Measurement of di-electron anisotropy
for strong magnetic field search in Cu+Cu√
collisions at sN N = 200 GeV at PHENIX

Tomoya Hoshino (M110713)
Quark Physics Laboratory
Department of Physics
Graduate School of Science
Hiroshima University

Supervisor: Associate Prof. Kenta Shigaki
Primary examiner: Associate Prof. Kenta Shigaki
Examiner: Associate Prof. Tsunefumi Mizuno

February 28th, 2013

Abstract
Strong magnetic field is created in heavy ion collisions. This intense magnetic field is caused by non-central collisions of high energy
nuclei, for example gold or copper, in highly narrow space, about
10f m. And the maximum strength amounts to 1014 tesla at the energy region of the Relativistic Heavy Ion Collider(RHIC). This strengthis higher than Magnetar that is a type of neutron star with extremely
intense magnetic field, 1011 tesla. This magnetic field strength andspace-time evolution is calculated by some theoretical models.
Strong field physics is attracted because this has very interesting
physics such as vacuum birefringence, Schwinger mechanism and synchrotron radiation of quarks. Strong magnetic field physics is mainly
researched by theoretical side, but it had not detected in any experimental research. Thus, experimental confirmation of strong magnetic
field in heavy ion collisions is really important and meaningful. This
thesis is pioneering measurement of di-electron anisotropy and porlarization for magnetic field search in heavy ion collisions.
Creation of magnetic filed in heavy ion collisions is appeared in the
primary stage of collisions. And this strong field can have eﬀect on
decay of photons and its yield of azimuth dependence and polarization.
In addition, since photons and leptons are not operated by strong
interaction, these are good probe to detect this strong magnetic field’s
eﬀects.
In this thesis, Cu+Cu 200 GeV Minimum Bias data set is usedbecause this is one of the best data set in PHENIX to analyze with
di-electron analysis. In this analysis, asymmetry of di-electrons withrespect to the reaction plane and porlarization are showed. One isasymmetry of number of electron pairs, it is like elliptic flow. And
another is polarization, it means that di-electron opening angle correlation with reaction plane. Moreover centrality dependence of theseanisotropy and polarization are shown and discussed these plots forthe purpose of strong magnetic field search.
In conclusion, we measured anisotropy and polarization of di-electron
for selected centralities and selected mass region. The measured dielectron anisotropy is consistent with the calculated anisotropy using
elliptic flow (v2). We measure two di-electron polarization, one is thepolarization for reaction plane side, and the other is for magnetic fieldside. In order to cancel detector acceptance, we used double ratio using both side polarization. The double ratio value are consistent withunity in error bar.

2

Contents
1 Introduction
7
1.1 Relativistic heavy ion collisions . . . . . . . . . . . . . . . . . 7
1.2 The strong magnetic field in heavy ion collisions . . . . . . . . 7
1.3 Time evolutions . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.3.1 Time evolution of heavy ion collisions . . . . . . . . . . 8
1.3.2 Time evolution of the magnetic field . . . . . . . . . . 9
1.4 Photons and leptons . . . . . . . . . . . . . . . . . . . . . . . 11
2 Experimental setup
2.1 The RHIC accelerator . . . . . . . . . . . . . . . . . . . . . .
2.2 The PHENIX experiment . . . . . . . . . . . . . . . . . . . .
2.3 Grobal detectors . . . . . . . . . . . . . . . . . . . . . . . . .
2.3.1 Beam-Beam Counters(BBC) . . . . . . . . . . . . . . .
2.3.2 Zero Degree Calorimeters(ZDC) . . . . . . . . . . . . .
2.4 PHENIX Magnets . . . . . . . . . . . . . . . . . . . . . . . . .
2.5 Central arm detectors . . . . . . . . . . . . . . . . . . . . . . .
2.5.1 Ring Imaging Cherenkov Counter(RICH) . . . . . . . .
2.5.2 Electro-Magnetic Calorimeter (EMCal) . . . . . . . . .
2.5.3 Drift Chamber (DC) . . . . . . . . . . . . . . . . . . .
2.5.4 Pad Chamber (PC) . . . . . . . . . . . . . . . . . . . .

12
12
13
14
14
15
15
16
16
17
19
19

3 Analysis
3.1 Run and event selection . . . . . . . . . . . . . . . . . . . . .
3.1.1 Data sets . . . . . . . . . . . . . . . . . . . . . . . . .
3.1.2 Global cuts . . . . . . . . . . . . . . . . . . . . . . . .
3.1.3 Centrality determination . . . . . . . . . . . . . . . . .
3.1.4 Reaction Plane determination . . . . . . . . . . . . . .
3.1.5 Electron identification . . . . . . . . . . . . . . . . . .
3.1.6 Pair selection . . . . . . . . . . . . . . . . . . . . . . .
3.2 Invariant mass . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.3 Anisotropy . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.4 Polarization . . . . . . . . . . . . . . . . . . . . . . . . . . . .

21
21
21
21
22
22
24
24
27
28
29

4 Results and Discussion
32
4.1 Anisotropy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
4.1.1 Comparison with diﬀerent mass region . . . . . . . . . 32

3

4.2

4.1.2 Comparison with v2 . . . . . . . . . . . . . . . . . . . 32
Polarization . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

5 Conclusion

36

4

List of Figures
1.1
1.2
1.3

Time evolution of heavy ion collisions [3] . . . . . . . . . . . .
Time evolution of the magnetic field [4] . . . . . . . . . . . . .
Schematic image of strong magnetic field in heavy ion collisionand decay of di-electrons from virtual photons. . . . . . . . . .
2.1 Relativistic Heavy Ion Collider(RHIC) complex [5] . . . . . . .
2.2 Overview of the PHENIX Detector [6] . . . . . . . . . . . . .
2.3 Beam Beam Counter [7] . . . . . . . . . . . . . . . . . . . . .
2.4 Schematic view of the location of Zero Degree Calorimeters [8]2.5 The PHENIX Central and Muon Magnets [9] . . . . . . . . . .
2.6 A cut through view of RICH detector [10] . . . . . . . . . . .
2.7 View of a lead-scintillator module [11] . . . . . . . . . . . . . .
2.8 View of a lead-glass super module [11] . . . . . . . . . . . . .
2.9 The layout of wire position of Drift Chamber [12] . . . . . . .
2.10 The layout of wire position of Drift Chamber [12] . . . . . . .
3.1 ZVertex distribution measured by BBC . . . . . . . . . . . . .
3.2 Impact Parameter . . . . . . . . . . . . . . . . . . . . . . . . .
3.3 Reaction plane and almond shape in non-central collision . . .
3.4 Ψrp distribution measured by BBC . . . . . . . . . . . . . . .
3.5 Electrons/Positrons transverse momentum distribution . . . .
3.6 Invariant e+ e− mass spectrum . . . . . . . . . . . . . . . . . .
3.7 Schematic image of RP(reaction plane) and MF(magnetic field)
region . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.8 Definition of polarization analysis . . . . . . . . . . . . . . . .
4.1 Anisotropy (0.12 ≤ Mee ≤ 0.3GeV /c2 ) . . . . . . . . . . . . . .
√4.2 Cu+Cu sN N 200 GeV v2 10% Centrality bins . . . . . . . .
4.3 Anisotropy comparison . . . . . . . . . . . . . . . . . . . . . .
4.4 Polarization (0.12 ≤ Mee ≤ 0.3GeV /c2 ) . . . . . . . . . . . . .

5

9
10
11
12
13
14
15
16
17
18
18
19
20
21
22
23
23
25
27
29
31
32
33
34
35

List of Tables
12Comparison with the strength of magnetic fields . . . . . . . . 8
ϕv cut parameters . . . . . . . . . . . . . . . . . . . . . . . . . 26

6

1

Introduction

1.1

Relativistic heavy ion collisions

High-energy heavy-ion collision experiment is the only way to reproduce theQuark-Gluon Plasma. The Quark-Gluon Plasma(QGP) is the new phase inquantum chromodynamics(QCD) phase diagram. This phase only exists atextremely high temperature and high energy density. In QGP phase, quarks
and gluons are released from nucleons. To research on the nature of QGP,
the Relativistic Heavy Ion Collider was constructed at Brookhaven National
Laboratory(the United States of America).
RHIC provides collisions of heavy ion, such as gold and copper, at the√center of mass energy per√nucleon( sN N ) up to 200 GeV and proton at
the center of mass energy( s) up to 500 GeV. The Pioneering High Energy
Nuclear Interaction eXperiment(PHENIX) is the one of four experiment in
RHIC. PHENIX is specialized for the measurement of leptons and photons.

1.2

The strong magnetic field in heavy ion collisions

The possibility of creation of the strong magnetic field in non-central heavyion collisions is discussed in about 30 years ago. The pioneering theoreticalwork [1] estimated the strength of magnetic field in heavy ion collisions.
In recent years, the strong magnetic field in heavy ion collisions is notedand discussed the space-time evolution of the magnetic field and physics
phenomena in the field, for example vacuum birefringence [?], synchrotron
radiation, spitting and chiral magnetic eﬀect [4].
The strength of created magnetic field in heavy ion collision is estimatedabout 1018 Gauss at the energy region of RHIC. This strength is stronger
than magnetars one and this is thought the strongest magnetic field in theuniverse. Table 1 shows the comparison with the strength of some magnetic
fields.
The principle of the magnetic filed creation in heavy ion collisions is very
primitive. Strength of magnetic fields created by charged particles can be
described by Lienard-Wiechert equation.
−
→
−
→
→
−
→−
e [−
(1 − v2 /c2 )v × R]→B ( r , t) =.
4π cR3 [1 − (v/c)2 sin2 ϕRv ]3/2

7

(1)

−
−
→ → →
→
−
−
Here R = −r − r′ and ϕRv is the angle between R and →v.
In heavy ion collisions, charged particles (nucleus) are accelerated to light
speed and collided with nucleus in very narrow space, about 10 fm. This is
the reason why the strong magnetic field is created.
Strength [Gauss] Realized as
0.6
100
8.3×104
4.5 ×105∼ 1012
4 ×1013∼ 1015∼ 1018∼ 1019The Earth’s magnetic fieldA typical hand-held magnet
Super conducting magnets in LHC
Strongest-steady artificial magnets
Magnetic filed on surface of the neutron stars
Critical magnetic field of electrons
Magnetic filed on surface of the magnetars√
Non-central heavy-ion collisions at sN N = 200GeV (RHIC)√
Non-central heavy-ion collisions at sN N = 5.5TeV (LHC)

Table 1: Comparison with the strength of magnetic fields
The magnetic field in heavy ion collisions is very strong and it has very
interesting physics. However the field is not yet directly detected in experiments.

1.3

Time evolutions

In this section, time evolution of heavy ion collisions are described. Figure
1.1 shows the space-time evolution in heavy ion collisions at RHIC.
1.3.1

Time evolution of heavy ion collisions

Time evolution of high-energy heavy-ion collisions is explained step by stepin this section.
Pre equilibrium: τ = 0 ∼ τ0
Accelarated to an energy of 200 GeV per nucleon that they are Lorentz
contracted in longitudinal directions and it looks like pan-cakes shape in thelab frame. They are collided in narrow space and create patrons and gluons.
It becomes local thermal-equilibrium state shortly.
8

QGP in thermal equilibrium: τ = τ0 ∼ τC
After that, thermal-equilibrium state transforms to QGP phase (phasetransition). QGP phase can be described by hydrodynamics. QGP is expanding and lowering the heat.
Mixed phase between QGP and hadrons: τ = τC ∼ τH
When surpass the critical temperature, hadronization become.
Hadron gas: τ = τH ∼ τF
The temperature lower the heat as the system expands. Finally, QGP
phase transforms to hadron phase.

Figure 1.1: Time evolution of heavy ion collisions [3]1.3.2Time evolution of the magnetic field

Time evolution of the magnetic field is explained step by step in this section.
Figure 1.2 shows time evolution of the magnetic field in heavy ion collisions that is estimated by one theoretical model. The strength of magnetic
field is the strongest at the moment of collision and it is rapidly decreasing. Its time evolution shows this magnetic field is dominant in the preequilibrium region and still very strong even after a few fm. Thus This QGP
9

is formed in the strong magnetic field and the primary stage of QGP can be
influenced by the field.
On the one hand, QGP is a marked in central collisions, on the other hand,the strength of magnetic field is a marked in non-central collisions(∼ centrality 60%). In addition, electro-magnetic probes such as electrons and photons
are adequate for the search of the magnetic field, because their probes are
not operated by the strong interaction. Therefore the measurement electromagnetic probes used is to know the preceding stage of QGP state and its
evolution.

Figure 1.2: Time evolution of the magnetic field [4]

10

1.4

Photons and leptons

Leptons and photons are very good probe to search the magnetic field inheavy ion collisions because they are not interact with strong interaction
which is dominant in QGP phase. Thus these probes are expected to haveinformations of the magnetic field even after hadronization.
New approaches to detect the field are discussed in recent years. According to calculation of vacuum polarization tensor, virtual photons (lowmass di-electrons e+ e− ) have anisotropy and polarization with respect to thereaction plane.
Figure 1.3 is a schematic image of the field in heavy ion collisions and it
shows decays of virtual photons to di-electrons. Virtual photons which moveparallel to reaction plane are aﬀected by the magnetic field. Thus the numberof electron pairs asymmetry between the magnetic field and reaction plane
occur and they are polarized when they move perpendicular to the magnetic
field.

Figure 1.3: Schematic image of strong magnetic field in heavy ion collisionand decay of di-electrons from virtual photons.

11

2

Experimental setup

In this section, the RHIC accelerator, the PHENIX experiment and its detectors are described.

2.1

The RHIC accelerator

The Relativistic Heavy Ion Collider(RHIC) in Brookhave National Laboratory(BNL) in the United States was built to research the high energy heavy
ion physics (Fig.2.1).
The maximum energy of RHIC is 100 GeV per nucleon for heavy ion
and that is 250 GeV for proton. The heavy ions and protons produced at
the source are accelerated at Tandem Van de Graaﬀ and LINAC, respectively, and injected into the Booster. The Booster accelerate them up to 95
MeV per nucleon. They are transported to the Alternating Gradient Synchrotron(AGS) and more accelerated to 8.86 GeV. After that, heavy ions andprotons are injected to RHIC.

Figure 2.1: Relativistic Heavy Ion Collider(RHIC) complex [5]12The RHIC ring has a circumference of 3.8 km and consists of two rings,
one(”Blue Ring”) for clockwise and the other(”Yellow Ring”) for counterclockwise. These rings cross at six interaction points and experiments suchas PHENIX and STAR are build in each one of six points.

2.2

The PHENIX experiment

The PHENIX detector is described at Fig.2.2. The PHENIX, Pioneering
High Energy Nuclear Interaction eXperiment, is one of the experiment in
RHIC collider. The PHENIX detector are composed of global detectors andcentral arm detectors. And this experiment is specialized for measurementof leptons and photons.

Figure 2.2: Overview of the PHENIX Detector [6]The PHENIX experiment is a detector system which consists of threedetector group. First, central arm, two central arms have |η| < 0.35 pseudorapidity coverage and 180◦ azimuthal angle. Second, two muon arms, northand south arm cover 1.15 < η < 2.44 and −1.15 > η > −2.45, respectively,
and both are covering full azimuth 2π. The last, global detectors, these arelocated near the beam pipe.

13

2.3

Grobal detectors

In this section, the globe detectors such as BBC and ZDC are described.
The main purpose of global detectors is make triggers and to measure theluminosity and centrality in heavy ion collisions.
2.3.1

Beam-Beam Counters(BBC)The Beam-Beam counters(BBC) are one of the PHENIX global detector(Fig.2.3).
This detectors located on north and south side at 144.35cm along beam pipe
from the collision point and covers the pseude-rapidity frm 3.0 to 3.9. Eachof BBC is composed of 64 quartz Cherenkov counters. BBC have following
five major tasks; 1) to trigger the Minimum Bias events, 2) to measure thecollision vertex, 3) to obtain the collision time, 4) to determine centrality, 5)
to determine reaction plane.
The collision time and vertex are determined by the average time anddiﬀerence to BBC North and South:
collision time =TSouth + TN orth − (2 × L)/c2

(2)

TSouth − TN orth×c(3)2
where TSouth and TN orth are the average hit time, c is the light velocity
and L is the distance between z = 0 and both BBCs, L = 144.35 cm.
collision vertex =Figure 2.3: Beam Beam Counter [7]142.3.2

Zero Degree Calorimeters(ZDC)

The Zero Degree Calorimeters(ZDC) are hadron calorimeter which are located at 18 m north and south side along beam pipe from the collision
point(Fig.2.4). Since both ZDCs are located at charged particles are deflected by the DX dipole magnet, only neutrons hit the ZDCs. The tasks of
ZDCs are to measure the energy of spectator neutrons and to monitor the
beam luminosity.

Figure 2.4: Schematic view of the location of Zero Degree Calorimeters [8]2.4

PHENIX Magnets

PHENIX has two type magnets; One is Central Magnet and the other is
Muon Magnets(North and South side). The Central Magnet provides a magnetic field around the collision point. The momentum of charged particle isdetermined by the bending of their track in this magnetic field. The Central
Magnet covers a polar angle range of 70◦ < θ < 110◦ and pseudo-rapidity
range of |η| < 0.35.
The Central Magnet consists of inner and outer Helmholtz coil. Bothmagnets can be operated with the same(++) or opposite(+−) polarity. The
magnetic field is 1.15 Tesla in the same polarity configuration. In the opposite
polarity configuration, it leads to a cancellation of the magnetic field to about
zero field integral inside of 50 cm around the vertex.

15

Figure 2.5: The PHENIX Central and Muon Magnets [9]2.5Central arm detectors

In this section, detectors in central arms are described. The central arm
detectors measure for charged particles tracking, particle identification andtheir momentum. They cover |ϕ| < π/2 of azimuth angle and |η| < 0.35 of
pseudo-rapidity.
2.5.1

Ring Imaging Cherenkov Counter(RICH)

The Ring Imaging Cherenkov Counter(RICH) is detector for identifying withelectrons and positrons. Charged particles traveling in a medium with a velocity βc and when these are greater than the speed of light in the medium,
Cherenkov radiation is emitted under angle cos θ = 1/nβ. The RICH detector provides discrimination between electrons and pions by using theirdiﬀerence Cherenkov threshold.
Fig.?? is shown a cut through view of RICH detector in PHENIX. Cherenkov
light produced in the radiator gas(CO2 ) are focused and reflected on two
spherical mirrors and are detected by each photon multiplier tubes(PMTs).

16

Figure 2.6: A cut through view of RICH detector [10]2.5.2Electro-Magnetic Calorimeter (EMCal)

The Electro-Magnetic Calorimeter(EMCal) measures the energy and positionof photons and electrons. In addition, The EMCal serves as trigger on rare
events with high momentum photons and electrons. The EMCal consists of
two type calorimeter, one is the lead-scintillator(PbSc) calorimeter and theother is lead-glass(PbGl) calorimeter. The PbSc provides six sectors for all
four sector in the West Arm and for the two top sector in East Arm. And
the PbGl provides two sectors for the bottom sector in the East Arm.
The PbSc is a sampleing calorimeter made of alternating tile of Pb andscintillator as shown in Fig.2.7. This calorimeter contains a total if 15,552√
individual towers. The energy resolution of the PbSc is σEE = 8.1%
⊕ 2.1%.
E
The PbGl is a Cherenkov type calorimeter which made of lead glass. The
PbGl calorimeter comprises 192 super modules(SMs) and each SM comprises
24 modules. Fig.2.8 shows the interior view if one super module of PbGl√
⊕ 0.76%.
calorimeter. The energy resolution of the PbGl is σEE = 5.9%E17

Figure 2.7: View of a lead-scintillator module [11]Figure 2.8: View of a lead-glass super module [11]18

2.5.3Drift Chamber (DC)The Drift Chambers(DCs) are covering both Central Arms. This detector isthe main tracking device for charged particles in PHENIX. The DCs reconstruct the tracking of charged particles in r − ϕ plane in order to determine
transverse momentum of each particles. The Wast and West DCs are cover3
5
ing |η| = ±0.35 in pseudo-rapidity, − 16
π < ϕ < 16
π and 11π < ϕ < 19
π in
16
16
azimuth, respectively. The detectors are filled with mixed gas of 50% Argon
and 50% Ethane.

Figure 2.9: The layout of wire position of Drift Chamber [12]2.5.4

Pad Chamber (PC)The Pad Chambers(PCs) are multi-wire promotional chambers. This detectors consists of two cathode pad readout. The PCs determine space point
along the straight trajectory of charged particles in oder to determine polar
angle θ which allows to determine the z component of the momentum vector.

19

Figure 2.10: The layout of wire position of Drift Chamber [12]20

3Analysis

3.1

Run and event selection

In this section, some basic information on this analysis such as data sets, run
and track selection variables and analysis used runs are described.
3.1.1

Data sets√
Minimum Bias data set in Cu+Cu collisions at sN N = 200 GeV are usedin this analysis. This data set is collected in year 2005 at PHENIX. The
run number in Cu+Cu collisions is from to . All runs are separated into two
diﬀerent groups as ++ field and −− field runs. In this analysis, ++ field(magnetic field on) data set are used that consists of 264 runs.
3.1.2

Global cuts

The following global cuts are used in order to select events that used in thisanalysis.
• Zvertex-25 < zvtx < 25 cm
Zvertex means the collision points.

Figure 3.1: ZVertex distribution measured by BBC

21• Track quality
dc quality == 31 or 51 or 63
This parameter is Drift Chamber tracks quality. The value of 31 means
the tracks are required the hit of X1, X2 wire and unique hit of UV
wire. The value of 51 means the tracks are• Fiducial cuts
-1.37 < phi0 < 1 || 2.14 < phi0 < 4.51
This cut remove the region of the MVD service pipes in order to reject
conversion electrons. phi0 is the track’s ϕ direction at the vertex.
3.1.3

Centrality determination

Copper nuclei are not point-like particle. Thus, in heavy ion collisions, thecenter of each two nuclei are shifted by a distance that called the impact pa→−
rameters b as showed Fig.3.2. Centrality is decided by charge sum detectedby the BBC detectors.

Figure 3.2: Impact Parameter

3.1.4

Reaction Plane determination

Reaction Plane Ψrp (n) is defined the following equation.
∑()
ωi sin nϕi
1−1iΨrp (n) =tan ∑ni ωi cos nϕi(4)Here ωi is the charged sum detected in azimuthal angle ϕi . Figure 3.4
shows the Reaction Plane distribution measured by BBC.
22

Figure 3.3: Reaction plane and almond shape in non-central collision

Figure 3.4: Ψrp distribution measured by BBC

23

3.1.5

Electron identification

The cut parameters are used in order to identify electrons and positrons.
• RICH2 > n0The n0 is the number of fired phototubes in nominal ring radius.
χ 2 /npe0 ≥ 25
χ2 -like shape variable of the RICH ring associated with the track divided by the benumbed of photo-electrons(npe0).
disp ≤5
The disp is the displacement between the ring center reconstructedfrom the fired phototubes and the track projection position in the RICH
phototubes array.
• EMCalprob > 0.01
This prob parameter is the probability that the particle shower is Electromagnetic.
0.5 ≤ ecore/mom ≤ 2.0
The ecore is the shower core energy detected at EMCal and the momis the magnitude of the momentum detected at Drift Chamber.
√emcsdphi2e + emcsdze2 ≤ 4.0
The emcsdphie is track matching in ϕ direction at EMCal surface andthe emcsdzz is track track matching in z direction at EMCal surface.

3.1.6

Pair selection

In order to select di-electrons, the following cut variables are used.
• pT thresholdpT ≥ 0.3 GeV/c
This cut parameter is single electron transverse momentum (pT ) threshold.
24

Figure 3.5: Electrons/Positrons transverse momentum distribution• DC ghost track cut|∆dcphi | ≥ 0.02 && |∆dczed | ≥ 0.5
This cut is applied to reject ghost tracks in Drift Chamber. If any
two tracks fulfill above region, the one with worse EMCal matching is
rejection, because it is likely to be a ghost track.
• RICH ring sharing rejection|∆crossphi | ≥ 0.01 && |∆crossz | ≥ 25
This cut is applied in order to reject a hadron. When two tracks areparallel to each other while traveling through the RICH gas, they sharethe same Cherenkov ring. In this case, one of them is most likely amisidentified hadron. Thus, such tracks should be rejected.
• ϕv cut
The pairs from gamma conversions can be identified by their orientation
angles relative to the PHENIX magnetic field called as ϕv . This cut
is used in order to reject beam pipe conversions and it can be remove> 98% of all conversions pairs.
25

The ϕv cut can be described as follows equations.
−→→p e− + −
p e+
−
→
u = −
→
−
→
| p e− + p e+ |

(5)

−
→
→
−
v =−
p e− × →
p e+

(6)

→
−
→
−
ω =−
u ×→
v

(7)

−
→
ez = (0, 0, 1)

(8)

−
→
−
u ×→
ez
−
→
ua = −→→−| u × ez |

(9)

(−
)
→
→
ω ·−
ua
ϕv = arccos −
−
|→
ω ||→
ua |

(10)

−
→
Here →
p e− and −p e+ are 3-momentum vector of electron and positron,
respectvely. And ez is a unit vector correctional with z axis.
The applied ϕv cut parameters that is used in this analysis are following(Table. 2).
Mass region [MeV]ϕv cut[rad]0 < Mee ≤ 10
10 < Mee ≤ 30
30 < Mee ≤ 90
90 < Mee ≤ 140
140 < Mee

No cutϕv ≥ 0.08 π
ϕv ≥ 0.035 π
ϕv ≥ 0.03 π
ϕv ≥ 0.02 π

Table 2: ϕv cut parameters

26

3.2

Invariant mass

The invariant mass Me+ e− is calculated by following equation.
√→−
Me+ e− = (Ee+ + Ee− )2 − (−p e+ + →p e− )2→
where E is the energy of particle and −p is particle momentum.
√
√
−
→
→
2
2
2
p 2e− )2
(Ee+ + Ee− ) = ( me+ + p e+ + m2e− + −

(11)

(12)

−
→
p e+ = (pe+ x , pe+ y , pe+ z )

(13)

→
−
p e− = (pe− x , pe− y , pe− z )(14)

The invariant e+ e− mass distribution showed in fig.3.6. In this anisotropy
and polarization analysis, 0.12 < Mee < 0.3GeV /c2 is used because thisregion is include about 20% virtual photon component.

Number of pairs

Massee
104

103

102

10

1
0

0.5

1

1.5

2

2.5

3

3.5
4
Massee[GeV/c2]

Figure 3.6: Invariant e+ e− mass spectrum. The blue band is 0.12 < Mee <0.3GeV /c2 . This region is include about 20% virtual photon component.

27

3.3

Anisotropy

The anisotropy is calculated by the following equations.
First, two electrons momentum are added. This vector means the direction of di-electron in XY plane.
−
→
−
→
−
→
P e− e+ xy = P e− + P e+

(15)

Second, in order to determine the reaction plane in XY plane, the reaction
plane vector is defined.
−→
P rp xy = (cos Ψrp , sin Ψrp )

(16)

−
→
−
→
Third, the angle of between these two vectors P e− e+ and P rp are measured.
−→−→
P e− e+ · P rpcos φanisotropy = →−−→| P e− e+ || P rp |(17)

φanisotropy = arccos φanisotropy(18)• 0 ≤ φanisotropy < 41 π and 34 π < φanisotropy ≤ π
The electron pairs are in in-plane. This means the momentum of dielectrons direction are reaction plane.
• 14 π ≤ φanisotropy < 43 π
The electron pairs are in out-plane. This means the momentum of
di-electrons direction are magnetic field.
Finally, the number of electron pairs are compared between in-plane andout-plane. The anisotropy is defined as follows:
NRP − NM F
(19)
NRP + NM F
NRP and NM F are the number of electron pairs in the reaction plane sideand the magnetic field side, respectively.
Anisotropy =28

Figure 3.7: Schematic image of RP(reaction plane) and MF(magnetic field)
region3.4

Polarization

The polarization is calculated by the following equations.
First, the 3-momentum vector which is the perpendicular to the momentum vector of electron and positron are defined.
−
→
−
→
−
→
P e = P e− × P e+

(20)

Second, the vector of magnetic field is defined by using Ψrp and unit
vector of the z axis.
→−
P rp = (cos Ψrp , sin Ψrp , 0)

(21)

−
→
−
→
→
P mf = P rp × −
ez
(22)
−
→
−
→
Third, calculate the angle between P e and P mf . It means the angle
between two plane, reaction plane and ”electron pair plane”. This value isused for the polarization of in-plane electron pair.
29

−
→ −
→
P e · P mf
cos φpolarizationRP = −→ −→| P e || P mf |(23)

φpolarizationRP = arccos φpolarizationRP(24)

For the calculation of the polarization of out-plane electron pairs, thefollowing angle are used. This means that the angle between ”electron pairplane” and the plane which is composed by magnetic field and z unit vector.
→− −→P e · P rpcos φpolarizationM F = −→ →−| P e || P rp |(25)

φpolarizationM F = arccos φpolarizationM F(26)• 0 ≤ φpolarization < 14 π and 34 π < φpolarization ≤ πThis φpolarization region means that di-electron momentum vector isparallel to the reaction plane for in-plane polarization or magnetic field
for out-plane polarization.
• 14 π ≤ φpolarization < 43 πThis φpolarization region means that di-electron momentum vector is perpendicular to the reaction plane for in-plane polarization or magnetic
field for out-plane polarization.
The number of electron pairs are compared between the parallel andperpendicular plane to the reaction plane(magnetic field).
()
N∥ − N⊥P olarizationRP =(27)
N∥ + N⊥ RP()
N∥ − N⊥P olarizationM F =(28)
N∥ + N⊥ M F
Finally, the double ratio is calculated with the double ratio of the bothside polarization.
) /()(
N∥ − N⊥N∥ − N⊥(29)
D.R =N∥ + N⊥ RP
N∥ + N⊥ M FThis is because in order to cancelation of detector geometrical acceptance.
30

Figure 3.8: Definition of polarization analysis

31

4

Results and Discussion

In this section, the results of anisotropy and polarization are showed.

4.1

Anisotropy

4.1.1

Comparison with diﬀerent mass region

Anisotropy

The anisotropy as a function of centrality is showed in Figure 4.1. The plotis zero consistent within error range.

0.4
0.3
0.2
0.1
0
-0.1
-0.2
-0.3
10

20

30

40

50

60

70

80

90
Centrality[%]

Figure 4.1: Anisotropy (0.12 ≤ Mee ≤ 0.3GeV /c2 )4.1.2

Comparison with v2

The azumithal dependence of the particle yield can be described in the form
if a Fourier series as follows:
(
)
∞
3
2
∑
1 dN
dN
E 3 =
1+
2vn (pT ) cos[n(ϕ − ΨR )](30)
dp
2π pT dpT dyn=1
32

where E is the energy of the particle, p the momentum, pT the transverse momentum, ϕ the azimuthal angle, y the rapidity and ΨR the angle of reaction
plane.
In order to compare with elliptic flow (v2) and my anisotropy results,
calculate the following equation with following two integral region.
∫(1 + 2v2 cos(2ϕ)) dϕ = ϕ + v2 sin(2ϕ) + Constant(31)
The in-plane and out-plane ϕ range are following, respectively.
• 0 ≤ ϕ ≤ π4 , 3π
≤ ϕ ≤ 5π
, 7π
≤ ϕ ≤ 2π : the region of Reaction Plane44
4side (RP), 5π≤ ϕ ≤ 7π
: the region of Magnetic Field side (MF)• π4 ≤ ϕ ≤ 3π
4
4
4

Figure 4.2: Cu+Cu√
sN N 200 GeV v2 10% Centrality bins

The both region are integrated by dϕ and calculate the anisotropy.
M F − RP(32)
M F + RP√
The v2 value are used the result of hadron elliptic flow at sN N = 200
GeV Cu+Cu in PHENIX as showed in Figure 4.2.
Anisotorpy =33

Figure 4.3 shows the comparison with anisotropy. The red dot is theanisotropy results of this thesis and black dot with gray band is the anisotropy
calculated by using the values of elliptic flow. These two results are consistent.

Anisotorpy

Graph
0.2

0.15
0.1
0.05
0

-0.05
-0.1
-0.15
-0.2

10

15

20

25

30

35

40

45

Figure 4.3: Anisotropy comparison

34

50
Centrality[%]4.2

Polarization

Double Ratio

The first measurement of di-electron double ratio of polarization in heavy
ion collisions ia showed in Figure 4.4 (0.12 ≤ Mee ≤ 0.3GeV /c2 ). The resultof double ratio is consistent with unity in error range.

1.4

1.3

1.2

1.1

1

0.9

0.8
10

20

30

40

50

60

70

80

90
Centrality[%]

Figure 4.4: Polarization (0.12 ≤ Mee ≤ 0.3GeV /c2 )35

5Conclusion

We have measured the di-electron anisotropy and polarization for selected√
centralities in sN N = 200 GeV Cu+Cu collisions.
We identified e+ and e− tracks from large other charged particle tracks
and reconstructed the invariant mass spectrum of e+ e− pairs. We selected
low mass region, 0.12 < Mee ≤ 0.3GeV /c2 . This region has about 20%
virtual photon component.
At anisotropy measurement, the anisotropy is consistent with zero and it
is good agreement with an anisotropy calculated from elliptic flow.
We measured di-electron polarization. This is the first result of di-electron
polarization in heavy ion collisions. In order to cancel PHENIX detector acceptance, we used double ratio in order to cancellation of detector acceptance.
The double ratio is consistent with unity in statistic error. This measurement
did not detect polarization.
We will analyze with Au+Au data which has more statistics than Cu+Cu
and we are going to analyze back ground subtraction such as combinatorialand Dallitz decay in order to get virtual photon component. After that, we
will measure anisotropy and polarization of virtual photons for the search of
the magnetic field.

36

Acknowledgements
I would like to gratefully acknowledge my superbisor Associate Professor
Kenta Shigaki for his support and guidance. I gratefully thank the staﬀ
member of Quark Physics Lab. Professor Toru Sugitate and Assistant Professor Kensuke Homma for them great suggestions and advices. I would liketo acknowledge Dr. Yasuyuki Akiba and and Dr. Yorito Yamaguchi for them
help and good advices.
I would like to thank all PHENIX collaborators. I had a great experience
as a PHENIX collaborator and Beam-Beam Counter expert during RHIC
Run- 12.
I would like to extend my thanks to M.Nihashi-san and Y.Nakamiya-san.
Nihashi-san gave me a great support of my BNL stay. Nakamiya-san taught
me detail of electron analysis and discussed with me about my analysis. I
also want to thank all students of Quark Physics Lab.

37

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http://www.hepl.hiroshima-u.ac.jp/phx/bnl/bnl.html

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site

[8] Nuclear Instruments and Methods A 499 (2003) 433 - 436[9] PHENIX website http://www.phenix.bnl.gov/[10] Nuclear Instruments and Methods A 499 (2003) 143 - 148[11] L. Aphecetche et al. (PHENIX Collaboration), PHENIX calorimeter,
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[12] K. Adcox et al. (PHENIX Collaboration), PHENIX central arm tracking
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38