Measurements of Azimuthal Angle Dependence of
HBT radii with respect to the event plane√in sNN = 2.76 TeV Pb-Pb collisionsat LHC-ALICE

Naoto Tanaka

March 2018

Measurements of Azimuthal Angle Dependence of
HBT radii with respect to the event plane√in sNN = 2.76 TeV Pb-Pb collisionsat LHC-ALICE

Naoto Tanaka
Doctoral Progam 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
Quark is a fundamental particle of matter and normally confined in composite particles
called hadrons such as protons and neutrons which are the constituents of all kinds of atoms
and molecules. On the other hand, for a few millionth seconds after the Big Bang, universe
was in extremely high temperature and quarks and gluons were almost not bound together andfreely moving at the limitted space. It is like "plasma" state in quark and gluon matter so-called
Quark Gluon Plasma (QGP). Therefore, precise understanding of the properties of QGP is a key
to access how our universe evolves.
Experimentally, relativistic heavy ion collision is an unique tool to recreate QGP on theearth. The study of nucleus-nucleus collisions at ultra-relativistic energies aims to characterize
the properties of QGP. This requires a precise understanding of spatio-temporal evolution.
Hanbury-Brown and Twiss interferometry is an unique tool to access the geometrical source
size, freeze-out time, and emission duration of the system, which is an essential study for research of QGP.
Due to the large pressure gradient, QGP is largely expanded toward the surface, and initialelliptic shape which is determined by collision geometry leads to an anisotropy of pressure
gradient. If the local thermal equilibrium is established, system evoluion can be describedwith hydrodynamics, and initial geometrica anisotropy is converted as azimuthal anisotropy inmomentum space. Event plane (Ψ2 ) is an experimentally measured direction of the short axis
of initial elliptic shape based on the charactristics of azimuthal anisotropy itself. Basically HBTinterferometry is the method to measure the source size. By combining the HBT measurements
and event plane method, HBT interferometry is extended to the method to measure not only thesource size but also the source shape at freeze-out. That is measurements of azimuthal angle
dependence of HBT radii with respect to event plane. At RHIC, a relation between the initialeccentricity and the final eccentricity was extensively studied with this method. This relation issensitive to flow profile, expansion time, and viscosity of the source. Results indicate the initialelliptic shape is significantly diluted by collective expansion, but out-plane elongated ellitic
shape still remains at freeze-out.
If a huge number of nucleons exist in the nucleus, the shape of initial overlap region would
be perfectly elliptic shape. Since intial density distribution largely fluctuations, Initial geometry
can have higher order geometrical anisotropy (triangular, quadrangle shape, and higher orderanisotropic shape). This triangular shape could be also preserved at freeze-out and obserbed
with HBT measurements. Model simulation suggests that finite oscillations of HBT radii withrespect to Ψ3 (an direction perpendicular to the side of a initial triangular shape) can be observed, and first measurements of the azimuthal angle dependence of HBT radii with respect toΨ3 are performed at PHENIX. However the oscillation sign of HBT radii with respect to Ψ3
could not be determined, due to large uncertainties. In order to extract the triangular shape atfreeze-out and constrain theoretical model, it is important to disentangle the oscillation sign of
HBT radii relative to Ψ3 . In LHC energy, large multiplicity and excellent event plane resolution
in ALICE detector allows us to measure the azimuthal angle dependence of HBT radii withrespect to Ψ3 much more precisely. Furthermore, hydrodynamical model predicts oscillatons of
HBT radii with respect to Ψ2 in LHC energy could be different to those in RHIC. Measurementof azimuthal angle dependence of HBT radii with respect to Ψ2 and Ψ3 in LHC energy is oneof the important studies to understand the system evolution of heavy ion collisions.

Since ellipticity of participant becomes larger from central to peripheral collisions, "centrality" is one of the good probes for initial geometry. Also a triangular shape slightly changes
with centrality due to the difference of the number of participating nucleons. However, whencentrality changes from central to peripheral, not only initial geometry but also system size,
freeze-out temperature and flow velocity changes simultaneously. In order to separate the difference of system size and system shape, another probe to initial geometry is indispensable.
Recently Event Shape Engineering technique is developed, which is method to select initialgeometry within a certain centrality This technique gives us the new insight of relation between
the initial geometry and other observables.
In this thesis, measurements of azimuthal angle dependence of pion HBT radii with respect√
to Ψ2 and Ψ3 in Pb-Pb collisions at sNN = 2.76 TeV with ALICE detector is performed andthe space time evolution of heavy ion collisions is discussed with the relation between the initial collision geometry and geometrical source shape at freeze-out. Explicit oscillation can beobserved in azimuthal angle dependence of HBT radii with respect to Ψ2 . Final source eccentricity was extracted with relative amplitudes of HBT radii and a relation to initial eccentricity
calculated with Glauber simulation. it indicates that ,in heavy ion collisions, large collective
flow strongly expands the source along the short axis of elliptic shape during QGP state andfinal source eccentricity are significantly diluted. For azimuthal angle dependence of HBT radiiwith respect to Ψ3 , no significant oscillation of HBT radii can be found in the direction of beamaxis. However, in the azimuthal plane, we can determine the oscillation sign of HBT radii withrespect to Ψ3 .
Also, Event Shape Engineering technique is applied to the measurements of azimuthal angle dependence of pion HBT radii with respect to Ψ2 . This is the first measurements. Relation between initial geometrical source shape and final source shape at freeze-out within acertain centrality are measured as a function of centrality. Oscillation amplitudes of HBT radiiin azimuthal plane with respect to Ψ2 are significantly modified with Event Shape selection.
The difference of oscillation amplitudes of HBT radii within a certain centrality is scaled with
2nd -order azimuthal anisotropy. Blast wave model is an analytical tool to extract freeze-out
paramaters by fitting spectra of identified hadrons, azimuthal anisotropy, and azimuthal angle
dependence of HBT radii with respect to event plane. Blast wave fitting is performed to HBT
measurements with Event Shape Engineering in order to interpret the difference of oscillation
amplitudes of HBT radii.
Event Shape Engineering is also applied to the measurements of azimuthal angle dependence of pion HBT radii with respect to Ψ3 . This is also the first measurements. Contrary to
HBT measurement with respect to Ψ2 , no significant modification can be found in Event ShapeEngineering selection to azimuthal angle dependence of HBT radii with respect to Ψ3 , though
3rd -order azimuthal anisotropy is explicitly modified with Event Shape selection.

Contents
1 Introduction

1

1.1

Quantum Chromodynamics and Quark Gluon Plasma . . . . . . . . . . . . . .

1

1.2

Relativistic Heavy Ion Collisions . . . . . . . . . . . . . . . . . . . . . . . . .

6

1.2.1

Participant Spectator Picture . . . . . . . . . . . . . . . . . . . . . . .

8

1.2.2

Space Time Evolution . . . . . . . . . . . . . . . . . . . . . . . . . .

10

Experimental Observables . . . . . . . . . . . . . . . . . . . . . . . . . . . .

13

1.3.1

Transverse momentum spectra and radial flow . . . . . . . . . . . . . .

13

1.3.2

Azimuthal anisotropy . . . . . . . . . . . . . . . . . . . . . . . . . . .

17

1.3.3

Event Shape Engineering . . . . . . . . . . . . . . . . . . . . . . . . .

21

1.3.4

Hanbury-Brown and Twiss Interferometry . . . . . . . . . . . . . . . .

25

Thesis Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

36

1.3

1.4

2 Experiment

37

2.1

Large Hadron Collider (LHC) . . . . . . . . . . . . . . . . . . . . . . . . . .

37

2.2

ALICE experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

41

2.2.1

ALICE magnet system . . . . . . . . . . . . . . . . . . . . . . . . . .

43

2.2.2

Time Projection Chamber . . . . . . . . . . . . . . . . . . . . . . . .

45

2.2.3

Inner Tracking System . . . . . . . . . . . . . . . . . . . . . . . . . .

47

2.2.4

Time Of Flight . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

50

2.2.5

TZERO . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

53

2.2.6

VZERO . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

55

2.2.7

Forward Multiplicity Detector . . . . . . . . . . . . . . . . . . . . . .

57

3 Data Analysis
3.1

59

Event characterization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
i

59

3.2

3.3

3.4

3.5

3.6

3.7

3.1.1

Event trigger and data set . . . . . . . . . . . . . . . . . . . . . . . . .

59

3.1.2

Centrality determination . . . . . . . . . . . . . . . . . . . . . . . . .

60

3.1.3

Event selection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

63

3.1.4

Track selection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

63Azimuthal anisotropy and Event plane . . . . . . . . . . . . . . . . . . . . . .

63

3.2.1

Azimuthal anisotropy of emitting particles . . . . . . . . . . . . . . . .

64

3.2.2

Event plane determination . . . . . . . . . . . . . . . . . . . . . . . .

66

3.2.3

Event plane calibration . . . . . . . . . . . . . . . . . . . . . . . . . .

68

3.2.4

Measurements of azimuthal anisotropy . . . . . . . . . . . . . . . . .

73

Event Shape Engineering . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

76

3.3.1

Event Shape Engineering . . . . . . . . . . . . . . . . . . . . . . . . .

76

3.3.2

Event plane resolution with ESE . . . . . . . . . . . . . . . . . . . . .

80

Particle identification(PID) . . . . . . . . . . . . . . . . . . . . . . . . . . . .

81

3.4.1

Energy loss(dE/dx) in TPC . . . . . . . . . . . . . . . . . . . . . . . .

81

3.4.2

Particle identification in TOF . . . . . . . . . . . . . . . . . . . . . . .

83

3.4.3

TPC-TOF combined PID . . . . . . . . . . . . . . . . . . . . . . . . .

84

Pair selection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

86

3.5.1

Two Track Resolution . . . . . . . . . . . . . . . . . . . . . . . . . .

86

3.5.2

Pair Cut . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

88

3.5.3

Fraction of shared TPC cluster . . . . . . . . . . . . . . . . . . . . . .

88

3.5.4

Angular distance in ∆η -∆ϕ ∗ . . . . . . . . . . . . . . . . . . . . . . .

88

HBT analysis method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

93

3.6.1

Fitting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

95

3.6.2

Corrections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

96

3.6.3

Event Plane Resolution Correction . . . . . . . . . . . . . . . . . . . . 101

Systematic Uncertainties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
3.7.1

Systematic Uncertainties for HBT analysis . . . . . . . . . . . . . . . 105

3.7.2

Systematic Uncertainties for vn analysis . . . . . . . . . . . . . . . . . 115

4 Results
4.1

120

Azimuthal angle dependence of HBT radii with respect to Ψ2 . . . . . . . . . . 120
ii

4.2

4.1.1

1D projection of 3D Correlation functions . . . . . . . . . . . . . . . . 120

4.1.2

Centrality dependence of HBT radii with respect to Ψ2 . . . . . . . . . 123

Azimuthal anisotropy of charged hadrons . . . . . . . . . . . . . . . . . . . . 125
4.2.1

Second order azimuthal anisotropy and Event Shape Engineering (q2 )selection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125

4.2.2

Third-order azimuthal anisotropy and Event Shape Engineering (q3 ) selection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136

4.3

Azimuthal angle dependence of HBT radii with respect to Ψ2 with Event ShapeEngineering q2 selection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146

4.4

4.5

Azimuthal angle dependence of HBT radii with respect to Ψ3 . . . . . . . . . . 152
4.4.1

1D projection of 3D Correlation functions . . . . . . . . . . . . . . . . 152

4.4.2

Centrality dependence of HBT radii with respect to Ψ3 . . . . . . . . . 154

Azimuthal angle dependence of HBT radii with respect to Ψ3 with Event ShapeEngineering q3 selection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154

4.6

Consistency check of HBT radii with the previous results from ALICE . . . . . 162

4.7

Consistency check of v2 and v3 with the previous results from ALICE . . . . . 162

5 Discussion
5.1

168

Final source Eccentricity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168
5.1.1

Centrality dependence of final source eccentricity . . . . . . . . . . . . 168

5.1.2

kT dependence of final source eccentricity . . . . . . . . . . . . . . . . 172

5.1.3

Relative amplitude of HBT radii with respect to Ψ2 with ESE q2 selection173

5.1.4

kT dependence of relative amplitude of HBT radii with respect to Ψ2with ESE q2 selection . . . . . . . . . . . . . . . . . . . . . . . . . . 174

5.2

v2 scaling of Final source eccentricity . . . . . . . . . . . . . . . . . . . . . . 178

5.3

Interpretation of initial eccentricity difference with Blast-wave Model . . . . . 181
5.3.1

5.4

Extracted parameters of freeze out configuration with Blast-wave model 183

Final source triangular shape . . . . . . . . . . . . . . . . . . . . . . . . . . . 189
5.4.1

Centrality dependence of relative amplitude of HBT radii with respect
to Ψ3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189

5.4.2

kT dependence of final source eccentricity . . . . . . . . . . . . . . . . 190

iii

5.4.3

Relative amplitudes of HBT radii relative to Ψ3 with q3 selection . . . 192

5.4.4

kT dependence of relative amplitude of HBT radii with respect to Ψ3with ESE q3 selection . . . . . . . . . . . . . . . . . . . . . . . . . . 193

5.5

v3 scaling of Final source eccentricity . . . . . . . . . . . . . . . . . . . . . . 198

6 Conclusion

200

iv

List of Figures
1.1

The running strong coupling constant αs as a function of momentum transfer Q, which is determined from τ decay, deep inelastic scattering process, radiative
ϒ decay, hadronic final states of e+ e− annihilation, hadronic Z decay width andinclusive jet cross section in p p̄ [1]. Data is compared with pQCD calculation
shown in black line.

1.2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4

Static quark potential as a function of distance between quark and anti-quarkin 2+1 flavor Lattice QCD calculation [2]. Simulation result (black marker) is
fitted with Eq. 1.6. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1.3

4

Energy density and 3 times the pressure normalized by T 4 calculated on Lattice
QCD as a function of temperature [3]. εSB /T 4 denotes the Stefan-Boltzmann
limits where the interaction of two particles are ignored. The vertical band
indicates the transition region 185 MeV < T < 195 MeV. The black bars at
high temperatures indicate the systematic shift of data that would arise frommatching to a hadron resonance gas at T = 100 MeV. . . . . . . . . . . . . . .

1.4

5

Schematic illustration of colliding nuclei before and after the collision [4]. Impact parameter is defined as the distance between the centers of the two colliding
nuclei in a transverse plane to the beam axis. In relativistic heavy ion collisioncolliding nuclei is divided into participant ,which participate to collisions, andSpectator that does not participate to collisions. . . . . . . . . . . . . . . . . .

1.5

9Schematic diagram of space-time evolution in relativistic heavy ion collisions.
Two colliding nuclei is approaching from t < 0 shown as "beam". Collision timeis defined as t = 0, z = 0. QGP phase is shown orange-yellow gradation and after
the hadronization hadron phase is shown in blue gradation. . . . . . . . . . . .

v

11

1.6

pT spectra of π , K and p as a function of centrality, for positive (red circle) and√negative (blue square) hadrons measured in sNN = 2.76 TeV Pb-Pb collisions[9]. Each panel shows central (0-5%) to peripheral (80-90%); spectra scaledby factors 2n (n denotes the centrality bin, which most central collisions arecorresponding to n = 0 and n becomes larger from central to peripheral). Two
different fits are performed, individual fit to each particles (dashed lines) andsimultaneous fit to π , K, p (dotted lines).

1.7. . . . . . . . . . . . . . . . . . . .

15

The correlation between the two extracted parameters, kinetic freeze-out temperature Tkin and average expansion velocity ⟨βT ⟩ by Blast-wave fits in Pb-Pb√√
collisions at sNN = 2.76 TeV (Black contours) and Au-Au collisions at sNN= 200 GeV (Blue contours) [9].

1.8

. . . . . . . . . . . . . . . . . . . . . . . . .

Illustrations of non-central relativistic heavy ion collisions in geometrical space(a) and momentum space (b) . . . . . . . . . . . . . . . . . . . . . . . . . . .

1.9

16

19

Azimuthal anisotropy v2 , v3 , v4 and v5 as a function of centrality percentile in√sNN = 2.76 TeV Pb-Pb Collisions from ALICE. vn is calculated with two particle correlation method using pT > 0.2 GeV/ccharged particles are integrated.
ALICE data (black marker) is compared with viscous hydro dynamical calculation (solid and dashed lines) (left). pT dependence of azimuthal anisotropy incentrality 30-40% via Event Plane method by ATLAS collaboration(right). For
both hydrodynamical simulations, shear viscosity to entropy ratio η /s is set to
0.2 [11].

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

19

1.10 pT integrated azimuthal anisotropy v2 and v3 as a function of number of par√ticipant with two different pT ranges. in Au-Au collisions at sNN = 200 GeV
from PHENIX. Experimental data (black marker) are compared with theoretical predictions of two different initial state models MC-KLN and Glauber anddifferent viscosity [12].

. . . . . . . . . . . . . . . . . . . . . . . . . . . . .

20

1.11 Azimuthal anisotropy v2 , v3 , v4 and v5 as a function of transverse momentum
for centrality 30-40% measured with a two particle correlation method of two
different rapidity gaps, ∆η > 0.2 (open symbol) and ∆η > 1.0 (closed symbol),√in Pb-Pb collisions at sNN = 2.76 TeV from ALICE. Data are compared withhydrodynamical simulations of ideal fluid and viscous fluid (η /s = 0.08). . . .
vi

20

1.12 The probability distribution of event-by-event v2 (left), v3 (middle) and v4 (right)√
in 5 or 6 centrality in Pb-Pb collisions at sNN = 2.76 TeV from ATLAS collaboration [14]. Error bars represent statistical uncertainties and Systematic
uncertainties are shown as shaded bands. Solid line denotes the distributions
assuming the vn are radial projections of 2D Gaussian distributions. . . . . . .

21

1.13 Azimuthal anisotropy v2 as a function of transverse momentum in the centrality
20-30% (top left) for events selected on q2 measured in Pb-Pb Collisions at√
sNN = 2.76 TeV from ATLAS. Bottom left panel shows the ratio of v2 forq2 -selected to that without q2 selection, and a correlation of v2 between pT:0.5-2GeV/c and pT :3-4GeV/c in 7 centrality bins. Data points represent v2
correlation in various q2 selection and overlaid gray band denotes that without
q2 selection [17]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

23

1.14 The ratio of pT distributions of π + + π − , K + + K − , and p + p̄ with large (small)q2 selection to those without q2 selection in centrality 30-40% are shown in left√
(right) panel. Results are measured in sNN = 2.76 TeV Pb-Pb Collisions from
ALICE collaboration. Data are compared with the Blast-wave model which is
the hydrodynamical inspired model. Based on the Blast-wave model, an average boost velocity ⟨βT ⟩ is 0.41% enhanced (0.22% suppressed) with ESE selection [19]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

24

1.15 The schematic figure of HBT interferometry. Two identical particles emitted
from a certain source. The emission points of each particles are defined as r1
and r2 . A distance between r1 and r2 is expressed as R. Two particles areobserbed by detectors at x1 and x2 . The distance between emission point anddetected position is L. If R is much smaller than L, route 1 and route 2 can not
be identified with quantum mechanics.

. . . . . . . . . . . . . . . . . . . . .

26

1.16 Extracted 3D HBT radii (Rside , Rout , and Rlong ) and λ as a function of mT for√
4 centralities measured in Au-Au collisions at sNN = 200 GeV from PHENIX[25]. Data are compared with hydrokinetic simulation (HKM model) and viscoushydrodynamic model (Bozek) . . . . . . . . . . . . . . . . . . . . . . . . . .

vii

29

1.17 Azimuthal angle dependence of charged pion HBT radii (Rout , Rside , Rlong , and√Ros ) with respect to Ψ2 for 3 centralities measured in Au-Au collisions at sNN= 200 GeV from STAR (left). Initial source eccentricity calculated with Glauber
simulation v.s. source eccentricity at freeze-out obtained with azimuthal angle
dependence of HBT radii (right) [28]. The dashed line indicates that εinitial =ε f inal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

30

1.18 Azimuthal angle dependence of charged pion HBT radii (Rout , Rside , Rlong , andRos ) with respect to Ψ2 for 6 kT classes in LHC energy, calculated with hydrodynamical model [30]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

31

1.19 Azimuthal angle dependence of charged pion HBT radii (Rout , Rside , Rlong , andRos ) with respect to Ψ2 (top 4 panels) and Ψ3 (bottom 4 panels) for 2 centralities√
measured in Au-Au collisions at sNN = 200 GeV from PHENIX [32]. First row
from the left denotes Rside , second row is Rout , third row is Rlong , and fourth
row indicates Ros .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

33

1.20 Relative amplitude of squared HBT radii (Rout , Rside , and Ros ) with respect toΨ3 as a function of initial eccentricity which is calculated with Glauber model.
√
The results are calculated with data in Au-Au collisions at sNN = 200GeV
from PHENIX [32]. Pair transverse momentum kT is integrated from 0.2 to
2.0 GeV/c. Dashed line indicates relative amplitude of HBT radii = 0 and εn =|2R2µ ,n /R2µ ,0 |. Boxes represent the systematic uncertainties. . . . . . . . . . . .

34

1.21 Relative amplitude of squared HBT radii(Rout , Rside , and Ros ) for charged pion
pairs with respect to Ψ3 as a function of pair transverse momentum kT for two√centralities measured in Au-Au collisions at sNN = 200GeV from PHENIX[32]. Data is compared with two different Gaussian source distribution model.
Solid line indicates that flow dominated case which is spherical spatial distribution superimposed on an large triangular flow, and dashed line shows geometry
dominant case which is the geometric triangular distribuion superimposed on

2.1

an azimuthally symmetric radial flow. . . . . . . . . . . . . . . . . . . . . . .

35

CERN accelerator complex . . . . . . . . . . . . . . . . . . . . . . . . . . . .

40

viii

2.2

3D schematic of ALICE experiment at the CERN LHC. ITS, TPC, TRD, TOF,EMCAL, PHOS and HMPID are installed in 0.5 Tesla magnetic field which isapplied with solenoid magnet to measure the midrapidity particles. Forward
detectors (VZERO, TZERO, FMD, PMD and ZDC) are also in the magnet andused for triggering and event characterization. In order to trigger the cosmic-ray,
ACORDE is placed on the top of solenoid magnet. Muon arm which consistsof Muon tracker (MCH) and Muon trigger (MTR) is located in the forward∫

rapidity in the dipole magnet with Bdz=3Tm after the front absorber. . . . . .

42

2.3

3D schematic of L3 magnet . . . . . . . . . . . . . . . . . . . . . . . . . . . .

44

2.4

3D schematic of Dipole magnet . . . . . . . . . . . . . . . . . . . . . . . . .

44

2.5

3D schematic of TPC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

46

2.6

A cartoon illustrating of TPC wire in the readout chamber . . . . . . . . . . . .

46

2.7

Two track efficiency for particle density dNch /dy=8300 as a function of absolutevalue of generated momentum difference of two particle with HIJING+GEANT
simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2.8

46

Front view of SPD layers layout. It is a modular structure with 10 sectors madeof light carbon-fibre in φ . 6 staves (2 ladders and 4 ladders) are fixed on eachcarbon-fibre sectors. [41] . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2.9

48

A cartoon illustrating of how to measure hit position of charged particles with
SDD. Electrons created by charged particles drift in a silicon and signals areread in n type substrate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

48

2.10 SDD (third layer) has 14 ladders and 6 detectors in each ladders for 3rd layerand 22 ladders and 8 detectors for 4th layer[42] . . . . . . . . . . . . . . . . .

49

2.11 Side view of SSD. SSD has 34 and 38 ladders for layer5 and layer6, respectively. Each ladder in layer 5 and 6 is made of 38 and 26 SSD modules, respectively [43]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
√2.12 Primary vertex resolution in Pb-Pb collisions at sNN = 2.76TeV as a function

49of half of the tracklets multiplicity of the event. In this figure, the resolutionof X and Z coordinate primary vertex resolution are shown. The resolution is
obtained dividing the tracks of the event in two random samples.[44] . . . . . .

ix

49

2.13 Construction details of a double-stack MRPC modules. 4 resistive inner glasses
are inserted between external glass plate. Width of 5 gas Gaps are 250 µ m,
having fishing lines between the glass. High voltage are applied to the speciallydeveloped acrylic paint loaded on the external glass.[46] . . . . . . . . . . . .

51

2.14 (Left) Time distribution of MRPC with respect to scintillator start time after
the slewing calibration. Then 50.8ps time resolution can be obtained Gaussian
fitting. (Right) Efficiency, time resolution and the ratio of streamer as a functionof applied voltage across 5 gas gaps. Typical efficiency and time resolution flatas a function of applied high voltage is more than 11kV with 99.9% efficiency
before the streamer is getting large.[47] . . . . . . . . . . . . . . . . . . . . .

52

2.15 Distribution of the difference between measured and expected arrival time on
TOF for selected pions (p:0.95-1.05GeV/c) in Pb-Pb collisions at 2.76 TeV. The
width of the signal of charged pions at 1.5GeV/c. Red line shows the Gauss
function to data.[48] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

52

2.16 T0 detector C-side has 12 cherenkov counters made of fused cherenkov radiator(quartz) and photomultiplier located on the opposite side of IP.[47] . . . . . . .

54

2.17 Time of flight distribution obtained in the 2004 test run with 6 GeV/c particles. Both start and stop time is measured with T0 (Cherenkov radiator+PMT)
modules. The FWHM for this distribution is 94 ps which corresponds to 28 ps
R.M.S. [49]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

54

2.18 V0 detector Aside consists of 32 scintillator (4 rings and 8 segments of 45 degrees in φ ) and WLS fibers. For V0A detector, WLS fibres are spaced by 1cm
and each segments are divided with "magtile" construction method. Signals arecollected with WLS fibres and sent to connected PMT. [47] . . . . . . . . . . .

56

2.19 Schematic figure of coupling of the scintillator and WLS fibre for V0C. WLS
fibre is embedded along the two radial edges of the scintillator piece. [49]. . . .

56

2.20 Location of overall FMD detector and ITS. Orange point denotes IP and ITS
depicted yellow and green region surrounds the IP. On the right side of ITS,
FMD3 module (two rings) is shown in red. Two rings on the left side of ITS
indicate FMD2 module. Distantly located from ITS, FMD2 and FMD3, FMD1
module is represented as grey octagon. [49] . . . . . . . . . . . . . . . . . . .
x

58

2.21 Assembly of FMD inner ring and FMD1 from 10 modules (left) and FMD outerring from 20 modules (right). Each modules are subdivided into two segments
in azimuthal angle. [49] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.1

58

Geometry and time alignment of V0A, V0C and hadron absorber for muon
spectrometer. Cyan trapezoid denotes hadron absorber. Interaction Point isshown in black full circle. [49] . . . . . . . . . . . . . . . . . . . . . . . . . .

3.2

60

Distribution of sum of V0A+C amplitude. Data is fitted with NBD Glauber fit
shown in red line. The centrality classes obtained by fitting are indicated withgrey and white region. Inserted figure denotes a zoom of the most peripheral
events. [50] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3.3

62

Centrality resolution with V0A, V0C, V0A+C, TPC(|η |<0.8), SPD(|η |<1.4) asa function of centrality percentile. resolution is calculated with 6 all estimation
detectors. [50]3.4

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

η v.s. ϕ 2D distribution measured via FMD. Z axis denotes multiplicity of eachsilicon strip channel. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3.5

62

67

2nd, 3rd and 4th harmonic q vector x distribution which is determined via FMD
A side + C side combined in centrality 0-50%. Red line shows uncorrecteddistribution. Blue line indicates event plane distribution with recentering calibration. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3.6

69

2nd, 3rd and 4th harmonic event plane distribution which is determined via
FMD A side + C side combined in centrality 0-50%. Red line shows uncorrected distribution. Orange line indicates event plane distribution with recentering calibration. Blue line represents recentering+flattening calibrated event
plane distribution.

3.7. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

70

2nd, 3rd and 4th harmonic event plane resolution which is determined via TPC(|η |<1.0),
V0 A side, V0 C side, V0 A+C combined, FMD A side, FMD C side, FMD
A+C combined , TPC(0.5 < η < 1.0) and TPC(-1.0 < η < -0.5) as a functionof centrality. Blue line indicates event plane distribution with recentering calibration. Since number of segments in V0 is not sufficient, 4 th harmonic event
plane resolution via V0 is not shown.

xi. . . . . . . . . . . . . . . . . . . . . .

72

3.8

2nd, 3rd and 4th harmonic anisotropy vn as a function of centrality. vn is measured via Event plane method and compared with previous ALICE results(twoparticle correlation). Event plane is determined with 4 different detectors. . . .

3.9

74

2nd, 3rd and 4th harmonic anisotropy vn as a function of pT for the 6 centrality
classes, measured via event plane method. Black markers are previous ALICE
results via event plane method. . . . . . . . . . . . . . . . . . . . . . . . . . .

75

3.10 The magnitude of q2 (left) and q3 (right) flow vector distribution in centrality
0-10%. qn is determined with FMD A+C combined. . . . . . . . . . . . . . . .

77

3.11 Cumulated q2 (left) and q3 (right) distribution as a function of centrality. flow
vector qn is determined via FMD A+C combined. Contour(5%) maps are plotted simultaneously. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

78

3.12 Cumulated q2 (left) and q3 as a function of q2 (left) and q3 distribution in centrality 0-1%(red) and 30-31%(blue). Spline fitting is performed to each slice of
qn distribution. Cumulated q-vector distribution is rebeined (merged 100 bins
in one) for visibility. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

78

3.13 q2 (left) and q3 distribution for each 20% q2 and q3 in centrality 10-20%. Bothq2 and q3 are determined with FMD A+C combined. . . . . . . . . . . . . . .

79

3.14 2nd(left) and 3rd(right) harmonic event plane resolution as a function of centrality for each 20% q2 and q3 event classes. q2 and q3 selection and event
plane are determined with FMD A+C combined. Inclusive(without q2 and q3
cut ) event plane resolutions are also depicted as open marker. Event plane resolution is extracted with 3-sub event method and the combination is FMD A+C,
TPC(-1.0<η <-0.5) and TPC(0.5<η <1.0). . . . . . . . . . . . . . . . . . . . .

xii

80

3.15 π /K(left) and K/p(right) separation power in the ITS, TPC, TOF and HMPID
as a function of transverse momentum at mid rapidity. Y axis denotes the distance between the peaks divided by the resolution for the pion and the kaon,
respectively. The left (right) panel shows the separation of pions and kaons(kaons and protons), expressed as the distance between the peaks divided by the
resolution for the pion and the kaon, respectively. For the TPC, an additional
dashed line is depicted in a edge of rapidity coverage. The lower panels show
the range over which the different ALICE detector systems have a separation
power of more than 2σ [55]. [54]. . . . . . . . . . . . . . . . . . . . . . . .

81

3.16 Energy loss (dE/dx) in TPC as a function of charge z × particle momentum in
Pb-Pb 2.76 TeV collisions. pions, electrons, kaons and protons are identified
with the difference of energy loss especially in low momentum region.

. . . .

82

3.17 Distribution of β measured by TOF detector as a function of track momentumin Pb-Pb 2.76 TeV collisions . . . . . . . . . . . . . . . . . . . . . . . . . . .

83

3.18 TPC dE/dx Nσ of π ± as a function of particle transverse momentum shownin left and Right figure represents TOF Nσ for pion as a function of particle
momentum transverse. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

85

3.19 Cartoon illustrating of track merging and splitting effect. Horizontal lines denotes the TPC read-out pad rows. Two tracks shown in red and blue dashedlines are spatially close. Measured TPC clusters of these two tracks are shown
as orange markers. Elliptic markers in yellow gradation denote the shared
TPC clusters. Due to finite two track resolution, two tracks are falsely reconstructed as one track depicted as black solid line(left), while tracks composedof shared TPC clusters(black dashed line) can be falsely reconstructed as two
tracks shown in orange and blue solid line(right). . . . . . . . . . . . . . . . .

87

3.20 Track splitting & Track merging effect in correlation function C2 as a functionof one dimensional relative momentum(Qinv ). Blue marker indicates correlation
function without any pair cut. Red marker denotes correlation function after paircut. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

xiii

87

3.21 Two dimensional ratio in ∆η -∆ϕ ∗ in different kT at the TPC radii(R=1.1). kT
ranges are 0.2-0.3, 0.3-0.4, 0.4-0.5, 0.5-0.6, 0.6-0.7, 0.7-1.0 GeV/c. In all kTranges, one can find the broad peak and narrow dip at small ∆η -∆ϕ ∗ .

. . . . .

90

3.22 ∆η projection of the ∆η -∆ϕ ∗ 2D ratio for each kT range, at R = 1.1m Projected∆η distributions are fitted with double Gaussian function shown in black dashedlines.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

91

3.23 ∆ϕ ∗ projection of two dimensional ration in ∆η -∆ϕ ∗ at R = 1.1m. Double
Gaussian fitting is performed to all distributions.

. . . . . . . . . . . . . . . .

91

3.24 Width of Gaussian fit function(narrow dip) to ∆η distribution as a function of
extrapolated TPC radius R for each kT bins in centrality 0-50%. Closed circle
and open square marker denote π + π + and π − π − pairs, respectively. Width
grows from low kT to high kT . . . . . . . . . . . . . . . . . . . . . . . . . . .

92

3.25 Width of Gaussian fit function(narrow dip) to ∆ϕ ∗ distribution as a function of
extrapolated TPC radius R. Closed circle and open square marker denote π + π +and π − π − pairs, respectively. Width grows from low kT to high kT . . . . . . .

92

3.26 Relative momentum distribution of real pairs(red) and mixed pairs(blue) are
simultaneously depicted in top panel. Mixed pair distribution is scaled to thereal pair in qinv range 0.17-0.34 (GeV/c). Correlation function is extracted with
event mixing technique and shown in red marker on the bottom panel. . . . . .

94

3.27 1D projection of 3D Relative momentum distribution of real pairs(red) andmixed pairs(blue) are simultaneously depicted in top three panels. Left, Middle and Right panel are out, side and long direction, respectively. Mixed pair
distribution is scaled to the real pair in qout,side,long range 0.15-0.30 (GeV/c).
Correlation function is extracted with event mixing technique and shown inblue marker on the bottom panel.

. . . . . . . . . . . . . . . . . . . . . . . .

94

3.28 Coulomb correction factor F (q) calculated by Coulomb wave function assuming Gaussian distribution at pair transverse momentum kT :0.2-0.3GeV/c for 3
centralities as a function of one dimensional relative momentum qinv . . . . . .

xiv

98

3.29 Correlation function of pure pions for centrality 20-30% calculated with HIJING + GEANT (left panel). Correlation function of generated pions that detector effect is not included is shown in closed marker and Correlation function of reconstructed pions that detector effects are included is plotted in open
marker. (open marker). Ratio of generated correlation function to reconstructed
correction function shown in right panel.

. . . . . . . . . . . . . . . . . . . . 100

3.30 Extracted 3D HBT radii of charged pions as a function of azimuthal pair angle
with respect to Ψ3 . Comparison of before and after the event plane resolution
correction,. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

3.31 Extracted 3D HBT cross term of charged pions as a function of azimuthal pairangle with respect to Ψ3 . Comparison of before and after the event plane resolution correction. All points are shifted along the y-axis for visibility. . . . . 104

3.32 Extracted 3D HBT radii (λ , R2out , R2side , R2long ) of charged pions as a functionof azimuthal pair angle relative to Ψ2 . Figures in top rows represents centrality 0-5%, while 5-10% are shown in bottom panels. Default value is positive
and negative pions combined results shown in black marker. Comparison withpositive and negative charged pion pairs are depicted as orange circle and blue
square, respectively.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

3.33 Extracted 3D HBT radii (λ , R2out , R2side , R2long ) of charged pions as a functionof azimuthal pair angle relative to Ψ2 . Figures in top rows represents centrality
0-5%, while 5-10% are shown in bottom panels. Default value is positive andnegative B field combined results shown in black marker. Positive and negative
B field results are separately analyzed and depicted in green circle and yellow
square, respectively.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

3.34 Extracted 3D HBT radii (λ , R2out , R2side , R2long ) as a function of azimuthal pairangle relative to Ψ2 , varying fit range from 150GeV/c (Default) to 200GeV/c
for each 10GeV/c bin. Figures in top rows represents centrality 0-5%, while 510% are shown in bottom panels. Default fit range is 150GeV/c shown in black
marker. Comparison with positive and negative charged pion pairs are depicted
as orange circle and blue square, respectively. . . . . . . . . . . . . . . . . . . 108

xv

3.35 Extracted 3D HBT radii (λ , R2out , R2side , R2long ) as a function of azimuthal
pair angle relative to Ψ2 with two different pair selection. Figures in top rows
represents centrality 0-5%, while 5-10% are shown in bottom panels. Default
pair selection is plotted in black marker. Comparison with the result in tighter
pair selection are depicted as orange circle. . . . . . . . . . . . . . . . . . . . 109
3.36 Extracted 3D HBT radii (λ , R2out , R2side , R2long ) as a function of azimuthal pairangle relative to Ψ2 with two different event plane via FMD A+C and VZERO
A+C. Figures in top rows represents centrality 0-5%, while 5-10% are shown inbottom panels. FMD A+C combined results is plotted in black marker. VZERO
A+C combined results is plotted in blue marker.

. . . . . . . . . . . . . . . . 110

3.37 charged pion, kaon and proton(anti-proton) pT integrated v2 as a function ofcentrality with two different TPC number of clusters for track reconstruction.
pT is integrated from 0.15-1.5 GeV/cwhich is corresponding to same selectionfor HBT analysis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
3.38 charged pion, kaon and proton(anti-proton) pT integrated v2 as a function ofcentrality with two different tracking modes using TPC and ITS. pT is integrated from 0.15-1.5 GeV/cwhich is corresponding to same selection for HBT
analysis.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116

3.39 Systematic difference of centrality determination detectors V0 amplitude andnumber of clusters in second layer of SPD for charged pion, kaon and proton(antiproton) pT integrated v2 as a function of centrality. pT is integrated from 0.151.5 GeV/cwhich is corresponding to same selection for HBT analysis. . . . . . 117
3.40 Systematic difference of primary vertex position along the beam axis tight(|zvtx|<5cm)
and loose(|zvtx|<10cm) for charged pion, kaon and proton(anti-proton) pT integrated v2 as a function of centrality. pT is integrated from 0.15-1.5 GeV/cwhichis corresponding to same selection for HBT analysis. . . . . . . . . . . . . . . 117
3.41 Systematic difference of Event plane determination detector for charged pion,kaon and proton(anti-proton) pT integrated v2 as a function of centrality. pT
is integrated from 0.15-1.5 GeV/cwhich is corresponding to same selection for
HBT analysis.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118

xvi

3.42 Systematic difference of probability of Bayesian approaches 75% loose cut (redopen circle) and tight (blue open square) for charged pion, kaon and proton(antiproton) pT integrated v2 as a function of centrality. pT is integrated from 0.151.5 GeV/cwhich is corresponding to same selection for HBT analysis. . . . . . 118
4.1

Projection of 3D correlation function to 1D (outward, sideward, and longitudinal axis) of charged pions in 0.2 < kT < 1.5 GeV/c for two azimuthal bins(|φ pair − Ψ2 | < π /16 (Black marker) and 7π /16 < |φ pair − Ψ2 | < π /2 (Redmarker)) at centrality 0-50%. The projection range of other q components arewithin 50 GeV/c . Solid lines denote the fitting function of 3D correlation function. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122

4.2

Extracted HBT parameters (Rout , Rside , Rlong λ , Ros , Rol , and Rsl ) of charged
pions in 0.2 < kT < 1.5 GeV/c as a function of azimuthal pair angle with respect
to 2nd -order event plane for 6 different centrality bins. The data points at φ pair −Ψ2 = π are same value at φ pair − Ψ2 = 0. Systematic uncertainties are shownwith transparent bands. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124

4.3

Charged hadron and identified hadron (π , K and p) pT integrated v2 as a function of centrality. Both Ψ2 and q2 are determined via FMD A+C and pT are
integrated from 0.15 to 1.5GeV/c. Each 20% ESE q2 selection is applied toflow measurements(closed markers). Open black markers denote no ESE selected v2 . Systematic uncertainties are depicted as transparent bands.

4.4. . . . . 127

Ratio of v2 with each 20% q2 selection to without q2 selection (unbiased sample) for unidentified and identified charged hadrons (π , K and p). Both Ψ2
and q2 vector are determined via FMD A+C and pT are integrated from 0.15 to
1.5GeV/c. Systematic uncertainties are depicted as transparent bands. . . . . . 128

4.5

Measurement of charged pion v2 as a function of pT for 6 centrality classes witheach 20% q2 selection and no q2 selected samples. Systematic uncertainties areplotted as transparent bands. . . . . . . . . . . . . . . . . . . . . . . . . . . . 129

4.6

Measurement of charged kaon v2 as a function of pT for 6 centrality classes witheach 20% q2 selection and no q2 selected samples. Systematic uncertainties areplotted as transparent bands. . . . . . . . . . . . . . . . . . . . . . . . . . . . 130

xvii

4.7

Measurement of protons and anti-protons v2 as a function of pT for 6 centrality
classes with each 20% q2 selection and no q2 selected samples. Systematic
uncertainties are plotted as transparent bands. . . . . . . . . . . . . . . . . . . 131

4.8

Ratio of v2 with each 20% q2 selection to without q2 selection (unbiased sample) for charged pions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132

4.9

Ratio of v2 with each 20% q2 selection to without q2 selection (unbiased sample) for charged kaons. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133

4.10 Ratio of v2 with each 20% q2 selection to without q2 selection (unbiased sample) for protons.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134

4.11 Transverse momentum dependence of v2 ratio with each 20% q2 selection towithout q2 selection (unbiased sample) for unidentified and identified charged
hadrons (π , K and p) for 6 centrality bins. . . . . . . . . . . . . . . . . . . . . 135
4.12 Centrality dependence of v3 for unidentified and identified charged hadrons (π ,K and p) with and without q3 selection. Results for each 20% q3 selection
are shown and pT are integrated from 0.15 to 1.5. Systematic uncertainties areplotted as transparent bands. . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
4.13 Ratio of v3 with each 20% q3 selection to without q3 selection (unbiased sample) for unidentified and identified charged hadrons (π , K and p). Both Ψ3
and q3 vector are determined via FMD A+C and pT are integrated from 0.15 to
1.5GeV/c. Systematic uncertainties are depicted as transparent bands. . . . . . 138
4.14 Measurement of charged pion v3 as a function of pT for 6 centrality classes witheach 20% q3 selection and no q3 selected samples. Systematic uncertainties areplotted as transparent bands. . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
4.15 Measurement of charged kaon v3 as a function of pT for 6 centrality classes witheach 20% q3 selection and no q3 selected samples. Systematic uncertainties areplotted as transparent bands. . . . . . . . . . . . . . . . . . . . . . . . . . . . 140
4.16 Measurement of protons and anti-protons v3 as a function of pT for 6 centrality
classes with each 20% q3 selection and no q3 selected samples. Systematic
uncertainties are plotted as transparent bands. . . . . . . . . . . . . . . . . . . 141
4.17 Ratio of v3 with each 20% q3 selection to without q3 selection (unbiased sample) for charged pions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142
xviii

4.18 Ratio of v3 with each 20% q3 selection to without q3 selection (unbiased sample) for charged kaons. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143
4.19 Ratio of v3 with each 20% q3 selection to without q3 selection (unbiased sample) for protons.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144

4.20 Transverse momentum dependence of v3 ratio with each 20% q3 selection towithout q3 selection (unbiased sample) for unidentified and identified charged
hadrons (π , K and p) for 6 centrality bins. . . . . . . . . . . . . . . . . . . . . 145
4.21 Extracted HBT parameters (Rout , Rside , Rlong , λ , Ros , Rol , and Rsl ) of charged
pions in 0.2 < kT < 1.5 GeV/c as a function of azimuthal pair angle with respect
to 2nd -order event plane for 6 different centrality bins. Bottom 20% q2 selection
is applied to HBT measurements. The data points at φ pair − Ψ2 = π are samevalue at φ pair − Ψ2 = 0. Systematic uncertainties are plotted as transparent
bands. All points of Ros are shifted along the y-axis for visibility.

. . . . . . . 147

4.22 Extracted HBT parameters (Rout , Rside , Rlong , λ , Ros , Rol , and Rsl ) of charged
pions in 0.2 < kT < 1.5 GeV/c as a function of azimuthal pair angle with respect
to 2nd -order event plane for 6 different centrality bins. 20-40% q2 selection isapplied to HBT measurements. The data points at φ pair −Ψ2 = π are same value
at φ pair − Ψ2 = 0. Systematic uncertainties are plotted as transparent bands. All
points of Ros are shifted along the y-axis for visibility.

. . . . . . . . . . . . . 148

4.23 Extracted HBT parameters (Rout , Rside , Rlong , λ , Ros , Rol , and Rsl ) of charged
pions in 0.2 < kT < 1.5 GeV/c as a function of azimuthal pair angle with respect
to 2nd -order event plane for 6 different centrality bins. 40-60% q2 selection isapplied to HBT measurements. The data points at φ pair −Ψ2 = π are same value
at φ pair − Ψ2 = 0. Systematic uncertainties are plotted as transparent bands. All
points of Ros are shifted along the y-axis for visibility.

. . . . . . . . . . . . . 149

4.24 Extracted HBT parameters (Rout , Rside , Rlong , λ , Ros , Rol , and Rsl ) of charged
pions in 0.2 < kT < 1.5 GeV/c as a function of azimuthal pair angle with respect
to 2nd -order event plane for 6 different centrality bins. 60-80% q2 selection isapplied to HBT measurements. The data points at φ pair −Ψ2 = π are same value
at φ pair − Ψ2 = 0. Systematic uncertainties are plotted as transparent bands. All
points of Ros are shifted along the y-axis for visibility.
xix. . . . . . . . . . . . . 150

4.25 Extracted HBT parameters (Rout , Rside , Rlong , λ , Ros , Rol , and Rsl ) of charged
pions in 0.2 < kT < 1.5 GeV/c as a function of azimuthal pair angle with respect
to 2nd -order event plane for 6 different centrality bins. Top 20% q2 selection isapplied to HBT measurements. The data points at φ pair −Ψ2 = π are same value
at φ pair − Ψ2 = 0. Systematic uncertainties are plotted as transparent bands. All
points of Ros are shifted along the y-axis for visibility.

. . . . . . . . . . . . . 151

4.26 Projection of 3D correlation function to 1D (outward, sideward, and longitudinal directions) of charged pions in 0.2 < kT < 1.5 GeV/c for two different
azimuthal angle classes, |φ pair − Ψ3 | < π /24 (Black marker) and 7π /24 <|φ pair − Ψ3 | < π /3 (Red marker) in centrality 0-50%. The projection rangeof the other q components are within 50 GeV/c . Solid line denotes the fitting
function of 3D correlation function. . . . . . . . . . . . . . . . . . . . . . . . 153
4.27 Extracted HBT parameters (Rout , Rside , Rlong , λ , Ros , Rol , and Rsl ) of charged
pions in 0.2 < kT < 1.5 GeV/cas a function of azimuthal pair angle with respect
to 3rd -order event plane for 6 different centrality bins. The data points at φ pair −Ψ3 = 2π /3 is same value to those at φ pair − Ψ3 = 0. Systematic uncertainties
are shown as transparent bands. . . . . . . . . . . . . . . . . . . . . . . . . . . 156
4.28 Extracted HBT parameters (Rout , Rside , Rlong , λ , Ros , Rol , and Rsl ) of charged
pions in 0.2 < kT < 1.5 GeV/c as a function of azimuthal pair angle with respect
to 3rd -order event plane for 6 different centrality bins. Bottom 20 % q3 selection
is applied to HBT measurements. The data points at φ pair − Ψ3 = 2π /3 issame value to those at φ pair − Ψ3 = 0. Systematic uncertainties are plotted astransparent bands. All points of Ros are shifted along the y-axis for visibility. . 157
4.29 Extracted HBT parameters (Rout , Rside , Rlong , λ , Ros , Rol , and Rsl ) of charged
pions in 0.2 < kT < 1.5 GeV/c as a function of azimuthal pair angle with respect
to 3rd -order event plane for 6 different centrality bins. 20-40% q3 selection
is applied to HBT measurements. The data points at φ pair − Ψ3 = 2π /3 issame value to those at φ pair − Ψ3 = 0. Systematic uncertainties are plotted astransparent bands. All points of Ros are shifted along the y-axis for visibility. . 158

xx

4.30 Extracted HBT parameters (Rout , Rside , Rlong , λ , Ros , Rol , and Rsl ) of charged
pions in 0.2 < kT < 1.5 GeV/c as a function of azimuthal pair angle with respect
to 3rd -order event plane for 6 different centrality bins. 40-60% q3 selection
is applied to HBT measurements. The data points at φ pair − Ψ3 = 2π /3 issame value to those at φ pair − Ψ3 = 0. Systematic uncertainties are plotted astransparent bands. All points of Ros are shifted along the y-axis for visibility. . 159
4.31 Extracted HBT parameters (Rout , Rside , Rlong , λ , Ros , Rol , and Rsl ) of charged
pions in 0.2 < kT < 1.5 GeV/c as a function of azimuthal pair angle with respect
to 3rd -order event plane for 6 different centrality bins. 60-80% q3 selection
is applied to HBT measurements. The data points at φ pair − Ψ3 = 2π /3 issame value to those at φ pair − Ψ3 = 0. Systematic uncertainties are plotted astransparent bands. All points of Ros are shifted along the y-axis for visibility. . 160
4.32 Extracted HBT parameters (Rout , Rside , Rlong , λ , Ros , Rol , and Rsl ) of charged
pions in 0.2 < kT < 1.5 GeV/c as a function of azimuthal pair angle with respect
to 3rd -order event plane for 6 different centrality bins. Top 20% q3 selection
is applied to HBT measurements. The data points at φ pair − Ψ3 = 2π /3 issame value to those at φ pair − Ψ3 = 0. Systematic uncertainties are plotted astransparent bands. All points of Ros are shifted along the y-axis for visibility. . 161
4.33 3D HBT radii (Rout ) of charged pions as a function of pair transverse momentum kT for 6 centrality bins. My results are plotted as closed circles which is
obtained with azimuthal dependence of HBT radii with respect to Ψ3 . Opensquared markers are results of azimuthal differential pion HBT with respect toΨ2 [60]. Open circle markers are results of azimuthal integrated pion HBTanalysis [61]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163
4.34 3D HBT radii (Rside ) of charged pions as a function of pair transverse momentum kT for 6 centrality bins. My results are plotted as closed circles which is
obtained with azimuthal dependence of HBT radii with respect to Ψ3 . Opensquared markers are results of azimuthal differential pion HBT with respect toΨ2 [60]. Open circle markers are results of azimuthal integrated pion HBTanalysis [61]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164

xxi

4.35 3D HBT radii (Rside ) of charged pions as a function of pair transverse momentum kT for 6 centrality bins. My results are plotted as closed circles which is
obtained with azimuthal dependence of HBT radii with respect to Ψ3 . Opensquared markers are results of azimuthal differential pion HBT with respect toΨ2 [60]. Open circle markers are results of azimuthal integrated pion HBTanalysis [61]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165
4.36 Identified hadron (π , K and p) v2 as a function of pT for 6 centrality classes.
Ψ2 is determined via FMD A+C. My results are plotted as closed markers andsystematic uncertainties of my calculation are depicted as transparent bands.
Published results are plotted as opened markers [62]. . . . . . . . . . . . . . . 166
4.37 Identified hadron (π , K and p) v3 as a function of pT for 6 centrality classes.
Ψ2 is determined via FMD A+C. My results are plotted as closed markers andsystematic uncertainties of my calculation are depicted as transparent bands.
Published results are plotted as opened markers [62]. . . . . . . . . . . . . . . 167
5.1

Relative amplitudes of squared HBT radii (Rout , Rside , Rlong , Ros , Rol and Rsl )for charged pion pairs with respect to Ψ2 as a function of centrality measured in√
Pb-Pb collisions at sNN = 2.76 TeV. Pair transverse momentum kT is integrated
from 0.2 to 1.5 GeV/c. Dashed line indicates relative amplitude of HBT radii =0. Transparent red boxes represent the systematic uncertainties. . . . . . . . . . 170

5.2

Relation between initial source eccentricity(Glauber Model Calculation[13])and final source eccentricity extracted with azimuthal angle dependence of charged
pion HBT radii (2R2side,2 /R2side,0 ) in Pb-Pb 2.76 TeV. kT is integrated from 0.2
to 1.5 GeV/c. Dashed line is 2R2side,2 /R2side,0 =0 and dotted line denotes ε initial =ε f inal . Boxes represent the systematic uncertainties and statistical uncertainties
are smaller than marker size . . . . . . . . . . . . . . . . . . . . . . . . . . . 171
5.3

Relative amplitudes of squared HBT radii (Rout , Rside , and Ros ) for charged
pion pairs with respect to Ψ2 as a function of pair transverse momentum kT for√
3 centrality bins measured in Pb-Pb collisions at sNN = 2.76 TeV. Dashed line
indicates relative amplitude of HBT radii = 0. Transparent boxes represent thesystematic uncertainties. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172

xxii

5.4

Each 20% q2 selection is applied to relative amplitude of squared HBT radii(Rout , Rside , Rlong , Ros , Rol and Rsl ) for charged pion pairs with respect to Ψ2 asa function of centrality measured in Pb-Pb 2.76TeV collisions. Pair transverse
momentum kT is integrated from 0.2 to 1.5 GeV/c. Dashed line indicates relative amplitude of HBT radii = 0. Transparent boxes represent the systematic
uncertainties. All points are shifted along the x-axis for visibility.

5.5. . . . . . . 174

Each 20% q2 selection is applied to relative amplitudes of squared HBT radii(Rout , Rside , and Ros ) of charged pion pairs with respect to Ψ2 for kT 0.2-0.3
GeV/cas a function of centrality measured in Pb-Pb 2.76TeV collisions. Dashedline indicates relative amplitude of HBT radii = 0. Transparent boxes represent
the systematic uncertainties. All points are shifted along the x-axis for visibility. 175

5.6

Each 20% q2 selection is applied to relative amplitudes of squared HBT radii(Rout , Rside , and Ros ) of charged pion pairs with respect to Ψ2 for kT 0.3-0.4
GeV/cas a function of centrality measured in Pb-Pb 2.76TeV collisions. Dashedline indicates relative amplitude of HBT radii = 0. Transparent boxes represent
the systematic uncertainties. All points are shifted along the x-axis for visibility. 176

5.7

Each 20% q2 selection is applied to relative amplitudes of squared HBT radii(Rout ,Rside , and Ros ) of charged pion pairs with respect to Ψ2 for kT 0.4-1.5 GeV/casa function of centrality measured in Pb-Pb 2.76TeV collisions. Dashed line indicates relative amplitude of HBT radii = 0. Transparent boxes represent thesystematic uncertainties. All points are shifted along the x-axis for visibility.

5.8

. 177

Each 20% q2 selection, where 0-20% (80-100%) corresponds to the smallest(largest) q2 bin, is applied to relative amplitude of squared HBT radii(Rout , Rside ,Rlong , Ros , Rol and Rsl ) for charged pion pairs with respect to Ψ2 as a functionof charged pion v2 measured in Pb-Pb 2.76TeV collisions. Pair transverse momentum kT is integrated from 0.2 to 1.5 GeV/c. Dashed line indicates relative
amplitude of HBT radii = 0. Transparent boxes represent the systematic uncertainties. Relative amplitude of HBT radii without q2 selection also dipicted as
open black circle. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180

xxiii

5.9

Blast-wave fitting to identified particle(π , K and p) pT spectra as a functionof centrality[9]. Difference of panels denote centrality. Error bar indicates
quadratic sum of static and systematic uncertainties. Black lines shows theactual fit range and red, green and blue lines are extrapolated line of fitting
functions for π , K and p, respectively. . . . . . . . . . . . . . . . . . . . . . . 183

5.10 Blast-wave fitting to identified particle(π , K and p) v2 with 40-60% q2 selection
as a function of pT . Difference of panels denote centrality. Error bar indicates
quadratic sum of static and systematic uncertainties. Black lines shows theactual fit range and red, green and blue lines are extrapolated line of fitting
functions for π , K and p, respectively. . . . . . . . . . . . . . . . . . . . . . . 184
5.11 Blast-wave fitting to azimuthal angle dependence of Rout , Rside , Rlong and Ros
with 40-60% q2 selection as a function of centrality. Systematic uncertainties
are shown as transparent bands and statistical uncertainties are smaller thanmarker size. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186
5.12 Extracted freeze out parameters with Blast Wave fitting to spectra, v2 and HBT
as a function of average number of participant calculated with Glauber model
without q2 selection. Tf and ρ0 are extracted by fitting π , K and p spectra, andthe other parameters are obtained with simultaneous fitting π , K and p v2 andcharged pion HBT radii (Rout , Rside , Rlong , and Ros ) with respect to Ψ2 . . . . 187
5.13 Extracted freeze out parameters with Blast Wave fitting to spectra, v2 and HBT
as a function of average number of participant calculated with Glauber modelwith each 20% q2 selection. No q2 selected results are simultaneously dipicted
as open black circle. Tf and ρ0 are extracted by fitting π , K and p spectra, andthe other parameters are obtained with simultaneous fitting π , K and p v2 andcharged pion HBT radii (Rout , Rside , Rlong , and Ros ) with respect to Ψ2 . . . . 188
5.14 Relative amplitude of squared HBT radii(Rout , Rside , Rlong , Ros , Rol and Rsl )for charged pion pairs with respect to Ψ3 as a function of centrality measured
in Pb-Pb 2.76TeV collisions. Pair transverse momentum kT is integrated from
0.2 to 1.5 GeV/c. Dashed line indicates relative amplitude of HBT radii = 0.
Transparent blue boxes represent the systematic uncertainties. . . . . . . . . . 190

xxiv

5.15 Relative amplitudes of squared HBT radii (Rout , Rside , and Ros ) for charged
pion pairs with respect to Ψ3 as a function of pair transverse momentum kT for√
3 centrality bins measured in Pb-Pb collisions at sNN = 2.76 TeV. Dashed line
indicates relative amplitude of HBT radii = 0. Transparent boxes represent thesystematic uncertainties. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191
5.16 Each 20% q3 selection is applied to relative amplitude of squared HBT radii(Rout ,Rside , Rlong , Ros , Rol , and Rsl ) for charged pion pairs with respect to Ψ3 as afunction of centrality measured in Pb-Pb 2.76TeV collisions. Pair transverse
momentum kT is integrated from 0.2 to 1.5 GeV/c. Dashed line indicates relative amplitude of HBT radii = 0. Transparent boxes represent the systematic
uncertainties. All points are shifted along the x-axis for visibility.

. . . . . . . 194

5.17 Each 20% q2 selection is applied to relative amplitudes of squared HBT radii(Rout , Rside , and Ros ) of charged pion pairs with respect to Ψ3 for kT 0.2-0.3
GeV/cas a function of centrality measured in Pb-Pb 2.76TeV collisions. Dashedline indicates relative amplitude of HBT radii = 0. Transparent boxes represent
the systematic uncertainties. All points are shifted along the x-axis for visibility. 195
5.18 Each 20% q3 selection is applied to relative amplitudes of squared HBT radii(Rout , Rside , and Ros ) of charged pion pairs with respect to Ψ3 for kT 0.3-0.4
GeV/cas a function of centrality measured in Pb-Pb 2.76TeV collisions. Dashedline indicates relative amplitude of HBT radii = 0. Transparent boxes represent
the systematic uncertainties. All points are shifted along the x-axis for visibility. 196
5.19 Each 20% q3 selection is applied to relative amplitudes of squared HBT radii(Rout ,Rside , and Ros ) of charged pion pairs with respect to Ψ3 for kT 0.4-1.5 GeV/casa function of centrality measured in Pb-Pb 2.76TeV collisions. Dashed line indicates relative amplitude of HBT radii = 0. Transparent boxes represent thesystematic uncertainties. All points are shifted along the x-axis for visibility.

xxv

. 197

5.20 Each 20% q3 selection, where 0-20% (80-100%) corresponds to the smallest(largest) q3 bin, is applied to relative amplitude of squared HBT radii(Rout , Rside ,Rlong , Ros , Rol , and Rsl ) for charged pion pairs with respect to Ψ3 as a functionof charged pion v3 measured in Pb-Pb 2.76TeV collisions. Pair transverse momentum kT is integrated from 0.2 to 1.5 GeV/c. Dashed line indicates relative
amplitude of HBT radii = 0. Transparent boxes represent the systematic uncertainties. Relative amplitude of HBT radii without q3 selection also dipicted as
open black circle. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199

xxvi

List of Tables
1.1

Summary of relativistic heavy ion collisions . . . . . . . . . . . . . . . . . . .

7

2.1

Parameters for CERN’s accelerator complex[35] . . . . . . . . . . . . . . . . .

38

2.2

Beam parameters for LHC[37] . . . . . . . . . . . . . . . . . . . . . . . . . .

38

2.3

Performance of ITS (position and two track resolution for azimuthal and beam
directions). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2.4

48

Table indicates nominal distance in z from the IP to the detector plane, pseudorapidity coverage of each detector modules, number of azimuthal sectors distributed in each ring and number of read out detector strip . . . . . . . . . . . .

57

3.1

Pion selection for HBT analysis with TPC and TOF nσ . . . . . . . . . . . . .

85

3.2

Event classification of event mixing . . . . . . . . . . . . . . . . . . . . . . .

93

3.3

Input source size of Coulomb interaction . . . . . . . . . . . . . . . . . . . . .

98

3.4

Systematic table for λ w.r.t. Ψ2 unbiased(No q2 selection) . . . . . . . . . . . 111

3.5

Systematic table for Rout w.r.t. Ψ2 unbiased(No q2 selection) . . . . . . . . . . 112

3.6

Systematic table for Rside w.r.t. Ψ2 unbiased(No q2 selection) . . . . . . . . . . 113

3.7

Systematic table for Rlong w.r.t. Ψ2 unbiased(No q2 selection) . . . . . . . . . . 114

3.8

Systematic table for the identified charged hadron v2 as a function of centrality

5.1

Fit ranges for identified hadron spectra, v2 , and HBT radii . . . . . . . . . . . . 182

xxvii

119

Acknowledgement
This thesis would not have been possible if not for Prof. Yasuo Miake. He gave me an opportunity to have great experiences in the field of heavy ion collisions. His extensive knowledgesof heavy ion physics, detector, and electronics always stimulate my interest to this field. I am
deeply grateful to Prof. ShinIchi Esumi for daily discussion and a great deal of ideas which
not only enlightened my analysis, but also gave me an encouragement to never give up. I havelearned a lot of things from him, knowledge of heavy ion physics, experimental techniques, andhow interesting physics on heavy ion collisions is. I would like to thank Prof. Tatsuya Chujo forgiving me comments and many opportunities to talk at workshops. I would like to thank Prof.
Motoi Inaba. His knowledge and experience of hardwares always help me to solve some problems of detectors and electronics. I would also like to thank Prof. Oliver Busch, Prof. Shingo
Sakai, Prof. Yosuke Watanabe, Prof. Kyoichiro Ozawa, and Prof. Hiroyuki Sako for giving me
a lot of comments and suggestions of this analysis. I would like to express my gratitude to Prof.
Kazuhiro Yabana for his comments and advices for my thesis. Comfort and useful computing
sytems in our group would never been possible without Mr. Sumio Kato.
This work was also supported by Takafumi Niida. I am deeply grateful to him for his carefulreading of my thesis and always giving me new insights, useful commets, and advices for myanalysis. I have learned knowledge and tecnique of HBT from him. I also thank Dr. Dosatsu
Sakata and Dr. Juhyun Bohm for their support.
I would like to thank Dr. Sanshiro Mizuno, Dr. Hiroshi Nakagomi, Dr. Daisuke Watanabe,
Ms. Tomo Nakajima, Ms Kaoru Gunji, and Mr. Satoshi Horiuchi for encouragement andinteresting conversations.
I also thank all the members of High Energy Nuclear Physics Group at University of Tsukuba,
Mr. Hiroki Yokoyama, Dr. Toshihiro Nonaka, Mr. Ritsuya Hosokawa, Mr. Ryo Aoyama, Mr.
Tetsuro Sugiura, Mr. Lee Joonil, Mr. Byungchul Kim, Mr. Kazuki Sato, Mr. Yoshimi Rebaza, Mr. Taichi Ichisawa, Mr. Hiroki Kato, Mr. Yota Kawamura, Mr. Daichi Kawana, Mr.
Kazuya Nakagawa, Mr. Toma Suzuki, Ms. Mina Hatakeyama, Ms. Yukiko Hoshi, Ms. Kana
Nakagawa, Mr. Hiroshi Saito, Mr. KenIchi Tadokoro, Mr. Masahiro Takamura.
Finally, I would like to thank my family Hideo, Kimiyo, and Ayano. This thesis would havenever been completed without their countinuos supports and understanding.

xxviiiChapter 1
Introduction
A quark is a fundamental particle of matter and normally confined in composite particles called
hadrons such as protons and neutrons which are the constituents of all kinds of atoms andmolecules. On the other hand, for a few millionth seconds after the Big Bang, universe was inextremely high temperature and quarks and gluons were almost not bound together and freely
moving at the limitted space. It is just like "plasma" state in quark and gluon matter so-called
Quark Gluon Plasma(QGP). That is to say, precise understanding of the properties of Quark
Gluon Plasma is a key to access how our universe evolves.
In this chapter, we introduce Quantum Chromodynamics, where the QGP is predicted, andrelativistic heavy ion collisions.

1.1 Quantum Chromodynamics and Quark Gluon Plasma
Strong interaction between quarks and gluons are described in Quantum Chromodynamics(QCD),
where gluons mediate strong interaction of quarks. It is analogous to Quantum Electrodynamics(QED) which describes the electro-magnetic interaction between two charged particles. In
QED, photons mediate the electro-magnetic interaction. Unlike photons, gluons themselves
have color charge and participate in the strong interaction.
For a quark with invariant mass m f , the classical Lagrangian density can be expressed by:() 1 a aL = ∑ q̄ f iγ µ Dµ − m f − Fµν
Fµν ,
4
f
Nf

(1.1)

where q f denotes quark field in three different flavor f (= 1, 2, 3), γ µ is the Dirac matrix, Dµ
is the co-variant derivative of QCD and gluon field strength tensor is presented as Fµν . Dµ andFµν are given by:
1

λa a
A ,
2 µ
= ∂µ Aaν − ∂ν Aaµ − g fabc Abµ Acν ,Dµ = ∂µ + ig

(1.2)

a
Fµν

(1.3)

where Aaµν is gluon field in eight different flavour a (= 1, ..., 8), λa is eight Gell-Mann matrices,g denotes the dimensionless coupling constant in QCD, which indicates the interaction strengthof quark to quark and quark to gluon. Coupling constant g can be defined with fine structure
constant αs by:g≡√
4παs .

(1.4)

Using perturbative QCD (pQCD) theory, αs can be expressed as a function of momentum transfer Q given by:( )α Q2 =

1

β0 ln (Q2 /ΛQCD )

,

(1.5)

where ΛQCD is called QCD scale parameter, which denotes the coupling strength in QCD, β0
represents the first term of β -function. Equation 1.5 indicates pQCD suggests the coupling
constant αs is not "constant" any more but the function of Q. Figure 1.1 shows the running
coupling constant αs as a function of momentum transfer Q, which is measured from various
reaction such as τ decay, deep inelastic scattering [1], ϒ decay, e+ e− annihilation, Z decay, andpT dependence of jet cross section in p p̄. pQCD calculation shows excellent agreement to experimental results. One can find the important characteristic of strong interaction in QCD from
Fig. 1.1. Coupling constant αs becomes rapidly weaker from small Q to large Q. It indicateswhen the momentum scale increases, interaction of quark-gluon and quark-quark significantly
decreases, called "asymptotic freedom". Therefore, at the limit as Q approaches infinity, quarks
and gluons can be free to move as if they were "free particle".
At small momentum scale Q, however, pQCD calculation is no longer able to be relied on,
as perturbative approximation does not converge at small Q (< 1 GeV/c) due to large αs . A
Lattice QCD is an ideal tool of non-perturbative QCD calculation to describe the interaction ofquarks and gluons at small momentum scale. Figure 1.2 shows the static quark potential as afunction of the distance of quark to anti-quark in 2+1 flavor Lattice QCD calculation [2]. Resultof quark potential V (r) in Lattice QCD is fitted with the following equation :V (r) = V0 −2α+ σ r,
r(1.6)

where V0 , α , and σ are free parameter for fitting and the simulated results are well reproducedwith the fitting function. Equation 1.6 indicates strong interaction can be described as 1/r such
like Coulomb potential at short range and are proportional to the distance r at long range. At
the limit as the distance r approaches to infinity, quark potential V (r) becomes infinity, whichindicates quarks are confined in the hadrons and can not be isolated from them. This is called"color confinement" and also important characteristics of strong interaction, as is the case withthe asymptotic freedom.
Phase transition is frequently used to describe transitions between solid, liquid, and gaseousstates of matter. It is well known that first-order phase transition of water (liquid-vapor andliquid-solid) occurs when its temperature and pressure exceed the transition line. Based on
the asymptotic freedom, phase transition in QCD is predicted. In extemely high temperatureor high baryon density, quarks and gluons are de-confined from the hadrons into a deconfinedmatter, so-called Quark Gluon Plasma. Lattice QCD suggests one of the numerical calculationsof phase transition in QCD matter, though Lattice QCD can be applicable at the limit as baryon
density approaches to zero.
Figure 1.3 shows energy density and 3 times pressure normalized by temperature to thefourth calculated on Lattice QCD [3]. Both ε /T 4 and 3p/T 4 rapidly rise at the vertical band
dipicted in Fig. 1.3 (185 MeV < T < 195 MeV). ε /T 4 indicates the degrees of freedom ofquarks and gluons. This significant rise of degrees of freedom indicates the phase transition of
hadrons-Quark Gluon Plasma. Horizontal line on εSB denotes the Stefan Boltzmann limit where
the interaction of quarks and gluons are ignored. Excess from the Lattice QCD results(red andblue) comes from the interaction of quarks and gluons.

39. Quantum chromodynamics

131

M S suitable(9.24a)(9.24b)(9.24c)f the newnd classes ofwhen leavingnd(9.25a)(9.25b)(9.25c)(9.25d)ec. fit).(9.25e)Figure 1.1:
Q,
The running
coupling
constant α as a function
Figure9.4: strong
Summaryof measurementsof α asof amomentum
function transferofsswhich is thedeterminedfrom τenergy
decay, deep
inelastic
scattering
process,degreeradiative
decay, hadronic
respective
scale
Q. TherespectiveofϒQCDrage quoted final states of e+ e− annihilation, hadronic Z decay width and inclusive jet cross section in p p̄
perturbation theory used in the extraction of α is indicated ins, which has [1]. Data is compared with pQCD calculation shown in black line. s
brackets (NLO: next-to-leading order; NNLO: next-to-next-tove average ofof the (small)

he various
esult froms. Expressing
d the world
urement, theng to 3.5 andand [278],
etween somet a specificexplanations
o definitely

1 FLAVORTOWARD
THE PHYSICALPOINT
PHYSICAL
leading2 þorder;
res. LATTICE
NNLO: QCD
NNLO
matchedwith resummed
3
next-to-leading
logs; N LO: next-to-NNLO).
2.0
TABLE XIII. PACS-CS andlattice units at ð#ud ; #s Þ ¼ ð0:1
are the same as in Table XI.

V(r)C. Quigg, Am. J. Phys. 78, 1081 (2010).
6. A.S. Kronfeld andlattice[arXiv:1002.5032[hep-ph]].
1.5
PACS-CS323 (7. R. Stock (Ed.), Relativistic Heavy Ion Physics, Springer-Verlag
CP-PACS/JLQCD
203 (Berlin, Heidelberg, 2010.
8. Proceedings of the Quark Matter 2011 Conference, to appear J.
1.0 G.
Phys.
0.22
9. T. van Ritbergen, J.A.M. Vermaseren, and S.A. Larin, Phys.
Lett. B400, 379 (1997) [arXiv:hep-ph/9701390].
1/r0
0.21
10. M. 0.5
Czakon, Nucl. Phys. B710, 485 (2005) [arXiv:hepph/0411261].
0
10
5
15
0.20
11. Y. Schroder and M. Steinhauser,
JHEP
0601,
051 (2006)r[arXiv:hep-ph/0512058].
and wellFIG.quark
26 (color
online).
Static quark
potentialbetweenVðrÞ at #Figure1.2:
Staticpotential
as aKuhn,functionof distancequark
in
ud ¼ and anti-quark
0.19
on of current
12. K.G.
Chetyrkin,
J.H.
and
C.
Sturm,
Nucl.
Phys. B744,
0:13770
as a representativecase. Solidline (black
denote marker)
the fit result
resultis fitted with Eq. 1.6.
2+1 flavor Lattice
QCD calculation[2]. Simulation
121with(2006)[arXiv:hep-ph/0512060].
Eq. (79).
0.18
13. A.G. Grozin et al., arXiv:1107.5970 [hep-ph].
sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
14. K.G. Chetyrkin, B.A. Kniehl,
1:65" ! M. Steinhauser, Nucl. Phys.
4 and
:
(81)
r
¼
0
B510, 61 (1998) [arXiv:hep-ph/9708255].
0.17"
the energy
0
0.01
15. See for example section 11.4 of M.E. Peskin and D.V. Schroeder,
ediction of
In Introduction
Table XII we list
results for Field
r0 including
the system“AnTotheQuantum
Theory,”Reading,USA:
where results

out in the ratio

due to the exponential Boltzmann suppression of the states.
One could also use the hadron resonance gas value for thepressure at T0 ¼ 100 MeV as the starting point for the
integration. The two HRG model calculations in Fig. 3
show that at this temperature the pressure is insensitive tothe exact value of the cutoff mmax . We have used both
approaches as well as linear interpolations between thetemperatures at which we calculated !"" =T 4 to estimate
systematic errors arising in the calculation of the pressure.
The actual results for p=T 4 and other thermodynamic
observables shown in the following have been obtained

light quark condennge quark contribuproportional to thetively. However, thems , only holds trueures larger than 1.5
he transition region,
quark sector reaches
bution.

1

1.2

1.4

1.6
εSB/T

Tr0

0.4

4

0.6

0.8

1.2
εSB/T 4

Tr0

16
14

1

ε/T 4

12
10
4

8

ε/T : Nτ=46
83p/T 4: Nτ=4
6
8

6
4
2

T [MeV]

400

500

600

p4
asqtadp4
asqtad3p/T 4700

T [MeV]0
100 150 200 250 300 350 400 450 500 550
41.3: Energy density and 3 times the pressure normalized by T calculated on Latticed 3 times theFigurepressurecalculated on lattices with temporal extent N! ¼limits
4, 6where[4],theand 8 using
QCD as a function of temperature [3]. εSB /T 4 denotes the Stefan-Boltzmann
8 lattices.
withompares results
obtainedtheareasqtad
actions
the Nthe! ¼
interactionof two withparticles
ignored.and
Thep4
verticalband onindicatestransition
region Crosses
185
T < 195from
MeV. different
The black bars
at high temperaturesindicate
the systematic
dataThe black
he pressureMeV
that<arises
integration
schemes
as discussed
in shiftthe oftext.
that ofwould
arisethatfrom would
matchingarise
to a hadronresonance
gas at to
T = a100
MeV. resonance gas at T ¼
stematic shift
data
from
matching
hadronn region 185 MeV < T < 195 MeV. It should be emphasized that these data have not been

014504-8

5

1.2 Relativistic Heavy Ion Collisions
In order to study the properties of Quark Gluon Plasma, sufficient energy density and temperature to bring the QCD phase transition is necessary. Experimentally, relativistic heavy ion
collision is the unique tool to recreate QGP on the earth. Two massive ions such as gold andleads are relativistically accelerated at nearly speed of light with the accelerator complex andtwo ions have a head-on collisions at the energy of upwards of a few TeV. In order to recreate
QGP, powerful heavy ion accelerators are necessary.
Currently, high energy heavy ion experiments are performed with accelerator complex at
European Organization for Nuclear Research (CERN) and Brookhaven National Laboratory(BNL). Large Hadron Collider (LHC) and Relativistic Heavy Ion Collider(RHIC) are built in
CERN and BNL, respectively. At LHC, in particular studies on energy loss in QCD matter√and detailed bulk properties are ongoing through world top energy sNN = 5.02 TeV Pb-Pb
collisions. On the other hand, study of phase diagram in QCD and to find the critical end point
are performed with beam energy scan at RHIC.
Table 1.1 shows the summary of relativistic heavy ion collisions experiments. Data used√in this analysis is sNN = 2.76 TeV Pb-Pb collision events obtained by A Large Ion Collider
Experiment (ALICE) at LHC.
In order to recreate Quark Gluon Plasma, filling the large area with high baryon density stateis important. therefor massive ions are used to study Quark Gluon Plasma. Pb and Au nucleus
are often used and their radius are approximately 7 fm. Recently, it is expected that QGP might
can be regenerated even in small system collisions, such as p-Pb, d-Au and related analysis isactively performed.

6

Table 1.1: Summary of relativistic heavy ion collisions√
Year Accelerators Location
SpeciessNN Energy(GeV)
16 O, 28 Si
1986
AGS
BNL
5.4
197 Au
1992
4.8
16
32
1986
SPS
CERN
O, S
19.4
208
1994
Pb
17.4
197
2000
RHIC
BNL
Au
130
197 Au
2001
200
197
2003
d- Au
200
197
2004
Au
200, 62.4
63 Cu
2005
200, 62.4, 22.4
200 Au
2007
200
197
2008
d- Au
200, 62.4
197 Au2010
200, 62.4, 39, 11.5, 7.7
197 Au2011
200, 19.6, 27
238
2012
U
193
63 Cu-197 Au
2012
200
197
2014
Au
200, 14.6
3
197
2014
He- Au200
2015
p-197 Au
200
197
2015
p- Al
200
197
2016
Au
200
2016
d-197 Au
200, 62.4, 19.6, 39
197 Au
2017
54
208
2010
LHC
CERN
Pb
2760
208
2011
Pb
2760
2013
p-208 Pb
5020
208
2015
Pb
5020
208
2016
p- Pb
5020, 8160
129 Xe
2017
5440

7

1.2.1 Participant Spectator Picture
Figure 1.4 shows the schematic illustration of colliding nuclei before and after a collision. Since
two colliding nuclei are accelerated at the nearly speed of light, The longitudinal size shrinks
due to Lorentz contracted shown in Fig. 1.4(left). Longitudinal size is expressed as 2R/γ , where
R is a radius of each nucleus and γ denotes Lorentz factor.
The degree of overlap area is defined as impact parameter b, which is the distance between
the centers of the two colliding nuclei in a transverse plane to the beam axis.
Since the time scale in relativistic heavy ion collisions is much shorter than Fermi motion of
nucleons in two colliding nuclei, "Participant Spectator picture" can be applicable to describe
the collision geometry. After the collision, nucleons in two colliding nuclei are categorized
into two types, participant and spectator. Participant is the nucleons which participate to thecollision shown as red, blue, and green particles in Fig. 1.4(right). Spectator is the nucleons
which does not participate to the collision shown as white particles in Fig. 1.4(right).
In order to understand the property of QGP, initial collision geometry, such as impact parameter b, number of participant N part and participant shape is essential. However these value
cannot be directly measured in experiment. Instead of measuring impact parameter b, "centrality" is estimated by measured multiplicity. The other valuables are determined with Glaubermodel simulation [5].
Based on the Wood-Saxon potential, density distribution of nucleons are expressed by:ρA (r) =ρ0,
1 + exp ([r − RA ] /a)(1.7)

where r denotes radius of each nucleon, RA is nucleus radius, a is diffusion parameter in nucleus
surface. ρ0 represents the normalization factor where∫ 3 A
d rρ (r) = A. Density function of

nucleon is given with the integration of Eq 1.7 in the direction of z axis, which is given by:∫ ∞TA (x, y) =∞dzρA (x, y, z) .

Number of participant in a certain x, y coordinate is represented as n part and given by:

8

(1.8)

Figure 1.4: Schematic illustration of colliding nuclei before and after the collision [4]. Impact
parameter is defined as the distance between the centers of the two colliding nuclei in a transverse plane to the beam axis. In relativistic heavy ion collision colliding nuclei is divided into
participant ,which participate to collisions, and Spectator that does not participate to collisions.

[(n part = TA (x + b/2, y) 1 − 1 −[σ0TB (x − b/2, y)(+ TB (x − b/2, y) 1 − 1 −

)B ]

σ0
TA (x + b/2, y)

)A ]

(1.9)
,

where the centers of two colliding nuclei are shifted by impact parameter b, A and B are mass of
two nuclei, σ0 denotes the cross section of an inelastic p-p collisions at corresponding collisionenergy. Then the number of participant N part is expressed as integration of Eq 1.9 :∫N part (b) =dxdyn part (x, y; b) ,9

(1.10)

1.2.2 Space Time Evolution
After the collision, if the energy and baryon density are sufficient, phase transition to Quark
Gluon Plasma occurs in overlap region of two colliding nuclei. System cools down as it expands
significantly and eventually phase transition to hadron state takes place. Figure 1.5 shows theschematic diagram of space-time evolution in relativistic heavy ion collisions. Two nuclei areapproaching from z > 0 (< 0) and t < 0. Two nuclei collide at z = 0 and t = 0.
In Fig. 1.5, space time evolution of heavy ion collisions is classified into 4 stages.
• Parton Cascade (Pre-equilibrium)• QGP state• Chemical freeze-out• Kinetic freeze-out
Parton Cascade (Pre-equilibrium)
After the collision, huge number of partons are created by hard scattering and large energy
density, which is deposited by the nucleus collision, in initial overlap region in two colliding
nuclei. Some of the models are predicted to describe this state, such as color string model[6] and Color Glass Condensate (CGC) [7]. But the real mechanism of pre-equilibrium is still
open question in heavy ion collisions. As parton production and parton scattering occurs one
after another, energy density and entropy increases. Partonic matter reaches the local thermal
equilibrium at proper time τ0 .
QGP stateOnce phase transition occurs and local thermal equilibrium is established in a QCD matter,system drastically expands due to the pressure gradient. Space time evolution of Quark Gluon
Plasma can be described by relativistic hydrodynamics based on the conservation laws of energymomentum tensor and baryon number, which is given by:∂µ T µν = 0,

(1.11)∂ j µ = 0,(1.12)

10

6

CHAPTER 1. INTRODUCTION

Figure 1.5: Schematic diagram of space-time evolution in relativistic heavy ion collisions. Twocolliding nuclei is approaching from t < 0 shown as "beam". Collision time is defined as t = 0,z = 0. QGP phase is shown orange-yellow gradation and after the hadronization hadron phaseis shown in blue gradation.

Figure 1.4: A schematic idea of the space-time evolution of relativistic heavy-ion collisions [4].
For number of binary collisions, the density function ncoll is given by the product of
TB as

0 , TA , andwhere T µν denotes the energy-momentum tensor, j µ is baryon number current. Suppose no vis-n(x, y; b) =T (x + b/2, y)T (x + b/2, y).

A
B
coll
cosity of its hydro model(perfect fluid), 0energy
momentum tensor
and baryon number current

are given
by: collisions N
The numberof binarycoll is expressed as an integral of ncoll
Z
Ncoll (b)= dxdyn(x, y; b).
µνµ νcoll µνT= (ε + P) u u − Pg,

(1.13)

ji = ni uν ,(1.14)uν = γ (1, vx , vy , vz ) ,(1.15)µ

1.2.2Space-Time Evolution(1.9)(1.10)Space-time evolution of relativistic heavy ion collisions, assuming it creates QGP, undergoes
where ε asdenotes
local
density,
local pressure,nB represents
baryonofnumber
and uν heavy ion
multiple phases
shown
inenergy
Fig.1.4.
ThisP issectionintroduces
the phases
relativisticµν is Minkowski
fluid four
velocity.
metric tensor.
collisions iselapsedsince
the ginitialcollisions.
In Eq. 1.14, five unknown parameters exists, P, ε , nB and velocity parameters : vx , vy ,0 < ⌧ < ⌧v0z, ,Parton
Cascadeand Thermalization
while 4 unknown
parameterswhich are nB and velocity parameters are contained. If the
The mechanics of initial nucleon collisions i.e. 11
parton cascades are described by several models
such as color-string models[7], a color glass condensate[8], and a perturbative QCD model[9].
The real mechanics of initial parton cascade have not become clear so far, which is one of main
topics in heavy ion physics. The entropy increases in parton cascade and QCD matter reachesequation of state related to local energy density and local pressure are given, all parameters are
determined.
Once all parameters in hydrodynamics are fixed, space-time evolution of Quark Gluon
Plasma can be calculated until the freeze-out (hadronization).
Chemical and Kinetic freeze-out
During QGP stages, mean free path of partons are shorter than the dimensions of the system. As
the system strongly expands due to internal pressure, temperature of the system decreases andmean free path is longer and longer. At the time when mean free path of the partons becomescomparable with the system size, QGP break up into individual hadrons. This is called "freezeout". Freeze-out is categorized into 2 stages, chemical freeze-out and kinetic freeze-out.
Species of hadrons are determined at the "Chemical freeze-out". Suppose the uniform fireball in chemical equilibrium, the number of particle density ni is expressed with the simple
statistical model :∫ni = did3p1,38π exp [(Ei − µi ) /T ] ± 1(1.16)

where di is spin degeneracy, p is momentum, Ei denotes total energy, µi is chemical potential
and T is temperature of the system.
After the chemical freeze-out, no more hadrons are produced from the system, thoughhadron-hadron scattering still ongoing and they exchange their momentum and energy. Whenmean free path becomes equal to the system size, momentum and energy of hadrons are fixedand fly away without interaction each other. This is called "Kinetic freeze-out".

121.3 Experimental Observables
In this section, Experimental observables related to this analysis is presented.

1.3.1 Transverse momentum spectra and radial flow
Transverse momentum (pT ) of generated hadrons are one of the important tools to extract
property of QGP because transverse momentum of produced particles originated from only
collision dynamics. It is thought transverse momenta are generated by two processes, softand hard components. Hard process is quark-quark and quark-gluon hard scattering with large
momentum transfer and high transverse momentum quarks fragment into high pT hadrons. HighpT component is expressed by:
dN= Ap−nT .
d pT(1.17)The other soft process is thermal particles production. Low pT (pT < 2 GeV/c) particle
production is expressed as invariant cross section given by:

d3σ
E 3
d p

1 d2σ
=
,
2π pT d pT dy(1.18)

1d2σ,
2π mT dmT dy)(−mT≈ exp −,
T=

where y is rapidity, mT is transverse mass represented as mT =(1.19)(1.20)√

E 2 − p2z , and T is an in-verse slope parameter of charged hadron pT spectrum, so-called effective temperature. In pp
collisions, this inverse slope parameter T is independent of particle mass, which is called mT
scaling [8]. On the other hand, in nucleus-nucleus collisions, the inverse slope parameter T
is proportional to particle mass as if all particles are emitted with common velocity. This effect is induced by the pressure gradient of QGP and called as "radial flow" which indicates theisotropic expansion in the azimuthal plane. The inverse slope parameter is given by:
1
T = T f + m0 ⟨β 2 ⟩,213

(1.21)

where T f denotes the kinetic freeze-out temperature, ⟨β 2 ⟩ is average expansion velocity and m0
is mass.
Kinetic freeze-out parameter and radial flow velocity can be analytically extracted by theBlast-wave model. Blast-wave model is phenomenological hydrodynamic model and each parameter T f and ⟨β ⟩ can be estimated with fitting the spectra of hadrons.
Figure 1.6 shows the centrality dependence of positive and negative π , K, p spectra in√sNN = 2.76 TeV Pb-Pb collisions from ALICE [9]. Positive and negative particle spectra
are comparable within the systematic uncertainties. Each spectra are fitted with the following
Blast-wave parametrization:
1 dN∝
pT dpT(∫ R
0

rdrmT I0pT sinh ρ
Tkin

)

(
K1

mT cosh ρ
Tkin

)
,

(1.22)

where I0 and K1 are modified Bessel functions, r is the radial distance in the azimuthal plane, R
is the radius of fire ball, Tkin is kinetic freeze-out temperature, ρ is the velocity profile , and mT√
is the transverse mass which is given by mT = p2T + m2 . The velocity profile ρ is given by:ρ = tan−1 βT ,(1.23)

where βT denotes the expansion velocity in transverse plane.
Fit range is shown as horizontal line int the top-left pale of Fig. 1.6. π , K, p combined fitting
is well described with Blast-wave function. The extracted parameters, Tkin and ⟨βT ⟩, are shown
as a function of centrality in Fig. 1.7. Average transverse velocity increases from peripheral tocentral collisions ,while the kinetic freeze-out temperature decreases with increasing centrality.
√
ALICE results are compared with the results in Au-Au collisions at sNN = 200GeV Au-Au
from STAR. For central collisions, 10% stronger radial flow can be observed at LHC energy
than RHIC energy. For central collisions, the radial flow in LHC energy is 10% stronger thanthat in RHIC energy.

14

106
10

103

0-5%

102

0-5%

10
1

10-1

10-2

10

T

102

K
(b)

T

T

103

T

ALICE Collaboration

Range of combined fit

104

(a)

104

sNN = 2.76 TeV

105

πRange of combined fit

5p

1/N ev 1/(2πp ) d2N /(dp dy ) [(GeV/ c )-2]

1/N ev 1/(2πp ) d2N /(dp dy ) [(GeV/ c )-2]

Centrality Dependence of p, K, p in Pb–Pb at

1

10-1

positive
negativecombined fit
individual fit-3

10

positive
negativecombined fit
individual fit

10-2

10-3

80-90%

80-90%

-4

0

0.5

1

1.5

2

10

2.5 3
pT (GeV/c )

0

0.5

1

1.5

2

2.5 3
pT (GeV/c )1/N ev 1/(2πp ) d2N /(dp dy ) [(GeV/ c )-2]

104
Range of combined fit

103

(c)

10

0-5%

1

T

T

102

p

10-1

10-2

10-3
10-4
10-5

0

positive
negativecombined fit
individual fit

180-90%

2

3

4
5
pT (GeV/c )Figureof π , K
and p as(paT )function
of ofcentrality,
positive(red
circle) ofFig. 4:1.6:(color ponline)
Transverse
momentum
distribution(a) p, (b)for
K, and(c) p asa function
T spectra√= 2.76Each
TeVpanel
Pb-Pb
collisions
Eachandnegativesquare)hadronsmeasuredin sNNcentrality,for(bluepositive(circles)and negative(squares)hadrons.
shows
central[9].
to peripheral
n
n
panel
to 2peripheral(80-90%);factorsblast-wave
2 (n denotes
data; shows
spectracentral
scaled (0-5%)
by factors(peripheral
data not spectra
scaled). scaled
Dashedbycurves:fits tothe
indicentrality
bin,
which
most
centralcollisions
arecorresponding
to
n
=
0
and
n
becomeslargervidual particles; dotted curves: combined blast-wave fits (see text for details). Statistical (error bars)
fromcentral to (boxes)
peripheral).
Two different
fitsadditional
are performed,
individual
fit to (Table
each particles
and systematic
uncertainties
plotted. An
normalization
uncertainty3) has to beπ,K,p(dottedlines).
(dashedlines)
andsimultaneous
fit
toadded in quadrature.

1511

0.1
0.05

ALICE, Pb-Pb, sNN = 2.76 Te
BRAHMS, Au-Au, sNN = 200
PHENIX, Au-Au, sNN = 200 G
STAR, Au-Au, sNN = 200 GeV

(b)

0
10

103

102

dN

0.2

ALICE, Fit Range
π: 0.5 < p T < 1 GeV/c
K: 0.2 < p T < 1.5 GeV/c
p: 0.3 < p T < 3.0 GeV/c

0.18
0.16

80-90%

0.14

70-80%

0.08
0.06

0.22
0.2
0.18
0.16
0.14

0.12
0.1

T kin (GeV)

T kin (GeV)

Fig. 9: (color online) (a) p/p = (p + p̄)/(p + + p ) and (b) K/p =pfunction of dNch /dh, compared to previous results at sNN = 200 G

0-5%STAR, Fit Range
π: 0.5 < p T < 0.8 GeV/c
K: 0.2 < p T < 0.75 GeV/ c
p: 0.35 < p T < 1.2 GeV/c

0.12
0.1

0-5%

0.08

(a)

0.06

0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7

Stand
High
Low p

0.25 0.3 0.3〈βT〉

Figure 1.7: Fig.
The correlation
betweenthe two extracted
parameters,
kinetic freeze-out
tempera10: (color
online)(a) Resultsof blast-wave
fits,√compared to siture Tkin and average expansion velocity ⟨βT ⟩ by Blast-wave fits in Pb-Pb collisions at sNN =√uncertainty
contours
includeeffect of the bin-by-bin systematic
2.76 TeV (Black
contours) andAu-Au collisions
at sthe
NN = 200 GeV (Blue contours) [9].

represents the full systematic uncertainty (see text for details), the ST
uncertainties. (b) Comparison of fit results for different fit ranges; thof the bin-by-bin systematics (see text for details).

20

16

1.3.2 Azimuthal anisotropy
Azimuthal anisotropy of emitting particles gives us the key information on initial geometry andviscosity of Quark Gluon Plasma. In non-central nucleus-nucleus collisions, the overlap areaof two nuclei has elliptic shape like an almond due to collision geometry which is sketched inFig. 1.8. If the local thermal equilibrium is established at the overlap region, mean free pathis much shorter than system size and system evolution can be described by hydrodynamics.
In hydrodynamical picture, pressure gradient generates collective flow. Pressure gradient is
considered to be steeper in the direction of reaction plane and collective flow is much moredeveloped in this direction. Finally particle production is strongly biased with this collective
motion, and more particles are produced along the short axis of the system than the long axis.
Therefore azimuthal distribution at freeze-out is expected to be well reflected by an azimuthal
anisotropy in the initial coordinate space (Fig. 1.8).
Azimuthal anisotropy of particle distribution can be extracted by Fourier expansion givenby:∞dN∝ 1 + ∑ vn cos (n [φ − Ψn ]) ,dφ
n=1(1.24)

where Ψn is nth -order event plane determined by the azimuthal distribution of emitting particles
in experiment, φ is azimuthal angle of particles, vn represents the coefficients in the Fourier decomposition and indicates the magnitude of azimuthal anisotropy. Especially 2nd -order Fourier
coefficient is called elliptic flow and has been studied to probe the early stage of Quark Gluon
Plasma.
If a huge number of nucleons exist in the nucleus and initial density distribution are smooth,the shape of initial overlap region would be perfectly elliptic shape. However the number ofnucleons is finite and the density distribution of nucleons largely fluctuates event-by-event.
This event by event participant fluctuation also generates pressure gradient and higher order
flow coefficients.
Figure 1.9 shows azimuthal anisotropy v2 , v3 , v4 , and v5 measured in Pb-Pb collisions at√
sNN = 2.76 TeV. Left panel in Fig. 1.9 shows centrality dependence from ALICE. v2 has large
centrality dependence and v2 significantly becomes larger from central to peripheral collisions.
It is considered that the initial elliptic shape strongly contributes to this behaviour. On the other17

hand, higher order harmonic flow coefficients (v3 , v4 , and v5 ) gently increase with increasing
centrality percentile because fluctuation of the initial density distribution depends weakly on thecentrality. Right panel in Fig. 1.9 shows pT dependence of azimuthal anisotropy from ATLAS[11]. Both 2nd -order and higher order harmonic flow becomes larger from low pT to high pT ,and ALICE and ATLAS data are compared with viscous hydrodynamical simulations depictedwith bands, shear viscosity to entropy ratio η /s is set to 0.2 in this model. Hydrodynamical
simulation fully reproduces the experimental results up to 5th -order flow coefficients with verysmall viscosity, this results suggest that QGP is nearly perfect fluid with small viscosity.
Figure 1.10 shows pT -integrated azimuthal anisotropy v2 , v3 as a function of average Npart√
in Au-Au collisions at sNN = 200 GeV by PHENIX collaboration [12]. Experimental data
are compared to hydrodynamical simulations with different initial state models and viscosity.
For the results on v2 , almost all hydrodynamical models can successfully reproduce the data.
However, MC-KLN model cannot describe v3 well with the same parameter as that used for v2 ,though Glauber model simulation can reproduce both v2 and v3 . Therefore higher order flow
harmonics give us stronger constraining power for the initial state.
Figure 1.11 shows pT dependence of azimuthal anisotropy v2 , v3 , v4 , and v5 for central√
ity 30-40% calculated with two particle correlation method in sNN = 2.76 TeV by ALICE
collaboration [13]. Experimental data are compared with the models of ideal and viscous hydrodynamical simulations with η /s = 0.0 and η /s = 0.08, respectively. Theoretical calculationof viscous hydrodynamical model can describe experimental data up to pT = 2.0GeV/c, anddiscrepancy between ideal and viscous hydrodynamic simulation is much larger for v3 than v2 .
Thus measurements of azimuthal anisotropy of emitting particle is powerful probe for initialgeometry and hydrodynamical properties of Quark Gluon Plasma, and in particular higher harmonic flow (v3 ) has a strong constraining power of not only the initial geometry but also theviscosity of the source.

18

the collision, we expect a greater effect on photon anisotropic flow; this will be examined in a subsequent work.
We emphasize that preequilibrium dynamics that is not
fully accounted for may still influence the amount of initialtransverse flow.
1.3. EXPERIMENTAL OBSERVABLES
13
The effect of changing the switching time from !switch ¼
0:2 fm=c to !switch ¼ 0:4 fm=c is shown in Fig. 5. Results
(a)
(b)
agree within statisticalerrors, but tend to be slightly lower
for the later switching time. The nonlinear interactions ofclassical fields become weaker as the system expands andtherefore Yang-Mills dynamics is less effective than hydrodynamics in building up flow at late times. Yet it is reassury that there is a window in timepy
ing
where both descriptionsx
px
produceequivalent results.
zpz
Because a constant "=s is at best a rough effective meaFigure(a)Initial overlapof suretwo nuclei
innonevolving
centralcollisions
in coordinate
ofrelativistic
the
shear
viscosity
to(b)Collective
entropydensityratio,
Figure1.8: 1.8:
Illustrationsof non-central
heavy
ion
collisions
inspace.
geometrical
space
(a)
flow into the direction of reaction plane in momentum space.
and momentum space (b)we present results for a parametrized temperature dependent"=s, following [38]. We use the same parametrization (HHhave a elliptic flow v2,q , the azimuthal
distribution
mesons is withgiven byHQ) asin Ref.of[38,39]
a minimum of ð"=sÞðTÞ ¼ 0:08
d2at
NM T ¼ 180 MeV,2 approximately at the crossover from≈ (1 + 2v2,q cos 2φ) ≃ 1 + 4v2,q .
(1.38)
dφpquark-gluon
T dpT
plasma to hadron gas in the used equation of

Therefore the elliptic flow for the meson and baryon have the following relation to the elliptic flow
quarks and anti-quarks.P H Y S I C A L R E V I E W L E T T E R S
PRL 110, 012302of(2013)

0.08

0.02

1.3.4

0.06

〈vn 〉

0.04

energy defined as KET = mT −m. Unscaled v2 in the left panel a) shows clearly
distinction between
0.1
η/s =0.2
mesons and baryons although the mass ordering vanishes
by KET scaling. When the number of
constituent quarks scaling is applied for them, 0.1
v2 for all particle species show good agreement asshown in the right panel b) of Fig. 1.9.
0.05

〈vn 〉

〈vn 〉

2 1/2

0.1

(1.39)

v2
ATLAS 30%–40%,
EP
v2
v2
ALICE datavn{2},forpTexperimental>0.2 GeV
If these relations
are trueobservables,vit
would indicate the collective
flowATLAS
at par 20%–30%, EP3v3v3 = 0.4 fm/c
η/s = 0.2
narrow: τswitch
0.2
narrow: η/s(T)tonic level, that is the existence of quark-gluon plasma.v
0.15
v4
4
wide:
τswitchv=4the0.2scaling
fm/c wide:η/s=0.2
Right two panels in Fig. 1.9 show v2 for identified particles
with andwithout
by thevv5v5
5number of constituent quarks nq as functions 0.15
of KET and KET /nq , where KET is the transverse
2 1/2

0.12

0.25 v2,B (pT ) ≃ 3v2,q (pT /3).
v2,M (pT ) ≃ 2v2,q (pT /2),
0.2

2 1/2

0.14

week
4 JANU

Higher-order Harmonic Flow0.05

0
The initial geometry of the overlap region in heavy ion collisions has been treatedas the00.5collisions 1
1.5
0
of nuclei
with20smooth 30
density profile
so far.
recent study,0.5
it is consideredthatthespatial0104050 In 01
1.5 pT [GeV] 2
density distributionof the participating nucleons fluctuates geometricallypand
it leads to highercentrality percentile[GeV]
T
order harmonic deformation. This spatial fluctuation is converted to momentum
space by theFIG.
6(coloronline). Comparison√ of vn ðpT Þ usin
collective
expansionandthenhigher-orderharmonicflowv
iscreated.
nFIG. 4 (color online).
anisotropic
Figure 1.9:Root-mean-square
Azimuthal anisotropy
v2 , v3 , vflow
v5 as a functionof centrality
percentile insNN
4 andco"=s
¼
0:2
and
a
temperaturedependentð"=sÞðTÞ a
Azimuthal distribution of emitted
particlesis online).
rewritten as thefollowing:FIG.
5(colorComparison
of
v
ðp
Þ
using
two differnT,
computedasafunctionofcentrality,
comparedefficients hv2n i1=2
= 2.76 TeV Pb-Pb Collisions from ALICE. vn is calculatedwith
twoparticlecorrelation
methodRef. [38]. Experimental data by the!
0:2(blackfm=cmarker)(wide)is and
0:4 fm=c
entbyswitchingtimestrized!switchin¼
dN
n 2 f2; 3; 4g,
to experimental
datapT of> 0.2
vn f2g,using
GeV/ccharged
particles
integrated.
∝ the
1 are
+ ALICE
2vn cos(n(φCollaboration−ALICEΨn )), datausing(1.40)comthe EP method[4] (points). Banddφ calculation
Experimental
data bylines)
the (left).
ATLAS
Collaboration using
Collaboration [3](points).
Resultshydroare fordynamical
200 (narrow).
eventsper centralparedwith viscous(solid and dashedp dependence0

n=1T
statistical errors.
vn EP= 30-40%⟨cos(n(φ−Ψ(1.41)
ity with bands ofindicating
errors.
n ))⟩,(points).
themethod[4]
Bands byindicate
errors.
azimuthalstatisticalanisotropy
in centrality
via
Event
Plane method
ATLASstatisticalcollaboration(right). For both hydrodynamical simulations, shear viscosity to entropy ratio η /s is set to01230
state. The result, compared to "=s ¼ 0:2
is s
the collision, 0.2
we[11].
expect a greater effect on photon aniso-

tropic flow; this will be examined in a subsequent work.
We emphasize that preequilibrium dynamics that is not
fully accounted for may still influence the amount of initialtransverse flow.
The effect of changing the switching time from !switch ¼
19
0:2 fm=c to !switch ¼ 0:4 fm=c is shown in Fig. 5. Results
agree within statistical errors, but tend to be slightly lower
for the later switching time. The nonlinear interactions ofclassical fields become weaker as the system expands and20%–30% central collisions in Fig. 6. The results
tinguishable when studying just one collision en
insensitivity of our results to two very different f
forms may suggest that the development of flow iaffected at intermediate times when "=s is very sm
since second order viscous hydrodynamics brea
when !#$ is comparable to the ideal terms, our fr
may be inadequate for too large values of "=s.
We compare results for top RHIC energies,

vn

Figure 1.10: pT integrated azimuthal anisotropy v2 and v3 as a function of number of partic√
ipant with two different pT ranges. in Au-Au collisions at sNN = 200 GeV from PHENIX.
Experimental data (black marker) are compared with theoretical predictions of two different
initial state models MC-KLN and Glauber and different viscosity [12].

(a)Centrality 30-40%
v2{2}
v3{2}
v4{2}
v5{2}
v2 (η/s = 0.0)
v2 (η/s = 0.08)
v3 (η/s = 0.0)
v3 (η/s = 0.08)

0.3

0.2

0.1

0
0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

pt (GeV/ c )Figure 1.11: Azimuthal anisotropy v2 , v3 , v4 and v5 as a function of transverse momentum
for centrality 30-40% measured with a two particle correlation method of two different rapidity√
gaps, ∆η > 0.2 (open symbol) and ∆η > 1.0 (closed symbol), in Pb-Pb collisions at sNN =2.76 TeV from ALICE. Data are compared with hydrodynamical simulations of ideal fluid andviscous fluid (η /s = 0.08).

20

1.3.3 Event Shape Engineering
For a further understanding of the spatio-temporal evolution of Quark Gluon Plasma, one ofessential studies is initial collision geometry. In order to extract the initial collision geometry
experimentally, collision centrality which is sensitive to impact parameter has been used until
now. Recently, however, it is thought the initial geometry of nuclear overlap region largely
fluctuates even at a fixed and narrow centrality window.
Figure 1.12 shows probability distributions of event-by-event v2 , v3 and v4 as a function of√centrality measured in Pb-Pb collisions at sNN = 2.76 TeV from ATLAS collaboration [14].
The coefficients of azimuthal anisotropy vn are largely fluctuated within a certain centrality
bin and these fluctuations becomes larger from central to peripheral collisions. The azimuthal
anisotropy is sensitive probe to the initial geometry, thus these fluctuations come mostly fromthe fluctuations of the initial geometry at initial stage of the collisions.

Figure 1.12: The probability distribution of event-by-event v2 (left), v3 (middle) and v4 (right)√
in 5 or 6 centrality in Pb-Pb collisions at sNN = 2.76 TeV from ATLAS collaboration [14].
Error bars represent statistical uncertainties and Systematic uncertainties are shown as shaded
bands. Solid line denotes the distributions assuming the vn are radial projections of 2D Gaussian
distributions.

Recently a new approach to select these event-by-event initial geometry fluctuations, socalled the Event Shape Engineering (ESE), was developed utilizing these large fluctuations ofazimuthal anisotropy [15]. The ESE technique offers to select the initial geometrical source

21

shape with the event-by-event flow vector qn . The event-by-event flow vector qn is given by:
M

Qn,x = ∑ cos (nϕi ) ,(1.25)

Qy,x = ∑ sin (nϕi ) ,(1.26)

i
M
i

(
) √
qn = Q2n,x + Q2n,y / M.

(1.27)

where n is the harmonics of Fourier coefficients, M is the multiplicity of an event, and ϕi is theazimuthal angles of emitting particles. The initial geometrical source shape can be calculatedwith nth -order participant eccentricity εn which is given by:

εn =

√

2 + ε2 ,εn,xn,y(1.28)

εn,x = ⟨rn cos (nϕ )⟩,(1.29)

εn,y = ⟨rn sin (nϕ )⟩,(1.30)

where x and y represent space coordinates of nucleons and ϕ is an azimuthal angle of nucleons.
By selecting the magnitude of an event-by-event flow vector qn , we can control event-by-event
initial geometrical source shape, i.e. larger (smaller) qn tends to have larger (smaller) εn .
One of the big advantages on ESE technique is extracting how initial geometrical shape contributes to the other observables. Figure 1.13(left) shows azimuthal anisotropy v2 as a functionof pT for various q2 selections and no q2 selection by ATLAS collaboration [17]. Elliptic flow
coefficients v2 largely changes with q2 selection and this enhancement(suppression) does notdepends on pT . Right panel of Fig. 1.13 shows the correlation between low momentum v2 (0.5
< pT <2 GeV/c) and high momentum v2 (3 < pT < 4 GeV/c) in 7 centrality bins for 6 q2 classes.
The gray band represents a correlation between low pT v2 and high pT v2 without ESE selection, which shows the "boomerang-like" structure that can be understood by viscous-damping
effects to different pT ranges according to hydrodynamical model calculations [18]. For various
q2 selections, however, the correlation of v2 in two different pT ranges dramatically changes toa linear correlation. This linearity indicates that hydrodynamical viscous effects are determinedby not the geometrical source shape but the source size.
Left(Right) panel in Fig. 1.14 shows the ratio of pT distributions of π + + π − , K + + K − , andp + p̄ with top (bottom) 10% q2 selection to those without q2 selection. Transverse momentum
22

spectra of identified hadrons are enhanced (suppressed) with large (small) q2 selection, andthe ratio of pT spectra becomes larger (smaller) from low pT to high pT with large (small) q2
selection. Mass ordering can be explicitly found for the ratio of pT spectra. Data are comparedwith the Blast-wave model and an average boost velocity can be extracted with this model.
Large (small) q2 selection enhances (suppresses) not only the azimuthal anisotropy v2 but also
the average boost velocity of the system.

Figure 1.13: Azimuthal anisotropy v2 as a function of transverse momentum in the centrality√
20-30% (top left) for events selected on q2 measured in Pb-Pb Collisions at sNN = 2.76 TeVfrom ATLAS. Bottom left panel shows the ratio of v2 for q2 -selected to that without q2 selection, and a correlation of v2 between pT :0.5-2GeV/c and pT :3-4GeV/c in 7 centrality bins.
Data points represent v2 correlation in various q2 selection and overlaid gray band denotes thatwithout q2 selection [17].

23

1Statistical uncertainties
Systematic uncertainties
0

1

2

3

4

p (GeV/c )

T

1.05

T

Blast Wave model (π++π-)
Blast Wave model (K++K )Blast Wave model (p+p)π++πK++Kp+p(1/N ev) d2N /(dp dη) (unbiased)

T

T

1.1

2
(1/N ESEev ) d N /(dp dη) (ESE-selected)

T(1/N ev) d2N /(dp dη) (unbiased)2(1/N ESEev ) d N /(dp dη) (ESE-selected)

ALICE 30-40% Pb-Pb sNN = 2.76 TeV∆〈β 〉 = 0.41%

ALICE 30-40% Pb-Pb sNN = 2.76 TeV∆〈β 〉 = -0.22%
T

1.05

π++π+
K +K
p+p

Blast Wave model (π++π-)+
Blast Wave model (K +K )Blast Wave model (p+p)

1

0.95

Statistical uncertainties
Systematic uncertainties
0

1

2

3

4

p (GeV/c )T

T

Figure 1.14: The ratio of pT distributions of π + + π − , K + + K − , and p + p̄ with large (small)q2 selection to those without q2 selection in centrality 30-40% are shown in left (right) panel.
√
Results are measured in sNN = 2.76 TeV Pb-Pb Collisions from ALICE collaboration. Data arecompared with the Blast-wave model which is the hydrodynamical inspired model. Based on
the Blast-wave model, an average boost velocity ⟨βT ⟩ is 0.41% enhanced (0.22% suppressed)with ESE selection [19].

241.3.4 Hanbury-Brown and Twiss Interferometry
There are several parameters which describe the bulk properties of Quark Gluon Plasma, suchas freeze-out temperature, boost velocity, viscosity, system size, freeze-out time, and emission
duration. Basically the freeze-out temperature, boost velocity, and viscosity can be extractedwith measurements of hadron spectra and azimuthal anisotropy. The other parameters (thesystem size, freeze-out time, and emission duration) can be extracted with Hanbury-Brown
and Twiss interferometry (HBT interferometry). HBT measurement is an unique and essential
method to address system size at freeze-out, emission duration, and system life time.
HBT is a method to measure the source size with two identical particles [20, 21]. Here weassume a simple model that two identical particles are emitted from a certain source, which isshown in Fig. 1.15. Emission points of two identical particles are defined r1 and r2 , and twoparticles are observed by two detectors which are located at x1 and x2 . We define r as the lengthfrom emission point to observed position, and R as the length from r1 to r2 . Two particles cantake two routes to two detectors, route1 and route2 which are depicted as red lines and blue lines
in Fig. 1.15). If two particles are identical and L is much larger than R, quantum mechanics
can be applied, then we cannot identify route. Hence, we can write the wave function of two
identical particles in the from:1
Ψ12 = √ [Ψ1 (r1 ) Ψ2 (r2 ) ± Ψ2 (r1 ) Ψ1 (r2 )] ,2]
1 [= √ A1 A2 e−ip1 (x1 −r1 ) e−ip2 (x2 −r2 ) ± A1 A2 e−ip1 (x2 −r1 ) e−ip2 (x1 −r2 ) ,2(1.31)

where Ψ is a wave function of the signle particle with momentum p emitted from r, A is amplitude of wave function. The Sign of second term in Eq. 1.31 is determined with particle species.
Bosons have positive and fermions have negative sign.Then the probability density is expressedby:|Ψ12 |2 = |A1 |2 |A2 |2 [1 ± cos (∆x∆p)](1.32)

where ∆x = x1 − x2 and ∆p = p1 − p2 . The cosine term in Eq. 1.32 represents HBT correlation
term. Therefore an effect of quantum interference are stronger in small ∆x or ∆p, and This correlation can be found as enhancement of probability density for bosons, thus HBT interferenceis also called Bose-Einstein enhancement.

25

x1

Detector1

route1

Sourcer1

route2R

r2
x2
L

Detector2

Figure 1.15: The schematic figure of HBT interferometry. Two identical particles emitted froma certain source. The emission points of each particles are defined as r1 and r2 . A distance
between r1 and r2 is expressed as R. Two particles are obserbed by detectors at x1 and x2 . The
distance between emission point and detected position is L. If R is much smaller than L, route
1 and route 2 can not be identified with quantum mechanics.

We define the correlation function C2 with the following equation:
C2 =P (p1 , p2 )P (p1 ) P (p2 )(1.33)

where P (p1 , p2 ) denotes the probability to observe two particles with momentum p1 and p2
simultaneously, and P (p1 ) (P (p2 )) is the probability to observe single particle with momentum
p1 (p2 ) indipendently. If we assume the density distribution of source as a Gaussian function,
the correlation function C2 of two identical particles are given by()
C2 = 1 ± λ exp −R2 q2 ,(1.34)

where λ is chaoticity (incoherence) parameter which is sensitive to incoherence of the system
and resonance. The chaoticity parameter λ takes 0 to 1, and R is so-called "HBT radius" and indicates a standard deviation of the source distribution. The relative momentum of two identical26

particles is represented as q = p1 − p2 .
Bertsch Pratt parametrization
For 1-dimensional HBT analysis, correlation function is given by√
qinv =

q2x + q2y + q2z + q20 ,(1.35)

q0 = E1 − E2 .

(1.36)

Bertsch-Pratt parametrization are frequently utilized to extend the one dimensional to 3 dimensional HBT analysis [22, 23]. In this parametrization, one dimensional relative momentum
q is decomposed into outward (qout ), sideward (qside , and long (qlong ). Long denotes a termalong the direction of the beam axis and azimuthal plane is represented as outward and sideward, where outward is parallel to pair transverse momentum kT and sideward is perpendicular
to kT . The pair transverse momentum kT is given by:

kT =

−
−→
p→
T1 + pT2.
2(1.37)

When we apply the Bertsch-Pratt parametrization in the Longitudinal Co-Moving System(LCMS) where pz1 + pz2 = 0, the correlation function C2 is re-written as()
C2 = 1 + λ exp −R2 q2 ,()= 1 + λ exp −R2x q2x − R2y q2y − R2z q2z − σt2 q20 ,)(= 1 + λ exp −R2out q2out − R2side q2side − R2long q2long ,(1.38)(1.39)(1.40)

where σt represents an emission duration and σt is included in Rout in Eq. 1.40, therefore
Rside is driven by a purely geometrical information on the system.
HBT measurements in dynamical system
In a static source, measured HBT radii represent a standard deviation of whole system size. In
dynamically expanding source such as our universe (Hubble flow), however, HBT radii is equal
not to a standard deviation of whole system size but the "length of homogeneity region" [24].
The length of homogeneity region depends on the expansion velocity of the source and pair27

transverse momentum kT . HBT radii calculated with larger kT pairs are corresponding to more
surface part of the source.
Therefore, in order to understand the space time evolution of the dynamical source with
HBT measurements, study of kT dependence is important.
Extraction of geometrical source size and freeze-out time
Figure 1.16 shows the extracted 3D HBT radii (Rside , Rout and Rlong ) and λ as a function√
pair transverse mass mT = kT2 + m2 for various centrality bins measured in Au-Au collisions√
at sNN = 200 GeV from PHENIX collaboration [25]. All 3D HBT radii strongly decrease
with increasing mT . This explicit mT (kT ) dependence indicates the dynamical expansion of thesystem, and this mT is qualitatively described with hydrodynamical model. If the difference in
HBT radii between pions and kaons is due to particle mass, i.e. if the freeze-out time, emission
duration, and subsequent system evolution are same, HBT radii of pions and kaons are scaledwith mT . However Rout and Rlong of kaons are larger than that of pions, though Rside of kaon is√comparable to those of pions. This behaviour can be also found in Pb-Pb collisions at sNN 2.76
TeV from ALICE collaboration [26]. Based on the hydrodynamics, kT dependence of Rout issensitive to transverse velocity and that of Rlong is sensitive to freeze-out time. The geometricalsource size, system life time, and emission duration can be extracted with analytical formula[27]. It suggests that maximal emission time for kaons is larger than the one for pions. Therefore
it could indicate that pions and kaons have different space-time correlations.
Azimuthal angle dependence of HBT radii with respect to Ψ2
Basically HBT is the method to measure the source "size", but this technique is extended withcombining the study of azimuthal anisotropy. Based on the hydrodynamical model, short axis
of initial elliptic shape can be obtained with 2nd -order event plane Ψ2 as described in detail in
Sec. 3.2.2. Measurement of azimuthal angle dependence of HBT radii with respect to Ψ2 gives
us the information on "source elliptic shape" at freeze-out [28, 25, 29].
Left panel in Fig. 1.17 shows the azimuthal angle dependence of charged pion HBT radii√with respect to Ψ2 for 3 centralities measured in Au-Au collisions at sNN = 200 GeV fromSTAR [28]. No significant oscillation can be found in Rlong , but, in azimuthal plane, Rout , Rside
and Ros have explicit oscillations. This indicates that final source has a finite eccentricity. Final28

Figure 1.16: Extracted 3D HBT radii (Rside , Rout , and Rlong ) and λ as a function of mT for√
4 centralities measured in Au-Au collisions at sNN = 200 GeV from PHENIX [25]. Data
are compared with hydrokinetic simulation (HKM model) and viscous-hydrodynamic model(Bozek)source eccentricity is calculated by fitting azimuthal angle dependence of HBT radii whichis shown in left panel of Fig. 1.16. Right panel in Fig. 1.17 shows relation between initialeccentricity and final eccentricity. The dashed line in Fig. 1.17 indicates εinitial = ε f inal . One canfind that final source eccentricity is much smaller than the dashed line. It indicates that initialelliptic shape is significantly diluted but out-plane elongated shape still remains at freeze-out.
This information should constrain the space time evolution of the system.
Figure 1.18 shows hydrodynamic simulation of the azimuthal angle dependence of charged
pion HBT radii with respect to Ψ2 for 6 kT calculated for LHC energy [30]. No significant
oscillation can be found in Rlong and the oscillation of Rout and Rside is out of phase (Rout isconvex upward and Rside is concave up) in top 4 kT classes, which is consistent to RHIC results.

29

However, in kT : 0-0.2 GeV, a clearly different behaviour can be found, i.e. the oscillation of
Rout is concave up and the one of Rside is convex upward in kT = 0 GeV, but both the oscillationof Rout and Rside are convex upward in kT = 0.2 GeV.

Figure 1.17: Azimuthal angle dependence of charged pion HBT radii (Rout , Rside , Rlong , and√Ros ) with respect to Ψ2 for 3 centralities measured in Au-Au collisions at sNN = 200 GeV
from STAR (left). Initial source eccentricity calculated with Glauber simulation v.s. source
eccentricity at freeze-out obtained with azimuthal angle dependence of HBT radii (right) [28].
The dashed line indicates that εinitial = ε f inal .

30

Figure 1.18: Azimuthal angle dependence of charged pion HBT radii (Rout , Rside , Rlong , andRos ) with respect to Ψ2 for 6 kT classes in LHC energy, calculated with hydrodynamical model
[30].

31

Azimuthal angle dependence of HBT radii with respect to Ψ3
As mentioned in Sec. 1.3.2, the initial participant fluctuation makes higher order anisotropic
flow, and recently higher order azimuthal anisotropy is actively measured to determine the initialgeometry and the viscosity of the source.
Concerning HBT analysis, the possibility of 3rd -order oscillation of HBT radii with respect
to Ψ3 is suggested with Blast-wave model calculation and AMPT simulation [31]. Measurementof azimuthal angle dependence of HBT radii with respect to Ψ3 should be an unique and directprobe for triangular shape at freeze-out. In order to understand the space time evolution of hot
dense matter, investigating not only the elliptic shape but also triangular shape at freeze-out isimportant.
Figure 1.19 shows the first measurement of the azimuthal angle dependence of charged pion
HBT radii with respect to Ψ3 (Ψ2 dependence of HBT radii is plotted simultaneously) was√
performed in Au-Au collisions at sNN = 200 GeV. Rside oscillation relative to Ψ3 is very weak,
but Rout has a visible oscillation with respect to Ψ3 .
Figure 1.20 shows the relative amplitude of squared HBT radii with respect to Ψ3 (bluemarkers) and Ψ2 (red markers) obtained by Fig. 1.19. Contrary to relative amplitude of Rside
with respect to Ψ2 , relative amplitude of Rside with respect to Ψ3 has negative or zero value,
whereas both relative amplitude of Rout with respect to Ψ2 and Ψ3 has positive value. Also
relative amplitude of Ros with respect to Ψ2 and Ψ3 has positive value.
However triangular shape at freeze-out cannot be obtained directly from relative amplitude
of HBT radii as is the case with eccentricity. In order to understand 3rd order oscillation of
HBT radii, some of the model calculations (Gaussian toy model and hydrodynamical simulation) were performed [33, 34]. Figure 1.21 shows the relative amplitude of HBT radii withrespect to Ψ3 as a function of pair transverse momentum kT , compared with two extreme case
Gaussian model. Solid line indicates triangular flow dominated case which is spherical spatial distribution superimposed on an large triangular flow, and dashed line indicates triangular
geometry dominated case which is the geometric triangular distribuion superimposed on an azimuthally symmetric radial flow. Data prefer flow dominated case, but relative amplitude of
Rside in high kT can not be described by this model. In order to constrain the ε3 and v3 in themodel, more precise measurements of oscillation is important.

32

Figure 1.19: Azimuthal angle dependence of charged pion HBT radii (Rout , Rside , Rlong , andRos ) with respect to Ψ2 (top 4 panels) and Ψ3 (bottom 4 panels) for 2 centralities measured in√
Au-Au collisions at sNN = 200 GeV from PHENIX [32]. First row from the left denotes Rside, second row is Rout , third row is Rlong , and fourth row indicates Ros .

33

Figure 1.20: Relative amplitude of squared HBT radii (Rout , Rside , and Ros ) with respect toΨ3 as a function of initial eccentricity which is calculated with Glauber model. The results are√calculated with data in Au-Au collisions at sNN = 200GeV from PHENIX [32]. Pair transverse
momentum kT is integrated from 0.2 to 2.0 GeV/c. Dashed line indicates relative amplitude of
HBT radii = 0 and εn = |2R2µ ,n /R2µ ,0 |. Boxes represent the systematic uncertainties.

34

Figure 1.21: Relative amplitude of squared HBT radii(Rout , Rside , and Ros ) for charged pion
pairs with respect to Ψ3 as a function of pair transverse momentum kT for two centralities√
measured in Au-Au collisions at sNN = 200GeV from PHENIX [32]. Data is compared withtwo different Gaussian source distribution model. Solid line indicates that flow dominated casewhich is spherical spatial distribution superimposed on an large triangular flow, and dashed line
shows geometry dominant case which is the geometric triangular distribuion superimposed onan azimuthally symmetric radial flow.

351.4 Thesis Motivation
Investigating a relation between initial and final source shape is quite important to understandthe dynamics of the QGP and to provide feedbacks to theoretical models to further constrain
the space time eveolution of heavy ion collisions. Azimuthal angle dependence of HBT radii isan unique and direct probe to access the final source shape. Measurements of azimuthal angle
dependence of pions and kaons HBT radii with respect to Ψ2 and Ψ3 are extensively performedin Au-Au collisions at RHIC, but there is a remaining questions. Due to the large uncertainties,
the oscillation sign of HBT radii relative to Ψ3 is still not understood at this point. Furthermore,hydrodynamical model predicts the oscillaton of HBT radii with respect to Ψ2 in LHC energy
could be different to that in RHIC. Solving these problems will help to understand the scenariosof the system evolution.
In this thesis, measurements of azimuthal angle dependence of pion HBT radii with respect√
to Ψ2 in Pb-Pb collisions at sNN = 2.76 TeV with ALICE detector is performed. Hydrodynamical model predicts oscillation of HBT radii relative to Ψ2 shows the different behaviour to
RHIC results due to extremely large collective flow [30]. Therefore we present the oscillation
amplitude of HBT radii with respect to Ψ2 and discuss the space time evolution of system withcomparison of initial and final eccentricity.
Also measurements of azimuthal angle dependence of pion HBT radii with respect to Ψ3
is performed. In LHC energy, large multiplicity and excellent event plane resolution allows
us to measure azimuthal angle dependence of pion HBT radii with respect to Ψ3 much more
precisely to disentangle relative amplitude of HBT radii is positive or negative value to constrain
the theoretical model.
Centrality is one of the probes for the initial geometry. However, when centrality changes
from central to peripheral collisions, not only the initial geometry but also system size, freezeout temperature, and flow velocity change simultaneously. Recently ESE technique is proposed
to select the initial geometry within a certain centrality window, and it gives us the new insightof relation between the initial geometry and other observables separately from the system size.
In this thesis, ESE q2 and q3 selection are applied to the measurements of azimuthal angle
dependence of pion HBT radii with respect to Ψ2 and Ψ3 . Relation between initial geometricalsource shape and final source shape at freeze-out are discussed with Blast-wave model.

36

Chapter 2
Experiment
In this chapter, we introduce the performance of Large Hadron Collider where the hot anddense matter is created with relativistic heavy ion collisions and ALICE detector to measure theemitting particles. Also the method to accelerate the ions with LHC and to detect the particles
with ALICE detector is described.

2.1 Large Hadron Collider (LHC)
Large Hadron Collider (LHC) is the world’s largest particle accelerator, which is designed for
proton-proton, proton-Lead and Lead-Lead collisions. LHC is built at European Organization of Nuclear Research (CERN) and located in French-Switzerland border. It consists of a
27-kilometer ring of superconducting magnets to boost the energy of the particles along theway. Inside the LHC beam pipe, particles circulate in the two tubes kept at ultrahigh vacuum(10−13 atm). They are manipulated with electromagnetic devices. Dipole magnets keep thebeam to the circular orbits and quadrupole magnets focus the beam. Summary of the CERN’s
accelerator complex parameter is shown in Table2.1.
Table2.2 is beam parameters for LHC.The number of collision events per second (Nevent ) isexpressed with the following equation.
Nevent = L σevent ,(2.1)

where σevent is the cross section and L is the machine luminosity which denotes the machine
performance. Higher collision rate is the key for the precise understanding of high energy
physics. L can be described under the Gaussian beam distribution by

37

L =

Nb2 nb frev γF,
4πεn β ∗(2.2)

where Nb is the number of particles per bunch and nb is the number of bunches per beam. frev
is the number of bunch crossing per unit of time. γ is the relativistic gamma factor. εn is thenormalized transverse beam emittance. β ∗ denotes the width of the beam F is the geometricluminosity reduction factor due to the crossing angle at the IP.
Table 2.1: Parameters for CERN’s accelerator complex[35]Quantity
unit
value
Circumferencem
26.659m
Dipole operating temperature K 1.9K (-271.3oC)
Peak magnetic dipole filed
T
8.33T
Number of magnets
9593
Number of main dipoles
1232
Number of main quadrupoles
392
Number of RF cavities
8 per beam

Table 2.2: Beam parameters for LHC[37]Quantity
unit
Protons
Top collision energy
TeV (TeV/u for ions)14
Design luminosity
cm−2 s−1
1.0 × 1034
Number of bunch
2808
Number of particles per bunch
1.15 × 1011
RMS bunch length
cm
7.55

Ions
5.5
1.0 × 1027592
7.0 × 107
7.94

Figure2.1 shows the scenario of protons and lead ions acceleration[36]. Lead ions are accelerated to high energy with a succession of the machines : ECR (Electron Cyclotron Resonancesource) , LINAC3 (LINear ACcelerator 3), LIEIR (Low Energy Ion Ring), PS (Proton Synchrotron), SPS (Super Proton Synchrotron) and LHC. At first, lead ions are produced with
ECR ion source. A highly purified sample of solid lead (208 Pb) is heated to 550o C, then solid
lead (208 Pb) sample becomes a vapor. Evaporated lead (208 Pb) are injected into ECR. ECR
is a plasma generator with solenoid magnets and Fe-Nd-B permanent magnet sextupole. The
plasma electrons are confined with two magnet and accelerated with the electric field. Throughthe inelastic collisions between evaporated lead sample and hot electrons, lead sample are ionized (electron impact ionization)[38]. In this stage, Many different charge states are mixed
38

between Pb25+ and Pb29+ . Pb27+ at 2.5keV/u are selected and injected into LINIAC3. LINAC3
is the linear accelerator which consists of two machines, RFQ (Radio Frequency Quadrupole)and IH Linac. 100MHz RFQ accelerates the lead ions to 250keV/u and IH Linac increases thebeam energy up to 4.2MeV/u. After the acceleration of Linac, lead (Pb27+ ) ions are strippedby a 100 µ g/cm2 carbon foil and Pb53+ are provided to LEIR. LEIR also accelerates the ions,but the most important function of LEIR is electron cooling. Each long pulse from LINAC3 issplit into 4 shorter bunches which contains 2.2 × 108 lead ions. Beam energy is accelerated to
72MeV/u in LEIR for 2.5 seconds. The nominal number of bunches per beam in LHC is 592,
so it takes around 10 minutes for LEIR to fulfill this requirement. The beam in LEIR is transferred into the PS and accelerated to 5.9 MeV/u. Then Pb53+ beam is fully stripped by a thin
0.8 mm aluminum foil at the PS exit. Pb82+ beam is provided into SPS. The SPS accelerates it
to 177GeV/u. Eventually ion beam is injected into LHC and accelerated up to 2.76 TeV/u.
Heavy ion collisions are provided at four interaction points in LHC and four experiments
are investigating the physics of Quark Gluon Plasma : ALICE, ATLAS, CMS and LHCb whichis installed in the huge underground caverns.

39

Figure 2.1: CERN accelerator complex

40

2.2 ALICE experiment
ALICE (A Large Ion Collider Experiment) is one of the large experiment at LHC-CERN. Morethan 1500 physicists, engineers from 41 countries are devoted to research into the property of
Quark Gluon Plasma with heavy ion collisions at LHC. In heavy ion collisions, huge number ofparticles are produced which contains the clue of hot dense matter. ALICE detectors are optimized to measure the these particles with high granularity. In particular, charged particle can bereconstructed from low transverse momentum (pminT ≈ 0.15(GeV /c)) and particle identification
can be performed with wide momentum range up to 20 GeV/c.
Figure 2.2 is the 3D schematic of ALICE detector apparatus. Central barrel detectors
are installed in the L3 solenoid magnet with 0.5T magnetic field and consist of 7 detectors Inner Tracking system (ITS), Time Projection Chamber (TPC), Transition Radiation Detector(TRD), Time Of Flight (TOF), ElectroMagnetic CALorimeter (EMCAL), PHOton Spectrometer (PHOS) and High Momentum Particle Identification Detector (HMPID). Triggering andevent characterization are performed with forward detectors (VZERO, TZERO, FMD, PMD
and ZDC). ACORDE is located on the top of L3 magnet for triggering the cosmic-ray to collecthigh muon multiplicity events. In order to measure light vector mesons (ρ , ω ) and quarkonium(J/ψ , ϒ) with µ + µ − decay channel, Muon Spectrometer are placed in the forward rapidity.
In this section, the detectors related to this thesis is introduced.

41

Figure 2.2: 3D schematic of ALICE experiment at the CERN LHC. ITS, TPC, TRD, TOF,EMCAL, PHOS and HMPID are installed in 0.5 Tesla magnetic field which is applied withsolenoid magnet to measure the midrapidity particles. Forward detectors (VZERO, TZERO,
FMD, PMD and ZDC) are also in the magnet and used for triggering and event characterization.
In order to trigger the cosmic-ray, ACORDE is placed on the top of solenoid magnet. Muonarm which consists of Muon tracker(MCH) and Muon trigger (MTR) is located in the forward∫
rapidity in the dipole magnet with Bdz=3Tm after the front absorber.

42

2.2.1 ALICE magnet system
ALICE is equipped with two large magnets (L3 magnet and dipole magnet) to measure thecharged particle momentum and particle identification.
Figure2.3 shows the general layout of ALICE L3 magnet which is inherited from LEP-L3
experiment[39]. L3 magnet is a solenoid magnet consists of octagonal iron barrel yoke aroundthe coil and two pole caps at the end of barrel. In order to make homogeneity magnetic field
especially in TPC, it is important to limit the stray flux at the end of magnet. Two pole caps
which is in the form of two semicircle per each side are build for it. Each sector is equipped
as a hinged "door". the coil of L3 magnet consists of 168 octagonal turns constructed from 60
mm×890 mm aluminum plates. typical current for the solenoid coil is 30 kA and each turns
has a water cooling circuit. L3 magnets covers Central-barrel detectors and forward detectors
except for ZDC and provide 0.5T magnetic filed in the direction of parallel to the beam.
Second magnet is dipole magnet for muon spectrometer located in the forward rapidity at
7m from interaction point. It’s one of the biggest warm dipoles in the world. The schematic ofdipole magnet assembly is denoted in Figure2.4. ALICE dipole magnet provide 0.7T nominal
magnetic field (3 T · m field integral) in a perpendicular direction of beam axis and typicalcurrent is 6kA. A water cooling system is designed to manage the heat of joule effect in coil.
This magnet consists of two semi-circular coils and the vertical poles of the rectangular yoke.

43

Figure 2.4: 3D schematic of Dipole
magnetFigure 2.3: 3D schematic of L3 magnet

44

2.2.2 Time Projection Chamber
Time Projection Chamber (TPC) is one of the main tracking detector for charged particles which
covers mid rapidity. TPC has a lot of roles to provide the momentum of charged particles,
particle identification with energy loss, collision vertex determination and two track separation(it’s very important for this HBT correlation analysis. Because Bose-Einstein enhancement canbe found in "closed" two pair).
This detector allows us to extract 3 dimensional trajectory of charged particles. ALICE TPC
is designed of cylinder filled with Ne CO2 (90:10) mixed gas. Inside of TPC, filed cage keeps
the uniform electric filed in parallel direction to the beam axis. Charged particles traveling in
the TPC ionize the gas and produce election-ion pairs along their trajectories. Electrons drift
along electric field toward read-out pad due to the central high voltage electrode. Vicinity thegrid of anode wire, drifting electrons make avalanche and their signals are amplified and readout from pad. x-y position of charged track can be obtained with their signal on the read-out
pad and z axis coordinate can be extracted from the drift time of ionized electrons.
TPC acceptance is full azimuthal angle and |η |<0.9 (Though TPC drift volume covers muchwider η acceptance, its acceptance is limited to combine the other detectors and to get good trackreconstruction quality). The schematic of TPC is shown in Figure2.5. TPC inner and outerradius is about 80cm and 250cm respectively. Overall length in the beam direction is 510cm.
Drift volume is 88m3 and divided in two regions by the central electrode that applied HV is
100kV. Filed cage keeps uniform electric filed with the voltage gradients 400V/cm. TPC signals
are obtained with readout chambers on the end cap of TPC. 18 readout chambers per each side
are installed on the end cap. Each readout chambers are trapezoidal shape and each covers 20
degree in azimuthal angle. Figure 2.6 shows the schematic of wire planes in a TPC. Drifting
electrons are amplified with avalanche around the anode wire (1450-1720V). A lot of ions arealso produced with that avalanche and it cause a noticeable reduction in electron transmission.
Cathode wire can collect these ions and separate the drift volume from the amplification region.
Gating grid is power full tool to control the elections and ions go through with applied voltage
and there are mainly two important roles. First one is to prevent ions created in the previous
event from escaping to drift volume. escaping ions can cause distortions of the drift field.
Second one is to prevent electrons from entering the avalanche region if there’s no valid triggers.

45

In this analysis, two track resolution and dE/dx resolution are important. HIJING+GEANT
study of two track separation is plotted in Figure2.7. Pericles are generated with HIJING indNch /dy=8300. After the reconstruction with GEANT, two track efficiency is calculated asa ratio of the number of reconstructed pairs to that of generated ones. One can see that pairefficiency goes down if their momentum differs by less than 0.015GeV /c[40]. dE/dx resolution
depends on particle density produced in the event. dE/dx resolution is 7.0% (dNch /dy=1300),
8.6% (dNch /dy=4300) and 17.3% (dNch /dy=8300).

Figure 2.6: A cartoon illustrating of
TPC wire in the readout chamber

Figure 2.5: 3D schematic of TPC

Figure 2.7: Two track efficiency for particle density dNch /dy=8300 as a function of absolutevalue of generated momentum difference of two particle with HIJING+GEANT simulation46

2.2.3 Inner Tracking System
Precise vertex position determination and secondary vertices of charm and hyperon decay canbe derived with Inner Tracking System (ITS). Also ITS improves momentum and angle measurement by TPC-ITS combined tracking. Particle identification of low momentum can beobtained with energy loss in ITS.
ITS is composed of six cylindrical layers of silicon detectors which is the surrounding theinteraction point. The two innermost layers are Silicon Pixel Detectors (SPD), the middle twolayers are Silicon Drift Detectors (SDD) and the two outermost layers are Silicon Strip Detectors(SSD). Figure 2.8 shows the front view of SPD layout (view from the beam axis). SPD consistsof two silicon pixel detector modules at radius 3.9cm and 7.6cm. 6 staves (2 for inner and
4 for outer) are fixed on the each lightweight carbon-fiber sectors. Each stave has 4 ladders,each ladder consisting of 256 × 160 cell matrix silicon pixel sensor. In third and forth layers
are SDD which is precise position resolution silicon detector. charged particles create electron
hole-pair by ionization traveling inside the silicon. Electrons drifts with the constant speed
towards n type silicon substrate by parallel drift cathodes. The drift time and read-out position
allows us to unambiguous determination of both x-y coordinates with low readout channels andhigh position resolution. Figure2.10 shows the picture of SDD inner layer[42]. SDD consistsof 14 ladders and 6 detectors for 3rd layer at radius 14.9cm and 22 ladders and 8 detectors
for 4th layer at radius 23.8cm. 5th and 6th outermost layers SSD are double sided silicon
strip detectors.Both p-side and n-side has strip structure and provides x-y coordinate with highposition resolution and small dead time.SSD is composed of 34 ladders at radius 38.4cm and38 ladders at radius 43.4cm. Each ladders have 23 and 26 silicon strip detector modules for 5
and 6 layers (Figure2.11).
The acceptance of ITS is full azimuthal angle and |η | < 0.9 except for the inner most pixel
layer. In order to extend acceptance for multiplicity measurement, SPD first layer η acceptanceis much wider |η | < 1.75. Position and angle resolution is written in Table2.3.
Figure2.12 shows primary vertex resolution in Pb-Pb collisions at 2.76 TeV as a functionof half of the event tracklets multiplicity. Obtained vertex resolution is fitted with the equation
spotted in the figure and extrapolated to most central collisions (0-5%) corresponding to orange
region[44].

47Table 2.3: Performance of ITS (position and two track resolution for azimuthal and beam directions).
Parameter
Spatial precision rϕSpatial precision z
Two track resolution rϕ
Two track resolution z

Unit Silicon Pixel Detector
µm
12
µm
70
µm
100
µm
600

Figure 2.8: Front view of SPD layers layout. It
is a modular structure with 10 sectors made oflight carbon-fibre in φ . 6 staves (2 ladders and 4
ladders) are fixed on each carbon-fibre sectors.
[41]

Silicon Drift Detector
38
28
200
600

Silicon Strip Detector
20
830
300
2400

Figure 2.9: A cartoon illustrating of how tomeasure hit position of charged particles with
SDD. Electrons created by charged particles
drift in a silicon and signals are read in n type
substrate48

Figure 2.10: SDD (third layer) has 14 ladders
and 6 detectors in each ladders for 3rd layer and22 ladders and 8 detectors for 4th layer[42]Figure 2.11: Side view of SSD. SSD has 34 and38 ladders for layer5 and layer6, respectively.
Each ladder in layer 5 and 6 is made of 38 and
26 SSD modules, respectively [43].

√Figure 2.12: Primary vertex resolution in Pb-Pb collisions at sNN = 2.76TeV as a function ofhalf of the tracklets multiplicity of the event. In this figure, the resolution of X and Z coordinate
primary vertex resolution are shown. The resolution is obtained dividing the tracks of the eventin two random samples.[44]492.2.4 Time Of Flight
ALICE Time Of Flight (TOF) detector is made of Multigap Resistive Plate Chamber (MRPC).
Measuring time of flight of the emitting particle is important to identify particle species by thedifference of its mass (Although PID can be applied with energy loss in TPC, π /K and K/pseparation could not reach over 1 GeV/c and 2 GeV/c, respectively). The requirement for TOF
is 3σ separation of π /K and K/p for momenta up to 2.5 GeV/c and 4 GeV/c, respectively.
The time resolution of TOF should be better than 100ps to fulfill it[45]. Scintillates and phototubes are the representative of TOF detector with good time resolution and used in high energy
physics frequently. But the cost we envisage is prohibitive to cover the large acceptance with
scintillator/phototube type detector. Thus MRPC is selected in ALICE to offer the excellent
time resolution and ≈ 176m2 coverage driving down producing cost. ALICE TOF is of 1593
double-stack MRPC covering full azimuthal angle and |η | < 0.9 acceptance. It is a modular
structure called Super Module (SM) with 18 sectors in azimuthal angle and each of SM are
divided into 5 modules (a central, 2 intermediate and 2 outer modules). 19 strips are installed
in intermediate and outer modules. Central module is of 15 strips.
Figure 2.13 shows the schematic of a double-stack MRPC module. MRPC is a stuck of 4
highly resistive (≈ 1013 Ωcm) soda-lime glass plates (400µ m thick). These internal glasses are
separated with fishing line to hold the width of gas gap to be 250µ m. External glasses (550µ m
thick glass plate) are on the either side of internal glasses. High voltage are applied to speciallydeveloped acrylic paint loaded with metal oxides on the external glasses. Non-flammable gas
are used in ALICE and it’s mixture of 97% C2 H2 F4 and 3% SF6 Applied electric filed is greaterthan 100kV/cm2 on the each gas gaps. Charged particles going through gas gaps makes theavalanche and image charge are readout by 2.5 × 3.5 cm2 size cathode pick up pads (48×2
pads per strip).
The performance of TOF detector is shown in figure2.14. Time distribution with respect tostart time defined with scintillator is plotted in left figure. Data is fitted with Gauss function andtime resolution is 50.8ps. Right figure represents TOF efficiency, time resolution and the ratioof streamer as a function of applied voltage between 5 gas gaps. Efficiency becomes plateau
from ∼11kV to reach 99.9%. time resolution doesn’t have significant applied high voltage
dependence and the value is 50ps.

50

Figure2.15 is distribution of difference between measured and expected time of flight for
charged pions measured in Pb-Pb 2.76TeV collisions. It indicates TOF PID resolution. The
width of this distribution contains time resolution of start counter. Obtained overall time resolution is 86 ps.

Figure 2.13: Construction details of a double-stack MRPC modules. 4 resistive inner glasses
are inserted between external glass plate. Width of 5 gas Gaps are 250 µ m, having fishing lines
between the glass. High voltage are applied to the specially developed acrylic paint loaded on
the external glass.[46]51Figure 2.14: (Left) Time distribution of MRPC with respect to scintillator start time after theslewing calibration. Then 50.8ps time resolution can be obtained Gaussian fitting. (Right)
Efficiency, time resolution and the ratio of streamer as a function of applied voltage across 5
gas gaps. Typical efficiency and time resolution flat as a function of applied high voltage ismore than 11kV with 99.9% efficiency before the streamer is getting large.[47]Figure 2.15: Distribution of the difference between measured and expected arrival time on TOF
for selected pions (p:0.95-1.05GeV/c) in Pb-Pb collisions at 2.76 TeV. The width of the signalof charged pions at 1.5GeV/c. Red line shows the Gauss function to data.[48]522.2.5 TZERO
TZERO (T0) detector is of two arrays detector composed by 24 Cherenkov counters depicted
in figure2.16. Each cherenkov counters are of quartz + fine mesh photomultiplier. FEU − 187from Russian firm Electron is selected for photomultiplier as good timing resolution in magnetic
field of 0.5T and large radiation dose (up to 500krad). Size of quartz radiator is 30mm diameterand 30mm long. charged particles passing through quartz radiator and if the speed of charged
particles is greater than that of light in the medium, cherenkov light is emitted. Cherenkov lightis amplified with photomultiplier. T0 detector provide multiplicity by measured number ofphotons with good time resolution. T0 detector two arrays (T0C and T0A) are on the opposite
side of interaction point along with the beam axis. Each arrays are located at 0.7m and 3.6m
from the IP for T0C (backward rapidity) and T0A (forward rapidity), respectively. So two arraysof T0 detector covers asymmetric rapidity coverage, T0A covers full azimuthal angle and 4.5 <η 5.0 and T0C covers backward rapidity -3.3 < η < -2.9. We can estimate approximate vertex
position with the difference of signal arrival time between T0Aside and T0Cside.
Main roles of T0 detector are following.
• To provide the main signal for ALICE L0 trigger• To give the start time for the Time Of Flight (TOF) detectors with good time resolution• To supply the early "wake up" trigger for Transition Radiation Detector (TRD)
In order to fulfill the required performance, T0 must have 50ps time resolution (it indicates±1.5m vertex position resolution). Total dead time should be below the 25 ns.
Figure 2.17 shows the time of flight distribution obtained in the 2004 test run. Test beam
was performed in CERN PS with 6GeV/c negative pion and kaon beams. Both start and stoptime are determined with T0 detector. Obtained Time resolution of T0 detector is 28 ps withfollowing equations (Eq. 2.4 and 2.4) where σT OF is sigma of Time of flight distribution andσdet denotes time resolution of a detector.

FWHM
,
2.35
σTOF
σdet = √ .
2

σT OF ≈

53

(2.3)
(2.4)

Figure 2.16: T0 detector C-side has 12
cherenkov counters made of fused cherenkov
radiator (quartz) and photomultiplier located on
the opposite side of IP.[47]Figure 2.17: Time of flight distribution obtained in the 2004 test run with 6 GeV/c particles. Both start and stop time is measured with
T0 (Cherenkov radiator+PMT) modules. The
FWHM for this distribution is 94 ps which corresponds to 28 ps R.M.S. [49].

54

2.2.6 VZERO
VZERO (V0) detector is of two arrays scintillation counters installed on the opposite side of IP.
Each arrays are called V0 Aside (V0A) and V0 Cside (V0C). V0 A side is located at 340cm
from IP and covers forward rapidity (2.8 < η < 5.1). V0 C side is installed at the distance of
90cm from IP and covers backward rapidity (-3.7 < η < -1.7). Each array is composed of 32
segment scintillation detector distributed in 4 rings (Figure2.18). Each rings have 8 segments
and 0.5 and 0.6 unit of pseudo-rapidity coverage for Aside and Cside, respectively. One segmentof VZERO detector are made of photomultiplier tube (PMT) + BC404 plastic scintillator from
Bicron with 2.5 and 2.0cm for V0A and V0C, respectively. Light from MIPs in scintillators
are collected with WLS (WaveLength Shifter) fibre and transferred to fine mesh PMT R594670 from Hamamatsu which can be operated in 0.5T magnetic field. The different couplings ofscintillator and WLS fibre is applied for V0A and V0C due to the limited space at C side (shownin Figure2.18, 2.19).
Number of charged particles traveling the each segment are measured with MIPs (Minimum
Ionizing Particles). The difference of arrival time between V0A and V0C provide approximate
vertex position.
V0 detector has several function written in the following.
• To provide Minimum Bias trigger for the central detectors in both pp and Pb-Pb collisions• To give the centrality with the multiplicity by measuring MIPs in V0• To reject p-gas events, caused by beams interacting with the residual gas in the beam pipe,in Minimum Bias pp collision• Determination of Event plane with large rapidity coverage
In central Pb-Pb collisions total number of MIPs reaches 4000 (8000) for V0A (V0C) whensecondary particles are included. Dynamic range of each segment in V0 is required to be atleast 1-500 and 1-1000 for A and C side, respectively.
In Pb-Pb collisions, trigger efficiency depends on centrality. Except for very peripheral
collisions, trigger efficiency is almost 100% and Pb-gas and Pb-halo collisions can be removed
with V0 arrival time. Time resolution is about 450 ps and 350 ps for V0A and V0C.
55

Figure 2.18: V0 detector Aside consists of 32
scintillator (4 rings and 8 segments of 45 degrees in φ ) and WLS fibers. For V0A detector,
WLS fibres are spaced by 1cm and each segments are divided with "magtile" construction
method. Signals are collected with WLS fibres
and sent to connected PMT. [47]Figure 2.19: Schematic figure of coupling of
the scintillator and WLS fibre for V0C. WLS
fibre is embedded along the two radial edges ofthe scintillator piece. [49].

56

2.2.7 Forward Multiplicity Detector
Forward Multiplicity Detector (FMD) is silicon strip detector to measure charged particles.
FMD is of 3 modules (FMD1, FMD2 and FMD3). Figure2.20 represents the location of overall
FMD detector and ITS. FMD3 is located on the right side of ITS at the distance of 320cm. On
the left side of ITS, two grey rings indicate FMD2 module located at 75.2 (83.4) cm from IP.
Distantly located from ITS, FMD3 is located at 62.8 (75.2) cm from IP. FMD2, 3 is composed of
two rings (inner and outer) geometry similar to PHOBOS experiment. FMD1, FMD2 inner and
FMD3 inner rings consist of 40 segments silicon strip detector. each silicon sensor is subdivided
into two azimuthal sectors and 512 strips. FMD2 outer and FMD3 outer rings are composed of
20 segments silicon strip detector. each silicon sensor is subdivided into two azimuthal sectors
and 216 strips.
Location, segmentation and acceptance of FMD detector is summarized in table2.4.

Table 2.4: Table indicates nominal distance in z from the IP to the detector plane, pseudorapidity coverage of each detector modules, number of azimuthal sectors distributed in eachring and number of read out detector strip
Modulesz (cm)
η coverage
Azimuthal sectors Radial strips
FMD1
320.0 3.68 < η < 5.03
12512
FMD2 inner 83.4
2.28 < η < 3.68
20
512
40
256
FMD2 outer 75.2
1.70 < η < 2.29
FMD3 inner -75.2 -2.29 < η < -1.70
20
512
40
256
FMD3 outer -62.8 -3.40 < η < -2.01

FMD has several function written in the following. Especially 20 (40) segmentations inazimuthal angle allow us the precise measurement of event plane with excellent event plane
resolution, in particular for higher order event plane resolution.
• To provide charged particle multiplicity distribution in large rapidity acceptance (-3.4<η <1.7, 1.7<η <5.0).
• To give precise determination of higher order event planes with 20 (40)segmentations inazimuthal angle

57

Figure 2.20: Location of overall FMD detector and ITS. Orange point denotes IP and ITS
depicted yellow and green region surrounds the IP. On the right side of ITS, FMD3 module(two rings) is shown in red. Two rings on the left side of ITS indicate FMD2 module. Distantly
located from ITS, FMD2 and FMD3, FMD1 module is represented as grey octagon. [49]Figure 2.21: Assembly of FMD inner ring and FMD1 from 10 modules (left) and FMD outerring from 20 modules (right). Each modules are subdivided into two segments in azimuthal
angle. [49]58Chapter 3
Data Analysis
In this chapter, we introduce data set and analysis method which is event characterization, track
selection, pair selection, particle identification of charged hadrons. Also fitting and correctionfor HBT analysis are described here.

3.1 Event characterization
The following event characterization method is written in this section.
• Event trigger and data set• Centrality determination• Event selection and track selection• Event plane determination

3.1.1 Event trigger and data set
Event trigger is important to identify the beam-beam collisions in nuclear and particle physics
using accelerator.
Minimum bias trigger is defined with V0 multiplicity that at least one signal can be foundon the both V0A and V0C segment in coincidence with at least one beam at the ALICE IP. Multiplicity triggers (Central and Semi-central triggers) are also defined with V0 detector. V0A andV0C charge are integrated and compared with the defined threshold programmed in FPGA[49].
Central and Semi-central trigger are corresponding 0-10% and 10-50% online trigger, respectively.
59

In this analysis, 30 million events in√
sNN = 2.76 TeV Pb-Pb collisions collected with the

ALICE detector in 2011 using Minimum Bias, Semi-central and Central trigger. Beam-gas
and beam-halo collisions are the machine induced background of beam-beam collisions (about
10% of all triggered data[50]). These events can be rejected with offline event selection using
particles arrival time on V0A and V0C. Figure 3.1 shows the geometry and time alignment of
V0A and V0C. In beam-beam collisions, particles should be measured in 11ns and 3ns after thecollision at V0A and V0C, respectively.

Figure 3.1: Geometry and time alignment of V0A, V0C and hadron absorber for muon spectrometer. Cyan trapezoid denotes hadron absorber. Interaction Point is shown in black fullcircle. [49]3.1.2 Centrality determination
Initial volume and source shape are important probe for studying the hot dense matter andsignificantly correlated with impact parameter. But impact parameter can not be directly measured in experiment. So the concept of "centrality" which is defined as the overlap percentile
of initial source is incorporated Initial source volume becomes larger when centrality becomes
peripheral to central in participant-spectator model and the volume of the initial overlap region
can be expressed via the number of participating nucleons(N part ) and the number of binary
collisions(Ncoll ).
In this analysis, centrality is estimated via Glauber fitting with VZERO multiplicity distribution based on Negative Binomial Distribution(NBD)[50]. In this model, all emitting particles
are generated with a given N part and Ncoll value and Both N part and Ncoll are defined as theconcept of "ancestors" expressed by Nancestors = f N part + (1 − f )Ncoll . This is two component
models that nucleus-nucleus collisions are decomposed into soft and hard interactions, where60soft and hard interactions are proportional to N part and Ncoll , respectively. Each particles areproduced based on negative binomial distribution. The probability of measuring n hists in eachancestor is expressed with the following equation.

Pµ ,k (n) =(µ /k)nΓ(n + k)Γ(n + 1)Γ(k) (µ /k + 1)n+k(3.1)

where µ is the mean multiplicity in each ancestor and k denotes the width. Figure 3.2 shows
the VZERO multiplicity distribution with 3-out-of-3 trigger which is defined by signals in V0A
and V0C and at least two hits in the outer layer of SPD. Z-vertex cut |Vz |< 10cm is applied. The
distribution is fitted with NBD Glauber shown in red line and centrality can be extracted withthis model.
In ALICE, centrality can be extracted various detectors, V0A+C, V0A, V0C, SPD, TPC and
ZDC. We can estimate the centrality resolution via 6 different centrality value with them. The
average centrality value is calculated for each event.

⟨c⟩ =

∑N
i=0 ciN(3.2)

where ci is the centrality via each estimator and i denotes each estimator running over all
detectors (N = 6). Here we define the difference between average and each centrality value
via each estimator ∆i = ci - ⟨c⟩. Then average value is iteratively calculated with following
equation, replacing ⟨c⟩ until ⟨c⟩ is converged.
2∑N
i=0 ci /∆i⟨c⟩ = N∑i=0 1/∆2i(3.3)

Finally centrality resolution is obtained with the R.M.S. of ∆i distribution via each detector.
Figure 3.3 shows the centrality resolution of V0A+V0C, SPD(|η |<1.4), TPC(|η |<0.8), V0A,
V0C, ZDC-ZEM. The best centrality resolution is achieved with V0A+C combined estimator.
Centrality resolution via V0A and V0C combined is 0.5 to 2 % from most central to peripheral
collisions.

61

Figure 3.2: Distribution of sum of V0A+C amplitude. Data is fitted with NBD Glauber fit
shown in red line. The centrality classes obtained by fitting are indicated with grey and white
region. Inserted figure denotes a zoom of the most peripheral events. [50]Figure 3.3: Centrality resolution with V0A, V0C, V0A+C, TPC(|η |<0.8), SPD(|η |<1.4) as afunction of centrality percentile. resolution is calculated with 6 all estimation detectors. [50]623.1.3 Event selection
Centrality is estimated with the amplitude of VZERO A and C side combined which has the best
centrality resolution[50] and centrality 0-50% events are analyzed in this analysis. Minimum
Bias trigger, Semi-central and Central triggers are required. Primary vertex is determined with
SPD. Vertex position along the beam axis(zvtx ) is important because the detector acceptancelargely changes if z vertex position shifts from the nominal interaction point. |zvtx | < 8.0cm cut
is applied.

3.1.4 Track selection
In this analysis, charged tracks are reconstructed with TPC, constrained with primary vertex via
SPD. TPC clusters are fitted with Kalman filter algorithm[51]. Decay particles are also included
and they are the background of this analysis. Thus the appropriate track selection is necessary to
be applied. Fitting quality for track finding is provide with χ 2 per number of degree of freedom.
There are 159 pad rows in TPC readout chamber and at most 159 TPC clusters are used for trackreconstruction. In this analysis, χ 2 /NDF is required to be below 4.0 and tracks which is reconstructed with at least 80 clusters are used. This selection allows us to remove the tracks which is
not originated from primary vertex. In addition to this track selection, kink tracks are rejected.
Kink structure is the feature of secondary particles. Moreover the contamination of daughter
tracks from weak decay can be reduced with Distance of Closest Approach(DCA). tracks are
extrapolated into primary vertex and then minimum distance between track and primary vertex
is defined as DCA. DCA is divided into DCAxy (DCA in transverse direction) and DCAz (DCA
in longitudinal direction). Absolute value of DCAxy and DCAz is constrained to be below 2.4cm
and 3.2cm, respectively. Also two dimensional DCA cut DCAxy 2 /(2.4)2 + DCAz 2 (3.2)2 < 1.0
is applied. For HBT study, the selection of 0.15 < pT < 1.5 GeV/c is applied (HBT correlationin high pT particles is too small). Tracks in |η | < 0.8 were used for both HBT and flow analysis.

3.2 Azimuthal anisotropy and Event plane
Azimuthal anisotropy of emitting particles and the method to extract event plane are describedin this section.

63

3.2.1 Azimuthal anisotropy of emitting particles
The modulation of azimuthal angle of emitting particles are extracted with Fourier-expansionof the distribution in azimuthal angle ϕ .

x0 1 ∞r (ϕ ) =+ ∑ (xn cos (nϕ ) + yn sin (nϕ ))2π π n=1(3.4)

d ϕ r (ϕ ) cos (nϕ )(3.5)

d ϕ r (ϕ ) sin (nϕ )

(3.6)

xn =

∫ 2π
0

∫ 2π

yn =

0

where r and ϕ denote the azimuthal distribution and azimuthal angle of emitting particles.
Harmonics of Fourier series is described as n. xn and yn are the nth oscillation components ofr (ϕ ) for x and y direction and calculated with integrated r (ϕ ) over all particles. In experiments,
number emitting particles are finite. So integral of xn and yn are expressed as sum:xn =∑ ri (ϕi) cos (nϕi)(3.7)∑ ri (ϕi) sin (nϕi)

(3.8)

i

yn =

i

where i is the index of emitting particles and each cosine and sine of azimuthal angles ϕiweighted with azimuthal distribution ri are summed over all particles. Here we define n-th order"Event plane"(Ψrn ) where each even plane angle is corresponding to the short axis of n-th order
polygon(n = 2 : elliptic shape and n = 3 : triangular shape). Azimuthal distribution r (ϕ ) isrewritten with respect to event planes:r (ϕ ) =)x0 1 ∞ ( ′+ ∑ xn cos (n [ϕ − Ψrn ]) + y′n sin (n [ϕ − Ψrn ])2π π n=1(3.9)

In symmetric nucleus-nucleus collisions, sine term of Fourier series should be zero. Therefore
sine term is vanished :64

)x0 1 ∞ ( ′+ ∑ xn cos (n [ϕ − Ψrn ])2π π n=1
(
))
∞ ( ′
x0
xn
cos (n [ϕ − Ψrn ])
=
1+2 ∑
2π
x
0
n=1
(
)
∞
x0
1 + 2 ∑ (vn cos (n [ϕ − Ψrn ]))
=
2π
n=1

r (ϕ ) =(3.10)(3.11)(3.12)

where the strength of azimuthal anisotropy can be expressed with vn = ⟨cos (n [ϕ − Ψrn ])⟩.
In this equation, ⟨ ⟩ denotes the average of running over all emitting particles. But "true" event
plane (Ψrn ) cannot be directly measured in experiment. Measured event plane (Ψn ) is defined
based on the assumption of azimuthal anisotropy with respect to event plane.
√

vobs
n
Ψn

xn2 + y2n
x0
( ) (
)
1 −1 yn2π=tan
0 ≤ Ψn ≤nxnn=(3.13)(3.14)

where vobs
n is observed azimuthal anisotropy with respect to measured event plane (Ψn ).
Azimuthal distribution of emitting particles(Eq. 3.5) can be expressed with vobsn and Ψn :
(
)
∞
x0
xn
xn
r (ϕ ) =1 + 2 ∑ cos (nϕ ) + sin (nϕ )(3.15)
2π
x
x
0
0
n=1
)
(
∞
x0
x
n
=
1 + 2 ∑ vobssin (nϕ ) sin (nΨn )(3.16)
n cos (nϕ ) cos (nΨn ) +
2π
x
0
n=1
(
)
∞
x0
=
1 + 2 ∑ vobs
(3.17)
n cos (n [ϕ − Ψn ])
2π
n=1
r
vobs
n are rewritten with true event plane Ψn , measured event plane Ψn and azimuthal anisotropy

vn :

65

vobs
= ⟨cos (n [ϕ − Ψn ])⟩n(3.18)= ⟨cos {n [ϕ − Ψrn ] − n [Ψn − Ψrn ]}⟩(3.19)= ⟨cos (n [ϕ − Ψrn ]) cos (n [Ψn − Ψrn ]) + sin (n [ϕ − Ψrn ]) sin (n [Ψn − Ψrn ])⟩ (3.20)= ⟨cos (n [ϕ − Ψrn ]) cos (n [Ψn − Ψrn ])(3.21)= vn ⟨cos (n [Ψn − Ψrn ])(3.22)= vn · Res {Ψn }(3.23)

where the sine terms are vanished due to the symmetric distribution with respect to event
planes. Here we define cosine of the difference between true and observed event plane as"resolution" of event plane (Res {Ψn }).

3.2.2 Event plane determination
In experiment, event plane can be extracted with flow vector(Q vector) expressed byNM =Qx,n =
Qy,n

∑

wi
i=0
∑N
i cos (nϕi )i=0 w√

MN∑i=0 wi sin (nϕi )√=M(3.24)(3.25)(3.26)

where wi is weight. In this analysis, Event plane is determined via 3 detectors, FMD,VZERO and TPC.
For TPC event plane, weight wi is pT up to pT =2.0GeV/c and wi = 2.0 for the other particles(pT >2.0). N denotes the number of charged particles reconstructed in TPC at one event and ϕiis the azimuthal angle of each particle.
For VZERO and FMD event plane, wi denotes the multiplicity in each PMT and silicon
strip channel and N is the total number of segment. Figure 3.4 shows the η -ϕ 2D distribution
in FMD. The acceptance of FMD A + C side combined is -3.4<η <-1.7 and 1.7<η <5.1 andazimuthal angle divided into 20 silicon strip channel. Z axis denotes the multiplicity in each
strip channel. azimuthal angle ϕi for FMD event plane is given by the center position of each66

strip channel. ϕi for VZERO event plane is represented as ϕi = π /4 × (0.5 + nseg %8) where nseg
indicates the index of each segment(0-64 channel).

Figure 3.4: η v.s. ϕ 2D distribution measured via FMD. Z axis denotes multiplicity of eachsilicon strip channel.

Using nth order Q vector in Eq. 3.26, 3.26, event plane can be expressed by:()
1 −1 Qx,nΨn = tan
n
Qy,n

67

(3.27)

3.2.3 Event plane calibration
In nucleus-nucleus collisions, reaction planes and event planes are randomly determined because the collision geometry cannot be controlled by accelerator technique. Event plane distribution should be flat. But measured event plane distribution is in fact not flat. it’s because event
plane determination detectors have dead and non-uniform gain channels. Also non-central beam
position can cause the non flat event planes. These effects are corrected with 2 step event planecalibrations, re-centering and flattening. Each calibration parameters are extracted in each run
based on the assumption of stability during one run.
Re-centering calibration
Mean of Qx,n , Qy,n vector distribution should be 0 and width of q vector in each x, y direction
should have same value if event plane is flat. Re-centering calibration is the correction of mean
and R.M.S. value of Qx,n , Qy,n given as

Qx,n − ⟨Qx,n ⟩σxQy,n − ⟨Qy,n ⟩=σy()
1 −1 Qcorr
y,n
=
tan
n
Qcorr
x,n

Qcorr
=
x,n

(3.28)

Qcorr
y,n(3.29)

Ψrecn(3.30)

where ⟨Qx,n ⟩, ⟨Qy,n ⟩ denote the average of Qx,n , Qy,n and σx , σy represent the width of Qx,n ,Qy,n in a certain run. Figure 3.5 shows 2nd, 3rd and 4th order harmonic Qx vector before/after
re-centering. Red, green solid line are raw Qx,n , Qy,n distribution and blue solid, black dashedline denote re-centering Qx,n , Qy,n distribution. After re-centering calibration, The mean valueof Q vector distribution is zero and width is perfectly same.
In Figure 3.6, Blue and black line show the raw event plane and event plane after recentering calibration. Event planes are largely flattened with this correction because the nonuniform detector channel and non-central beam position can be corrected with this calibration.
Residual non-uniform components are corrected with flattening correction.

68

Figure 3.5: 2nd, 3rd and 4th harmonic q vector x distribution which is determined via FMD A
side + C side combined in centrality 0-50%. Red line shows uncorrected distribution. Blue line
indicates event plane distribution with recentering calibration.

Flattening calibration
Residual non-uniform components especially higher harmonics oscillations are corrected with
Flattening calibration defined bynΨnf lat = nΨrec
n + n∆Ψn(3.31)

Nn∆Ψn =rec∑ Ak cos (knΨrecn ) + Bk sin (knΨn )(3.32)

k=1

where Ak , Bk represent the Fourier coefficients of event plane distribution. In flattening
f lat

calibration, average cosine and sine of nΨn

are corrected to be zero. This calibration forces

event planes to be flat distribution with Fourier expansion. Ak and Bk can be determined as

2
Ak = − ⟨sin (nkΨrec
n )⟩
k
2
Bk =
⟨cos (nkΨrecn )⟩k(3.33)(3.34)In this analysis, flattening calibration was calculated up to 8th order Fourier coefficients.
Eventually, flat event plane distribution can be obtained in orange line shown in Figure 3.6.
Event plane resolution
Analytically, nth harmonic event plane resolution can be expressed as the following equation :69

Figure 3.6: 2nd, 3rd and 4th harmonic event plane distribution which is determined via FMD
A side + C side combined in centrality 0-50%. Red line shows uncorrected distribution. Orange line indicates event plane distribution with recentering calibration. Blue line represents
recentering+flattening calibrated event plane distribution.

√
(
) [
( 2)
( 2 )]πχm2
χm
χm⟨cos [km (Ψm − Ψr )]⟩ = √ χm exp −× I(k−1)/2+ I(k+1)/2(3.35)
4
4
4
2 2√
where ⟨cos [km (Ψm − Ψr )]⟩ represents event plane resolution and χm ≡ vm 2N. vm is theFourier coefficient of azimuthal anisotropy. N is the number of particles used for event plane
calculation Iν is the modified Bessel function of ν [52]. This equation represents event plane
resolution is expressed with multiplicity and strength of azimuthal anisotropy. The more multiplicity is used to determine event plane and strength of anisotropic flow itself is larger, the better
event plane resolution can be obtained.
The event plane correlation between detector A and B can be expressed as:

( [
])
( [
]
)
⟨cos n Ψn,A − Ψn,B ⟩ = ⟨cos n Ψn,A − Ψn,r − n [Ψn,B − Ψn,r ] ⟩( [])= ⟨cos n Ψn,A − Ψn,r ⟩⟨cos (n [Ψn,B − Ψn,r ])⟩{}= Res Ψn,A Res {Ψn,B }(3.36)(3.37)(3.38)

When two detectors A and B cover symmetric rapidity acceptance and have same multiplic{}{}
ity, event plane resolution Res Ψn,A is equal to Res Ψn,A if we assume same vn is measured
70

at detector A and B. Because event plane resolution is determined with vn value and multiplicity
N. Therefore event plane resolution can be obtained by two sub-event method given as :
√
{
}
( [
])
Res Ψn,A = Res {Ψn,B } = ⟨cos n Ψn,A − Ψn,B ⟩(3.39)

This method is used to calculate TPC event plane resolution in this analysis.
However ALICE forward detectors basically have "asymmetric" rapidity coverage. For
asymmetric rapidity detectors, event plane resolution is usually calculated with "3-sub event
method". Here we think 3 different rapidity detectors a, b and c. Event plane correlations canbe expressed by :
( [
])
{
}
⟨cos n Ψn,a − Ψn,b ⟩ = Res {Ψn,a } Res Ψn,b
( [
])
{
}
⟨cos n Ψn,b − Ψn,c ⟩ = Res Ψn,b Res {Ψn,c }⟨cos (n [Ψn,c − Ψn,a ])⟩ = Res {Ψn,c } Res {Ψn,a }Using Eq. 3.41, 3.42 and 3.42, event plane can be given as :√( [])⟨cos n Ψn,a − Ψn,b ⟩⟨cos (n [Ψn,c − Ψn,a ])⟩( [])Res {Ψn,a } =⟨cos n Ψn,b − Ψn,c ⟩(3.40)(3.41)(3.42)(3.43)In the same way, event plane resolution of b and c detectors can be obtained with Eq. 3.41, 3.42
and 3.42.
In this analysis, FMD A side, FMD C side, FMD A+C combined, V0 A side, V0 C side andV0 A+C combined event plane is calculated with 3-sub event method.
Figure 3.7 is 2nd, 3rd and 4th harmonic event plane resolution via VZERO, FMD and TPC.
Blue and green markers represent event plane via VZERO and FMD. Black markers denote
TPC event plane resolution. TPC rapidity is divided into 4 regions, -1.0 < η < -0.5, -0.5 < η <0.0, 0.0 < η < 0.5 and 0.5 < η < 1.0. FMD(V0) A+C combined resolution can be extracted with
TPC event plane resolution and event plane correlation between TPC and FMD A+C combined
by :
( [
])
{
} ⟨cos n Ψn,T PC − Ψn,FMD A+C ⟩{}Res Ψn,FMD A+C =Res Ψn,T PC(3.44)In this analysis, main event plane determination detector is FMD. The best resolution value
is 0.87 at mid central collisions for 2nd harmonic event plane. Initial collision geometry and71multiplicity distributions determine this structure. Third and fourth harmonic event plane has
different behavior compared to 2nd order. The best resolution value for 3rd harmonic event
plane is 0.5 at central collisions and resolution value becomes smaller from central to peripheral. Compared to VZERO event plane resolution, FMD performs better event plane resolution
especially in higher harmonic orders owing to larger rapidity acceptance. This excellent resolution helps us to understand the detail structure in higher order harmonic anisotropy.

Figure 3.7: 2nd, 3rd and 4th harmonic event plane resolution which is determined via
TPC(|η |<1.0), V0 A side, V0 C side, V0 A+C combined, FMD A side, FMD C side, FMD
A+C combined , TPC(0.5 < η < 1.0) and TPC(-1.0 < η < -0.5) as a function of centrality. Blueline indicates event plane distribution with recentering calibration. Since number of segmentsin V0 is not sufficient, 4 th harmonic event plane resolution via V0 is not shown.

72

3.2.4 Measurements of azimuthal anisotropy
Charged hadron vn can be extracted with vobsn and event plane resolution Res {Ψn }, using Eq.
3.23 :

vtrue
=
n

vobs
n
Res {Ψn }(3.45)

Figure 3.8 shows higher harmonic azimuthal anisotropy vn as a function of collision centrality. Event planes are determined via 4 different detectors, V0 A+C combined, FMD A+C
combined, TPC C2 and TPC A2. Here we define TPC C2 and TPC A2 as TPC(-1.0 < η < -0.5)
and TPC(0.5 < η < 1.0), respectively. pT is integrated from 0.2 to 2.0 GeV/c.
Rapidity gap between event plane and vn measurement has to be at least 0.9, Since, forsmaller rapidity gap, non flow effect such as resonance decay particles and jet enhance or suppress the measured vn value.
Higher order anisotropy vn via event plane method is compared with previous ALICE results
for the purpose of consistency check[13]. Charged hadron vn measured via two particle correlation method is also depicted as black marker in Figure 3.8. 2nd, 3rd and 4th order anisotropy
via two different methods are fully consistent within the systematic uncertainty.
pT dependence of higher harmonic anisotropy vn for the 6 centrality classes is presented in
Figure 3.9. Event plane is determined with 3 different detectors and the difference between
them is so small. Previous ALICE results measured via event plane method are also shown in
Figure 3.9 to check the consistency. Previous ALICE results and results in this analysis are
consistent within the systematic uncertainty for all centralities.

73

Figure 3.8: 2nd, 3rd and 4th harmonic anisotropy vn as a function of centrality. vn is measured
via Event plane method and compared with previous ALICE results(two particle correlation).
Event plane is determined with 4 different detectors.

74

Figure 3.9: 2nd, 3rd and 4th harmonic anisotropy vn as a function of pT for the 6 centrality
classes, measured via event plane method. Black markers are previous ALICE results via event
plane method.
75

3.3 Event Shape Engineering
Event Shape Engineering (ESE) is method to select the event-by-event flow fluctuation by themagnitude of flow vector [15]. In this section, analysis method of ESE technique is described.

3.3.1 Event Shape Engineering
nth order flow vector is decomposed into x and y axis and represented as Qx,n and Qy,n . These
vectors are measured "event by event" and the length of them is sensitive to "event by event
flow amplitude".
Here we describe the definition of Q vector again. Using Qx,n and Qy,n flow vectors, themagnitude of flow vector is given as :

M =

∑ wi

(3.46)

i=0∑i=0 wi cos (nϕi )√M∑i=0 wi sin (nϕi )
√
=
M
√
=
Q2x,n + Q2y,n

Qx,n =(3.47)

Qy,n(3.48)

qn(3.49)

qn indicates the length of nth harmonic flow vector. Figure 3.10 shows the length of 2nd and3rd harmonic flow vector distributions in Pb-Pb 2.76 TeV collisions for 0-10% centrality. flow
vector is determined with FMD A+C combined.
Left and right figure is 2nd and 3rd harmonic flow vector distribution, respectively. Meanvalue of q2 is larger than that of q3 and width of q3 is slightly larger than that of q2 . Flow vector
distribution is determined by 3 components.
• Event by event flow fluctuations• Statistical fluctuations due to finite particle multiplicity used to determine flow vector• Event plane resolution
The signal of event by event flow fluctuations are smeared with other two components.
Statistical fluctuations and event plane resolution can broaden the q vector distribution.
76

One can obtain the event by event flow amplitude by dividing qN distribution into flow vector
event class. In order to determine the flow vector event class, cumulated qN distribution as afunction of centrality is measured in Figure 3.11. Cumulated qn vector distribution is projected
to Y axis for each 1% centrality. Spline fit is performed to each slice of cumulated qn vector
distribution. In this method, for each qn vector value can be evaluated from the corresponding
percentile.
For each 20 % of q2 and q3 classes are determined with FMD A+C combined q-vector inthis thesis. qn class 0 % means smallest vn events and 100% denotes largest vn events. Figure
3.13 shows q2 (left) and q3 (right) distribution for each q-vector classes in centrality 10-20%.

Figure 3.10: The magnitude of q2 (left) and q3 (right) flow vector distribution in centrality 010%. qn is determined with FMD A+C combined.

77

Figure 3.11: Cumulated q2 (left) and q3 (right) distribution as a function of centrality. flow vector
qn is determined via FMD A+C combined. Contour(5%) maps are plotted simultaneously.

Figure 3.12: Cumulated q2 (left) and q3 as a function of q2 (left) and q3 distribution in centrality 0-1%(red) and 30-31%(blue). Spline fitting is performed to each slice of qn distribution.
Cumulated q-vector distribution is rebeined (merged 100 bins in one) for visibility.

78

Figure 3.13: q2 (left) and q3 distribution for each 20% q2 and q3 in centrality 10-20%. Both q2
and q3 are determined with FMD A+C combined.

79

3.3.2 Event plane resolution with ESE
nth order event plane resolution is determined with amplitude of vn and multiplicity used todetermine the event plane. ESE is the selection for event by event flow amplitude. Therefore amplitude of vn changes in a same multiplicity(centrality) events and one can assume thatlarger(smaller) qn selection makes event plane resolution better(worse).

Figure 3.14: 2nd(left) and 3rd(right) harmonic event plane resolution as a function of centrality
for each 20% q2 and q3 event classes. q2 and q3 selection and event plane are determinedwith FMD A+C combined. Inclusive(without q2 and q3 cut ) event plane resolutions are also
depicted as open marker. Event plane resolution is extracted with 3-sub event method and the
combination is FMD A+C, TPC(-1.0<η <-0.5) and TPC(0.5<η <1.0).

Figure 3.14 shows the 2nd(left) and 3rd(right) order event plane resolution as a functionof centrality for each 20% q2 and q3 event classes, respectively. q2 , q3 selection and event
planes(Ψ2 and Ψ3 ) are determined with FMD A+C combined. Inclusive(without q2 and q3cuts) event plane resolutions are also shown as open marker. Event plane resolution is extracted with 3-sub event method that the combination is FMD A+C, TPC(-1.0<η <-0.5) andTPC(0.5<η <1.0). For both 2nd and 3rd harmonic event plane resolutions are explicitly enhanced(suppressed) with larger(smaller) qn selection , compared to inclusive events. Ψ2 resolution is about 0.25 to nearly 1.0 for centrality 0-50%, while Ψ3 resolution is about 0.1 to nearly
0.73 for centrality 0-50%.

80

3.4 Particle identification(PID)
In ALICE, hadrons and leptons are identified with the combination of several detectors depending on the particle transverse momentum. Figure 3.15 shows π /K(left) and K/p(right)
separation power with the ITS, TPC, TOF and HMPID. The separation power can be evaluatedby the Gaussian width of σ for pions and kaons, respectively.

Figure 3.15: π /K(left) and K/p(right) separation power in the ITS, TPC, TOF and HMPID
as a function of transverse momentum at mid rapidity. Y axis denotes the distance between
the peaks divided by the resolution for the pion and the kaon, respectively. The left (right)
panel shows the separation of pions and kaons (kaons and protons), expressed as the distance
between the peaks divided by the resolution for the pion and the kaon, respectively. For the
TPC, an additional dashed line is depicted in a edge of rapidity coverage. The lower panels
show the range over which the different ALICE detector systems have a separation power ofmore than 2σ [55]. [54]In this analysis, low momentum charged pions (0.15 < pT (GeV/c) < 1.5) are used for HBT
analysis and π ± , K ± and p + p̄ are analyzed for azimuthal anisotropy with ESE measurements.
One can find that the identification of π ± , K ± and p + p̄ are covered with TPC and TOF in
ALICE (Figure ??). Thus TPC and TOF combined PID is applied to measure the particle
species.

3.4.1 Energy loss(dE/dx) in TPC
As we introduced in section 2.2.2, TPC is the main tracking detector in ALICE. Particle identification information can be also extracted besides tracking. PID is performed by measuring81

specific energy loss(dE/dx) in gas, charge and momentum simultaneously. The energy loss ofcharged particle in material are described with Bethe-Bloch formula which is given as :

−

dE
z2
≈ 2 ln γdx
β(3.50)

where z denotes the charge, β is the ratio of particle velocity to the speed of light and γ isthe Lorentz factor. If the particle momentum is measured in TPC, difference of particle massand charge makes the separation of dE/dx depending of its particle species.
Energy loss (dE/dx) distribution as a function of charge z × particle momentum are presented in Figure 3.16. In low momentum region, dE/dx distribution is definitely separated for
different particle species. But, in pT higher than 0.5 GeV/c, energy loss of pions and kaons isalmost converged. Therefore, in this momentum range, TOF is the main PID detector.

Figure 3.16: Energy loss (dE/dx) in TPC as a function of charge z × particle momentum in
Pb-Pb 2.76 TeV collisions. pions, electrons, kaons and protons are identified with the differenceof energy loss especially in low momentum region.

82

3.4.2 Particle identification in TOF
Time of Flight is a good probe to identify the particle species and measured with the difference
between start time ( which is mainly determined with TZERO detector in ALICE) and stop timewith TOF. Particle velocity is given by :

L
ct
p
= √
p2 + m2

β =(3.51)(3.52)

where L is the distance between start and stop counter, which can be extrapolated of reconstructed track in TPC, t represents time of flight and c denotes the speed of light in vacuum.
In addition β can be expressed with particle momentum p and mass m. In a given momentum p, velocity β is driven only from particle mass. Figure 3.17 is the distribution of TOF β
as a function of a track momentum. each species are explicitly separated due to the differenceof their mass. Combinatorial background comes from the miss-matched track of TPC-TOF inhigh multiplicity heavy ion collisions.

Figure 3.17: Distribution of β measured by TOF detector as a function of track momentum in
Pb-Pb 2.76 TeV collisions833.4.3 TPC-TOF combined PID
The separation power of π /K and K/p can be improved with the complementrarity of thedifferent detector system. In this analysis 0.15-2.0 GeV/c pions, kaons and protons are used.
Since TOF efficiency is not sufficient at low momentum(pT < 0.5 MeV/c), TPC is the main PID
detector for this pT region. At intermediate pT range, TOF performs more than 3 σ separation
power up to 3.0GeV/c for π /K and 5GeV/c for K/p, respectively.
For HBT analysis, charged pion identification was performed with Nσ which is the Gaussian
description of the response function of the detector given as :nσ (π , TPC) =nσ (π , TOF) =π
dE/dxmeasured − dE/dxexpected

σTπPC(timehit − startTime) − timeπexpected
σPID(T OF)(3.53)(3.54)

where dE/dxmeasured is measured dE/dx in TPC and energy loss expected as pion is repπ
. Expected signal is calculated with Bethe-Bloch formula taking into
resented as dE/dxexpectedaccount the η dependence and the multiplicity dependence. Distance to expected signal is divided by TPC dE/dx resolution(σ π ).
timehit denotes the hit time measured in TOF. startTime is the collision time which is measured by means of the following (listed in order of priority) :1. TZERO detector (sum of the time signal from A and C side)
2. TOF detector itself with a combinatorial algorithm based on χ 2 minimization
3. Average TOF start time for the run
TOF Expected time is measured from measured track length to reach the TOF and momentum in the pion mass hypothesis. The TOF PID resolution σPID(T OF) is expressed with quadratic
sum of intrinsic time resolution of TOF detector, time resolution of start counter and tracking
capability of ALICE.
Figure 3.18 shows nσ TPC(left) and TOF(right) for charged pions. Charged pions are identified with the following selection in Table 3.1.

84

Table 3.1: Pion selection for HBT analysis with TPC and TOF nσ
Momentum (GeV/c) TOF enable PID estimator
0.15-0.65⃝|nσ (π , TPC)| < 3.0 ∩ |nσ (π , TPC)| < 3.0
0.15-0.5×|nσ (π , TPC)| < 3.0
0.5-0.65
×|nσ (π , TPC)| < 2.0
0.65-1.5⃝|nσ (π , TPC)| < 5.0 ∩ |nσ (π , TPC)| < 3.0

Figure 3.18: TPC dE/dx Nσ of π ± as a function of particle transverse momentum shown in left
and Right figure represents TOF Nσ for pion as a function of particle momentum transverse.

For azimuthal anisotropy with ESE study, charged pion, kaon and proton identified with isBayesian approach[56]. In this analysis, the probability of π , K and p is estimated via TPC andTOF combined and required at least 90% for each species. Simultaneously TPC and TOF nσ
are required within 1.0. PID purity of pions, kaons and protons is better than 95% where pT up
to 2.0GeV/c.

85

3.5 Pair selection
3.5.1 Two Track Resolution
HBT correlation is of the interference effect between two identical particles in low momentum
range. Therefore in order to measure the HBT interferometry, precise selection of pairs in low
relative momentum is necessary. However it’s not easy to identify those pairs with tracking
detector due to the high multiplicity in heavy ion collisions. Pairs where two tracks have similar
momenta and small angular distance may have the following reconstruction effects due to thefinite two track resolution of tracking detector.
• Track merging• Track splitting
track merging
Track merging is when two track is falsely reconstructed as one track or one of them are notreconstructed shown in Figure 3.19(left). Horizontal lines denotes the TPC pad rows in read-out
chamber. Suppose, for instance, two tracks are traveling spatially close in the tracking detectors.
Due to the finite spatial resolution, two tracks can be falsely reconstructed as one track. This
result in a depletion of close pairs and can cause the suppression of correlation function.
track splitting
Track splitting is when one track is falsely reconstructed as two tracks that are spatially close.
Figure 3.19 (right) shows the cartoon illustrating of track splitting. Some of TPC clusters areshared between spatially close two tracks. These clusters, in particular, can cause track splitting
effect, since it assumes as if two tracks makes shared clusters by tracking algorithm. This resultin an enhancement of tracks with close pairs and correlation function. These tracks are also
known as "ghost track".

86

Figure 3.19: Cartoon illustrating of track merging and splitting effect. Horizontal lines denotes
the TPC read-out pad rows. Two tracks shown in red and blue dashed lines are spatially close.
Measured TPC clusters of these two tracks are shown as orange markers. Elliptic markers inyellow gradation denote the shared TPC clusters. Due to finite two track resolution, two tracks
are falsely reconstructed as one track depicted as black solid line(left), while tracks composedof shared TPC clusters(black dashed line) can be falsely reconstructed as two tracks shown inorange and blue solid line(right).

Figure 3.20: Track splitting & Track merging effect in correlation function C2 as a function of
one dimensional relative momentum(Qinv ). Blue marker indicates correlation function without
any pair cut. Red marker denotes correlation function after pair cut.

87

Track merging and splitting on correlation function
Figure 3.20 shows the track merging and splitting effect on correlation function. in ideal casewhere only Bose-Einstein enhancement exists, correlation function is expressed as Gaussian
and does not exceed two. Measured correlation function is a quit different shape from Gaussianand exceed two. These modification of correlation function can be removed by pair selection.

3.5.2 Pair Cut
In this analysis, pair selection is applied with Number of shared TPC clusters and angulardistance in ∆η and ∆ϕ ∗ .

3.5.3 Fraction of shared TPC cluster
Due to high multiplicity events in Pb-Pb collisions, reconstructed tracks shares same TPC Pad
channel in Figure 3.19. This leads to generate the ghost tracks. In order to remove this effect,
Fraction of shared TPC clusters are used. Number of shared TPC clusters means the numberof TPC pad row which is shared with other tracks shown. Fraction of shared TPC clusters arecalculated with following equation.

Fshare =NshareNhits(3.55)

where Fshare is fraction of shared TPC clusters and Nhits is the number of TPC clusters toreconstruct a certain track. Nshare is the number of shared TPC clusters to reconstruct it. Whenthe pair of the fraction of shared TPC cluster is larger than 5%, this pair is removed.

3.5.4 Angular distance in ∆η -∆ϕ ∗The other pair cut is Angular distance in ∆η and ∆ϕ ∗ . Two track resolution strongly depends onthe distance of two tracks. the relative angle ∆η and ∆ϕ in a certain radius of TPC is calculated
in consideration of the magnetic fields. ∆η is not affected by magnetic filed. ∆ϕ is extrapolated
as following equation.

∗−1∆ϕ = ϕ1 − ϕ2 + sin

(

)
(
)
−0.015 · e · Bz · R−1 −0.015 · e · Bz · R
− sin
pT 1
pT 2

88

(3.56)

where ∆ϕ ∗ is extrapolated azimuthal angle of the tracks in a certain radius R. Bz [T] is themagnetic field in z direction and e is the elementary charge. phi1 and phi2 are the azimuthal
angle of the tracks at the vertex and pT 1 and pT 2 are the transverse momentum of the tracks.
The angular distance in ∆η and ∆ϕ ∗ distribution can be calculated various TPC radii R and pairtransverse momentum kT .
In order to optimize the cut value of ∆η and ∆ϕ ∗ . ∆η v.s. ∆ϕ ∗ distribution are calculatedwith TPC radii R bin (R = 0.8, 1.0, 1.1, 1.2, 1.3, 1.4[m]) and kT bin (0.2-0.3, 0.3-0.4, 0.4-0.5,
0.5-0.6, 0.6-0.7, 0.7-1.0[GeV /c]).
Figure 3.21 illustrates the ratio of real and mixed two dimensional distribution in ∆η and∆ϕ ∗ of each kT bin. Y-axis is the extrapolated ∆ϕ ∗ at the TPC radii(R = 1.1[m]). The acceptanceand efficiency effect are corrected by event mixing. So the ratio of real and mixed event should
be unity, if there is no physics correlation in this ∆η and ∆ϕ ∗ region. But a depletion canbe seen near ∆η = 0 and ∆ϕ ∗ = 0. This depletion is coming from the inefficiency effect bytrack merging. This track merging effect is getting stronger with increasing the pair transverse
momentum kT .
In order to study this effect in detail, we projected this two dimensional distribution to ∆ηdirection in Figure 3.22 and ∆ϕ ∗ in Figure 3.23. Projected histograms are fitted with double
Gaussian function as dashed line in order to quantify the width of this depletion. The meanvalue of fit function is fixed to 0.
The sigma of narrower Gaussian as a function of extrapolated TPC radius R is shown in
Figure 3.24 and 3.25. 3.24 is ∆η direction and 3.25 is ∆ϕ ∗ direction. Both ∆η and ∆ϕ ∗ direction, σ∆η takes a minimum value at R = 1.0[m] and R = 1.1[m] for σ∆ϕ ∗ . The extrapolated
radius dependence of σ is larger in ∆ϕ ∗ direction. We determined to use the extrapolated radius
R = 1.1[m]. Eventually the following ∆η and ∆ϕ ∗ cut is applied for my analysis.
• ∆ϕ ∗ < 0.066 (R = 1.1[m])• ∆η < 0.018
This values are determined by the fit result (3σ of narrower Gaussian in lowest kT range
which has widest Gaussian σ ).

89

Figure 3.21: Two dimensional ratio in ∆η -∆ϕ ∗ in different kT at the TPC radii(R=1.1). kT
ranges are 0.2-0.3, 0.3-0.4, 0.4-0.5, 0.5-0.6, 0.6-0.7, 0.7-1.0 GeV/c. In all kT ranges, one canfind the broad peak and narrow dip at small ∆η -∆ϕ ∗ .

90Figure 3.22: ∆η projection of the ∆η -∆ϕ ∗ 2D ratio for each kT range, at R = 1.1m Projected∆η distributions are fitted with double Gaussian function shown in black dashed lines.

Figure 3.23: ∆ϕ ∗ projection of two dimensional ration in ∆η -∆ϕ ∗ at R = 1.1m. Double
Gaussian fitting is performed to all distributions.

91

Figure 3.24: Width of Gaussian fit function(narrow dip) to ∆η distribution as a function of
extrapolated TPC radius R for each kT bins in centrality 0-50%. Closed circle and open square
marker denote π + π + and π − π − pairs, respectively. Width grows from low kT to high kT .

Figure 3.25: Width of Gaussian fit function(narrow dip) to ∆ϕ ∗ distribution as a function of
extrapolated TPC radius R. Closed circle and open square marker denote π + π + and π − π −pairs, respectively. Width grows from low kT to high kT .

92

3.6 HBT analysis method
Theoretically correlation of Bose-Einstein enhancement can be extracted with the ratio of two
particles probability distribution to single particle probability distribution. Basically pairs insame events (real pair) contain the physical correlation such as HBT and flow correlation. However, in experiment, pair distribution is biased by the limited detector acceptance and finite
particle efficiency. These effects are corrected with event mixing technique.
In event mixing technique, one dimensional correlation function is given by :C2 (q) =

qreal
qmix

(3.57)

where the numerator qreal is the relative momentum distribution of real pairs and denominator qmix denotes the relative momentum distribution of mixed pairs which mean pair distribution
in different events. Mixed pairs have no physical correlations. But mixed pairs are also affectedwith similar acceptance and efficiency effect. when mixed event have similar event properties
centrality and z vertex position.
As a consequence, the detector effect is canceled out and correlation function C2 is driven
from only the physical correlations.
In order to subtract the detector effect correctly, we define event class(Table 3.2 of event
mixing technique. The classification of event plane angle Ψ2 and Ψ3 is only applied for studyof HBT w.r.t. Ψ2 and Ψ3 , respectively.
Relative momentum distribution of real pairs and mixed pairs are simultaneously shown in
Figure 3.26, 3.27 (top panel). Shape of these distributions mainly comes from pT distribution
and detector effect. The small difference between real and mixed pairs at low relative momentum region is the HBT correlation. Correlation function is calculated with event mixing methodin the bottom panel of Figure 3.26, 3.27.
Table 3.2: Event classification of event mixing
Class nameRange
Bin width
Centrality
0-50%
5%
z vertex|zvertex | < 8.0 cm 2 cm
π /30 radΨ2
Ψ2 < π /2 radΨ3
Ψ3 < π /3 rad
π /30 rad93

Figure 3.26: Relative momentum distribution of real pairs(red) and mixed pairs(blue) are
simultaneously depicted in top panel. Mixed pair distribution is scaled to the real pair in qinv
range 0.17-0.34 (GeV/c). Correlation function is extracted with event mixing technique andshown in red marker on the bottom panel.

Figure 3.27: 1D projection of 3D Relative momentum distribution of real pairs(red) and mixed
pairs(blue) are simultaneously depicted in top three panels. Left, Middle and Right panel areout, side and long direction, respectively. Mixed pair distribution is scaled to the real pairin qout,side,long range 0.15-0.30 (GeV/c). Correlation function is extracted with event mixing
technique and shown in blue marker on the bottom panel.

943.6.1 Fitting
χ 2 test is commonly used to examine the goodness-of-fit test. Suppose a histogram of theobserved x values with N bins and the number of entries in bin i is ni . When we perform fit
to this histogram with function f (x). Goodness-of-fit test is examined based on Pearson’s χ 2statistic,(ni − f (x))2
χ =∑
f (x)
i=1
2

N

(3.58)

This statistical test works if the data ni (i = 1, 2, 3,..., N) are Poisson distributions and thenumber of entries in each bin is not too small(e.g. ni ≥ 5). But measured correlation function
which is expressed by the ratio of two Poisson distribution is not itself Poisson distributed,
especially when taking the ratio of small numbers. Simple χ 2 examination is inappropriate for
fitting the correlation function. As a consequence, maximum log-likelihood is used to examine
the goodness-of-fit. A log-likelihood minimization function is given by :
2
χPML
= −2

[
(
)
(
)]
C (A + B)
A+B
A ln+ B lnA (C + 1)
B (C + 1)(3.59)

where A and B are relative momentum distribution for real and mixed pairs, respectively. C
is the ratio of A to B. This maximum log-likelihood equation assumes that Both real and mixed
are distributed as Poisson.

953.6.2 Corrections
Compared to the ideal case where the correlation function C2 can be simply expressed by BoseEinstein enhancement, analyzing experimentally measured correlation function is more complicated. We have to perform appropriate corrections for measured correlation functions in order toextract correct source radii. In this analysis, applied corrections to the correlation functions fall
into three categories, Coulomb interaction, momentum resolution correction and event plane
resolution correction.
Coulomb Interaction
First correction concerns Coulomb interaction for charged particles. One dimensional Correlation function (Figure 3.26) is not simple Gauss function. There is dip structure at small qinv .
This is Coulomb-induced correlations. In order to extract HBT correlation, Coulomb correlation must be analyzed at small qinv , where HBT correlation can be also found as well.
Experimentally we have two correction method of Coulomb interaction. First one is Gamow
correction, where the correlation function itself is corrected with Gamow factor[57]. Gamow
factor is given by :2πηe2πη − 1
µ e2
η ≡
Z1 Z2
h̄q

G (η ) =(3.60)(3.61)

where µ is reduced mass and Z1 e, Z2 e are charge. For realistic source, however, Gamow factor
procedure over-corrects the correlation function due to the very long-lived-resonances such asη and ω .
Second one is Bowler-Sinyukov fit, where Coulomb interaction is included in the fit function[58].
In this method, Core-Halo picture is employed. Correlation function is decomposed into Coreand Halo term. Long-lived-resonances and misidentified particles are referred to as Halo which
has neither Femtoscopic correlation nor Coulomb induced correlation. The other pion pairs
which directly emanates from the source are described as Core. The ratio of Core to Halo isdetermined with the empirical parameter λ . In this analysis Bowler Sinyukov fit is used. Fit
function of Bowler Sinyukov fit can be expressed as :96

C2 (q) = C2Core (q) +C2Halo= N [λ {1 + G (q) F (q)} + (1 − λ )](3.62)(3.63)

where C2Core (q) and C2Halo (q) are Core and Halo term of correlation function, respectively.
N is normalization factor, F (q) is Coulomb component, G (q) is Gaussian estimated HBT correlation component. For one dimensional HBT analysis, G (q) is described as :()
G (qinv ) = exp −R2inv q2inv(3.64)

and for three dimensional out-side-long coordinates, G (q) is expressed by :

(

(

)

G qout , qside , qlong = exp−R2out q2out − R2side q2side − R2long q2long − 2R2os qout qside − 2R2ol qout qlong − 2R2sl qside qlong(3.65)

Coulomb component F (q) is given by Coulomb wave function given by :

−
−
Pc (→
p1 , →
p2 ) =

∫

−
−
Pc (→
p1 , →
p2 )
Fc =
→
−
−
P12 ( p1 , →p2 )(3.66)

d 3 rρ (⃗r) |Ψc,sym |2(3.67)

where Fc is the strength of Coulomb interaction, Ψc,sym denotes the symmetrized Coulomb−−
wave function, P (→p ,→p ) is the probability to observe two particles with coulomb interaction,
c

1

2

−
−
Pc (→
p1 , →
p2 ) is the probability to observe two particles without coulomb interaction and ρ (r) is
the spatial distribution of the distance between two particles.
Since Eq.3.67 requires the spatial coordinate of two particles, We have to assume the particle
distribution of source. In this analysis, source distribution is assumed to be Gaussian. For eachpion pairs, 20 particles are randomly distributed according to Gaussian whose width is σinput .
and Coulomb-induced correlation is calculated by the average of 361 pairs(from 20 particles).
In order to calculate the correct Coulomb correlation strength, the determination of σinput isimportant. We performed the second times iteration procedure and input source size for iterationis shown in Table 3.3.

97)

Table 3.3: Input source size of Coulomb interaction
Centrality Source size
0-5%
11 fm
5-10%
10 fm
10-20%
9 fm
20-30%
8 fm
30-40%
7 fm
40-50%
6 fm

Figure 3.28: Coulomb correction factor F (q) calculated by Coulomb wave function assuming
Gaussian distribution at pair transverse momentum kT :0.2-0.3GeV/c for 3 centralities as afunction of one dimensional relative momentum qinv

Figure 3.28 shows Coulomb correction factor F (q) calculated by Coulomb wave function
assuming Gaussian distribution at pair transverse momentum kT :0.2-0.3GeV/c for 3 centralities
as a function of one dimensional relative momentum qinv .

98

Momentum Resolution Correction
Second correction is momentum resolution correction. Finite resolution effects in the relative
momentum artificially smear the correlation function especially at small relative momentum,
where quantum interference becomes important. The effect of momentum resolution is estimated by using Monte Carlo event generator HIJING[59] and full simulation of the ALICE
detectors with GEANT. Correction factor is calculated by the double ratio of the ideal correlation function divided by smeared correlation function. Here we define the correction factor
CF (qinv ) given by :CF (q) =C2 (qgen )C2 (qrec )(3.68)

where C2 (qgen ) denotes correlation function generated from HIJING with perfect momentum resolution. Generated particles are propagated through detector simulation GEANT andsmeared correlation function is described as C2 (qrec ). Each correlation function is expressed as( gen )A (qgen )C2 qinv =B (qgen )A (qrec )C2 (qrecinv ) =
B (qrec )(3.69)(3.70)

where A and B are real and mixed relative momentum distribution obtained by event mixing
technique. Event, track and pair selections are exactly same condition to experimental analysis and resonance decay particles are rejected to exclude these effect on correlation function.
Pair selection such as fraction of shared TPC clusters and angular distance in ∆ϕ ∗ -∆η cut isapplied on both generated and reconstructed particles. For generated particles, corresponding
reconstructed track information was used to these selections. But geometrical information is
not included in HIJING. Therefore HBT correlation is inserted into real pair distribution as theweight iteratively until the extracted HBT radii converge.
Figure 3.29 shows the comparison of the generated correlation function and reconstructed
correlation function (left panel) and the correction factor CF (qinv ) (right panel).
Obtained correction factor are included in the fitting function expressed by
C (q) = N [λ {1 + G (q) F (q)} + (1 − λ )] /CF (q)99(3.71)

generated
1.2

reconstructed
1.1

C2 (gen) / C2 (rec)

C2

1.3

1

1

0.9
0

1.01

0.1

0.2

0.3

0

q inv (GeV/c )

0.1

0.2

0.3

q inv (GeV/c )Figure 3.29: Correlation function of pure pions for centrality 20-30% calculated with HIJING +GEANT (left panel). Correlation function of generated pions that detector effect is not included
is shown in closed marker and Correlation function of reconstructed pions that detector effects
are included is plotted in open marker. (open marker). Ratio of generated correlation function
to reconstructed correction function shown in right panel.

1003.6.3 Event Plane Resolution Correction
Finite event plane resolution smears the oscillation of azimuthally sensitive HBT with respect
to event plane. A model independent correction is applied in this analysis. Real and mixed
q-distribution are corrected with the following equation.

N(q, Ψ j ) = Nexp (q, Ψ j ) +nbin /2

∑

[ exp
]
exp
ζn,m (∆) Nc,n(q) cos(nΨ j ) + Ns,n(q) sin(nΨ j )(3.72)

n=m,2m,...

where N(q, Ψ j ) is the measured relative momentum distribution of real and mixed pairs,
nbin is the number of azimuthal bins and m is the order of the measured event plane. Ψ j denotes
the center of jth azimuthal bin which corresponds to azimuthal pair angle with respect to theexpexpmeasured event plane. Nc,n(q), Ns,n(q) and ζn,m (∆) are expressed as⟨⟩
1 nbinNexp (q, Ψ) cos(nΨ) =∑ Nexp(q, Ψ) cos(nΨ j ),nbin j=1(3.73)⟨⟩
1 nbinexpNs,n(q) = Nexp (q, Ψ) sin(nΨ) =∑ Nexp(q, Ψ) sin(nΨ j ),nbin j=1(3.74)n∆/2− 1,
sin(n∆/2) ⟨cos(n(Ψm − ΨR ))⟩(3.75)

exp
Nc,n
(q) =

ζn,m (∆) =where ⟨cos(n(Ψm − ΨR ))⟩ is well known correction factors as event plane resolution. Ψm andΨR are measured and real event plane. ∆ denotes the width of azimuthal angular bins. When
we calculate the pairs relative to 3rd -order event plane, only odd value of n = 3 and above aresummed over. In this analysis, only n = 3 case are calculated.
Equation 3.73 correct the smeared oscillation amplitude which is affected by finite event
plane resolution and event plane binning.
Figure 3.30 shows the comparison between event plane resolution corrected and uncorrected
result of extracted 3D HBT radii of charged pions as a function of ϕ pair − Ψ3 . Event plane isdetermined with FMD AC. As can be seen, event plane resolution correction does not change
the average value of azimuthal angle dependence. But the oscillation amplitude became a little
larger due to the event plane resolution correction. Figure 3.31 illustrates comparison of thecross term before and after the event plane resolution correction. Due to the average values ofcross term are almost zero, All data points are shifted along the y-axis for visibility. Ros shows
the clear sine curve in all centrality.
101Solid lines shows the fit function by cosine function. Both corrected and uncorrected fit
functions are plotted as red and black lines, respectively.
There is an another method to correct the oscillation amplitude of azimuthal HBT radiiwith respect to event plane. This method is much more simple one compared with bin-by-bin
method, which is expressed as
R2µ ,n,true = R2µ ,n,measured / ⟨cos(n(Ψm − ΨR ))⟩n∆/2,sin (n∆/2)(3.76)

where R2µ ,n,measured denotes the measured oscillation amplitude of squared HBT radii. R2µ ,n,trueis the corrected oscillation amplitude of squared HBT radii. ⟨cos(n(Ψm − ΨR ))⟩ is nth order
event plane resolution. Pair angle with respect to Ψn are divided into nbins classes and eachn∆/2represents finite azimuthal bin widthbins can be expressed as ∆ = π /nbins . The term sin(n∆/2)

correction. Oscillaion amplitude of HBT radii are smeared with finite number of divisions inpair angles with respect to event plane.
For QA of event plane resolution correction, corrected oscillation functions are comparedwith results with different correction method with Equation 3.76 in Figure 3.30. Red solidline is fit function of bin-by-bin event plane correction and Blue line is obtained by applying
Equation 3.76 to uncorrected black line. Two different method is fully consistent (the differenceis about 0.1%).

102

25

20

20

15

15

10

R 2long (fm2)

R 2side (fm2)

25

30

0

0.5

ϕ

1
pair

10

1.5
2
- ΨEP, 3 (rad)

0

0.5

0.7

50

λ

R 2out (fm2)

30

pair

1.5
2
- ΨEP, 3 (rad)0.2 < k T < 2.0GeV/c
Uncorrected 10-20%
Uncorrected 20-30%
Uncorrected 30-40%
Uncorrected 40-50%
Corrected 10-20%
Corrected 20-30%
Corrected 30-40%
Corrected 40-50%

0.6

40

ϕ

1

0.5
30
0.4
20
10

0.3
0

0.5

ϕ

1
pair

1.5
2
- ΨEP, 3 (rad)

0.2

0

0.5

ϕ

1
pair

1.5
2
- ΨEP, 3 (rad)Figure 3.30: Extracted 3D HBT radii of charged pions as a function of azimuthal pair angle
with respect to Ψ3 . Comparison of before and after the event plane resolution correction,

103

Rsl2 (fm2)

Ros2 (fm2)

1

0

0

−1

−1
0

Rol2 (fm2)

1

0.5

ϕ

1
pair

1.5
2
- ΨEP, 3 (rad)

0

0.5

ϕ

1
pair

1.5
2
- ΨEP, 3 (rad)0.2 < kT < 2.0GeV/c

1

Uncorrected 10-20%
Uncorrected 20-30%
Uncorrected 30-40%
Uncorrected 40-50%

0Corrected 10-20%−1Corrected 20-30%
Corrected 30-40%

0

0.5

ϕ

1
pair

1.5
2
- ΨEP, 3 (rad)Corrected 40-50%

Figure 3.31: Extracted 3D HBT cross term of charged pions as a function of azimuthal pairangle with respect to Ψ3 . Comparison of before and after the event plane resolution correction.
All points are shifted along the y-axis for visibility

1043.7 Systematic Uncertainties
In this section, the effect of various source systematic uncertainties on HBT analysis and vn
analysis are presented.

3.7.1 Systematic Uncertainties for HBT analysis
Systematic uncertainties related to HBT analysis are listed as following.
• Systematic difference of positive pion pairs and negative pion pairs• Effect of magnetic field polarity• Various fitting range of relative momentum• Effect of different pair selection• Effect of different event plane determination detector
Systematic Uncertainties of Charge
Positive and negative pions are combined for this analysis. The difference of each positive andnegative pion pairs for Azimuthally sensitive HBT studies are estimated as systematic uncertainty. Since Ros significantly changes with respect to charge difference, which comes from thedifference of pair selection effect to positive and negative particles. More positive pairs are subtracted in qout qside > 0, while more negative pairs are subtracted in qout qside < 0. This effect canbe canceled out by summing positive and negative charged particles. Therefore the differenceof charged pion pairs are excluded from systematic uncertainties for Ros . Figure 3.32 shows
the azimuthal angle dependence of squared HBT radii relative to Ψ2 in 0-5, 5-10% centrality
with positively and negatively charged pions and combined of them.
Systematic uncertainty of different charge is defined by the standard deviation of HBT radii
R in each azimuthal angle bins given by:v(
u 2uchargeσ=t
Rµ ,i − Rµ∑)µ ,combined/2(3.77)iwhere µ denotes the HBT radii in each directions at Bertsch-Pratt frame, namely Rµ = Rout
, Rside , Rlong , λ , Rol and Rsl .
105

0.5

50

λ 3D

50

R out

50

R sideR long

0-5%
0.4
40

40

40

30

30

30

0.3
Default
positive
negative

0.2
0.1

0

1

ϕ

pair

0.5

2

3

20

0

1

- ΨEP,2 (GeV/c )

ϕ

pair

50

λ 3D

2

3

20

0

- ΨEP,2 (GeV/c )

1

ϕ

pair

50

R out

2

3

20

0

- ΨEP,2 (GeV/c )

1

ϕ
50

R side2pair

3- ΨEP,2 (GeV/c )R long

5-10%
0.4
40

40

40

30

30

30

0.3

0.2
0.1

0

1

ϕ

pair

2

3

20

0

1

- ΨEP,2 (GeV/c )

ϕ

pair

2

3

20

0

- ΨEP,2 (GeV/c )

1

ϕ

pair

2

3

- ΨEP,2 (GeV/c )

20

0

1

ϕ

pair

2

3

- ΨEP,2 (GeV/c )Figure 3.32: Extracted 3D HBT radii (λ , R2out , R2side , R2long ) of charged pions as a function of
azimuthal pair angle relative to Ψ2 . Figures in top rows represents centrality 0-5%, while 510% are shown in bottom panels. Default value is positive and negative pions combined results
shown in black marker. Comparison with positive and negative charged pion pairs are depicted
as orange circle and blue square, respectively.

Systematic Uncertainties of Charge and Magnetic Field Polarity
In ALICE, two different polarity of the solenoid magnetic field are applied. This analysis was
performed in both positive and negative magnetic field combined results. The difference in two
magnetic field polarity is analyzed separately.
Figure 3.34 shows the azimuthal angle dependence of squared HBT radii relative to second
harmonic event plane in centrality 0-5, 5-10% with three patterns of magnetic field polarity,
positive, negative and combined them.
Systematic uncertainty of different magnetic field polarity is defined by the standard deviation of HBT radii R in each azimuthal angle bins given by:
v(
u 2
u
mag
σ
=t
Rµ ,i − R
µ

∑
i

106

)
µ ,combined/2(3.78)

0.5

50

λ 3D

50

R out

50

R sideR long

0-5%

0.4
40

40

40

30

30

30

0.3
Default(combined)
B filed + +B filed - -

0.2

0.1

0

1

ϕ

pair

0.5

2

3

20

0

- ΨEP,2 (GeV/c )

1

ϕ

pair

50

λ 3D

2

3

20

0

- ΨEP,2 (GeV/c )

1

ϕ

pair

50

R out

2

3

20

0

- ΨEP,2 (GeV/c )

1

ϕ
50

R side2pair

3- ΨEP,2 (GeV/c )R long

5-10%

0.4
40

40

40

30

30

30

0.3
0.2

0.1

0

1

ϕ

pair

2

3

20

0

- ΨEP,2 (GeV/c )

1

ϕ

pair

2

3

20

0

- ΨEP,2 (GeV/c )

1

ϕ

pair

2

3

- ΨEP,2 (GeV/c )

20

0

1

ϕ

pair

2

3

- ΨEP,2 (GeV/c )Figure 3.33: Extracted 3D HBT radii (λ , R2out , R2side , R2long ) of charged pions as a functionof azimuthal pair angle relative to Ψ2 . Figures in top rows represents centrality 0-5%, while
5-10% are shown in bottom panels. Default value is positive and negative B field combined
results shown in black marker. Positive and negative B field results are separately analyzed anddepicted in green circle and yellow square, respectively.

Systematic Uncertainties of Fit Range
3 dimensional fit is performed to correlation function at the relative momentum range 0-150
GeV/c . Systematic difference with varying the fitting range is studied for each 10 GeV/c bin
up to 200 GeV/c .
Figure 3.34 shows the azimuthal angle dependence of squared HBT radii relative to second
harmonic event plane in centrality 0-5, 5-10% with three patterns of magnetic field polarity,
positive, negative and combined them.
Systematic uncertainty of different magnetic field polarity is defined by the standard deviation of HBT radii R in each azimuthal angle bins given by:v(
u 5u
Rµ ,i − Rσ range = t
µ

∑
i

107

)
µ ,150MeV/c/5(3.79)

0.5

50

λ 3D

50

R out

50

R sideR long

0-5%

0.4
40

40

40

30

30

30

0.3
150MeV/ c (Default)
140MeV/ c
130MeV/ c

0.2

0.1

0

1

ϕ

pair

0.5

2

3

20

0

- ΨEP,2 (GeV/c )

1

ϕ

pair

50

λ 3D

2

3

20

0

- ΨEP,2 (GeV/c )

1

ϕ

pair

50

R out

2

3

20

0

- ΨEP,2 (GeV/c )

1

ϕ
50

R side2pair

3- ΨEP,2 (GeV/c )R long

5-10%

0.4
40

40

40

30

30

30

0.3
0.2

0.1

0

1

ϕ

pair

2

3

- ΨEP,2 (GeV/c )

20

0

1

ϕ

pair

2

3

20

- ΨEP,2 (GeV/c )

0

1

ϕ

pair

2

3

- ΨEP,2 (GeV/c )

20

0

1

ϕ

pair

2

3

- ΨEP,2 (GeV/c )Figure 3.34: Extracted 3D HBT radii (λ , R2out , R2side , R2long ) as a function of azimuthal pairangle relative to Ψ2 , varying fit range from 150GeV/c (Default) to 200GeV/c for each 10GeV/cbin. Figures in top rows represents centrality 0-5%, while 5-10% are shown in bottom panels.
Default fit range is 150GeV/c shown in black marker. Comparison with positive and negative
charged pion pairs are depicted as orange circle and blue square, respectively.

Systematic Uncertainties of Pair Cut
The analysis is repeated at tighter Pair Cut selection definition with angular distance in ∆ϕ ∗ ∆η .
Default cut value is determined by fitting ∆ϕ ∗ ∆η with double Gaussian and 3 σ of narrower
Gauss function. Effect of tighter pair selection with 3.5 σ is studied as shown in the followinglist.
• ∆ϕ ∗ : 0.066(default), 0.077(tight)• ∆η : 0.018(default), 0.021(tight)
Figure 3.35 shows the azimuthal angle dependence of squared HBT radii relative to second
harmonic event plane in centrality 0-5, 5-10% with two different pair selection.
Systematic uncertainty of pair selection is defined by the absolute value of HBT radii R ineach azimuthal angle bins given by:108

σµpair = |Rµ ,tight − Rµ ,de f ault |
0.5

50

λ 3D

50

R out

(3.80)
50

R sideR long

0.4
40

40

40

30

30

30

0.3
Default

0.2
0.1

Tight

0

1

ϕ

pair

0.5

2

3

20

0

- ΨEP,2 (GeV/c )

1

ϕ

pair

50

λ 3D

2

3

20

0

- ΨEP,2 (GeV/c )

1

ϕ

pair

50

R out

2

3

20

0

- ΨEP,2 (GeV/c )

1

ϕ
50

R side2pair

3- ΨEP,2 (GeV/c )R long

0.4
40

40

40

30

30

30

0.3

0.2
0.1

0

1

ϕ

pair

2

3

- ΨEP,2 (GeV/c )

20

0

1

ϕ

pair

2

3

20

- ΨEP,2 (GeV/c )

0

1

ϕ

pair

2

3

- ΨEP,2 (GeV/c )

20

0

1

ϕ

pair

2

3

- ΨEP,2 (GeV/c )Figure 3.35: Extracted 3D HBT radii (λ , R2out , R2side , R2long ) as a function of azimuthal pairangle relative to Ψ2 with two different pair selection. Figures in top rows represents centrality 05%, while 5-10% are shown in bottom panels. Default pair selection is plotted in black marker.
Comparison with the result in tighter pair selection are depicted as orange circle.

Systematic Uncertainties of Event Plane Determination Detector
Systematic study of different event plane is performed with VZERO detector which has different
event plane resolution and rapidity gap between HBT measurements. Estimation of systematic
uncertainty is evaluated via VZERO A+C combined event plane.
Figure 3.36 shows the azimuthal angle dependence of squared HBT radii relative to second harmonic event plane in centrality 0-5, 5-10% with two different event plane, FMD A+C
combined and VZERO A+C combined.
Systematic uncertainty of different event plane is defined by the absolute value of HBT radii
R in each azimuthal angle bins given by:

109

0.5

50

λ 3D

50

R out

50

R sideR long

0-5%

0.4
40

40

40

30

30

30

0.3
FMDAC(Default)
V0AC

0.2

0.1

0

1

ϕ

pair

0.5

2

3

20

0

- ΨEP,2 (GeV/c )

1

ϕ

pair

50

λ 3D

2

3

20

0

- ΨEP,2 (GeV/c )

1

ϕ

pair

50

R out

2

3

20

0

- ΨEP,2 (GeV/c )

1

ϕ
50

R side2pair

3- ΨEP,2 (GeV/c )R long

5-10%

0.4
40

40

40

30

30

30

0.3
0.2

0.1

0

1

ϕ

pair

2

3

20

0

- ΨEP,2 (GeV/c )

1

ϕ

pair

2

3

20

- ΨEP,2 (GeV/c )

0

1

ϕ

pair

2

3

- ΨEP,2 (GeV/c )

20

0

1

ϕ

pair

2

3

- ΨEP,2 (GeV/c )Figure 3.36: Extracted 3D HBT radii (λ , R2out , R2side , R2long ) as a function of azimuthal pairangle relative to Ψ2 with two different event plane via FMD A+C and VZERO A+C. Figures
in top rows represents centrality 0-5%, while 5-10% are shown in bottom panels. FMD A+C
combined results is plotted in black marker. VZERO A+C combined results is plotted in blue
marker.

σµEP = |Rµ ,tight − Rµ ,de f ault |(3.81)

Total systematic uncertainty is calculated by quadratic sum of each systematic error givenby :σµtot =√(σ charge)2+ (σ mag )2 + (σ range )2 + (σ pair )2 + (σ EP )2 +(3.82)

Ratio of total systematic uncertainty to HBT radii (or relative amplitude of squared HBTradii) is shown in Table 3.43.53.63.7.

110

Table 3.4: Systematic table for λ w.r.t. Ψ2 unbiased(No q2 selection)
centrality 0-5%∆ϕ0
π /8 π /4 3π /8 π /2 5π /8 3π /4 7π /8charge(%)
0.03 0.66 0.48 0.02 0.03 0.27 0.02 0.19
B field(%)
0.09 0.24 0.21 0.05 0.05 0.49 0.39 0.54
Event plane(%)
0.41 0.15 0.57 0.07 0.37 1.13 0.37 0.60
Pair cut(%)
3.96 3.87 4.08 4.15 3.89 3.95 3.97 3.71
Fit range(%)
1.72 1.79 1.76 1.82 1.76 1.69 1.76 1.74
Quadratic sum (%) 4.34 4.33 4.51 4.53 4.29 4.48 4.37 4.19
centrality 5-10%
charge(%)
0.09 0.38 0.03 0.40 0.25 0.34 0.07 0.10
B field(%)
0.16 0.14 0.25 0.14 0.27 0.22 0.11 0.39
Event plane(%)
0.44 0.52 0.22 0.11 0.11 0.35 0.61 0.00
Pair cut(%)
3.99 3.88 4.05 3.90 3.67 3.87 3.92 4.02
Fit range(%)
1.25 1.24 1.28 1.33 1.31 1.22 1.29 1.26
Quadratic sum (%) 4.21 4.13 4.26 4.15 3.92 4.10 4.18 4.23
centrality 10-20%
charge(%)
0.31 0.2 0.16 0.81 0.18 0.36 0.64 0.24
B field(%)
0.84 0.76 0.54 1.01 0.76 0.23 0.22 0.18
Event plane(%)
0.65 0.29 0.37 0.31 0.12 0.00 0.56 0.17
Pair cut(%)
3.54 3.67 3.29 3.47 3.20 4.01 3.54 3.66
Fit range(%)
0.79 0.84 0.83 0.94 0.87 0.94 0.90 0.84
Quadratic sum (%) 3.79 3.86 3.46 3.84 3.41 4.14 3.76 3.77
centrality 20-30%
charge(%)
0.18 0.12 0.19 0.56 0.29 0.46 0.33 0.26
B field(%)
0.14 0.68 0.67 0.03 0.63 0.48 0.53 0.41
Event plane(%)
0.06 0.16 0.36 0.63 0.47 0.08 0.68 0.40
Pair cut(%)
2.95 3.33 2.68 2.62 3.39 2.78 2.45 3.19
Fit range(%)
1.02 0.97 1.06 1.05 1.06 0.95 1.02 0.98
Quadratic sum (%) 3.13 3.54 2.99 2.95 3.65 3.02 2.81 3.40
centrality 30-40%
charge(%)
0.51 1.15 0.25 1.49 0.24 1.31 0.55 1.66
B field(%)
0.68 0.13 1.01 0.25 1.09 1.04 0.35 0.24
Event plane(%)
0.91 0.80 0.98 0.39 1.63 0.62 0.43 1.19
Pair cut(%)
2.27 2.32 1.84 2.34 2.11 2.17 2.11 2.32
Fit range(%)
1.14 1.09 1.00 0.97 1.12 0.94 1.12 1.15
Quadratic sum (%) 2.83 2.92 2.54 2.98 3.10 2.96 2.51 3.31
centrality 40-50%
charge(%)
0.29 0.10 1.15 0.08 0.27 0.13 0.61 1.45
B field(%)
2.03 0.49 0.72 0.68 1.26 0.61 2.22 1.51
Event plane(%)
1.68 0.92 0.45 0.18 0.21 0.99 0.52 0.00
Pair cut(%)
1.78 1.78 2.26 2.21 1.89 1.67 2.57 1.47
Fit range(%)
1.26 1.09 1.14 1.27 1.31 1.44 1.37 1.38
Quadratic sum (%) 3.43 2.33 2.91 2.64 2.65 2.50 3.75 2.90
111

Table 3.5: Systematic table for Rout w.r.t. Ψ2 unbiased(No q2 selection)
centrality 0-5%∆ϕ0
π /8 π /4 3π /8 π /2 5π /8 3π /4 7π /8charge(%)
0.38 0.27 0.11 0.09 0.21 0.18 0.22 0.07
B field(%)
0.94 0.37 1.11 0.97 0.67 0.86 0.58 0.67
Event plane(%)
0.13 0.30 0.14 1.03 0.39 0.26 0.14 0.45
Pair cut(%)
1.75 1.59 1.67 1.83 1.69 1.91 1.87 1.70
Fit range(%)
1.12 1.18 1.15 1.17 1.11 1.02 1.11 1.12
Quadratic sum (%) 2.31 2.06 2.32 2.59 2.17 2.35 2.27 2.19
centrality 5-10%
charge(%)
0.69 0.32 0.37 0.64 0.60 0.62 0.54 0.54
B field(%)
0.21 0.24 0.20 0.16 0.70 0.13 0.11 0.33
Event plane(%)
0.00 0.50 0.36 0.48 0.11 0.00 0.45 0.23
Pair cut(%)
1.41 1.36 1.45 1.30 1.22 1.36 1.33 1.40
Fit range(%)
1.36 1.30 1.32 1.37 1.36 1.30 1.39 1.38
Quadratic sum (%) 2.09 1.99 2.04 2.06 2.05 1.99 2.05 2.08
centrality 10-20%
charge(%)
0.47 0.35 0.75 1.15 0.27 0.31 0.59 0.36
B field(%)
1.47 1.41 1.21 2.54 1.74 1.88 1.03 0.87
Event plane(%)
1.05 0.03 0.08 0.04 0.34 1.24 0.58 0.19
Pair cut(%)
1.32 0.99 1.33 1.54 1.37 1.94 1.25 1.61
Fit range(%)
0.94 1.00 0.97 1.06 0.99 1.10 1.05 1.02
Quadratic sum (%) 2.48 2.03 2.18 3.35 2.47 3.19 2.10 2.13
centrality 20-30%
charge(%)
0.45 0.32 0.12 0.51 0.34 0.21 0.56 0.19
B field(%)
0.91 0.70 0.29 0.19 0.70 0.78 0.24 0.92
Event plane(%)
0.10 0.76 0.81 0.19 0.12 1.48 0.26 0.46
Pair cut(%)
0.70 1.59 0.87 1.67 2.12 1.05 0.83 1.36
Fit range(%)
1.60 1.52 1.61 1.48 1.46 1.41 1.51 1.50
Quadratic sum (%) 2.02 2.45 2.02 2.31 2.69 2.44 1.85 2.28
centrality 30-40%
charge(%)
0.50 1.10 0.15 0.22 0.05 0.36 0.84 1.59
B field(%)
0.50 0.28 0.64 0.16 0.38 0.42 1.13 0.43
Event plane(%)
1.20 0.53 0.83 0.39 1.76 1.41 0.31 0.40
Pair cut(%)
0.93 1.20 0.96 0.90 1.18 0.48 0.77 0.50
Fit range(%)
2.04 1.91 1.73 1.64 1.89 1.55 1.95 2.00
Quadratic sum (%) 2.64 2.58 2.24 1.93 2.87 2.22 2.55 2.67
centrality 40-50%
charge(%)
0.84 1.09 0.71 0.16 0.50 1.67 1.17 1.38
B field(%)
2.81 1.67 1.69 0.40 0.57 2.45 2.80 1.30
Event plane(%)
1.16 0.52 2.81 0.86 0.51 0.20 3.41 1.30
Pair cut(%)
0.93 0.90 1.63 0.39 0.12 1.42 1.06 0.64
Fit range(%)
2.63 2.23 2.23 2.42 2.56 2.80 2.72 2.82
Quadratic sum (%) 4.21 3.16 4.35 2.63 2.72 4.32 5.42 3.69
112

Table 3.6: Systematic table for Rside w.r.t. Ψ2 unbiased(No q2 selection)
centrality 0-5%∆ϕ0
π /8 π /4 3π /8 π /2 5π /8 3π /4 7π /8charge(%)
0.08 0.30 0.47 0.18 0.06 0.60 0.20 0.17
B field(%)
1.27 1.23 1.15 0.58 0.49 1.21 1.08 1.61
Event plane(%)
0.56 0.29 0.34 0.12 0.12 0.37 0.06 0.16
Pair cut(%)
0.13 0.05 0.09 0.09 0.03 0.03 0.05 0.01
Fit range(%)
1.72 1.79 1.78 1.86 1.81 1.70 1.76 1.74
Quadratic sum (%) 2.21 2.21 2.20 1.96 1.88 2.20 2.07 2.38
centrality 5-10%
charge(%)
0.21 0.47 0.20 0.31 0.17 0.35 0.87 0.45
B field(%)
0.32 0.60 0.75 0.35 0.32 0.47 0.38 0.29
Event plane(%)
0.90 0.06 0.26 0.31 0.43 0.50 0.53 0.23
Pair cut(%)
0.20 0.23 0.36 0.43 0.15 0.31 0.13 0.11
Fit range(%)
1.44 1.40 1.44 1.51 1.49 1.40 1.48 1.46
Quadratic sum (%) 1.75 1.61 1.69 1.67 1.60 1.63 1.84 1.57
centrality 10-20%
charge(%)
0.60 0.54 0.46 0.47 0.45 0.46 1.49 0.54
B field(%)
0.56 0.47 0.73 0.28 1.13 0.36 0.28 0.56
Event plane(%)
0.41 1.06 0.16 1.07 0.62 0.82 0.64 0.51
Pair cut(%)
0.14 0.31 0.07 0.56 0.69 0.69 0.35 0.34
Fit range(%)
0.96 1.03 1.03 1.18 1.12 1.21 1.13 1.04
Quadratic sum (%) 1.33 1.67 1.36 1.78 1.90 1.72 2.03 1.44
centrality 20-30%
charge(%)
0.16 0.14 0.90 0.80 1.03 0.23 1.44 0.61
B field(%)
1.06 0.14 0.56 0.29 0.20 0.74 0.58 0.49
Event plane(%)
0.36 0.73 0.53 0.07 0.57 0.38 0.01 0.75
Pair cut(%)
0.60 0.53 0.56 0.70 1.00 0.16 0.59 0.52
Fit range(%)
1.54 1.46 1.62 1.60 1.61 1.50 1.56 1.47
Quadratic sum (%) 2.00 1.73 2.08 1.94 2.24 1.74 2.28 1.90
centrality 30-40%
charge(%)
1.61 0.89 0.28 2.29 1.57 0.91 0.55 2.05
B field(%)
1.11 0.29 0.74 1.81 0.84 1.17 0.59 0.34
Event plane(%)
0.24 1.20 2.28 0.15 1.01 0.20 0.78 1.57
Pair cut(%)
0.33 0.04 0.18 0.87 1.32 0.84 0.63 0.30
Fit range(%)
1.86 1.74 1.70 1.71 2.00 1.63 1.94 1.84
Quadratic sum (%) 2.73 2.31 2.96 3.49 3.15 2.36 2.33 3.20
centrality 40-50%
charge(%)
0.23 1.05 0.60 0.41 0.59 0.54 1.91 0.34
B field(%)
1.63 0.12 0.21 1.44 0.88 0.68 0.92 0.51
Event plane(%)
0.27 0.31 0.10 1.85 0.89 2.79 0.22 0.80
Pair cut(%)
0.51 0.20 1.45 1.61 1.29 1.43 1.33 0.17
Fit range(%)
2.29 1.99 2.13 2.47 2.64 2.79 2.55 2.50
Quadratic sum (%) 2.88 2.28 2.65 3.79 3.25 4.28 3.58 2.70
113

Table 3.7: Systematic table for Rlong w.r.t. Ψ2 unbiased(No q2 selection)
centrality 0-5%∆ϕ0
π /8 π /4 3π /8 π /2 5π /8 3π /4 7π /8charge(%)
0.10 0.60 0.29 0.17 0.11 0.39 0.05 0.27
B field(%)
0.77 0.62 1.19 0.66 0.42 1.31 1.15 1.14
Event plane(%)
0.41 0.41 0.12 0.24 0.16 0.81 0.45 1.04
Pair cut(%)
3.28 3.25 3.40 3.44 3.17 3.24 3.29 3.08
Fit range(%)
2.15 2.22 2.16 2.24 2.18 2.10 2.21 2.18
Quadratic sum (%) 4.01 4.05 4.21 4.17 3.88 4.17 4.15 4.09
centrality 5-10%
charge(%)
0.55 0.56 0.14 0.26 0.14 0.36 0.27 0.63
B field(%)
0.34 0.16 0.69 0.17 0.26 0.24 0.23 0.29
Event plane(%)
0.13 0.72 0.19 0.28 0.39 0.35 0.45 0.19
Pair cut(%)
3.57 3.59 3.68 3.56 3.29 3.49 3.55 3.58
Fit range(%)
1.46 1.43 1.48 1.53 1.47 1.37 1.46 1.45
Quadratic sum (%) 3.92 3.97 4.04 3.90 3.64 3.79 3.88 3.93
centrality 10-20%
charge(%)
0.75 0.35 1.55 0.42 0.63 0.61 0.84 0.52
B field(%)
0.66 1.59 0.19 2.18 1.00 0.50 0.69 2.37
Event plane(%)
0.66 0.33 0.44 0.63 0.32 0.23 1.01 0.60
Pair cut(%)
3.25 3.34 3.09 3.20 2.64 3.30 3.49 3.59
Fit range(%)
1.08 1.16 1.14 1.28 1.20 1.32 1.25 1.18
Quadratic sum (%) 3.63 3.91 3.67 4.15 3.15 3.65 4.00 4.53
centrality 20-30%
charge(%)
0.56 1.19 0.17 0.87 1.75 0.08 0.85 0.19
B field(%)
0.10 1.61 0.40 0.29 1.98 0.10 0.31 1.19
Event plane(%)
0.29 0.04 0.61 0.03 1.68 0.91 0.15 0.89
Pair cut(%)
2.92 3.18 2.84 2.80 3.14 2.67 2.73 3.31
Fit range(%)
1.76 1.67 1.81 1.76 1.74 1.65 1.73 1.68
Quadratic sum (%) 3.47 4.11 3.45 3.43 4.77 3.27 3.36 4.00
centrality 30-40%
charge(%)
1.19 1.20 1.85 0.15 0.13 1.01 0.66 0.17
B field(%)
0.60 0.73 0.21 1.21 0.53 0.59 0.09 0.12
Event plane(%)
0.26 1.54 0.04 1.60 0.69 1.70 1.32 1.07
Pair cut(%)
2.05 2.54 1.88 1.67 2.17 2.49 1.89 2.50
Fit range(%)
2.21 2.09 1.93 1.88 2.18 1.76 2.20 2.21
Quadratic sum (%) 3.31 3.89 3.28 3.22 3.20 3.68 3.25 3.51
centrality 40-50%
charge(%)
0.25 1.18 1.42 0.40 0.97 0.50 0.88 1.57
B field(%)
2.05 0.96 2.38 0.98 1.78 0.40 0.97 2.00
Event plane(%)
1.33 1.79 1.84 1.84 3.30 0.98 1.89 0.75
Pair cut(%)
2.01 1.64 2.74 1.81 2.10 1.97 2.24 1.74
Fit range(%)
2.72 2.32 2.50 2.73 2.86 3.16 3.01 2.98
Quadratic sum (%) 4.18 3.69 4.98 3.90 5.25 3.90 4.40 4.35
114

3.7.2 Systematic Uncertainties for vn analysis
In this analysis 6 variables are considered for the systematic uncertainties and they are listed asfollowing.
• Effect of number of TPC clusters• Systematic uncertainties of centrality estimator• Various track reconstruction mode
• Effect of primary vertex position in the direction of beam axis
• Systematic uncertainties of Event Plane determination detectors• Systematic uncertainties related to particle identification selection with TPC and TOF
Effect of number of TPC clusters
For track reconstruction of charged particles, at least 80 TPC clusters are required in this analysis. 2 different number of TPC cluster 50 (loose) and 90 (tight) selection are considered for
estimation of systematic uncertainties (Total number of TPC pad rows are 159, therefore this
TPC cluster selection is corresponding to 30-56% of all TPC pad rows).
Figure 3.37 are centrality dependence of charged pion, kaon and proton(anti-proton) v2 withtwo different number of TPC clusters for track reconstruction.
proton

v2

kaon

v2

v2

pion
0.2

0.2

TPC number of clusters
0.2

0.15

0.15

0.15

0.1

0.1

0.1

0.05

0.05

0.05

TPC # of clusters : 80 (default)

0

0
0

10

20

30

40

50

centrality (%)

TPC # of clusters : 50 (loose)

0
0

10

20

30

40

50

0

centrality (%)

10

20

30

40

50

TPC # of clusters : 90 (tight)

centrality (%)

Figure 3.37: charged pion, kaon and proton(anti-proton) pT integrated v2 as a function ofcentrality with two different TPC number of clusters for track reconstruction. pT is integrated
from 0.15-1.5 GeV/cwhich is corresponding to same selection for HBT analysis.

115

Systematic Uncertainties of Tracking mode
In this analysis, TPC clusters and primary vertex determined with ITS are used for track reconstruction. However we have several different track reconstruction algorithms. Uncertainties
originating from two different tracking modes (Global track and hybrid track). Global track andHybrid track are defined by track reconstruction with the combination of TPC and ITS.
Figure 3.38 are azimuthal anisotropy of charged pion, kaon and proton(anti-proton) v2 as afunction of centrality with two different tracking mode using TPC and ITS combined. Compared to pions and kaons, the systematic difference of protons and anti-protons are larger especially in peripheral collisions.
proton

v2

kaon

v2

v2

pion
0.2

0.2

0.15

0.15

0.15

0.1

0.1

0.1

0.05

0.05

0.05

0

0
0

10

20

30

40

50

centrality (%)

Tracking Mode

0.2

TPC only track (default)

Global track(SDD hit)

0
0

10

20

30

40

50

0

centrality (%)

10

20

30

40

50

Hybrid trackcentrality (%)

Figure 3.38: charged pion, kaon and proton(anti-proton) pT integrated v2 as a function ofcentrality with two different tracking modes using TPC and ITS. pT is integrated from 0.15-1.5
GeV/cwhich is corresponding to same selection for HBT analysis.

Systematic Uncertainties of Centrality Estimator
The contribution from centrality estimator is estimated with changing centrality determination
detector from VZERO signal to the number of clusters in SPD outer layer.
Figure 3.39 are azimuthal anisotropy of charged pion, kaon and proton(anti-proton) v2 as afunction of centrality with different centrality estimator using second layer in SPD.
Effect of primary vertex position in the direction of beam axis
Primary vertex position affects the detector acceptance in particular to the direction of beamaxis. The effect of primary vertex position in the direction of beam axis(zvtx) is studied byvarying the different z-vertex selection, |zvtx| < 5cm (tight) and |zvtx| < 10cm (loose) event
selection.

116

proton

v2

kaon

v2

v2

pion
0.2

0.2

0.15

0.15

0.15

0.1

0.1

0.1

0.05

0.05

0.05

0

0
0

10

20

30

40

50

Centrality determination

0.2

V0A+C Multiplicity (default)

Second layer of ITS

0
0

10

20

centrality (%)

30

40

50

0

10

20

centrality (%)

30

40

50

centrality (%)

Figure 3.39: Systematic difference of centrality determination detectors V0 amplitude andnumber of clusters in second layer of SPD for charged pion, kaon and proton(anti-proton) pT
integrated v2 as a function of centrality. pT is integrated from 0.15-1.5 GeV/cwhich is corresponding to same selection for HBT analysis.

Figure 3.40 are azimuthal anisotropy of charged pion, kaon and proton(anti-proton) v2 as afunction of centrality with two different Z vertex selection (tight and loose).
proton

0.2

v2

kaon

v2

v2

pion
0.2

0.15

0.15

0.15

0.1

0.1

0.1

0.05

0.05

0.05

0

0
0

10

20

30

40

50

centrality (%)

Zvertex0.2|zvtx| < 8 cm(default)|zvtx| < 10 cm (loose)

0
0

10

20

30

40

50

0

centrality (%)

10

20

30

40

50

|zvtx| < 5 cm (tight)

centrality (%)

Figure 3.40:
Systematic difference of primary vertex position along the beam axis
tight(|zvtx|<5cm) and loose(|zvtx|<10cm) for charged pion, kaon and proton(anti-proton) pT
integrated v2 as a function of centrality. pT is integrated from 0.15-1.5 GeV/cwhich is corresponding to same selection for HBT analysis.

Systematic Uncertainties of Event Plane Determination Detector
In this thesis, azimuthal anisotropy is calculated with Event plane method and default event
plane determination detector is FMD A+C combined which covers wide rapidity range with
rapidity gap |∆η | > 0.9 between FMD and TPC. Systematic uncertainties of event plane determination detector is estimated with 7 different Event plane determination detector, V0C, V0A,
V0AC, FMDC, FMDA, FMDAC, TPCC2(-1.0 < η < -0.5), TPCA2(0.5 < η < 1.0). When Event
plane is determined with TPCC2(TPCA2), tracks in 0.5 < η < 1.0(-1.0 < η < -0.5) are used to
calculate vn with |∆η | > 1.0.
Figure 3.41 are azimuthal anisotropy of charged pion, kaon and proton(anti-proton) v2 as a117

function of centrality with 7 different Event plane determination detectors.
proton

v2

kaon

v2

v2

pion
0.2

0.2

0.2

0.15

0.15

0.15

0.1

0.1

0.1

0.05

0.05

0.05

0

0
0

10

20

30

40

50

0
0

10

20

centrality (%)

30

40

50

0

10

20

centrality (%)

30

40

50

centrality (%)

E.P. detector
FMDAC (default)
V0C
V0A
V0AC
FMDC
FMDA
TPCC2
TPCA2

Figure 3.41: Systematic difference of Event plane determination detector for charged pion,kaon and proton(anti-proton) pT integrated v2 as a function of centrality. pT is integrated from
0.15-1.5 GeV/cwhich is corresponding to same selection for HBT analysis.

Systematic Uncertainties related to particle identification selection with TPC and TOF
Charged pions, kaons and protons are identified with the probability of Bayesian approacheswith TPC and TOF. Systematic uncertainties associated with particle identification is studied
with changing the value of minimum probability of Bayesian approaches.
Figure 3.42 are azimuthal anisotropy of charged pion, kaon and proton(anti-proton) v2 as afunction of centrality. In this study, two different PID selection are estimated with minimum
probability of Bayesian approaches 75%(loose) and 90%(tight).
proton

0.2

v2

kaon

v2

v2

pion
0.2

0.15

0.15

0.15

0.1

0.1

0.1

0.05

0.05

0.05

0

0
0

10

20

30

40

50

centrality (%)

PID probability

0.2

80% (default)
75% (loose)

0
0

10

20

30

40

50

0

centrality (%)

10

20

30

40

50

centrality (%)

90% (tight)

Figure 3.42: Systematic difference of probability of Bayesian approaches 75% loose cut (redopen circle) and tight (blue open square) for charged pion, kaon and proton(anti-proton) pT
integrated v2 as a function of centrality. pT is integrated from 0.15-1.5 GeV/cwhich is corresponding to same selection for HBT analysis.

118

Table 3.8: Systematic table for the identified charged hadron v2 as a function of centrality
π + and π −centrality(%)
0-5
5-10 10-20 20-30 30-40 40-50
TPC ncls
3.99
3.23
2.36
1.68
1.23
0.95
Centrality estimator
3.99
0.84
3.25
0.16
0.25
0.43
Tracking Mode
0.99
0.56
0.09
0.28
0.35
0.44
Primary vertex position 2.79
0.92
0.40
0.25
0.15
0.15
Event plane detector
2.99
2.37
2.34
1.80
1.84
1.83
PID selection
0.21
0.16
0.08
0.03
0.02
0.04
Quadratic sum
7.04
4.24
4.66
2.50
2.26
2.17
+
−
K and KTPC ncls
4.20
3.55
2.75
2.17
1.68
1.40
Centrality estimator
4.59
1.05
3.42
0.01
0.14
0.58
Tracking Mode
2.30
2.03
1.50
1.19
0.80
0.56
Primary vertex position 2.78
0.82
0.31
0.21
0.07
0.10
Event plane detector
5.13
4.19
3.80
2.73
2.42
2.01
PID selection
0.63
0.60
0.41
0.30
0.24
0.20
Quadratic sum
8.86
6.04
6.02
3.70
3.07
2.59
p and p̄
TPC ncls
2.93
2.71
2.19
1.69
1.35
1.12
Centrality estimator
6.25
1.17
4.51
0.13
0.23
0.92
Tracking Mode
12.57 11.21 10.70 9.45
8.10
6.26
Primary vertex position 7.44
2.41
1.06
0.42
0.47
0.46
Event plane detector
13.79 11.27 9.19
5.22
3.81
1.84
PID selection
1.96
1.57
1.05
0.59
0.29
0.17
Quadratic sum
21.33 16.42 15.04 10.95 9.07
6.71

119

Chapter 4
Results
In this chapter, the results of extracted 3D HBT radii for charged pions as a function of azimuthal
pair angle with respect to 2nd -order and 3rd -order event plane measured in Pb-Pb collisions at√
sNN = 2.76 TeV are presented. We also report on the results obtained with the Event ShapeEngineering technique (q2 and q3 cut) applied to azimuthal anisotropy v2 and v3 and azimuthal
angle dependence of HBT radii with respect to 2nd -order and 3rd -order event plane which issensitive to source shape at freeze-out.

4.1 Azimuthal angle dependence of HBT radii with respect
to Ψ2
4.1.1 1D projection of 3D Correlation functions
Azimuthal pair angle of pions with respect to 2nd -order event plane is divided into 8 bins. Eachazimuthal angle bin width is π /8(rad).
Figure 4.1 shows correlation functions of charged pions measured in 0.2 < kT < 1.5 GeV/c
for two azimuthal bins (|φ pair − Ψ2 | < π /16 and 7π /16 < |φ pair − Ψ2 | < π /2) corresponding
to in-plane and out-plane directions of Ψ2 angle, respectively) in centrality 0-50% after the binby-bin correction on the event plane resolution. Three-dimensional correlation functions areprojected along to each axis (out, side, long). When making the projection of the 3D correlation
function to a specific q direction, the projections over the other q components was performed
within 50 GeV/c for each numerator and denominator in Eq. 3.65. Left columns show theC2 projected into the outward direction, middle columns for the sideward direction, and rightcolumns for the longitudinal direction. Top to bottom panels are corresponding to the projected120

correlation function in central (0-5%) to peripheral (40-50%) collisions. Black and red solidlines represent fit functions to the projected correlation functions measured in in-plane and outplane directions.
The 3D fitting to correlation function for all centrality and both azimuthal angle are wellsucceeded (chi2 /NDF is smaller than 1.0). For longitudinal axis, correlation functions at inplane and out-plane directions are almost same in all centrality. In most central collisions 0-5%
and 5-10%, there is no significant difference between correlation function measured in in-planeand out-plane of Ψ2 directions. On the other hand, difference of correlation functions between
in-plane and out-plane grows from central to peripheral collisions. Width of correlation function for outward in in-plane is explicitly larger than that in out-plane in centrality 30-40%. In
addition to that, this behaviour is opposite in sideward. Width of correlation function indicates
extracted HBT radii. Therefore, the significant difference of HBT radii in outward and sideward
can be seen and becomes larger in particular at peripheral collisions.
This difference of source size is discussed in detail with centrality dependence of extracted
HBT radii with respect to Ψ2 in the next section.

121

C2

C2

C2

in-plane : ϕ -Ψ2=0
pair
out-plane: ϕ -Ψ2=π/2

1.2

0-5%

1.2

1.2

1.1

1.1

pair

1.1

1

1

q

1
−0.1

0

0.1

out

q

−0.1

1.2

1.2

1.1

1.1

1.1

1

q

0

0.1

out

q

−0.1

1.2

1.1

1.1

1.1

1
0

−0.1

0.1

q

1.2

1.2

1.1

1.1

1.1

1

1

1
−0.1

0

0.1

out

q

−0.1

1.2

1.1

1.1

1.1

1
0

0.1

out

q

−0.1

1.2

1.2

1.1

1.1

1.1

1
−0.1

0

q

out

q long

1
−0.1

0.1

0.1

40-50%

1.2

1

0

side

C2

q

q long

1
−0.1

C2

0.1

C2

0

0.1

30-40%

1.2

−0.1

0

side

1.2

1

q long

C2

q

C2

0.1

C2

0

0.1

20-30%

1.2

−0.1

0

side

C2

out

C2

q

q long

1
−0.1

0.1

C2

0

0.1

10-20%

1.2

−0.1

0

side

1.2

1

q long

1
−0.1

C2

0.1

C2

0

C2

−0.1

0.1

5-10%

1.2

1

0

side

C2

0.1

C2

0

C2

−0.1

0

−0.1

0.1

q

side

0

0.1

q longFigure 4.1: Projection of 3D correlation function to 1D (outward, sideward, and longitudinalaxis) of charged pions in 0.2 < kT < 1.5 GeV/c for two azimuthal bins (|φ pair − Ψ2 | < π /16(Black marker) and 7π /16 < |φ pair − Ψ2 | < π /2 (Red marker)) at centrality 0-50%. The projection range of other q components are within 50 GeV/c . Solid lines denote the fitting functionof 3D correlation function.
1224.1.2 Centrality dependence of HBT radii with respect to Ψ2
Figure 4.2 shows the extracted 3D HBT radii for charged pions as a function of azimuthal
pair angle with respect to 2nd -order event plane Ψ2 for 6 different centralities. Data points at

φ pair − Ψ2 = π is same value to those at φ pair − Ψ2 = 0 for symmetry with respect to the event
plane. Charged pions for HBT analysis is measured in mid rapidity (|η | < 0.8) and event planeΨ2 is determined via FMD A+C combined at forward rapidity (1.7 < η < 5.0 and -3.4 < η <-1.7). Smeared oscillation of HBT radii due to finite event plane resolution is corrected withmodel independent bin-by-bin event plane resolution correction [63]. Systematic uncertainties
are plotted as transparent band and statistical uncertainties are smaller than marker size.
For boost-invariant system, the azimuthal angle dependence of HBT radii should be described with cosine and sine series functions [63].

R2µ (∆ϕ ) = R2µ ,0 + 2 ∑ R2µ ,n (∆ϕ ) cos(n∆ϕ ),(4.1)

R2os (∆ϕ ) = 2 ∑ R2os,n (∆ϕ ) sin(n∆ϕ ),

(4.2)

n

n

where R2µ ,n is the nth -order Fourier coefficients, µ is each direction in the Bertsch-Pratt parametrization (µ = out, side and long). ∆ϕ = φ − Ψn . In this thesis, the summation over n takes n = 2
for the case of 2nd -order event plane dependence. The parameter R2µ ,0 indicates the averageof squared HBT radii, while R2µ ,2 represents the oscillation of azimuthal angle dependence of
HBT radii. Azimuthal angle dependence of HBT radii are fitted with Eq.4.1. To extract theparameters, the azimuthal angle dependence of HBT radii shown in Fig. 4.2 are fitted with Eq.
4.2, 4.2.
As can be seen in Fig. 4.2, Rout and Rside have the explicit oscillations with respect toΨ2 and the oscillations in outward are out of phase with the one in sideward by π /2 radian.
Compared to the oscillation amplitude of Rout , Rside oscillation amplitude is much smaller for
all centralities. The out-side cross term Ros shows sine oscillation and its oscillation amplitude
grows from central to peripheral collisions. The average HBT radii become smaller from central
to peripheral collisions due to the geometry of initial overlap region. Contrary to average HBTradii, oscillation amplitude of Rout and Rside increase from central to peripheral collisions. λ ,Rlong , Rol , and Rsl have no significant oscillation.
123

0.4

0.3

0.2

R 2side (fm2)

R 2out (fm2)

λ

0.5

40
35

30

25

25

20

20

15

15

π
pair

0

- Ψ2 (rad)50

R 2os (fm2)R 2long (fm2)

ϕ

10
π
2

40

π

ϕ

pair

0

- Ψ2 (rad)R 2ol (fm2)

0

π
2

35

30

10
0.1

40

4

2

30

0

0

20

−2

−2

−4

−4

0

π
2

π

ϕ

pair

π
2

0

- Ψ2 (rad)

π

ϕ

pair

- Ψ2 (rad)

π

ϕ

pair

- Ψ2 (rad)pair- Ψ2 (rad)

4

2

10

π
2

0

π
2

π

ϕ

R 2sl (fm2)

Pb-Pb sNN=2.76 TeV
π+π+ and π-π- combined4

k T : 0.2-1.5 GeV/c2centrality
0-5%

0

5-10%
−2

10-20%−4

20-30%
0

π
2

ϕ

pair

π

30-40%- Ψ2 (rad)40-50%

Figure 4.2: Extracted HBT parameters (Rout , Rside , Rlong λ , Ros , Rol , and Rsl ) of charged pions in
0.2 < kT < 1.5 GeV/c as a function of azimuthal pair angle with respect to 2nd -order event planefor 6 different centrality bins. The data points at φ pair −Ψ2 = π are same value at φ pair −Ψ2 = 0.
Systematic uncertainties are shown with transparent bands.

124

4.2 Azimuthal anisotropy of charged hadrons
In this section, at first, event shape engineering technique is applied to azimuthal anisotropy
measurements which is sensitive to initial geometry.

4.2.1 Second order azimuthal anisotropy and Event Shape Engineering(q2 ) selection
Event shape engineering technique is the selection of event-by-event flow amplitude with themagnitude of flow vector qn described in Eq. 3.49 [15]. Based on the model simulation, it could
be one of the methods to select initial source shape [16]. Since ESE is the selection of eventby event flow amplitude, large or small q2 selection should affect to azimuthal anisotropy of
momentum space, and measured v2 should become large (small) with large (small) q2 selection
Figure 4.3 shows charged pions, charged kaons, proton (anti-proton) and charged particle v2
measured with event plane method as a function of centrality with each 20% q2 selection. Bothevent plane and q2 vector are determined via FMD A+Ci and rapidity gap between flow measurement and event plane determination is |∆η | > 0.9. Transverse momentum (pT ) is integrated
from 0.15-1.5GeV/c which is corresponding to same pT range to HBT measurements.
For three particle species, larger (smaller) azimuthal anisotropy v2 can be selected with ESE
selection. Unbiased v2 is comparable to v2 with 40-60% q2 selection. Difference between ESE
selected v2 and unbiased v2 increases from central to peripheral collisions.
In order to qualitatively estimate the ESE effect, the ratio of v2 with and without q2 selection
is calculated as shown in Fig. 4.4. For all three particle species, azimuthal anisotropy v2 isenhanced (suppressed) with q2 selection. One can find that v2 ratio does not show significant
centrality dependence. However, in most central collisions 0-5%, the effect of ESE selection issmaller than that in the other centrality classes. This behaviour is consistent to previous results[19], and it could be due to smaller v2 signal and Ψ2 resolution in most central collisions. By
applying q2 selection, azimuthal anisotropy v2 is enhanced by 26% for q2 :80-100%, 8% for
q2 :60-80%, and suppressed by 3% for q2 :40-60%, 14% for q2 :20-40%, and 26% for q2 :0-20%.
Figure 4.5, 4.6 and 4.7 shows pT dependence of charged pions, charged kaons and protons(anti-protons) as a function of centrality(0-50%). Each 20% q2 selection is applied to flow
measurements up to pT 4GeV/c. As is the case of pT integrated v2 measurements (Fig. ??),

125

ESE q2 selection effect (enhancement and suppression) of v2 can be found for all three particle
species. Difference of v2 with and without q2 selection can be seen up to pT 4 GeV/c, and it
depends on the value of v2 .
Transeverse momentum dependence of the ratio of q2 selected to no q2 selected v2 are shownin Fig. 4.8, 4.9 and 4.10. Effect of q2 selection does not have explicit pT dependence for pions,
kaons, and protons. However, in the largest q2 class at centrality 5-40%, the ratio of v2 with andwithout q2 selection slightly decreases with increasing pT , and this behaviour is also found forthe v2 ratio in the smallest q2 . But pT dependence of v2 ratio is opposite to the one in the largest
q2 selection. The ratio of v2 in the smallest q2 selection becomes larger from low pT to high pT ).
This small pT dependence might be interpreted that anisotropic flow mostly originates from low
momentum particles and also indicates that the magnitude of q2 vector is a global property ofthe event which is not biased by jet.
Figure 4.11 denotes the comparison of the v2 ratios with and without q2 selection amang
three particle species in centrality 0-50%. In all centrality and all q2 classes, the enhancement
and suppression with q2 selection does not depends on the particle species.

126

Figure 4.3: Charged hadron and identified hadron (π , K and p) pT integrated v2 as a functionof centrality. Both Ψ2 and q2 are determined via FMD A+C and pT are integrated from 0.15 to
1.5GeV/c. Each 20% ESE q2 selection is applied to flow measurements(closed markers). Open
black markers denote no ESE selected v2 . Systematic uncertainties are depicted as transparent
bands.

127

Figure 4.4: Ratio of v2 with each 20% q2 selection to without q2 selection (unbiased sample)
for unidentified and identified charged hadrons (π , K and p). Both Ψ2 and q2 vector are determined via FMD A+C and pT are integrated from 0.15 to 1.5GeV/c. Systematic uncertainties
are depicted as transparent bands.

128

Figure 4.5: Measurement of charged pion v2 as a function of pT for 6 centrality classes witheach 20% q2 selection and no q2 selected samples. Systematic uncertainties are plotted astransparent bands.

129

Figure 4.6: Measurement of charged kaon v2 as a function of pT for 6 centrality classes witheach 20% q2 selection and no q2 selected samples. Systematic uncertainties are plotted astransparent bands.

130

Figure 4.7: Measurement of protons and anti-protons v2 as a function of pT for 6 centrality
classes with each 20% q2 selection and no q2 selected samples. Systematic uncertainties areplotted as transparent bands.

131

Figure 4.8: Ratio of v2 with each 20% q2 selection to without q2 selection (unbiased sample)
for charged pions.

132

Figure 4.9: Ratio of v2 with each 20% q2 selection to without q2 selection (unbiased sample)
for charged kaons.

133

Figure 4.10: Ratio of v2 with each 20% q2 selection to without q2 selection (unbiased sample)
for protons.

134

Figure 4.11: Transverse momentum dependence of v2 ratio with each 20% q2 selection towithout q2 selection (unbiased sample) for unidentified and identified charged hadrons (π , K
and p) for 6 centrality bins.

1354.2.2 Third-order azimuthal anisotropy and Event Shape Engineering (q3 )selection
If higher-order azimuthal anisotropy (such as v3 , v4 and so on) originates from the initial density fluctuations and subsequent hydrodynamical evolution, Event Shape Engineering techniqueapplied to the higher-orders to possibly study a response of such initial density fluctuations to
the hydrodynamic evolution and an effect to final state spatial anisotropies. In this thesis, westudy a sensitivity of 3rd -order azimuthal anisotropy v3 to q3 .
Figure 4.12 shows v3 of charged pions, charged kaons, proton(anti-proton) and charged
particle which are measured with event plane method as a function of centrality with each 20%
q3 selection. In the same way to q2 study, event plane and q3 vector are determined via FMD
A+C and a gap of pseudorapidity for v3 measurements and event plane reconstruction is |∆η |> 0.9. Transverse momentum is integrated from 0.15-1.5 GeV/c which is the same pT range asthat for HBT measurements.
For charged pions, charged kaons, and protons (anti-protons), v3 is enhanced (suppressed)by q3 selection. Triangular flow v3 without q3 selection is comparable to the one with 60-80%
q3 selection. Contrary to q2 selection to v2 , the difference of v3 with and without q3 selection
seems to slightly depend on centrality.
For the qualitative estimation of ESE effect, ratio of v3 with q3 selection to the one without
q3 selection is shown in Fig. 4.13. In top 20 % q3 selection, v3 ratio explicitly depends on
centrality. The ratio of v3 is enhanced 20% in most central collisions (0-5%) , but it decreases
down to 9% in peripheral collisions (40-50%). On the other hand, no significant centrality
dependence can be seen in the ratio of v3 for the other q3 classes.
Basically, selectivity of large (small) q3 selection depends on the Ψ3 resolution and Ψ3
resolution is worse than Ψ2 resolution and it decreases from central to peripheral collisions,this centrality dependence in the q3 selection could be explained by the worse Ψ3 resolution.
However not only Ψ3 resolution but also Ψ3 resolution depends on centrality. Therefore, if
the correlation of event plane resolution and ESE selectivity is linear, centrality dependence
should be also seen in the ratio of v2 . it might be explained by the correlation of event plane
resolution and ESE selectivity is "non-linear" and selectivity rapidly decreases in smaller event
plane resolution.

136

Figure 4.13 is fitted with 0th -order polynomial function to extract the value of enhancement(suppression) of v3 ratio. The ratio of v3 is enhanced by 2% for q3 :60-80% and suppressed by
5% for q3 :40-60%, 10% for q3 :20-40%, and 13% for q3 :0-20%.
Figure 4.14, 4.15, and 4.16 shows pT dependence of charged pions, charged kaons, andprotons (anti-protons) for 6 different centrality bins. Each 20% q3 selection is applied to v3
measurements up to pT 4GeV/c. The enhancement (suppression) can be found for π + (π − ), K +(K − ), and p ( p̄). The defference of the v3 ratio with and without q3 selection depends on v3
signal size.
Transverse momentum dependence of the ratios of v3 with and without q3 selection are
shown in Fig. 4.17, 4.18, and 4.19. Similarly to v2 ratio, no significant pT dependence can beseen in v3 ratio with and without q3 selection.
Figure 4.20 shows the comparison of v3 ratio of charged pions, kaons, and protons (antiprotons) for 6 different centrality bins and two q3 classes (60-80% and 80-100%). As well as
q2 selection, an effect of q3 selection to v3 does not depend on the particle species.

Figure 4.12: Centrality dependence of v3 for unidentified and identified charged hadrons (π , K
and p) with and without q3 selection. Results for each 20% q3 selection are shown and pT are
integrated from 0.15 to 1.5. Systematic uncertainties are plotted as transparent bands.

137

Figure 4.13: Ratio of v3 with each 20% q3 selection to without q3 selection (unbiased sample)
for unidentified and identified charged hadrons (π , K and p). Both Ψ3 and q3 vector are determined via FMD A+C and pT are integrated from 0.15 to 1.5GeV/c. Systematic uncertainties
are depicted as transparent bands.

138

Figure 4.14: Measurement of charged pion v3 as a function of pT for 6 centrality classes
with each 20% q3 selection and no q3 selected samples. Systematic uncertainties are plotted astransparent bands.

139

Figure 4.15: Measurement of charged kaon v3 as a function of pT for 6 centrality classes
with each 20% q3 selection and no q3 selected samples. Systematic uncertainties are plotted astransparent bands.

140

Figure 4.16: Measurement of protons and anti-protons v3 as a function of pT for 6 centrality
classes with each 20% q3 selection and no q3 selected samples. Systematic uncertainties areplotted as transparent bands.

141

Figure 4.17: Ratio of v3 with each 20% q3 selection to without q3 selection (unbiased sample)
for charged pions.

142

Figure 4.18: Ratio of v3 with each 20% q3 selection to without q3 selection (unbiased sample)
for charged kaons.

143

Figure 4.19: Ratio of v3 with each 20% q3 selection to without q3 selection (unbiased sample)
for protons.

144

Figure 4.20: Transverse momentum dependence of v3 ratio with each 20% q3 selection towithout q3 selection (unbiased sample) for unidentified and identified charged hadrons (π , K
and p) for 6 centrality bins.

1454.3 Azimuthal angle dependence of HBT radii with respect
to Ψ2 with Event Shape Engineering q2 selection
Study of pion, kaon and proton v2 and v3 measurement with Event Shape Engineering techniquegives us the results that larger and smaller v2 and v3 can be selected.
Study on v2 and v3 of unidentified and identified hadrons (π , K and p) with ESE shows thatevent-by-event v2 and v3 can be selected, though the enhancement (suppression) varies fromsmall to large.
In order to investigate the correlation between initial and final source shape, Event ShapeEngineering q2 selection is applied to azimuthal angle dependence of HBT radii.
Figure 4.21, 4.22, 4.23, 4.24, and 4.25 shows the extracted 3D HBT radii for charged pions
as a function of azimuthal pair angle with respect to Ψ2 for 6 different centrality bins with
0-20%, 20-40%, 40-60%, 60-80%, and 80-100% q2 selection, respectively. Azimuthal angle
dependence of HBT radii with respect to Ψ2 without q2 selection are simultaneously plotted as
open circles.
In Fig. 4.25, no significant modification can be found in λ , Rlong , Rol , and Rsl with q2
selection. But oscillation amplitudes of HBT radii in Rout and Ros are explicitly enhanced, as is
the case with large q2 selected v2 . Also the oscillation amplitude of Rside is slightly enhancedwith this q2 selection.
In Fig. 4.22, 4.23 and 4.23, oscillation amplitudes of Rout , Rside , and Ros are slightly modified with 20-40%, 40-60%, and 60-80% q2 selections, respectively.
In Fig. 4.21, oscillation amplitudes of Rout and Ros are slightly suppressed with 0-20% q2
selection. Although q2 selection modifies the oscillation amplitudes of Rout and Ros , sign of
their oscillations does not change. The oscillation amplitude of Rside are also slightly suppressedwith 0-20% q2 selection. However, in most central 0-5% collisions, Rside has changed oscillation sign from concave up to convex up with 0-20% q2 selection. Therefore, in this q2 range,
Rout and Rside has same oscillation sign (convex up).

146

0.4

0.3

0.2

0

π
2

35

pair

25

25

20

20

15

15

10

10
π
2

0

R 2os (fm2)

40

35
30

- Ψ2 (rad)

50

40

30

π

ϕ

R 2long (fm2)

40

π

ϕ

pair

0

- Ψ2 (rad)10

R 2ol (fm2)

0.1

R 2side (fm2)

R 2out (fm2)

λ

0.5

5

π
2

π

ϕ

pair

- Ψ2 (rad)pair- Ψ2 (rad)

4

2
30

0

20

−5

−2
−4

10
0

π
2

π

ϕ

R 2sl (fm2)

0

pair

2

0

−2
−4
π
2

π

ϕ

pair

π
2

0

- Ψ2 (rad)

4

0

−10

- Ψ2 (rad)

π

ϕ

pair

- Ψ2 (rad)q : 0-20%
2
centrality
0-5%
5-10%
10-20%
20-30%
30-40%
40-50%

0

π
2

π

ϕ

No q selection
2
centrality
0-5%
5-10%
10-20%
20-30%
30-40%
40-50%

Figure 4.21: Extracted HBT parameters (Rout , Rside , Rlong , λ , Ros , Rol , and Rsl ) of charged pions
in 0.2 < kT < 1.5 GeV/c as a function of azimuthal pair angle with respect to 2nd -order event
plane for 6 different centrality bins. Bottom 20% q2 selection is applied to HBT measurements.
The data points at φ pair − Ψ2 = π are same value at φ pair − Ψ2 = 0. Systematic uncertainties
are plotted as transparent bands. All points of Ros are shifted along the y-axis for visibility.

147

0.4

0.3

0.2

0

π
2

35

pair

25

25

20

20

15

15

10

10
π
2

0

R 2os (fm2)

40

35
30

- Ψ2 (rad)

50

40

30

π

ϕ

R 2long (fm2)

40

π

ϕ

pair

0

- Ψ2 (rad)10

R 2ol (fm2)

0.1

R 2side (fm2)

R 2out (fm2)

λ

0.5

5

π
2

π

ϕ

pair

- Ψ2 (rad)pair- Ψ2 (rad)

4

2
30

0

20

−5

−2
−4

10
0

π
2

π

ϕ

R 2sl (fm2)

0

pair

2

0

−2
−4
π
2

π

ϕ

pair

π
2

0

- Ψ2 (rad)

4

0

−10

- Ψ2 (rad)

π

ϕ

pair

- Ψ2 (rad)q : 20-40%
2centrality
0-5%
5-10%
10-20%
20-30%
30-40%
40-50%

0

π
2

π

ϕ

No q selection
2
centrality
0-5%
5-10%
10-20%
20-30%
30-40%
40-50%

Figure 4.22: Extracted HBT parameters (Rout , Rside , Rlong , λ , Ros , Rol , and Rsl ) of charged pions
in 0.2 < kT < 1.5 GeV/c as a function of azimuthal pair angle with respect to 2nd -order event
plane for 6 different centrality bins. 20-40% q2 selection is applied to HBT measurements. The
data points at φ pair − Ψ2 = π are same value at φ pair − Ψ2 = 0. Systematic uncertainties areplotted as transparent bands. All points of Ros are shifted along the y-axis for visibility.

148

0.4

0.3

0.2

0

π
2

35

pair

25

25

20

20

15

15

10

10
π
2

0

R 2os (fm2)

40

35
30

- Ψ2 (rad)

50

40

30

π

ϕ

R 2long (fm2)

40

π

ϕ

pair

0

- Ψ2 (rad)10

R 2ol (fm2)

0.1

R 2side (fm2)

R 2out (fm2)

λ

0.5

5

π
2

π

ϕ

pair

- Ψ2 (rad)pair- Ψ2 (rad)

4

2
30

0

20

−5

−2
−4

10
0

π
2

π

ϕ

R 2sl (fm2)

0

pair

2

0

−2
−4
π
2

π

ϕ

pair

π
2

0

- Ψ2 (rad)

4

0

−10

- Ψ2 (rad)

π

ϕ

pair

- Ψ2 (rad)q : 40-60%
2centrality
0-5%
5-10%
10-20%
20-30%
30-40%
40-50%

0

π
2

π

ϕ

No q selection
2
centrality
0-5%
5-10%
10-20%
20-30%
30-40%
40-50%

Figure 4.23: Extracted HBT parameters (Rout , Rside , Rlong , λ , Ros , Rol , and Rsl ) of charged pions
in 0.2 < kT < 1.5 GeV/c as a function of azimuthal pair angle with respect to 2nd -order event
plane for 6 different centrality bins. 40-60% q2 selection is applied to HBT measurements. The
data points at φ pair − Ψ2 = π are same value at φ pair − Ψ2 = 0. Systematic uncertainties areplotted as transparent bands. All points of Ros are shifted along the y-axis for visibility.

149

0.4

0.3

0.2

0

π
2

35

pair

25

25

20

20

15

15

10

10
π
2

0

R 2os (fm2)

40

35
30

- Ψ2 (rad)

50

40

30

π

ϕ

R 2long (fm2)

40

π

ϕ

pair

0

- Ψ2 (rad)10

R 2ol (fm2)

0.1

R 2side (fm2)

R 2out (fm2)

λ

0.5

5

π
2

π

ϕ

pair

- Ψ2 (rad)pair- Ψ2 (rad)

4

2
30

0

20

−5

−2
−4

10
0

π
2

π

ϕ

R 2sl (fm2)

0

pair

2

0

−2
−4
π
2

π

ϕ

pair

π
2

0

- Ψ2 (rad)

4

0

−10

- Ψ2 (rad)

π

ϕ

pair

- Ψ2 (rad)q : 60-80%
2centrality
0-5%
5-10%
10-20%
20-30%
30-40%
40-50%

0

π
2

π

ϕ

No q selection
2
centrality
0-5%
5-10%
10-20%
20-30%
30-40%
40-50%

Figure 4.24: Extracted HBT parameters (Rout , Rside , Rlong , λ , Ros , Rol , and Rsl ) of charged pions
in 0.2 < kT < 1.5 GeV/c as a function of azimuthal pair angle with respect to 2nd -order event
plane for 6 different centrality bins. 60-80% q2 selection is applied to HBT measurements. The
data points at φ pair − Ψ2 = π are same value at φ pair − Ψ2 = 0. Systematic uncertainties areplotted as transparent bands. All points of Ros are shifted along the y-axis for visibility.

150

0.4

0.3

0.2

0

π
2

35

pair

25

25

20

20

15

15

10

10
π
2

0

R 2os (fm2)

40

35
30

- Ψ2 (rad)

50

40

30

π

ϕ

R 2long (fm2)

40

π

ϕ

pair

0

- Ψ2 (rad)10

R 2ol (fm2)

0.1

R 2side (fm2)

R 2out (fm2)

λ

0.5

5

π
2

π

ϕ

pair

- Ψ2 (rad)pair- Ψ2 (rad)

4

2
30

0

20

−5

−2
−4

10
0

π
2

π

ϕ

R 2sl (fm2)

0

pair

2

0

−2
−4
π
2

π

ϕ

pair

π
2

0

- Ψ2 (rad)

4

0

−10

- Ψ2 (rad)

π

ϕ

pair

- Ψ2 (rad)q : 80-100%
2
centrality
0-5%
5-10%
10-20%
20-30%
30-40%
40-50%

0

π
2

π

ϕ

No q selection
2
centrality
0-5%
5-10%
10-20%
20-30%
30-40%
40-50%

Figure 4.25: Extracted HBT parameters (Rout , Rside , Rlong , λ , Ros , Rol , and Rsl ) of charged pions
in 0.2 < kT < 1.5 GeV/c as a function of azimuthal pair angle with respect to 2nd -order event
plane for 6 different centrality bins. Top 20% q2 selection is applied to HBT measurements.
The data points at φ pair − Ψ2 = π are same value at φ pair − Ψ2 = 0. Systematic uncertainties
are plotted as transparent bands. All points of Ros are shifted along the y-axis for visibility.

1514.4 Azimuthal angle dependence of HBT radii with respect
to Ψ3
4.4.1 1D projection of 3D Correlation functions
Azimuthal pair angle of pions with respect to 3rd -order event plane is divided into 8 bins. Eachbin width is π /12(rad).
Figure 4.26 shows the correlation function of charged pions measured for 0.2 < kT < 1.5
GeV/cfor two azimuthal bins (|φ pair − Ψ3 | < π /24 and 7π /24 < |φ pair − Ψ3 | < π /3) corresponding to in-plane and out-plane directions of Ψ3 angle, respectively) at centrality 0-50%
after the bin-by-bin correction on the event plane resolution. As is the case with azimuthal
angle dependence of HBT radii with respect to Ψ2 , three-dimensional correlation functions areprojected along to each axis (outward, sideward, and longitudinal directions). The other q components within 50 MeV/c are projected. To make the projection of the 3D correlation function
to a specific q direction, the projections over the other q components was performed within
50 GeV/c for each numerator and denominator in Eq. 3.65. Left columns show the correlation
function C2 in the outward direction, middle columns for C2 in the sideward directions, and rightcolumns for C2 in the longitudinal directions. Difference of row is centrality (top row figures
are central (0-5%) and bottom row figures indicate peripheral (40-50%) collisions). Dashedlines are fit function which is also projected to each directions. Top to bottom panels show the
projected correlation function in central (0-5%) to peripheral (40-50%) collisions. Black andred solid lines represent fit functions to the projected correlation functions measured in in-planeand out-plane of the Ψ3 directions.
Contrary to 2nd -order event plane dependence of HBT correlation function, no significant
difference between correlation function in-plane and out-plane directions with respect to Ψ3 inall centrality for outward, sideward, and longitudinal axis.

152

ϕ -Ψ3=0pair
ϕ -Ψ3=π/3

C2

C2

C2
1.2

0-5%

1.2

1.2

1.1

1.1

pair

1.1

1

1

q

1
−0.1

0

0.1

out

q

−0.1

1.2

1.2

1.1

1.1

1.1

1

q

0

0.1

out

q

−0.1

1.2

1.1

1.1

1.1

1
0

−0.1

0.1

q

1.2

1.2

1.1

1.1

1.1

1

1

1
−0.1

0

0.1

out

q

−0.1

1.2

1.1

1.1

1.1

1
0

0.1

out

q

−0.1

1.2

1.2

1.1

1.1

1.1

1
−0.1

0

q

out

q long

1
−0.1

0.1

0.1

40-50%

1.2

1

0

side

C2

q

q long

1
−0.1

C2

0.1

C2

0

0.1

30-40%

1.2

−0.1

0

side

1.2

1

q long

C2

q

C2

0.1

C2

0

0.1

20-30%

1.2

−0.1

0

side

C2

out

C2

q

q long

1
−0.1

0.1

C2

0

0.1

10-20%

1.2

−0.1

0

side

1.2

1

q long

1
−0.1

C2

0.1

C2

0

C2

−0.1

0.1

5-10%

1.2

1

0

side

C2

0.1

C2

0

C2

−0.1

0

−0.1

0.1

q

side

0

0.1

q longFigure 4.26: Projection of 3D correlation function to 1D (outward, sideward, and longitudinal
directions) of charged pions in 0.2 < kT < 1.5 GeV/c for two different azimuthal angle classes,|φ pair − Ψ3 | < π /24 (Black marker) and 7π /24 < |φ pair − Ψ3 | < π /3 (Red marker) in centrality
0-50%. The projection range of the other q components are within 50 GeV/c . Solid line denotes
the fitting function of 3D correlation function.
153

4.4.2 Centrality dependence of HBT radii with respect to Ψ3
Figure 4.27 shows the extracted 3D HBT radii for charged pions as a function of azimuthal pairangle with respect to 3rd -order event plane Ψ3 for 6 different centralities. The data points at

φ pair − Ψ3 = π /3 are same value to those at φ pair − Ψ3 = 0 based on the symmetry with respect
to event plane. Charged pions for HBT analysis are measured at mid rapidity and event planeΨ3 is determined via FMD A+C combined. Systematic uncertainties are plotted as transparent
bands and statistical uncertainties are smaller than marker size.
The azimuthal angle dependence of HBT radii is fitted with Eq.4.1. But, in 3rd order event
plane case, the summation over n takes n = 3 in Eq.4.1.
As is the case with azimuthal angle dependence of HBT radii with respect to Ψ2 , λ , Rlong ,Rol , and Rsl have no explicit oscillation in all centrality. On the other hand, Rout and Rside havefinite oscillations. For Ψ2 case, oscillations of Rout are convex upward and those of Rside areconcave upward. However, for Ψ3 case, both Rout and Rside are convex upward. This behaviourof the azimuthal angle dependence Rside in Ψ3 is explicitly different to Ψ2 . However, as we
mentioned in Sec. 4.3, in bottom 20% q2 and most central 0-5% collisions, oscillation signs of
Rout and Rside are same and concave up. Two different analysises are quit similar at the point ofsame oscillation signs in outward and sideward. Contrary to Ψ2 case, no significant oscillation
relative to Ψ3 in Ros can be found.

4.5 Azimuthal angle dependence of HBT radii with respect
to Ψ3 with Event Shape Engineering q3 selection
Figure 4.28, 4.29, 4.30, 4.31, and 4.32 shows the extracted 3D HBT radii for charged pions
as a function of azimuthal pair angle with respect to 3rd -order event plane Ψ3 for 6 different
centralities with each 20% q3 selection. The results without q3 selection are simultaneously
plotted as open circles.
For v3 measurement with q3 selection, measured v3 is enhanced (suppressed) with large(small) q3 selection and the difference between q3 selected v3 and inclusive v3 is remarkable at
central collisions.
But there is no significant difference in azimuthal angle dependence of HBT radii withrespect to Ψ3 between with and without q3 selection even in top 20% and bottom 20% q3154selection, while the explicit differences can be found in azimuthal angle dependence of HBTradii with respect to Ψ2 with q2 selection.

155

0.4

0.3

0.2

R 2side (fm2)

R 2out (fm2)

λ

0.5

40
35

30

25

25

20

20

15

15

2π
3
pair

0

- Ψ3 (rad)50

R 2os (fm2)R 2long (fm2)

ϕ

10
π
3

40

30

ϕ

2π
3

pair

0

- Ψ3 (rad)R 2ol (fm2)

0

π
3

35

30

10
0.1

40

1.5
1

π
3

ϕ

2π
3

pair

- Ψ3 (rad)pair- Ψ3 (rad)

1.5
1

0.5

0.5

0

0

−0.5

−0.5

−1

−1

−1.5

−1.5

20

10
0

π
3

2π
3

ϕ

pair

- Ψ3 (rad)

ϕ

pair

- Ψ3 (rad)ϕR 2sl (fm2)

Pb-Pb sNN=2.76 TeV
1.5π+π+ and π-π- combined1

k T : 0.2-1.5 GeV/ccentrality

0.5
0

0-5%

−0.5

5-10%
10-20%−1

0

20-30%

−1.5

π
3

2π
3

0

π
3

2π
3

30-40%ϕpair- Ψ3 (rad)40-50%

Figure 4.27: Extracted HBT parameters (Rout , Rside , Rlong , λ , Ros , Rol , and Rsl ) of charged pions
in 0.2 < kT < 1.5 GeV/cas a function of azimuthal pair angle with respect to 3rd -order event
plane for 6 different centrality bins. The data points at φ pair − Ψ3 = 2π /3 is same value to thoseat φ pair − Ψ3 = 0. Systematic uncertainties are shown as transparent bands.
0

π
3

2π
3

156

0.5

R 2side (fm2)

R 2out (fm2)

λ

0.6

40
35

40
35

30

30

25

25

20

20

15

15

10

10

0.4

0.3

0.2

0

π
3

2π
3
pair

π
3

0

- Ψ3 (rad)50

R 2os (fm2)R 2long (fm2)

ϕ

40

ϕ

2π
3

pair

0

- Ψ3 (rad)R 2ol (fm2)

0.1

4

π
3

ϕ

2π
3

pair

- Ψ3 (rad)pair- Ψ3 (rad)

4

2

2

30

0

0

20

−2

−2

−4

−4

10
0

π
3

2π
3

R 2sl (fm2)

ϕ

pair

4
2
0
−2
−4

0

π
3

ϕ

2π
3

pair

π
3

0

- Ψ3 (rad)- Ψ3 (rad)

ϕ

2π
3

pair

- Ψ3 (rad)q : 0-20%
3
centrality
0-5%
5-10%
10-20%
20-30%
30-40%
40-50%

0

π
3

ϕ

2π
3

No q selection
3
centrality
0-5%
5-10%
10-20%
20-30%
30-40%
40-50%

Figure 4.28: Extracted HBT parameters (Rout , Rside , Rlong , λ , Ros , Rol , and Rsl ) of charged pions
in 0.2 < kT < 1.5 GeV/c as a function of azimuthal pair angle with respect to 3rd -order event
plane for 6 different centrality bins. Bottom 20 % q3 selection is applied to HBT measurements.
The data points at φ pair − Ψ3 = 2π /3 is same value to those at φ pair − Ψ3 = 0. Systematic
uncertainties are plotted as transparent bands. All points of Ros are shifted along the y-axis forvisibility.

157

0.5

R 2side (fm2)

R 2out (fm2)

λ

0.6

40
35

40
35

30

30

25

25

20

20

15

15

10

10

0.4

0.3

0.2

0

π
3

2π
3
pair

π
3

0

- Ψ3 (rad)50

R 2os (fm2)R 2long (fm2)

ϕ

40

ϕ

2π
3

pair

0

- Ψ3 (rad)R 2ol (fm2)

0.1

4

π
3

ϕ

2π
3

pair

- Ψ3 (rad)pair- Ψ3 (rad)

4

2

2

30

0

0

20

−2

−2

−4

−4

10
0

π
3

2π
3

R 2sl (fm2)

ϕ

pair

4
2
0
−2
−4

0

π
3

ϕ

2π
3

pair

π
3

0

- Ψ3 (rad)- Ψ3 (rad)

ϕ

2π
3

pair

- Ψ3 (rad)q : 20-40%
3centrality
0-5%
5-10%
10-20%
20-30%
30-40%
40-50%

0

π
3

ϕ

2π
3

No q selection
3
centrality
0-5%
5-10%
10-20%
20-30%
30-40%
40-50%

Figure 4.29: Extracted HBT parameters (Rout , Rside , Rlong , λ , Ros , Rol , and Rsl ) of charged pions
in 0.2 < kT < 1.5 GeV/c as a function of azimuthal pair angle with respect to 3rd -order event
plane for 6 different centrality bins. 20-40% q3 selection is applied to HBT measurements.
The data points at φ pair − Ψ3 = 2π /3 is same value to those at φ pair − Ψ3 = 0. Systematic
uncertainties are plotted as transparent bands. All points of Ros are shifted along the y-axis forvisibility.

158

0.5

R 2side (fm2)

R 2out (fm2)

λ

0.6

40
35

40
35

30

30

25

25

20

20

15

15

10

10

0.4

0.3

0.2

0

π
3

2π
3
pair

π
3

0

- Ψ3 (rad)50

R 2os (fm2)R 2long (fm2)

ϕ

40

ϕ

2π
3

pair

0

- Ψ3 (rad)R 2ol (fm2)

0.1

4

π
3

ϕ

2π
3

pair

- Ψ3 (rad)pair- Ψ3 (rad)

4

2

2

30

0

0

20

−2

−2

−4

−4

10
0

π
3

2π
3

R 2sl (fm2)

ϕ

pair

4
2
0
−2
−4

0

π
3

ϕ

2π
3

pair

π
3

0

- Ψ3 (rad)- Ψ3 (rad)

ϕ

2π
3

pair

- Ψ3 (rad)q : 40-60%
3centrality
0-5%
5-10%
10-20%
20-30%
30-40%
40-50%

0

π
3

ϕ

2π
3

No q selection
3
centrality
0-5%
5-10%
10-20%
20-30%
30-40%
40-50%

Figure 4.30: Extracted HBT parameters (Rout , Rside , Rlong , λ , Ros , Rol , and Rsl ) of charged pions
in 0.2 < kT < 1.5 GeV/c as a function of azimuthal pair angle with respect to 3rd -order event
plane for 6 different centrality bins. 40-60% q3 selection is applied to HBT measurements.
The data points at φ pair − Ψ3 = 2π /3 is same value to those at φ pair − Ψ3 = 0. Systematic
uncertainties are plotted as transparent bands. All points of Ros are shifted along the y-axis forvisibility.

159

0.5

R 2side (fm2)

R 2out (fm2)

λ

0.6

40
35

40
35

30

30

25

25

20

20

15

15

10

10

0.4

0.3

0.2

0

π
3

2π
3
pair

π
3

0

- Ψ3 (rad)50

R 2os (fm2)R 2long (fm2)

ϕ

40

ϕ

2π
3

pair

0

- Ψ3 (rad)R 2ol (fm2)

0.1

4

π
3

ϕ

2π
3

pair

- Ψ3 (rad)pair- Ψ3 (rad)

4

2

2

30

0

0

20

−2

−2

−4

−4

10
0

π
3

2π
3

R 2sl (fm2)

ϕ

pair

4
2
0
−2
−4

0

π
3

ϕ

2π
3

pair

π
3

0

- Ψ3 (rad)- Ψ3 (rad)

ϕ

2π
3

pair

- Ψ3 (rad)q : 60-80%
3centrality
0-5%
5-10%
10-20%
20-30%
30-40%
40-50%

0

π
3

ϕ

2π
3

No q selection
3
centrality
0-5%
5-10%
10-20%
20-30%
30-40%
40-50%

Figure 4.31: Extracted HBT parameters (Rout , Rside , Rlong , λ , Ros , Rol , and Rsl ) of charged pions
in 0.2 < kT < 1.5 GeV/c as a function of azimuthal pair angle with respect to 3rd -order event
plane for 6 different centrality bins. 60-80% q3 selection is applied to HBT measurements.
The data points at φ pair − Ψ3 = 2π /3 is same value to those at φ pair − Ψ3 = 0. Systematic
uncertainties are plotted as transparent bands. All points of Ros are shifted along the y-axis forvisibility.

160

0.5

R 2side (fm2)

R 2out (fm2)

λ

0.6

40
35

40
35

30

30

25

25

20

20

15

15

10

10

0.4

0.3

0.2

0

π
3

2π
3
pair

π
3

0

- Ψ3 (rad)50

R 2os (fm2)R 2long (fm2)

ϕ

40

ϕ

2π
3

pair

0

- Ψ3 (rad)R 2ol (fm2)

0.1

4

π
3

ϕ

2π
3

pair

- Ψ3 (rad)pair- Ψ3 (rad)

4

2

2

30

0

0

20

−2

−2

−4

−4

10
0

π
3

2π
3

R 2sl (fm2)

ϕ

pair

4
2
0
−2
−4

0

π
3

ϕ

2π
3

pair

π
3

0

- Ψ3 (rad)- Ψ3 (rad)

ϕ

2π
3

pair

- Ψ3 (rad)q : 80-100%
3
centrality
0-5%
5-10%
10-20%
20-30%
30-40%
40-50%

0

π
3

ϕ

2π
3

No q selection
3
centrality
0-5%
5-10%
10-20%
20-30%
30-40%
40-50%

Figure 4.32: Extracted HBT parameters (Rout , Rside , Rlong , λ , Ros , Rol , and Rsl ) of charged pions
in 0.2 < kT < 1.5 GeV/c as a function of azimuthal pair angle with respect to 3rd -order event
plane for 6 different centrality bins. Top 20% q3 selection is applied to HBT measurements.
The data points at φ pair − Ψ3 = 2π /3 is same value to those at φ pair − Ψ3 = 0. Systematic
uncertainties are plotted as transparent bands. All points of Ros are shifted along the y-axis forvisibility.

1614.6 Consistency check of HBT radii with the previous resultsfrom ALICE
Figure 4.33, 4.34, and 4.34 shows 3D HBT radii (Rout , Rside , and Rlong ) of charged pions as afunction of pair transverse momentum kT for 6 centrality bins. My results are plotted as closed
circles which is obtained with azimuthal dependence of HBT radii with respect to Ψ3 . Opensquared markers are results of azimuthal differential pion HBT with respect to Ψ2 [60]. Opencircle markers are results of azimuthal integrated pion HBT analysis [61]. Error bars of myanalysis and open squared markers are quadratic sum of sytematic and statistical uncertainties.
Shaded bands are systematic uncertainties of open circles. All three results are calculated with√data measured in Pb-Pb collisions at sNN = 2.76 TeV collisions.
Rout of my calculations are fully consistent to two published results within systematic uncertainties for all centralities and kT . Rside of my calculations in central 0-10% collisions areslightly smaller than the other calculations. But all results are consistent within systematic uncertainties. Rlong of my calculations in kT 0.3-0.5 GeV/c are slightly smaller than the othercalculations. But all results are consistent within systematic uncertainties.

4.7 Consistency check of v2 and v3 with the previous resultsfrom ALICE
Figure 4.36 and 4.37 show Identified hadron (π , K and p) v2 and v3 as a function of pT for 6
centrality classes. My results are compared with previous results from ALICE.
Charged pion and kaon v2 and v3 are fully consistent with published results. Proton v2 andv3 of my calculation are systematically smaller than published results in especially smaller pT .
But both results are consistent within systematic uncertainties.

162

8

centrality 5-10 %6

R out (fm)6

R out (fm)centrality 0-5 %8centrality 10-20 %

6

4

4

20.2 0.3 0.4 0.5 0.6 0.7
k T (GeV/c)2
0.2 0.3 0.4 0.5 0.6 0.7
k T (GeV/c)

20.2 0.3 0.4 0.5 0.6 0.7
k T (GeV/c)86centrality 20-30 %My analysis
centrality
0-5 %5-10 %
10-20 %
20-30 %
30-40 %
40-50 %

8

6

centrality 30-40 %PRL.118.(2017).222301
centrality
0-5 %5-10 %
10-20 %
20-30 %
30-40 %
40-50 %

R out (fm)4

R out (fm)

R out (fm)
R out (fm)

8

8

6

centrality 40-50 %Phys.Rev.C 93 (2016) 024905
centrality
0-5 %5-10 %
10-20 %
20-30 %
30-40 %
40-50 %

4

4

4

20.2 0.3 0.4 0.5 0.6 0.7
k T (GeV/c)2
0.2 0.3 0.4 0.5 0.6 0.7
k T (GeV/c)

20.2 0.3 0.4 0.5 0.6 0.7
k T (GeV/c)

Figure 4.33: 3D HBT radii (Rout ) of charged pions as a function of pair transverse momentum
kT for 6 centrality bins. My results are plotted as closed circles which is obtained with azimuthal
dependence of HBT radii with respect to Ψ3 . Open squared markers are results of azimuthal
differential pion HBT with respect to Ψ2 [60]. Open circle markers are results of azimuthal
integrated pion HBT analysis [61].

163

8

centrality 5-10 %6

R side (fm)6

R side (fm)centrality 0-5 %8centrality 10-20 %

6

4

4

20.2 0.3 0.4 0.5 0.6 0.7
k T (GeV/c)2
0.2 0.3 0.4 0.5 0.6 0.7
k T (GeV/c)

20.2 0.3 0.4 0.5 0.6 0.7
k T (GeV/c)86centrality 20-30 %My analysis
centrality
0-5 %5-10 %
10-20 %
20-30 %
30-40 %
40-50 %

8

6

centrality 30-40 %PRL.118.(2017).222301
centrality
0-5 %5-10 %
10-20 %
20-30 %
30-40 %
40-50 %

R side (fm)4

R side (fm)R side (fm)R side (fm)

8

8

6

centrality 40-50 %Phys.Rev.C 93 (2016) 024905
centrality
0-5 %5-10 %
10-20 %
20-30 %
30-40 %
40-50 %

4

4

4

20.2 0.3 0.4 0.5 0.6 0.7
k T (GeV/c)2
0.2 0.3 0.4 0.5 0.6 0.7
k T (GeV/c)

20.2 0.3 0.4 0.5 0.6 0.7
k T (GeV/c)

Figure 4.34: 3D HBT radii (Rside ) of charged pions as a function of pair transverse momentum
kT for 6 centrality bins. My results are plotted as closed circles which is obtained with azimuthal
dependence of HBT radii with respect to Ψ3 . Open squared markers are results of azimuthal
differential pion HBT with respect to Ψ2 [60]. Open circle markers are results of azimuthal
integrated pion HBT analysis [61].

164centrality 5-10 %8

R long (fm)R long (fm)8centrality 10-20 %

8

6

6

4

4

4

20.2 0.3 0.4 0.5 0.6 0.7
k T (GeV/c)2
0.2 0.3 0.4 0.5 0.6 0.7
k T (GeV/c)

20.2 0.3 0.4 0.5 0.6 0.7
k T (GeV/c)centrality 20-30 %8
6My analysis
centrality
0-5 %5-10 %
10-20 %
20-30 %
30-40 %
40-50 %

centrality 30-40 %8
6

PRL.118.(2017).222301
centrality
0-5 %5-10 %
10-20 %
20-30 %
30-40 %
40-50 %

R long (fm)6

R long (fm)R long (fm)R long (fm)centrality 0-5 %centrality 40-50 %8
6

Phys.Rev.C 93 (2016) 024905
centrality
0-5 %5-10 %
10-20 %
20-30 %
30-40 %
40-50 %

4

4

4

20.2 0.3 0.4 0.5 0.6 0.7
k T (GeV/c)2
0.2 0.3 0.4 0.5 0.6 0.7
k T (GeV/c)

20.2 0.3 0.4 0.5 0.6 0.7
k T (GeV/c)

Figure 4.35: 3D HBT radii (Rside ) of charged pions as a function of pair transverse momentum
kT for 6 centrality bins. My results are plotted as closed circles which is obtained with azimuthal
dependence of HBT radii with respect to Ψ3 . Open squared markers are results of azimuthal
differential pion HBT with respect to Ψ2 [60]. Open circle markers are results of azimuthal
integrated pion HBT analysis [61].

1655-10%

My calculation

JHEP 1609 (2016) 164

pion
0.3

0.4

pion
0.3

kaon

0.3

kaon

protonproton

0.2

0.2

0.2

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2.5

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3.5

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0

30-40%

v2

v2

0.2

0.2

0.1

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0.1

0

0

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3

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2

2.5

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T

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0.4

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0.4

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T

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T

20-30%
0.4

00.5

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v2

10-20%

0.4

v2

v2

v2

0-5%
0.4

T

0

0.5

1

1.5

2

2.5

3

3.5

4

p (GeV/c )T

Figure 4.36: Identified hadron (π , K and p) v2 as a function of pT for 6 centrality classes. Ψ2 isdetermined via FMD A+C. My results are plotted as closed markers and systematic uncertainties
of my calculation are depicted as transparent bands. Published results are plotted as opened
markers [62].

1665-10%

My calculation

JHEP 1609 (2016) 164

pion
0.3

0.4

pion
0.3

kaon

0.3

kaon

protonproton

0.2

0.2

0.2

0.1

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0

0

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v3

v3

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2

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T

3

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40-50%

0.4

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2

T

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p (GeV/c )

0.3

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1

T

20-30%
0.4

00.5

p (GeV/c )T

v3

10-20%

0.4

v3

v3

v3

0-5%
0.4

T

0

0.5

1

1.5

2

2.5

3

3.5

4

p (GeV/c )T

Figure 4.37: Identified hadron (π , K and p) v3 as a function of pT for 6 centrality classes. Ψ2 isdetermined via FMD A+C. My results are plotted as closed markers and systematic uncertainties
of my calculation are depicted as transparent bands. Published results are plotted as opened
markers [62].

167Chapter 5
Discussion
5.1 Final source Eccentricity
In order to understand space-time evolution of the system, one of the important probes is arelation between the initial geometrical shape and final source shape at the time of kinetic
freeze-out. Blast wave model suggests that an eccentricity at the freeze-out can be extractedwith relative amplitude of azimuthal angle dependence of HBT radii with respect to Ψ2 at thelimit of kT = 0 [63]. Based on the assumption of Blast wave approach, a final source eccentricity
is given by

ε f inal = 2

R2side,2R2side,0= −2R2out,2R2side,0=2R2os,2R2side,0.

(5.1)

where R2µ ,2 denotes the second-order oscillation amplitude of HBT radii with respect to Ψ2 ,and R2µ ,0 represents average HBT radii. Both two parameters R2µ ,2 and R2µ ,0 are obtained byfitting azimuthal angle dependence of HBT radii with respect to Ψ2 using Eq.(4.1). It should
be noted that the parameters Rout and Ros terms include temporal information and thus relative
amplitude of Rout and Ros tends to be much affected by radial and anisotropic flow than that in
Rside . Therefore, in order to extract geometrical source shape, the relative amplitude of Rside ismore suitable to study a geometrical shape of the source.

5.1.1 Centrality dependence of final source eccentricity
Figure 5.1 shows relative amplitudes of squared HBT radii for charged pion pairs with respect
to Ψ2 as a function of centrality obtained by fitting Figure 4.2 with Eq.(4.1). Pair transverse
momentum kT is integrated from 0.2 to 1.5 GeV/c, where the mean kT is ∼ 0.4 GeV/c. The
168

relative amplitudes of Rlong , Rol and Rsl are almost zero for all centrality. On the other hand,
the relative amplitudes of Rout , Rside and Ros have explicitly non-zero value and they grow from
central to peripheral collisions. This behaviour is similar to centrality dependence of v2 which
is sensitive to the initial eccentricity. Therefore it is likely that centrality dependence of initialeccentricity still remains at the freeze-out. The relative amplitudes of Rout and Ros are larger
than that of Rside . It indicates that Rout and Ros oscillations include the temporal term, thus they
are biased by collective flow and enhaced compared to geometrical term Rside .
In order to study the relation between the initial eccentricity and the final eccentricity, Finalsource eccentricity (relative amplitude of squared HBT radii) as a function of initial eccentricity
obtained with Glauber model simulation[13] is shown in Figure 5.2. Dashed line represents

ε final (2R2side,2 /R2side,0 ) = 0 and dotted lined indicates ε initial = ε final which means that the initialout-plane elongated elliptic shape remains even if the source size enlarges with the systemevolution.
The final source eccentricity is much smaller than dotted line. This indecates that the initial elliptic shape is strongly diluted in the final state throuhg the system evolution because oflarge radial and elliptic flow. However initial elliptic shape can not be reversed by collective
expansion even in LHC energy.
The final source eccentricity almost linearly increases with increasing centrality, whichmeans centrality dependence of initial overlap region still remains at freeze-out.

169

0.2

2R 2long,2 / R2long,02R 2side,2 / R2side,0-2R 2out,2 / R2out,0

0.3

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40

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30

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0.2

π+π+ and π-π- combined

0
0

10

20

30

40

50

k T : 0.2-1.5 GeV/ccentrality (%)

Figure 5.1: Relative amplitudes of squared HBT radii (Rout , Rside , Rlong , Ros , Rol and Rsl ) for
charged pion pairs with respect to Ψ2 as a function of centrality measured in Pb-Pb collisions at√
sNN = 2.76 TeV. Pair transverse momentum kT is integrated from 0.2 to 1.5 GeV/c. Dashed line
indicates relative amplitude of HBT radii = 0. Transparent red boxes represent the systematic
uncertainties.

170

50

centrality (%)

Pb-Pb sNN=2.76 TeV

0.1

50

0.2

0.1

10

20

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0.1

0

10

centrality (%)

2R 2ol,2 / R2side,0

0.3

40

centrality (%)

2R 2os,2 / R2side,0centrality (%)-2R 2out,2 / R2side,0

0.2

0.1

0

2R 2sl,2 / R2side,00.3

2R 2side,2 / R2side,0≈εfinal

Pb-Pb sNN=2.76 TeV- -π+π+ and π π combined

0.4

k T : 0.2-1.5 GeV/ c

0.2

0
0

0.2

0.4

εinitial
Figure 5.2: Relation between initial source eccentricity(Glauber Model Calculation[13]) andfinal source eccentricity extracted with azimuthal angle dependence of charged pion HBT radii(2R2side,2 /R2side,0 ) in Pb-Pb 2.76 TeV. kT is integrated from 0.2 to 1.5 GeV/c. Dashed line is
2R2side,2 /R2side,0 =0 and dotted line denotes ε initial = ε f inal . Boxes represent the systematic uncertainties and statistical uncertainties are smaller than marker size

171

5.1.2 kT dependence of final source eccentricity
Figure 5.3 shows relative amplitudes of squared HBT radii for charged pions with respect to Ψ2
as a function of pair pair transverse momentum kT . Pair transverse momentum is divided for 3
bins (0.2-0.3, 0.3-0.4, 0.4-1.5GeV/c).
Relative amplitude of Rside , Rout , and Ros (2R2side,2 /R2side,0 , -2R2out,2 /R2out,0 , -2R2out,2 /R2side,0 ,2R2os,2 /R2side,0 ) increases with increasing pair transverce momentum kT for all 3 centrality bins

0.3
0.2

-2R2out,2/R 2side,02R2side,2/R 2side,0-2R2out,2/R 2out,0and slope of kT dependence of 2R2side,3 /R2side,0 becomes slightly larger from central to peripheral.
0.3
0.2

0.2

0.1

0.1

0.1

0

0

0

−0.1

0.2

0.4

0.6

−0.1

0.2

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k T (GeV/c )2R2os,2/R 2side,0

0.3

0.6

k T (GeV/c )

−0.1

0.2

0.4

0.6

k T (GeV/c )0.3

Pb-Pb sNN=2.76 TeV

0.2
0.1
0

−0.1

π+π+ and π-π- combined
0.2

0.4

0.6

centrality
0-10%
10-30%
30-50%k T (GeV/c )Figure 5.3: Relative amplitudes of squared HBT radii (Rout , Rside , and Ros ) for charged pion
pairs with respect to Ψ2 as a function of pair transverse momentum kT for 3 centrality bins√
measured in Pb-Pb collisions at sNN = 2.76 TeV. Dashed line indicates relative amplitude of
HBT radii = 0. Transparent boxes represent the systematic uncertainties.

1725.1.3 Relative amplitude of HBT radii with respect to Ψ2 with ESE q2
selection
Studying the centrality dependence of relative amplitude of HBT radii gives us the relation
between initial and final source eccentricity. However when the centrality changes from central
to peripheral collisions, not only the eccentricity but also a volume size, freeze-out temperature,and flow velocity change simultaneously. Event Shape Engineering technique, i.e. q2 selection
allows us to select the events which have more elliptical shape in the initial state within a certain
centrality window. When we apply ESE selection to azimuthal angle dependence of HBT radiiwith respect to Ψ2 , more details study of the relation between initial and final source eccentricity
can be expected.
Figure 5.4 shows relative amplitude of charged pion HBT radii (Rout , Rside , Rlong , Ros , Rol
and Rsl ) with respect to Ψ2 with each 20% q2 selection as a function of centrality , where 020% (80-100%) corresponds to the smallest (largest) q2 bin. No significant modification with
q2 selection is found in relative amplitude of Rlong , Rol and Rsl . The Relative amplitudes ofRlong , Rol and Rsl don’t depend on centrality and initial eccentricity. On the other hand, relative
amplitude of Rout and Ros significantly changes with q2 selection. Relative amplitude of Routand Ros becomes larger from small q2 to large q2 and the variation by q2 selection becomeslargest at 30-40% centrality bin.
Also the relative amplitude of Rside , which is most sensitive to final source shape, shows asimilar trend to that of Rout and Ros . But in most central 0-5% and smallest q2 event selection,
relative amplitude of Rside seems to show a negative value although it is consistent with zero
within the systematic uncertainties. It might be a hint that the small elliptical shape by thesmallest q2 selection has vanished or even has been reversed with strong radial flow and elliptic
flow.
Another interesting feature of q2 dependence can be found in Rside oscillation in mid central
collisions. The q2 dependence of the relative amplitude of Rside shows a smilar behaviour tothose of Rout and Ros except 20-40% centrality. Results for 20-40% centrality show a slightdifferent trend although the uncertainties are large.

173

0.2

2R 2long,2 / R2long,02R 2side,2 / R2side,0-2R 2out,2 / R2out,0

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Pb-Pb sNN=2.76 TeV

0.2

π+π+ and π-π- combined

0.1

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k T : 0.2-1.5 GeV/c

40

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centrality (%)

2R 2ol,2 / R2side,0

0.3

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2R 2os,2 / R2side,0centrality (%)-2R 2out,2 / R2side,0

0.2

0.1

0

2R 2sl,2 / R2side,0

0.3

0

10

20

30

40

50

centrality (%)

q class2
80-100%60-80%
40-60%
20-40%
0-20%

centrality (%)

Figure 5.4: Each 20% q2 selection is applied to relative amplitude of squared HBT radii (Rout ,Rside , Rlong , Ros , Rol and Rsl ) for charged pion pairs with respect to Ψ2 as a function of centrality
measured in Pb-Pb 2.76TeV collisions. Pair transverse momentum kT is integrated from 0.2
to 1.5 GeV/c. Dashed line indicates relative amplitude of HBT radii = 0. Transparent boxes
represent the systematic uncertainties. All points are shifted along the x-axis for visibility.

5.1.4 kT dependence of relative amplitude of HBT radii with respect to Ψ2with ESE q2 selection
In dynamically expanding source, study of kT dependence of HBT radii is important to understand the system evolution.
Figure 5.5, 5.6, and 5.6 shows relative amplitudes of squared HBT radii (Rout , Rside , and Ros )of charged pion pairs with respect to Ψ2 as a function of centrality measured in Pb-Pb 2.76TeV
collisions for kT 0.2-0.3, 0.3-0.4, and 0.4-1.5 GeV/c, respectively.
One can find that relative amplitude of Rout and Ros explicitly changes with q2 selection. For
all centrality bins and all kT bins, relative amplitudes of Rout and Ros becomes larger from small

174

q2 to large q2 and the sensitivity to q2 selection is largest at results of highest kT .
In centrality 0-10 and 30-50%, relative amplitude of Rside shows the similar q2 dependence
to those of Rout and Ros . As is the case with Sec. 5.1.3, relative amplitude of Rside does notchange with q2 selection in mid-central collisions (10-30%) in low kT 0.2-0.4 GeV/c. However
relative amplitude of Rside of largest kT becomes larger from small q2 to large q2 even in midcentral collisions.
Based on the Blast-wave model study, final source eccentricity can be extracted with relative
amplitude of Rside at the limit as kT approaches 0, which means that HBT radii in larger kT
becomes smaller with system expansion, and the obtained results shows that no q2 dependence
was found in smallest kT . Therefore, in mid-central collisions (10-30%), it might indicatethat final source eccentricity does not depend on the initial eccentricity due to the correlation

0.3
0.2

-2R2out,2/R 2side,02R2side,2/R 2side,0-2R2out,2/R 2out,0

between initial eccentricity and radial and elliptic flow.
0.3
0.2

0.2

0.1

0.1

0.1

0

0

0

−0.1

20

40

−0.1

20

centrality (%)
2R2os,2/R 2side,0

0.3

40

centrality (%)

0.3

Pb-Pb sNN=2.76 TeV
0.2
0.1

π+π+ and π-π- combined

0
−0.1

20

40

k T : 0.2-0.3 GeV/c

−0.1

20

40

centrality (%)

q 2 class80-100%60-80%
40-60%
20-40%
0-20%

centrality (%)

Figure 5.5: Each 20% q2 selection is applied to relative amplitudes of squared HBT radii (Rout ,Rside , and Ros ) of charged pion pairs with respect to Ψ2 for kT 0.2-0.3 GeV/cas a function ofcentrality measured in Pb-Pb 2.76TeV collisions. Dashed line indicates relative amplitude of
HBT radii = 0. Transparent boxes represent the systematic uncertainties. All points are shiftedalong the x-axis for visibility.

175

0.2

-2R2out,2/R 2side,02R2side,2/R 2side,0-2R2out,2/R 2out,0

0.3

0.3
0.2

0.2

0.1

0.1

0.1

0

0

0

−0.1

20

40

−0.1

20

centrality (%)
2R2os,2/R 2side,0

0.3

40

centrality (%)

0.3

Pb-Pb sNN=2.76 TeV
0.2
0.1

π+π+ and π-π- combined

0
−0.1

20

40

k T : 0.3-0.4 GeV/c

−0.1

20

40

centrality (%)

q 2 class80-100%60-80%
40-60%
20-40%
0-20%

centrality (%)

Figure 5.6: Each 20% q2 selection is applied to relative amplitudes of squared HBT radii (Rout ,Rside , and Ros ) of charged pion pairs with respect to Ψ2 for kT 0.3-0.4 GeV/cas a function ofcentrality measured in Pb-Pb 2.76TeV collisions. Dashed line indicates relative amplitude of
HBT radii = 0. Transparent boxes represent the systematic uncertainties. All points are shiftedalong the x-axis for visibility.

176

0.2

-2R2out,2/R 2side,02R2side,2/R 2side,0-2R2out,2/R 2out,0

0.3

0.3
0.2

0.2

0.1

0.1

0.1

0

0

0

−0.1

20

40

−0.1

20

centrality (%)
2R2os,2/R 2side,0

0.3

40

centrality (%)

0.3

Pb-Pb sNN=2.76 TeV
0.2
0.1

π+π+ and π-π- combined

0
−0.1

20

40

k T : 0.4-1.5 GeV/c

−0.1

20

40

centrality (%)

q 2 class80-100%60-80%
40-60%
20-40%
0-20%

centrality (%)

Figure 5.7: Each 20% q2 selection is applied to relative amplitudes of squared HBT radii(Rout ,Rside , and Ros ) of charged pion pairs with respect to Ψ2 for kT 0.4-1.5 GeV/cas a function ofcentrality measured in Pb-Pb 2.76TeV collisions. Dashed line indicates relative amplitude of
HBT radii = 0. Transparent boxes represent the systematic uncertainties. All points are shiftedalong the x-axis for visibility.

1775.2 v2 scaling of Final source eccentricity
Difference of the initial geometry within a certain centrality bin can not be selected by centrality.
Therefore, in order to understand relation between initial eccentricity and final eccentricity,
another probe for initial eccentricity is indispensable.
Figure 5.4 shows relative amplitude of charged pion HBT radii (Rout , Rside , Rlong , Ros , Rol
and Rsl ) with respect to Ψ2 with each 20% q2 selection as a function of charged pion v2 . No
q2 selected results are simustaneously plotted as open markers. Azimuthal anisotropy (v2 ) is
sensitive probe to initial geometry and more preferable to use for x-axis than centrality.
Basically q2 selection is event by event flow fluctuation selection and v2 is strongly reflectedby 20 % q2 selection in Figure 4.3, thus x-axis is significantly changed even in a fixed centrality
from Figure 5.4 to Figure 5.8. The relative amplitude of Rout , Rside and Ros with q2 selection
has been observed to scale with v2 . Based on the Blast-wave model, relative amplitudes of Rside, Rout and Ros are sensitive to the eccentricity in the final state. Therefore it indicates that azimuthal anisotropy in geometrical space (2R2side,2 /R2side,2 , -2R2out,2 /R2side,2 and 2R2os,2 /R2side,2 ) andmomentum space (v2 ) are significantly correlated, and this correlation does not depend on eventby event flow (initial geometry) fluctuation. One can find that -2R2out,2 /R2side,2 is almost proportional to v2 , and intersept seems not to be zero. On the other hand, 2R2out,2 /R2side,2 has quadraticlike function rather than linear linear function and intersept is almost zero. 2R2os,2 /R2side,2 hassimilar shape to -2R2out,2 /R2side,2 .
13(energy denThe relation of azimuthal anisotropy v2 , eccentricity ε2 , and system size Npartsity dNd η ) are explored in various collision energy and geometry, and empirically elliptic flow canbe defined by(
v2 = ε2 × f

dN
dη

)
.

(5.2)

The amplitude of elliptic flow can be determined with initial geometry and system size.
ESE q2 selection is applied with in a "fixed centrality", i.e. system size does not changewith q2 selection. Second order flow vector q2 dependence of HBT radii modulation in a fixed
centrality is driven by only the initial eccentricity.
Second order flow vector q2 dependence of -2R2out,2 /R2out,2 and 2R2out,2 /R2side,2 (slope as wellas intercept) does not depend on centrality. Therefore relation between initial eccentricity
and -2R2out,2 /R2out,2 and 2R2out,2 /R2side,2 does not depend on centrality. Also q2 dependence of178

2R2side,2 /R2side,2 does not depend on centrality in centrality 0-20% and 40-50%. However slopeof 2R2side,2 /R2side,2 seems to be vanished in centrality 20-40$. It indicates that relation between
initial eccentricity and 2R2side,2 /R2side,2 ( final eccentricity) might change with centrality. But
this effect is negligible within the systematic uncertainties, thus more precise measurements are
required.

179

0.2

2R 2long,2 / R2long,02R 2side,2 / R2side,0-2R 2out,2 / R2out,0

0.3

0.3

0.2

0.1

0.1

0

0

0

0.1

0.15

0.05

0.1

0.2

0.05

0.3

0.2

0.2

0.1

0.1

0

0

0

0.1

0.15

0.05

0.1

v2

0.3

0.15

0.2

π+π+ and π-π- combined

0.1

0
0.05

0.1

0.15

v2

0.05

0.1

v2

Pb-Pb sNN=2.76 TeV

k T : 0.2-1.5 GeV/c

0.15

0.3

0.1

0.05

0.1

v2

2R 2ol,2 / R2side,0

0.3

0.15

v2

2R 2os,2 / R2side,0v2-2R 2out,2 / R2side,0

0.2

0.1

0.05

2R 2sl,2 / R2side,0

0.3

0.15

v2

centrality
No q2 selection
0-5%
5-10%
10-20%
20-30%
30-40%
40-50%

Figure 5.8: Each 20% q2 selection, where 0-20% (80-100%) corresponds to the smallest(largest) q2 bin, is applied to relative amplitude of squared HBT radii(Rout , Rside , Rlong , Ros ,Rol and Rsl ) for charged pion pairs with respect to Ψ2 as a function of charged pion v2 measured in Pb-Pb 2.76TeV collisions. Pair transverse momentum kT is integrated from 0.2 to 1.5
GeV/c. Dashed line indicates relative amplitude of HBT radii = 0. Transparent boxes represent
the systematic uncertainties. Relative amplitude of HBT radii without q2 selection also dipicted
as open black circle.

1805.3 Interpretation of initial eccentricity difference with Blastwave Model
In order to understand the effect of q2 selection to relative amplitude of HBT radii with respect
to Ψ2 , Blast-wave model approach which is extended to HBT correlation[63] is applied.
Blast-wave model is analytical method to extract the parameters of freeze out configuration to fit the particle spectra and azimuthal anisotropy. In this thesis, extended Blast-wave
model suggested in [63] is applied. In this model, freeze out configuration is expressed with 8
parameters listed below.
• Tf : Freeze out temperature• ρ0 : Transverse flow velocity• ρ2 : 2nd -order oscillation of transverse flow
• Rx : Source size along the event plane• Ry : Source size perpendicular to the event plane• αs : Surface diffuseness of the emission source• τ0 : Freeze out time• ∆τ : emission duration
Transverse source size is determined with Rx and Ry , and weighting function of source shapeis given by1Ω (r, ϕs ) =,(5.3)
1 + e(r̃−1)/αs
where αs is surface diffuseness of the emission source and ϕs indicates the spatial azimuthal
angle of emission point which is given by(tan (ϕs ) =

Ry
Rx

)2
tan (ϕb ) ,(5.4)

where ϕb indicates the azimuthal direction of the boost. In this model, the boost angle is perpendicular to the elliptical sub-shell on which the source element is found. r̃ is normalized ellipticalradius expressed by181

Table 5.1: Fit ranges for identified hadron spectra, v2 , and HBT radii
Spectra++−π and πK and K −p and p̄pT (GeV/c)
0.5 - 1.13
0.4 - 1.4
0.6 - 1.69v2++−
π and πK and K −p and p̄pT (GeV/c)
0.5 - 1.13
0.4 - 1.4
0.6 - 1.69
HBT radii+−+
π and πK and K −p and p̄⟨kT ⟩ (GeV/c)

0.415

√
r̃ (r, ϕs ) ≡












(r cos (ϕs ))2 (r sin (ϕs ))2
+
.
R2x
R2y

(5.5)

Transverse flow velocity profile is expressed as follows :ρ (r, ϕs ) = r̃ (ρ0 + ρ cos (2ϕb )) .

(5.6)

In this thesis, identified hadron(π , K, p) spectra, identified hadron(π , K, p) v2 and azimuthal
angle dependence of charged pion HBT radii relative to Ψ2 are used to Blast-wave fit. The
sensitivity of spectra, v2 and HBT radii to determine freeze-out parameters are quit different.
For example, spectra shape is determined with Tf and ρ0 , and v2 is sensitive to ρ2 and Rx /Ry
in particular. HBT radii are sensitive to all parameters, but Rside is independent of τ0 and ∆τ .
Therefore in order to constrain the fitting, Tf and ρ0 are determined with spectra fitting and theother parameters are extracted with simultaneous fitting v2 and azimuthal angle dependence of
HBT radii with respect to Ψ2 , fixing the parameters Tf and ρ0 obtained with spectra fitting. The
surface diffuseness parameter is set to be 0 as default. The fit ranges of spectra, v2 and azimuthal
angle dependence of HBT radii are shown in Table. 5.1.
Figure 5.9 shows the Blast-wave fitting to identified hadron spectra. Positive and negative
pions, kaons, and protons are fitted simultaneously. The fitting functions reproduce the data foreach particle species well.
Figure 5.10 shows the Blast-wave fitting to identified hadron (pions, kaons, and protons)
v2 with 40-60% q2 selection applied as a function of pT . In all centrality, pions and kaons v2
are well reproduced with this model up to pT 1-2GeV/cdepending on the particle species. But
182

protons v2 at low pT is underestimated with Blast-wave model.
Figure 5.11 shows the Blast-wave fitting to azimuthal angle dependence of Rout , Rside ,Rlong and Ros with 40-60% q2 selection applied as a function of centrality. In all centrality, Routand Rlong are well described with Blast-wave model for both average HBT radii and oscillation

p

10−3

1

2

3

4
p (GeV/c )

0

1

2

3

4
p (GeV/c )

103

10−3

1

2

3

4
p (GeV/c )

1

2

3

4
p (GeV/c )
T

T

1

10−3

0

0

103

40-50%

T

30-40%

T
T

1

1

T

d2N /(Nev2πp dp dy) (GeV/c )-1TT

20-30%

10-20%

10−3

T

103

103

T
T

1

10−3

0

d2N /(Nev2πp dp dy) (GeV/c )-1

d2N /(Nev2πp dp dy) (GeV/c )-1

+

5-10%

d2N /(Nev2πp dp dy) (GeV/c )-1

T

K

103

T

π+
1

T

0-5%

d2N /(Nev2πp dp dy) (GeV/c )-1

103

T

d2N /(Nev2πp dp dy) (GeV/c )-1

amplitude, while oscillation amplitude of Rside and Ros in Blast-wave model is overestimated.

1

10−3

0

1

2

T

3

4
p (GeV/c )
T

0

1

2

3

4
p (GeV/c )T

Figure 5.9: Blast-wave fitting to identified particle(π , K and p) pT spectra as a function ofcentrality[9]. Difference of panels denote centrality. Error bar indicates quadratic sum of staticand systematic uncertainties. Black lines shows the actual fit range and red, green and blue lines
are extrapolated line of fitting functions for π , K and p, respectively.

5.3.1 Extracted parameters of freeze out configuration with Blast-wave
model
Figure 5.12 shows the extracted freeze out parameters (Tf , ρ0 , rho2 , R2x , R2x /R2y , τ , and ∆τ ) as afunction of average number of participant calculated with Glauber model. No q2 selections areapplied to spectra, v2 , and HBT measurements.
The freeze-out temperature (Tf ) slightly decreases with increasing ⟨N part ⟩ and flow velocity(ρ0 ) becomes larger from small ⟨N part ⟩ to large ⟨N part ⟩. Source size at freeze out (R2x ) andfreeze out time (τ ) becomes larger from small ⟨N part ⟩ to large ⟨N part ⟩. Also emission duration∆τ becomes slightly larger from small ⟨N part ⟩ to large ⟨N part ⟩. Eccentricity of final source
183

v2

v2

v2

0-5%

q :40-60%
2

5-10%

10-20%

π+
0.2

0.2

K+

0.2

p

0

0

0

1

2

3
4
p (GeV/c )

0

0

1

2

0.2

0

1

2

3
4
p (GeV/c )34p (GeV/c )40-50%

0.2

0

2

T

30-40%

0.2

1v2

20-30%

0

0

T

v2

v2

T

3
4
p (GeV/c )

0

0

1

2

T

3
4
p (GeV/c )

0

1

T

2

3
4
p (GeV/c )T

Figure 5.10: Blast-wave fitting to identified particle(π , K and p) v2 with 40-60% q2 selection
as a function of pT . Difference of panels denote centrality. Error bar indicates quadratic sumof static and systematic uncertainties. Black lines shows the actual fit range and red, green andblue lines are extrapolated line of fitting functions for π , K and p, respectively.

(R2x /R2y ) increases with increasing ⟨N part ⟩.
Figure 5.13 shows the extracted freeze out parameters (Tf , ρ0 , rho2 , R2x , R2x /R2y , τ and ∆τ )as a function of average number of participant calculated with Glauber model. For each 20%
q2 selections are applied to v2 and HBT measurements. Spectra also changes slightly with q2
selection, but the difference is expected to be smaller than systematic uncertainties. In thisthesis, we assumes that temperature and ρ0 do not depend on q2 selection.
Second order modulation of flow radipidity ρ2 becomes explicitly larger (smaller) withlarger (smaller) q2 selection. By applying Event Shape Engineering q2 selection, v2 is largely
enhanced or suppressed, and ρ2 is sensitive to v2 . Thus this behavior can be understood thatsuch a correlation between v2 and ⟨N part ⟩ dependence of ρ2 is very similar to that of v2 . But
No significant modification to R2x , τ and ∆τ by q2 selection is found. System life time (tau) andellipticity (R2x /R2y ) slightly changes with q2 selection, i.e. tau and R2x /R2y in large q2 selection
tends to have larger value than those in small q2 selection. It indicates that not only velocity
field but also eccentricity at freeze-out and system life time could be modified with different
initial shape. But it is negligible within the systematic uncertainties.
184

However this Blast Wave model could not reproduce small oscillation of Rside , thus in orderto understand geometrical information R2x and R2x /R2y , more realistic model is necessary.

185

R 2side (fm2)

R 2out (fm2)

60

40

40

20

20

0

1

2
ϕ

pair

3
- Ψ2 (rad)60

0

R 2os (fm2)R 2long (fm2)

60

1

2
ϕ

pair

3
- Ψ2 (rad)pair3- Ψ2 (rad)

5

40
0

20

0

1

2
ϕ

pair

3
- Ψ2 (rad)q 2:40-60%
Pb-Pb sNN=2.76 TeV
π+π+ and π-π- combined
k T : 0.2-1.5 GeV/ ccentrality
0-5%
5-10%
10-20%
20-30%
30-40%
40-50%

−5

0

1

2
ϕ

Blast wave fit
centrality
0-5%
5-10%
10-20%
20-30%
30-40%
40-50%

Figure 5.11: Blast-wave fitting to azimuthal angle dependence of Rout , Rside , Rlong and Ros
with 40-60% q2 selection as a function of centrality. Systematic uncertainties are shown astransparent bands and statistical uncertainties are smaller than marker size.

186

2

1.5

ρ

0

ρ

Tf(GeV)

0.2

0.2

1
0.1

0.1
0.5

100

200

300

0

400
<Npart>20

0

Ry / Rx

R x (fm)

0

100

200

300

0

400
<Npart>1.50

τ(fm/c )

0

100

200

300

400
<Npart>

100

200

300

400
<Npart>18

12.667

10

7.333
1

∆ τ(fm/c )

0

100

200

300

400
<Npart>

0

100

200

300

400
<Npart>2

0

10

Pb-Pb sNN=2.76 TeV
5π+π+ and π-π-00k T : 0.2-1.5 GeV/c
100

200

300

400
<Npart>Figure 5.12: Extracted freeze out parameters with Blast Wave fitting to spectra, v2 and HBT as
a function of average number of participant calculated with Glauber model without q2 selection.
Tf and ρ0 are extracted by fitting π , K and p spectra, and the other parameters are obtained withsimultaneous fitting π , K and p v2 and charged pion HBT radii (Rout , Rside , Rlong , and Ros )with respect to Ψ2 .

187

2

1.5

ρ

0

ρ

Tf(GeV)

0.2

0.2

1
0.1

0.1
0.5

100

200

300

0

400
<Npart>20

0

Ry / Rx

R x (fm)

0

100

200

300

0

400
<Npart>1.50

τ(fm/c )

0

100

200

300

400
<Npart>

100

200

300

400
<Npart>18

12.667

10

7.333
1

∆ τ(fm/c )

0

100

200

300

400
<Npart>

0

100

200

300

400
<Npart>

2

0

10

5

0

0

100

200

300

400
<Npart>Pb-Pb sNN=2.76 TeV
π+π+ and π-πk T : 0.2-1.5 GeV/cq selection
2
0-20%
20-40%
40-60%
60-80%
80-100%0-100%

Figure 5.13: Extracted freeze out parameters with Blast Wave fitting to spectra, v2 and HBT as
a function of average number of participant calculated with Glauber model with each 20% q2
selection. No q2 selected results are simultaneously dipicted as open black circle. Tf and ρ0 are
extracted by fitting π , K and p spectra, and the other parameters are obtained with simultaneous
fitting π , K and p v2 and charged pion HBT radii (Rout , Rside , Rlong , and Ros ) with respect to
Ψ2 .

188

5.4 Final source triangular shape
PHENIX performed the first measurement of azimuthal angle dependence of charged pion HBTradii with respect to Ψ3 [32], and relative amplitude of Rout has positive or zero and relative
amplitude of Rside has negative or zero value. In order to extract the final source triangularshape, it is important to determine whether the relative amplitude of HBT radii with respect toΨ3 is positive or negative (or even zero).
Compared to Ψ2 dependence, initial source triangular shape and triangular flow signal ismuch smaller. Therefore extraction of final source triangular shape is much more difficult. In
LHC-ALICE experiment, owing to large multiplicity and excellent event plane resolution at
forward detector, more detailed study of the final source triangular shape can be performed. In
this section, the first measurement of azimuthal angle dependence of charged pion HBT radiimeasured in Pb-Pb collisions at 2.76 TeV are presented.

5.4.1 Centrality dependence of relative amplitude of HBT radii with respect to Ψ3
Figure 5.14 shows relative amplitude of squared HBT radii for charged pion pairs with respect
to Ψ3 as a function of centrality obtained with Figure 4.27. Pair transverse momentum kT is
integrated from 0.2 to 1.5 GeV/c, where the mean kT is approximately 0.4 GeV/c. Relative amplitude of Rlong , Rol , and Rsl is almost zero for all centrality within the systematic uncertainty.
Relative amplitude of Rout has explicitly positive value and slightly increases from central to peripheral collisions. Positive oscillation amplitude and this centrality dependence is very similar
to Rout oscillation with respect to Ψ2 , though oscillation amplitude of Ψ3 is much smaller thanthat of Ψ3 , and relative amplitude of Rside has negative value in all centrality and center values
slightly increase from central to peripheral. Relative amplitude of Ros cross term has positive orzero value and no significant centrality dependence can be found. In ALICE, explicit oscillation
signals with respect to Ψ3 and small centrality dependence are obtained.
Contrary to HBT measurement with respect to Ψ2 , -2R2out,3 /R2out,0 (-2R2out,3 /R2side,0 ) is positiveand 2R2side,3 /R2side,0 is negative. This feature can also be found in the relative amplitude of HBTradii with respect to Ψ2 in most central 0-5% collisions and smallest q2 class in Figure 5.8.
Common point between two different measurement are "small eccentricity in the initial state".

189

Therefore, when the initial geometrical source shape is almost round shape, relative amplitude

0.1
0.05
0

−0.05

2R 2long,3 / R2long,02R 2side,3 / R2side,0-2R 2out,3 / R2out,0of Rout and Rside has same negative sign.
0.1
0.05
0

−0.05

−0.1
10

20

30

40

50

10

20

30

−0.05

0.1
0.05
0

−0.05

−0.1
10

20

30

40

50

0

10

20

30

40

0.1
0.05
0

−0.1
0

10

20

30

centrality (%)

40

50

centrality (%)

0

10

20

30

40

0.05

π+π+ and π-π- combined

0

−0.05
−0.1
0

10

20

30

40

50

k T : 0.2-1.5 GeV/ccentrality (%)

Figure 5.14: Relative amplitude of squared HBT radii(Rout , Rside , Rlong , Ros , Rol and Rsl ) for
charged pion pairs with respect to Ψ3 as a function of centrality measured in Pb-Pb 2.76TeV
collisions. Pair transverse momentum kT is integrated from 0.2 to 1.5 GeV/c. Dashed line
indicates relative amplitude of HBT radii = 0. Transparent blue boxes represent the systematic
uncertainties.

5.4.2 kT dependence of final source eccentricity
Figure 5.15 shows relative amplitudes of squared HBT radii for charged pions with respect toΨ3 as a function of pair pair transverse momentum kT . Pair transverse momentum is divided
for 3 bins (0.2-0.3, 0.3-0.4, 0.4-1.5GeV/c).
Relative amplitude of Rout (-2R2out,3 /R2out,0 and -2R2out,3 /R2side,0 ) slightly increases with in190

50

centrality (%)

Pb-Pb sNN=2.76 TeV

0.1

50

centrality (%)

−0.05

−0.1
0

2R 2sl,3 / R2side,050

2R 2ol,3 / R2side,0040centrality (%)

2R 2os,3 / R2side,0-2R 2out,3 / R2side,0

0.05

0

−0.1
0

centrality (%)

0.1

0.05

−0.05

−0.1
0

0.1

creasing pair transverce momentum kT in centrality 0-30%.
Relative amplitude of Rside (-2R2side,3 /R2side,0 ) slightly decreases from low kT to high kT incentrality 0-10%, but no significant kT dependence can be found in the other centrality bins
(10-50%).
Relative amplitude of Ros (-2R2os,3 /R2side,0 ) increases from low kT to high kT in all centrality

0.1-2R 2out,3/ R 2side,02R 2side,3/ R 2side,0-2R 2out,3/ R 2out,0bins and the slopes become larger from central to peripheral collisions.
0.1

0.1

0

0

0

−0.1

−0.1

−0.1

0.2

0.4

0.6

0.2

0.4

2R 2os,3/ R 2side,0k T (GeV/c )0.6

k T (GeV/c )0.1

Pb-Pb sNN=2.76 TeV
0−0.1

π+π+ and π-π- combined
0.2

0.4

0.6

0.2

0.4

0.6

k T (GeV/c )centrality
0-10%
10-30%
30-50%k T (GeV/c )Figure 5.15: Relative amplitudes of squared HBT radii (Rout , Rside , and Ros ) for charged pion
pairs with respect to Ψ3 as a function of pair transverse momentum kT for 3 centrality bins√
measured in Pb-Pb collisions at sNN = 2.76 TeV. Dashed line indicates relative amplitude of
HBT radii = 0. Transparent boxes represent the systematic uncertainties.

191

5.4.3 Relative amplitudes of HBT radii relative to Ψ3 with q3 selection
The results of v3 measurements with each 20% q3 selection shows an explicit difference fromv3 without q3 selection, which means the initial triangular shape can be selected with this q3
selection. It indicates that event by event triangular flow fluctuation could be selected withq3 selection. Model comparison with relative amplitude of HBT radii with respect to Ψ3 at
PHENIX suggests that oscillation of HBT radii comes mostly from triangular flow. But the relation between triangular flow and oscillation of HBT radii is still not observed experimentally.
Measurements of azimuthal angle dependence of HBT radii relative to Ψ3 with q3 selection
give us the direct approach to the relation between triangular flow and 3rd -order modulation of
HBT radii.
Figure 5.16 shows relative amplitudes of charged pion HBT radii (Rout , Rside , Rlong , Ros ,Rol , and Rsl ) with respect to Ψ3 with each 20% q3 selection as a function of centrality. No
significant modification with q3 selection is found in relative amplitude of all HBT radii within
the systematic uncertainties, though v3 changes with q3 selection.
Simulation result from a Gaussian toy model suggests oscillation amplitude of HBT radiiwith respect to Ψ3 are dominated by triangular flow[33]. If q3 cut can select amplitude oftriangular flow, oscillation amplitude of HBT radii should change.
Three possibilities can be considered to interpret the q3 dependence. First possibility is q3
selectivity is not sufficient to modify the oscillation amplitude of HBT radii with respect toΨ3 . Sensitivity of initial triangular shape can be different between HBT measurement and flow
measurement. In central collisions, Ψ3 resolution is good, but the signal of v3 itself is smallest,
whereas the signal of v3 is relatively large and Ψ3 resolution is not so good in peripheral. Thus
our experimental precision might be insufficient to see the variation.
Second possibility is oscillation amplitude of HBT radii with respect to Ψ3 are not dominated by triangular flow. Third possibility is triangular flow is not originated from hydrodynamical expansion.
In order to reveal relation between v3 and relative amplitude of HBT radii with respect toΨ3 , comparison with realistic model simulation are necessary.

192

5.4.4 kT dependence of relative amplitude of HBT radii with respect to Ψ3with ESE q3 selection
Figure 5.17, 5.18, and 5.18 shows relative amplitudes of squared HBT radii (Rout , Rside , and Ros )of charged pion pairs with respect to Ψ3 as a function of centrality measured in Pb-Pb 2.76TeV
collisions for kT 0.2-0.3, 0.3-0.4, and 0.4-1.5 GeV/c, respectively.
For the lowest kT bin (0.2 - 0.3 GeV/c), no significant q3 dependence can be found in relative amplitude of Rout and Ros (-2R2out,3 /R2out,0 , and -2R2out,3 /R2side,0 , and 2R2os,3 /R2side,0 ). Relative
amplitudes of Rout and Ros are positive or almost zero. In centrality 0-30%, relative amplitudeof Rside decreases with increasing q3 .
In mid and high kT bins (0.3 - 1.5 GeV/c), no significant q3 dependence can be found for
relative amplitudes of Rout , Rside , and Ros ).

193

0.2
0.1
0
−0.1

2R 2long,3 / R2long,02R 2side,3 / R2side,0-2R 2out,3 / R2out,0

0.3

0.3
0.2
0.1
0
−0.1

−0.2
10

20

30

40

50

10

20

30

0

50

0

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0.1
0

−0.2

40

50

0

10

20

30

0.3

40

Pb-Pb sNN=2.76 TeV

0.1

π+π+ and π-π- combined

−0.1
−0.2
0

10

20

30

40

50

0

10

20

30

centrality (%)

0.2

0

50

k T : 0.2-1.5 GeV/c40

50

centrality (%)

q class
380-100%60-80%
40-60%
20-40%
0-20%

centrality (%)

Figure 5.16: Each 20% q3 selection is applied to relative amplitude of squared HBT radii(Rout ,Rside , Rlong , Ros , Rol , and Rsl ) for charged pion pairs with respect to Ψ3 as a function of centrality
measured in Pb-Pb 2.76TeV collisions. Pair transverse momentum kT is integrated from 0.2 to
1.5 GeV/c. Dashed line indicates relative amplitude of HBT radii = 0. Transparent boxes
represent the systematic uncertainties. All points are shifted along the x-axis for visibility.

194

50

0

−0.2

centrality (%)

40

0.1

−0.2
30

30

0.2

−0.1

20

20

0.3

−0.1

10

10

centrality (%)

−0.1

0

2R 2sl,3 / R2side,040

2R 2ol,3 / R2side,00.1

0centrality (%)

2R 2os,3 / R2side,0-2R 2out,3 / R2side,0

0.2

0.1

−0.2
0

centrality (%)

0.3

0.2

−0.1

−0.2
0

0.3

0.2

-2R2out,2/R 2side,02R2side,2/R 2side,0-2R2out,2/R 2out,0

0.3

0.3
0.2

0.2

0.1

0.1

0.1

0

0

0

−0.1

20

40

−0.1

20

centrality (%)
2R2os,2/R 2side,0

0.3

40

centrality (%)

0.3

Pb-Pb sNN=2.76 TeV
0.2
0.1

π+π+ and π-π- combined

0
−0.1

20

40

k T : 0.2-0.3 GeV/c

−0.1

20

40

centrality (%)

q class
380-100%60-80%
40-60%
20-40%
0-20%

centrality (%)

Figure 5.17: Each 20% q2 selection is applied to relative amplitudes of squared HBT radii(Rout , Rside , and Ros ) of charged pion pairs with respect to Ψ3 for kT 0.2-0.3 GeV/cas a functionof centrality measured in Pb-Pb 2.76TeV collisions. Dashed line indicates relative amplitude of
HBT radii = 0. Transparent boxes represent the systematic uncertainties. All points are shiftedalong the x-axis for visibility.

195

0.2

-2R2out,2/R 2side,02R2side,2/R 2side,0-2R2out,2/R 2out,0

0.3

0.3
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0.1

0.1

0.1

0

0

0

−0.1

20

40

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20

centrality (%)
2R2os,2/R 2side,0

0.3

40

centrality (%)

0.3

Pb-Pb sNN=2.76 TeV
0.2
0.1

π+π+ and π-π- combined

0
−0.1

20

40

k T : 0.3-0.4 GeV/c

−0.1

20

40

centrality (%)

q class
380-100%60-80%
40-60%
20-40%
0-20%

centrality (%)

Figure 5.18: Each 20% q3 selection is applied to relative amplitudes of squared HBT radii(Rout , Rside , and Ros ) of charged pion pairs with respect to Ψ3 for kT 0.3-0.4 GeV/cas a functionof centrality measured in Pb-Pb 2.76TeV collisions. Dashed line indicates relative amplitude of
HBT radii = 0. Transparent boxes represent the systematic uncertainties. All points are shiftedalong the x-axis for visibility.

196

0.2

-2R2out,2/R 2side,02R2side,2/R 2side,0-2R2out,2/R 2out,0

0.3

0.3
0.2

0.2

0.1

0.1

0.1

0

0

0

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20

40

−0.1

20

centrality (%)
2R2os,2/R 2side,0

0.3

40

centrality (%)

0.3

Pb-Pb sNN=2.76 TeV
0.2
0.1

π+π+ and π-π- combined

0
−0.1

20

40

k T : 0.4-1.5 GeV/c

−0.1

20

40

centrality (%)

q class
380-100%60-80%
40-60%
20-40%
0-20%

centrality (%)

Figure 5.19: Each 20% q3 selection is applied to relative amplitudes of squared HBT radii(Rout ,Rside , and Ros ) of charged pion pairs with respect to Ψ3 for kT 0.4-1.5 GeV/cas a function ofcentrality measured in Pb-Pb 2.76TeV collisions. Dashed line indicates relative amplitude of
HBT radii = 0. Transparent boxes represent the systematic uncertainties. All points are shiftedalong the x-axis for visibility.

197

5.5 v3 scaling of Final source eccentricity
As is the case for q2 dependence of oscillation amplitude of HBT radii, it is important to investigate v3 dependence of relative amplitudes of HBT radii relative to Ψ3 .
Figure 5.20 shows relative amplitude of charged pion HBT radii (Rout , Rside , Rlong , Ros , Rol ,and Rsl ) with respect to Ψ3 with each 20% q3 selection as a function of charged pion v3 . No
q3 selected results are simustaneously plotted as open markers. Azimuthal anisotropy (v3 ) is
sensitive probe to initial geometry and more preferable to use for x-axis than centrality.
ESE q3 selection is applied with in a "fixed centrality", i.e. system size does not changewith q3 selection. Third order flow vector q3 dependence of HBT radii modulation in a fixed
centrality is driven by only the initial triangularity, if triangular flow is originated from initialgeometrical triangularity.
Third order flow vector q3 dependence of -2R2out,3 /R2out,0 and -2R2out,3 /R2side,0 do not show
monotonical increasing (decreasing), more like quadratic function. -2R2out,3 /R2out,0 and -2R2out,3 /R2side,0becomes small v3 to large v3 in small v3 (up to 0.025) and they are increasing with increasing
v3 in large v3 (0.025 - 0.04).
No significant q3 dependence can be found in all the other relative amplitudes of HBT radii(2R2side,3 /R2side,0 , 2R2long,3 /R2long,0 2R2os,3 /R2side,0 , 2R2ol,3 /R2side,0 , 2R2sl,3 /R2side,0 ).

198

0.1

0

0.2

0.1

0

−0.1
0.01

2R 2long,3 / R2long,02R 2side,3 / R2side,0-2R 2out,3 / R2out,0

0.2

0.03

0.04

0.05

0.01

−0.1
0.02

0.03

0.04

0.2

0.1

−0.1

−0.1

−0.1

0.04

0.05

0.01

0.02

0.03

0.04

v3

0.2

0.05

0.1

π+π+ and π-π- combined

−0.1
0.01

0.02

0.03

0.04

0.05

0.01

0.02

0.03

0.04

v3

Pb-Pb sNN=2.76 TeV

0

0.04

k T : 0.2-1.5 GeV/c

0.05

0.1

0

0.03

0.03

0.2

0

0.02

0.02

v3

0

0.01

2R 2sl,3 / R2side,00.01

2R 2ol,3 / R2side,0

0.1

0.05

v3

2R 2os,3 / R2side,0-2R 2out,3 / R2side,0

v3

0.2

0.1

0

−0.1
0.02

0.2

0.05

v3

centrality
No q3 selection
0-5%
5-10%
10-20%
20-30%
30-40%
40-50%

v3

Figure 5.20: Each 20% q3 selection, where 0-20% (80-100%) corresponds to the smallest(largest) q3 bin, is applied to relative amplitude of squared HBT radii(Rout , Rside , Rlong , Ros , Rol ,and Rsl ) for charged pion pairs with respect to Ψ3 as a function of charged pion v3 measured in
Pb-Pb 2.76TeV collisions. Pair transverse momentum kT is integrated from 0.2 to 1.5 GeV/c.
Dashed line indicates relative amplitude of HBT radii = 0. Transparent boxes represent thesystematic uncertainties. Relative amplitude of HBT radii without q3 selection also dipicted as
open black circle.

199

Chapter 6
Conclusion
The measurements of azimuthal angle dependence of charged pion HBT radii with respect to√second and third order event plane have been performed in Pb-Pb collisions at sNN = 2.76
TeV. Event Shape Engineering q2 and q3 selection are applied to identified hadron (π , K and p)
v2 , v3 and we has reported the new approach to relation between initial and final source shape by
applying ESE technique to azimuthal angle dependence of charged pion HBT radii with respect
to Ψ2 and Ψ3 .
Explicit oscillation can be observed in azimuthal angle dependence of HBT radii(Rout , Rside
and Ros ) with respect to Ψ2 . Final source eccentricity was extracted with relative amplitudeof Rside and relation to initial eccentricity calculated with Glauber simulation indicates that ,inheavy ion collisions, large collective flow strongly expands the source along the short axis ofelliptic shape during QGP state and final source eccentricity are significantly diluted. However
initial out-plane elongated elliptic shape and centrality dependence can be observed at freezeout.
By applying Event Shape Engineering technique q2 selection to measurement of v2 , identified hadron v2 is significantly enhanced or suppressed for all centrality. Enhancement(suppression)of v2 equally contributes to charged pions, kaons and protons(anti-protons), no particle species
dependence can be found. Effect of q2 on v2 depends indeed weak on pT , however slightly
larger effect can be found in lower pT . This could be interpreted anisotropic flow is driven
by the low momentum particles. Also q3 selection was applied to identified v3 measurement.
Contrary to q2 selection to v2 , enhancement(suppression) of v3 depends on centrality. Effectof q3 selection is stronger in central than in peripheral. This can be considered this centrality
dependence comes from insufficient Ψ3 event plane resolution to select the initial triangular200

shape. As is the case in q2 cut for v2 , the effect of q3 selection for v3 is independent of pT .
Oscillation amplitude of Rout and Ros with respect to Ψ2 significantly changes with q2 selection and relative amplitude of Rside is slightly enhanced(suppressed) with q2 selection. In mostcentral 0-5% collisions and smallest q2 selected events, relative amplitude of Rside has zero or
negative value, though Rout oscillation has same sign to the other centrality and q2 class. Thiscan be interpreted initial elliptic shape might be vanished or even reversed at freeze out due
to small initial eccentricity and large elliptic flow. In centrality 20-40% collisions, oscillation
amplitude of Rside does not depends on the q2 selection.
Blast wave approach has performed to azimuthal angle dependence of HBT radii relative toΨ2 with q2 selection to interpret the difference of oscillation amplitude. second order oscillationin transverse flow explicitly enhanced(suppressed) with larger(smaller) q2 selection, and final
source eccentricity is slightly modified with q2 selection. However freeze out time and emission
duration do not show significant changes with q2 cut.
For azimuthal angle dependence of HBT radii with respect to Ψ3 , no significant oscillation
can be found in relative amplitude of Rlong , Rol and Rsl , but -2R2out,3 /R2out,0 has positive valueand 2R2side,3 /R2side,0 is negative value for all centrality. -2R2out,3 /R2out,0 becomes larger from central
to peripheral, while 2R2side,3 /R2side,0 slightly decrease with increasing centrality. 2R2os,3 /R2side,0 has
positive or zero and no significant centrality dependence can be found.
Contrary to q2 selection to HBT measurement relative to Ψ2 , no significant modification
can be found in q3 selection to azimuthal angle dependence of HBT radii with respect to Ψ3 ,though v3 is explicitly enhanced(suppressed) with larger(smaller) q3 selection.
Three interpretation can be considered to this result. First one is q3 selectivity is not sufficient to change the relative amplitude of HBT radii. Sensitivity of initial triangular shape canbe different between HBT measurement and flow measurement. Second possibility is oscillation amplitude of HBT radii with respect to Ψ3 are not dominated by triangular flow. Third
possibility is triangular flow is not originated from hydrodynamical expansion.

201

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