au 200GeV
Hadron v3 |η|<0.35

200 400
dN/dη

600

0
0

200 400dN/dη

600

Figure B.5: Scaled v3 for charged hadrons at mid-rapidity with ϵ3 estimated in the three initial√condition models in Au+Au and Cu+Au collisions at sN N = 200GeV. Left:Nucleon base Monte
Carlo Glauber model, Middle:Quark base Monte Carlo Glauber model and Right:IPGlasma mode
models, a good agreement between all collision systems is seen. Conversely, the discrepancy
between Au+Au and Cu+Au is seen in IPGlasma model.
In Fig. B.5, the scaled v3 at mid-rapidity with ϵ3 estimated from nucleon base Monte Carlo
Glauber, quark base Monte Carlo Glauber and IPGlasma models in Au+Au and Cu+Au colli√
sions at sN N = 200GeV. For the peripheral point in Cu+Au collisions, the scaled v3 values in
Cu+Au collisions are consistent with those in Au+Au collisions in all initial condition models. In
IPGlasma model, the scaled v3 values in central Cu+Au collisions are closer to those in Au+Au
collisions compared to the results in the two Monte Carlo Glauber models.
We perform the eccentricity scaling of vn at forward/backward rapidity. Fig. B.6 shows thescaled v2 at forward/backward-rapidity with the ϵ2,A+B in the three initial condition models. In
all initial condition models, the forward and backward scaled v2 with ϵ2,A+B are consistent. For
system size comparison, the scaled Cu-going v2 in Cu+Au collisions is consistent with those in
the symmetric collision systems for whole centrality classes in the three initial condition models.
Although the Au-going v2 is scaled well with the three initial condition models in mid-central andB.1. INITIAL GEOMETRY MODEL DEPENDENCE

0.25

v2/ε2

0.15 MC-Glauber nucleon
0.1

0.05

0.2
v2/ε2,A+B

0.2

Au+Au 200 GeV
Cu+Au 200 GeV Au-going
Cu+Au 200 GeV Cu-going
Cu+Cu 200 GeV

0.25
Au+Au 200 GeV
Cu+Au 200 GeV Au-going
Cu+Au 200 GeV Cu-going
Cu+Cu 200 GeV

0.15 MC-Glauber quark
0.1

0.05

0.2
v2/ε2,A+B

0.25

135

0.15 IPGlasma
0.1

0.05

0
0 100 200 300 400 500
dNch/dη00 100 200 300 400 500
dNch/dη

Au+Au 200 GeV
Cu+Au 200 GeV Au-going
Cu+Au 200 GeV Cu-going0
0 100 200 300 400 500
dNch/dηFigure B.6: Scaled v2 for charged hadrons at forward/backward-rapidity with ϵ2 estimated in the√three initial condition models in Au+Au, Cu+Au and Cu+Cu collisions at sN N = 200GeV.
Left:Nucleon base Monte Carlo Glauber model, Middle:Quark base Monte Carlo Glauber modeland Right:IPGlasma mode

MC-Glauber nucleon

0.05

00 100 200 300 400 500
dNch/dη

0.1

Au+Au 200 GeV
Cu+Au 200 GeV Au-going
Cu+Au 200 GeV Cu-going

MC-Glauber quark

0.05

00 100 200 300 400 500
dNch/dη0.1
v3/ε3,A+B

Au+Au 200 GeV
Cu+Au 200 GeV Au-going
Cu+Au 200 GeV Cu-goingv3/ε3,A+B

v3/ε3,A+B

0.1

Au+Au 200 GeV
Cu+Au 200 GeV Au-going
Cu+Au 200 GeV Cu-going

IPGlasma

0.05

00 100 200 300 400 500
dNch/dηFigure B.7: Scaled v3 for charged hadrons at forward/backward-rapidity with ϵ3 estimated√in the three initial condition models in Au+Au and Cu+Au collisions at sN N = 200GeV.
Left:Nucleon base Monte Carlo Glauber model, Middle:Quark base Monte Carlo Glauber modeland Right:IPGlasma mode
peripheral collisions, the scaled Au-going v2 is somewhat worse in central collisions. However,the central Au-going v2 /ϵ2,A+B with IPGlasma models in Cu+Au collisioins is close to that in
Au+Au collisions compared to the scaled v2 in the other two models.
In Fig. B.7, the scaled v3 at forward/backward-rapidity with ϵ3 in nucleon base Monte Carlo
Glauber model, quark base Monte Carlo Glauber model and IPGlasma model. Like the scaled
v2 results, a good agreement between Au-going and Cu-going v3 /ϵ3 is seen in all initial condition
models, and the colliding system diﬀerence is seen in mid-central and central collisions. However
the scaled v3 with ϵ3,A+B in IPGlasma model in Cu+Au collisions is closer to that in Au+Au
collisions for mid-central and central collisions compared to the other initial condition models.

Appendix C

Eccentricity and Npart scaling
C.1

Weighted eccentricity and Npart scaling of forward/backward
vn

In this section, we show the weighted eccentricity scaling of vn as a function of weighted
Npart at forward/backward-rapidity in Cu+Au collisions. In Fig. C.1, C.2, C.3 and C.4, thescaled vn for Au-going side and Cu-going side are compared to the symmetric collision systems.
In all figures, the weighted εn changes from εn,Cu to εn,Au and the weighted Npart changes from
2Npart,Cu to 2Npart,Au from the left to the right column.

136C.1. WEIGHTED ECCENTRICITY AND NP ART SCALING OF FORWARD/BACKWARD VN 137

0.1

Au+Au

0.2 1.5Npart,Au+0.5Npart,Cu
1.00ε2,Au+0.00ε2,Cuχ2/ndf=13.17

v2/ε2

0.1

χ2/ndf=8.64

v2/ε2

χ2/ndf=5.00

v2/ε2

v2/ε2

χ2/ndf=2.11

0.2 1.0Npart,Au+1.0Npart,Cu
1.00ε2,Au+0.00ε2,Cu
0.1

0.2 2.0Npart,Au+0.0Npart,Cu
1.00ε2,Au+0.00ε2,Cuχ2/ndf=18.05

v2/ε2

0.2 0.0Npart,Au+2.0Npart,Cu0.2 0.5Npart,Au+1.5Npart,Cu
1.00ε2,Au+0.00ε2,Cu
1.00ε2,Au+0.00ε2,Cu

0.1

0.1

Cu+Au Au-going
Cu+Cu

0

0 100 200 300 4000
100 200 300+2.0Npart,Cu0.2 0.5Npart,Au+1.5N
Npart part,Cu
0.2 0.0Npart,AuN
part
0.50ε2,Au+0.50ε2,Cu
0.50ε2,Au+0.50ε2,Cu
0.1

0.1

0

χ2/ndf=0.70

0.1

0.1

0.1

0

0

100 200 300 4000

100

200

300

χ2/ndf=1.33

0.1

0

0

0

100 200 300 4000
Npart

χ2/ndf=4.02

0.1

200 300
Npart

4000

0

4000
100 200 300
Npart part,Cu
0.2 1.5Npart,Au+0.5N
0.25ε2,Au+0.75ε2,Cuχ2/ndf=0.97

4000
100 200 300
Npart part,Cu
0.2 2.0Npart,Au+0.0N
0.25ε2,Au+0.75ε2,Cuχ2/ndf=1.07

04000
100 200 300
0.2 1.5Npart,Au+0.5N
Npart part,Cu
0.00ε2,Au+1.00
ε2,Cu
χ2/ndf=2.81

200 300
Npart

4000

4000
100 200 300
0.2 2.0Npart,Au+0.0N
Npart part,Cu
0.00ε2,Au+1.00
ε2,Cu

400

χ2/ndf=2.10

0.1

0
100

400

0.1

0.1

0
100

χ2/ndf=2.83

04000
100 200 300
0.2 1.0Npart,Au+1.0N
Npart part,Cu
0.00ε2,Au+1.00
ε2,Cu

400

0.1

0.1

v2/ε2

χ2/ndf=5.62

v2/ε2

v2/ε2

0.1

4000
100 200 300
Npart part,Cu
0.2 1.0Npart,Au+1.0N
0.25ε2,Au+0.75ε2,Cu0part,Cu
0.2 0.5Npart,Au+1.5N
0.2 0.0Npart,AuN+2.0N
Npart part,Cu
part ε
0.00ε2,Au+1.00
0.00ε2,Au+1.00
ε2,Cu
2,Cu

4000
100 200 300
Npart part,Cu
0.2 2.0Npart,Au+0.0N
0.50ε2,Au+0.50ε2,Cu

0

0.1

0

χ2/ndf=7.90

χ2/ndf=1.55

0.1

v2/ε2

χ2/ndf=2.23

v2/ε2

χ2/ndf=3.87

04000
100 200 300
Npart part,Cu
0.2 1.5Npart,Au+0.5N
0.50ε2,Au+0.50ε2,Cu

00 100 200 300 4000
100 200 300+2.0Npart,Cu0.2 0.5Npart,Au+1.5N
Npart part,Cu
0.2 0.0Npart,AuN
part
0.25ε2,Au+0.75ε2,Cu
0.25ε2,Au+0.75ε2,Cu
v2/ε2

4000
100 200 300
Npart part,Cu
0.2 1.0Npart,Au+1.0N
0.50ε2,Au+0.50ε2,Cu

0

0.1

0

v2/ε2

χ2/ndf=0.52

v2/ε2

v2/ε2

χ2/ndf=1.12

0.1

0χ2/ndf=8.23

v2/ε2

0

400

v2/ε2

0.1

v2/ε2

0

χ2/ndf=5.37

4000
100 200 300
Npart part,Cu
0.2 2.0Npart,Au+0.0N
0.75ε2,Au+0.25ε2,Cu

v2/ε2

0.1

4000
100 200 300
Npart part,Cu
0.2 1.5Npart,Au+0.5N
0.75ε2,Au+0.25ε2,Cu

v2/ε2

0.1

χ2/ndf=2.90v2/ε2

χ2/ndf=1.23

v2/ε2

v2/ε2

χ2/ndf=0.33

4000
100 200 300
Npart part,Cu
0.2 1.0Npart,Au+1.0N
0.75ε2,Au+0.25ε2,Cu

0

v2/ε2

0 100 200 300 4000
100 200 300+2.0Npart,Cu0.2 0.5Npart,Au+1.5N
Npart part,Cu
0.2 0.0Npart,AuN
part0.75ε2,Au+0.25ε2,Cu
0.75ε2,Au+0.25ε2,Cu

0

v2/ε2

0

v2/ε2

0

0
100

200 300
Npart

4000

100

200 300
Npart

Figure C.1: Weighted eccentricity scaling of v2 as a function of weighted Npart for Au-going sidein Cu+Au collisions compared to the Au+Au and Cu+Cu results.

400

APPENDIX C. ECCENTRICITY AND NP ART SCALING

0.1

Au+Au

χ2/ndf=41.97

v2/ε2

0.1

χ2/ndf=31.05

v2/ε2

v2/ε2

χ2/ndf=20.67

0.2 1.0Npart,Au+1.0Npart,Cu
1.00ε2,Au+0.00ε2,Cu

0.2 1.5Npart,Au+0.5Npart,Cu
1.00ε2,Au+0.00ε2,Cuχ2/ndf=54.42

0.1

0.2 2.0Npart,Au+0.0Npart,Cu
1.00ε2,Au+0.00ε2,Cuχ2/ndf=67.24

v2/ε2

0.2 0.0Npart,Au+2.0Npart,Cu0.2 0.5Npart,Au+1.5Npart,Cu
1.00ε2,Au+0.00ε2,Cu
1.00ε2,Au+0.00ε2,Cu

v2/ε2

138

0.1

0.1

Cu+Au Cu-going
Cu+Cu

0

0 100 200 300 4000
100 200 300+2.0Npart,Cu0.2 0.5Npart,Au+1.5N
Npart part,Cu
0.2 0.0Npart,AuN
part
0.50ε2,Au+0.50ε2,Cu
0.50ε2,Au+0.50ε2,Cu
0.1

0.1

0

χ2/ndf=7.65

0.1

0.1

0.1

0

0

100 200 300 4000

100

200

300

χ2/ndf=1.47

0.1

0

0

0

100 200 300 4000
Npart

χ2/ndf=0.39

0.1

200 300
Npart

4000

0

4000
100 200 300
Npart part,Cu
0.2 1.5Npart,Au+0.5N
0.25ε2,Au+0.75ε2,Cuχ2/ndf=3.53

4000
100 200 300
Npart part,Cu
0.2 2.0Npart,Au+0.0N
0.25ε2,Au+0.75ε2,Cuχ2/ndf=6.02

0

4000
100 200 300
0.2 1.5Npart,Au+0.5N
Npart part,Cu
0.00ε2,Au+1.00
ε2,Cu
χ2/ndf=0.64

200 300
Npart

4000

4000
100 200 300
0.2 2.0Npart,Au+0.0N
Npart part,Cu
0.00ε2,Au+1.00
ε2,Cu

400

χ2/ndf=1.35

0.1

0
100

400

0.1

0.1

0
100

χ2/ndf=17.32

0

4000
100 200 300
0.2 1.0Npart,Au+1.0N
Npart part,Cu
0.00ε2,Au+1.00
ε2,Cu

400

0.1

0.1

v2/ε2

χ2/ndf=0.90

v2/ε2

v2/ε2

0.1

4000
100 200 300
Npart part,Cu
0.2 1.0Npart,Au+1.0N
0.25ε2,Au+0.75ε2,Cu0part,Cu
0.2 0.5Npart,Au+1.5N
0.2 0.0Npart,AuN+2.0N
Npart part,Cu
part ε
0.00ε2,Au+1.00
0.00ε2,Au+1.00
ε2,Cu
2,Cu

4000
100 200 300
Npart part,Cu
0.2 2.0Npart,Au+0.0N
0.50ε2,Au+0.50ε2,Cu

0

0.1

0

χ2/ndf=2.23

χ2/ndf=12.28

0.1

v2/ε2

χ2/ndf=0.38

v2/ε2

χ2/ndf=0.15

04000
100 200 300
Npart part,Cu
0.2 1.5Npart,Au+0.5N
0.50ε2,Au+0.50ε2,Cu

00 100 200 300 4000
100 200 300+2.0Npart,Cu0.2 0.5Npart,Au+1.5N
Npart part,Cu
0.2 0.0Npart,AuN
part
0.25ε2,Au+0.75ε2,Cu
0.25ε2,Au+0.75ε2,Cu
v2/ε2

4000
100 200 300
Npart part,Cu
0.2 1.0Npart,Au+1.0N
0.50ε2,Au+0.50ε2,Cu

0

0.1

0

v2/ε2

χ2/ndf=4.20

v2/ε2

v2/ε2

χ2/ndf=1.60

0.1

0χ2/ndf=37.13

v2/ε2

0

400

v2/ε2

0.1

v2/ε2

0

χ2/ndf=28.69

4000
100 200 300
Npart part,Cu
0.2 2.0Npart,Au+0.0N
0.75ε2,Au+0.25ε2,Cu

v2/ε2

0.1

4000
100 200 300
Npart part,Cu
0.2 1.5Npart,Au+0.5N
0.75ε2,Au+0.25ε2,Cu

v2/ε2

0.1

χ2/ndf=20.63

v2/ε2

χ2/ndf=13.94

v2/ε2

v2/ε2

χ2/ndf=7.99

4000
100 200 300
Npart part,Cu
0.2 1.0Npart,Au+1.0N
0.75ε2,Au+0.25ε2,Cu

0

v2/ε2

0 100 200 300 4000
100 200 300+2.0Npart,Cu0.2 0.5Npart,Au+1.5N
Npart part,Cu
0.2 0.0Npart,AuN
part0.75ε2,Au+0.25ε2,Cu
0.75ε2,Au+0.25ε2,Cu

0

v2/ε2

0

v2/ε2

0

0
100

200 300
Npart

4000

100

200 300
Npart

Figure C.2: Weighted eccentricity scaling of v2 as a function of weighted Npart for Cu-going sidein Cu+Au collisions compared to the Au+Au and Cu+Cu results.

400

C.1. WEIGHTED ECCENTRICITY AND NP ART SCALING OF FORWARD/BACKWARD VN 139

0.08

0.08
0.08
0.08
0.08
0.0Npart,Au+2.0Npart,Cu 0.5Npart,Au+1.5Npart,Cu
1.0Npart,Au+1.0Npart,Cu
1.5Npart,Au+0.5Npart,Cu
2.0Npart,Au+0.0Npart,Cu
0.06 1.00ε3,Au+0.00ε3,Cu 0.06 1.00ε3,Au+0.00ε3,Cu
0.06 1.00ε3,Au+0.00ε3,Cu
0.06 1.00ε3,Au+0.00ε3,Cu
0.06 1.00ε3,Au+0.00ε3,Cuχ2/ndf=3.95

v3/ε3

0.04

Au+Au 0.02

χ2/ndf=10.29

v3/ε3

0.04

0.02

χ2/ndf=20.16

v3/ε3

χ2/ndf=34.53

v3/ε3

v3/ε3

χ2/ndf=55.44

0.04

0.04

0.04

0.02

0.02

0.02

0

0

0

Cu+Au Au-going

000.080 100 200 300 0.08
4000
100 200 300 0.08
4000
100 200 300 0.08
4000
100 200 300 0.08
4000
100 200 300
0.0Npart,AuN+2.0Npart,Cu 0.5Npart,Au+1.5N
1.0Npart,Au+1.0N
1.5Npart,Au+0.5N
2.0Npart,Au+0.0N
N
N
Npart part,Cu
Npart part,Cu
part,Cupart,Cu
part
part
part
0.06 0.75ε3,Au+0.25ε3,Cu 0.06 0.75ε3,Au+0.25ε3,Cu
0.06 0.75ε3,Au+0.25ε3,Cu
0.06 0.75ε3,Au+0.25ε3,Cu
0.06 0.75ε3,Au+0.25ε3,Cu
0.04

χ2/ndf=16.26

χ2/ndf=8.23

v3/ε3

0.04

v3/ε3

χ2/ndf=27.42v3/ε3

χ2/ndf=42.69

v3/ε3

v3/ε3

χ2/ndf=64.00

0.04

0.04

0.04

0.02

0.02

0.02

0.02

0.02

0

0

0

0

0

0.080 100 200 300 0.08
4000
100 200 300 0.08
4000
100 200 300 0.08
4000
100 200 300 0.08
4000
100 200 300
0.0Npart,AuN+2.0Npart,Cu 0.5Npart,Au+1.5N
1.0Npart,Au+1.0N
1.5Npart,Au+0.5N
2.0Npart,Au+0.0N
N
N
Npart part,Cu
Npart part,Cu
part,Cupart,Cu
part
part
part
0.06 0.50ε3,Au+0.50ε3,Cu 0.06 0.50ε3,Au+0.50ε3,Cu
0.06 0.50ε3,Au+0.50ε3,Cu
0.06 0.50ε3,Au+0.50ε3,Cu
0.06 0.50ε3,Au+0.50ε3,Cu
0.04

χ2/ndf=23.46

0.04

0.04

0.02

0.02

0.02

0.02

0.02

0

0

0

0

0

0.080 100 200 300 0.08
4000
100 200 300 0.08
4000
100 200 300 0.08
4000
100 200 300 0.08
4000
100 200 300
0.0Npart,AuN+2.0Npart,Cu 0.5Npart,Au+1.5N
1.0Npart,Au+1.0N
1.5Npart,Au+0.5N
2.0Npart,Au+0.0N
N
N
Npart part,Cu
Npart part,Cu
part,Cupart,Cu
part
part
part
0.06 0.25ε3,Au+0.75ε3,Cu 0.06 0.25ε3,Au+0.75ε3,Cu
0.06 0.25ε3,Au+0.75ε3,Cu
0.06 0.25ε3,Au+0.75ε3,Cu
0.06 0.25ε3,Au+0.75ε3,Cu
0.04

χ2/ndf=31.66

0.04

400

χ2/ndf=21.40

v3/ε3

0.04

v3/ε3

χ2/ndf=44.33

v3/ε3

χ2/ndf=60.32

v3/ε3

v3/ε3

χ2/ndf=81.36

0.04

400

χ2/ndf=14.13

v3/ε3

0.04

v3/ε3

χ2/ndf=35.56

v3/ε3

χ2/ndf=51.37

v3/ε3

v3/ε3

χ2/ndf=72.73

0.04

400

0.04

0.02

0.02

0.02

0.02

0.02

0

0

0

0

0

χ2/ndf=69.23

0.04

χ2/ndf=53.40

0.04

χ2/ndf=40.53

v3/ε3

0.04

v3/ε3

χ2/ndf=89.60

v3/ε3

0.04

v3/ε3

v3/ε3

0.080 100 200 300 0.08
4000
100 200 300 0.08
4000
100 200 300 0.08
4000
100 200 300 0.08
4000
100 200 300 400
0.0Npart,AuN+2.0Npart,Cu 0.5Npart,Au+1.5N
1.0Npart,Au+1.0N
1.5Npart,Au+0.5N
2.0Npart,Au+0.0N
part,Cupart,CuN
N
Npart part,Cu
Npart part,Cu
part
part
part
ε3,Cu
0.06 0.00ε3,Au+1.00ε3,Cu
0.06 0.00ε3,Au+1.00ε3,Cu
0.06 0.00ε3,Au+1.00ε3,Cu
0.06 0.00ε3,Au+1.00
0.06 0.00ε3,Au+1.00ε3,Cu
0.04

0.02

0.02

0.02

0.02

0.02

0

0

0

0

0

0

100 200 300 4000
Npart

100

200 300
Npart

4000

100

200 300
Npart

4000

100

200 300
Npartχ2/ndf=29.73

4000

100

200 300
Npart

Figure C.3: Weighted eccentricity scaling of v3 as a function of weighted Npart for Au-going sidein Cu+Au collisions compared to the Au+Au result.

400

140

APPENDIX C. ECCENTRICITY AND NP ART SCALING

0.08

0.08
0.08
0.08
0.08
0.0Npart,Au+2.0Npart,Cu 0.5Npart,Au+1.5Npart,Cu
1.0Npart,Au+1.0Npart,Cu
1.5Npart,Au+0.5Npart,Cu
2.0Npart,Au+0.0Npart,Cu
0.06 1.00ε3,Au+0.00ε3,Cu 0.06 1.00ε3,Au+0.00ε3,Cu
0.06 1.00ε3,Au+0.00ε3,Cu
0.06 1.00ε3,Au+0.00ε3,Cu
0.06 1.00ε3,Au+0.00ε3,Cuχ2/ndf=18.05

v3/ε3

0.04

Au+Au 0.02

χ2/ndf=7.32

v3/ε3

0.04

0.02

χ2/ndf=1.91

v3/ε3

χ2/ndf=2.43

v3/ε3

v3/ε3

χ2/ndf=10.48

0.04

0.04

0.04

0.02

0.02

0.02

0

0

0

Cu+Au Cu-going

000.080 100 200 300 0.08
4000
100 200 300 0.08
4000
100 200 300 0.08
4000
100 200 300 0.08
4000
100 200 300
0.0Npart,AuN+2.0Npart,Cu 0.5Npart,Au+1.5N
1.0Npart,Au+1.0N
1.5Npart,Au+0.5N
2.0Npart,Au+0.0N
N
N
Npart part,Cu
Npart part,Cu
part,Cupart,Cu
part
part
part
0.06 0.75ε3,Au+0.25ε3,Cu 0.06 0.75ε3,Au+0.25ε3,Cu
0.06 0.75ε3,Au+0.25ε3,Cu
0.06 0.75ε3,Au+0.25ε3,Cu
0.06 0.75ε3,Au+0.25ε3,Cu
0.04

χ2/ndf=2.92

χ2/ndf=9.40

v3/ε3

0.04

v3/ε3

χ2/ndf=1.19

v3/ε3

χ2/ndf=4.88

v3/ε3

v3/ε3

χ2/ndf=15.60

0.04

0.04

0.04

0.02

0.02

0.02

0.02

0.02

0

0

0

0

0

0.080 100 200 300 0.08
4000
100 200 300 0.08
4000
100 200 300 0.08
4000
100 200 300 0.08
4000
100 200 300
0.0Npart,AuN+2.0Npart,Cu 0.5Npart,Au+1.5N
1.0Npart,Au+1.0N
1.5Npart,Au+0.5N
2.0Npart,Au+0.0N
N
N
Npart part,Cu
Npart part,Cu
part,Cupart,Cu
part
part
part
0.06 0.50ε3,Au+0.50ε3,Cu 0.06 0.50ε3,Au+0.50ε3,Cu
0.06 0.50ε3,Au+0.50ε3,Cu
0.06 0.50ε3,Au+0.50ε3,Cu
0.06 0.50ε3,Au+0.50ε3,Cu
0.04

χ2/ndf=1.32

0.04

0.04

0.02

0.02

0.02

0.02

0.02

0

0

0

0

0

0.080 100 200 300 0.08
4000
100 200 300 0.08
4000
100 200 300 0.08
4000
100 200 300 0.08
4000
100 200 300
0.0Npart,AuN+2.0Npart,Cu 0.5Npart,Au+1.5N
1.0Npart,Au+1.0N
1.5Npart,Au+0.5N
2.0Npart,Au+0.0N
N
N
Npart part,Cu
Npart part,Cu
part,Cupart,Cu
part
part
part
0.06 0.25ε3,Au+0.75ε3,Cu 0.06 0.25ε3,Au+0.75ε3,Cu
0.06 0.25ε3,Au+0.75ε3,Cu
0.06 0.25ε3,Au+0.75ε3,Cu
0.06 0.25ε3,Au+0.75ε3,Cu
0.04

χ2/ndf=2.32

0.04

400

χ2/ndf=2.03

v3/ε3

0.04

v3/ε3

χ2/ndf=6.28

v3/ε3

χ2/ndf=14.66

v3/ε3

v3/ε3

χ2/ndf=29.04

0.04

400

χ2/ndf=4.12

v3/ε3

0.04

v3/ε3

χ2/ndf=2.71

v3/ε3

χ2/ndf=9.02

v3/ε3

v3/ε3

χ2/ndf=21.85

0.04

400

0.04

0.02

0.02

0.02

0.02

0.02

0

0

0

0

0

χ2/ndf=21.54

0.04

χ2/ndf=11.63

0.04

χ2/ndf=5.66

v3/ε3

0.04

v3/ε3

χ2/ndf=36.91

v3/ε3

0.04

v3/ε3

v3/ε3

0.080 100 200 300 0.08
4000
100 200 300 0.08
4000
100 200 300 0.08
4000
100 200 300 0.08
4000
100 200 300 400
0.0Npart,AuN+2.0Npart,Cu 0.5Npart,Au+1.5N
1.0Npart,Au+1.0N
1.5Npart,Au+0.5N
2.0Npart,Au+0.0N
part,Cupart,CuN
N
Npart part,Cu
Npart part,Cu
part
part
part
ε3,Cu
0.06 0.00ε3,Au+1.00ε3,Cu
0.06 0.00ε3,Au+1.00ε3,Cu
0.06 0.00ε3,Au+1.00ε3,Cu
0.06 0.00ε3,Au+1.00
0.06 0.00ε3,Au+1.00ε3,Cu
0.04

0.02

0.02

0.02

0.02

0.02

0

0

0

0

0

0

100 200 300 4000
Npart

100

200 300
Npart

4000

100

200 300
Npart

4000

100

200 300
Npartχ2/ndf=2.88

4000

100

200 300
Npart

Figure C.4: Weighted eccentricity scaling of v3 as a function of weighted Npart for Cu-going sidein Cu+Au collisions compared to the Au+Au result.

400

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