Measurement of electron azimuthal anisotropy andimplications of heavy quark flow in Au+Au√
collisions at sNN = 200 GeV

Shingo Sakai

February 2007

Measurement of electron azimuthal anisotropy andimplications of heavy quark flow in Au+Au√
collisions at sNN = 200 GeV

Shingo Sakai(Doctoral Program in Physics)

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

Abstract
The goal of the Relativistic Heavy Ion Collider (RHIC) at Brookhaven National Laboratory is to produce ”Quark Gluon Plasma (QGP)”, which is a new state of matter of deconfined quarks and gluons and to investigate a property of the matter. Charm quark is believed
to be produced in initial collisions via gluon fusion therefore it is a powerful probe for studying the early stage of the medium created in the collisions. Elliptic flow, which is defined asa second harmonic parameter v2 of the Fourier expansion of the azimuthal distribution withrespect to the reaction plane. Elliptic flow is also sensitive to the initial stage of the collision,
since it originates from the initial geometrical overlap of colliding nuclei as well as the later
elliptical collective expansion.
Charm quark production has been studied by measuring electrons from their semi-leptonic
decays in the PHENIX experiment at RHIC. We define measured inclusive electron sample
into two components: (1)’non-photonic’ - primarily semi-leptonic decays of charmed- (andbottom-) mesons and (2)’photonic’ - Dalitz decays of light neutral mesons (π0 , η , η 0 , ω andφ ) and photon conversions in the detector material. Two independent techniques ’cocktail’
and ’photon converter’ methods are used to subtract the ’photonic’ electrons backgrounds
from the measured inclusive electron in order to extract the ’non-photonic’ electron signal from the heavy flavor decays. More than 50 % of measured inclusive electrons comefrom ’non-photonic’ sources above 1.5 GeV/c which is given by the relative ratio of those’photonic’ and ’non-photonic’ sources and materials in the PHENIX experiment.
In this thesis, we study the elliptic azimuthal anisotropy of charm quark via electron
measurement. We first measure the transverse momentum dependence of the azimuthal
anisotropy parameter v2 for inclusive electrons at mid-rapidity (|η | < 0.35). The data are√taken with the PHENIX detector in Au+Au collisions at sNN = 200 GeV during the fourthexperiment period at RHIC run (2004 - 2005). The v2 of electrons from heavy flavor decays
is then extracted by subtracting the v2 of ’photonic’ electrons. This ’photonic’ electron
v2 is determined with a Monte Carlo simulation using measured v2 and pT distributionsof these ’photonic’ sources, which is same as the ’cocktail’ method, and this has also been
confirmed by the ’converter’ method. The extracted v2 of the ’non-photonic’ electron clearlyshows positive value below pT = 3.0 GeV/c, which is slightly smaller to pion v2 though. The
v2 increases with pT up to about 1.5-2.0 GeV/c and then saturate or slightly reduces withincreasing pT .

The main source of the ’non-photonic’ electrons is D meson decay at relatively low pT
below 2-3 GeV/c. The non-zero v2 for the ’non-photonic’ electrons automatically gives a
non-zero v2 of D meson according to the decay kinematics. D meson v2 is estimated with aMonte Carlo simulation using the measured ’non-photonic’ electron v2 . Various pT dependent shape of D meson v2 are assumed in the simulation and the peak v2 value is obtained
to be 0.09 ± 0.03 at about 2-3GeV/c. This corresponds to a non-zero v2 measurement of D√
meson in Au+Au collisions at sNN = 200 GeV. Based on a quark coalescence model, thecharm quark v2 is finally estimated. The same pT dependent shape for v2 of charm quark
are assumed as for the light and strange quarks. The effective mass of charm quark is used
for coalescing them at similar velocity. Simultaneous fitting of measured v2 parameters forpion, Kaon, proton and ’non-photonic’ electron are done with 2 v2 parameters for light andheavy quarks including the quark coalescence effect and D meson decay kinematics. The
extracted charm quark v2 has similar magnitude compared with the light quark. Besides thebinary collision scaling of ’non-photonic’ electron yield at relatively low pT region below 2
GeV/c, the strong elliptic anisotropy of the heavy quark, which is similar to the light quark,
indicates a collectivity of different mass quarks during the quark phase of the high density
matter created in the ultra-relativistic heavy-ion collisions. The collectivity of the heavy
quark is one of the strong evidence of the quark gluon plasma. The presented results on v2
measurement of ’non-photonic’ electron give a strong support of this scenario.

Acknowledgement
This thesis work was supproted by many people. I would like to take this opportunity tothank the many people who have made this work possible.
I first would like to express my great gratitude to Prof. Yasuo Miake for providing me theexciting field of physics. His appropriate and valuable advice show me the way to promote
the present work, and I could learn a lot from him as an experimental physicist.
I express my heartfelt thanks to Prof. Shinichi Esumi. His appropriate and valuableadvice and disccusions for the analysis are very helpful for this work. I am much obliged
to Prof. Tatsuya Chujo for his adivice and discussion not only in the analysis but also workat BNL. I wish to express my thanks to Dr. Susumu Sato for leading me to the PHENIX
experiment. His approproate advice have helped me a lot to work at BNL. I would like also
thank Prof. Motoi Inaba. His advice of electronics for detectors is very helpful for me towork at BNL.
I wich to acknowledge for all the collaborations of the PHENIX experiment. I am grateful to the spokesperson Prof. W. A. Zajc for his various arrangemebts for my activity at
BNL. I express my appreciation to deputy spokesperson Dr. Yasuyuki Akiba for various
advice for the electron analysis. I wich to express my special thanks to Prof. M. Kaneta forhis helpful advice on the analysis. I would like to many thanks to Heavy-Light convenors,
Prof. R. Averbeck, Dr. M. L. Brooks, Dr. V. Cianciolo, Prof. A. D. Frawley, Prof. T. K.
Hemmick, Dr. A. Molov, Dr. D. Silvermyr and Dr. W. Xie, for thier kind helps and many
useful advice for the analysis.
I wich to acknowledge for all the members of the PHENIX-J. I gratefully acknowledge
to the PHENIX-J leader, Prof. Hideki Hamagaki for the continuous financial support. I
extend hearty thanks to Prof. Kyoichiro Ozawa for the financial support and various advice
for working at BNL. I acknowledge for Dr. F. Kajihara and Dr. T. Gunji for giving me a lotof information about electron analysis. I also acknowledge for Mr. T. Hachiya and Mr. T.
Nakamura for thier kind help and useful adivice at BNL.
Here I acknowledge all the members of the high energy nuclear experiment group at
University of Tsukuba. I would like to thank Mr. H. Masui for giving me many informationIof the reaction plane analysis. I would like to thank Mr. M. Konno, Mr. T. Takagi and
Ms. M. Shimomura for their help, freindship and kindness. I would like to thank Dr. A.
Kiyomichi for his kind helps. I would like to thank Mr. S. Kato and Mr. Y. Nagata forsupporting computer environment. I also acknowledge Mr. H. Tsuruoka, Mr. M. Ono, Mr.
Y. Kuroki, Mr. M. Oka, Mr. M. Narisawa, Mr. K. Miki, Mr. M. Sato, Mr. Y. Ikeda, Mr. R.
Tanabe, Mr. Y. Tomita, Mr. T. Niida, Mr. K. Watanabe, Mr. D. Sakata, Mr. A. Saito, Mr.
M. Sano, Mr. N. Shiohara and Ms. N. Seya for their freindship.
Finally, I would like to express my great appreciation to my parents, Kougi Sakai and
Emiko Sakai, for their support and encouragement to accomplish this work.

IIContents
1

Introduction
1.1 Quantum Chromodynamics and Quark Gluon Plasma . . . . . . . . . . . .
1.2 Quark Gluon Plasma and Relativistic Heavy Ion Collisions . . . . . . . . .
1.3 Thermalization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.3.1 Chemical Equilibrium . . . . . . . . . . . . . . . . . . . . . . . .
1.3.2 Kinematical equilibrium and the collective flow . . . . . . . . . . .
1.4 Suppression of high pT particle . . . . . . . . . . . . . . . . . . . . . . . .
1.5 Quark coalescence/recombination . . . . . . . . . . . . . . . . . . . . . .
1.5.1 Baryon excess . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.5.2 Quark coalescence model . . . . . . . . . . . . . . . . . . . . . .
1.6 Charm quark production in RHIC . . . . . . . . . . . . . . . . . . . . . .
1.7 Thesis motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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The PHENIX experiment
2.1 The Relativistic Heavy Ion Collider (RHIC) . . . . . . . . . . . . . . . . .
2.2 Overview of PHENIX experiment . . . . . . . . . . . . . . . . . . . . . .
2.3 Magnet system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.4 Global detectros . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.4.1 Beam-Beam counters (BBCs) . . . . . . . . . . . . . . . . . . . .
2.4.2 Zero-Degree Calorimeters (ZDCs) . . . . . . . . . . . . . . . . . .
2.5 Central arms detectros . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.5.1 Drift Chambers (DCs) . . . . . . . . . . . . . . . . . . . . . . . .
2.5.2 Pad Chambers (PC) . . . . . . . . . . . . . . . . . . . . . . . . . .
2.5.3 Ring Imaging Cherenkov Counters (RICHs) . . . . . . . . . . . . .
2.5.4 Electro-Magnetic Calorimeters (EMCals) . . . . . . . . . . . . . .
2.5.5 Time of flight (TOF) . . . . . . . . . . . . . . . . . . . . . . . . .

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Data analysis
3.1 Event selection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2 Centrality determination . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.3 Track Reconstruction and momentum determination . . . . . . . . . . . . .
3.4 Electron identification . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.5 Azimuthal anisotropy analysis . . . . . . . . . . . . . . . . . . . . . . . .
3.5.1 Fourier Expansion of Azimuthal Distribution . . . . . . . . . . . .
3.5.2 Reaction Plane method . . . . . . . . . . . . . . . . . . . . . . . .
3.5.3 Reaction Plane Determination . . . . . . . . . . . . . . . . . . . .
3.5.4 Reaction Plane Resolution . . . . . . . . . . . . . . . . . . . . . .
3.6 Determination of non-photonic electron at PHENIX . . . . . . . . . . . . .
3.6.1 Electron sources . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.6.2 Cocktail method . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.6.3 Photon converter method . . . . . . . . . . . . . . . . . . . . . . .
3.6.4 Ratio of the non-photonic to photonic electrons RNP . . . . . . . .

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Experimental Results
4.1 Inclusive Electron v2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2 Non-photonic electron v2 . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2.1 Method of non-photonic electron v2 determination . . . . . . . . .
4.2.2 Determination of photonic electron v2 by the cocktail method . . .
4.2.3 Kaon decay background subtraction . . . . . . . . . . . . . . . . .
4.3 Systematic uncertainties . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.4 Result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Discussions
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5.1 Non-photonic electrons . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
5.2 Kinematics of D meson decay and v2 . . . . . . . . . . . . . . . . . . . . . 87
5.3 Charm quark v2 assuming Quark Coalescence Model . . . . . . . . . . . . 91
5.4 Bottom quark contribution to the non-photonic electron v2 . . . . . . . . . 97
5.5 Comparison with charm quark thermalization model . . . . . . . . . . . . 98
5.6 Non-photonic electron RAA and v2 . . . . . . . . . . . . . . . . . . . . . . 101

6Conclusions

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A Data points

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IV

B Kinematics
B.1 Transverse Momentum and Transverse Mass . . . . . . . . . . . . . . . . .
B.2 Rapidity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
B.3 Centrality Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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List of Figures
1.1

A schematic phase diagram of nuclear matter as a function of temperature
T and baryon chemical potential µ . The figure is taken from [2]. . . . . . .
1.2 εB j τ as a function of N part measured at three RHIC energies. . . . . . . . .
1.3 Space-time evolution of a nucleus-nucleus collisions. The time and temperature are calculated by the hydrodynamical framework [6] . . . . . . . . .
√1.4 Ratios of measured yield of various particles in sNN = 130 GeV and 200
GeV, compared with the statistical thermal model [11]. . . . . . . . . . . .
1.5 Inverse slop parameter T as a function of particle mass. They are comparedat mid-rapidity in p + p, S+S and Pb+Pb collisions [15]. . . . . . . . . . .
1.6 An illustration of non-central Au-Au collisions. The plane defined as the
direction of the impact parameter (x) and direction of the beam axis (z) iscalled “reaction plane”. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.7 Beam energy dependence of v2 measured at AGS, SPS and RHIC. . . . . .
1.8 Identified particle (π , K, p) v2 as a function of pT [19]. The lines in the
figure are the hydrodynamical model calculation. [23] . . . . . . . . . . . .
1.9 Comparison of two-particle azimuthal angular correlations of charged particles for central Au + Au, central d + Au and p + p collisions, where NTrigger
is the number of high pT particles. . . . . . . . . . . . . . . . . . . . . . .
1.10 Nuclear modification factor measured for π 0 and charged particle in Au +√Au collisions (RAA ) and in d + Au collisions (RdAu ) in sNN = 200 GeV
as a function of pT [25]. The bands around the data points show systematic errors, while the shade band around unity indicates the normalization
uncertainty. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.11 p/π ratio as a function of pT for central (0-10 %), mid-central (20-40 %)and peripheral (60-92 %) collisions [30]. The dashed and dotted lines areproton and pion ratio in gluon and in quark jets [31]. . . . . . . . . . . . .
1.12 p/π ratio measured by PHENIX [30] and quark coalesnce model calculations
for the comparison. This picture is taken from [5]. . . . . . . . . . . . . . .
VI

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1.13 v2 /n as a function of pT /n where n is the number of constituent quarks plus
anti-quarks (n = 2 for meson and n = 3 for baryon). . . . . . . . . . . . . .
1.14 (a) Invariant differential cross section of electrons from heavy flavor, charm√and bottom, in p + p collisions at sNN = 200 GeV measured at PHEXNIX
experiment [39]. The curves are the FONLL calculations. (b) Ration of thedata and the FONLL. The upper and lower curves are theoretical upper andlower limit of the FONLL. . . . . . . . . . . . . . . . . . . . . . . . . . .
1.15 Theoretical prediction of pT distributions of initial (solid), prethermal (dotdashed) and thermal (dashed) charm production for central Au+Au colli√
sions at RHIC energy, sNN = 200 GeV [41]. . . . . . . . . . . . . . . . .
2.1
2.2

A picure of RHIC complex. . . . . . . . . . . . . . . . . . . . . . . . . . .
RHIC acceleration scenario for gold. The tandem Van de Graff accelerators,
the Booster synchrotron, and the Alternating Gradient Synchrotron (AGS)
serve as the injector for RHIC. . . . . . . . . . . . . . . . . . . . . . . . .
2.3 PHENIX experiment setup in the 4th year of RHIC run (Run4). (Top:Cross
section perpendicular to the beam pipe. Bottom: East side view of the crosssection along the beam pipe.) PHENIX consists of four parts of spectrometerarms and a set of global detectors. The spectrometers around mid-rapidity
are called central arms (west and east arm), and those around forward rapidity are called muon arms. . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.4 PHENIX acceptance of leptons and hadrons. . . . . . . . . . . . . . . . . .
2.5 Line drawings of the PHENIX magnets, shown in perspective and cut away
to show the interior structures. . . . . . . . . . . . . . . . . . . . . . . . .
2.6 (a) A picture of BBC array comprising 64 BBC elements. (b) A picture of
BBC element which consists of one-inch mesh dynode PMT mounted on a
3 cm quartz radiator. (c) The configuration of the 64 PMTs of each BBC. .
2.7 (a) Top view of the beam geometry and ZDC location. (b) A picture of ZDC.
2.8 (left) The layout of wire position of DCs. (right) Constriction of a DC frame.
2.9 A picture of the Pad Chamber system in PHENIX. . . . . . . . . . . . . .
2.10 A cutaway view of one arm of the PHENIX RICH detector. . . . . . . . . .
2.11 Interior view of a Pb-scintillator calorimeter module showing a stack of scintillator and lead plates, wavelength shifting fiber readout and leak fiber inserted in the central hole. . . . . . . . . . . . . . . . . . . . . . . . . . . .

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2.12 Schematic diagram of the components of a single TOF panel which consistsof 96 plastic scintillation counters with PMTs at both ends, light guides andsupports. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Correlation between the charge sum measured by BBC and energy of spectator neutrons measured by the ZDC. The lines in the figure present thedefinition of centrality classes in the ZDC vs. BBC plot. . . . . . . . . . .
3.2 Definition of the Hough transform parameters (φ and α ) for the drift chamber track reconstruction. The X1 and X2 hits in the drift chamber are shownas small circles. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.3 an illustration of electron identification by RICH. (a) Cherenkov photons are
emitted by the electrons at RICH. (b) The Cherenkov photons are reflectedby the mirrors and make a circle with 5.9 cm radius on the PMT arrays. . .
3.4 (E − p)/p/σ distributions in different pT slices. The red line is for electrons
after subtracting background (blue line) from electrons identified by RICH
(black line). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.5 E/p distributions for electrons (black solid line), swapped background (dotted line), and hadrons (red solid line). The figure is taken from [46] . . . . .
3.6 Signal to background ratio (B/(S + B)) for electron signals as a function of
pT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.7 Illustration of the reaction plane determination. The true reaction plane can
not measure directly therefore the plane is determined by using anisotropic
flow itself. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.8 Azimuthal angle correlation of the measured reaction plane North side andSouth side of BBC. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.9 Centrality dependence of the combined reaction plane resolution that is determined by the BBCs. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.10 pT distribution of the background electrons [54][55]. The background electrons are calculated by the hadron decay generator. . . . . . . . . . . . . .
3.11 Invariant yield specra of the photonic electron. The curve in the figure is
the specra from cocktail method and the closed circles is from the conveter
method. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.12 Invariant yield of the non-photonic electron measured in Au+Au collisions√
at sNN = 200 GeV [55]. . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.13 Transverse momentum dependence of the ratio of the number of non-photonic
non−γ γ/Ne ). . . . . .
electrons to the number of photonic electrons (RNP = Ne

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4.1

Transverse momentum dependence of the inclusive electron v2 measured
with 10 % step of the centrality bin. The v2 was already corrected for thereaction plane resolution. . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2 Inclusive electron v2 for the the minimum bias events as a function of pT . .
4.3 Relative contributions of electron sources (π 0 Dalitz, γ conversion, η Dalitz
and direct γ ) to the photonic electrons. The data is obtained from Fig 3.10. .
4.4 pT dependence of electron v2 from π 0 decay. . . . . . . . . . . . . . . . .
4.5 η v2 calculated by the Blast wave model. The dashed lines are the systematic
uncertainties from the systematic uncertainty of π , kaon, proton v2 . . . . . .
4.6 pT dependence of electron v2 from η decay. η v2 was assumed same askaon in this calculation. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.7 Direct γ v2 as a function of pT measured at PHENIX. The closed circles
are inclusive γ v2 and the open squares are background γ v2 mainly from π 0
decay. The open circles corresponds to the direct γ v2 [59]. . . . . . . . . .
4.8 pT dependence of photonic electron v2 obtained by the cocktail and converter method. The lines on the figure are photonic electron v2 calculated by
the cocktail method. The solid line is the mean of v2 and the dashed lines
are the systematic uncertainty of the cocktail method. The photonic electronv2 from converter is shown as boxes. . . . . . . . . . . . . . . . . . . . . .
4.9 Inclusive electron v2 measured with/without converter (Minimum bias event).
4.10 pT dependence of the non-photonic electron v2 (closed circles) together with
the inclusive electron v2 (open circles) and photonic electron v2 (solid line).
4.11 Contribution of kaon decays to the non-photonic electron yield. The contribution is 18% at pT = 0.5 GeV/c and decreases to less than 5% for pT > 1.0
GeV/c. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.12 Input kaon pT spectra for the kaon decay study simulation (the solid line).
The spectrum at pT > 2.0 GeV/c we assumed that the shape is same as π 0spectrum. The pT spectrum is obtained by the scaled π 0 spectra connectedaround pT = 2.0 GeV/c to the low pT kaon spectrum with the scale factor
0.55. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.13 pT shape dependence of electron v2 from kaon decay. The v2 which is calculated from the exponential shape (the dashed red line in Fig. 4.12) is shown
as red points and the scaled π 0 spectra (the blue line in Fig. 4.12) is shownas blue points. No shape dependence is seen in the electron v2 . . . . . . . .

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4.14 (Top) The inclusive electron v2 measured with three type of the reaction
plane. (Bottom) The relative ratio respect to the v2 measured with combinedN(S)reaction plane (v2 /vconb
2 ). . . . . . . . . . . . . . . . . . . . . . . . . .
4.15 Systematic uncertainty from the electron identification is estimated by measuring the inclusive electron v2 with slightly change the condition of thestandard electron identification cut (top). The relative ratio of the E/p isabout 2%, the n0 is about 2% and the χ 2 /npe0 is 1 % (bottom). . . . . . .
4.16 Transverse momentum dependence of non-photonic electron v2 in Au+Au√
collisions at sNN = 200 GeV for the minimum bias events. The verticalsolid lines mean statistical error and the brackets mean 1 σ systematic uncertainty of the non-photonic electron v2 . . . . . . . . . . . . . . . . . . .
5.1

5.2
5.3

5.4

5.5

5.6

(Left) Non-photonic electron yield (0.8 < pT < 4.0 GeV/c) measured in
Au+Au collisions at 200 GeV [42] scaled by Ncoll as a function of Ncoll . Normalizing by the nuclear overlap function obtained charm electron cross section per N + N collisions (right-hand scale). (Right) Invariant yields of electrons from heavy flavor decays for different Au+Au centrality [55], scaledby powers of ten for clarity. The solid lines are the result of a FOMLL calculation of heavy flavor decay normalized to the p + p data [39] for eachcentrality class. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
pT dependence of the confidence level for the non-zero non-photonic electron v2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Comparison of the non-photonic electron v2 with pion v2 . The dashed lineon the figure is the electron v2 from charm decay assuming charm has samev2 as u quark. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
(Left) Momentum correlation between D meson and the decay electrons.
The open circles represent the mean pT of D meson momentum distribution
for the given electron momentum and closed data points represent the peakof the D meson momentum distribution. (Right) pT dependence of the decay
angle correlation between D mesons and the decay electrons. . . . . . . . .
Results of the χ 2 test between measured non-photonic electron v2 and electron v2 calculated by the simulation assuming D meson v2 shape as a function of the scaled factor a(%) . . . . . . . . . . . . . . . . . . . . . . . . .
1 σ bands for each of the five assumed D v2 shapes. . . . . . . . . . . . . .

X

81

82

83

85
86

87

88

90
91

5.7

D meson v2 estimated by the measured non-photonic electron v2 . The closed
data points represent the most suitable D meson v2 (χ 2 minimum) from thecalculation. The dashed lines are χ 2 minimum+2 value for the determination. The open circles is π v2 . . . . . . . . . . . . . . . . . . . . . . . . . . 92
5.8 Universal v2 shape for quarks in the calculation. . . . . . . . . . . . . . . . 93
5.9 Fitting result of non-photonic electron v2 , kaon v2 and proton v2 . The open
circles mean decay electron v2 from the D meson. . . . . . . . . . . . . . . 94
5.10 Contour plot of v2 of c quark and u quark which are defined as v2,u = a × v2,q
and v2,c = b × v2,q . Here v2,q is a quark v2 determined by the measured
v2 . Parameters a and b are obtained by fitting the measured non-photonic
electron v2 , Kaon v2 and proton v2 . The contour lines are one sigma step. . 95
5.11 Charm quark v2 from the calculation based on the quark coalescence model.
The solid lines are obtained from 1 σ contour level, the dashed lines are
obtained 2 σ contour level and the dotted + dashed lines are obtained from4 σ contour level by the fitting. The open boxes mean u quark v2 . . . . . . 96
5.12 Electron v2 from D meson and B meson. The curve (1) is assumed no b quark
contributions to the non-photonic electron (only take into account charm
decay). The curve (2) is assumed that D meson and B meson v2 have samev2 . And the curve (3) is assumed that B meson v2 is zero. . . . . . . . . . . 97
5.13 The electron spectra from the model calculations. The dashed curves are the
PYTHIA calculation and solid curves are the results from the thermal hydrodynamic model. The data in the figure is non-photonic electron measured√
at PHENIX in Au+Au collisions at sNN = 130 GeV [51]. The picture istaken from [10]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
5.14 Comparison of the non-photonic electron v2 with two different charm flow
scenarios [62]. The solid line corresponds to the no rescattering of the initially produced charm quarks (without flow), while the dashed line reflects
the effect of complete thermalization (with flow). . . . . . . . . . . . . . . 100
5.15 RAA of the non-photonic electrons in 0-10 % central collisions comparedwith model calculations assuming the initial gluon densities is 3500 (dN g /dy= 3500). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
5.16 (a) RAA of the non-photonic electron in 0-10 % central collisions. (b) v2
for the non-photonic electron in minimum bias collisions. The curves in
the figure are model calculations assuming elastic scattering mediated byresonance excitation of D meson and B meson in QGP [65]. . . . . . . . . 104

XI

B.1 an illustration of the collision geometry for a non-head on nucleus-nucleus
collision. The nucleons in the overlap region of the two nuclei participate in
the collisions are called ”participants” and the rest nucleons which are notparticipate in the collisions are called ”spectator” . . . . . . . . . . . . . . 114

XII

Chapter 1
Introduction
The goal of high energy nucleus collisions at the Relativistic heavy ion collider (RHIC) it tocreate ”Quark-Gluon plasma” (QGP) which is a plasma state of quarks and gluons. In thischapter we introduce QGP and review the experimental results at RHIC. Finally, we present
the thesis motivation, why we study elliptic flow of charm quark.

1.1 Quantum Chromodynamics and Quark Gluon Plasma”What is the fundamental particle ?” is one of the longstanding questions for mankind. At
the present day, the fundamental particle is thought to be ”quark”. Quark was introducedby M. Gell-Mann and G. Zweig to classify hadrons. They proposed that all hadrons are
made from quarks and the model based on the proposition is called quark model. In themodel hadron made from two quark and anti-quark pair is called meson and made fromthree quarks is called baryon. One of the most interesting features of the quark is that they
are confined in hadron and they can not be observed outside hadron. The strong interaction,
one of the fundamental interactions in the nature, is the interaction which binds quarks
and makes hadrons. The remarkable feature of the strong interaction is that the interaction
is weak at shorter distances or large momentum transfer (Q2 ). The feature of the strong
interaction is called ”asymptotic freedom”. Due to the feature quarks move as free, noninteracting particles, in the depth of hadrons. Experimentally such feature has been observedin the deep inelastic scattering between leptons and hadrons in 1969. On the other hand, theinteraction is strong with distances or small momentum transfer therefore quarks are pulledback by the strong force if they try to move from hadrons. It is the reason why single quark
has not been observed alone.
The strong interaction is described by the field theory of Quantumchromo dynamics
1CHAPTER 1. INTRODUCTION

2(QCD). QCD is based on non-Abelian gauge theory and the gauge theory represents asymptotic freedom. In the theory, the filed is given as color charge and gluons are gauge bosons
that mediate strong color charge interactions between quarks. As results of QCD calculation, the effect of confinement for quarks is broken and a matter becomes a plasma state ofquark and gluons under the very high temperature and high density circumstance. Such thestate of the matter is called ”Quark-Gluon plasma” (QGP). Lattice QCD predicts the phasetransition to a QGP at a temperature T ∼ 170 MeV which corresponds to an energy density
ε ∼ 1.0 GeV/fm3 [1] at zero baryon density, nearly an order of magnitude larger than that ofnormal nuclear matter. It is thought that such a state of matter existed in the early universe,
a few µ s after ”Big Bang” which is thought the origin of the universe. Now we have alreadyknown that the universe has been expanding since Big-Bang. If we extrapolate the universe
to Big-Bang, dawn of the universe can be extremely high dense and temperature. Therefore
the research of QGP is an important topic not only testing QCD but also astrophysics.

1.2 Quark Gluon Plasma and Relativistic Heavy Ion Collisions
Figure 1.1 shows the phase diagram of nuclear matter as a function of temperature andbaryon chemical potential µ . Ordinary nuclear matter exist at T ∼ 0 and µB ∼ 1. It is
predicted that the nuclear matter becomes QGP state by raising temperature or density. Relativistic heavy ion collisions are powerful tool to achieve high temperature (or density) because a large amount of energy is deposited in the collision zone.
Since 1980 attempt to create QGP has been performed at AGS (Alternating GradientSynchrotron) in Brookhaven National Laboratory and Super Proton Synchrotron (SPS) in
European Laboratory for Particle Physics (Table 1.1). The AGS accelerates Si ions up to√
the center of mass energy per nucleon ( sNN ) 5.4 GeV and Au ions up to 4.8 GeV. The
SPS accelerates O ions up to 19.4 GeV and Pb ions up to 17.4 GeV. The relativistic heavy
ion collider (RHIC) at BNL is the world’s first head-on collider and the maximum centerof mass energy per nucleon pair is 200 GeV. RHIC has been operated from June 2000 andachieved the center of mass energy per nucleon 200 GeV in 2001. LHC is the future highenergy heavy ion collision program and it will be operated from 2007.
The initial energy density of the collision region can be estimated by the Bjorken energy
density [3] asεB j =dET (τ f orm )τ f rom A⊥
dy
1

(1.1)

CHAPTER 1. INTRODUCTION

3

Figure 1.1: A schematic phase diagram of nuclear matter as a function of temperature T andbaryon chemical potential µ . The figure is taken from [2].

where τ f orm is a proper time that the secondary particles can be consider ”formed”, A⊥ is
the nuclei transverse over rap area and dET /dy is the transverse energy per unit rapidity. In
PHENIX experiment, the value of dET /d η has been measured and obtained 600 GeV for√
0-5% centrality [4]. This leads to εB j τ f orm = 5.4 ± 0.6 GeV c−1 fm−2 for sNN = 200 GeV(A⊥ was 140 fm2 and a scale factor of 1.25 was used for the transformation from dET /d η todET /dy [4]). Figure 1.2 shows εB j τ f orm as a function of the number of participant nucleons(N part ) measured at PHENIX [4]. Historically, εB j has been calculated by choosing τ f orm= 1 fm/c. As described in [5], more realistic value of τ f orm from experiment is 0.35 fm/c
which is larger than the crossing time of the nuclei (2R/γ = 0.13 fm/c) and this leads to√εB j ∼ 15 GeV/fm3 in central Au+Au collisions at sNN = 200 GeV. This energy density is

CHAPTER 1. INTRODUCTION
Accelerator
AGS

Location
BNL

SPS

CERN

RHIC

BNL

LHC

CERN

4
Ion beam
16 O, 28 Si
197 Au16 O, 32 S
208 Pb197 Au+197 Au
197 Au+197 Au
208 Pb+208 Pb

√
sNN
5.4
4.8
19.4
17.4
130
200
5600

Start
Oct. 1986
Apr. 1992
Sep. 1986
Nov. 1994
2000
2001
2007(project)

Table 1.1: List of heavy ion accelerator facilities with the ion beams, the center of massenergy.
larger than 1 GeV/fm3 which was predicted by the lattice QCD, therefore the matter createdat RHIC might be above the threshold for QGP formation.
A key issue of QGP formation at the relativistic heavy ion collisions is the thermalizationof the matter created by the collisions. At the very early stage of the collisions among quarks
and gluons scatter each other in the collision zone. If they scatter over and over again, thematter made from quarks and gluons can be achieved a thermal equilibrium, i.e. quark
gluon plasma. Once thermal equilibrium is achieved, the evolution of the matter would
be described by the hydrodynamical framework. Figure 1.3 shows the illustration of thespace-time evolution of the medium created by heavy ion collisions at RHIC as a functionof temperatures and times calculated by the hydrodynamical framework [6]. At the center ofthe collision zone, pressure is larger than the surrounding area thus the medium expands andtemperature is drop. At the expanding process medium passes through the mixed phase of
QGP and hadron gas then the medium becomes hadron gas state. In the hadron gas, inelastic
scattering of hadrons continues until chemical freeze-out temperature where hadron species
and ratios are fixed (Chemical equilibrium). During this time the system is still expanding,
finally elastic scattering of hadrons finishes at kinetic freeze-out temperature where hadron
momentum are fixed (Kinematical equilibrium).
Various measurements have been proposed to study QGP and the property in the relativistic heavy ion collisions. Here we introduce some of them.
1. Particle ratio - chemical freeze-out
Measurement of particle ratios give us information about chemical equilibrium due toparticle species are fixed at the point of chemical freeze-out.
2. Elliptic flow

5

2

∈Bjτ [GeV/fm /c]

CHAPTER 1. INTRODUCTION

4

2200 GeV
130 GeV19.6 GeV
0

100

200

300

Np

Figure 1.2: εB j τ as a function of N part measured at three RHIC energies.

Elliptic flow, collective motion in azimuthal space, is thought sensitive to the earlystage of collisions. In the hydrodynamical frame work, elliptic flow is arised as aresult of pressure gradient in thermal equilibrium system. If strong elliptic flow isobserved, it indicates thermal equilibrium status is achieved at the RHIC energy.
3. Suppression of high pT particle- energy loss
QCD predicts that the partons (quark and gluons) lose their energy when they propagate the dense matter [7]. As the result, jet productions are suppressed. One of thegood methods to study parton energy loss in the matter is a comparison of the yieldfor high pT particles in Au+Au collisions with the yield in p + p collisions. The originof the high pT particle production is jet, therefore high pT particle production is suppressed compared with the production in p + p collisions if partons lose their energy
in the matter.
4. Charmed hadrons
Charm quark is believed to be produced in initial collisions via gluon fusion and it
propagates through hot and dense medium created in the collisions. At RHIC energies,
charm quark energy loss [8], charm quark coalescence [9] and charm quark flow [10]has been proposed to study the properties of the medium created by Au+Au collisions.
Such topics can be observed via charm hadron, i.e. D meson, mesurement.

CHAPTER 1. INTRODUCTION

6

Figure 1.3: Space-time evolution of a nucleus-nucleus collisions. The time and temperatureare calculated by the hydrodynamical framework [6].

1.3 Thermalization
1.3.1 Chemical Equilibrium
Measurement of particle ratios give us information about chemical equilibrium due to particle species are fixed at the point of chemical freeze-out. A simple statistical model which assumes local thermal equilibrium reproduces well the measured ratios. The statistical modelis based on a grand canonical ensemble to describe the partition function and hence thedensity of the particles of species i an equilibrated fireball is given as byni = gi

1d p3,
3(2π ) exp([(Ei − µi )/T ]) ± 1(1.2)

with particle density ni , spin degeneracy gi , momentum p, total energy E and chemical potential µi = µB Bi − µs Si − µI3 Ii3 . The quantities Bi , Si and Ii3 are baryon, strangeness andCHAPTER 1. INTRODUCTION

Figure 1.4: Ratios of measured yield of various particles incompared with the statistical thermal model [11].

7√sNN = 130 GeV and 200 GeV,

isospin quantum numbers of the particles i. In this model, the temperature T and the baryochemical potential µB are the two independent parameters. Figure 1.4 shows the particle√
ratios measured in Au+Au collisions at sNN = 130 GeV and 200 GeV. The results are well√
reproduced by the thermochemical model with T = 176 MeV and µ = 41 MeV for sNN =√130 GeV , T = 177 MeV and µ = 29 MeV for sNN = 200 GeV/c. As described in Chapter
1.1, the lattice QCD predicts that the phase transition from a hadronic phase to QGP phase,
which takes place at a temperature of approximately 170 MeV. The temperature obtained
from particle ratios with thermochemical models has a good agreement with the prediction.
It indicates that the medium created at RHIC is achieved chemical equilibrium.

CHAPTER 1. INTRODUCTION

8

1.3.2 Kinematical equilibrium and the collective flow
Radial flow
Transverse momentum spectra is one of the most basic measurement not only in heavy ion
collisions but also proton-proton or proton-nucleus collisions to study the particle productions in the collisions. If the system created by collisions is a thermal equilibrium state ofhadron gas, the momentum distribution of particles should follow Maxwell-Boatman distribution and the transverseq momentum distribution is given as an exponential function oftransverse mass mT (=p2T + m2 ) as
1 dN
−mT
= A exp(),
mT dmTT(1.3)

where A is a constant and T is the inverse slope parameter of the distribution. It is known thatthe transverse momentum spectra is well described with the equation in the region pT <2.0
GeV/c and it is called mT scailing. The inverse slope parameter T corresponds to the temperature of the system at the kinetic freeze-out. In high energy p + p collisions the slope
parameters are independent of particle species [12] (Tπ ∼ TK ∼ Tp ∼ 150 MeV). On the other
hand, it has been observed that these parameters depend on the particle mass in heavy ion
collisions [13] [14]. Fig.1.5 shows inverse slop parameter T as a function of particle massmeasured in p + p, S+S and Pb+Pb collisions [15]. As we see, the slope parameters measured in S+S and Pb+Pb collisions are not independent of particle species. The feature of
the slope parameters in heavy ion collisions is that the larger mass particle have larger slope
parameter (Tπ < TK < Tp ) and it also increases with mass of collided nucleus. The feature
indicates that the existence of the outward collective motion (collective radial flow). Most
successful description of the difference slope parameters in heavy ion collisions is given bya model including the common transverse expanding velocity (collective flow) field together
with a moderate temperature of a thermalized system. If the system is a expanding system,
particles have a common transverse expanding velocity, not only their thermal motion. In
the framework, the slope parameter is given as;
1
T ∼ T f o + mβ 2
2

(1.4)

where T f o is the thermal temperature, m is particle mass and β is the collective velocity. At
RHIC we have obtained T f o = 177.0 ± 1.2 and β = 0.48 ± 0.07 in most central Au+Au√
collisions at sNN = 200 GeV [16] [17] from the fitting of transverse momentum spectra for
identified particles (π , K, p).

CHAPTER 1. INTRODUCTION

9

Figure 1.5: Inverse slop parameter T as a function of particle mass. They are compared at
mid-rapidity in p + p, S+S and Pb+Pb collisions [15].

Azimuthal anisotropy
As describe in the previous section, the measurement of transverse momentum spectra gives
us information about the existence of the outward collective motion. The measurement ofparticle production in the azimuthal space can be more information about collective motionin heavy ion collisions. The azimuthal distribution of the particles is very influenced by
the mean free path. When the mean free path is larger than the size of the system, theparticle emmision is isotropic in the azimuthal space. On the other hand, the mean freepath is much shoter than the size of the system (thus hydrodynamical description apply), theemission pattern is affected by the shape of the system. In heavy ion collisions at RHIC, themean free path is about 10−1 f m and it is smaller than the system size (R = 6 f m). In thehydrodynamical framework, the driving force of the collective flow is the pressure gradient,∆ P. In the non-central collisions, the initial medium shape created by the collisions is analmond like shape (Fig. 1.6). If the medium becomes equilibrium (or local equilibrium),the pressure in x − z plane is larger than y direction. Therefore the particles produced bycollisions are emitted more to the x − z plane than to y direction and it is measured as ananisotropy in the azimuthal distribution. Thus the initial spatial anisotropy in the early stageCHAPTER 1. INTRODUCTION

10

Figure 1.6: An illustration of non-central Au-Au collisions. The plane defined as the direction of the impact parameter (x) and direction of the beam axis (z) is called “reaction
plane”.

of the collisions transfer to the momentum space anisotropy in the final stage. Therefore theazimuthal anisotropy is sensitive to the early stage of the collisions, in contrast to transverse
momentum spectra which is sensitive to the expanding velocity in the final stage of thecollision.
Experimentally the azimuthal anisotropy has been studied using Fourier expansion of
the azimuthal distribution. The azimuthal anisotropy is defined by½¾
dN= N0 1 + ∑ 2vn cos(n(φ − ΨR.P. )) ,
(1.5)
dφ
n
where N0 is a normalization constant, φ is the azimuthal angle of particles, and ΨR.P. isthe direction of the nuclear impact parameter (”reaction plane”) in a given collision. The
harmonic coefficients, vn , indicate the strength of the nth anisotropy. Especially the second
harmonic coefficient of the Fourier expansion of the azimuthal distribution is called ”elliptic
flow” and it is sensitive to the early stage of the collisions [18]. Indeed, v2 which is thestrength of the elliptic flow scales with the geometrical eccentricity which is defined as;ε=hy2 i − hx2 i.
hy2 i + hx2 i(1.6)

CHAPTER 1. INTRODUCTION

11

Figure 1.7: Beam energy dependence of v2 measured at AGS, SPS and RHIC.

Elliptic flow has been measured in the early stage of the experiment of heavy ion collisions. Figure 1.7 shows the energy dependence of v2 measured at AGS, SPS and RHIC
experiments. At RHIC, strong elliptic flow has been measured and the value of v2 is 2 times
larger than v2 measured at SPS. The transverse momentum (pT ) dependence of v2 has beenmeasured for identified particles at RHIC [19, 20, 21, 22]. Figure 1.8 shows the pT dependence of v2 for π , K, p. In low pT region (<1.5 GeV/c), v2 increases with pT and show theclear mass dependence,pvπ2 > vK
(1.7)
2 > v2 .
This behavior is consistent with a hydrodynamical model calculation [23] which is shown as
solid lines in Fig. 1.8. The model assumes that the initial time when the local equilibrium
is occurred is 0.6 f m/c and the phase transition is first-order phase transition with a freezeout temperature of 120 MeV. The agreement suggets as evidence that the collective motion
develops in the very early stages of the reaction. In addition, the model assumes that theviscosity of the matter is very low, therefore the property of the matter created in Au+Au

CHAPTER 1. INTRODUCTION

12

Figure 1.8: Identified particle (π , K, p) v2 as a function of pT [19]. The lines in the figure
are the hydrodynamical model calculation. [23]collisions is thought like a liquid. The other feature of v2 for identified particles is that v2
for meson, π and K, is saturated around pT = 2.0 GeV/c, on the other hand v2 for baryon,p, is still increasing. The difference between meson v2 and baryon v2 is well explained by aquark coalescence model. The detail is described in the 1.5.

1.4 Suppression of high pT particle
One of the good probes for studying the medium is to measure high pT particles. In the
RHIC energy range, the dominant process of the particle production with pT > 2.0 GeV/c
in nucleon-nucleon collisions is hard scattering. Typically, particles with pT > 2 GeV/c are
produced from states with two roughly back-to-back jets which are the result of scatteringof constituents of the nucleons (partons) as described by the perturbative QCD. The hard
scattering will occur before the medium creation, therefore the high pT particles propagate

CHAPTER 1. INTRODUCTION

13

Figure 1.9: Comparison of two-particle azimuthal angular correlations of charged particles
for central Au + Au, central d + Au and p + p collisions, where NTrigger is the number ofhigh pT particles.

through the medium. It has been predicted that the high pT particles are suppressed compared with p + p collisions at RHIC due to the energy loss of partons, which called ”jetquenching”.
One of the methods to study the suppression of high pT hadrons is to measure twoparticle azimuthal angluar correlations. Due to the high pT particles are produced fromback-to-back jets, we can expect peaks around ∆ φ ∼ 0◦ (nearside) and ∆ φ ∼ 180◦ (awayside). Figure 1.9 shows the two particles (4 < pT < 6) azimuthal anglular correlations (∆ φ )for inclusive charged hadrons measured in p + p, d+Au and Au+Au in 200 GeV collisions[24]. Two particles are chosen such that one ”trigger” particle and the other ”associated”
particles. In this plot, other sources of angular correlations, elliptic flow, are subtracted. We
can see clear peaks around at ∆ φ ∼ 0◦ in p + p, d+Au and Au+Au which is expected fromthe high pT particles production. On the other hand, the away side peak at ∆ φ ∼ 180◦ isCHAPTER 1. INTRODUCTION

14

Figure 1.10: Nuclear modification factor measured for π 0 and charged particle in Au + Au√
collisions (RAA ) and in d + Au collisions (RdAu ) in sNN = 200 GeV as a function of pT[25]. The bands around the data points show systematic errors, while the shade band aroundunity indicates the normalization uncertainty.

vanished in Au+Au collisions, while p + p and d+Au collisions make peaks at ∆ φ ∼ 180◦ .
The different results between Au+Au and d+Au indicate that the suppression of the awayside yields in Au+Au is due to the final state interactions with the medium created in Au+Au
collisions. Because the same suppression of the away side yields should be also observed in
d+Au if the it is due to the initial state interaction.
The other method to study the suppression is to compare yields between Au+Au collisions and p + p collisions. The hard scattering cross section usually have a small crosssection (point like), therefore the yields are proportional to the number of a binary scaled
nucleon-nucleon (N + N) collisions, Ncoll . The yield ratio between Au+Au and p + p, nu-

CHAPTER 1. INTRODUCTION

15

clear modification factor RAA , is expressed as;
RAA =d σAu+Au,
hNbinary i × d σ p+p(1.8)

where d σAu+Au and d σ p+p is the cross section in Au+Au and p + p collisions. hNbinary iis the number of primary nucleon-nucleon collisions which is estimated by using Glaubermodel. If there is no medium effect, the rate of hard processes scales linearly with thenumber of binary collisions, hNbinary i. Therefore RAA is unity (RAA = 1.0). Figure 1.10√
shows the RAA for charged hadron and neutral pion in Au+Au collisions at sNN = 200
GeV, together with the nuclear modification factor measured in d+Au collisions (RdAu ) at√sNN = 200 GeV [25]. The RdAu is consistent with unity for neutral pions and greater thanunity for charged particles at high pT region. The RdAu is greater than unity means the yieldis enhanced compared to that in p + p collisions. It is known that the particle production in
p+A collisions is enhanced compared to p + p collisions due to ”Cronin effect” [26]. On
the other hand, RAA is much smaller than unity in Au+Au collisions. It means that the yieldsof Au+Au are suppressed. Such the suppression is not observed in d+Au collisions, thesuppression in Au+Au is due to the final state interactions. The high pT particle suppressionis thought that partons lose their energy through gluon bremsstrahlung [27] in the dense
matter , rescuing the momentum and depleting the yield [28] [29].

CHAPTER 1. INTRODUCTION

16

1.5 Quark coalescence/recombination
1.5.1

Baryon excess

One of the most striking observations in RHIC is that the yield of proton (anti-proton) islarger than that of pion at intermediate pT region (2 < pT (GeV/c) < 5). Figure 1.11 shows
p/π ratio as a function of pT for central (0-10 %), mid-central (20-40 %) and peripheral(60-92 %) collisions [17] [30]. As we see, the ratios increase with centrality and the ratio in
the central collisions is enhanced by almost a factor of three compared with the peripheral
results. The dashed and dotted lines are proton and anti-proton to pion ratio in gluon and inquark jets [31]. In the high pT (pT > 3.0 GeV/c), these ratios are consistent with the ratio
measured in peripheral Au+Au collisions. Thus the p/π ratio at the peripheral Au+Au collisions can be well describe by the fragmentation. On the other hand, the ratio at most central
collisions is much larger than the ratio which is expected by the fragmentation. This result
indicates that there is an additional mechanism for hadron production at the intermediate pT
region.

Ratio

1.8

proton/pion

anti-proton/pion

1.6

Au+Au 0-10%
Au+Au 20-30%
Au+Au 60-92%
p+p, s = 53 GeV, ISR
e+e-, gluon jets, DELPHI
e+e-, quark jets, DELPHI

1.4
1.2
1
0.8
0.6
0.4
0.2
0
0



1

2



3

4
0
p T (GeV/c)

1

2

3

4
5
p T (GeV/c)

Figure 1.11: p/π ratio as a function of pT for central (0-10 %), mid-central (20-40 %) andperipheral (60-92 %) collisions [30]. The dashed and dotted lines are proton and pion ratio
in gluon and in quark jets [31].

CHAPTER 1. INTRODUCTION

1.5.2

17

Quark coalescence model

A quark coalescence (recombination) model is one of the models for hadron production. In
the model, hadrons are produced by valence quarks from the thermal medium when they areclose together in the phase space. In the framework, mesons are formed from coalescenceof a quark and anti-quark and baryon are formed of three quarks. If the model can explain
the data, it implies that there is a larger thermalized source of quarks and anti-quarks and it
is a strong evidence for a QGP formation at RHIC [32].
In a simplified version of the model, the production rates of mesons and baryons areevaluated as,
Z
d 3 NM
E 3 ∝ dxwa (xpT )w¯b ((1 − x)pT )|Ψab |2 ,
(1.9)
d p
d 3 NB
E 3 ∝
d p

Z

dxdxwa (xpT )wb (x0 pT )wc ((1 − x − x0 )pT )|Ψabc |2 ,(1.10)

where w(pT ) is a parton distribution function, x is a fraction of the momentum and Ψab (Ψabc )is a meson (baryon) wave function. In case of an equal momentum fraction i.e. the effective
masses of constituent quarks are similar, x = 1/2 for mesons and x = x0 = 1/3 for baryons
and above equations can be written as;p/ π

PHENIX proton/π ratio
1.6

proton/ π+
proton/ π0
Duke
OregonTAMU w/ shower
TAMU no shower








1.4
1.2
1
0.8
0.6
0.4
0.2
0


0

1

2



3

4


5



6

7



8

9
10
p (GeV/c)T

Figure 1.12: p/π ratio measured by PHENIX [30] and quark coalesnce model calculations
for the comparison. This picture is taken from [5].

CHAPTER 1. INTRODUCTION18

d 3 NM= CM wa (pT /2)2 ,d3 p(1.11)

d 3 NB= CB wa (pT /3)3 ,d3 p(1.12)

where CM and CB correspond to the coalescence probabilities for mesons and baryons, respectively [33]. The coalescing parton distribution is assumed to be exponential, thermal.
Under this assumption, lower momentum quarks are much more abundant than those for thehigh pT quarks. As the result, the chance for lower pT quarks to recombine is much higher
than the chance for higher pT quarks from a hard scattering to fragment into the final state
hadrons. Thus baryons at moderate pT are enhanced relative to mesons as their transverse
momentum is the sum of three quarks rather than two. Figure 1.12 shows several quark coalescence model calculations for proton and pion ratio compared to the data as a function ofpT measured at PHENIX [30]. These calculations well reproduce the strong enhancementof proton at intermediate pT .
The model also predicts that the elliptic flow of hadrons reflect the elliptic flow of constituent quarks. The parton distribution function (w(pT )) for v2 is given as;
w = 1 + 2v2,q cos(2φ ),(1.13)

Then we can obtain following equation for meson and baryon v2 as;v2,M = 2v2,q (pT /2),

(1.14)

v2,B = 3v2,q (pT /3).

(1.15)

where v2,q is quark v2 . The model predicts that elliptic flow for mesons and baryons are
scaled with the number of constituent quarks. Fig. 1.13 shows that v2 as a function of pT
scales via the number of quarks, that is, v2 /n as a function of pT /n where n is the numberof constituent quarks plus anti-quarks (n = 2 for meson and n = 3 for baryon). As we cansee, v2 of identified hadrons after scaling with number of quarks falls on same curve. Thissuggests that the v2 already develops in the partonic phase for hadrons which made of lightquarks.

CHAPTER 1. INTRODUCTION

19

Figure 1.13: v2 /n as a function of pT /n where n is the number of constituent quarks plus
anti-quarks (n = 2 for meson and n = 3 for baryon).

CHAPTER 1. INTRODUCTION

20

1.6 Charm quark production in RHIC
Charm quark is believed to be produced in initial collisions via gluon fusion. Due to the large
mass of charm quark (∼ 1.5 GeV/c2 ), the charm production rate is calculated by pQCD [34][35]. Measurements of charmed hadron, D mesons, are good tools for the study of charm
production. But the directly measurement and the reconstruction of D meson (D0 → K − π + )is difficult due to the small short lifetime (D0 ∼ 124 µ m), the small yield of charm quarks
and the large background of other hadrons. Table 1.2 shows the main decay modes of the Dmeson. In PHENIX experiment charm quark production has been studied by measuring single electrons from semileptonic decays of charmed particles at mid-rapidity region (|η | <0.35). Electrons from charm were observed at the CERN-ISR experiment in early 1970[36] [37] before charm discovery and later it has interpreted as a signal of open charm [38].
Thus the measurement of single electron is also a powerful tool to study the charm production in heavy ion collisions. PHENIX has two independent techniques, “cocktail” methodand “conversion” method to remove electron backgrounds from other sources, mainly conversions and light hadron decays. In the cocktail method, background electrons are calculated by a Monte Carlo simulation and they are subtracted from inclusive electrons. In theconverter method, background electrons are experimentally determined with and without aconverter that increases/decreases background electrons. Figure 1.14 shows the invariant
differential cross section of electrons from heavy flavor, charm and beauty, in p + p colli√
sions at sNN = 200 GeV measured at PHEXNIX experiment [39]. The curves in the figure
are a fixed-order-plus-next-leading-log (FONLL) pQCD calculation [40]. The bottom figure
shows the ratio of the data and the FONLL. The upper and lower curves are the theoretical
upper and lower limits of the FONLL. The pQCD calculation describes the cross sections ofheavy flavor decay electrons within uncertainties.
Decay mode
D0 → K − π +D0 → e+ + X
D± → e± + X

branching ratio
3.80 %
6.87 %
17.2 %

Table 1.2: The branching ratio of D meson decay

In the relativistic heavy ion collisions there would be three stage of charm quark productions if the matter is thermalized: (1) the first stage is the initial collisions which is sameas the production in p + p collisions, (2) the second stage is the pre-equilibrium production
from secondary parton cascade, and (3) the third stage is the thermal production. Figure

3-2

E d3σ/dp (mb GeV c3)

CHAPTER 1. INTRODUCTION

21

10-1

(a)

+

10-2

p+p → (e + e-)/2 + X at s=200 GeV-310

PHENIX data
FONLL(total)
FONLL(c → e)
FONLL(b → e)
FONLL(b → c → e)

-4

10

-5

10

-6

10

10-7
-8

10

10-9

DATA/FONLL

10-10

3
2.5
2
1.5
1
0.5
0

(b)

1

2

3

4

5

6

7

8

9
10
pT (GeV/c)

Figure 1.14: (a) Invariant differential cross section of electrons from heavy flavor, charm and√bottom, in p + p collisions at sNN = 200 GeV measured at PHEXNIX experiment [39].
The curves are the FONLL calculations. (b) Ration of the data and the FONLL. The upperand lower curves are theoretical upper and lower limit of the FONLL.

1.15 shows the theoretical prediction for charm quark productions in Au+Au collisions at√
sNN = 200 GeV [41]. The theory predicts that the most largest contribution is also initialcollisions in Au+Au collisions and the thermal production is very small even if the matter isthermalized. In Au+Au collisions the total yield of heavy-flavor decay electrons was foundto scale with the number of nucleon-nucleon collisions as expected for pQCD [42]. Thus thedominant process of charm quark production in Au+Au collisions is also initial collisions
via gluon fusion.

CHAPTER 1. INTRODUCTION

22

Figure 1.15: Theoretical prediction of pT distributions of initial (solid), prethermal (dotdashed) and thermal (dashed) charm production for central Au+Au collisions at RHIC en√ergy, sNN = 200 GeV [41].

1.7 Thesis motivation
As reviewed in this chapter, there are various hints for QGP formation in Au+Au collisions√
at sNN = 200 GeV. The initial energy density of the medium created in Au+Au collisions
is about 15 GeV/fm3 and the value is large enough to create the QGP, which 1.0 GeV/fm3
predicted by the lattice QCD. The yield of high pT particles is strongly suppressed withregard to the yield in p + p collisions (RAA < 1.0). The origin of the suppression is thoughtthat energy loss of partons in the dense medium. Thus the density of the medium would bevery high. The particle ratios are well reproduced by a statistical thermal model calculation,
which assumed chemical freeze-out temperature of 170 MeV. Collective flow, especially alarge elliptic flow has been observed. The strength of elliptic flow, v2 , for various particles
has been measured. The v2 for identified particles shows a clear mass dependence and it iswell explained by the hydrodynamical model calculation which assumes a very early time
thermal equilibrium (τ = 0.6 fm/c) and very low viscosity. These v2 are scaled with theCHAPTER 1. INTRODUCTION

23

number of quarks. It indicates that partonic collectivity has been developed for light quarks
in Au + Au collision at RHIC. If charm quarks also flow same as light quarks, it indicatesa quark level thermalization is achieved at RHIC energies. Because it is needed that charm
quarks are scattered over and over again due to their large mass. Thus it would be a strong
evidence of QGP formation at RHIC.
In this thesis we study elliptic flow, which is defined as a second harmonic of the Fourier
expansion of the azimuthal distribution, of charm quark via electron measurement. We
measured the transverse momentum dependence of the azimuthal anisotropy parameter v2
for electrons at mid-rapidity (|η | < 0.35) with the PHENIX detector in Au+Au collisions at√
sNN = 200 GeV during the fourth experiment period at RHIC run (2004 - 2005). From
the result, we have measured the v2 of electrons from heavy flavor decay after subtractionof the v2 of electrons from other sources such as photon conversions and Dalitz decay fromlight neutral mesons. Based on the result, we discuss charm quark v2 in Au+Au collisions√
at sNN = 200 GeV.

Chapter 2
The PHENIX experimentThis analysis is based on the data measured at the RHIC-PHENIX experiment. In this chapter, the Relativistic Heavy Ion Collider complex and PHENIX detector system are presented.

2.1 The Relativistic Heavy Ion Collider (RHIC)

Figure 2.1: A picure of RHIC complex.

The Relativistic Heavy Ion Collider (RHIC) at BrookHaven National Laboratroy is the24CHAPTER 2. THE PHENIX EXPERIMENT
Circumference of rings (km)No. Intersection Regions
No. Bunches/ring
Bunch Spacing(nsec)
No. Particles/Bunch
Top Energy(GeV/u)
Luminosity, average(cm−2 sec−1 )Lifetime(h)

25
3.834
6
60
213
1× 109100
2×1026
10

Table 2.1: RHIC performance for Au ionsworld’s first head-on collider for heavy ions. RHIC consists of two rings which are madeup of 1,740 superconduction magnets and each ring has about 3.8 km circumference. Any
nuclear species from proton to Au can be accelerated and collided at interaction points at
RHIC. The maximum energy is 100 GeV per nucleon for Au ions and the center of mass√energy ( sNN ) is 200 GeV. The stored beam lifetime for Au is expected to be approximately
10 hours. The designed luminosity of RHIC for Au+Au is about 2 × 1026 cm−2 sec−1 at
top energy. A total of 60 bunches are injected into each ring. Table 5.1 shows the summaryof RHIC parameters.
Figure 2.2 shows the RHIC acceleration scenario for Au. Tandem Van de Graff accelerators, the Booster synchrotron, and the Alternating Gradient Synchrotron (AGS) serve as theinjector for RHIC. The negative Au ions (-1) injected to Tandem Van de Graff are accelerated at the kinematic energy of 1MeV/u and with Q=+12 charge state. Exiting from Tandem
Van de Graf, ions are further stripped to the charge state +32. Then they are injected into
the Booster synchrotron through heavy ion transfer line (550m). The Booster synchrotron
accelerates the beam up to 95 MeV/u. In the Booster to AGS line, the ions are strippedto charge state +77 and injected to AGS. At AGS the beam is accelerated to 10.8 GeV/u.
Then the ions are stripped to charge state +79 at the start of AGS to the RHIC transfer line
tunnel. The intensity of the beam injected to RHIC is 1 × 109 ions/bunch. Finally, the beamis accelerated to 100.0 GeV/u at RHIC. Four experiments (PHOBOS, BRAHMS, STAR and
PHENIX) are instrumented at the collision point. The detail of PHENIX is described in thenext section.

CHAPTER 2. THE PHENIX EXPERIMENT

26

Figure 2.2: RHIC acceleration scenario for gold. The tandem Van de Graff accelerators, theBooster synchrotron, and the Alternating Gradient Synchrotron (AGS) serve as the injector
for RHIC.

CHAPTER 2. THE PHENIX EXPERIMENT
Component
Magnet: central(CM)muon (MMS)
muon (MMN)
BBC
ZDC
DC

∆η
|η | <0.35-1.1 to -2.2
1.1 to 2.4
3.0< |η | <3.9
±2 mrad|η | <0.35

∆φ
360◦
360◦
360◦
360◦
360◦
90◦ × 2PC

|η | <0.35

90◦ × 2RICH
TOF
PbSc EMCal|η | <0.35|η | <0.35|η | <0.35

90◦ × 245◦
90◦ + 45◦

PbGl EMCal|η | <0.3545◦µ tracker:(µ TS)µ identifier:(µ IDS)(µ IDN)-1.15 to -2.25-1.15 to -2.25-1.15 to -2.44

360◦
360◦
360◦

27
Purpose and Special Feature
Up to 1.15 T・m.
0.72 T・m for η =20.72 T・m for η =2
start timing, fast vertex
Minimum bias trigger
Good momentum and mass resolution∆m/m = 0.4% at m=1.0GeV
Pattern recognition,
tracking for nonbend direction
Electron identification
Good hadron identification, σT OF = 100ps
For both calorimeters, photon andelectron detection
Good e± /π ± separation p > 2.0GeV /cby EM shower and p < 0.35GeV/c by ToF
K ± /π ± separation up to 1 GeV/c by ToF
Tracking for muons
Steel absorbers and Iarocci tubes formuon/hadron separation

Table 2.2: Summary of PHENIX detector subsystems

2.2 Overview of PHENIX experiment
A prime goal of PHENIX experiment is to detect QGP and study its physical properties. In
order to achieve the goal, PHENIX is designed to measure various particle species producedby the collisions especially photons, leptons and hadrons. These measurement are done with
11 independent sub-systems.
Figure 2.3 shows the setup of the PHENIX experiment in Run4. The acceptance of
PHENIX is shown in Fig. 2.4. PHENIX consists of four parts of spectrometers arms anda set of global detectors. The spectrometers around mid-rapidity are called central arms
(west and east arms) and those around forward rapidity are called muon arms. Table 2.2
shows the summary of the PHENIX detectors. The global detectors consist of Beam-BeamCounters (BBCs) and Zero-Degree Calorimeters (ZDCs). These detectors provide a timeof beam-beam collision, collision vertex, event trigger and centrality. The central arms are

CHAPTER 2. THE PHENIX EXPERIMENT

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&('





 ",)*  ! ! % )"*

.-&

& 

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01  228

0  12//

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

PC1

!" $#% "  ")*  ! + ! % ),"*/&/3. 5&# *76

.3 54 # *76
Figure 2.3: PHENIX experiment setup in the 4th year of RHIC run (Run4). (Top:Cross
section perpendicular to the beam pipe. Bottom: East side view of the cross section along
the beam pipe.) PHENIX consists of four parts of spectrometer arms and a set of global
detectors. The spectrometers around mid-rapidity are called central arms (west and eastarm), and those around forward rapidity are called muon arms.

CHAPTER 2. THE PHENIX EXPERIMENT

29

designed to track particles emitted from collisions vertex, reconstruct momentum, and identify charged and neutral particles. The central arms cover the pseudo rapidity range |η | <0.35 and 90◦ in azimuthal angle. The central arms consist with several subsystems. The
Drift Chamber (DC) measure charged particle trajectories in the r-φ plane. The Pad Chambers (PC) measure 3-D space position along the straight line particle trajectories. The Ring
Imaging Cherenkov Counter (RICH) is the primary detector for electron identification, isinstalled in each central arm. The Electro-Magnetic Calorimeter (EMCal) is used to measure the spatial position and energy of electrons and photons. The combination of the RICH
and the EMCal allows separation of π /e to less than 10−4 at p < 4.7 GeV/c [43]. The
time-of flight detector (TOF) is the primary detector for charged hadron identification. The
performance of the RICH, TOF and EMCal is summarized in Table 2.3.

Acceptance

PH ENIX
Muo n

157.5°φ (angle)

120°

0°

Electron+ Photon

Muo n

-120°

Hadron
-2.0

0.0

2.0

Rapidity
Figure 2.4: PHENIX acceptance of leptons and hadrons.

2.3 Magnet system
The PHENIX magnet system [44] is composed of three spectrometer magnets, Central Magnet (CM) and Muon Magnets (MMN and MMS), with warm iron yokes and water-cooled
copper coils. The transverse momentum of each particle is determined from its curvature

CHAPTER 2. THE PHENIX EXPERIMENTElectrons

Photons
Hadrons30

π /e < 10−4 at p≤4.7GeV/c
RICH for <4.7 GeV/c
EMCal for > 0.5 GeV/cpt ≤ 1 GeV/c for 0.5 sr with PbGl.
pt ≤ 1 GeV/c for 1.5 sr with PbSc.
≤ 2.3 GeV/c π for 0.38 sr≤ 1.6 GeV/c K for 0.38 sr≤ 5.0 GeV/c p for 0.38 sr
TOF with σ < 100 ps.

Table 2.3: Performance of the RICH, TOF and EMCal at Year-2 [43].
bend in the magnetic field provided in the central arms. The CM is energized by two pairs ofconcentric coils and provides a magnetic field around the interaction vertex that is parallelto the beam. The filed covers pseudo rapidity region |η | < 0.35 and it allows momentum
analysis of charged particles in polar angle range from 70◦ < θ < 110◦ . The magnetic field
for the central arms is axially symmetric around the beam axis. Its component parallel to thebeam axis has an approximately Gaussian dependence on the radial distance from the beamaxis, dropping from 0.48 T at the center to 0.096 T (0.048 T) at the inner (outer) radius of
the DC. The total field integral is about 0.8 Tesla-meters and that is minimized for the radiusof R > 2 m (the radius of the DC).

CHAPTER 2. THE PHENIX EXPERIMENT

31

Figure 2.5: Line drawings of the PHENIX magnets, shown in perspective and cut away toshow the interior structures.

CHAPTER 2. THE PHENIX EXPERIMENT

32

2.4 Global detectros
2.4.1

Beam-Beam counters (BBCs)

The main role of BBCs is to provide the time of beam-beam collision, to produce a signal
for the PHENIX LVL1 trigger, to measure the collision vertex point, to determine collisioncentrality with ZDCs and determined a reaction plane.

Figure 2.6: (a) A picture of BBC array comprising 64 BBC elements. (b) A picture of BBC
element which consists of one-inch mesh dynode PMT mounted on a 3 cm quartz radiator.
(c) The configuration of the 64 PMTs of each BBC.
Two sets of BBC are installed on North and South side of the collision point along thebeam axis. The BBCs are placed 144 cm from the center of the interaction region andsurround the beam pipe. This corresponds to a pseudo rapidity range from 3.0 to 3.9 overthe full azimuth. Each BBC is composed 64 BBC elements. A BBC element consists of
one-inch mesh dynode photomultiplier tubes mounted on a 3 cm quartz radiator. The BBC

CHAPTER 2. THE PHENIX EXPERIMENT

33

elements are grouped inner, middle and outer ring. The outer ring of the BBC is 30 cm
and inner ring is 10 cm with clearance between the beam pipe and the BBC of 1 cm. Thisconfiguration is satisfied following requirements.
• Work in a high magnetic field environment due to the BBC is placed just behind theCM.
• Radiation hard due to the BBC is placed in a very high-level radiation area aroundbeam pipe.
• The BBC have a large dynamic range from 1 Minimum Ionizing Particles (MIP) to√
30 MIP’s due to the BBC are operated with not only Au-Au collisions at sNN = 200√
GeV but also p-p collisions up to sNN = 500 GeV
The BBCs measure the time and charge of leading charged particles from beam collisions. The time information provides the time of beam-beam collision and the collisionvertex. The charge information is used for determinations of the centrality and the reaction
plane angles.

2.4.2 Zero-Degree Calorimeters (ZDCs)
The purpose of the ZDCs is to detect neutral beam fragments from collisions. The ZDCs
are installed in the four RHIC experiments and each ZDC covers 2 mrad of forward angularcone which corresponds to η >6.0. In PHENIX two ZDCs are located 18 m downstream
and upstream from the interaction point along the beam axis. In front of the ZDCs, the DX
dipole magnets are installed. Charged particles emitted from collisions are deflected out of
the ZDC acceptance by the magnets. The ZDCs are hadron calorimeters. A single ZDC
consists of three modules and each module consists of Tungsten alloy plates and ribbonsof commercial optical fibers in a sampling layer. The coincidence signal from the ZDCs isused for the minimum bias trigger of beam interactions, and it uses as an other type of event
triggers and the luminosity monitor. The total energy (multiplicity) of neutrons measured
by the ZDCs is to be correlated with event geometry therefore the information is useful todetermine the centrality in each events.

CHAPTER 2. THE PHENIX EXPERIMENT

34

Figure 2.7: (a) Top view of the beam geometry and ZDC location. (b) A picture of ZDC.

CHAPTER 2. THE PHENIX EXPERIMENT

35

2.5 Central arms detectros
2.5.1

Drift Chambers (DCs)

The main role of DCs is to measure charged particle’s momentum and provide position
information that is used to link tracks through the various PHENIX detector subsystems.
The DCs are located in the region from 2 to 2.4 m from the z axis and 2 m along the beam
direction.
In order to have good tracking efficiency for the highest multiplicities at Au-Au collisions, the DC is designed to satisfy the below requests.
• single wire resolution better than 150 in r-φ• single wire two track separation better than 1.5 mm.
• single wire efficiency better than 99%• spatial resolution in the z direction better than 2 mm.
Each DC consists of two independent gas volumes located in the west and east arms. A
gas mixture of 50 % Ar and 50 % Ethan is chosen for the operation based on (1) uniform drift
velocity at the electric field E ∼ 1 kV/cm, (2) high gas gain (3) low diffusion coefficients.
The volume of the each detector is defined by a cylindrical titanium frame defining theazimuthal and beam-axis limits of the detector volume. Each frame consists of 20 identical
keystones. The lay out of one of the keystone is shown in Fig. 2.8. Each keystone has six
types of wire modules: X1, U1, V1, X2, U2 and V2. The modules contain 4 sense planes
and 4 cathode planes forming cells with a 2-2.5 cm drift space in the φ direction. The X1
and X2 wire modules give the track measurements in r − φ . In addition, the modules have
two sets of small angle U,V wire planes used in the pattern recognition. U1, V1, U2, andV2 measure the z coordinate. These wires have stereo angles of 6 relative to the X wires tominimize track ambiguities by matching the z resolution of the pad chamber.
Following performances are obtained.
• single wire efficiency 95-98 %• spatial resolution 100-120 mkm• angular resolution d α /α ∼ 1 mrad

CHAPTER 2. THE PHENIX EXPERIMENT

36

Figure 2.8: (left) The layout of wire position of DCs. (right) Constriction of a DC frame.

CHAPTER 2. THE PHENIX EXPERIMENT

2.5.2

37

Pad Chambers (PC)

PCs are multi-wire proportional chambers which consist of a single plane of wires insidea gas volume bounded by two cathode pad planes. One cathode pad is separated into anarray of pixels. Nine pixels are connected together electronically to form the pad, the basic
element of the detector. A cell is defined by three pixels and the area is 0.84 × 0.84 cm2 .
This provides a position resolution of ± 1.7 mm along z and 2.5 mm in r − φ .
The central arms have three layers of PCs (called PC1, PC2 and PC3). PC1 is locatedbetween the DC and RICH on the East and West arms. Also on both arms, PC3 is installed
just in front of EMCal. PC2 behind the RICH is in West arm only. PC1 determined thethree-dimensional momentum vector by providing the z coordinate at the exit of the DC.
The information of the DC and PC1 provides direction vectors through the outer detectorsof subsystems (RICH, PC2, PC3, EMCal and TOF). PC2 and PC3 are used to reject particles
produced by either secondary interactions and particle decays outside the aperture of the DC
and PC1 or low-momentum primary tracks that curve around PC1 in the magnetic field and
hit PC2 and PC3. The recognition of three points on a track through the whole detector
ensures that the response from electron identifying detectors (RICH and EMCal) and themomentum from the DC is correlated for particle identification.

2.5.3 Ring Imaging Cherenkov Counters (RICHs)
The RICH is the primary device for identifying electrons among a very large number ofcharged particles. The basic idea of RICH is a Cherenkov radiation. Cherenkov radiationarises when a charged particle in a material medium moves faster than the speed of light inthat same medium. The speed of light in a medium is given by
v = c/n,(2.1)

where c is light velocity and n is the index of the medium. Therefore a particle emits
Cherenkov radiation when it has a velocity
v particle > c/n.

(2.2)

Cherenkov photons are emitted as a circular cone shape with an angle θc defined as;cos θc =

1
βn

(2.3)

CHAPTER 2. THE PHENIX EXPERIMENT

38

Figure 2.9: A picture of the Pad Chamber system in PHENIX.
where β is v particle /c and it is a threshold for Cherenkov radiation. A particle which has afaster velocity than the light emits photons, and the emitted photons are optically focusedon the photon detectors, on which Cherenkov photons are detected on a ring in the RICH.
Figure 2.10 shows a cutway drawing of the RICH. Each RICH has a volume of 40 m3 ,with an entrance window area of 8.9 m2 and an exit window area of 21.6 m2 . Each detector
contains 48 composite mirror panels which are formed two intersecting spherical surfaces.
The total reflecting area is 20 m2 . The reflectivity of the mirrors is about 83 % at 200 nm,
rising to 90 % at 250 nm. Cherenkov light radiated in the RICH is focused onto two arrays of1280 UV photomultipliers (Hamamats H3171S). The phototubes are fitted with 2 diameter
Winston cones and have magnetic shields. The phototube UV glass windows absorb photons
of wavelengths below 200 nm. Each phototube array is located on either side of the RICH
entrance window.
During Run4, CO2 (n = 1.000410) was used as the Cherenkov radiator. It has a pion
Cherenkov threshold of 4.65 GeV/c and produces an average of 12 photons per ring for
β = 1 particle. The ring diameter for CO2 gas is about 11.8 cm.
The π /e discrimination capabilities of the RICH are determined by three factors.
• the value of the pion Cherenkov threshold

CHAPTER 2. THE PHENIX EXPERIMENT

39• the statistical fluctuations in the number of photoelectrons produced by an electron in
RICH• the background counting rates in the RICH
The combination of the RICH and EMCal allows a separation of π /e to less than 10−4 at
p < 4.7 GeV/c.

Figure 2.10: A cutaway view of one arm of the PHENIX RICH detector.

CHAPTER 2. THE PHENIX EXPERIMENT

2.5.4

40

Electro-Magnetic Calorimeters (EMCals)

The EMCal is used to measure the spatial position and energy of electrons and photons.
It covers the full central arm acceptance of 70◦ < θ < 110◦ with each of the two walls
subtending 90◦ in azimuth. There are eight sectors of the EMcal in the East and Westarm. The West arm has four sectors of Pb-scintillator sampling calorimeter, and the Eastarm has two sectors of Pb-scintillator and two of Pb-glass Cherenkov calorimeter. The
Pb-scintillator electromagnetic calorimeter is a shashlik type sampling calorimeter made ofalternating tiles of Pb and scintillator consisting of 15,552 individual towers. Figure 2.11
shows a Pb-scintillator electromagnetic calorimeter module consisting of four towers. The√
Pb-scintillator electromagnetic calorimeter has a nominal energy resolution of 8.1 %/ E
(GeV)⊕2.1%. A Pb-glass calorimeter sector consists 192 super modules. Each super module comprises 24 Pb-glass modules. The Pb-scintillator electromagnetic calorimeter has a√nominal energy resolution of 6 %/ E (GeV).
The EMCal is designed to identify and measure the total energy of electrons and photons.
Electrons deposit all of their energy in the the EMCals , therefore the energy E measured
by the EMCasl and the momentum p measured by the DCs should match, i.e. E/p 1.0 for
electrons. On the other hand hadrons deposit only a small fraction of their total energy inthe calorimeter.

2.5.5 Time of flight (TOF)
The TOF provides flight time of particles. Using the flight time of particles, the particles are
identified by calculating the mass as;
m2 = p2 ((T f light c 2) − 1),
L(2.4)

where L is a flight path length which is the distance between TOF and vertex position in themagnetic field and T f light is a flight of the particles. It is designed to have a timing resolutionof 100 ps in order to separate particles in the high momentum range, i.e. π /K separation up
to 2.4 GeV/c and K/p separation up to 4.0 GeV/c.
The TOF is located at a radial distance of 5.06 m from the interaction point, between
PC3 and the EMCal in the east arm, and it covers |η | < 0.35 and ∆ φ = 45 in azimuthal
angle. The TOF system consists of 10 panels of TOF walls and the wall consists of 96 TOF
elements. One TOF element consists a plastic scintillator slat and PMTs are equipped at
the both ends of the scintillator. The Schematic diagram of the components of a single TOF
panel are illustrated in Fig. 2.12.

CHAPTER 2. THE PHENIX EXPERIMENT

41

Figure 2.11: Interior view of a Pb-scintillator calorimeter module showing a stack of scintillator and lead plates, wavelength shifting fiber readout and leak fiber inserted in the centralhole.

CHAPTER 2. THE PHENIX EXPERIMENT

42

Figure 2.12: Schematic diagram of the components of a single TOF panel which consists of
96 plastic scintillation counters with PMTs at both ends, light guides and supports.

Chapter 3
Data analysis
The analysis of the elliptic flow for electrons originating from heavy quarks is based on the√experimental data of Au+Au collisions at sNN = 200 GeV in the fourth year of RHIC run(2004 ∼ 2005) at RHIC. About 700M minimum bias events are used in this analysis after
the vertex cut is applied. In this chapter, we present the event selection, track reconstruction,
electron identification, reaction plane determination and determination of the electrons fromheavy quarks.

3.1 Event selection
The collisions which meet following conditions are used in this analysis. The first conditionis the number of hits in the BBC and the ZDC. For the BBC, a coincidence between thenorth and south BBC with at least two PMTs fired in each BBC is required. For the ZDC,
at least one forward neutron has to be register in each of two ZDCs. In addition, a cut forthe collision vertex point along the beam line is also required. The collision vertex point isdetermined by the timing difference of the BBCs. A event vertex position along the beamaxis (Zvertex ) is reconstructed from the time information from BBCs as;
Zvertex =(T1 − T2 )c
,
2

(3.1)

where c is the light velocity, T1 and T2 are the observed values of arrival time for prompt
particles at the each side of BBC. The resolution is about ± 0.6 cm. Based on a simulation√of the BBC, the efficiency for above condition is 92±3 % in Au+Au sNN = 200 GeV.
We required |zvertex | < 20 cm to eliminate electrons originating from the conversions at thecentral magnets [42].

43CHAPTER 3. DATA ANALYSIS

44

3.2 Centrality determination
In collisions with non-zero impact parameter, charged particles produced by the collisions
are detected by the BBC, and spectator neutrons which do not participate in the collisions
are detected by the ZDC. For central collisions charged particles detected by the BBC are
more than spectator neutrons detected by the ZDC. On the other hand, spectator neutrons
measured by the ZDC are more than the charged particles detected by the BBC in peripheral
collisions. Therefore combining information of the spectator neutrons measured by the ZDC
and the charge sum information measured by the BBC provides the event centrality. Figure
3.1 shows the correlation between the charge sum of charged particles and the energy sumof spectator neutrons. For more peripheral collision, the spectator neutrons measured by
the ZDC are decreased, i.e. spectator neutrons miss the ZDC. It is due to the intrinsic pT
from their Fermi motion inside the Au nuclei or they may bound in deuterons of heavier
fragments and thus swept away by the magnet. It has been reported by the NA49 experimentthat such spectator neutrons missing has an approximately liner relationship as a function of
the impact parameter [45], and the missing is larger in peripheral collisions. The lines in the
figure present the definition of centrality classes in the ZDC vs. BBC plot. The centrality
classes are obtained by calculating the angle φ of a given (BBC,ZDC) point with relative
to (BBC0 ,ZDC0 ) = (0.2,0.0) on the ZDC vs. BBC plane. The full φ range which covers
centrality 0 % to 92 % is divide to 92 with the same number of counts in each bins. The
first bin is defined as centrality 0 %, therefore the n th bin represents centrality n − 1 %.
Thus we obtain the one-to-one correspondence between φ cut and the centrality ranges inpercentages.

3.3 Track Reconstruction and momentum determination
Charged particle tracks are reconstructed by using the hit information at the DCs (X1, X2,
UV1 and UV2) and the PC1. In the process of the reconstruction, tracks that traverse both
X1 and X2 wires are looked then the remaining tracks that traverse the X1 or X2 regions
are looked. Using a combinatorial Hough transform (CHT), tracks are found from the hits
information of X1 and X2 wires. In this method all possible combinations of hits at X1 andX2 wires are transferred into a feature space defined by the polar angle at the intersectionof the track with a reference radius near the mid-point of the drift chamber, φ , and theinclination of the track at that point α which is proportional to the inverse of the transverse
momentum. Figure 3.2 provides a schematic illustration of these variables. Assuming
tracks are straight line in the DCs, tracks are appear as peaks in the Hough transform feature

CHAPTER 3. DATA ANALYSIS

45

Figure 3.1: Correlation between the charge sum measured by BBC and energy of spectator
neutrons measured by the ZDC. The lines in the figure present the definition of centrality
classes in the ZDC vs. BBC plot.

CHAPTER 3. DATA ANALYSIS

46

space, because all hit pairs for a given track will have same φ and α .

Figure 3.2: Definition of the Hough transform parameters (φ and α ) for the drift chambertrack reconstruction. The X1 and X2 hits in the drift chamber are shown as small circles.
After the reconstruction of the track by using CHT method, then the z coordination of
tracks is determined by using PC1 and UV wires. If there is a PC1 association, then astraight line connecting zver and PC1 z fixes the direction of the track in z. If there is morethan one PC1 association, the one with more associated UV hits is accepted to be the correct
track. As a result of the reconstruction, variety possible tracks are made. These tracks are
identified by the quality bits which are defined by the hit information at X1, X2, UV and
PC1.
Tracks reconstructed by the DC-PC1 are then associated outer detectors (detectors afterPC1). These detectors provide hits information at the detectors and the one with the closed
distance to the track intersection point is identified as the hit associated with the track (track
matching). Then the distance in φ and z directions between the track projection point andthe associated hit position are calculated. In this analysis, following the track matching isrequired.
q2• EMCdz2electrons + EMCd φelectrons< 2.0,

CHAPTER 3. DATA ANALYSIS

47

here EMCdzelectrons (EMCd φelectrons ) is a 1 σ deviation of the distance in z (φ ) direction
between the track projection point and the associated hit position at the EMC, after electron
identification (See Chapter 3.4).
Momentums for charged particles are determined by measuring the track’s angular deflections from a straight line ,α . The α measured in the drift chamber is closely related to
the field integral along the track trajectory. For tracks emitted perpendicular to the beamaxis, this relation can be approximated by

α=

K
,
p

(3.2)

where K=101 mrad GeV is the field integral.

3.4 Electron identification
After reconstructing the tracks, electrons are identified with following processes. First the
RICH PMTs associated with the track were searched. This is done by first reflecting the
track about the RICH mirror as if it were a light ray, and then projecting the reflected trackon the RICH PMT. The projection point is the center of the ring which was made by the
Cherenkov radiation, therefore PMT hits near the projection point in the region (3.4 cm< r < 8.4 cm) are searched (the expected ring radius is 5.9 cm). The region is determinedfrom the position resolution of the PMTs. If the hits are found, following parameters arecalculated from the projection point and PMT hit information.
• Distance between ring center and track projection (disp)• Number of hit PMTs (n0)• Difference from the expected ring shape (chi2/npe0)
The minimum and maximum values of these parameters for electron identification are summarized in Table 3.1. In addition to the RICH parameters, following a parameter is calculated for the electron identification.
• shower shape in EMC (prob)
To reduce the background from hadrons and photon conversions far from the vertex, energy measured in the EMCal and momentum matching (E/p) is required. Electrons deposit
all of their energy in the EMCal; therefore the E/p is approximately 1.0 though the E/pof hadrons have a mip peak at 〜 0.3 by hadronic interactions in EMCal. The momentum

CHAPTER 3. DATA ANALYSIS

48

Figure 3.3: an illustration of electron identification by RICH. (a) Cherenkov photons are
emitted by the electrons at RICH. (b) The Cherenkov photons are reflected by the mirrors
and make a circle with 5.9 cm radius on the PMT arrays.
of electrons produced by the photon conversion outer of the vertex are overestimated their
momentum, therefore the E/p is smaller than 1.0. Fig. 3.4 shows the Energy and momentum matching ((E − p)/p/σ ). Here the σ means a standard deviation of (E − p)/p. In thisanalysis, we require E/p matching is larger than −2σ . The dashed line in the figure is the
background caused by the accidental association of tracks with RICH hits. The backgroundis estimated by the ”flip-and-swap” technique;• destroy position correlations between tracks and rings bynates

inverting

RICH coordi-• perform electron ID cuts with inverted RICH coordinates
The swapped background shape of E/p distributions agrees with the distribution for theidentified hadrons (Fig.3.5) [46]. Figure 3.6 shows the pT dependence of the ratio of theCHAPTER 3. DATA ANALYSIS

49

Type

parameter

Event

Event vertex |bbcz| < 20cm

Track matchingq2< 2.0σ
EMCdz2electrons + EMCd φelectrons

RICH

n0 ≥ 3chi2/npe0 < 10.0
disp < 5.0

other−2.0 < E−pp /σprob > 0.01

Table 3.1: Summary of electron ID
signal and background. As shwon in Fig. 3.6, the background is about 18 % at pT = 0.5
GeV/c and the contribution is getting smaller at higher pT region.

CHAPTER 3. DATA ANALYSIS

pt v.s. EoP vertex 20

50

pt v.s. EoP vertex 20

70000

0.5<pT<0.6

60000

45000
40000

pt v.s. EoP vertex 20

0.6<pT<0.7

30000

pt v.s. EoP vertex 20

0.7<pT <0.8

18000

25000
16000

35000
50000
30000
40000

14000

20000

12000

25000
15000

10000

20000

30000

8000

15000

10000

6000

20000
10000

4000

5000

10000

-2 -1 00.8<pT<0.9

20000

2000

5000

1 2

3

pt v.s. EoP vertex 20

0.9<pT<1.014000-2 -1 01 2

3

-2 -1 0pt v.s. EoP vertex 20

pt v.s. EoP vertex 20

18000

9000

16000

1.0<pT<1.2

8000

12000

14000

7000

10000

12000

6000

8000

10000

5000

8000

4000

6000

3000

4000

2000

2000

1000

1 2

3

-2 -1 01 2

3

pt v.s. EoP vertex 20

1.2<pT <1.4

6000

1.4<pT<1.7

5000
4000

3000

6000
2000

4000
1000

2000

-2 -1 01 2

3

pt v.s. EoP vertex 20-2 -1 01 2

3

pt v.s. EoP vertex 20

2500

1.7<pT<2.0

1400

1 2

3

pt v.s. EoP vertex 20

2.0<pT<2.5

1200

2000

0

-2 -1 0
450
400

300

800

250

1000

3



2.5<pT <3.0

180
160

3.0<pT<3.6

140
120

1500

100

200

80

150

60

100

40

200

50

20

0

0

600

1 2

pt v.s. EoP vertex 20

350

1000

0

-2 -1 0

400
500

0

-2 -1 01 2

3

-2 -1 01 2

3

-2 -1 00

1 2

3

-2 -1 01 2

3

pt v.s. EoP vertex 20100


3.6<pT<5.0

80

60

40

20

0

-2 -1 0

1 2

3

Figure 3.4: (E − p)/p/σ distributions in different pT slices. The red line is for electrons
after subtracting background (blue line) from electrons identified by RICH (black line).

CHAPTER 3. DATA ANALYSIS

51

Figure 3.5: E/p distributions for electrons (black solid line), swapped background (dottedline), and hadrons (red solid line). The figure is taken from [46]

Bg/(Sig+Bg)

CHAPTER 3. DATA ANALYSIS

52

0.18
0.16
0.14
0.12
0.1
0.08
0.06
0.04
0.02
0



0.5

1

1.5

2

2.5



3

3.5

4
4.5
pT (GeV/c)

Figure 3.6: Signal to background ratio (B/(S + B)) for electron signals as a function of pT .

CHAPTER 3. DATA ANALYSIS

53

3.5 Azimuthal anisotropy analysis
3.5.1

Fourier Expansion of Azimuthal Distribution

The azimuthal anisotropy has been studied by using Fourier expansion of the azimuthal
distributions. The azimuthal distribution of the produced particles, r(φ ), can be written in
the form of Fourier expansion [47];r(φ ) =

x0 1 ∞+ ∑ [xn cos(nφ ) + yn sin(nφ )],2π π n=1(3.3)

For the case of a finite number of particles, the integrals become simple sums over particles:
xn =
yn =

Z 2π
0

Z 2π
0

r(φ ) cos(nφ )d φ = ∑ ri cos(nφi ),(3.4)

r(φ ) sin(nφ )d φ = ∑ ri sin(nφi ),

(3.5)

i

i

where i runs over all particles generated by collisions, and φi is the azimuthal angle of i-th
particle. Here two variables, vn and ψn , are defined as;qvn = xn2 + y2n ,
(3.6)
yn
)/n.
(3.7)
xn
Here vn means the strength of the n-th harmonic and ψn is called ”reaction plane” corresponding to the direction of the nuclear impact parameter in a given collision (Fig. 1.6).
Using the azimuthal angle of the reaction plane (ψn ) and strength of n-th harmonic (vn ),the coefficients xn and yn are written as;

ψn = tan−1 (xn = vn cos(nψn ),(3.8)

yn = vn sin(nψn ).

(3.9)

From Eq. 3.8 and 3.9, Eq. 3.3 is written by using the vn and ψn ;x0 1 ∞+ ∑ [xn cos(nφ ) + yn sin(nφ )]r(φ ) =
2π π n=1==

x0 1 ∞+ ∑ [vn cos(nψn ) cos(nφ ) + vn sin(nψn ) sin(nφ )]2π π n=1
x0 1 ∞+ ∑ [vn cos(n(φ − ψn ))].
2π π n=1(3.10)(3.11)(3.12)

Especially the first harmonic (n=1) and second harmonic (n=2) coefficients called “directedflow” and “elliptic flow”, respectively.

CHAPTER 3. DATA ANALYSIS

3.5.2

54

Reaction Plane method

The azimuthal angle distribution of the particle emission measured with respect to the reaction plane is called the “reaction plane method”. In the reaction plane method the azimuthal
anisotropy is defined by½¾
dN= N0 1 + ∑ 2vn cos(n(φ − ΨR.P. )) ,(3.13)
dφnwhere N0 is a normalization constant, φ is the azimuthal angle of particles, and ΨR.P. is theazimuthal angle of the reaction plane. The harmonic coefficients, vn , indicate the strength of
the nth anisotropy. The value of vn is calculated by
vn = hcos(n(φ − ΨR.P. ))i.

(3.14)In experiment the true reaction plane is not able to determine directly, therefore the reaction
plane is determined by using the anisotropy flow itself [48]. It means that the event plane
can be determined independently for each harmonic of the anisotropic flow. Figure 3.7 is an
illustration of the reaction plane determination. The flow vector Qn and the reaction planeangle Ψ are defined as
Qn cos(nΨn ) = ∑ wi cos(nφi ),(3.15)i

Qn sin(nΨn ) = ∑ wi sin(nφi ).

(3.16)i

Therefore the azimuthal angle of the reaction plane for the nth harmonic is determined by
µ
¶
meas.
−1 ∑i wi sin(nφi )
ψn
= tan
/n,
(3.17)∑i wi cos(nφi )where φi is the azimuthal angle of each particle used in the reaction plane determination andwi is the corresponding weight. Due to the finite reaction plane resolution, coefficients in
the Fourier expansion of the azimuthal distribution with respect to the “measured” reaction
plane (vmeas.
) are smaller than coefficients measured with respect to the “real” reaction planen(vn ). The relation between v2 measured with respect to the “measured” reaction plane and“real” reaction plane isvmeas.
=< cos(n(φi − ψmeas. )) >n=< cos(n(φi − ψmeas. + ψtrue − ψtrue )) >=< cos(n(φi − ψtrue ) cos(n(ψmeas. − ψtrue )) >+ < sin(n(φi − ψtrue ) sin(n(ψmeas. − ψtrue )) >= vn < cos(n(ψmeas. − ψtrue )) > .

(3.18)

CHAPTER 3. DATA ANALYSIS

55

Figure 3.7: Illustration of the reaction plane determination. The true reaction plane can notmeasure directly therefore the plane is determined by using anisotropic flow itself.

Therefore the value of vn is given as;vn =vmeas.
n.
< cos(n(ψmeas. − ψtrue )) >(3.19)The value of < cos(n(ψmeas. − ψtrue )) > is characterized as the “resolution” of the reaction
plane [47].

3.5.3Reaction Plane Determination

In this analysis the v2 is estimated by using the reaction plane determined by the second
harmonic (n = 2), since a better accuracy of vn is obtained by using the same harmonic’s
reaction plane [48]. The reaction plane is determined by using both BBCs. In the PHENIX
experiment the reaction plane is also determined by using the central arm detectors. One of
the key issues of the reaction plane determination is non-flow effects such as jets, resonancedecays and HBT. This can be avoided by taking two remote rapidity intervals [49]. Since
each BBC is roughly three units of pseudo rapidity away from the central arms, it is expectedthat the non-flow effects are smaller there than that in the central arm detectors [19].
Using the BBC information, the reaction plane is given by
µ
¶
−1 Qy
ψ = tan/2,(3.20)Qx

CHAPTER 3. DATA ANALYSIS

5664

Qx = ∑ qi cos(2φi ),(3.21)i=1
64

Qy = ∑ qi sin(2φi ),(3.22)

i=1

where φi is the azimuthal angle of each PMT and qi is the charge information of each PMT.
Due to the random distribution of the impact parameter direction in the collisions, the reacψsouth

bbcn_bbcs
1.5

18000

1

17500
0.5
17000
0
16500
-0.5
16000
-1
15500
-1.5
-1.5

-1

-0.5

0

0.5

1

1.5
ψnorthFigure 3.8: Azimuthal angle correlation of the measured reaction plane North side and Southside of BBC.
tion plane should have an isotropic azimuthal distribution. Because of the possible azimuthal
asymmetries in the BBC response, however, the measured reaction plane distribution is
anisotropic. We use the following two step methods to correct the reaction plane. First the
distribution of Qx and Qy are recentered by subtracting hQx i and hQy i over all events:=
Qcorr.
xQx − hQx i,
σ Qx(3.23)

Qcorr.
=yQy − hQy i,
σQy(3.24)

here hQx,y i is the mean of Qx,y and σQx ,Qy is the width of Qx,y . This method does notremove higher harmonic components for the reaction plane determination, and we apply an

CHAPTER 3. DATA ANALYSIS

57

additional correction method, the reaction plane flattening [50]. In this method, a flattening
reaction plane is accomplished by using a shift,nψ flat = nψobs. + ∆ψ ,(3.25)

where ∆ψ is the correction factor for the reaction plane. ∆ψ is determined by∆ψ = ∑ An cos(2nψobs. ) + Bn sin(2nψobs. ).

(3.26)n

An and Bn are defined by requiring the nth Fourier moment of the new reaction plane (ψ flat )to vanish,2
An = − hsin(2nψobs. )i,n2
Bn =hcos(2nψobs. )i.
n(3.27)(3.28)

Figure 3.8 shows the azimuthal angle correlation of the reaction plane between measured
North side and South side of BBC. The plane is after applying the flattening corrections. A
combined reaction plane, which is defined by weighted averaging the reaction plane angles
obtained by the south side BBC and the north side BBC, is used to measure the v2 in thisanalysis.

3.5.4

Reaction Plane Resolution

As described in 3.5.2, v2 measured with respect to the “measured” reaction plane is correctedwith the reaction plane resolution (Eq. 3.19). The true reaction plane can not be measured
directly, therefore the reaction plane resolution is necessary. The reaction plane resolution[48, 49] is expressed as;√π< cos(n(ψmeas. − ψtrue )) >= √ χm exp(−χ 2 /4)[I0 (χm2 /4) + I1 (χm2 /4)].
(3.29)
2 2√
where χm = vm /σ (= vm 2N ) and I is the modified Bessel function. Figure 3.9 is shown
the centrality dependence of the combined reaction plane resolution.

CHAPTER 3. DATA ANALYSIS

58



σR.P.

0.5



0.45


0.4


0.35


0.3


0.25


0.2


0.15


0.1


0.05


0
0




10

20



30

40



50



60

70



80

90
100
centrality

Figure 3.9: Centrality dependence of the combined reaction plane resolution that is determined by the BBCs.

CHAPTER 3. DATA ANALYSIS

59

3.6 Determination of non-photonic electron at PHENIX
3.6.1

Electron sources

As shown in Fig. 3.4, clear signals of electrons are obtained. The electrons produced by thecollisions come from various sources.
1. Dalitz decays of π 0 , η , ω , φ
2. Di-electron decays of ρ , ω , φ
3. photon conversion
4. thermal di-leptons
5. Kaon decays (Ke3 )6. Heavy flavor decays (charm and beauty)
In this analysis, electrons from sources (1)-(4) are called “photonic” electron and (5),(6) arecalled “non-photonic” electron. The sources (1)-(5) are considered to be background. The
key issue of the charm study via electron is the background subtraction. PHENIX has used
two independent techniques “cocktail” [51] and “photon converter” [42] methods to subtractthe electron backgrounds and get electrons from heavy flavor decays.

3.6.2 Cocktail method
In the cocktail method, background electrons are determined as a cocktail of electrons frombackground sources. Those background electrons are calculated with a Monte Carlo event
generator of hadron decay. The most important background is the π 0 decay (π 0 Dalitz andphoton conversion from π 0 decay). For the study, the measured π 0 pT spectra by PHENIX
was used as an input for the generator. The other light mesons’s contribution, η , ρ , ωand φ , are also calculated. The spectral shapesof those light hadrons (h) are obtainedqfrom the pion spectra by mT scaling (pT →p2T + Mh2 − Mπ2 ). In addition, spectra of lightmesons are normalized respected to the π 0 as: η /π 0 = 0.48 ± 0.03 [52], ρ 0 /π 0 = 1.0 ± 0.3,
ω /π 0 = 0.90 ± 0.06 [53], φ /π 0 = 0.25 ± 0.08. The background from η is the second largest
contribution and the spectra determined by the above method is consistent with measured η
spectra at PHENIX above 2.0 GeV/c. The internal and external conversion of direct photons
are included in the cocktail using the meausred direct photon spectrum at PHENIX. Kaonbackgounds in non-photonic electrons are also included in the cocktail. The backgrounds

CHAPTER 3. DATA ANALYSIS

60

are obtained by using measured Kaon spectrum. Figure 3.10 shows the background electron
spectrum which is calculated by the hadron decay generator.

Figure 3.10: pT distribution of the background electrons [54][55]. The background electrons
are calculated by the hadron decay generator.

3.6.3 Photon converter method
In the photon converter method, electrons from photonic sources are experimentally determined by using a photon converter. The photon converter is a brass which has 1.7 % radiation length and is installed near the interaction point. The converter increases the yields ofbackground by a fixed factor (Rγ ∼ 2.3), therefore we can separate signal and backgroundelectrons by comparing the electron spectra with and without the converter. The yield ofelectrons with and without converter can be written as,
Neconv−in = Rγ Neγ + Nenon−γ ,Neconv−out = Neγ + Nenon−γ ,(3.30)

where Rγ is the ratio of the number of photonic electrons with and without the converter.
γ
The value of Ne is the number of photoni electrons which come from Dalitz decays of

CHAPTER 3. DATA ANALYSIS

61

Figure 3.11: Invariant yield specra of the photonic electron. The curve in the figure is the
specra from cocktail method and the closed circles is from the conveter method.
non−γ

light neutral mesons and photon conversions, and Ne
is the number of the non-photonic
electrons mainly from heavy flavor decays. From Eq. 3.31 the yield of the photonic electron
is obtained as;
N conv−in − Neconv−out.
(3.31)Neγ = eRγ − 1√Figure 3.11 shows the invariant yield of the photonic electrons in Au+Au collisions at sNN= 200 GeV in the minimum bias events. The curve in the figure is the specra from cocktail
method. The photonic electron spectrum obtained both methos are consistent. Thus photonic
backgrounds in the PHENIX is well understood.

CHAPTER 3. DATA ANALYSIS

3.6.4

62

Ratio of the non-photonic to photonic electrons RNP

1 d2N (GeV/c)-22πpTdpTdy

The yield of non-photonic electrons is determined by subtracting the yield of photonic elecγ
tron from the inclusive electron yield, Neconv−out − Ne . Figure 3.12 shows the invariant yieldof the non-photonic electron [55]. At low pT (pT < 1.6 GeV/c), photonic electrons determinded by the converter method was subtracted, and determinded with cocktail method
was subtracted at high pT . Figure 3.13 shows the transverse momentum dependence of thenon−γγ/Ne ). Above pT = 1.5 GeV/c,ratio of photonic and non-photonic electron (RNP = Ne
RNP is larger than 1.0. It means that more than 50 % of electrons are originated from thenon-photonic sources above pT = 1.5 GeV/c at PHENIX.
1

10-1

10-2

10-3

10-4

10-5

10-6

10-7

10-8
0

0.5

1

1.5

2

2.5

3

3.5

4
4.5
5
pT (GeV/c)

Figure 3.12: Invariant yield of the non-photonic electron measured in Au+Au collisions at√
sNN = 200 GeV [55].

CHAPTER 3. DATA ANALYSIS

63

R NP

4


3.5


3
2.5
2
1.5
1


0.5


0
0






0.5

1

1.5

2

2.5



3

3.5

4



4.5
5
pT (GeV/c)

Figure 3.13: Transverse momentum dependence of the ratio of the number of non-photonic
non−γ γ/Ne ).
electrons to the number of photonic electrons (RNP = Ne

Chapter 4
Experimental Results
In this chapter, we present a method to calculate the non-photonic electron v2 and show its√
transverse momentum dependence at mid-rapidity in sNN = 200 GeV Au+Au collisions.

4.1 Inclusive Electron v2
The measurement of inclusive electron v2 is the first step for the non-photonic electron v2
determination. Using the measured reaction plane with the BBCs, the inclusive electron v2
was calculated from Eq. 3.14, v2 = hcos(2(φ − Ψmeas.
R.P. ))i. As is described in the previous
chapter, v2 measured with the reaction plane determined by experiment is smeared due tothe finite reaction plane resolution. In this analysis, the inclusive electron v2 is measured
with 10 % step of centrality bins and corrected for the reaction plane resolution as shown in
Fig.3.9. Figure 4.1 shows the transverse momentum dependence of the inclusive electron v2
measured with 10 % step of centrality bins. The v2 was already corrected for the reaction
plane resolution. As also described in the previous section, less than 10 % of the backgroundremains due to accidental RICH associations. Such background was subtracted as;ee
dNcanddNbackdN=−,d(φ − ΨR.P. ) d(φ − ΨR.P. ) d(φ − ΨR.P. )

(4.1)

e
e
where Ncand
is the number of electrons identified by RICH and Nback
is the number of the
backgrounds. The number of the backgrounds are obtained by ”flip-and-swap” techniquedescribed in Chapter 3.4. The transverse momentum dependence of the inclusive electron
v2 in Fig. 4.1 is obtained after subtracting the backgrounds. The inclusive electron v2 forthe minimum bias events is calculated by the weighted average of the v2 measured with 10

64

CHAPTER 4. EXPERIMENTAL RESULTS

65% step of centrality bin as;
vmim.bias(pT ) =
2∑cent w(cent, pT )v2 (cent, pT ),∑cent w(cent, pT )(4.2)

here v2 (cent, pT ) means the v2 measured with 10 % step of the centrality bin and wcent (pT )is the corresponding weight. The weight is determined by using the electron signals in eachpT bin, after the background subtractions. Figure 4.2 shows the inclusive electron v2 for theminimum bias event.

CHAPTER 4. EXPERIMENTAL RESULTS

10<centrality<20

0.25

v2

0<centrality<100.25

pt v.s. v2v2

pt v.s. v2v2

pt v.s. v2

66

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

-0.05

-0.05

-0.05

1

1.5

2

2.5

3

3.5

4

4.5

5

p (GeV/c)

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

p (GeV/c)

pt v.s. v2v2

pt v.s. v2-0.1
030<centrality<40

0.25

-0.1
0

40<centrality<50

0.25
0.2

0.2
0.15

0.1

0.1

0.1

0.05

0.05

0.05

0

0

0

-0.05

-0.05

-0.05

2

2.5

3

3.5

4

4.5

5

p (GeV/c)

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

p (GeV/c)

60<centrality<70

0.25

-0.1
0

70<centrality<80

0.25

0.2

0.15

0.15

0.15

0.1

0.1

0.1

0.05

0.05

0.05

0

0

0

-0.05

-0.05

-0.05

1

1.5

2

2.5

3

3.5

4

4.5

5

p (GeV/c)

-0.1
0

0.5

1

1.5

2

2.5

3

3.5



0.2

0.5

3.5

4

4.5

5

p (GeV/c)50<centrality<60

0.25

0.2

-0.1
0

3

4

4.5

5

p (GeV/c)

pt v.s. v2v2

pt v.s. v2v2

pt v.s. v2

-0.1
0

2.5

v2

1.5

2



0.15

1

1.5

0.25

0.2

0.5

1

pt v.s. v2

0.15

-0.1
0

0.5

v2

0.5

v2

-0.1
0

20<centrality<30

0.25

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

p (GeV/c)

-0.1
0

0.5

1

1.5

2

2.5

3

80<centrality<90

3.5

4

4.5

5

p (GeV/c)v2

pt v.s. v2


90<centrality<93

0.25
0.2
0.15
0.1
0.05
0
-0.05
-0.1
0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

p (GeV/c)

Figure 4.1: Transverse momentum dependence of the inclusive electron v2 measured with 10% step of the centrality bin. The v2 was already corrected for the reaction plane resolution.

CHAPTER 4. EXPERIMENTAL RESULTS

v2

67



0.25

inclusive electron v2
Minimum bias (cenrality ; 0 - 92 %)



0.2


0.15


0.1


0.05


0
-0.05
0






0.5

1

1.5

2

2.5



3

3.5

4



4.5
5
pT(GeV/c)

Figure 4.2: Inclusive electron v2 for the the minimum bias events as a function of pT .

CHAPTER 4. EXPERIMENTAL RESULTS

68

4.2 Non-photonic electron v2
4.2.1

Method of non-photonic electron v2 determination

The azimuthal distribution of electrons (dN e /d φ ) is the sum of the azimuthal distributionsof photonic electrons (dN γ /d φ ) and non-photonic electrons (dN non−γ /d φ ):
γ

non−γ

dNe
dNe dNe
=
+
dφ
dφ
dφ

.

(4.3)

The second harmonic of the Fourier expansion of each azimuthal distribution is given as;
´
´
³
³
γ
non−γ
cos(2φ )Ne (1 + 2v2e cos(2φ )) = Neγ 1 + 2v2e cos(2φ ) + Nenon−γ 1 + 2v2e
Ã!
γ γ
non−γ non−γ
N
+
N
v
v
e
e
2e
2e
= (Neγ + Nenon−γ ) 1 + 2
cos(2φ ) ,(4.4)γ
non−γ
Ne + Neγ

non−γ

where v2e is the v2 of the inclusive electron, v2e is the v2 of the photonic electrons and v2e
is the v2 of the non-photonic electrons. From Eq. 4.4, the relation between the number ofelectrons and the v2 are given as;
γ

non−γ

Ne v2e = Neγ v2e + Nenon−γ v2e
γ

non−γ

γ

.

(4.5)

non−γ

Here Ne is sum of Ne and Ne(Ne = Ne + Ne).
In this analysis the non-photonic electron v2 is calculated by two methods. The first
method calculates the photonic electron v2 as a cocktail of contributions from photonic
sources and subtracts it from inclusive data (Cocktail method). From Eq. 4.5, the nonphotonic electron v2 can be expressed asγ(1 + RNP )v2e − v2e
non−γ
=
,
v2e
RNP
non−γ

where RNP is difined as Ne
as;

(4.6)

γ

/Ne . In the cocktail method, the photonic electron v2 is given

NX→e
(4.7)
γ v2X→eNe
where X represents species of the parent particle, NX→e is the number of electrons decayedfrom the parent particle and v2X→e is the decay electron v2 . The decay electron v2 (v2X→e ) is
calculated by using the measured parent pT distributions and v2 .
The second method calculates the non-photonic electron v2 by using the inclusive electron v2 measured with and without the thin converter material (Converter method). The yieldγ

v2e = Σ

CHAPTER 4. EXPERIMENTAL RESULTS

69

of electrons with and without the converter can be written as,
Neconv−in = Rγ Neγ + Nenon−γ ,Neconv−out = Neγ + Nenon−γ ,(4.8)

where Rγ is the ratio of the number of photonic electrons with and without the converter.
From Eq. 4.8, the relation between the number of electrons and the v2 are given as;
γ

non−γ

Neconv−in vconv−in= Rγ Neγ v2e + Nenon−γ v2e
2e
γ

non−γ

Neconv−out vconv−out= Neγ v2e + Nenon−γ v2e
2e

,

,

(4.9)

where vconv−inis the inclusive electron v2 measured with the converter and vconv−out
is the2e2enon−γ
inclusive electron v2 measured without the converter. From Eq. 4.9, v2e
is obtained as;
Rγ (1 + RNP )vconv−out− (Rγ + RNP )vconv−in
non−γ
2e
2e
=
v2e
.
RNP (Rγ − 1)

(4.10)

γ

The photonic electron v2 , v2e , is also obtained as;
γ

v2e =

(1 + RNP )vconv−out− (Rγ + RNP )vconv−in2e2e(1 − Rγ ).

(4.11)

4.2.2 Determination of photonic electron v2 by the cocktail method
In cocktail method, the photonic electron v2 is calculated as a cocktail of decay electronv2 from photonic sources. The most largest contribution to the photonic electrons is π 0Dalitz decay. It is about 60% contribution for the photonic electrons. The second largest
contribution is γ conversion, mainly from π 0 → γγ , in material within the acceptance. The
contribution from γ conversion is about 30%. The other important component is η Dalitz
decay and direct γ decay. η Dalitz decay contribution is about 10% of the photonic component and the v2 is smaller than π 0 v2 at low pT , therefore it is important to determinedphotonic electron v2 at low pT region . Direct γ decay is negligibly small at low pT region
but more than 10% of the photonic electrons come from the direct γ at pT > 3.5 GeV/c. It
is larger contribution than η Dalitz decay at the high pT region. The reason why direct γ isalso important in the cocktail calculation is the direct γ v2 is much smaller than other v2 . In
this calculation, π 0 , η and direct γ decays were taken into account. The other component
is about a few %, therefore the contributions were ignored. The contribution of π 0 Dalitz, γ
conversion, η Dalitz decay and direct γ for photonic electron as a function of pT is shownin Fig 4.3. The contributions are calculated from Fig.3.10. The details of decay electron v2
calculation from π 0 , η and direct γ decays are described in following sections.

Relative ratio to photonic

CHAPTER 4. EXPERIMENTAL RESULTS

70

1
0.9
0.8
0.7
0.6

π0 → γ ee

0.5
0.4
0.3
0.2
0.1
0
0

γ conversion
η → γ eedirect γ contribution


0.5

1

1.5

2

2.5



3

3.5

4



4.5
5
pT (GeV/c)

Figure 4.3: Relative contributions of electron sources (π 0 Dalitz, γ conversion, η Dalitz anddirect γ ) to the photonic electrons. The data is obtained from Fig 3.10.

v2

CHAPTER 4. EXPERIMENTAL RESULTS

71

0.2

π0 v2
π0 → e v2


0.18



0.16
0.14
0.12
0.1
0.08
0.06
0.04
0.02
0
0



1

2



3

4



5

6
7
pT (GeV/c)

Figure 4.4: pT dependence of electron v2 from π 0 decay.
electron v2 from π 0
To calculate the decay electron v2 from π 0 , pT spectra and v2 for the parent particles areneeded. The PHENIX has been measured π 0 pT spectrum [56] and we used this spectrum
as an input for the π 0 simulation. The input π 0 v2 was obtained from combined measured
v2 for charged π [57] and π 0 [58]. The simulation result of the π 0 Dalitz decay is shownin Fig. 4.4. At low pT , the decay electron v2 is larger than the parent v2 . Due to the factthat the decay opening angle of the π 0 decay is small, the electron has about the sameazimuthal angle as the parent, while at the same time the electron pT is smaller than the π 0
pT . Therefore the electron v2 at a given pT corresponds to the larger v2 of the π 0 at higher
pT . Thus the electron v2 is larger than the parent v2 . The v2 of conversion electrons ofπ 0 decay photon should be almost exactly the same. Their v2 is originated from the v2 of
π 0 . And the angular correlation between the π 0 direction and electron direction should bealmost the same for Dalitz and the conversion photons. Therefore this result was used when
we calculated the electron v2 from the photon conversion.

CHAPTER 4. EXPERIMENTAL RESULTS

72

Component T f (MeV)
ρ0
ρa
s2
mean
168 ± 19 0.794 ± 0.026 0.067 ± 0.008 0.071 ± 0.007
upper sys. 140 ± 19 0.776 ± 0.026 0.056 ± 0.008 0.067 ± 0.007
lower sys. 225 ± 22 0.856 ± 0.026 0.094 ± 0.011 0.080 ± 0.007
Table 4.1: Blastwave fitting result

electron v2 from η Dalitz decay
For η → e simulation, the input η spectrum for the simulation was obtained by using themeasured η spectrum at PHENIX experiment [52]. The measurement
q is above 1.0 GeV/ctherefore low pT spectrum shape is evaluated by mT scaling (pT → p2T + Mη2 − Mπ2 0 ) of theπ 0 spectrum , and the absolute normalization is scaled to match the measured η spectrum.
In this analysis kaon v2 is used as input η v2 because η v2 has not been measured in RHIC.
The reason why kaon v2 is used as an input for η simulation is described following. At
low pT region, v2 shows the clear mass dependence [19]. Therefore we estimated the masseffect by using a simple hydrodynamical mode, Blast wave model, which well describes themeasured identified charged particle v2 at low pT region [20]. In the Blast wave model, v2
is given as
R 2π
d φb cos(2φb )I2 (αt )K1 (βt )(1 + 2s2 cos(2φb )),(4.12)
v2 (pT ) = 0 R 2π
0 d φb I0 (αt )K1 (βt )(1 + 2s2 cos(2φb ))where I0 , I2 and K1 are the modified Bessel functions, and αt (φb ) = (pt /T f ) sinh(ρ (φb )) andβt = (mt /T f ) cosh(ρ (φb )). ρ (φb ) is defined by ρ (φb ) = ρ0 + ρa cos(2φb ) where φb is theazimuthal angle. The parameters (ρ0 , ρα and s2 ) were obtained by fitting measured π , K,
p v2 at in PHENIX [19]. The fitting parameters are shown in Table 4.1. The result of η v2
calculated by the Blast wave model is shown as a red line in Fig. 4.5. The dashed lines are
uncertainty of η v2 which is calculated by the Blast wave model. It was obtained by fitting
the systematic uncertainties of π 0 , K and p v2 with Eq. 4.12. The mass effect between kaonand η v2 is small and consistent within the uncertainty. At high pT region, meson v2 can besame from the quark coalescence model prediction therefore η v2 assumes to be the samev2 as kaon v2 at high pT ration.
The transverse momentum dependence of charged kaon v2 has measured up to aroundpT = 3.0 GeV/c and KS0 v2 has measured up to around pT = 6.0 GeV/c [22]. The K and KS0
v2 is consistent around 3.0 GeV/c, therefore the combined kaon and KS0 v2 was used as an
input in this simulation. The result of electron v2 from η Dalitz is shown in Fig. 4.6.

CHAPTER 4. EXPERIMENTAL RESULTS

73

pT (GeV/c)v2

Figure 4.5: η v2 calculated by the Blast wave model. The dashed lines are the systematic
uncertainties from the systematic uncertainty of π , kaon, proton v2 .

0.22

η v2 (K v2)
η → e v2

0.2
0.18
0.16
0.14
0.12
0.1
0.08
0.06
0.04
0.02
0
0



1

2



3

4



5

6
7
pT (GeV/c)

Figure 4.6: pT dependence of electron v2 from η decay. η v2 was assumed same as kaon inthis calculation.

CHAPTER 4. EXPERIMENTAL RESULTS

74

Figure 4.7: Direct γ v2 as a function of pT measured at PHENIX. The closed circles areinclusive γ v2 and the open squares are background γ v2 mainly from π 0 decay. The open
circles corresponds to the direct γ v2 [59].

electron v2 from direct γ
As shown in Fig. 4.3, direct γ contribution is larger contribution than η Dalitz decay at highpT region. At pT > 3.5 GeV/c, more than 10% photonic electrons come from the direct
γ . The direct γ v2 is expected zero because they do not interact with medium created by
the collisions. The direct γ v2 has been measured at PHENIX [59]. Figure 4.7 shows theresult of direct γ v2 measurement at PHENIX. In the analysis direct γ v2 was obtained by
the same method, the cocktail method, of the non-photonic electron v2 measurement. First
inclusive γ v2 was measured then background v2 , γ v2 mainly from π 0 decay, calculated by
the simulation was subtracted from the inclusive γ v2 . The open symbols in the figure arecorresponded to the direct γ v2 . The current result of direct γ v2 has large error bars but theresult is consistent with zero. As shown previous in the simulations, the decay electron v2
correlates with the parent v2 . Therefore electrons from direct γ which has zero v2 , shouldhave also zero v2 .

CHAPTER 4. EXPERIMENTAL RESULTS

75

photonic electron v2
Using the decay electron v2 calculated from above simulations and the relative contributions
to photonic electron, photonic electron v2 was obtained. The result is shown in Fig. 4.8 aslines. The solid line is the mean of v2 and the dashed lines are 1 σ systematic uncertainof the cocktail method. The uncertainty from photonic electron v2 includes the systematic
uncertainty of all parent v2 for the calculation. The π v2 and K v2 has about 5 % uncertainty
without the reaction plane uncertainty. We apply this value to the uncertainty of the photonic
electron v2 . For direct γ v2 , we applied ∆v2 = ±0.1 for the uncertainty.

v2

0.25

photonic electron v2 (converter)
photonic electron v2 (cocktail))

0.2

0.15

0.1

0.05

0


0

0.5

1

1.5

2

2.5



3

3.5

4



4.5
5
pT (GeV/c)

Figure 4.8: pT dependence of photonic electron v2 obtained by the cocktail and converter
method. The lines on the figure are photonic electron v2 calculated by the cocktail method.
The solid line is the mean of v2 and the dashed lines are the systematic uncertainty of the
cocktail method. The photonic electron v2 from converter is shown as boxes.

CHAPTER 4. EXPERIMENTAL RESULTS

76

Comparison of photonic electron v2 by converter method
The photonic electron v2 was also determined by the converter method. Non-photonic electron and photonic electron v2 can be separated by using the inclusive electron v2 measured
with (vconv−in) and without (vconv−out) converter as;
2e
2e
non−γ

v2e

γ
v2e

=

Rγ (1 + RNP )vconv−out− (Rγ + RNP )vconv−in
2e
2e

=

(1 + RNP )vconv−out− (Rγ + RNP )vconv−in
2e2e(4.13)

RNP (Rγ − 1)
(1 − Rγ ).

(4.14)

v2

Figure 4.9 shows the inclusive electron v2 with/without the converter. If the photonic electron v2 and non-photonic electron v2 is same, the v2 measured with/without the converter
should be same. Due to the small statistics of the converter run, v2 measured with converter
has a large statistical uncertainty. But it seems like v2 measured with th converter is larger
than v2 measured without the converter. It indicates that non-photonic electron v2 is different
from the photonic electron v2 . The photonic electron v2 obtained by the converter method isshown as open boxes in Fig. 4.8. The lines on the figure are photonic electron v2 calculatedby the cocktail method. The photonic electron v2 obtained by the converter method has alarge statistical error due to the small statistics of the converter run but the photonic electron
v2 obtained by two methods are well consistent.



0.25

inc. electron (w/o converter)inc. electron (w/ converter)



0.2


0.15


0.1


0.05


0
-0.05
0






0.5

1

1.5

2

2.5



3

3.5

4



4.5
5
pT (GeV/c)

Figure 4.9: Inclusive electron v2 measured with/without converter (Minimum bias event).

CHAPTER 4. EXPERIMENTAL RESULTS

4.2.3

77

Kaon decay background subtraction

v2

Figure 4.10 shows the transverse momentum dependence of the non-photonic electron v2
after subtraction of the v2 of the photonic electron v2 . Due to the large statistical error of
the photonic electron meausred by the converter method, we used the v2 measured by the
cocktail method in this analysis. At low pT region the non-photonic electron contribution issmall, therefore the inclusive electron v2 is similar to the background v2 . On the other hand,
the inclusive electron v2 is getting similar to the non-photonic electron v2 with increasing
pT due to the good signal to background ratio at high pT .

inc. electron v2
non-photonic electron v2
photonic electron v2



0.25


0.2


0.15


0.1


0.05


0
-0.05
0






0.5

1

1.5

2

2.5



3

3.5

4



4.5
5
pT (GeV/c)

Figure 4.10: pT dependence of the non-photonic electron v2 (closed circles) together with
the inclusive electron v2 (open circles) and photonic electron v2 (solid line).
The background from kaon decays (K → π eν ) still remains in the non-photonic electron
v2 (Fig. 4.11). The contribution of kaon decays to the non-photonic electron yield is 18%
at pT = 0.5 GeV/c and decreases to less than 5% for pT > 1.0 GeV/c [42]. In the analysis,
the kaon decay contribution is also subtracted. The input kaon v2 is obtained from measured
kaon v2 and KS0 v2 as described in the η v2 simulation study. The input kaon spectrum for thesimulation is shown as a solid line in Fig. 4.12. At low pT we used measured kaon spectrum

78e

NK->e/Nnon-γ

CHAPTER 4. EXPERIMENTAL RESULTS



0.25



0.2



0.15



0.1



0.05



0
0






0.5

1

1.5

2

2.5



3

3.5

4



4.5
5
pT (GeV/c)

Figure 4.11: Contribution of kaon decays to the non-photonic electron yield. The contribution is 18% at pT = 0.5 GeV/c and decreases to less than 5% for pT > 1.0 GeV/c.

[60] as the input for the simulation. At high pT kaon spectrum has not been measured inheavy ion collisions, we assumed that the spctrum is same as π 0 spectrum. We scaled theπ 0 spectra to connect around pT = 2.0 GeV/c with the measured kaon spectrum. To study
the shape dependence of decay electron v2 , we also calculated the electron v2 assumed thekaon spectrum shape is an exponential function at pT > 2.0 GeV/c. The results are shownin Fig.4.13. The v2 which was calculated from the exponential shape is shown as blue points
and the scaled π 0 spectra is shown as red points. The difference between the decay electron
v2 in the different assumptions at high pT kaon spectrum is negligibly small.

1/(2πN)d /dp Tdy (GeV/c)-2

CHAPTER 4. EXPERIMENTAL RESULTS

10

79

2

2

10
1
-1

10

-2

10

-3

10

-4

10

-5

10

-6

10

-7

10

-8

10



0

1

2



3

4


5



6

7



8

9
10
pT (GeV/c)

Figure 4.12: Input kaon pT spectra for the kaon decay study simulation (the solid line). The
spectrum at pT > 2.0 GeV/c we assumed that the shape is same as π 0 spectrum. The pT
spectrum is obtained by the scaled π 0 spectra connected around pT = 2.0 GeV/c to the low
pT kaon spectrum with the scale factor 0.55.

pT (GeV/c)

Figure 4.13: pT shape dependence of electron v2 from kaon decay. The v2 which is calculated from the exponential shape (the dashed red line in Fig. 4.12) is shown as red points
and the scaled π 0 spectra (the blue line in Fig. 4.12) is shown as blue points. No shapedependence is seen in the electron v2 .

CHAPTER 4. EXPERIMENTAL RESULTS

80

4.3 Systematic uncertainties
The systematic uncertainties of non-photonic electron v2 are summarized below.
• Reaction plane
The systematic uncertainty of the reaction plane is estimated by the measurements of
the inclusive electron v2 with North side, South side and the combined reaction plane.
The top side in the Fig. 4.14 show the inclusive electron v2 measured with three typeof the reaction plane. The bottom plots is relative ratio respect to the v2 measured
with combined reaction plane. The relative ratio is about 5 % and we apply it as theuncertainty form the reaction plane.
• Electron identification
The systematic uncertainty from the electron identification is estimated by measuring
the inclusive electron v2 with the slightly changed the electron identification cut fromthe standard one. In this analysis we changed the parameters of E/p (E − p/p/σ ),n0 and χ 2 /npe0. Figure 4.15 shows the comparison with the inclusive electron v2
measured with the standard cuts and with the slightly changed those cuts condition(these v2 are not corrected by reaction plane resolution). We apply the relative ratio
respect to the standard cut as the uncertainty from the electron cut. The relative ratioof the E/p is about 2%, the n0 is about 2% and the χ 2 /npe0 is 1%. Therefore thetotal systematic uncertainty from electron identification is 3 %.
• Background v2
As described in the previous section, the uncertainties from photonic electron v2 comefrom π 0 (π ) v2 and kaon v2 . The π and K v2 has about 5 % uncertainty without thereaction plane. We apply this value to the uncertainty of the photonic electron v2 . We
also apply 5 % for the v2 for kaon decay subtraction uncertainty.
• RNP
The systematic uncertainty of the RNP comes from the uncertainties of inclusive electron spectra and the subtracted background spectra [55]. The systematic uncertaintyof the inclusive electron spectra includes the uncertainties in the geometrical acceptance, the reconstruction efficiency and the occupancy correction. The uncertainty inthe converter subtraction, Rγ determination, and the cocktail subtraction are mainly
originated from the systematic uncertainty of the input pion spectrum for the calculation.
The total systematic uncertainty is obtained by quadratic sum of above uncertainties.

v2

CHAPTER 4. EXPERIMENTAL RESULTS



comb R.P.
South R.P.
North R.P.

0.18


0.16


81

0.14


0.12


0.1


0.08


0.06


0.04


0.02


0




0

0.5

1

1.5

2

2.5

0.5

1

1.5

2

2.5





3

3.5

4

4.5

3.5

4

4.5



5
pT (GeV/c)

Nor(Sou)/v comb - 1.0
v2
2



0.5


0.4


0.3


0.2


0.1
-0

-0.1
-0.2
-0.3
-0.4
-0.5


0





3





5
pT (GeV/c)

Figure 4.14: (Top) The inclusive electron v2 measured with three type of the reaction
plane. (Bottom) The relative ratio respect to the v2 measured with combined reaction plane
N(S)(v2 /vconb).
2

CHAPTER 4. EXPERIMENTAL RESULTS

pT (GeV/c)pT (GeV/c)pT (GeV/c)pT (GeV/c)82

pT (GeV/c)pT (GeV/c)

Figure 4.15: Systematic uncertainty from the electron identification is estimated by measuring the inclusive electron v2 with slightly change the condition of the standard electron
identification cut (top). The relative ratio of the E/p is about 2%, the n0 is about 2% andthe χ 2 /npe0 is 1 % (bottom).

CHAPTER 4. EXPERIMENTAL RESULTS

83

4.4 Result

v2

Figure 4.16 shows the transverse momentum dependence of the non-photonic electron v2√
in Au+Au collisions at sNN = 200 GeV for the minimum bias events. The kaon decay
contributions are subtracted in the figure. The vertical solid lines mean statistical error andthe brackets mean 1 σ systematic uncertainty of the non-photonic electron v2 .

non-photonic electron v2
Minimum bias



0.15



0.1



0.05



0

-0.05
0






0.5

1

1.5

2

2.5



3

3.5

4



4.5
5
pT(GeV/c)

Figure 4.16: Transverse momentum dependence of non-photonic electron v2 in Au+Au col√lisions at sNN = 200 GeV for the minimum bias events. The vertical solid lines mean
statistical error and the brackets mean 1 σ systematic uncertainty of the non-photonic electron v2 .

Chapter 5
Discussions
5.1 Non-photonic electrons
Charm quark production has been studied via the non-photonic electron yields in the PHENIX
experiment. Since charm is only produced in the initial collisions via gluon fusions, the total
yield should be scaled by the number of nucleon-nucleon binary collisions (Ncoll ). The left
figure in Fig. 5.1 shows dN/d η (0.8 < pT < 4.0 GeV/c)/Ncoll of the non-photonic electron
vs. Ncoll for the minimum bias and five centrality bins together with p + p collisions [42].
α and it found α = 0.938 ± 0.075(stat.) ± 0.018(sys.). The cenThe data are fitted with ANcoll
trality dependence of the total non-photonic electron yield is consistent with Ncoll scaling,
as expected for the charm production. The right figure in Fig. 5.1 shows invariant yields of
non-photonic electrons for minimum bias events and in five centrality classes, 0-10 %, 10-20%, 20-40 %, 40-60 %, and 60-90 % [55]. The curves in the figure are the result of a FONLL
calculation from heavy flavor decay (c and b) [40] normalized to the p + p data [39]. For
all centralities, the non-photonic electron spectra measured in Au+Au collisions agree wellwith p + p difference. As we discuss later, a suppression with respect to p + p reference
develops towards high pT in central collisions suggests energy loss of charm quarks in thedense matter. The dominant source of the p + p reference is charm decay (Fig. 1.14). We
concluded that dominant source of electrons after background subtraction is ”charm” quark.
Thus the non-photonic electron v2 can be reflected to the azimuthal anisotropy of charm
quarks. As shown in the Fig. 4.16, a clear non-zero non-photonic electron v2 has been
observed. We calculated the confidence level for the non-zero v2 . We assumed that thedata of measured non-photonic electron v2 follows a Gaussian distribution and the σ was
obtained by calculating quadratic sum of the statistical and systematic errors of the nonphotonic electron v2 assumed these errors are independent. The result is shown in Fig.5.2
84

CHAPTER 5. DISCUSSIONS

-2

x10
0.35

0.8

centrality binned





0.7

min-bias

0.6

p + p at s = 200 GeV

0.3





0.25



0.5


0.2



0.4


0.15



0.3


0.1



0.2




0.05

0.1


0





0

103



200

400



600



800

0
1000 1200
Ncoll

1 d2N [(GeV/c) -2)]2π pT dpTdy

-4dNe /dy(0.8<pT <4.0) / T AA (mb)dNe /dy(0.8<pT <4.0)/ N Coll

x10

85

102
10
1

10-1

10-2
10-3
10-4
10-5
10-6
10-7
10-8
10-9

10-10

Min-Bias × 103

10-11

10-20% × 10

-12

10

10-13
10-14
0

2

0-10% × 10

1

20-40% × 100-140-60% × 10

-2

60-92% × 10
-2p+p × 10 /42mb


0.5

1

1.5

2

2.5

3

3.5

4 4.5 5
pT [GeV/c]

Figure 5.1: (Left) Non-photonic electron yield (0.8 < pT < 4.0 GeV/c) measured in Au+Au
collisions at 200 GeV [42] scaled by Ncoll as a function of Ncoll . Normalizing by the nuclear
overlap function obtained charm electron cross section per N + N collisions (right-handscale). (Right) Invariant yields of electrons from heavy flavor decays for different Au+Au
centrality [55], scaled by powers of ten for clarity. The solid lines are the result of a FOMLL
calculation of heavy flavor decay normalized to the p + p data [39] for each centrality class.

conf. level (%)

CHAPTER 5. DISCUSSIONS

86

100



80



60

40

non- γconfidence level for v2e

≠0

20





0



0.5

1

1.5

2

2.5



3

3.5

4



4.5
5
pT (GeV/c)

Figure 5.2: pT dependence of the confidence level for the non-zero non-photonic electronv2 .

as a function of pT . We found that the confidence level is more than 90 % for a non-zero
non-photonic electron v2 below pT < 2.7 GeV/c. Figure 5.3 shows the comparison with
non-photonic electron v2 and charged pion v2 . The trend of the non-photonic electron v2
is similar to the π v2 , increasing with pT up to about 1.5-2.0 GeV/c and then saturate orslightly reduces with increasing pT ,One of the interesting observation of the non-photonic electron v2 is that the v2 is smaller
than the pion in whole pT range. This feature of the non-photonic electron v2 is different
from the other identified particle v2 at high pT . As we showed in the introduction, theidentified particle v2 (π , K, p) shows the mass dependence, the heavier particle has smaller
v2 than the lighter particle, at low pT region. But identified particles have same v2 (v2 ∼0.15) around pT = 1.5 GeV/c. On the other hand, the maximum v2 of the non-photonic
electron v2 is about 0.1 and it does not have same v2 as pion, kaon and proton. The solid lineon the figure is a model calculation based on a quark coalescence model. 1 In the model,
charm quark v2 is assumed same as u quark v2 . The model predicts that the maximum v2 for
1 The solid line in the Fig. 5.3 is obtained with a = 1.0 and b = 1.0 in Eq. 5.7.

CHAPTER 5. DISCUSSIONS

87

v2

electron from charm decay is about 0.1 though charm quark has same v2 as u quark v2 . The
details about the model is described later in this chapter.

non-photonic e v2
π v2
D → e v2 (c v2=u v2)



0.25


0.2


0.15


0.1


0.05


0
-0.05
0






0.5

1

1.5

2

2.5



3

3.5

4



4.5
5
pT (GeV/c)

Figure 5.3: Comparison of the non-photonic electron v2 with pion v2 . The dashed line on
the figure is the electron v2 from charm decay assuming charm has same v2 as u quark.

5.2 Kinematics of D meson decay and v2
As discussed in the previous section, the main source of the non-photonic electron is D
meson. Thus the non-photonic electron v2 would be kinematically determined and allow
us to study D meson v2 . The issue of the D meson v2 via electron measurement is thatelectrons originating from semileptonic decays of D mesons might have a significant angulardeviation from the original D meson direction due to their large mass difference. If theangular deviation is large, the non-photonic electron does not reflect parent D meson v2
because electron v2 is largely smeared by the angular deviation. We studied the effect on v2
with a Monte Carlo simulation. The right hand figure in Fig. 5.4 shows the decay angle

CHAPTER 5. DISCUSSIONS

88

Figure 5.4: (Left) Momentum correlation between D meson and the decay electrons. The
open circles represent the mean pT of D meson momentum distribution for the given electron
momentum and closed data points represent the peak of the D meson momentum distribution. (Right) pT dependence of the decay angle correlation between D mesons and the decay
electrons.

CHAPTER 5. DISCUSSIONS

89

correlation between D mesons and the decay electrons (cos(φe − φD )) as a function of pT of
decay electrons. The result suggests that below pT = 1.0 GeV/c the angular deviation is large
so that the decay electrons are not emitted to the same direction of parent D mesons. On theother hand, above 1.0 GeV/c the angular deviation is very small so that the decay electrons
are emitted to the same direction of parent D mesons. Thus the non-photonic electron v2
above 1.0 GeV/c well reflects the v2 of D meson.
The corresponding electron momentum above 1.0 GeV/c is about 1.5 GeV/c to D meson(the left hand figure in Fig. 5.4). Thus a non-zero non-photonic electron v2 leads that the Dmeson v2 is also a non-zero at the intermediate pT region. Furthermore, v2 of D meson via
non-photonic electron v2 has been studied by a simple model calculations. We do not know
the real shape of D meson v2, therefore we assumed various pT depend shape of D meson
v2 . In this calculation we defined D meson v2 as;vD
2 (pT ) = a × f (pT ),(5.1)

where f (pT ) is the pT depend shape of D meson v2 and a is a scale factor for f (pT ). We
calculated v2 for the decay electrons from the D meson which have v2 defined as above
equation with a Monte Carlo simulation. Then we determined the best scale parameter withchi-squared test with the measured non-photonic electron v2 ;non−γ

χ 2 = Σ{(v2non−γ− vD→e)/σv2 }2 ,
2

(5.2)

where v2
is the measured non-photonic electron v2 , vD→e
is the electron v2 obtained from2
the Monte Carlo simulation of the D meson decay, and σ is the statistical and systematic
error of the measured non-photonic electron v2 . In the calculation π , K, p and φ v2 were used
for the pT depend shape of D meson v2 . These v2 are the measured v2 in Au+Au collisions√
at sNN = 200 GeV. Recently PHENIX has published the results that identified particles v2
are well scaled with mT − m [57]. We also tested D meson v2 with this transverse kinetic
energy scaling law in this study. Figure 5.5 shows the χ 2 /nd f (nd f = 13) as a functionof the scale factor (a) and Fig. 5.6 shows the 1 σ bands for each of the five assumed v2
shapes. The χ 2 test shows that D meson v2 shape, which shows the saturation around pT= 2.0 GeV/c for meson (π , K, φ ), well represent the shape for the non-photonic electron
v2 . Therefore D meson might be prefer to have a meson like v2 shape (saturate around 2.0
GeV/c). The best shape for the D meson v2 expected by the non-photonic electron v2 isa pion v2 shape. Figure 5.7 shows the D meson v2 obtained above method. The closed
data points represent the most suitable D meson v2 (χ 2 minimum) from the calculation, andthe shared band means χ 2 /nd f is less than 2.0. Based on the calculation, we found that D
meson v2 expected from the non-photonic electron v2 is 0.09 ± 0.03. The open circles onχ 2/ndf

CHAPTER 5. DISCUSSIONS

490

vD2 = a*vπ23.5

vD2 = a*vp2

vD2 = a*vK2





3
2.5
2
1.5
1


0.5


χ 2/ndf

0




0







20 40 60 80 100 120 140 160 180 0
a (%)

4



vD2 = a*vφ2




3.5



20 40 60 80 100 120 140 160 180 0
a (%)





20 40 60 80 100 120 140 160 180
a (%)

Dm

vD2 = a*v2 T



3
2.5
2
1.5
1


0.5


0




0





20 40 60 80 100 120 140 160 180 0
a (%)





20 40 60 80 100 120 140 160 180
a (%)

Figure 5.5: Results of the χ 2 test between measured non-photonic electron v2 and electronv2 calculated by the simulation assuming D meson v2 shape as a function of the scaled factora(%)

CHAPTER 5. DISCUSSIONS

91

v2

the figure is π v2 . The result indicates that D meson v2 is smaller than π v2 below pT < 3.5
GeV/c.
π v2 shape ; 1 σK v2 shape ; 1 σ
p v2 shape ; 1 σ
φ v2 shape ; 1 σscaled D v2 shape ; 1 σ

0.2

0.15

0.1

0.05

0


0

0.5

1

1.5

2

2.5



3

3.5

4

4.5
5
pT (GeV/c)

Figure 5.6: 1 σ bands for each of the five assumed D v2 shapes.

5.3 Charm quark v2 assuming Quark Coalescence Model
Previous measurements of v2 for hadrons made of light quarks, such as pions and kaons,suggests that the v2 already formed in the partonic phase. It would indicate that a partonic
collectivity has been developed for light quarks, such as u and d, in Au + Au collision at
RHIC. If v2 for charm quarks is also a non-zero v2 , it would be a strong indication of thepatonic thermalization not only for the light quarks but also for the heavy quarks in Au+Au
collisions at RHIC. As described in Introduction, various measurements in Au+Au collisions
at RHIC are well explained by the quark coalescence model. The model can be also usefulfor charm quark v2 study. As introduced in Chapter 1.5, v2 for mesons is expressed as;
v2,M (pT ) = v2,q (pT /2),

(5.3)

in the quark coalescence framework if the effective masses of constituent quarks are similar.
On the other hand, charm quark has much larger mass than the light quark (mc À mu ),CHAPTER 5. DISCUSSIONS

92

Figure 5.7: D meson v2 estimated by the measured non-photonic electron v2 . The closed
data points represent the most suitable D meson v2 (χ 2 minimum) from the calculation. The
dashed lines are χ 2 minimum+2 value for the determination. The open circles is π v2 .

therefore the most of D meson momentum is carried by the charm quark. As described in[61], the momentum fraction x in Eq. 1.9 is given as;
x=

mi
,
mM

(5.4)

here mM is a mass of meson and mi is an effective mass of quark. In case of D mesons, themomentum fraction x for u quark and c quark is given as xu ∼ 1/6 and xc ∼ 5/6. The asymmetric momentum configuration arises because coalescence requires the constituents to have
similar velocities, not momentum. Thus D meson v2 in the quark coalescence framework is
written as;
5
1
(5.5)
v2D (pT ) = v2,u ( pT ) + v2,c ( pT ).
66Here v2,u is u quark v2 and v2,c is c quark v2 . A same pT dependent shape for v2 of charm
quark are assumed as for the light and strange quarks. Then we assumed v2 of c quark andv2

CHAPTER 5. DISCUSSIONS

93

0.1

0.09

universal v2 for quark

0.08
0.07
0.06
0.05
0.04
0.03
0.02
0.01
0
0



1

2

3

4



5
pT (GeV/c)

Figure 5.8: Universal v2 shape for quarks in the calculation.
u quark as;
v2,u (pT ) = a × v2,q (pT ),(5.6)

v2,c (pT ) = b × v2,q (pT ),where v2,q (pT ) means the pT dependent shape for quarks v2 . The shape is shown in Fig.
5.8. Thus D meson v2 is given as;
¡
¢ ¡
¢
1
5
v2D (pT ) = a × v2,q ( pT ) + b × v2,q ( pT ) .
6
6

(5.7)

We calculated the decay electron v2 from the D meson v2 with a Monte Carlo simulation anddetermined the parameters a and b to reproduce the measured non-photonic electron v2 . The
parameters a and b are obtained by a simultaneously fitting not only for the non-photonic
electron v2 but also for kaon and proton v2 . The fitting function for kaon and proton v2 isgiven as;v2,K (pT ) = 2a × v2,q (pT /2),

(5.8)

v2,p (pT ) = 3a × v2,q (pT /3).
If the charm quark v2 is zero, the parameter b is also zero. Figure 5.9 show the fitting resultof non-photonic electron v2 , kaon v2 and proton v2 . The two parameter a and b obtained by
the fitting are shown in Fig. 5.10. The contour lines are one sigma step. The most innerCHAPTER 5. DISCUSSIONS

v2

94

non-photonic e

0.2

proton

Kaon

0.15

0.1

0.05

0


0

0.5

1

1.5

2

2.5



3



3.5

4 4.5 0pT GeV/c






0.5

1

1.5

2

2.5



3

3.5

4

4.5
pT GeV/c



0



0.5

1

1.5

2

2.5



3

3.5

4 4.5
pT GeV/c

Figure 5.9: Fitting result of non-photonic electron v2 , kaon v2 and proton v2 . The open
circles mean decay electron v2 from the D meson.

line means 1 σ contour level and the most outer line means 4 σ contour level. The best
fitting parameter set is; a = 1.0 and b = 0.96 with χ 2 /ndf = 21.85/27. From the contour plot
the parameter b for charm quark is consistent with the parameter a for u quark within the1 σ contour level. Thus v2 of charm quark is same as u quark within the 1 σ contour level
and the data suggest a non-zero charm quark v2 as regards assuming the universal quark v2
and applying the quark coalescence model. The strong elliptic anisotropy of charm quarks,
which is similar to the light quark, indicates a collectivity of different mass quarks during thequark phase of the high density matter created in the ultra-relativistic heavy-ion collisions.
Figure 5.11 shows the charm quark v2 , v2,c = b × v2,q (pT ), from the calculation. The solidline is obtained from the 1 σ contour level, and the dashed line is obtained 2 σ contour level
and the dotted + dashed line is obtained from 4 σ contour level from the fitting.

CHAPTER 5. DISCUSSIONS

95

b ; c quark

1.8
1.6

2σ

1.4
1.2

1σ

1

102



0.8


0.6

4σ



0.4


0.2




0.6



0.7



0.8

0.9

1

1.1

1.2

1.3
1.4
a ; u quark

Figure 5.10: Contour plot of v2 of c quark and u quark which are defined as v2,u = a × v2,q
and v2,c = b × v2,q . Here v2,q is a quark v2 determined by the measured v2 . Parameters a andb are obtained by fitting the measured non-photonic electron v2 , Kaon v2 and proton v2 . The
contour lines are one sigma step.

CHAPTER 5. DISCUSSIONS

v2

96

u quark v2
charm v2 ; 1 σcharm v2 ; 2 σcharm v2 ; 4 σ

0.14
0.12
0.1
0.08
0.06
0.04
0.02
0


0

0.5

1

1.5

2

2.5



3

3.5

4



4.5
5
pT (GeV/c)

Figure 5.11: Charm quark v2 from the calculation based on the quark coalescence model.
The solid lines are obtained from 1 σ contour level, the dashed lines are obtained 2 σ contour
level and the dotted + dashed lines are obtained from 4 σ contour level by the fitting. The
open boxes mean u quark v2 .

CHAPTER 5. DISCUSSIONS

97

5.4 Bottom quark contribution to the non-photonic electron v2

v2

In the reported momentum region, the dominant source of the non-photonic electron ischarm quark decay. On the other hand, bottom quark contributions to the non-photonic
electrons would be larger than the charm quark at high pT region. Current bottom quark
contribution to the non-photonic electron is model dependent. Based on a pQCD (FONLL),
electron from bottom quark decay is calculated and it predicts that the contribution is larger
than that of charm quarks above 4.5 GeV/c (Nb→e /Nc→e > 1.0 above pT = 4.5 GeV/c). As
shown in Fig. 1.14, the prediction consistents with the non-photonic electron measured inp + p collisions at PHENIX experiment.

0.2

D → e ; (1)B

D → e + B → e (v2 = vD2) ; (2)

0.15

B

D → e + B → e (v2 = 0) ; (3)
0.1

0.05

0



0

0.5

1

1.5

2

2.5



3

3.5

4



4.5
5
pT (GeV/c)

Figure 5.12: Electron v2 from D meson and B meson. The curve (1) is assumed no b quark
contributions to the non-photonic electron (only take into account charm decay). The curve(2) is assumed that D meson and B meson v2 have same v2 . And the curve (3) is assumed
that B meson v2 is zero.

CHAPTER 5. DISCUSSIONS

98

We studied bottom quark contributions to the non-photonic electron v2 . The non-photonic
electrons originating from bottom quarks mainly decay from B mesons which contain a bottom quark. In this study, we considered three extreme models for B meson. The first model
assumes no B meson (no bottom quark) contribution to the non-photonic electron v2 (curve1). The second model assumes that D meson and B meson have same v2 (curve 2). And thethird model assumes that B meson v2 is zero (curve 3). This assumption arises from that B
meson has much larger mass than D meson, thus the v2 of B is much smaller than D meson
v2 . For the model (2) and (3), a ratio of electrons from charm and bottom (Nb→e /Nc→e ) is
obtained from the pQCD calculation [40] in Fig. 1.14. The D (B) meson v2 was used the v2
determined from the non-photonic electron v2 in Chapter 5.2 for the model (2). Figure 5.12
shows results of the model calculations. As we see, bottom quark contributions reduce thenon-photonic electron v2 at high pT region. The reason why the second model is smaller
than the first model is that the decay electron v2 from B meson decay is largely smeared dueto the large mass difference between B meson and electron. Current measured non-photonic
electron v2 has large errors at high pT and it consistent with the three models.

5.5 Comparison with charm quark thermalization model
One of the big questions in heavy ion collisions at RHIC is the thermalization of the charm
quark. As we discussed, the measured non-photonic electron spectrum is consistent with
pQCD calculation. On the other hand, it was pointed out in [10], electron spectrum also
reproduce by the hydrodynamical models assuming the complete thermal equilibrium forcharm hadrons. The curves in fig. 5.13 are the electron pT spectra from the model calculations. The data in the figure is non-photonic electron spectrum measured at PHENIX in√
Au+Au collisions at sNN = 130 GeV [51]. The solid curves are electron spectrum obtained by the initial pQCD charm production without final state interactions (PYTHIA) andthe dashed curves are the spectrum by the complete thermal equilibrium for charm hadrons(hydro model). The difference between two models are very small at low pT region. Thus
the spectrum is not sensitive to the thermalization of charm quarks.
On the other hand, it has been predicted that the elliptic flow of electron from charmis very sensitive to the thermalization of charm quarks [62]. In the calculation D mesons
are formed by the charm quark coalescence with thermal light quarks at the hadronization
stage. For charm quark momentum spectra, two extreme scenarios are considered. The firstscenario assumes no reinteractions after the production of charm-anticharm quark pairs at
the initial state hard processes (calculated from PYTHIA). The second scenario assumes
complete thermalization with the transverse flow of the bulk matter. Both scenarios pro-CHAPTER 5. DISCUSSIONS

99

Figure 5.13: The electron spectra from the model calculations. The dashed curves are the
PYTHIA calculation and solid curves are the results from the thermal hydrodynamic model.
The data in the figure is non-photonic electron measured at PHENIX in Au+Au collisions at√
sNN = 130 GeV [51]. The picture is taken from [10].

duce electron spectra from D meson decays which are consistent with non-photonic single
electron spectra as published by PHENIX. In both scenarios, B meson contribution is nottaken into account. Fig. 5.14 shows the v2 of decay electrons from D mesons in the ”noreinteraction” scenario as a dashed line, while the solid line represents the ”thermalization”
scenario. Even if the charm quark v2 is zero, electron v2 for ”no reinteraction” scenario
is not zero due to the v2 of light quark component. The model predicts that the electrons
from the thermalized model has two times larger v2 than the pQCD model. Thus the nonphotonic electron v2 is sensitive the charm quark thermalization. The red data points are themeasured non-photonic electron v2 values as presented in this thesis. The “thermalization +flow” scenario well reproduces the measured non-photonic electron v2 below 1.5 GeV/c.

CHAPTER 5. DISCUSSIONS

v2

100

w/ charm flow
w/o charm flow



0.15



0.1



0.05



0

-0.05
0






0.5

1

1.5

2

2.5



3

3.5

4



4.5
5
pT (GeV/c)

Figure 5.14: Comparison of the non-photonic electron v2 with two different charm flow scenarios [62]. The solid line corresponds to the no rescattering of the initially produced charm
quarks (without flow), while the dashed line reflects the effect of complete thermalization(with flow).

CHAPTER 5. DISCUSSIONS

101

5.6 Non-photonic electron RAA and v2
As introduced in Chapter 1.4, neutral pions and charged hadrons are largely suppressedat high pT region in Au+Au collisions compared with that in p + p collisions. Such the
suppression is not observed in d+Au collisions, thus the suppression in Au+Au collisions
is due to the final state interactions. This suppression is well represented a model whichassumes partons energy loss in the dense matter. The suppression has been also observed
the non-photonic electrons in Au+Au collisions. As we shown in the Fig. 5.1, the invariant
pT spectra of the non-photonic electron in Au+Au collisions agree well with p + p reference
at low pT in all centrality. On the other hand, a suppression with respect to p + p collisions
develop at high pT . The suppression is quantified the nuclear modification factor, RAA ,which is defined as;
d σAu+Au
RAA =
.
(5.9)
hNbinary i × d σ p+p
Figure 5.15 shows the RAA for the non-photonic electrons as a function of pT in most central√
collisions (centrality 0 - 10 %) measured in Au+Au collisions at sNN = 200 GeV. We see
the RAA is unity (RAA = 1.0) at low pT . This means that the yield of the non-photonic electronis scaled by the number of collisions (Ncoll ). As we discussed, charm quarks are produced
at the initial collisions and the total yield is scaled by the number of collisions (Ncoll ). The
result is consistent with charm quark production. On the other hand, the RAA for the nonphotonic electron is smaller than unity (RAA < 1.0) at high pT region and it is same as the
RAA for the neutral pions (RAA ∼ 0.2). Thus the suppression suggests that charm and bottom
quarks also lose their energy in the dense matter same as light quarks. It should be notedthat the energy loss of charm have little effect on the total charm yield.
This result leads that a very dense partonic matter, even if heavy quarks (c and b) are
stopped, is created in Au+Au collisions at RHIC energy. The curve in the figure is model
calculation includes gluon radiations as a mechanism of the parton energy loss. The model
assumes that an intial gluon density is 3500 ( dNg /dy = 3500) [63]. Thus the RAA for the
non-photonic electron suggests that a large initial gluon density is achieved ( dNg /dy >3500) in Au+Au collisions. In such the dense partonic matter, a mean free path of charm
quark would be smaller than the system size. We briefly estimated the mean free path ofcharm quark in the dense matter. Applying Bjorken formula Eq. 1.1, the parton density pervolume would be given as;
1 dN p.
(5.10)n=τ A⊥ dyIn this calculation, we used life time of QGP (∼ 5 fm/c) for τ and assumed an initial partondensity per rapidity is dN p /dy ∼ dNg /dy = 3500. Under the assumptions, the parton density

R AA

CHAPTER 5. DISCUSSIONS

102

1.8




centrality ; 0 - 10 %

1.6

g


dN /dy = 3500

1.4
1.2
1
0.8
0.6
0.4
0.2
0
0



1

2



3

4



5



6

7



8

9
10
pT (GeV/c)

Figure 5.15: RAA of the non-photonic electrons in 0-10 % central collisions compared withmodel calculations assuming the initial gluon densities is 3500 (dN g /dy = 3500).

per volume is n = 7 (fm−3 ). The perturbative QCD estimated charm rescattering crosssection is about 3 mb [64]. Then we got the mean free path of charm quark is about 0.5 fm.
This value is smaller than the size of the system (R ∼ 6 fm). The length of the mean freepath is important parameter for v2 . If the path is larger than the system, the emission patternin the azimuthal space is isotropic. On the other hand, the emission pattern in the azimuthal
space is anisotropic if the path is smaller than the system. Thus we expect the azimuthal
anisotropy of charm quarks. This result consistent with the non-zero charm quark v2 as we
discussed in this chapter.
Measurements of v2 and RAA for the non-photonic electron suggest that charm quark
flow and energy loss in the matter created in Au+Au collisions. Large charm quark energy
loss suggests that the matter created Au+Au collisions are very dense. Charm flow suggeststhat charm quarks are frequently scattered in the dense partonic matter. Thus we expectthat thermalization of charm quark is achieved. If charm quark is really thermalized in

CHAPTER 5. DISCUSSIONS

103

the matter, thermalization time of charm quark must be comparable to existence time of
QGP (< 5 fm/c). Information about thermalization time of charm quark is gained from thenon-photonic electron v2 and RAA compared with a model prediction. The curves on the
figures are results of the model [65]. The main assumption in the model is elastic scattering
mediated by resonance excitation of D meson and B meson in QGP. Theoretical evidence forthe resonance state is provided by QCD calculations [66] [67]. The model simultaneously
reproduce the measured non-photonic electron RAA and v2 . It was pointed in [68], resonancestate of D meson and B meson reduce charm quark thermalization times by a factor of ∼3 compared to pQCD and it is about a few fm/c. The thermalization time is comparable
existence time of QGP. Thus the measured non-photonic electron RAA and v2 suggest thatcharm quark thermalization is achieved in Au+Au collisions.

CHAPTER 5. DISCUSSIONS

104

RAA

1.8
1.6

(a)

0-10% central

1.4
1.2
1
0.8
0.6
0.4
0.2


v2

0.18


0.16

(b) minimum bias



0.14


0.12


0.1


0.08


0.06


0.04


0.02


0

-0.02

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

pT [GeV/c]
Figure 5.16: (a) RAA of the non-photonic electron in 0-10 % central collisions. (b) v2 forthe non-photonic electron in minimum bias collisions. The curves in the figure are model
calculations assuming elastic scattering mediated by resonance excitation of D meson and B
meson in QGP [65].

Chapter 6
Conclusions
In this thesis, we have studied the elliptic azimuthal anisotropy of charm quark via electron
measurement. We first measured the transverse momentum dependence of the azimuthal
anisotropy parameter v2 , the second harmonic of the azimuthal distribution, for electrons at√
mid-rapidity (|η | < 0.35) with the PHENIX detector in Au+Au collisions at sNN = 200
GeV. From the result, we have extracted electron v2 from heavy flavor decays by subtracting v2 of electrons from other sources such as photon conversions and Dalitz decay. This
photonic electron v2 is determined with a Monte Carlo simulation using measured v2 andpT distributions of these photonic sources. The yield of the non-photonic electron after subtracting the background is consistent with the expected yield from charm decay. Therefore
the non-photonic electron v2 can be reflected the charm quark v2 . We found that the nonphotonic electron v2 is clearly positive below 3.0 GeV/c in the minimum bias event sample.
The non-photonic electron v2 is smaller than the π v2 in whole pT range, but the trend issame as π v2 , increasing with pT up to about 1.5-2.0 GeV/c and then saturate or slightly
reduces with increasing pT .
The main source of the non-photonic electron is D meson decay. We found that the decay
angle is strongly correlated with the angle of the parent D meson and the decay electron v2
well reflects the parent D meson v2 . Therefore the non-zero v2 for the non-photonic electron
gives a non-zero D meson v2 . We have extracted the D meson v2 from the non-photonic
electron v2 by assuming various pT dependent shape of D meson v2 . D meson prefers a pion
like v2 shape which saturates around 2.0 GeV/c and the peak v2 value is obtained to be 0.09
± 0.03 at about 2-3GeV/c.
Based on a quark coalescence model, we finally estimated charm quark v2 from the nonphotonic electron v2 . In the calculation, the same pT dependent shape for v2 of charm quark
is assumed as for the light and strange quarks, and the effective mass of charm quark is used

105

CHAPTER 6. CONCLUSIONS

106

for coalescing them at similar velocity. Simultaneous fitting of measured v2 parameters forpion, Kaon, proton and the non-photonic electron are done with 2 v2 parameters for lightand heavy quarks including the quark coalescence effect and D meson decay kinematics.
We found that the extracted charm quark v2 has similar magnitude compared with the lightquark. The strong elliptic anisotropy of charm quarks, which is similar to the light quark,
indicates a collectivity of different mass quarks during the quark phase of the high density
matter created in the ultra-relativistic heavy-ion collisions. The strong anisotropy of charm
quarks also indicates that charm quarks are frequently scattered in the dense partonic matter
due to their large mass, and we expect thermalization of charm quarks. We compared our
result with a model assuming complete thermalization with the transverse flow of the bulk
matter. Below 1.5 GeV/c the non-photonic electron v2 is in good agreement with the model.
The energy loss of charm quarks is also observed via electron measurement at RHIC. The
theoretical model predicts that the early time thermalization of charm quarks, which is comparable to QGP life time, is required to reproduce the non-photonic electron v2 and RAA .
Thus presented results on v2 measurement of the non-photonic electron could be one of thestrong evidences of QGP formation at RHIC.

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Data points
pT
v2
stat.errsys. err
0.546386 0.0401248 0.00635844 0.0188964
0.645676 0.0408199 0.00711689 0.0187716
0.746343 0.0385246 0.00742423 0.0168286
0.847174 0.0625494 0.00767092 0.0144594
0.947197 0.0624677 0.00860714 0.0136772
1.08757 0.0562807 0.00699467 0.0117929
1.28803 0.0697922 0.00922824
0.011447
1.52502 0.0919859
0.0102673
0.0112271
1.82760.0869942
0.0152229
0.0106938
2.19729 0.0691944
0.018114
0.00931277
2.70457 0.0706206
0.0307665 0.00841521
3.24475 0.0307921
0.0465096 0.00789024
4.05063 0.00985709 0.0644943
0.0095675
Table A.1: Non-photonic electron v2 (minimum bias).

111

Appendix B
Kinematics
B.1 Transverse Momentum and Transverse Mass
We take a beam line to be z-axis of a frame. Consider a particle which has momentum p= (px ,py ,pz ) and mass m. The momentum component along the beam direction is called
the longitudinal momentum pz . The perpendicular to the beam direction are combined andcalled the transverse momentum given as;qpT = p2x + p2y(B.1)
The transverse momentum is Lorentz invariant in any frame moving parallel to the z direction. Using the transverse momentum, the transverse mass is defined asq
mT = p2T + m2 .
(B.2)

B.2 Rapidity
Rapidity, y, which is defined asy=

1 E + pzln
2 E − pz

(B.3)

is used as a target for the position of produced particles on z axis. The rapidity has a property
in that it is additive under Lorentz transformation, just like a Galilean velocity under Galileantransformation. In the relativistic limit (p À m), the rapidity becomes as;θy ∼ − ln(tan( )) ≡ η
2where θ is polar angle. The η is called pseudo-rapidity.
112

(B.4)

APPENDIX B. KINEMATICS

113

B.3 Centrality Variables
In heavy ion collisions due to the large size of the nucleus, their geometry aspects plays
important role in collision dynamics. Figure B.1 shows an illustration of the collision geometry for a non-head on nucleus-nucleus collision. The vector that connects the center ofthe nuclei is called ”impact parameter”. Therefore the impact parameter is good guide forthe collision geometry. In heavy ion collision impact parameter is represent as ”centrality”.
The head on nucleon-nucleon collision is defined as centrality 0% and the variable increase
with non-head on collisions. As is shown B.1, only the nucleons in the overlap region of
the two nuclei participate in the collisions. These nucleons are called ”participants” and therest nucleons which are not participate in the collisions are called ”spectator”. The variablewhich means how many nucleons participate in the collision is called ”number of participants (N part )” and the number of collisions of nucleons in the participant region is called”number of binary collisions (Ncoll )”.
Centrality variables are calculated by the Glauber model [69] which is based on a simple geometrical picture of a nucleus-nucleus collision. The model treats nucleus-nucleus
collision as multiple nucleon-nucleon collisions assuming• nucleus travel in straight line trajectories• not deflected after collisions• secondary particles and possible excitation is neglected• cross section is same in the vacuum
The nucleons in nucleus are randomly distributed according to Woods-Saxon distribution,
the density profile ρ (r) is defined as;

ρ (r) =

ρ0
1 + exp( r−R
a )

(B.5)

where R is radius of nucleus and a is surface diffuseness parameter. In case of Au+Au√
collisions at sNN = 200 GeV, the parameters are R = 6.38 fm, a = 0.54 fm ,ρ0 = 0.169
fm−3 and σNN = 42 mb.

APPENDIX B. KINEMATICS

114

Figure B.1: an illustration of the collision geometry for a non-head on nucleus-nucleus
collision. The nucleons in the overlap region of the two nuclei participate in the collisions
are called ”participants” and the rest nucleons which are not participate in the collisions arecalled ”spectator”