An evaluation of diﬀerent coordinate frames
in order to detect polarized virtual photons
in relativistic heavy ion collisions

Quark Physics Laboratory Department of Physics
Hiroshima University
M146985
Yosuke Ueda

February 10th, 2016
Supervisor : Associate Prof. Kenta Shigaki
Primary Examiner : Associate Prof. Kenta Shigaki
Examiner : Prof. Yasushi Fukazawa

1

Abstract
In relativistic heavy ion collisions, a strong magnetic field is generated. Whenthe nuclei having large charges which are accelerated to nearly the light speed collide oﬀ-centered in a small space of ˜10 fm, the intense magnetic field is created.
The strength of magnetic field amounts to ˜1019 Gauss. It is the strongest in our
universe. In the magnetic field, interesting eﬀects, such as chiral magnetic eﬀects,
synchrotron radiation of quark, and photon decay, may occur. Detections of theintense magnetic field, however, has not been realized. In this thesis, we discuss thefeasibility and methods of its detection.
In an intense magnetic field, the electron and positron from a virtual photon decay are asymmetric due to eﬀects of electron loops of virtual photon in a magnetic
field, and this electron-positron polarization depends on the direction of the magnetic field. We can hence detect the magnetic field by detection of virtual photon
polarization. We use polarization frames used in J/Ψ polarization measurement todetect the virtual photon polarization.
First, I evaluate the detection feasibility of virtual photon polarization in the
PHENIX experiment based on the statistics. We use the AuAu collision data at√
sN N = 200GeV in 2004 for this evaluation. As a result, the feasibility to detect
virtual photon polarization is ˜0.2σ . Also the estimated feasibility using the data
in 2010 is ˜0.6σ , and it using the data in 2014 is more order.
Next, I estimate the detection feasibility of virtual photon polarization using
three diﬀerent polarization frames, detection Helicity (HX) frame, Gottfried-Jackson(GS) frame, and Collins-Soper (CS) frame. I tested three polarization cases, nopolarization, perpendicular, and parallel with respect to the direction of magnetic
field. I compare these and decide that CS frame is the most useful frame for detectionof polarization. Also I simulate the polarization measurement with the polarization
rate to estimate the feasibility.

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Contents
1 Introduction
1.1 Relativistic Heavy Ion collisions . . . . . . . . . . . . . . . . . . . . . . . .
1.1.1 Space-Time development of QGP . . . . . . . . . . . . . . . . . . .
1.2 Generation of a intense magnetic field . . . . . . . . . . . . . . . . . . . . .
1.2.1 Physics phenomenon in a intense magnetic field . . . . . . . . . . .
1.3 Virtual photon polarization by a intense magnetic field . . . . . . . . . . .
1.4 Virtual photon in to measure the polarization . . . . . . . . . . . . . . . .
1.5 Theoretical feasibility of magnetic field detection . . . . . . . . . . . . . . .
1.6 Purpose . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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2 Experiment
2.1 RHIC accelerator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.1.1 PHENIX experiment . . . . . . . . . . . . . . . . . . . . . . . . . .
2.1.2 PHENIX detectors . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2 LHC accelerator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2.1 ALICE experience . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2.2 ALICE detectors . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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3 Method
3.1 Definition of polarization frames . . . . . . . . . . . . . . . . . . . . . . . .
3.1.1 Feature of each polarization frame . . . . . . . . . . . . . . . . . . .
3.1.2 Definition of angle on polarization frame . . . . . . . . . . . . . . .
3.2 Calculation of polarization parameter . . . . . . . . . . . . . . . . . . . . .

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4 Simulation
4.1 Method of simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2 Hijing event . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2.1 Same Hijing simulation . . . . . . . . . . . . . . . . . . . . . . . . .
4.2.2 Mixed Hijing simulation . . . . . . . . . . . . . . . . . . . . . . . .
4.3 Signal event . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.4 Signal-Hijing event . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.5 Subtraction of background eﬀect . . . . . . . . . . . . . . . . . . . . . . . .

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5 Result
5.1 Selection of good frame . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.2 Polarization measurement with detectors . . . . . . . . . . . . . . . . . . .
5.2.1 ALICE detector . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.2.2 PHENIX detector . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.3 With polarization rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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6 Conclusion

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7 Acknowledgement

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3List of Figures
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Schematic of heavy ion collision [1] . . . . . . . . . . . . . . . . . . . . . .
Accelerated nuclei collision [2] . . . . . . . . . . . . . . . . . . . . . . . . .
Space-time evolution of a heavy ion collision [3] . . . . . . . . . . . . . . .
Generation of magnetic field [5] . . . . . . . . . . . . . . . . . . . . . . . .
Diagram of virtual photon loop . . . . . . . . . . . . . . . . . . . . . . . .
Invariant mass distribution of electron-positron pair for AuAu collision(M.B., 1.0 < pT < 1.5 GeV /c) [7] . . . . . . . . . . . . . . . . . . . . . . .
Fraction of the direct photon component as a function of pT [7] . . . . . . .
RHIC accelerator [11] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
PHENIX detector [12] . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
EMC detector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
RICH detector [14] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
DCH detector [15] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
LHC accelerator [1] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
ALICE detector [1] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
ITS detector [16] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
TPC detector [16] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
TOF detector [16] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Polarization frames . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Definition of HX frame on lepton pair rest frame . . . . . . . . . . . . . . .
Definition of GJ frame on lepton pair rest frame . . . . . . . . . . . . . . .
Definition of CS frame on lepton pair rest frame . . . . . . . . . . . . . . .
Definition of zenith angle and azimuth angle . . . . . . . . . . . . . . . . .
Aanisotropy of zenith angle [18] . . . . . . . . . . . . . . . . . . . . . . . .
Anisotropy of azimuth angle [18] . . . . . . . . . . . . . . . . . . . . . . . .
Deviation from good frame [18] . . . . . . . . . . . . . . . . . . . . . . . .
Polarized lepton pair as polarization case . . . . . . . . . . . . . . . . . . .
cosθ distribution by Same Hijing event . . . . . . . . . . . . . . . . . . . .
Image of pT selection in cut region [7] . . . . . . . . . . . . . . . . . . . . .
cosθ distribution as each polarization frames . . . . . . . . . . . . . . . . .
λθ of Hijing event . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
cosθ distribution of signal as each polarization case . . . . . . . . . . . . .
cosθ distribution on each polarization case . . . . . . . . . . . . . . . . . .
Subtracted cosθ distribution on each polarization case . . . . . . . . . . . .
Corrected cosθ distribution on each polarization case . . . . . . . . . . . .
λθ , λϕ and λθϕ at all acceptance . . . . . . . . . . . . . . . . . . . . . . . .
λθ at ALICE acceptance . . . . . . . . . . . . . . . . . . . . . . . . . . . .
λθ at PHENIX . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
λθ at all, ALICE and PHENIX acceptance . . . . . . . . . . . . . . . . . .

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List of Tables
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Magnet field strength [4] . . . . . . . . . . . . . . . . . . . . . . . . . . . .
List of used value to estimate the theoretical feasibility . . . . . . . . . . .
List of λθ , λϕ and λθϕ at all acceptance . . . . . . . . . . . . . . . . . . . .
List of λθ at ALICE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
List of λθ at PHENIX acceptance . . . . . . . . . . . . . . . . . . . . . . .
List of λθ at with acceptance cut . . . . . . . . . . . . . . . . . . . . . . .

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1

Introduction

1.1

Relativistic Heavy Ion collisions

Quarks and gluons are confined in a hadron such as a proton or a neutron by strong
interaction in the present universe. In the early universe at a high temperature withinhundreds of thousandth second from the big bang, however, quarks and gluons are deconfined. Deconfined quarks and gluons are called the“Quark Gluon Plasma (QGP)”.
To know the property of the elementary particles, the research of QGP is extremely important. The way to do this is relativistic heavy ion collisions at Relativistic Heavy Ion
Collider (RHIC) at Brookhaven National Laboratory (BNL) and at Large Hadron Collider(LHC) at European Organization for Nuclear Research (CERN).
1.1.1

Space-Time development of QGP

The space-time development is the below.

Figure 1: Schematic of heavy ion collision [1]1. The nuclei are accelerated to near the light velocity. At this time, nuclei look likedisk-shaped with Lorentz contraction.
2. The accelerated nuclei collide. The nuclei do not always collide at the center, collisions is often slipped (these are called the non-central collision). In the case ofnon-central collisions, the overlap region of the nuclei is interacted. This interacted part is called participants, and the non-interacted and pass part is called thespectator (Figure 2).
The degree of central collision is called impact parameter. It is defined as the
distance between the centers of the nuclei when the colliding nuclei are passing,namely the distance of closest approach of nuclei. Also, the plane consisting of the
impact parameter and the beam axis, is called the reaction plane.
3. The temperature and energy density of the participants are increasing. When thetemperature of the participants is over the phase transition temperature, the quarks
6Figure 2: Accelerated nuclei collision [2]and gluons (these are called “parton”) are deconfined. The partons are scatteredwithout bound and collide one another, and new partons are generated. QGP is
this state.
4. QGP expands at near the light velocity with the high pressure inside. This rapid
expansion decreases the temperature and density, and re-connecting partons form
hadrons. The scattering and interaction, however, occur between the partons, andthe hadrons transfer their energy with inelastic scattering. Low energy particles
receiving energy with this inelastic scattering excite to higher energy levels. The
high energy particles transit the original energy with radiation of light particles ordecay, and new particles are continually generated.
5. The inelastic scattering of hadrons and the interaction between partons decrease thetemperature and density. When new hadrons are no longer generated, the specie
and amount of hadrons are fixed. It is called “chemical freezeout”. The hadrons
still collide one another with elastic collisions.
6. Distance between particles get over the mean free path with the consecutive expansion. The elastic collisions no longer occur, and the momenta of the particles are
fixed. It is called “kinematic freezeout”.
7. The generated particles are spread in a stable status. These particles are detectedusing detectors.

1.2

Generation of a intense magnetic field

The intense magnetic field is generated with relativistic heavy ion collision. The maximum
strength of it is 1019 Gauss. This strength far better than artificial stable magnetic fields(4.5 × 105 Gauss), and it get even over the magnetic field of magnetor which is a neutronstar and get attention to have a strong magnetic field (1015 Gauss). Therefore the magnetic
field with heavy ion collision is considered that it is the strongest magnetic field in theuniverse (Table 1).
7

Figure 3: Space-time evolution of a heavy ion collision [3]SystemThe Earth magnetic field on N magnet pole
A common magnet
The magnetic field in strong sunspots
The most strong and sustained magnetic field in laboratory
The most strong and only briefly magnetic field in laboratory
Magnetor
The generated magnetic field with relativistic heavy ion collision on RHIC
The generated magnetic field with relativistic heavy ion collision on LHC

Strength [Gauss]
0.6
100
4000
4.5 × 1051071014−15
1018
1019

Table 1: Magnet field strength [4]The equation (1) shows that the charged particles with velocity v creates the magnetic
field at time t, position r, r ′ . quote the Lienard-Wiechert Potential,
B(r, t) =1 − v 2 /c2
eµ0 v × R.
4π R3 [1 − (v/c)2 sinϕRv ]3/2

(1)

In this equation, R = r −r ′ , µ0 is the magnetic permeability in the vacuum, ϕRv is theangle between R and v. Z is the amount of charge, e is the elementary charge and c is thelight velocity. This equation is applied to the magnetic field generation with heavy ion
collision. Also, the life time of generated magnetic field is diﬀerence between the spectator
part and participant part. In the case that the beam axis is z axis, and the vertical axis
to z axis and a reaction plane is y axis, spectators have the z axis momentum and shift
the x axis direction with passing another. At that time, the magnetic field is generated
to the vertical direction to the reaction plane. In short, the direction of magnetic field isalways on xy plane. It is more strong than a magnetic field of participant, but spectators
runs away at near the light velocity because spectators do not collision. Therefore this8magnetic field is rapidly decreased.
On the other hand, the participant do not run away like spectators. Collisional particles is rolling around the vertical direction to the reaction plane with angular momentum.
The life time of it is more longer than the magnetic field of spectator. The angularmomentum is resisted, so the rolling speed decreases with passing time. Finally, thisparticipant runs away at the same time as QGP disappearing. Therefore, the magnetic
field of participant has the same life time as life time of QGP.

Figure 4: Generation of magnetic field [5]1.2.1

Physics phenomenon in a intense magnetic field

In a intense magnetic field, unnormal eﬀects occur. In this section, I describe some ofthese eﬀects.
Synchrotron radiation
When a intense magnetic field is generated in QGP, quarks and anti-quarks, these
are charged particles, wind the magnetic field. The winding quarks and anti-quarks
lose energy with radiation of photons, gluons and leptons pair like that electrons
winding a magnetic field radiate photons. In general, energy loss of quarks andanti-quarks occurs during particles pass though QGP. Therefore, this eﬀect causes
new energy loss.
Also, on synchrotron eﬀects, radiated particles are radiated toward a tangential
direction of charged particle’s momentum. A magnetic field with heavy ion collisions
is always generated at a perpendicular direction as reaction plane, so the radiationby the magnetic field concentrates the reaction plane and the azimuth dependence
is appeared.
Photon decay and double refraction in magnetic field9In a intense magnetic field, the refraction of photon is changed by the vacuum
polarization of electron-positron pair. This reason is that it is diﬀerence between
the component of velocity on perpendicular and parallel with respect to a magnetic
field with the anisotropy of response rate of electrons and positrons. This eﬀect iscalled the vacuum double refraction.
Also, the photon with enough amount of energy can decay to the fermion and antifermion. This eﬀect cannot occur in vacuum.
These eﬀects change the refraction of perpendicular photon with respect to magnetic
field. It may cause that photon yields have dependence on the azimuth.
Photon division
In the vacuum, the photon division is prohibited for Furry theorem. In a magnetic
field, however, the three-point interaction occurs and it may permit to divide thephoton. It permits the opposite process to this process; two low energy photons
may fuse into one high energy photon. This eﬀect can shift the photon energy
distribution to milder it.
The others, the chiral magnetic eﬀect, which the electric current flows along the parallel
to a magnetic field with topological transition, and Schwinger mechanism is predicted.
These are interesting and important to know physics at heavy ion collisions.

1.3

Virtual photon polarization by a intense magnetic field

The virtual photon is the photon having the mass for the uncertainty principle. The
virtual photon has electron-positron loops, and finally decays to the electron and positron(Figure 5).

Figure 5: Diagram of virtual photon loop

If the virtual photon, it is direct photon because the life time of a magnetic field is as
short as QGP, goes through a intense magnetic field, the magnetic field eﬀects electronpositron loops of a virtual photon. A electron and positron from a virtual photon depends
on the magnetic field. Therefore the detection of a virtual photon polarization proves theexistence of intense magnetic field.
The degree of polarization P is defined as below equation 2.
10

P =

R∥ − R⊥.
R∥ + R⊥(2)

R∥ is the probability that decays of a virtual photon is parallel to a direction ofmagnetic field, and R⊥ is the probability that decays a virtual photon is perpendicular toa direction of magnetic field. The order of polarization is 10% [6].

1.4

Virtual photon in to measure the polarization

Electron-positron pairs in experimental data include the pairs from the expectation ofvirtual photons such as pions. Therefore a electron-positron pair from virtual photons isneeded to select for the polarization measurement.
Figure 6 is the invariant mass distribution of electron pairs. The dot line shows themass component of each particle by the cocktail simulation. The blue line is the total
yield of dot lines. It is the background. The orange line is the expected direct photon
mass distribution. The invariant mass distribution of Dalitz decay and direct photon isshown by Kroll-Wada formula[8].
√1 dNe+ e−
4m2
2m2
1
2α
1 − 2 e (1 + 2 e )=S,
Nγ dMe+ e−
3πM e+ e−Me+ e− Me+ e−{
M 2+ − 3e e)(Hadron decay),|F (Me2+ e− )|2 (1 − M
2S=
1 (Direct virtual photon : pT ≫ Me+ e− ).

(3)
(4)

Me+ e− is the invariant mass of electron-positron pairs, me is the mass of electron, M is
the mass of hadron, and F (Me2+ e− ) is the shape factor. pT is the transverse momentum.
The orange line and blue line is normalized by the experiment data in the low mass region.
Then these are fitted by r. r is the ratio of direct photon and all photon. In Figure 6, it isfrom the AuAu collision minimum Bias (M.B.) data, the Dalitz decay of π0 is dominatedin Me+ e− < 120 M eV /c2 , and the ratio of direct photon Signal and Background (S/B) is
too small. On the other hand, in 120 < Me+ e− < 300 M eV /c2 , the S/B ration is good.
Figure 7 is the pT distribution of direct photon as pT function. (a) is pp collisions
data and (b) is AuAu collision data. Also the error bars and the error band represent
the statistical and systematic uncertainties. Curves are expected from a Next-to-LeadingOrder perturbative QCD (NLO pQCD) calculation. In this figure(b), the r is 0.15 - 0.2
in all pT range. This value is enough large.
For the polarization measurement from a magnetic field, I use the method of J/Ψ
polarization measurement. This method is discussed in details later.

1.5

Theoretical feasibility of magnetic field detection

I discuss about the estimate of theoretical feasibility of a magnetic field. The estimatedfor ALICE experience have already been done[6] and it is about 1σ with 2.76 TeV PbPb
collisions in 2011. This value is also expected to be larger with the LHC upgrade in 2018.
Hence I estimate it for PHENIX experience.
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Figure 6: Invariant mass distribution of electron-positron pair for AuAu collision (M.B.,
1.0 < pT < 1.5 GeV /c) [7]Figure 7: Fraction of the direct photon component as a function of pT [7]12The theoretical feasibility is shown by equation 6. Neall+ e− is all the combinations ofelectron-positron pairs. These pairs include non-correlated pairs. It is called the combinatorial background Bcb . The ration of polarization signal S and Bcb is S/Bcb . And fhadronicshows the rate of electron-positron pairs from hadrons. RE.P. is the resolution of reaction
plane, and P is the degree of virtual photon polarization. In this estimation, P is assumed
to 0.1. Also the momentum of all the virtual photons is perpendicular to a magnetic field,
and the polarized pattern of a virtual photon decay is perfect perpendicular or parallel toa magnetic field.
signal,signal + backgroundNeall+ e− /2 × S/Bcb × (1 − fhadronic ) × RE.P. × P
=
.
Neall
+ e− /2

statistical signif icance = √

(5)
(6)

The experimental data set for the estimation of theoretical feasibility is for AuAu
collisions in 2004. The region of data is in 0.12 < Me+ e− < 0.3 GeV /c2 and 1.0 < pT <2.0 GeV /c. Table 2 is the list of used value.
Neall
+ e−
S/Bcb
fhadronic
RE.P.
P

Values∼ 1.46 × 105 [9]∼10% [9]∼80% [9]∼30% [10]O(10−1 )Table 2: List of used value to estimate the theoretical feasibility

When these values are substituted, the theoretical feasibility is about 0.2σ. The M.B.
event is 8.1 × 108 in 2004, on the other hand, M.B. data is 7.3 × 109 in 2010. When otherconditions is the same in 2004 and 2010, the theoretical feasibility is about 0.6σ. M.B.
data is larger in 2014, hence the theoretical feasibility is enough large for the detection ofa magnetic field. Therefore I discuss the method of polarization measurement.

1.6

PurposeThe detection of a intense magnetic field is much important to understand physics eﬀects
at heavy ion collisions. In this thesis, I discuss the feasibility of a magnetic field detectionand evaluate the detection method using the J/Ψ polarization measurement. For evaluation, I divide the polarization case into the non-polarization, perpendicular polarizationand parallel polarization with respect to the magnetic field, then compare results using
the methods detecting polarization of each case. Then I simulate the photon polarization to conclude that the method gives the most diﬀerent result by diﬀerent polarization
scenarios, in short, to decide the most useful method.

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2

Experiment

In this study, eﬀects of detectors at relativistic heavy ion collision experiments are considered. I discuss that two detector in this thesis, PHENIX and ALICE detector. These
are shown in this section.

2.1

RHIC accelerator

Relativistic Heavy Ion Collider (RHIC) at BrookHeaven Natioal Laboratory (BNL) to
the fundamental study for relativistic heavy ion collisions. It is composed by double ring
(blue ring and yellow ring), and the circumferential length of its is about 3.8km. This
accelerator is the first relativistic heavy ion collider and it specializes in the heavy ion
acceleration. Therefore it can accelerate many types of atoms such as U, Au, Al, He3 anddeuterium. It means that this accelerator can verify many physic phenomenons which
generated by heavy ion collisions. RHIC accelerator can accelerate a proton to 250 GeV
and a heavy ion to 100GeV per nucleon.
RHIC has 4 experiment: PHENIX, STAR, BRAHMS, PHOBOS. I focus PHENIX
experience in this thesis.

Figure 8: RHIC accelerator [11]2.1.1

PHENIX experiment

The purpose of PHENIX experiment is the generation of QGP and the elucidation of QGP
characterization. This experience is with 430 scientists, 11 countries and 43 organizations.
14

From Japan, The University of Tokyo, University of Tsukuba, Hiroshima University and
RIKEN are participated. PHENIX experiment specializes in the measurement of photons
and leptons, so it is very suitable for the measurement of polarized photons. The Figure
9 is the overview PHENIX detector.
The PHENIX detector is separated as bellow.

Figure 9: PHENIX detector [12]1. Central Arm detector (-0.35 < η < 0.35)
The Central Arm detector, that is divided into East Arm and West Arm, measures
momentums and tracks of charged particles. Also charged particles are identified
by it.
2. Muon Detector ( -2.45 < η < -1.15, 1.15 < η < 2.44)
The Muon Detector measures forward muons. Muons is the lepton which does notinteract with QGP by the strong force. Therefore the measurement of muons isimportance to get the information about the initial state.
3. Grovel detector (3.4 < |η|)The Grovel detector triggers the measurement of events. This detector measures
the the centrality, luminosity, collision timing and collision point.
152.1.2

PHENIX detectors

I introduce some PHENIX detectors to measure the polarized photon detector below.
Electro Magnetic Calorimeter (EMC)
EMC measures the energy and potential of photons and charged particles. It covers
the pseudo rapidity < η < 0.375, and azimuth 30◦ < |ϕ| < 120◦ . In EMC, charged
particles generate photons by the bremsstrahlung. These photons generate electronsand positrons by the electron-positron pair generation, then these electrons andpositrons generate photons. This process, it is called the electro-magnetic shower,
is repeated until energy is under the critical energy point of electron-positron pair.
EMC detects the energy from this process and reconstructs generated particles.
EMC is composed by the lead scintillator (PbSc) calorimeter and lead glass (PbGl)
calorimeter. PbSc is at four sectors in the West Arm, and PbSc is at two sectors
and PbGl sectors is at two sectors in the East Arm.

EMC of lead-scintillator module [13]

EMC of lead-glass super module [13]Figure 10: EMC detector
Ring Imaging CHerenkov counter (RICH)
RICH identifies charged particles. The speed of particle is diﬀerence by each massin the medium. Also, charged particles in the medium scat the light when the speedof these is greater then the speed of light in the medium. The radiation angle θof this scattering light, it is called Cherenkov light, is cosθ = 1/nβ. Hence themeasurement of radiation angles informs the mass of charged particle, therefore
charged particles are identified. The medium gas (radiator gas) in RICH is CO2 ,and RICH detects the radiation light by Photon Multiplier Tubes (PMT). Also itcovers the pseudo rapidity |η| < 0.35 and azimuth 30◦ < |ϕ| < 120◦ .
DCH (Drift CHamber)
DCH, this wire chamber covers both the Central Arms, is the main detector for
tracking charged particles. It covers the pseudo rapidity |η| < 0.7, and azimuth−34◦ < ϕ < 56◦ on the west central arm and azimuth 125◦ < ϕ < 215◦ on the east
central arm. This detector measures tracks of charged particles by connecting detect
16Figure 11: RICH detector [14]points of charged particles which are bended tracks by the artificial magnetic field.
Also the transverse momentum is reconstructed by the measurement of curvatureof charged particle’s track.

Figure 12: DCH detector [15]2.2

LHC accelerator

Large Heavy ion Collider (LHC) is at European Organization for Nuclear Research (CERN).
It is at about 100m underground, and the circumferential length of its is about 27km. This
accelerator started up on 2008 with over about 10,000 scientists and about 100 countries
and organizations. Also LHC is the largest collider in the world, and it can accelerate theproton beam to 7.0 TeV. This accelerated energy is also the highest in the world.
17

LHC have 6 experiment: ATLAS, CMS, ALICE, LHC-b, TOTEM, LHC-f. These experiments have diﬀerence purpose and diﬀerence detector for each purpose. For example,
the purpose of ATLAS and CMS is the discovery of Higgs boson, generation of dark matter (supersymmetric particle: SUSY) and the consideration of beyond standard model.
In this thesis, I discuss the ALICE experience for relativistic heavy ion collision.

Figure 13: LHC accelerator [1]2.2.1

ALICE experience

ALICE experience is an abbreviation of “A Large Ion Collider Experiment”. The purposeof ALICE experience is the generation of QGP with PbPb collisions and the elucidationof QGP characterization. This experience is with 1,300 scientists, 35 countries and 120
organization. The University of Tokyo, University of Tsukuba, Hiroshima University
and RIKEN are collaborated from Japan. This experience only specializes in measure
relativistic heavy ion collisions at LHC experience. Therefore ALICE detector is needed
to detect many physics phenomenon which is generated with heavy ion collisions into wide
energy area. The Figure 14 is the overview of ALICE detector.
The ALICE detector is separated as bellow.
1. Central Barrel (-0.9 < η < 0.9)
The Central Barrel covers near the collision point. It is in the dipole magnet for
applying 5,000 Gauss. This magnetic field bends charged particles generated byheavy ion collisions, and momentums of these particles can be measured.
18

Figure 14: ALICE detector [1]2. Dimuon Spectrometer (-4 < η < 2.5)
The Dimuon Spectrometer measures the forward muon. The muon measurement isimportant as with PHENIX.
3. Grovel detector (3.4 < |η|)The Grovel detector selects collision events like PHENIX. For this selection, thecentrality, luminosity, collision timing and collision point is known.
2.2.2

ALICE detectors

I introduce some ALICE detectors to measure the polarized photon like PHENIX detector
below.
Inner Tracking System (ITS)ITS is at nearest the beam collision point. It covers the pseudo rapidity −0.9 < η <0.9 and azimuth 2π. This detector measures tracks of charged particles and energy
loss per a unit length (dE/dx). Also it can measure the number, collision point anddecay point of particles. The identification of particle species is decided using it.
ITS consists of SPD (Silicon Pixel Detector), SDD (Silicon Drift Detector) and SSD(Silicon Strip Detector). SPD is the innermost two layers, SSD is the following twolayers, and SPD is the two outer layers. These are the front line of silicon detector.

19

Figure 15: ITS detector [16]Time Projection Chamber (TPC)
TPS is the wire chamber at 85-250cm from the beam collision point. This detector
is much superior to measure tracks of charged particles and dE/dx, and the main
detector to measure charged particles. It covers the pseudo rapidity −0.9 < η < 0.9
and azimuth 2π. The mixed gas (Ne/CO2/N2 : 85.7%/9.5%/4.8%) fills in thisdetector. Energy loss and ion pair generation by ionization with charged particles
occur passing through this gas. Generated ionize electrons are drifted by innerelectric field, then tracks of charged particles are reconstructed using the position
and time information of this electron. Also, for measurements of charged particles
dE/dx, it can identify particle species.

Figure 16: TPC detector [16]20Time of Flight (TOF)
TPC measures the flight time of charged particles. The flight time is diﬀerent foreach particles, so particles are identified by the combination of the flight distance,
the momentum and the flight time. It has the high time resolution, about 50ps, itmeans that it can divide particles which has the momentum 1.5GeV/c to kaon andpion.

Figure 17: TOF detector [16]21

3MethodTo measure polarized photons, I use below three polarization frames. These frames areused to observe the J/ψ polarization. In this section, features of these frames are described.

3.1

Definition of polarization framesThis method is the measurement on the frames which are defined by the decayed leptons
pair (in this thesis, electron and positron) from the polarized particle (in this thesis,
virtual photon). The frames are called “Helicity (HX) frame”, “Gottfried-Jacson (GJ)
frame” and “Collins-Soper (CS) frame”. All the frames is on the rest frame of the leptons
pair. Also, all the frames is the same frame when the velocity of polarized particles iszero. To use three polarization frames, x, y and z axis should be defined. The belows
are the definition of z axis in each polarization frames (Figure 18, [17]). In these plot, yaxis is the perpendicular axis to production plane including z axis and collision beams.
y axis is the same on all polarization frames. x axis is defined as y axis and z axis onright-handed system in each polarization frames.

q rest frame

collisions center of mass frame
Figure 18: Polarization frames

3.1.1

Feature of each polarization frame

1. Helicity (HX) framez axis is that the momentum direction of leptons pair on the laboratory system.
The direction of magnetic field is always the vertical direction to the reaction plane
4. Also the reaction plane is the vertical plane to the beam axises. Thus themomentum direction of the polarized particle, this particle is the parent particle
which decays into leptons pair, to the magnetic field direction depends on the beamaxises, that is z axis on the laboratory system.
22

On HX frame, the polarization is measured by observing the momentum correlation
between the decayed positive lepton and the polarized parent particle. The purely
momentum polarization from magnetic field eﬀect can be measured on this frame.
However, The polarization measurement on HX frame just only uses the correlation
between the momentum direction of decayed lepton and parent particle. Hence themomentum deviation except from a magnetic field eﬀect, such as miss reconstruction, is too eﬀective.

Figure 19: Definition of HX frame on lepton pair rest frame

2. Gottfried-Jackson (GS) framez axis is that the beam momentum (beam1 or beam2 in Figure 18) on the rest frameof leptons pair.
On GS frame, the polarization is measured by observing the correlation between abeam axis and the positive lepton. It reflects the correlation between a beam axis,
it is the vertical axis to the direction of magnetic field, and the polarized particle.
Also, the zx plane and y axis on the laboratory system and on GS frame is the same
when the transverse momentum of parent particle is enough small.
However, this frame is defined by using only one of two beam axises. This encompasses the weakening of the correlation between magnetic field direction andpolarized particle, because the direction of a magnetic field is vertical to the bothbeam axises. This eﬀect is more eﬀective when the the transverse momentum of parent particle is not neglected, in other wards, the both beam axises on the laboratory
system and on GJ frame is not on the same plane.
3. Collins-Soper (CS) framez axis is that the bisector of beams (beam1 and -beam2 or -beam1 and beam2 in
Figure 18) on the rest frame of leptons pair.
On CS frame, the polarization is measured by observing the correlation between
the both beam axises and the positive lepton. The eﬀect of momentum direction
23

Figure 20: Definition of GJ frame on lepton pair rest frameof polarized parent particle is small on this frame. The zx plane and y axis on thisframe, on the laboratory system and on GJ frame is the same when the transverse
momentum of parent particle is neglected. And CS frame is rotated by 90◦ aroundy axis when the transverse momentum of parent particle is enough high and thelongitudinal momentum is neglected.
However, this frame uses two beam axises. Hence the correlation between the parent
particle and the beam axises is weaker than other frames.

Figure 21: Definition of CS frame on lepton pair rest frame

These frames have the advantage and disadvantage. Therefore the good frame selection
as measured target is needed.
24

3.1.2

Definition of angle on polarization frame

The zenith angle θ is defined as the angle between z axis and the angle of positive charged
lepton. The azimuth angle is defined as the angle between x axis and the vector of positive
charged lepton projected on xy plane (Figure 22).

Figure 22: Definition of zenith angle and azimuth angle

3.2

Calculation of polarization parameter

The angle distribution W (cosθ, ϕ) of polarized event is the below equation 8. i is thecontributing production processes andf i is the weighting function.

W (cosθ, ϕ) =n∑f (i) W (i) (cosθ, ϕ),

(7)

i=1

∝

1
(1 + λθ cos2 θ + λϕ sin2 θcosϕ + λθϕ sin2θcosϕ).
3 + λθ

(8)

Also, The each angle distribution can be calculated to integrate equation refeq:W overcosθ and ϕ (equation 9, 10).
W (cosθ) ∝ 1 + λθ cos2 θ,
25(9)W (ϕ) ∝ 1 +2λϕcos2 θ.
3 + λθ

(10)

λθ,ϕ,θϕ is the polarization parameter. To calculate λθϕ , the below definition (equation
12) of ϕ̃ is needed. The equation of ϕ̃ distribution is shown by equation 11. If the ϕ̃ is out
from 0 to 2π, the ϕ̃ is needed to properly add or subtract the 2π to become 0 < ϕ̃ < 2π.
The λθϕ is calculated by it.
{
ϕ − 43 ϕ(cosθ < 0),ϕ̃ ∝ϕ − 41 ϕ(cosθ > 0).

(11)

√
2λθϕ
cosϕ̃.
W (ϕ̃) ∝ 1 +3 + λθ

(12)

These polarization parameters show the asymmetry of the number of particles onangle distribution. Therefore, these parameters can be shown by the below equation 15.
P (|cosθ|) is the probability distribution of positive charged lepton.
P (|cosθ| > 1/2) − P (|cosθ| < 1/2)
3 λθ=,
P (|cosθ| > 1/2) + P (|cosθ| < 1/2)
4 3 + λθP (|cos2ϕ| > 0) − P (|cos2ϕ| < 0)
2 2λϕ
=
,
P (|cos2ϕ| > 0) + P (|cos2ϕ| < 0)π 3 + λθP (|sin2θcosϕ| > 0) − P (|sin2θcosϕ| < 0)2 2λθϕ=.
P (|sin2θcosϕ| > 0) + P (|sin2θcosϕ| < 0)π 3 + λθ

(13)
(14)
(15)

λθ is shown by the anisotropy of zenith angle. Therefore λθ = 0 is no-polarization,
λθ > 0 is parallel polarization and λθ < 0 is perpendicular polarization (Figure 23).

Figure 23: Aanisotropy of zenith angle [18]λϕ is shown the anisotropy of azimuth angle. Therefore λθ = 0 is no-polarization,
λθ > 0 is parallel polarization and λθ < 0 is perpendicular polarization (Figure.24).
26

Figure 24: Anisotropy of azimuth angle [18]Also, λθϕ is used to define the “better” polarization frame. The “best polarization
frame” is that λθ,ϕ have the best significance and the symmetry is the most shape. The
degree of symmetry is expressed by δ and this equation is below (equation 16).
1
2λθϕ
δ = arctan().
2λϕ − λθ(16)

δ shows the angle of rotation around y axis. The polarization frame having most small
δ is the most symmetric shape (Figure 25). In this figure, x’y’z’ system is good frame.
xyz system deviate delta from x’y’z’ system.

Figure 25: Deviation from good frame [18]274Simulation

4.1

Method of simulation

To think the polarization, I thought in the three cases that virtual photons have nopolarization, perpendicular polarization and parallel polarization with respect to a magnetic field (Figure 26).
Also, to estimate the precise polarization parameter, the background estimation isneeded. I used Heavy Ion Jet INteraction Generator (Hijing) to estimate the background. Hijing is Monte Carlo simulation to simulate the pp and AA collision. This
generator reproduce the heavy ion collision events which have the jet correlation and the
no-polarization. Therefore it is suitable to estimate the background for this simulation.
The particles in events that is generated by Hijing have the jet correlation. Also, particles
which is generated by Hijing is combined to reconstruct parent particles and reproduce
physics phenomena from heavy ion collisions. The event having the correlated particles iscalled “same Hijing event” in the following, because the correlated particles is combinedin the same collision event. Also, the event having the non-correlated particle is called“mixed Hijing event” in the following. Because the no-correlated particles is reconstructed
the combination of particles in the diﬀerent collision events, in other words, the diﬀerent
events is mixed.

Figure 26: Polarized lepton pair as polarization case

4.2

Hijing event

The Hijing events includes electron pairs from the expectation of virtual photons, I limit
the invariant mass and transverse momentum. In this thesis, the limit of reconstructedpair is 0.12 < Me+ e− < 0.3 GeV /c2 and 1.0 < pT < 5.0 GeV /c.
4.2.1

Same Hijing simulation

The cosθ distribution of positron in same Hijing event is Figure 27.
28

Figure 27: cosθ distribution by Same Hijing event

This cosθ distribution is normalized by entry, so this shows the probability distributionof positron. This distribution has the correlation. This reason is that when the high pTparticle exists, the pair in cut region is reconstructed with one high pT particle and many
low pT particle that is near the high pT particle (Figure 28). Hence the opening angle
between high pT particle and low pT particle is small. Therefore the deviation on the plot
is appeared.

Figure 28: Image of pT selection in cut region [7]4.2.2

Mixed Hijing simulation

To reject the correlation expect from the polarization of magnetic field eﬀect, I use themixed Hijing events. This mixed Hijing events also shows the probability distributionof positron by normalization like the same Hijing events. The cosθ distribution, it issubtracted the cosθ distribution of the mixed Hijing events from the cosθ distribution of
the same Hijing event, is Figure 29, and the λθ from this cosθ distribution is Figure 30.
The λθ is zero within uncertainties. It means that the λθ becomes zero on all the frames
when the case is no-polarization.
29

cosθ distribution on HX frame cosθ distribution on GJ frame cosθ distribution on CS frame
Figure 29: cosθ distribution as each polarization frames

Figure 30: λθ of Hijing event

30

4.3

Signal event

I created the electron and positron from virtual photon by myself, then the electron andpositron are polarized depending on three polarization cases. This polarized electronpositron pair is called “signal event”. The cosθ distribution is Figure 31. All the simulated electron pairs have the polarization. cosθ distribution is obvious diﬀerence as eachdiﬀerent polarization. I mix this signal event to the same Hijing event. When particles
is reconstructed by combining the same event to reconstruct the reproduce physics phenomena like the same Hijing event, this events is called “same signal-Hijing event”. Also,
when particles is reconstructed by combining the diﬀerent events like the mixed Hijing
event, this events is called “mixed signal-Hijing event”

No-polarization

Perpendicular polarization tomagnetic field

Parallel polarization tomagnetic field

Figure 31: cosθ distribution of signal as each polarization case

4.4

Signal-Hijing event

The cosθ distribution by same signal-Hijing event is Figure 32. In this distribution,
the parallel polarization can be distinguished from no-polarization and perpendicularpolarization. In contrast, the perpendicular polarization can not be distinguished fromno-polarization. This reason is that the shape of cosθ distribution of signal event, that isperpendicular polarization, and the same Hijing event is almost the same.

cosθ distribution on HX frame cosθ distribution on GJ frame cosθ distribution on CS frame
Figure 32: cosθ distribution on each polarization case

31

4.5

Subtraction of background eﬀect

To emphasize the diﬀerence of cosθ distribution for each polarization case, the normalizedcosθ distribution of mixed signal-Hijing event is subtracted from the normalized cosθ
distribution of same signal-Hijing event. This result is Figure 33.

Subtracted cosθ distribution
on HX frame

Subtracted cosθ distribution
on GJ frame

Subtracted cosθ distribution
on CS frame

Figure 33: Subtracted cosθ distribution on each polarization case

By this subtraction, the probability distribution of positron is negative on some region.
Hence I correct the cosθ distribution so that the smallest value is 0 (Figure 34). The cosθ
distribution means only the deviation of positron distribution, so this correction influence
the deviation from polarization.

Corrected cosθ distribution on Corrected cosθ distribution on Corrected cosθ distribution on
HX frame
GJ frame
CS frame
Figure 34: Corrected cosθ distribution on each polarization case32

5Result5.1

Selection of good frame

The λθ , λϕ and λθϕ is Table 3 and Figure 35. The used event set is that one polarized pairis in one collision event for the purpose that the polarization influence is more eﬀective
to select the best frame. Also the acceptance covers all region. The number of event is
10,000.
polarization caseno-polarization

perpendicularparallelpolarization frame
HX frame
GJ frame
CS frame
HX frame
GJ frame
CS frame
HX frame
GJ frame
CS frameλθ−0.40 ± 0.032−0.40 ± 0.032
1.7 ± 0.17
−0.35 ± 0.039−0.35 ± 0.039
2.4 ± 0.22
−0.69 ± 0.031−0.68 ± 0.031
1.0 ± 0.17

λϕ
−0.26 ± 0.018−0.19 ± 0.015−0.27 ± 0.012
0.81 ± 0.025
0.79 ± 0.026−0.24 ± 0.025−0.42 ± 0.015−0.41 ± 0.013−0.41 ± 0.018

λθϕ
0.005 ± 0.022−0.26 ± 0.015
0.06 ± 0.089−0.01 ± 0.015−0.21 ± 0.014−0.07 ± 0.089
0.013 ± 0.025−0.27 ± 0.032−0.003 ± 0.039

Table 3: List of λθ , λϕ and λθϕ at all acceptance

λϕ

λθ

λθϕ

Figure 35: λθ , λϕ and λθϕ at all acceptanceIn this figure, the λθ value is the same on HX frame and on GJ frame when thepolarization case is the no-polarization and the perpendicular polarization. And when
the case is the parallel polarization, λθ is nearly the other value on HX frame and on GJ
frame. On the other hand, λθ on CS frame is the enough diﬀerent for each polarization
cases. Also, λθ is not zero on CS frame at no-polarization. Because the correlation ofelectron-positron pair in the same signal-Hijing event is not completely subtracted fromsame signal-Hijing event. It is show that this subtraction method is needed to be more
optimized, however, this eﬀect from the correlation similarly influences all the polarization
frames for each polarization case. Thus the relativistic λθ value to the values on the otherframe for each other polarization case is useful. Therefore the CS frame is that thediscrimination ability to observe the polarization is high.
It can also be referred to from λθϕ and λϕ . On GJ frame, λθϕ is clearly non-zero. On
the other hand, on HX frame and on CS frame, λθϕ is zero within uncertainties at all the33polarization cases. λθϕ shows the deviation from the good frame, therefore HX frame andCS frame are good frames.
In addition, on HX frame and on GJ frame, λϕ is not zero, nevertheless the polarizationfrom magnetic field eﬀect does not have the division of phi direction. This value of λϕis considered to be from the momentum division of the leptons pair. By contrast, λϕ on
CS frame has nearly the same value at all the polarization cases because CS frame isinsulated from the influence of momentum distribution of the leptons pair.
From the above, CS frame is the best frame to measure the virtual photon polarizationfrom magnetic field eﬀect.

5.2

Polarization measurement with detectors

Next, I discuss the appearance at relativistic heavy ion collision experiments. In this thesis, the discussed experiment is ALICE experiment and PHENIX experiment. I simulate
the polarization measurement with the limited rapidity η and ϕ according to the detectorof each experiment.
5.2.1

ALICE detector

I simulate the polarization value λθ on CS frame at ALICE detector at first. This simulation is limited the rapidity η from -0.9 to 0.9. Because ALICE experiment mainly uses
TPC to measure the charged particles. This result is Table 4 and Figure 36. Also, λθ on
HX frame and GJ frame is described as reference.
polarization caseno-polarization

perpendicularparallelpolarization frame
HX frame
GJ frame
CS frame
HX frame
GJ frame
CS frame
HX frame
GJ frame
CS frameλθ−0.41 ± 0.030−0.41 ± 0.031
0.99 ± 0.085−0.40 ± 0.040−0.39 ± 0.040
1.7 ± 0.13
−0.75 ± 0.032−0.72 ± 0.032
1.2 ± 0.29

Table 4: List of λθ at ALICE

At parallel polarization on CS frame, λθ becomes nearly the case of no-polarization.
When the polarized parent particle goes to the beam axis, the z axis on CS frame isx axis on the laboratory system and the direction of a magnetic field is on yz planeon the laboratory system. In this situation, cosθ is 0 on CS frame. Also, at the parallel
polarization, the direction of decayed particles is the same as the direction of the magnetic
field. Thus, By cutting the rapidity region around the z axis, the probability that cosθ
is 0 and nearly 0 is smaller. Therefore, the λθ varies and reaches nearly the value at
no-polarization case at ALICE acceptance by this eﬀect.

34

Figure 36: λθ at ALICE acceptance

5.2.2

PHENIX detector

I simulate the polarization value λθ at PHENIX detector next. This simulation is limited
the rapidity |η| < 0.375 and 30◦ < |ϕ| < 120◦ . PHENIX experiment mainly uses DCH
to measure the track of charged particles, and EMC and RHIC to identify the charged
particles. Therefore the most smallest DCH acceptance in three detectors is applied. This
result is Table 5 and Figure 37.
polarization caseno-polarization

perpendicularparallelpolarization frame
HX frame
GJ frame
CS frame
HX frame
GJ frame
CS frame
HX frame
GJ frame
CS frameλθ−0.43 ± 0.053−0.43 ± 0.056
1.2 ± 0.20
−0.6 ± 0.10
−0.6 ± 0.1
2.1 ± 0.26
−0.93 ± 0.068−0.96 ± 0.070−0.7 ± 0.47

Table 5: List of λθ at PHENIX acceptance
At PHENIX detector, λθ at parallel polarization on CS frame goes away from theothers. The probability that cosθ is 0 and nearly 0 is smaller when the rapidity region iscut around the z axis. However, when the polarized parent particle goes to y axis on thelaboratory system, the direction of the both beam axises and z axis on CS frame is on
yz plane in the laboratory frame. Also, the direction of a magnetic field on xy plane isboosted by Lorentz transition and becomes nearly y axis on CS frame. In this situation,
The cosθ is about 1.0. Therefore, cosθ exceeds 0.5, that is the reference point to measurepolarization, at a certain point when the scattering direction of parents particle transports
z axis to y axis. And the trend of λθ transition is reversed. This reverse is started when the
rapidity is about 0.8. The PHENIX acceptance is the rapidity |η| < 0.375, this reversed
35

Figure 37: λθ at PHENIX

point is exceeded. Thus λθ at the parallel polarization on CS frame leaved from the value
at the no-polarization.

5.3

With polarization rate

In this section, I discuss the polarization measurement in consideration of polarization
rate of virtual photon. In the region that 0.12 < Me+ e− < 0.3 GeV /c2 and 1.0 < pT <5.0 GeV /c, the direct virtual photon is about one in 20 event[9] and the polarization rateof direct virtual photon is assumed to be 10%. Therefore, the polarized virtual photon isabout one in 200 event. I simulate the polarization measurement with this condition likeabove.
These results is Table 6 and Figure 38. The λθ is nearly the same and zero within theuncertainty at all acceptance. The devision at ALICE and PHENIX is considered to be
the eﬀect of acceptance cut. Thus it means that the polarization measurement is diﬃcult.
polarization caseno-polarization

perpendicularparallelpolarization frame
HX frame
GJ frame
CS frame
HX frame
GJ frame
CS frame
HX frame
GJ frame
CS frame

λθ at all acceptance−0.1 ± 0.39
−0.1 ± 0.44
0.2 ± 0.25
−0.09 ± 0.41
−0.06 ± 0.45
0.2 ± 0.27
−0.2 ± 0.41
−0.2 ± 0.46
0.08 ± 0.27

λθ at ALICE
0.3 ± 0.73
0.3 ± 0.65
−0.08 ± 0.83
0.6 ± 0.94
0.5 ± 0.74
−0.3 ± 0.98
0.2 ± 0.98
0.2 ± 0.65
−0.4 ± 0.66

Table 6: List of λθ at with acceptance cut

36

λθ at PHENIX
0.2 ± 0.58
0.3 ± 0.43
−0.4 ± 0.51
0.2 ± 0.65
0.3 ± 0.56
−0.3 ± 0.50
0.03 ± 0.48
0.1 ± 0.43
−0.5 ± 0.46

λθ at all acceptanceλθ at ALICE acceptanceλθ at PHENIX acceptance

Figure 38: λθ at all, ALICE and PHENIX acceptance37

6Conclusion

In this thesis, the theoretical feasibility of polarization measurement is discussed andthe best frame to observe polarization is found. Also, the polarization measurement issimulated with the polarization rate, one polarized pair in 200 event. From these results,
the theoretical feasibility is enough large, 0.6σ using the data in 2010 and more using
the data in 2014, and CS frame is the best frame to do it. On the other hand, thesimulation with the polarization rate shows that the polarization measurement is diﬃcult
because the polarization rate is low. However, it does not mean that polarization frames
are meaningless to measure the virtual photon polarization. In this thesis the methodfor direct virtual photon selection is pT and mass cut only. Therefor the possibility ofpolarization measurement exists when more eﬀective selection is used and the polarization
rate becomes high.
To measure the polarization, it needs to develop use the better method of direct virtualphoton selection. Also it is important that the statistics are increased, and the subtraction
method of the correlation of signal pair is optimizes. When these succeed, the polarization
measurement will succeed to use the polarization frame.

38

7

Acknowledgement

I would like to thank associate Prof. K.Shigaki. He purposed to me the theme of thisthesis, and he gave me many advises. I could not have done this thesis without his
support. Also, I would like to appreciate Prof. T.Sugitate. He taught me the method ofpresentation and advised for this thesis. These are important and valuable. Assistant Prof.
K.Homma discussed with me about the physics for this thesis, and gave me suggestions.
And I am grateful to Assistant Prof. T.Miyoshi for his comments at the meeting. I would
like to express my gratitude.
And I would like to appreciate whom I worked together in the Hiroshima group.
Hoshino-san and Nagashima-san taught me about the detectors and physics. Matsuurasan discussed with me the thesis, physics and anything. I had enjoyed working withher. Yamakawa-kun and Nobuhiro-kun also discussed with me anything. Especially,
Yamakawa-kun gave me suggestions for this thesis.
I am grateful to other people. I can not appreciate all of them enough. Thank you
everyone, again.

39

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