Longitudinal Double-Spin Asymmetry ofElectrons from Heavy Flavor√ Decays in
Polarized p + p Collisions at s = 200 GeV

Katsuro Nakamura
Department of Physics, Kyoto University
January, 2013

i

Longitudinal Double-Spin Asymmetry ofElectrons from Heavy Flavor√ Decays in
Polarized p + p Collisions at s = 200 GeV

A Dissertation Presented
byKatsuro Nakamura
to
The Graduate Schoolin Partial Fulfillment of the Requirements
for the Degree ofDoctor of Philosophy
inPhysics

Kyoto University
Jan 2013

ii

abstract
Polarized proton-proton collisions are performed on Relativistic Heavy Ion Collider(RHIC) at Brookhaven National Laboratory for a goal of determination of gluon polarization in a polarized proton. Currently, the gluon polarization ∆g(x) is well constrainedby measurements of double-spin asymmetries for π 0 and jet production at RHIC andpolarized deeply-inelastic scattering measurements. Especially, constraints of the gluon
polarization in small Bjorken x region is provided by the π 0 double-spin asymmetry
measurements. However, achievable Bjorken x region with these measurements is up
to the limit of 2×10−2 < x, because of requirement of minimum transverse momentumfor π 0 to satisfy an energy-scale of perturbative chromodynamics (pQCD) technique
for theoretical calculations. Due to the lack of measurement, an uncertainty of thepolarized gluon distribution in small Bjorken x region, x < 5×10−2 , still causes a large
uncertainty on the gluon polarization.
We performed the world-first measurement of longitudinal double-spin asymmetryof electrons from decays of hadrons containing√ heavy flavor (heavy flavor electrons) in
longitudinally polarized p + p collisions at s = 200 GeV. The measurement was performed at mid-rapidity (|η| < 0.35) of the PHENIX detector at RHIC. The heavy flavor
quarks are produced by mainly gluon-gluon scattering in the partonic-level interaction.
From pQCD calculations of partonic cross section, it is also known that the heavy flavor
measurement has large sensitivity for the polarized gluon distribution. Therefore, thedouble-spin asymmetry is excellent probe to measure |∆g/g(x)|2 . In addition, because
large mass of produced heavy quark supports a minimum energy-scale of the production process, ∼mc = 1.4 GeV (mc : charm quark mass), the pQCD technique is reliablewithout transverse momentum requirement in the measurement of the heavy quark
electron. Therefore, the measurement enables√us to explore the gluon polarization in
unexplored Bjorken x region around x∼2mc / s = 1.4 × 10−2 , and provides essential
information for the constraint on ∆G.
The measurement of the heavy flavor electron suﬀers large electron background,which increases both of statistical and systematic uncertainties on the measured doublespin asymmetry. The dominant background in this measurement is electron pair from
Dalitz decay of neutral mesons, especially π 0 and η mesons, and γ conversion in materials. To overcome this challenge, the background was eﬀectively rejected with Hadron
Blind Detector (HBD), which was newly installed in PHENIX at 2009. HBD is aposition-sensitive gas Čerenkov counter, which not only can identify electron, but also
can distinguish between the signal and background electrons by the amount of the produced Čerenkov light around the electron track. Using this feature, a new analysis
method with the HBD was developed.
With the HBD, we succeeded to improve the signal purity of the heavy flavor electrons by a factor of about 1.5 compared with a previous result which did not employ
HBD. Under the improved purity, the double-spin asymmetry of the heavy flavor electrons with transverse momentum ranging 0.5 < pT < 3.0 GeV/c was deduced. The
measured asymmetries are consistent with zero within statistical uncertainties. Fromthe measured spin asymmetry, a constraint on the polarized gluon distribution in the

iii
−2 <
−2
c
<
proton around log10 x = −1.6+0.5∼ x ∼ 8×10 ), µ = m = 1.4 GeV was ob−0.4 (10tained based on a leading order pQCD model as |∆g/g(x, µ)|2 < 3.0 × 10−2 (1σ).

Contents
1 Nucleon Spin Physics
1.1 Outline of Physics Motivation . . . . . . . . . . . . . . . . . . . . . . .
1.2 Nucleon Spin Structure and Parton Model . . . . . . . . . . . . . . . .
1.2.1 Parton Model . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2.2 Polarized Parton Distribution Function . . . . . . . . . . . . . .
1.3 Experiments for PDF Measurement . . . . . . . . . . . . . . . . . . . .
1.3.1 Deep Inelastic Scattering (DIS) experiment . . . . . . . . . . . .
1.3.2 Semi-Inclusive DIS (SIDIS) experiment . . . . . . . . . . . . . .
1.3.3 Hadron collision experiment . . . . . . . . . . . . . . . . . . . .
1.4 DGLAP Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.5 Current Status of Nucleon Spin Structure . . . . . . . . . . . . . . . . .
1.5.1 Determination of Unpolarized PDF . . . . . . . . . . . . . . . .
1.5.2 First Moment of Polarized PDF . . . . . . . . . . . . . . . . . .
1.5.3 ∆g(x) Measurements . . . . . . . . . . . . . . . . . . . . . . . .
1.5.4 Global Analysis for ∆g(x) . . . . . . . . . . . . . . . . . . . . .
1.6 Heavy Flavor Production in p + p Collisions . . . . . . . . . . . . . . .
1.6.1 Challenge for ∆g(x) Determination . . . . . . . . . . . . . . . .
1.6.2 Production Mechanism for Heavy Flavor Electron . . . . . . . .
1.6.3 Background for Heavy Flavor Electron Measurement . . . . . .

1
1
4
4
6
7
7
9
11
15
17
17
17
24
25
30
30
32
38

2 Experimental Setup
2.1 Relativistic Heavy Ion Collider (RHIC) . . . . . . . . . . . . . . . . . .
2.2 PHENIX Detector System . . . . . . . . . . . . . . . . . . . . . . . . .
2.2.1 Beam-beam counter (BBC) . . . . . . . . . . . . . . . . . . . .
2.2.2 Zero degree calorimeter (ZDC) and shower max detector (SMD)
2.2.3 Central magnet (CM) . . . . . . . . . . . . . . . . . . . . . . . .
2.2.4 Drift chamber (DC) . . . . . . . . . . . . . . . . . . . . . . . . .
2.2.5 Pad chamber (PC) . . . . . . . . . . . . . . . . . . . . . . . . .
2.2.6 Ring-imaging Čerenkov (RICH) Detector . . . . . . . . . . . . .
2.2.7 Electromagnetic calorimeter (EMCal) . . . . . . . . . . . . . . .
2.2.8 Hadron blind detector (HBD) . . . . . . . . . . . . . . . . . . .
2.3 PHENIX Data Acquisition . . . . . . . . . . . . . . . . . . . . . . . . .
2.3.1 Triggers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3.2 Data acquisition system . . . . . . . . . . . . . . . . . . . . . .

41
41
45
48
49
49
51
54
56
57
58
63
63
66

v

vi

CONTENTS

3 Electron Analysis
3.1 Conventional Techniques for Electron Analysis . . . . . . . . . . . . . .
3.1.1 Vertex position determination . . . . . . . . . . . . . . . . . . .
3.1.2 Track reconstruction by DC and PC . . . . . . . . . . . . . . .
3.1.3 Electron ID by RICH . . . . . . . . . . . . . . . . . . . . . . . .
3.1.4 Electron ID by EMCal . . . . . . . . . . . . . . . . . . . . . . .
3.1.5 Electron ID by HBD . . . . . . . . . . . . . . . . . . . . . . . .
3.1.6 Heavy flavor electron estimation in previous measurements . . .
3.2 Heavy Flavor Electron Analysis with HBD . . . . . . . . . . . . . . . .
3.2.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2.2 Data selection . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2.3 Electron selection . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2.4 Detector stability . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2.5 Non-photonic electron analysis with HBD . . . . . . . . . . . .
3.2.6 Non-photonic background estimation . . . . . . . . . . . . . . .

69
69
69
70
73
73
75
77
79
79
79
79
80
82
91

4 Results
4.1 Heavy Flavor Electron Yield . . . . . . . . . . . . . . . . . . . . . . . .
4.1.1 Systematic uncertainty on raw yield . . . . . . . . . . . . . . . .
4.1.2 Raw yield and signal purity of heavy flavor electron . . . . . . .
4.2 Cross Section of Heavy Flavor Electron . . . . . . . . . . . . . . . . . .
4.2.1 Integrated luminosity . . . . . . . . . . . . . . . . . . . . . . . .
4.2.2 Detector acceptance and reconstruction eﬃciency . . . . . . . .
4.2.3 Trigger performance . . . . . . . . . . . . . . . . . . . . . . . .
4.2.4 Cross section of heavy flavor electron . . . . . . . . . . . . . . .
4.3 Spin Asymmetry of Heavy Flavor Electron . . . . . . . . . . . . . . . .
4.3.1 Beam polarization . . . . . . . . . . . . . . . . . . . . . . . . .
4.3.2 Spin asymmetry calculation . . . . . . . . . . . . . . . . . . . .
4.3.3 Background spin asymmetry . . . . . . . . . . . . . . . . . . . .
4.3.4 Helicity pattern dependence . . . . . . . . . . . . . . . . . . . .
4.3.5 Spin asymmetry of heavy flavor electron . . . . . . . . . . . . .

97
97
97
98
100
100
100
107
107
111
111
112
114
117
117

5 Discussion
121
5.1 Constraint on ∆g(x) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
5.2 Future Prospect for ∆g(x) Measurement . . . . . . . . . . . . . . . . . 123
6 Conclusion

129

A Deep Inelastic Scattering (DIS) Experiment133

B Polarization Measurement at RHIC

137

Chapter 1
Nucleon Spin Physics
1.1

Outline of Physics Motivation

For many years, human beings have attempted to reveal what material consist of andunderstand what elemental particles are. For this purpose, inner structure of material
has been approached over time, and molecule, atom, nucleus, nucleon, and quark werediscovered so far. At the same time, there was a discovery of a new interaction in each
step, because they are made up with interactions of diﬀerent range-scale.
Besides this feature of the discovery, the study of the internal structure plays also
an important role to understand the dynamics among the constituent particles. Theinternal dynamics of a nucleon, which all the materials have, is described as quantum
chromodynamics (QCD) among quarks and gluons. The Lagrangian of QCD is alreadywell established as color SU(3) gauge interaction,∑
1 a µν
LQCD =q̄ (iD/ − mq ) q − Fµν
Fa
(1.1)
q
(
) 4
1
Dµ ≡ ∂ µ + ig
λa Aµa2µνµ ν
ν µFa ≡ ∂ Aa − ∂ Aa − gf abc Aµb Aνc ,where q is quark field, mq is quark mass, Aµa (a = 1 ∼ 8) is gauge field, λa is the GellMann matrix in SU(3), and f abc (a, b, c = 1 ∼ 8) is the structure constant of SU(3).
However, whereas high energy QCD interaction is well established with the perturbative
QCD (pQCD) method, a lot of low energy phenomena, including the QCD dynamics
in the nucleon, are still not enough understood.
It is known that nucleon is comprised of three quarks in the naive constituent quark
model. However, diﬀerent from structures of molecule or atom, the dynamic structureof nucleon is not simple due to large strong coupling constant of QCD in the low
energy interaction. The quarks in a nucleon create gluons, and the gluons also create
quark and anti-quark pairs at every moment. As a result, nucleon can be represented
as an aggregation of the three valence quarks and a lot of quarks, anti-quarks, andgluons surrounding them. The microscopic structure of a nucleon is described with aparton model, which represents the quarks, anti-quarks, and gluons (called partons) as
distributions of momentum fraction x compared with nucleon momentum (Bjorken x).
1

2

CHAPTER 1. NUCLEON SPIN PHYSICS

The parton distributions have been studied to understand the nucleon structure in
the last 40 years. From successes of deep inelastic scattering (DIS) experiments, theparton distributions in an unpolarized nucleon are obtained with accuracy, and thenthe quark and gluon composition in the nucleon system was revealed. However, they
represent only an aspect of spin-independent QCD dynamics in the nucleon, and spindependent QCD dynamics is also important especially to build up the nucleon spin 1/2(in the unit of Planck constant h̄). For further understanding of the nucleon structure
including spin-dependent dynamics, it is important to reveal how the partons compose
the nucleon spin 1/2.
For the purpose, a lot of polarized DIS experiments were performed. The firstbreakthrough was achieved by the European Muon Collaboration (EMC) [1, 2] at 1988.
They revealed a discrepancy from the Ellis-Jaﬀe sum rule [3, 4], and suggested thatpolarization of sea quarks, such as s quark in a nucleon, takes sizable contribution to
the nucleon spin. They also suggested that the total polarization of all quark and antiquark in a proton, ∆Σ, is too small to explain the proton spin with the constituent quark
model, which expects ∆Σ = 1. After these discoveries, experimental eﬀorts [5, 6, 7, 8, 9]
focused on a detailed understanding of the spin structure of the proton. The proton
spin 1/2 can be decomposed as 12 = 12 ∆Σ + ∆G + Lz from conservation of angularmomentum. The measurements precisely determined ∆Σ to be only about 30% of the
proton spin. The remaining proton spin can be attributed to the other components,
the gluon spin contribution (∆G) and/or orbital angular momentum contributions (Lz ).
Empirically, it is diﬃcult for system in a ground state, like as nucleon, to obtain orbital
angular momentum. Therefore, we attempted to understand the gluon polarization ∆G
in this work.
The Relativistic Heavy Ion Collider (RHIC) at Brookhaven National Laboratory(BNL), which can accelerate√ polarized proton beams and collide them at up to thecenter-of-mass energy of s = 510 GeV, is a unique and powerful facility to study
the gluon polarization, because gluon interaction dominantly participates in various
production processes in p + p collisions, e.g. π 0 , η, jet, heavy quark, and direct γ
production. One of the main goals of the RHIC physics program is to determine thegluon polarization through measurements of longitudinal double-spin asymmetries,
ALL ≡

∆σ
.
σ

(1.2)

In the equation, σ and ∆σ denote spin-independent and spin-dependent cross sectionsof a specific process in the polarized p + p collisions. ALL has been measured previously
in several channels by the PHENIX and STAR experiments in RHIC, including inclusive
π 0 [10, 11, 12, 13], η [14], and jet [15, 16, 17] production.
The total gluon polarization is given by∫ 1∆G(µ) ≡dx∆g(x, µ),(1.3)
0where x and µ represent Bjorken x and factorization scale respectively. The challenge
for the ∆G(µ) determination is to precisely map the gluon polarization density ∆g(x, µ)over a wide range of x. Using the asymmetries measured of π 0 , η, and jet production

1.1. OUTLINE OF PHYSICS MOTIVATION

3

and the world-data on polarized inclusive and semi-inclusive DIS [5, 6, 7, 8, 9, 18, 19,
20, 21, 22], a global analysis based on pQCD calculations was performed at the next-toleading order level in the strong coupling constant αS [23]. The ALL measurements at
RHIC as well as the polarized DIS measurements well constrain the gluon polarization∆g(x). Especially, constraints of the gluon polarization in small Bjorken x region isprovided by the π 0 ALL measurement. However, because of requirement of minimum
π 0 transverse momentum pmin
= 2.0 GeV/c to satisfy an energy-scale of pQCD calT
culations, achievable Bjorken x region with the π 0 measurement is up to the limit ofabout 2×10−2 . Due to the lack of data, large uncertainty of ∆g(x, µ) in small Bjorken
x region is still remaining, and it causes a substantial uncertainty of the total gluon
polarization ∆G.
To overcome above problems, heavy flavor production in p+p collisions, pp→QQ̄+X,
is an ideal measurement. The heavy quarks are produced dominantly by the gluon-gluon
interaction [24]. And it is known that this process has large sensitivity to ∆g(x) fromspin-dependent cross section calculated with leading order (LO) or next-to-leading order(NLO) pQCD [25, 26, 27]. Therefore, the double-spin asymmetry of the heavy flavor
production is excellent probe to measure (∆g/g(x))2 , because∫()2
dx1 dx2 ∆σ̂gg→QQ̄+X ∆g(x1 )∆g(x2 ) ∆σ̂gg→QQ̄+X∆g(x)ALL ∼ ∫
∼⟨
⟩
,
(1.4)
σ̂gg→QQ̄+Xg(x)dx1 dx2 σ̂gg→QQ̄+X g(x1 )g(x2 )where σ̂gg→QQ̄+X and ∆σ̂gg→QQ̄+X denote spin-independent and spin-dependent crosssections of the heavy flavor production in the parton interaction, respectively. In addition to the sensitivity for the gluon polarization, this process extends the explored
Bjorken x region of ∆g(x) as follows. In the heavy flavor production, the large massof the quark (charm: mc ∼ 1.4 GeV/c2 and bottom: mb ∼ 4.8 GeV/c2 ) requires large
center-of-mass energy (> 2m) in the partonic interaction and it supports large energyscale for the reliable pQCD cross section calculations at the whole pT region. Hence,
diﬀerent from π 0 and jet production, any minimum limits of p√T are not required in theheavy flavor production. For the heavy flavor production in s = 200 GeV p + p collisions, charm quark production pp→cc̄ + X is dominant compared with bottom quark
production pp→bb̄ + X. The Bjorken√ x region covered with the charm production iscentered at as small as around 2mc / s = 1.4 × 10−2 . Therefore, ALL of this process
enable the world-first approach on the gluon polarization in small Bjorken x regionx < 2×10−2 , which is important to reduce the uncertainty on ∆G.
At PHENIX, hadrons containing heavy flavors are identified through their semileptonic decays to electrons and positrons, which are called “heavy flavor electrons”(HFe) [28, 29]. Hence, we measured the double-spin asymmetry of the heavy flavor
electrons at the PHENIX detector.
The measurement of the heavy flavor electron suﬀers large electron background. To
obtain the double-spin asymmetry of the heavy flavor electron, the double-spin asymmetry of the background process is to be subtracted from the measured double-spin
asymmetry of inclusive electron production. Hence, the background enlarges uncertainties of the double-spin asymmetry measurement, and introduces other uncertainties
from double-spin asymmetry of the background. The dominant background of theheavy flavor electron is electron pair from Dalitz decay of π 0 and η, and γ conversion in

4

CHAPTER 1. NUCLEON SPIN PHYSICS

materials. The electron pair has small pair angle because they are produced by real andvirtual photon conversions. To remove the background eﬃciently, Hadron Blind Detector (HBD), which was newly installed in PHENIX at 2009, plays an important rolein the present measurement. HBD is a position-sensitive gas Čerenkov counter, which
identifies electron using cluster signal created by its Čerenkov light. Whereas heavy flavor electron creates a single cluster on the HBD, electron pair with small opening angle<(∼
0.1 rad) creates a merging cluster which has about twice amplitude of signal charge.
Using the diﬀerence of cluster charges between the heavy flavor electron and the electron
pair, the HBD can distinguish them from inclusive reconstructed electrons. The HBD
also play important role to reduce K decay background. The K decay (Ke3 : K→eνπdecay) is other main background of the heavy flavor electron measurement especially at
low transverse momentum region pT < 1.0 GeV/c. The double-spin asymmetry of theKe3 background is not well known, and therefore it is diﬃcult to be subtracted fromthe measured asymmetry with accuracy. Since the HBD is installed around the beamvertex and detects only electrons created within radius of R∼10 cm, the Ke3 electrons
are significantly rejected by the HBD. The HBD enable us to measure the double-spin
asymmetry of the heavy flavor electrons.
In this paper, we describe the first measurement of this asymmetry with the newdetector HBD, and a resulting constraint on the gluon polarization with a leading order
pQCD calculation. For the sake of simplicity, we use the word “electron” to include
both electron and positron throughout this paper, and distinguish by charge where
necessary. In the following section, we summarize the physics framework for the spin
structure of nucleon and the current status of the gluon polarization in the nucleon,
which are also described elsewhere [30, 31, 32, 33, 34, 35, 36, 37].

1.2

Nucleon Spin Structure and Parton Model

1.2.1

Parton Model

Whereas a nucleon in a constituent quark picture is drawn as simple composition of
three quarks, namely uud for a proton and udd for a neutron, the one in a microscopic
picture is drawn as a complex of uncountable quarks, anti-quarks and gluons which
are distributed in momentum space in addition to the three valence quarks. This
microscopic picture is called “parton model”. Figure 1.1(a) and 1.1(b) show the two
diﬀerent pictures of a proton. The u-quark and d-quark appeared in the constituent
quark picture are not point-like particles, but they are dressed with quarks, anti-quarks
and gluons. Surprisingly, the dressed quarks have almost same physical characteristics(spin, charge, flavor, color, and so on) as the point-like u-quark and d-quark other thantheir mass and magnetic moment.
The parton model describes nucleon in high energy phenomena, where the nucleon
has relativistic velocity. The quark, anti-quark and gluon in the nucleon, which havethe same relativistic velocity, are called “partons” and represented as probability distributions of finding the particles of longitudinal momentum fraction x in the parent
nucleon. Namely a parton of momentum fraction x in a nucleon of momentum P has
momentum xP . The fraction x is called Bjorken x and varies in the range of 0 < x < 1.

1.2. NUCLEON SPIN STRUCTURE AND PARTON MODEL(a) Constituent quark model.

5(b) Parton model.

Figure 1.1: Comparison of (a) the constituent quark model and (b) the parton model.
The large spherical body represents a proton and the small spheres in the proton are
the quarks and anti-quarks. In the parton model, dashed lines represent gluons.

parton: i

xp

nucleon(1 - x)pp
Figure 1.2: A picture of the parton of momentum fraction x in the nucleon of momentump.
In the parton model, understanding of nucleon structure can be translated into understanding distributions of the quarks, anti-quarks and gluons in the nucleon as functionsof Bjorken x. In a primitive parton model, the transverse momentum is ignored. Figure 1.2 displays a picture of a parton of Bjorken x in a nucleon. Considering that quark
and anti-quark simultaneously appear (disappear) via a pair creation (annihilation), thenet amount of the quarks can be defined as a diﬀerence of the amounts of the quarks
and anti-quarks. The net quarks are called “valence quarks”, and the other quarks arecalled “sea quarks”.
The probability distributions of the partons are generally denoted asfi (x) ≡

dPi
,
dx

(1.5)

where i = u, d, s, ..., g and Pi is the probability of finding the parton of Bjorken x. The

6

CHAPTER 1. NUCLEON SPIN PHYSICS

distribution for a specific quark or gluon is also denoted as¯
quark, anti-quark : u(x), ū(x), d(x), d(x),s(x), s̄(x), ...
gluon: g(x).

(1.6)

These distributions are called parton distribution functions (PDFs).
The momentum conservation requires a sum relation of∑∫ 1
idx xfi (x) = 1.

(1.7)

0

As explained above, diﬀerence between the numbers of quarks and anti-quarks shouldbe the number of valence quarks. Then, requirements for charge of +1, baryon numberof +1, strangeness of 0 in the proton deduces the following sum rules,
∫1
∫1
dx
[u(x)
−
ū(x)]=
dx uv (x) = 2,]
∫ 01 [∫01¯
dx d(x) − d(x) = 0 dx dv (x) = 1,
∫01
∫1
dx
[q(x)
−
q̄(x)]=
dx qv (x) = 0
0
0

(1.8)
for q = s, c, b, t,

where qv (x) represents a PDF for the valence quark.
The PDFs were studied by various deep inelastic scattering (DIS) experiments andtheir functional shapes were already determined with good accuracy. The experimental
determination of the PDFs are described in Sec. 1.3. One of the interesting discoveries
from the measurements is that the quarks and anti-quarks carry only about half of
momentum of the nucleon. Another half is attributed to gluons. Thus the gluons,
which are just radiations from the quarks and anti-quarks, play an important role todescribe the proton structure. Before explaining the experimental determination of
the PDFs, it is worth to introduce a polarized parton distribution function (polarizedPDF), which is a main subject of this work.

1.2.2

Polarized Parton Distribution Function

The next step to understand the proton structure is understanding how the partons form
the proton spin sz /h̄ = 1/2. For this purpose, we introduce the polarized PDF. In thispolarized parton model, new PDFs which represent spin-dependent parton distributions
in a proton polarized along direction of n are defined asdPi+(x; n),dx
dPi−≡ dx (x; n),fi+ (x; n) ≡fi− (x; n)(1.9)

where Pi± (x; n) represents the probability of finding a parton with Bjorken x and spin
±1/2 for quarks or ±1 for gluons along the direction n. By definition, unpolarized PDF
can be written asfi (x) = fi+ (x; n) + fi− (x; n)(1.10)

1.3. EXPERIMENTS FOR PDF MEASUREMENT

7

for any n. Especially when n is oriented to the proton momentum direction, longitudinalpolarized PDF is defined as net polarization of the parton, namely∆fi (x) ≡ fi+ (x; p/|p|) − fi− (x; p/|p|).

(1.11)

Here, fi± (x; p/|p|) is interpreted as PDF of finding helicity ± parton in helicity + proton,
respectively.1
Because the partons are bound together by means of QCD interaction, the parity
conservation of QCD requiresfi+ (x; −p/|p|) = fi+ (x; p/|p|),fi− (x; −p/|p|) = fi− (x; p/|p|).

(1.13)(1.14)

The first equation represents that the PDF of helicity − parton in helicity − proton issame as the PDF of helicity + parton in helicity + proton and the second equation isfor opposite helicity parton.

1.3

Experiments for PDF Measurement

1.3.1

Deep Inelastic Scattering (DIS) experiment

To investigate the functional shape of the quark PDF, DIS is an ideal method. The
DIS is lepton-nucleon inelastic scattering, l N → l + X, where time-scale of leptonparton interaction is enough shorter than one of bounding interaction among partons(impulse approximation). The diagram of the DIS is shown in Fig. 1.3(a). In the partonmodel, the DIS can be interpreted as interaction between the lepton and a parton in
the nucleon at a leading order diagram as shown in Fig. 1.3(b). Since gluon can notinteract with the lepton in the leading order, the interacting parton in the diagram isonly quark.
We denote four-dimensional momentum of the nucleon as p and four-dimensional
momentum of the incident and outgoing lepton as k and k ′ , respectively. Especiallyin a fixed target experiment with a lepton beam, p can be represented to be (M, 0) in
the laboratory frame, where M is the nucleon mass. The virtual photon momentumis q = k − k ′ , which can be easily measured in the DIS experiment by detecting theincident and outgoing lepton momenta. q gives energy-scale of the DIS experiment as
Q2 ≡ − q 2 . The impulse approximation requires large energy-scale such as,
Q2 ≫M 2 .

(1.15)

1

On the other hand, when n is perpendicular to the proton momentum direction, we can also define
transverse polarized PDF as
δfi (x) ≡ fi+ (x; r) − fi− (x; r),(1.12)

where r denotes perpendicular direction to the proton momentum. However, it is beyond the scope of
this paper to explain the transversely polarized PDF. For detailed description about the transversity,
please see elsewhere [38, 39].

8

CHAPTER 1. NUCLEON SPIN PHYSICS

k'leptonkk'
leptonkγ*γ*partonq=k–k'

q=k–k'

xp

nucleonnucleon

W

p

p

W

Time

Time
(a) Diagram of DIS.

(b) Diagram of DIS in the parton model.

Figure 1.3: (a) Diagram of DIS in a leading order interaction. (b) Diagram of DIS in
the parton model, which is represented as the lepton-parton scattering.
When we consider the virtual photon interacts with a parton of Bjorken x, and therefore
the parton has momentum of xp, the mass consistency of the parton before and after
the interaction requires(xp)2 = (xp + q)2 .

(1.16)

This relation can be translated to bex=

Q2,
2p · qwhere Q2 ≡ −q 2 .

(1.17)In a fixed target experiment, p · q is equal to M ν, where ν is energy transfer from thelepton in the laboratory frame, namely ν = k0 − k0′ . This means Bjorken x of theinteracting parton can be obtained by measuring only the kinematics of the lepton.
Figure 1.4 shows kinematics of the DIS experiment in the nucleon rest frame. Using
the kinematic variables in Fig. 1.4, unpolarized and polarized cross section of DIS canbe represented as()
4α2 E ′2 2ν1
d2 σDIS
′
2 θ
2
2 θ
2
(E, θ, E ) =sin F1 (x, Q ) + cos F2 (x, Q )
,
′
4
dΩdE
Q
M
2
2
ν
(1.18)
)
(
2
2
′
2
′
d ∆σDIS
2α E E + E cos θQ1(E, θ, E ′ ) =g1 (x, Q2 ) −g2 (x, Q2 )λN λl ,
′
2
dΩdE
Q E
M
Mν
ν
(1.19)
where λN and λl represent the initial helicities (±1) of the nucleon and lepton in thecenter-of-mass system. F1 , F2 , g1 , and g2 are called structure functions which describe
the nucleon structure.

1.3. EXPERIMENTS FOR PDF MEASUREMENT

9k'=(E',k')k=(E,k)leptonθ
q=(ν,q)N

p=(M,0)Figure 1.4: Kinematics of the DIS experiment in the nucleon rest frame.
According to Appendix A, the relations between the structure functions and thequark PDFs can be deduced as1∑ 2
e fq (x),2 q q∑
F2 (x, Q2 ) = x
e2q fq (x),F1 (x, Q2 ) =(1.20)(1.21)q1∑ 2
g1 (x, Q2 ) =e ∆fq (x),2 q q(1.22)

g2 (x, Q2 ) = 0.

(1.23)The measurement of the structure functions with the DIS experiment gives us information of the unpolarized and polarized PDFs.

1.3.2

Semi-Inclusive DIS (SIDIS) experiment

As discussed above, the DIS experiment can access to quark PDF. However, Eqs. 1.20 1.23 mean that the measurement can not separate the structure functions into eachflavor PDF. To investigate flavor-dependent quark PDF, semi-inclusive DIS (SIDIS)
measurement which detects high momentum π or K as well as the scattered lepton
in the final state is a useful tool and is performed at the SMC (CERN) [18], HERMES (DESY) [19, 20], and COMPASS (CERN) [21, 22] experiments. A diagram of
π/K
the SIDIS interaction is shown in Fig. 1.5. The process Di
in the figure represents
fragmentation of scattered particle i into π or K. In contrast to the inclusive DIS exper-

10

CHAPTER 1. NUCLEON SPIN PHYSICS

iment, the detected particles in the SIDIS experiment are high-energy2 charged π or K
in coincidence with the scattered lepton. For large energy fraction Eπ/K /Eγ ⋆ →1 where
Eπ/K (Eγ ⋆ ) is energy of the π or K (virtual photon), the most probable occurrence isthat the detected π and K contain the struck quark or anti-quark in their valence Fockstate. Therefore, they have diﬀerent sensitivity for the flavor of the struck quark andsolve the flavor-separated quark PDF [40].

k'leptonk

nucleonγ*

q=k–k'

parton: i xp

π/K

Diπ/K

p

Time
Figure 1.5: Diagram of SIDIS.
In leading order approximation, the structure functions after tagging a hadron h(= π ± , K ± ) can be represented as
∫ 1
1∑ 2
h
2
F1 (x, Q ) ∼e fq (x)dzDqh (z, Q2 ),(1.24)
2 q q
zmin
∫ 1
∑
h
2
2
F2 (x, Q ) ∼ x
eq fq (x)dzDqh (z, Q2 ),(1.25)
q

zmin

∫ 1
1∑ 2
h
2
eq ∆fq (x)dzDqh (z, Q2 ),g1 (x, Q ) ∼
2 q
zmin

(1.26)

g2h (x, Q2 ) ∼ 0,

(1.27)

where zmin ∼ 0.2. Here Dqh (z, Q2 ) [41] is the fragmentation function for the struck
quark or anti-quark q to produce the hadron h carrying energy fraction z = Eh /Eq in∫1
the target rest frame. Since zmin dzDqh (z, Q2 ) varies among the initial quark q and thefinal hadron h, the measurement of the structure functions with tagging several hadrons
enable us to solve the flavor-dependent PDFs.
2

typically greater than 20% of the energy of the incident virtual photon

1.3. EXPERIMENTS FOR PDF MEASUREMENT

1.3.3

11

Hadron collision experiment

Since DIS experiments use virtual photon to probe the nucleon, they can not access thegluon contribution in leading order process. Hence, hadron collisions play a role for thecomplement study of the gluon PDF, which can access gg, gq, and qq interactions at aleading order. In the hadron collisions, the gluon PDF can be studied by measuring crosssections of specific particle productions. Hadron collider experiments are performed at
Tevatron (pp̄) and LHC (pp), and polarized hadron collider experiment is at RHIC (⃗pp⃗),which is a world-only polarized hadron collider.
The factorization theorem [42, 43, 44] describes relations between the measured
cross sections and the parton PDF. It states that large momentum-transfer reactions
can be factorized into long and short-distance contributions. The long-distance components contain information of PDF. The short-distance interaction is a hard scatteringof partons and can be calculated from perturbation theory. While PDF describes universal properties of the nucleon and is same in diﬀerent reactions, the short-distanceinteraction varies depending on the hard scattering.
For an example, we consider the cross section of π production pp → π + X, where π
has large transverse momentum pT , ensuring large momentum transfer. The factorization theorem divides the process into three steps as shown in Fig. 1.6, and represents
the cross section as∫∫∫dσ(pp → π + X) = Σa,b,c dxa dxb dzc
fa (xa , µ)fb (xb , µ)dσ̂ab→c (xa pA , xb pB ,pπ, µ)Dcπ (zc , µ),zc(1.28)

where the sum is over all contributing partonic channels a+b→c+X with the associatedpartonic cross section dσ̂ab→c . fa,b describe the PDF of partons a and b respectively,
which include gluons as well as quarks. The outgoing particle c hadronizes into π withfragmentation function Dcπ . Any factorization of a physical quantity into contributions
associated with diﬀerent length scales will rely on a factorization scale that defines theboundary between what is referred to as short-distance and long-distance. In this case,this scale is represented by µ in Eq. 1.28. Essentially, µ is arbitrary and does not change
the resulting cross section. Then, the dependence on µ of the calculated cross section
is interpreted as an uncertainty in the theoretical prediction. This dependence on µ
decreases in higher-order perturbation theory. The partonic cross section dσ̂a+b→c canbe estimated with perturbation theory, and the fragmentation function Dcπ (z) has beenstudied at e+ e− colliders. Therefore, measuring the cross section, we can study the
PDF fa,b (x).
Using the same technique, longitudinally polarized PDF can be investigated by
performing longitudinally polarized pp collisions at RHIC. There are four helicity combinations of the colliding two protons. For these combinations, we can define four
helicity-dependent cross sections for specific particle production asdσ++ ,

dσ+− ,

dσ−+ ,

dσ−− ,(1.29)

12

CHAPTER 1. NUCLEON SPIN PHYSICS

proton: A
pA

pB

π
xa
parton: a
parton: bxb

σa+b→cπc

D c(z)

pπ

hard process
a+b→c+X

proton: B

Time
Figure 1.6: Diagram of π production in pp collisions. The process is factorized into
three parts: PDF, partonic hard scattering, and fragmentation into hadrons.
where two signs in the subscript represent helicity signs of the colliding two protons
in the center-of-mass system. If the interaction in the process involves only QCD and
QED, the cross section satisfies the parity conservation:dσ++ = dσ−− ,dσ+− = dσ−+ .

(1.30)The unpolarized cross section in Eq. 1.28 can be represented using the helicity-dependent
cross sections asdσ =1(dσ++ + dσ+− ) .
2(1.31)The polarized cross section is also defined using the helicity-dependent cross sections
as
d∆σ ≡

1
(dσ++ − dσ+− ) .
2(1.32)

This polarized cross section can be represented with polarized PDF like as Eq. 1.28.
For example, the polarized cross section of the π production can be represented as∫∫∫πd∆σ = Σa,b,c dxa dxb dzc∆fa (xa , µ)∆fb (xb , µ)d∆σ̂ab→c (xa pA , xb pB ,pπ, µ)Dcπ (zc , µ).
zc(1.33)

1.3. EXPERIMENTS FOR PDF MEASUREMENT

13d∆σ̂ab→c is partonic polarized cross section which is defined as)
1 ( +++−d∆σ̂ab→c ≡ dσ̂ab→c− dσ̂ab→c,(1.34)
2+±
where dσ̂ab→c
represents partonic cross section of a+b→c+X process when a and b havehelicities of + and ±, respectively. The partonic polarized cross section d∆σ̂ab→c can+±be also calculated from perturbation theory as well as dσ̂ab→c . dσ̂ab→cfrom LO pQCD
calculations are listed in Table 1.1 and partonic asymmetries âLL , which are definedas âLL ≡ d∆σ̂ab→c /dσ̂ab→c , are shown in Fig. 1.7. Using the calculated d∆σ̂ab→c , thepolarized PDF ∆fa,b can be studied by measuring d∆σ. Experimentally, the polarizedcross section is measured as the ratio to the unpolarized cross section, namelyd∆σ,(1.35)
dσ
because correlated systematic uncertainties on dσ and d∆σ are canceled out. The
asymmetry of the cross section ALL is called “longitudinal double-spin asymmetry” or“double-helicity asymmetry”.
ALL ≡Table 1.1: Table of QCD cross sections for definite helicity combinations from LO
pQCD calculations [45, 39]. The calculations ignore the quark mass. A common factor
παS2 /s2 is not included in all the cross section.
processqα qβ →qα qβ
qα q̄β →qδ q̄γdσ̂ ++ /dt
[
( 2
) ]
8 s2
s
2 s2
+ δαβ u2 − 3 tu
9 t2
(
)
s2
8
δ
δβγ
αδ
2
9
t

dσ̂ +− /dt
8
9
8
9

( 2

(

2
u
+ δαβ ut 2
t2

)

2

δαδ δβγ ut22+δαβ δδγ t +u
s2

2

2

(
qg→qg

s2
2s2
− 89 su
t2

gg→q q̄0

q q̄→gg

0

gg→gg

9
2

(

)

2s2
− su
− ust2
ut
t2

)

9
2

(

− 23 δαγ δαβ δδγ ust
( 2
)
2u
8 u2
−
2
t
9 su
( 2 2
)
1 u +t3 t2 +u2
−
3 ut
4 s2
( 2 2
)
64 t +u
16 t2 +u2
−
27 ut
3 s2
2

)

6 − 2sut − su
− ust2 − 2ut
t2
s2

)

Experimental Approach to ALL
From the definition of ALL represented as Eq. 1.35 and relations of Eq. 1.30, 1.31 and
1.32, the helicity-dependent cross section of specific particle production is parameterizedusing ALL as
dσλA λB (pT , y) = dσ0 (pT , y) [1 + λA λB ALL (pT , y)] ,(1.36)

CHAPTER 1. NUCLEON SPIN PHYSICS

a^LL

14
1

C

A

0.75

B

0.5

D

0.25
0

A gg→gg D qq→qq
B qq→qq E gg→qq
C qq’→qq’ E qq→gg
C qq’→qq’ E qq→gγ
C qg→qg E qq→q’q’
C qg→qγ E qq→ll

-0.25
-0.5
-0.75

E
-1
-0.8

-0.4

0

0.4

0.8

cosθ
Figure 1.7: The double helicity asymmetry of the partonic cross section (âLL ) as afunction of the scattering angle on the partonic center-of-mass system [46]. âLL can bederived from helicity-dependent cross sections listed in Table 1.1.
where dσ0 is the unpolarized cross section of this process, which can be obtained byaveraging spin of the initial protons, and λA,B denotes helicity (±) of the initial two
protons. In the equation, the cross section and spin asymmetry are represented asfunctions of transverse momentum pT and rapidity y of the produced particle.
Practically, the protons in the polarized beams are not fully polarized. When wepolarize the proton beam to a specific direction, quantized spin state of major protons
are in parallel to the direction, but ones of other protons are anti-parallel to the direction. The beam polarization P represents how much protons in the polarized beam arein the parallel spin state to the polarization direction, and is defined as
P ≡I+ − I−,
I+ + I−(1.37)

where I+ (I− ) is beam intensity for protons in the parallel (anti-parallel) spin state.
For the ALL measurement, the polarization direction is set to be longitudinal along thebeam axis. By definition, the beam polarization P varies from 0 to 1 since I+ is greaterthan I− , in general.
From Eq. 1.36, the experimentally detected yield of specific particle NhA hB can be1.4. DGLAP EQUATION

15

written as
NhA hB (pT , y) = ϵeﬀ (pT , y)LhA hB dσ0 (pT , y) (1 + hA hB PA PB ALL (pT , y)) ,(1.38)

where hA,B denotes the polarization direction of the beam A, B, parallel (+1) or antiparallel (−1) to the beam momentum, and LhA hB is integrated luminosity of pp collisions
crossing in polarization combination of (hA , hB ). ϵeﬀ is detection eﬃciency of the particle
including detector eﬃciency, acceptance, particle ID eﬃciency, and trigger eﬃciency.
Ideally, the eﬃciency ϵeﬀ does not depend on the beam polarization hA,B . However,
the eﬃciency can be slightly changed in diﬀerent polarizations due to the following
experimental reasons: (1) bunch-by-bunch diﬀerence in length of beam bunch and (2)readout electronics specific. These eﬀects are discussed in Sec. 4.3.2. The following
description ignores the eﬀects for the sake of simplicity.
The detected yields in the four polarization combination can be also written separately as
N++ (pT , y)N−− (pT , y)
N+− (pT , y)N−+ (pT , y)

=
=
=
=

N0 (pT , y) (1 + PA PB ALL (pT , y)) ,N0 (pT , y) (1 + PA PB ALL (pT , y)) /r−− ,N0 (pT , y) (1 − PA PB ALL (pT , y)) /r+− ,N0 (pT , y) (1 − PA PB ALL (pT , y)) /r−+ ,(1.39)

where N0 ≡ ϵeﬀ L++ dσ0 represents yield of the particle in unpolarized pp collisions underintegrated luminosity of L++ . r±± is relative luminosity, which is defined asr−− ≡ L++ /L−− ,r+− ≡ L++ /L+− ,r−+ ≡ L++ /L−+ .

(1.40)

Experimentally, the yields N±± are observable and the beam polarization PA,B andrelative luminosity r±± can also be measured as accelerator performance. Therefore,
the double spin asymmetry ALL can be obtained by fitting the observed yield N±±
according to Eq. 1.39 where ALL and N0 are set as two fitting parameters. It is worth
to mention that, if the eﬃciency ϵeﬀ does not depend on the direction of the beampolarization, Eq. 1.39 includes only the detected yields and the beam polarizations,and therefore any detection eﬃciencies do not aﬀect the ALL measurement.

1.4

DGLAP Equation

The parton model will be improved by introducing the quark and gluon interaction
described in QCD. As consequence of the Bjorken scaling violation, the description of
the unpolarized PDF q(x) and g(x) also depend on energy-scale of renormalization Q2 .
The Q2 evolution of those distributions is performed by DGLAP equations [47],)(∫αS (Q2 ) 1 dyxxdq(x, Q2 )22=q(y, Q )Pqq ( ) + g(y, Q )Pqg ( ) ,dlogQ2
2π
y
y
0 y
)
(
∫
1
dg(x, Q2 )αS (Q2 )
dy ∑
x
x
2
2
=
q(y, Q )∆Pgq ( ) + g(y, Q )∆Pgg ( ) , (1.41)
dlogQ2
2π
y
y
0 y
q

16

CHAPTER 1. NUCLEON SPIN PHYSICS

where Pij (z) are the splitting functions, representing the probability of finding a newparton i carrying a momentum fraction z(< 1) of a parent parton j. The splitting
functions are calculated from pQCD calculations as a probability that the real particle
i separates into the real particle j with a colinear momentum and a virtual particle.
For the polarized case, the Q2 evolution equations are drawn in analogous way withrespect to the unpolarized case. The Q2 dependence of the polarized PDF is given
(
)
x
x
2
2
∆q(y, Q )∆Pqq ( ) + ∆g(y, Q )∆Pqg ( ) ,
y
y
(
)
∫ 1
2
∑
x
αS
dy
x
d∆g(x, Q )∆q(y, Q2 )Pgq ( ) + ∆g(y, Q2 )Pgg ( ) . (1.42)
=
2
dlogQ
2π 0 yyyq,q̄d∆q(x, Q2 )αS=
dlogQ2
2π

∫ 1

dy
0 y

In the equation, the spin-dependent splitting functions ∆Pij (z) are defined as ∆Pij (z) =Pij+ (z) − Pij− (z), where Pij± (z) represents the splitting functions of finding a parton iwith helicity ± from a parent parton j with helicity +. It is convenient to separate theNSpolarized quark PDF in a flavor singlet ∆q S (x) and non-singlet ∆q3,8(x) defined as∆q S ≡ ∆U (x, Q2 ) + ∆D(x, Q2 ) + ∆S(x, Q2 )∑=∆q(x, Q2 ),q,q̄∆q3N S∆q8N S≡ ∆U (x, Q2 ) − ∆D(x, Q2 ),≡ ∆U (x, Q2 ) + ∆D(x, Q2 ) − 2∆S(x, Q2 ),(1.43)

where∆U (x, Q2 ) ≡ ∆u(x, Q2 ) + ∆ū(x, Q2 ),¯ Q2 ),∆D(x, Q2 ) ≡ ∆d(x, Q2 ) + ∆d(x,∆S(x, Q2 ) ≡ ∆s(x, Q2 ) + ∆s̄(x, Q2 ).
Hence, the spin-dependent DGLAP equations, which represent evolution of the polarized PDF, are represented as∫
NS(x, Q2 )d∆q3,8x
αS 1 dy N S
2
(y,
Q
)∆P
(
=
∆q
),
qq
3,8
dlogQ22π 0 yy)(∫αS 1 dyx
d∆q S (x, Q2 )
x
2
S
2
=
∆q (y, Q )∆Pqq ( ) + 2f ∆g(y, Q )∆Pqg ( ) ,dlogQ22π 0 y
y
y
)
∫ 1 (
2
d∆g(x, Q )
αS
dy
x
x
=
∆q S (y, Q2 )∆Pgq ( ) + ∆g(y, Q2 )∆Pgg ( ) , (1.44)2dlogQ
2π 0 yyywhere f is the number of quark flavors liberated into the final state (f = 3 below thecharm production threshold).

1.5. CURRENT STATUS OF NUCLEON SPIN STRUCTURE

1.5

Current Status of Nucleon Spin Structure

1.5.1

Determination of Unpolarized PDF

17

The unpolarized PDF can be determined from data of DIS and hadron hard-scatteringexperiments. Recently, the determination of the unpolarized PDF has been performedby several groups, CTEQ [48, 49], MSTW [50], NNPDF [51, 52], HERAPDF [53],ABDM [54], and GJR [55, 56]. These groups provide PDFs with NLO and next-to-next
leading order (NNLO) calculations, except CTEQ group which provides an NLO result.
Figure 1.8 shows the resulting unpolarized PDFs at a scale Q2 = 4 GeV2 from the
CTEQ group (CTEQ6). The functional from that CTEQ uses is()A5
xf (x, Q20 ) = A0 xA1 (1 − x)A2 eA3 x 1 + eA4 x,(1.45)

where Q0 is an initial scale of the evolution which is set to Q0 = 1.3 GeV, and A0,...,5
is an independent parameter. The PDF at all higher Q is determined from NLO
DGLAP equations. f (x, Q) represents PDFs of parton flavor combinations uv (x)≡u− ū,¯ g(x), and (ū + d)(x).
¯¯ To
dv (x)≡d − d,
CTEQ group assumes s(x) = s̄(x) = 0.2(ū + d).
¯ is parameterized as
distinguish the ū and d¯ distributions, the ratio d/ū¯ Q0 )d(x,= A0 xA1 (1 − x)A2 + (1 + A3 x) (1 − x)A4 .
ū(x, Q0 )(1.46)<As Fig. 1.8 shows, valence quarks are dominant in large Bjorken x region, 10−1 ∼x.
On the other hand, sea quarks and gluons are dominant and increasing in small Bjorken<
x region, x ∼10−1 . Especially, large fraction of the partons in this region consists ofgluons, which is about ten times larger than other sea quarks.

1.5.2First Moment of Polarized PDF

Angular momentum conservation
Considering angler momentum conservation as well as momentum conservation shownin Eq. 1.7, spin sz /h̄ = 1/2 of a longitudinally polarized proton is decomposed as
1∑
1
=
2
2 q

∫ 1

(

2

2

)

∫ 1
dx∆g(x, Q2 ) + Lq,g ,dx ∆q(x, Q ) + ∆q̄(x, Q ) +0(1.47)

0where q(x) and g(x) represent quark and gluon PDF respectively and Lq,g represents
orbital angular momentum of the quark and gluon. This equation is also written as11= ∆Σ(Q2 ) + ∆G(Q2 ) + Lq,g ,(1.48)
2
2∫1∫1
where ∆Σ(≡ 0 dx∆q S ) and ∆G(≡ 0 dx∆g) represent total spin polarizations of thequark and anti-quark, and the gluon, respectively.

CHAPTER 1. NUCLEON SPIN PHYSICS
x f(x)

18
1.2

u
d
u
d

1
0.8

s =s
c =c
gluon (× 0.2)

0.6
0.4
0.2
0
-4
10

-3

10

-2

-1

10

10

1
Bjorken x

Figure 1.8: Unpolarized PDF at Q2 = 4 GeV2 as a function of Bjorken x obtained by
CTEQ group [48] (CTEQ6). For gluon PDF, g(x) × 0.2 is displayed in this figure forconvenience.
Expectation of ∆Σ from the constituent quark model
In the constituent quark model, the simple SU(6) proton wave function1|p ↑⟩ = √ (2|u↑u↑d↓⟩ − |u↑u↓d↑⟩ − |u↓u↑d↑⟩ + cyclic)
18(1.49)

yields ∆Σ = 1, because the proton spin consists of only the constituent-quark spin. In
addition, when we take into account Bag models, the eﬀects of confinement and chiralsymmetry violation are to be considered [57]. From the eﬀects, Bag quarks obtain small
masses, and therefore relativistic eﬀect can not be ignored [58, 59, 60]. The relativistic
eﬀect creates orbital angular momentum even in the lowest partial wave. The orbital
angular momentum corrects the ∆Σ expectation into ∆Σ =∼ 0.65 [30, 31].

First moment of g1 (x)The first moment of the polarized structure function g1 (x), defined as
∫ 1

1∑ 2
eq
Γ1 ≡ dxg1 (x) =
q,q̄
2
0

∫ 1
dx∆q(x),(1.50)

0

has an important information: the quark helicity contribution to the nucleon spin.

1.5. CURRENT STATUS OF NUCLEON SPIN STRUCTURE
Defining ∆U , ∆D, and ∆S as

19

∫ 1

∆U ≡
∫ 01
∆D ≡

dx (∆u(x) + ∆ū(x)) ,(1.51)()¯
dx ∆d(x) + ∆d(x),(1.52)

dx (∆s(x) + ∆s̄(x)) ,(1.53)

∫ 01
∆S ≡
0

and neglecting the heavy quarks, the first moment for the proton can be written as
(
)
1 4
1
1
p
Γ1 =
∆U + ∆D + ∆S
(1.54)
2 9
9
9
1
1
1
=
(∆U − ∆D) +(∆U + ∆D − 2∆S) + (∆U + ∆D + ∆S) . (1.55)
1236
9Assuming the isospin symmetry, the first moment for the neutron can be also obtainedby ∆U ↔∆D.
Using Operator Product Expansion (OPE) method, the three terms in Eq. 1.55 are
related with the expectation values ai of the proton matrix element of a flavor SU(3)octet of quark axial-vector currents [3, 4]. ai is defined asi⟨P, S|J5µ|P, S⟩ = M ai Sµ ,i = 1...8,(1.56)iwhere M is related to the mass of the quarks. The currents J5µ
are given by the λi , theGell-Mann matrices as
 
u
λi
i
(1.57)
ψ =  d ,J5µ= ψ̄γµ γ5 ψ,
2swhere ψ represents annihilation operators of flavor SU(3) quarks in free quark field.
The element a0 is given by the singlet operator
0⟨P, S|J5µ|P, S⟩ = M a0 Sµ ,(1.58)0J5µ= ψ̄γµ γ5 ψ.

(1.59)

Finally the correspondence of the expectation values ai to the terms of Eq. 1.55 is asfollows
a3 = ∆U − ∆D,
a8 = ∆U + ∆D − 2∆S,
a0 = ∆U + ∆D + ∆S = ∆Σ.

(1.60)(1.61)(1.62)(1.63)The elements a3 and a8 are well known from the neutron β decay and the spin 1/2
hyperon decays in the SU(3) baryon octet. These can be expressed in terms of theparameters F and D, obtained from the aforementioned decays [33, 61],a3 = F + D = |gA | = 1.2694 ± 0.0028,
a8 = 3F − D = 0.585 ± 0.025,(1.64)(1.65)

20

CHAPTER 1. NUCLEON SPIN PHYSICS

where gA is the axial coupling constant.
The QCD improved parton model leads to some corrections [6] and modifies Eq. 1.55
to
((
)
)
1
1
4
p,n
2
2
Γ1
=
±a3 + a8 EN S (Q ) + a0 ES (Q ) ,(1.66)
1233()2
αS (Q2 )αS (Q2 )2EN S (Q ) ≡ 1 −− 3.58...,(1.67)ππ()2αS (Q2 )αS (Q2 )2...
(1.68)ES (Q ) ≡ 1 − 0.333− 1.10ππwhere Γn1 represents the first moment of g1 (x) for neutron.
Bjorken Sum Rule
Using Eq. 1.66, we can deduce the Bjorken sum rule,1Γp1 − Γn1 = a3 EN S (Q2 ).
6(1.69)The Bjorken sum rule provides the relation between isospin element a3 (Eq. 1.64)
and diﬀerence of the proton and neutron first moments. The Bjorken sum ruleassumes isospin SU(2) symmetry of quark dynamics in the nucleon.
Ellis-Jaﬀe Sum Rule
The Ellis-Jaﬀe sum rule assumes that ∆s = ∆s̄ = 0, and then a8 = ∆Σ(= a0 ).
Therefore, using Eq. 1.66, the Ellis-Jaﬀe sum rule is represented as
Γp,n
= ±
1

)
1
1 (
a3 EN S (Q2 ) + a8 EN S (Q2 ) + 4ES (Q2 ) ,1236(1.70))
1 (a8 EN S (Q2 ) + 4ES (Q2 ) .
18(1.71)orΓp1 + Γn1 =The validity of the Ellis-Jaﬀe sum rule corresponds to the validity of flavor SU(3)symmetry of quark dynamics in the nucleon and ∆S = 0.
Measurement of the sum rules
To check the validation of the sum rules, various polarized DIS experiments [5, 6, 7, 8, 9]
have been performed. These experiments measured g1 (x, Q2 ) for proton, deuteron, andneutron. The result for proton as function of Q2 and Bjorken x is shown in Fig. 1.9,
and the results for all particles as function of Bjorken x is shown in Fig. 1.10. The
deuteron g1 (x) can be interpreted as a combination of the proton and neutron g1 (x),
(
)
1 p
3
d
n
(1.72)
g1 (x) = (g1 (x) + g1 (x)) 1 − ωD ,22

g1p(x, Q2) + cn

1.5. CURRENT STATUS OF NUCLEON SPIN STRUCTURE

21

EMC

4

x=0.006

SMC
E143

3.5

E155

x=0.015HERMESx=0.025

CL A S

3

COMPASS

x=0.035
LSS 05x=0.0492.5

x=0.0772

x=0.120x=0.1701.5
x=0.2401x=0.340x=0.4800.5

x=0.740

0
1

10

2

2

10

Q [GeV2]

Figure 1.9: World data for g1 (x, Q2 ) or the proton with Q2 > 1 GeV2 and W ≡√(p + q)2 > 2.5 GeV. For clarity a constant cn = 0.28 × (11.6 − n) has been added
to the g1 (x) values within a particular x bin starting with n = 0 for x = 0.006. Also
shown is the QCD fit curves [62].

where ωD = 0.05 ± 0.01 takes into account the D-state admixture to the deuteron wave
function.
Using the g1 (x)∫ obtained from the COMPASS experiment, the evolution of the1
Bjorken sum rule xmin dxg1p−n (x) as a function of xmin as well as the Ellis-Jaﬀe sum∫1
rule xmin dxg1p+n (x) are estimated. The results are shown in Fig. 1.11. Note that thefirst moment of the g1p+n (x) saturates at x ∼ 0.05. Figure 1.11 represents that whereas
the Bjorken sum rule is satisfied well, the Ellis-Jaﬀe sum rule breaks. The violation of
the Ellis-Jaﬀe sum rule corresponds to polarization of the strange quarks in the nucleon
or flavor SU(3) violation in the quark dynamics in the nucleon.

CHAPTER 1. NUCLEON SPIN PHYSICS
1

xg p

22
0.08
EMC

0.07

SMC
E143

0.06
E155

0.05
0.04

HERMES
CL A SCOMPASS

0.03
0.02
0.01
0

1

xg d

0.04
SMC
E143

0.03

E155
HERMES

0.02

CLAS
COMPASS

0.01

0

1

xg n

-0.01
JLAB Hall A

0.02

E142
E154

0.01

HERMES

0
-0.01
-0.02
-0.03
10-2

10-1

x

1

Figure 1.10: World data on xg1 (x) as a function of x for the proton (top), deuteron(middle), and neutron(bottom) at the measurement Q2 . Only data points for Q2 >√1 GeV2 and W ≡ (p + q)2 > 2.5 GeV are shown.
∆Σ determination and remaining contribution
Using the measured first moments of proton and deutron and Eq. 1.66, COMPASS
obtained∆Σ(Q2 → inf) = 0.33 ± 0.03(stat) ± 0.05(syst).

(1.73)

1.5. CURRENT STATUS OF NUCLEON SPIN STRUCTURE

23

1∫ g (x) dx

x min

1

0.2

COMPASS data

Bjorkenp

g1 = g1 − g n1p

g1 = g1 + g n1
0.15

Ellis−Jaffe

0.1

0.05

0

10-2

10-1

x min

1

Figure 1.11: Convergence of the first moment integral of g1 as a function of the lower
integration limit xmin for the Bjorken integral (isospin non-singlet) and the Ellis-Jaﬀe
integral (iso-singlet) from the COMPASS proton and deuteron data at Q2 = 3 GeV2 .
The arrows indicate the theoretical expectations.

This value can not be explained by the expectation values ∆Σ = 1 (simple SU(6)) nor∆Σ =∼ 0.65 (SU(6) including the relativistic eﬀect). The remaining proton spin ∼ 70%
must be attributed to the gluon polarization ∆G and/or orbital angular momentum
Lz . Other than the orbital angular momentum from the relativistic eﬀect explained
above, from our empirical knowledge, system in the ground state basically does nothave its orbital angular momentum. 3 Therefore, it is natural to expect that the gluon
polarization ∆G in Eq. 1.48 contribute the remaining proton spin.4
3

Another reason is the technical diﬃculty for the orbital angular momentum measurement.
The interest to ∆G came also from an axial anomaly correction to ∆Σ [63, 64]. The axial anomaly,
which is a fundamental property in quantum field theory, produces an additional term in divergent
diﬀerential of the axial current as
4αS∂µ J5µ = nf8∑2π a=1ρσ
ϵµνρσ Gµνa Ga .

(1.74)

This anomaly term modifies the relation between singlet element a0 and quark polarization ∆Σ to bea0 = ∆Σ − nf

αS
∆G.
2π

(1.75)

Therefore, the violation of Ellis-Jaﬀe sum rule can be attributed to the gluon polarization ∆G. However
it requires very large ∆G, ∆G > 1, which is not preferable from recent experimental data.

24

1.5.3

CHAPTER 1. NUCLEON SPIN PHYSICS∆g(x) Measurements

The first ALL measurement attempting to look at gluon polarization was made by the
FNAL E581/704 Collaboration using a 200 GeV polarized proton beam and a polarized
proton target [65]. They measured ALL for inclusive multi-γ and π 0 π 0 production
consistent with zero within their sensitivities, which suggested ∆g/g(x) is not so large
in the region of 0.05 < x < 0.35.
DIS experiments can determine ∆g(x) via Q2 evolution of g1 (x, Q2 ) according to
DGLAP equations in Sec. 1.4. Other method to determine ∆g(x) with DIS is to tag
charmed mesons or high pT hadron pairs in the final state. Both tagging identifyγ ∗ g→q q̄ interaction process in the partonic level as shown in Fig. 1.12. The method offinding high pT hadron pair leads to large statistics, but these have larger backgroundcontributions from QCD Compton processes and fragmentation. The charmed meson
method is a clean process, but low statistics. The high-pT hadron tagging method
was employed at the HERMES [66], SMC [67], and COMPASS [68] experiments, andthe charmed meson tagging method was employed at the COMPASS experiment [69].
Results of ∆g/g(x) from these SIDIS measurements are shown in Fig. 1.13. The data
are compared to calculations using polarized PDF extracted from a global analysis topolarized world data from the DIS, SIDIS, and RHIC experiments (DSSV [70, 23],
which is described in Sec. 1.5.4, and LSS [71]).

k'
lepton
k
q

γ*

g

q

p
nucleonTime
Figure 1.12: Interaction of γ ∗ g→q q̄ in DIS, which is used for the ∆g measurement.
The ∆g(x) study with pp collisions is being performed in RHIC accelerator describedin Sec. 2.1. In RHIC, the ∆g is studied in the PHENIX and STAR experiments. The
PHENIX experiment performed ALL measurements for π 0 [10, 11, 12, 13], η [14], andjet [17] production and the STAR experiment performed ALL measurements for jet
production [72, 15]. The π 0 , η, and jet are mainly produced via gg and gq scattering∆g/g

1.5. CURRENT STATUS OF NUCLEON SPIN STRUCTURE

0.8

25

New COMPASS, high p , Q2>1 (GeV/c)2, 2002-2006
T

0.6
0.4

COMPASS, high p , Q2<1 (GeV/c)2, 2002-2003
T

COMPASS, open charm, 2002-2007
SMC, high p , Q2>1 (GeV/c)2
T
HERMES, high p , all Q2
T

0.2
0
-0.2
-0.4
-0.6

DSSV fit, µ 2=3(GeV/c)2
LSS fit with ∆G>0, µ 2=2.5(GeV/c)2
LSS fit with ∆G changing sign, µ 2=2.5(GeV/c)2

-0.8
10-2

10-1

xg

Figure 1.13: Gluon polarization ∆g/g(x) from leading-order analysis of hadron or
hadron-pair production as function of the probed gluon Bjorken x [68]. Also shown
are NLO pQCD fit curves (DSSV [70, 23] and LSS [71]).
√
process in the partonic level. The ALL of the π 0 production in s = 200 GeV pp
collisions measured by PHENIX from 2005, 2006, and 2009 (preliminary) is shownin Fig. 1.14 [73]. The data are compared to a calculation using DSSV
described in√Sec. 1.5.4. Figure 1.15 shows ALL of jet production measured in s = 200 GeV pp
collisions by STAR from 2006 and 2009 (preliminary) experiments [15].

1.5.4

Global Analysis for ∆g(x)As introduced in Sec. 1.5.3, there are a lot of data sets for ∆g(x) including the DIS,SIDIS, and RHIC experiments, which cover various Bjorken x regions and Q2 values.
To extract ∆g(x) and other polarized PDFs from these data sets, a global NLO pQCD
analysis, DSSV [70, 23], is performed. DSSV optimizes the agreement between themeasured spin asymmetries and corresponding theoretical calculations. The assumptionof the polarized PDF shapes in DSSV at an initial scale for the evolution of Q20 = 1 GeV2is()√
x∆fi (x, Q20 = 1 GeV2 ) = Ni xαi (1 − x)βi 1 + γi x + ηi x ,(1.76)
with free parameters Ni , αi , βi , γi , and ηi . The PDFs are evolved according to DGLAP
evolution formulae in Eq. 1.42. In DSSV, it is assumed ∆s(x) = ∆s̄(x), considering
the s-quark and s̄-quark are produced only via pair creation from gluon and they are
symmetric.
√0√ π ALL in s = 200 GeV pp collisions from the 2005 and 2006 PHENIX√ runs and ins = 62.4 GeV pp collisions from the 2006 PHENIX run, and jet ALL in s = 200 GeV

CHAPTER 1. NUCLEON SPIN PHYSICS

0

A πLL

π0

A LL

26

0.02
0.01

0.06

Preliminary

0

0.04

Preliminary
0.02

-0.01

0

-0.02

DSSV

-0.03

Run5+6+9

0.005
π0

A LL

-0.02
-0.04

Run-5 (9.7% Vertical Scale Unc.)

0

Run-6 (8.6% Vertical Scale Unc.)-0.005
1 1.5 2 2.5 3p (GeV/c)

-0.06

-0.04

Run-9 (8.8% Vertical Scale Unc.)

T

-0.08

-0.05
2

4

6

8

10
12
p (GeV/c)

2

4

6

10

12

T

T

(a) ALL (pT ) in π 0 production from PHENIX experiment.

8

p (GeV/c)(b) Combined result.

Figure 1.14: (a) The longitudinal double-spin asymmetry √in π 0 production at pseudorapidity ranging |η| < 0.35 as measured by PHENIX in s = 200 GeV polarized pp
collisions as function of π 0 transverse momentum. (b) Combined result compared to
the DSSV curve.
pp collisions from the 2005 and 2006 STAR runs are used in the DSSV analysis for the
RHIC data. The resulting∫ constraints on the∫polarized PDFs are shown in Fig. 1.16.
1
1
Table 1.2 shows resulting 0.001 dx∆fi (x) and 0 dx∆fi (x) for each PDF.
∫1
∫1
Table 1.2: Truncated first moments 0.001 dx∆fi (x) and 0 dx∆fi (x) at Q2 = 10 GeV2 .
∫1dx∆fi (x) has uncertainty corresponding to ∆χ2 = 1 in Fig. 1.16.
0.001
∫1
∫1
dx∆f
(x)
dx∆fi (x)
i
0.001
0
∆U
0.793+0.011
0.813−0.012∆D−0.416+0.011−0.458−0.009∆ū0.028+0.021
0.036−0.020+0.029¯∆d−0.089−0.029−0.115∆s̄
−0.006+0.010−0.057−0.012∆Σ

0.366+0.015−0.0180.242∆G0.013+0.106−0.120−0.084

Quark polarized PDFs are well determined from DIS and SIDIS data. The data
points from SIDIS experiments and the DSSV theoretical curves are shown in Fig. 1.17.
While ∆u is positive, ∆d has negative contribution to the proton spin. Also surprisingly,∆S = 2∆s̄ has about 10% negative contribution. The optimal DSSV fit satisfies the1.5. CURRENT STATUS OF NUCLEON SPIN STRUCTURE
0.08
2006 STAR Data
GRSV-std
GRSV ∆g=0
GRSV ∆g=-g
GS-C
DSSV

A LL
0.06

0.04

27

pp →jet+X-0.7<η<0.9

0.02

0±8.3% scale uncertainty frompolarization not shown

-0.02
5

10

15

20

25

30

35
p (GeV/c)
T(a) ALL (pT ) in jet production from 2006 STAR experiment.

(b) ALL (pT ) in jet production from 2009 STAR (c) ALL (pT ) in jet production from 2009 STAR
experiment at |η| < 0.5 (preliminary).
experiment at 0.5 < |η| < 1.0 (preliminary).

Figure 1.15:√ The longitudinal double-spin asymmetry in jet production as measured bySTAR in s = 200 GeV polarized pp collisions as function of jet transverse momentum.
(a) 2006 result at pseudorapidity ranging −0.7 < η < 0.9 and (b, c) 2009 preliminary
results at pseudorapidity ranging |η| < 0.5 and 0.5 < |η| < 1.0 are shown.
isospin and flavor symmetry relations,[]∆U − ∆D = (F + D) 1 + ϵSU (2) ,[]∆U + ∆D − 2∆S = (3F − D) 1 + ϵSU (3) ,(1.77)(1.78)

with discrepancy parameters from exact isospin SU (2) and flavor SU (3) symmetries ofϵSU (2) = 0.0011 and ϵSU (3) = −0.0035, where F + D = 1.269 and 3F − D = 0.586 areused.
The RHIC data put a strong constraint on the size of ∆g(x) for 0.05 < x < 0.2
as they mainly probe (∆g(x))2 . ∆g(x) comes out rather small in 0.05 < x < 0.2, andprefers to have a node. As can be seen from Table 1.2, the integral of ∆g(x) in theregion of 0.001 < x < 1 is found to be almost zero and the extrapolation to the regionof 0 < x < 1 is −0.084 at Q2 = 10 GeV2 .
The global analysis of DSSV does not include ALL of π 0 production from the28

CHAPTER 1. NUCLEON SPIN PHYSICS
0.4

–

0.05

–

x(∆u + ∆u)x(∆d + ∆d)

DSSV

0.3

0

∆χ2=1 (Lagr. multiplier)∆χ2=1 (Hessian)

0.2

-0.05

0.1

-0.1

0

-0.15
–

0.04

–

x∆u

x∆d

0.04

0.02

0.02

0

0

-0.02
-0.04

-0.02

DSSV
DNS
GRSV

DSSV ∆χ2=1
DSSV ∆χ2/χ2=2%

–

0.04

-0.04
0.3

x∆g

x∆s

0.2
0.02
0.1
0
0
-0.02
-0.1
-0.04

2

Q = 10 GeV
10-2GRSV max. ∆g
GRSV min. ∆g

2

10

-1

x

10

-2

-0.2
10

-1

x

Figure 1.16: DSSV polarized PDF. Sea quark and gluon PDFs are compared to GRSV
fits [74]. The shaded bands correspond to alternative fits with (green) ∆χ2 = 1 and(yellow) ∆χ2 /χ2 = 2%.

x ∆u(x)1.5. CURRENT STATUS OF NUCLEON SPIN STRUCTURE

0.4

29

HERMES
SMC

0.2

COMPASSx ∆u(x)x ∆d(x)

0
0

-0.2
0.1
0

x ∆d(x)

-0.1
0.1
0

x ∆s(x)

-0.1

0.1
0
10

-2

10

-1

1

x

Figure 1.17: Quark polarized PDF at a scale µ2 = 2.5 GeV2 . The theoretical curves
shown in this figure are LSS2010 [71], AAC2008 [75], and DSSV. Points represent
data from SMC [18], HERMES [19, 20], and COMPASS [21, 22] SIDIS experiments at¯
Q2 = 2.5 GeV2 . SMC results are extracted under the assumption that ∆ū(x) = ∆d(x).

PHENIX 2009 measurement (Fig. 1.14) and ALL of jet production from the STAR
2009 measurement (Fig. 1.15(b) and 1.15(c)), which have larger statistics. These figures also display theoretically calculated curves using DSSV. The theoretical curve in π 0
ALL looks consistent with the combined result of 2005, 2006, and 2009 measurements.
However, the theoretical curves in jet ALL has large deviations from the 2009 measurement. The diﬀerence of the behaviors between the π 0 and jet ALL results is attributed

30

CHAPTER 1. NUCLEON SPIN PHYSICS

to diﬀerence of sensitive Bjorken x coverage for the two measurements. A new global
analysis including these ALL data set and a new COMPASS DIS result [8, 9] is preformed, and called DSSV++. Figure 1.18(a) shows the fitting result of the DSSV++
compared with the π 0 and jet ALL results. Figure 1.18(b) shows the ∆g(x) result of
the DSSV++.
Especially, Fig. 1.18(b) represents unreliability of DSSV and DSSV++ ∆g(x). WhileDSSV expects a node structure in ∆g(x), DSSV++ does not have such node in 0.05 <x < 1. The node in DSSV ∆g(x) cancels the positive polarization
in 0.1 < x < 1 and∫1the negative polarization in x < 0.1, and expects the integral 0.001 dx∆g(x) to be small,
0.013+0.106
positive polarization in almost the−0.120 . On the other hand, DSSV++ shows the∫1whole Bjorken x region, and expects the integral 0.001 dx∆g(x) to be ∼ 0.3 as shownin Fig. 1.18(c).
The above unreliability on DSSV and DSSV++ ∆g(x) comes from the following
reasons:
Uncertainty in small Bjorken x region
Large uncertainty of ∆g(x) in extrapolation to x→0 is still remaining in the<
result. There is no data set in the small Bjorken x region of x ∼0.02, and the
behavior of ∆g(x) in this region is determined almost by the functional form in
the assumption. To constraint ∆G more precisely, the data points in the small
Bjorken x region is necessary.
Uncertainty in functional form of ∆g(x)ALL (pT ) in pp collisions gives an integral with respect to Bjorken x such like∫dx1 dx2 dz ∆fi (x1 )∆fj (x2 )∆σ̂(x1 , x2 , pT /z)D(z)∫.
(1.79)
dx1 dx2 dz fi (x1 )fj (x2 )σ̂(x1 , x2 , pT /z)D(z)To determine the functional shape of ∆g(x), we need measurements with otherprobes, e.g. heavy quark and direct γ production, which has sensitivity on diﬀerent Bjorken x region and sign of ∆g(x) from the current measurements.

1.6

Heavy Flavor Production in p + p Collisions

1.6.1

Challenge for ∆g(x) Determination

As discussed above, there is no ∆g(x) measurement in small Bjorken x region x ∼ 10−2 ,and then ∆g(x) in such small Bjorken x region resulting√from the global analyses is notenough reliable. ALL measurement of π 0 production in s = 200 GeV p + p collisions,which has sensitivity in the smallest Bjorken x region achieved by the current ∆g(x)measurements, reaches only up to the minimum limit of x ∼ 2 × 10−2 (log10 x ∼ −1.7).
This limit is originated by π 0 transverse momentum range of pT > 2.0 GeV/c, wherereliability of pQCD calculations is supported.
This lack of the data causes the large uncertainty on ∆G(µ). Under the current
experimental knowledge, a sizable gluon polarization in small Bjorken x region x ∼ 10−2

1.6. HEAVY FLAVOR PRODUCTION IN P + P COLLISIONS

31(a) Fitting result of DSSV++ for π 0 and jet ALL .

(b) Resulting DSSV++ ∆g(x).

(c)

∫1
min

dx∆g(x) in DSSV++.

Figure 1.18: (a) Fitting result of DSSV++ for (red circles) π 0 and (blue squares) jet ALL
compared with the measured values. (b) Resulting ∆g(x) from DSSV and DSSV++.
This plot also shows the ∆g(x) from DSSV+ which is a global analysis including only
new COMPASS DIS result [8, 9]. The red band represents the uncertainty calculated
from ∆χ2 /χ2 = 2% in the fitting, which corresponds to almost ∆χ2 = 8. (c) The
integral computed in the range from xmin to 1. The red band represents uncertainty
calculated from ∆χ2 /χ2 = 2% in the fitting.

32

CHAPTER 1. NUCLEON SPIN PHYSICS

compared to the proton spin 1/2 is not ruled out yet. Hence, a ∆g(x) measurement insuch small Bjorken x region is essential for the further constraint on ∆G(µ).

proton: A
pA

pB

HQ
p1 σ
a+b→Q+Q
parton: aH
Q DQ (z)k1
QQ
parton: bk2
p2 hard process
a+b→Q+Q

proton: B

Time
Figure 1.19: A diagram of heavy flavor production in pp collisions. Q (Q̄) in the
figure represents c (c̄) or b (b̄) quarks. HQ represents fragmented hadron from Q. In
this measurement, the heavy flavor production is detected by measuring electron fromsemi-leptonic decay of HQ (heavy flavor electron).
Heavy flavor quark (charm and bottom quarks) production in the polarized pp collisions is an ideal probe which overcomes above problems and measures gluon polarization
as discussed in later, Sec. 1.6.2. The diagram of the heavy flavor production is shownin Fig. 1.19. The PHENIX experiment in RHIC is a suitable facility to measure ALL ofthis process. Since the PHENIX detector has large spectrometers for electron detectionin the mid-rapidity region (|η| < 0.35), hadrons containing heavy flavors are measured
through their semi-leptonic decays to electrons and positrons, which are called “heavy
flavor electrons” (HFe) [28, 29]. In this work, ALL of the heavy flavor electron production is measured at the PHENIX experiment to access ∆g(x).

1.6.2

Production Mechanism for Heavy Flavor Electron

As shown in Eq. 1.28 and 1.33, the spin-independent and spin-dependent cross sectionsof the heavy quark production from partons i, j interaction in p+p collisions, σij→QQ̄+X
and ∆σij→QQ̄+X , can be calculated by using partonic cross sections, σ̂ij→QQ̄+X and1.6. HEAVY FLAVOR PRODUCTION IN P + P COLLISIONS

33(a) gg→QQ̄ + X
0.00060.00040.00020-0.0002
∆f(1)
gq
−
∆f(1)
gq

-0.0004-0.0006f(1)gq / 45−(1)
fgq / 45-0.0008-0.001
0.01

0.1

1

10

100

1000

10000

100000ξ =s/(4 m2) −1(b) gq(g q̄)→QQ̄ + X
0.05
0
-0.05
-0.1
∆f(1)
qq−
∆f(1)
qq-

-0.15

∆f(0)
qq-

-0.2
-0.25
1e-05

(1)
∆f(0)
qq- +2.7∆fqq

0.0001

0.001

0.01

0.1

1

10

100

1000

ξ =s/(4 m2) −1(c) q q̄→QQ̄ + X

Figure 1.20: Partonic cross sections (m2 /αS2 )σ̂ij and (m2 /αS2 )∆σ̂ij in LO and NLO(MS) calculations as a function of ξ≡s/(4m2 ) − 1, where we have set µ = 1.4 GeV for
simplicity and 4παS (µ = 1.4 GeV) = 2.7 as appropriate for charm production. It is(0)worth to note that there is no LO component in gq interaction, namely fgq = 0 and
(0)
(n)
(n)
∆fgq = 0, and helicities in q q̄ annihilation are conserved, namely fqq̄ = −∆fqq̄ .

34

CHAPTER 1. NUCLEON SPIN PHYSICS∆σ̂ij→QQ̄+X , as∫σij→QQ̄+X =∆σij→QQ̄+X =

∫

∫
dxi
dxi

∫

dxj σ̂ij→QQ̄+X fi (xi )fj (xj )dxj ∆σ̂ij→QQ̄+X ∆fi (xi )∆fj (xj ).

(1.80)The partonic cross sections of the heavy flavor quark are well studied with LO and NLO(MS) pQCD calculations with respect to order of αS [25, 26, 27]. The partonic cross(0)(1)
sections can be decomposed into LO and NLO components, fij and fij , as
(
)
2
m2
µ
(0)
(1)
(1)
2
σ̂ij (s, m , µ) = fij (ξ) + 4παS (µ) fij (ξ) + f¯ij (ξ) ln 2 ,(1.81)
αS2 (µ)
m
(
)
2
m2
µ
(0)
(1)
(1)
2
∆σ̂ij (s, m , µ) = ∆fij (ξ) + 4παS (µ) ∆fij (ξ) + ∆f¯ij (ξ) ln 2 ,(1.82)
αS2 (µ)m
where s is the available partonic center-of-mass energy squared, and β = (11NA −2nlf )/3 (NA : number of colors, nlf : number of light flavors). Figure 1.20 show thepartonic cross sections for gg, qg, q q̄ interactions. As these calculations show, partonic
cross sections for gg and q q̄ scattering are comparable, whereas ones for qg (q̄g) arerather small compared to them by a factor of ∼ 10−2 [26, 27]. Therefore, since theunpolarized gluon PDF is larger than the unpolarized quark PDF in small Bjorken x−2region around x ∼ 1.4×10
by a factor of ∼ 10, spin-independent cross section of∫∫
gg→QQ̄ + X process,dx1 dx2 σ̂gg g(x1 )g(x2 ), is predominant in the heavy flavor production especiallyfor small transverse momentum and contribution from q q̄→QQ̄ + X∫∫
process,
dx1 dx2 σ̂qq̄ q(x1 )q̄(x2 ), is a few percent [24]. In addition, when the polarized
gluon PDF in the small Bjorken x region is sizable compared with the polarized quark
PDF at Bjorken x of ∼ 1.4×10−2 as well as the unpolarized case, the ALL of heavy
quark production consist of only the spin-dependent cross section of gg interaction∆σgg . Therefore, ALL measurement of this process is an ideal probe for the ∆g/g(x)measurement.
Cross sections of the heavy flavor electron production can be calculated by using
the above cross sections of the heavy flavor production. Figure 1.21 shows comparison
between the unpolarized
and polarized cross sections of the heavy flavor electron pro√duction in the s = 200 GeV pp collisions estimated with the LO and NLO pQCD
calculations for the partonic interaction [28, 24]. The hadronization is simulated withfragmentation functions for c→D, b→B, and b→c→D. The electron pseudorapidity
in the calculation is ranging |η| < 0.35. The top panel corresponds to the unpolarizedcross section resulting from CTEQ6 unpolarized PDF and the middle panel corresponds
to the polarized cross section resulting from DSSV polarized PDF. The bottom panel
shows ratios of the NLO calculation to the LO calculation for the unpolarized andpolarized cross sections.
The two left panels in Fig. 1.22 represent the fractions of gg, q q̄, and qg contributions
in cross sections of the heavy flavor production from the NLO pQCD calculation, shownin Fig. 1.21. The top panel corresponds to unpolarized cross section and the bottom
panel corresponds to polarized cross section. The top panel represents that contribution

1.6. HEAVY FLAVOR PRODUCTION IN P + P COLLISIONS

10 2
10
1-1
10 -2
10 -3
10 -4
10
-2
10
10
10
10
10
4

35

e

dσ/dpT [µb/GeV]|η e| < 0.35e- d∆σ/dpT [µb/GeV]-3|η e| < 0.35

-4-5LO × 0.1-6NLO

dσ (NLO) / dσ (LO)20d∆σ (NLO) / d∆σ (LO)

0

2

4

e

pT [GeV]6

Figure 1.21: Comparison between LO and NLO calculated√ cross sections of heavy flavor
electron production at central rapidity |η| < 0.35 in s = 200 GeV pp collisions [24].
The top figure shows unpolarized cross sections and the middle figure shows polarized([] )1/2Q 2
Q̄ 22cross sections. Scale is varied simultaneously as µ = k mQ + (pT ) + (pT ) /2
in the range 1/2 < k < 2 which corresponds to the shaded band. The solid lines
correspond to the default scale k = 1. The bottom shows the ratio of NLO to LOpolarized and unpolarized cross sections.

from q q̄ is a few percent and qg is also less than 10% in electron transverse momentum
ranging pT < 1.5 GeV/c as mentioned above. The bottom figure for the polarizedcross section represents that DSSV expects small gg contribution to the polarized crosssection due to its small ∆g(x) distribution. The polarized cross section in transverse
momentum ranging 0.5 < pT < 1.5 GeV/c is determined almost by polarized gluon<<
8×10−2 where there are not enough data sets tox∼
PDF in Bjorken x ranging 10−2 ∼<2×10−2 ∆g(x) had never been explored before
constraint ∆g(x), and especially in x ∼this measurement.
The two right panels in Fig. 1.22 represent the fractions of charm and bottom
productions in the unpolarized (top-right) and polarized (bottom-right) cross sections.
As the unpolarized cross section, the heavy flavor production below pT ∼ 1.5 GeV/c

36

CHAPTER 1. NUCLEON SPIN PHYSICS

1

subprocess fractions

c, b, b → c fractions

σij / σσi / σ

CTEQ

1
c
b
b→c

gg
qq
qg

0.5

CTEQ

0.5

0

0

2

∆σi / ∆σ∆σij / ∆σ

DSSV

DSSV

1

1

0
0

-1
0

2

4

e

6

pT [GeV]

0

2

4

e

6

pT [GeV]Figure 1.22: Fractional amount of diﬀerent partonic subprocesses at NLO accuracy (leftcolumn) and of charm, bottom, and cascade (b→c) decays (right column) contributing
to the pT spectrum of the heavy flavor electron production [24]. Results are shownfor unpolarized (upper row) and polarized (lower row) pp collisions at RHIC using theCTEQ6 and DSSV set of PDF, respectively.
√
is dominated by charm quark in s = 200 GeV p + p collisions [28, 24]. The bottom
panel of Fig. 1.23 shows a distribution of Bjorken x contributing to the heavy flavor
electron production√ of transverse momentum 0.5 < pT < 1.5 GeV/c from the charmed
hadrons in the s = 200 GeV pp collisions at pseudorapidity ranging |η| < 0.35. For
comparison, the top panel of Fig. 1.23 shows a Bjorken x distribution of π 0 production
for several transverse momentum regions at the same pseudorapidity range, where theπ 0 ALL measurements at PHENIX have been performed [10]. Both distributions are
estimated by using PYTHIA simulation [76, 77]. Whereas the Bjorken x distribution of<x, the distribution of the heavy flavor production
the π 0 production ranges 2×10−2 ∼
−2
−2 <
<
ranges 10 ∼ x ∼ 8×10 , which covers an unexplored small Bjorken x region.
From the unpolarized and polarized cross sections shown√ in Fig. 1.21, we can obtain
ALL (pT ) of the heavy flavor electron production in the s = 200 GeV pp collisions
under the assumption of DSSV polarized PDF. DSSV expects ALL of ∼ 2 × 10−4 at
pT < 1.5 GeV/c. Therefore, the purpose of this measurement is to confirm such small<<
8×10−2 , where there are notx∼
asymmetry and obtain constraints of ∆g(x) in 10−2 ∼enough data sets for the constraint so far.

dN/d(log x)

1.6. HEAVY FLAVOR PRODUCTION IN P + P COLLISIONS

2<p <2.5 GeV/c

0.04

T

0.01

10

dN/d(Log x)

5

×102

4<p <5 GeV/c
T

0.03
0.02

×1

37

9<p T<12 GeV/c5×10

pp →π0+X
s =200GeV | η|<0.35
×10-3

+

pp→c c +X→(e +e-)/2+X
s =200GeV |ηe |<0.35
0.50<pe <1.25 GeV/c

4.5
4

T

3.5
3
2.5
2
1.5
1
0.5
0
-3

-2.5

-2

-1.5

-1

-0.5

0

Log x
10

Figure 1.23: Bjorken x distribution contributing
to (top) π 0 production [10] and (bot√tom) heavy flavor electron production in s = 200 GeV pp collisions at pseudorapidity
ranging |η| < 0.35 estimated by using PYTHIA simulation [76, 77]. Both distributions
do not include eﬃciency of the particle detection. Transverse momentum of the heavy
flavor electron ranges 0.5 < pT < 1.5 GeV/c and transverse momentum of π 0 ranges
2.0 < pT < 2.5 GeV/c, 4.0 < pT < 5.0 GeV/c, and 9.0 < pT < 12.0 GeV/c.

Advantages of heavy flavor electron measurement

We summarize the advantages of the heavy flavor electron measurement here. Due tothe large mass of the heavy quarks, the whole pT region of the heavy quark electron production satisfy large energy scale for pQCD calculations. This feature enables to extend
the sensitive Bjorken x region into the current unexplored region, x < 2×10−2 . The contributing Bjorken x for the heavy flavor electron measurement in 0.5 < pT < 1.5 GeV/c<<
8×10−2 . In addition, the heavy flavor electronx∼and |η| < 0.35 is ranging 10−2 ∼is dominantly produced by gg interaction and is sensitive to the gluon polarization∆g(x). Therefore, the ALL measurement of the heavy flavor electron production eﬃ<<
8×10−2 , and provide
x∼ciently constrains ∆g/g(x) at small Bjorken x region 10−2 ∼new information on ∆G.

38

CHAPTER 1. NUCLEON SPIN PHYSICS
0.001

0

-0.001eALL-0.002|η e| < 0.35
NLO

-0.003-0.004

DSSV

0

1

2

3

4

5

e

6

pT [GeV]7√
Figure 1.24: Theoretical expected ALL of the heavy flavor electron production in s =200 GeV pp collisions at a mid-rapidity region |η| < 0.35 with DSSV unpolarized PDFs
and NLO partonic cross section.

1.6.3

Background for Heavy Flavor Electron Measurement

A challenging point on the heavy flavor electron measurement is large backgroundelectrons in the pp collisions. Figure 1.25 shows a comparison of cross section spectra of the heavy flavor electron and background electron. The plotted cross sectionof the heavy flavor electron is a previously published result which was measured at2005 experiment [28]. The background electron spectra were estimated by a cocktail
method, which is described in Sec. 3.1.6. The cocktail method estimates the spectrawith a PHENIX-conventional event generator, which generates the hadrons or photons according to mT scaling cross sections [78] and simulates their decays with Monte
Carlo [29]. To estimate the background electrons from γ conversions and Ke3 decays,
GEANT3 simulation [79] was also used. Especially in a transverse momentum ranging
pT < 2.0 GeV/c, the yield of the background electrons is much larger than the heavy
flavor electrons.
As Fig. 1.25 shows, the dominant background sources are π 0 and η Dalitz decay(π 0 , η→e+ e− + γ), γ conversions in material, and Ke3 decay (K→eνπ). The first twosources create electron pair via virtual or real photon, and therefore the electron pairs
have small pair mass. These electron pairs are called “photonic electrons”. The electrons except the photonic electrons are called “non-photonic electrons”, which consistof the heavy flavor electrons for the large fraction, electrons from Ke3 decays, ande+ e− from light vector meson decays and Drell-Yan process. The Ke3 electrons significantly contribute only at small transverse momentum pT < 0.75 GeV/c. Ratio of theEd3σ/d3p [mb GeV-2/c3]

1.6. HEAVY FLAVOR PRODUCTION IN P + P COLLISIONS

39

p+p s =200GeV (05’ result)

2

10
10

05’ single e cross section
total BG electrons (cocktail)π0→γee1γ conversion

10-1

η→γee
η’→γee

-2

10

ρ→ee

-3

10

ω→ee and ω→π0ee

10-4

φ→ee and φ→ηee
direct γ contribution

-5

10

Ke3

10-6
10-7
10-8
10-9
10-100

1

2

3

4

5

pT [GeV/c]

Figure 1.25: Comparison of signal and background cross section spectra. Black points
with light-blue systematic uncertainty band represent cross section of heavy flavor electron production, which was measured at the 2005 PHENIX experiment. Several curves
represent cross sections of background electron production, which were estimated withan event generator [29]. Estimation of background electrons from γ conversions andKe3 decays used also GEANT3 simulation.
non-photonic electrons to the photonic background in PHENIX 2005 run is shown in
Fig. 1.26. In small pT region, 0.4 < pT < 2.0 GeV/c, where the statistics is significant
in the heavy flavor electron measurement, the ratio is less than 1.
Considering the background electrons (BG), the double spin asymmetry of inclusiveelectrons (S+BG) can be written as
BG
AS+BG(pT ) = D(pT )AHFe
LL (pT ) + (1 − D(pT )) ALL (pT ),LL(1.83)

where AHFe
LL represents double spin asymmetry of the heavy flavor electron production,
S+BG
ALL and ABG
LL represent double spin asymmetry of inclusive electrons and backgroundelectrons, and D(pT ) is a dilution factor which represents a fraction of the heavy flavor

CHAPTER 1. NUCLEON SPIN PHYSICS

nonphotonic/photonic

40
10

1

converter method
cocktail method
Ke3

10-1

ρ → eeω → eehadron

10-2

0

1

2

3

4

5

6

7

8

9
10
p T (GeV/c)

Figure 1.26: Ratio of non-photonic electrons to photonic background in PHENIX 2005
run [28]. The cocktail and converter methods are explained in Sec. 3.1.6. Error bars
are statistical errors and the error bands show the cocktail systematic errors. The solid,
dashed and dot-dashed curve are the remaining non-photonic backgrounds from Ke3 ,ρ→e+ e− , and ω→e+ e− , respectively. The blue dot curve is misidentified hadron.
electrons in the inclusive electrons, NeHFe /NeS+BG . Above equation can be transformed
into
AHFe
LL (pT ) =1
1 − D(pT ) BG
AS+BG
(pT ) −ALL (pT ).
LL
D(pT )
D(pT )(1.84)As this equation shows, D(pT ) dilutes the measured AS+BG(pT ) to derive AHFe
LL (pT ). And
LL
HFethen, D(pT ) also expands uncertainty on the ALL (pT ) propagated from uncertainty
on the measured AS+BG(pT ). Therefore, the purification of the heavy flavor electrons
LL
and the enhanced D(pT ) value is essential to reduce the uncertainty on the resulting
AHFe
LL (pT ) value.
In this analysis, we strongly suppressed the background with Hadron Blind Detector(HBD), which was newly installed in PHENIX at 2009. The HBD is a position-sensitive
gas Čerenkov counter, and separates the non-photonic electrons and the photonic electrons by using amplitude of produced cluster charge. We developed new analysis method
using this HBD feature as explained in Sec. 3.2.

Chapter 2
Experimental Setup
This analysis is based on the data which were taken at Relativistic Heavy Ion Collider(RHIC) with the PHENIX detector at Brookhaven National Laboratory (BNL) in the
United√ States during the 2009 run. In this run, longitudinally polarized p + p collisions
at s = 200 GeV were performed for 10 weeks (Apr. 19 − Jun. 29). The integrated
luminosity used in this analysis after quality assurance (QA) of data is ∼ 6.1 pb−1 andthe average beam polarizations during this measurement are ∼ 56% for the Blue beamand ∼ 57% for the Yellow beam.
For the heavy flavor electron measurement at PHENIX, a new detector called
Hadron Blind Detector (HBD), which started to be operated from this run, is the mostimportant detector to reject the photonic electron background in this analysis. Since it
was the first operation of HBD for physics measurement, a new analysis frameworks of
HBD was developed and was used in the measurement as described in Sec. 3.2.
In this chapter, RHIC is introduced in Sec. 2.1 and the PHENIX detector systemused in this analysis is explained in Sec. 2.2. The detailed description of the HBD isalso in Sec. 2.2. The PHENIX DAQ is described in Sec. 2.3.

2.1

Relativistic Heavy Ion Collider (RHIC)

RHIC provides high energy heavy ion collisions and polarized p + p collisions. Oneof the major goals of the heavy ion experiment is to investigate a new state of matter
which is referred to as Quark Gluon Plasma (QGP). RHIC can accelerate ions as heavy
as Au up to an energy of 100 GeV per nucleon, which results in heavy ion collisions√
at sN N = 200 GeV. RHIC can also accelerate and collide polarized proton beams forthe first time in the world, which provides us unique opportunities to study the spin
property of proton through strong and weak interactions. √The maximum energy forthe proton beam is 255 GeV which results in collisions at s = 510 GeV with design
luminosity of 2 × 1032 cm−2 s−1 .
Figure 2.1 shows an aerial view of RHIC accelerator complex and Fig. 2.2 represents
its schematic. The polarized proton beam is produced at optically-pumped polarized
ion source (OPPIS) [80] with the polarization of about 85%. Its intensity reaches 500 µA
in a single pulse of 300 - 400 µs, which corresponds to 9 - 12×1011 polarized protons.
41

42

CHAPTER 2. EXPERIMENTAL SETUP

Figure 2.1: Aerial view of RHIC accelerator complex in BNL.

For the first step, the pulse is accelerated by Linear Accelerator (LINAC) to a kinetic
energy of 200 MeV. This pulse is again accelerated by Booster up to 1.5 GeV, and also
by Alternating Gradient Synchrotron (AGS) up to 24.3 GeV. Then it is injected into
two independent rings at RHIC, via AGS-to-RHIC transfer line. Each beam travels in
opposite direction and collides each other at interaction points (IPs). Two independent
beams are called the Blue (clockwise) and Yellow (anti-clockwise) beams. RHIC has
six IPs and they are referred to IP12, IP2 IP4, IP6, IP8, and IP10 as in the case of aclock. The PHENIX detector is placed at IP8 and the STAR detector is placed at IP6
as Fig. 2.1 shows. Also at IP12, there is a polarimeter system which measures the beampolarization. Once RHIC was filled with beams, the beams are kept circulating in therings to provide collisions at the IPs. When the luminosity becomes too low, beams are
dumped and refilled. The sequence from injection to dump of the beam is called a fill.
The length of a fill is typically ∼ 8 hours.
The polarization of the stored beams in RHIC rings has horizontal direction except
around interaction points. Four spin rotators at the both sides of PHENIX IP in
Fig. 2.2 changes the polarization direction and enable longitudinally-polarized collisions
at PHENIX as shown in Fig. 2.3. The direction of the longitudinal beam polarization
at the IP is called beam-helicity.
The beam in RHIC has bunch structure and each ring contains 120 bunches ofpolarized proton beam, with a time interval of 106 nsec. Each bunch is filled withpredetermined beam-helicity pattern and this pattern is changed in diﬀerent fills inorder to confirm no pattern dependence in the spin asymmetry. There are basically fourtypes of the beam-helicity patterns, which are defined as repetitions of the following

2.1. RELATIVISTIC HEAVY ION COLLIDER (RHIC)
H-jet
polarimeter

Blue

Beam

w Be
Yello

am

43

RHIC
pC polarimeters

IP12

RHIC
Siberian
SnakesSpin Rotators
Polarized Proton
Source (OPPIS)Interaction Point (IP)Spin manipulating Magnet

Booster

Polarimeter
200MeV
Polarimeter

AGS

RF dipole
Cold snake

Warm snake

AGS pC Polarimeter

Figure 2.2: Schematic of RHIC accelerator complex.

transverse
spin rYELLOW beam(outgoing BLUE beam)

ota
t

or

longitudinal

PHENIX

spi

ota
nr

tor

transverse

BLUE beam(outgoing YELLOW beam)interaction point

Figure 2.3: Schematic drawing of beam spin rotation near the PHENIX IP with thespin rotators.

44

CHAPTER 2. EXPERIMENTAL SETUP

a)

b)
-

4
-

3 2

4 3

1
-

+ +

2

-

+ +

+

-

+

1

2

3

-

+

-

1

2

+

4

3

Yellow beamBlue beam1c)2

Yellow beam
3

1

-

4
Blue beam

4

Blue beam+ +--Yellow beam+-+-

Figure 2.4: An example of the beam-helicity pattern of RHIC polarized proton beams.
An arrow represents the direction of a beam. A box corresponds to a bunch and thesign (+ and −) in the box denotes the predetermined beam-helicity state of the bunch.
The colors of arrows and boxes represent the Blue and Yellow beams. a) Blue bunch
1 collides at Yellow bunch 1 and provide a helicity combination (+,+) collision. b)
One beam clock after, Blue bunch 2 and Yellow bunch 2 collide and provide a (+,−)
collision. c) The resulting beam-helicity combinations from the helicity patterns. The
helicity pattern provides the all four helicity combinations of collisions.
eight beam-helicity combinations:
{
P1 =
{
P2 =
{
P3 =
{
P4 =

B : + − + − − + −+Y : + + − − + + −−
B : − + − + + − +−Y : + + − − + + −−
B : + − + − − + −+Y : − − + + − − ++B : − + − + + − +−Y : − − + + − − ++,where the combinations appear from left to right. Figure 2.4a and 2.4b shows anexample of a helicity pattern for the first four bunches. In this example, the Blue beam
has a spin pattern “++−−” while the Yellow beam has a spin pattern “+−+−”. As theresult, we obtain helicity combinations of four bunch crossing as (+,+), (+,−), (−,+),and (−,−), where the left and right signs in the parentheses represent the Blue andYellow beam-helicities respectively, as shown in Fig. 2.4. These four combinations are2.2. PHENIX DETECTOR SYSTEM

45

the all possible helicity combinations. This alternative helicity changing greatly reduced
systematic uncertainty which comes from time dependence of the detector responses.
In 2009 RHIC run, 109 bunches out of 120 bunches are filled in each ring.
The beam polarizations for the Blue and Yellow beams are measured and monitoredby three polarimeters. Two of them are fast carbon ribbon polarimeter (pC polarimeter) [81] and polarized hydrogen gas jet target polarimeter (H-jet polarimeter) [82, 83],
installed in the RHIC ring, and another is PHENIX local polarimeter [84], installed at
the PHENIX experimental area. These three types of polarimeters measure a sizable
transverse single spin asymmetries for elastic scattering or specific particles production. Detailed information of the polarimeters and the polarization measurement aredescribed in Appendix B.

2.2

PHENIX Detector System

PHENIX [85, 86, 87, 88, 89, 90, 91] is one of the largest experimental facilities at RHIC.
PHENIX is designed to measure photons, leptons, and hadrons with excellent particle
identification capability and to deal with both high-multiplicity heavy-ion collisions andhigh event-rate p + p collisions.
A conventional Cartesian coordinate is defined in the PHENIX experimental area.
The PHENIX collision point is defined as the origin of the coordinate. z-axis is alongwith the beam pipe, pointing to north, and x and y-axes are pointing to west andvertical top respectively. The polar angle, θ, and the azimuthal angle, ϕ, are also
defined translating this Cartesian coordinate into the spherical coordinate. Rapidity
and pseudorapidity of particles are also defined asy =1 E + pzln
2 E − pz

(2.1)

η =

1 1 + cos θln,
2 1 − cos θ

(2.2)

and

where E and pz are energy and z component of momentum respectively.
PHENIX is composed of many sets of detectors. These detectors can be divided
into three groups: two central arms, two muon arms, and global detectors. Two east
and west central arms cover pseudorapidity range of |η| < 0.35 and half in azimuthal
angle. They are designed to detect photons, electrons, and hadrons. Two north andsouth muon arms cover 1.2 < η < 2.4 (north) and −2.2 < η < −1.2 (south) respectively
with a full azimuthal coverage. They are designed to detect muons.
Figure 2.5 shows a schematic of the PHENIX detector configuration in 2009. The
upper panel is a beam view of a cross section of the configuration at z = 0 whichshows the two central arms. The proton beams run perpendicular to the paper at thecenter of the detectors. The lower panel is a side view of a cross section at x = 0
which shows global detectors, beam-beam counters (BBCs), zero-degree calorimeters(ZDCs), and shower max detectors (SMDs). The PHENIX coordinate system is also

46

CHAPTER 2. EXPERIMENTAL SETUP

2009

PHENIX Detector
Y

EMCal

EMCal

PbSc

PbSc
CentralMagnet

PbSc

PbSc
DC

DC
RICH
PbSc

7.9 m = 26 ft

X

RICH
PbGl

BB
HBD

PC

PbSc

M

ne

n

uo

n

M

Central Magnet
ag
ne

No

t

10.9 m = 36 ft

South

East

Y

So

ZDC+SMD

PbGl

Beam View

West
ut
h

PC

rth

M

uo

M

t

ag

BBC
ZDC+SMD

Z

HBD

South

Side View

North

North

18.5 m = 60 ft

Figure 2.5: (Top) Beam view at z = 0 and (Bottom) side view at x = 0 of the PHENIX
detector configuration in 2009. The positions of the ZDCs and SMDs are not in same
scale as other detectors, but they are at 18 m away from the collision point.

shown in Fig. 2.5. The proton beams run along the z-axis and collide at the point of
y = 0 each other. These central arm spectrometers and global detectors are the main
detectors used in this measurement. The BBCs provide the collision point informationand the minimum bias (MB) trigger. The count of the MB trigger also provides theluminosity information for the cross section and the spin asymmetry analyses. The
electron measurements are made with the two central arm spectrometers that each cover2.2. PHENIX DETECTOR SYSTEM

47(a) Disassembled PHENIX detector system.

(b) PHENIX central arm (east).

(c) Assembled setup for the experiment.

Figure 2.6: (a) Disassembled PHENIX detector system. In this figure, HBD is removed.
(b) PHENIX east central arm. (c) Assembled setup for the experiment.

48

CHAPTER 2. EXPERIMENTAL SETUP

pseudorapidity |η| < 0.35 and azimuthal angle ∆ϕ = π/2. They consist of the central
magnet and the drift chamber (DC) and the multi-wire proportional pad chamber (PC)for charged particle tracking, the ring-imaging Čerenkov detector (RICH) for electron
identification, the electromagnetic calorimeter (EMCal) for energy measurement, andthe hadron blind detector (HBD) [92, 93, 94] for another electron identification andbackground rejection. The features of the detectors and the magnet are summarized asfollows.

2.2.1

Beam-beam counter (BBC)The BBCs consist of two identical sets of counters positioned at ±1.44 m from thenominal interaction point along the beam direction and pseudorapidity of 3.1 < |η| <3.9. One placed on +z (north) direction is named BBC-N and another placed on −z(south) direction is named BBC-S.
Figure 2.7(a) displays a picture of a BBC. The outer diameter of the BBC is 30 cm
and the inner diameter is 10 cm with a clearance of 1 cm between the BBC and beam
pipe. Each BBC consists of 64 hexagonal quartz Čerenkov radiators with a refractive
index of ∼ 1.5, each of which is attached to a one-inch Hamamatsu R6178 photomultiplier tube. Figure 2.7(b) shows a picture of a pair of a 3 cm quartz radiator anda photo-tube. They are sensitive to charged particles with β greater than 0.7.

(a) BBC module.

(b) A set of the quartz radiator and PMT.

Figure 2.7:
The electronics of the BBC readout consists of discriminators, shaping amplifiers,
time-to-voltage converters and flash ADCs (FADC). The timing and pulse height information is digitized online and is stored in Digital Memory Units (DMUs). The BBC hit
information is sent to BBC level-1 trigger board to provide BBC trigger as mentioned
in Sec. 2.3.1.
They measure the collision vertex along with beam axis by measuring the time
diﬀerence between the two counters and also provides the MB trigger defined by atleast one hit on each side of the vertex. The position resolution for the vertex is ∼ 2.0
cm in p + p collision after oﬄine slewing correction.

2.2. PHENIX DETECTOR SYSTEM

2.2.2

49

Zero degree calorimeter (ZDC) and shower max detector(SMD)

The ZDCs and SMDs, which are located at ±18.0 m away from the collision point
along the beam direction, detect neutral particles near the beam axis (θ < 2.5 mrad).
Figure 2.8 shows the location of the ZDCs and SMDs. They are placed behind the DX
magnets and between two beam pipes, thus most of the charged particles are swept
away and neutral particles with long life, which are mainly neutrons and photons, hit
the ZDCs and SMDs. The neutron identification is explained later. Protons, which
are produced with elastic or diﬀractive scatterings on the collisions, may hit the beam
pipes and induce showers. The resulting many charged particles may also hit the ZDCsand SMDs. A scintillation counter are placed in front of each ZDC for charged particle
veto. Figure 2.9(a) and 2.9(b) are pictures of the ZDC and SMD.
A ZDC consists of series three modules each of which has 1.7 interaction length or 51
radiation length. Figure 2.10 shows the mechanical design of the module. The module
consists of 27 layers of tungsten absorber plates and PMMA (Polymethylmethacrylate)
optical fibers. Neutron generates a hadronic shower in the tungsten plates, and charged
particles in the shower emit Čerenkov radiation. The radiation is detected through theoptical fibers with a photo-tube (Hamamatsu R329-02). The size of a tungsten plateis 10 cm wide, 18.7 cm high and 0.5 cm thick. The tungsten plates and the opticalfibers are tilted by 45 degrees to roughly match the direction of the Čerenkov radiation.
The energy resolution of the ZDC is obtained to be 21 % for neutrons at an energy of
100 GeV. Neutrons can be separated from photons with the energy deposit in the second
module of the ZDC, since the electromagnetic showers from photons cannot penetrate
the first module with 51 radiation length and do not reach the second module.
The SMD is scintillator hodoscopes which consist of 7 scintillator strips with a widthof 15 mm in the vertical direction to provide x-coordinate, and 8 strips with a width of
20 mm in the horizontal direction to provide y-coordinate. The SMD is placed between
the first and second ZDC modules. At the position, the induced shower by neutron
reaches maximum. The shower position is determined by weighted average of the SMD
hits. The position resolution of the SMD is ∼ 1 mm for 100 GeV neutrons.
Coincidence of hits in the ZDCs defines ZDC trigger. The ZDC trigger serves as
an independent luminosity measure as well as the BBC trigger. Comparison between
BBCs and ZDCs can be used to estimate the uncertainty on luminosity measure.

2.2.3

Central magnet (CM)The transverse momentum of each charged particle is determined by its bending curvature in the magnetic field provided by the PHENIX central magnet (CM) system [86].
The CM is energized by two pairs of concentric coils and provides an axial magnetic
field parallel to the beam direction.
During this measurement, the two coils of the CM were operated in the canceling(“+−”) configuration. This configuration is essential for the background rejection of
the heavy flavor electron measurement with the HBD as described later. A cross sectionof the magnetic field produced by the +− configuration is shown in Fig. 2.11, and the

50

CHAPTER 2. EXPERIMENTAL SETUP

Figure 2.8: (a) Location of the ZDC and SMD from the top view. (b) The view of thecross section in the figure (a). (c) The configuration of the three ZDC modules, SMD,
and scintillation veto counter.

SMD

(a)

ZDC
2nd module(b)

Figure 2.9: (a) Installed ZDC and SMD. (b) ZDC second module and SMD.

2.2. PHENIX DETECTOR SYSTEM

51

Figure 2.10: Mechanical design of ZDC module.

magnitude of the magnetic field as a function of distance from the collision point isshown in Fig. 2.12. In this configuration, the field is almost canceled out around thebeam axis in the radial region 0 < R < 50∫ cm, and has a peak value of ∼ 0.35 T aroundR ∼ 100 cm. The total field integral is | B × dl| = 0.43 Tm.1

2.2.4

Drift chamber (DC)The DCs are used to measure the charged particle trajectories
in x-y plane and deter√ 22mine the high resolution transverse momentum (pT ≡ px + py ). The DCs are locatedin the East and West arm at a radial distance of 2.02 < R < 2.46 m, and each covers
90 degrees in azimuth and |η| < 0.35 in pseudorapidity (corresponding to 1.8 m in1∫
| B × dl| = 0.78 Tm for the ++ configuration.

52

CHAPTER 2. EXPERIMENTAL SETUPPH ENIX

-4.0

-2.0

0.0

2.0

4.0

Z (m)

Magnetic field lines for the two Central Magnet coils in reversed (–) mode

Figure 2.11: Cross section of magnetic field in the +− CM configuration. The magnetic
field around the collision point is canceled out and the magnetic field exists only at
position with a ∼ 1.0 m radius from the collision point.
10000.0
Only Outer

8000.0Outer + Inner
6000.0Outer - Inner

4000.0Outer - Inner

2000.0

0.0

0

1

2

3

4

X (m) at Y=0

Figure 2.12: Field strength of the CM magnetic fields as a function of distance fromthe collision point. This plot shows three types of the configuration: using only outermagnet (+0), using both outer and inner magnets (++), and using outer and reversed
inner magnets (+−). The +− configuration is employed in this measurement.

2.2. PHENIX DETECTOR SYSTEM

53

R
.0 2
=2
m
90o

m

2.5 m

6
2.4
R=

Ti frame
mylar window

(a)

(b)

Figure 2.13: (a) Construction of DC frame. (b) Picture of DC.

z-direction). They use gas mixture of 50% Argon and 50% Ethane.
A schematic drawing of the DC frame and a picture of the DC is shown in Fig. 2.13.
Each DC consists of 20 sectors, each of which covers 4.5 degrees in azimuth. In eachsector, there are six types of wire modules stacked radially. They are called X1, U1,
V1, X2, U2, V2 plane from inner to outer.
The sketch of a sector and the layout of wire position are shown in Fig. 2.14. The X1
and X2 wire cells run in parallel to the beam to perform precise track measurements in
x-y plane. These wire cells are followed by two sets of small angle U, V wire planes usedin the patter recognition. U1, V1 U2, and V2 wires have stereo angle of about 6 degrees
relative to the X wires in order to measure the z-coordinate of the track. Each of X andU, V stereo cells contain 12 and 4 anode (sense) wires, respectively, which form cellswith a 2 - 2.5 cm drift space in ϕ direction. With the mixture gas of argon-ethane, theplateau drift velocity is 5.3 cm/µsec for field gradation from 800 V/cm to 1400 V/cm.
Therefore, the maximum drift time in a cell is approximately 470 nsec.
The DC signals are read out by ASD (pre-amplifier, shaping amplifier, discriminator)
chips and TMC (time memory cell) chips. The ASD chip contains 8 channels for preamplifier, and a 6 nsec shaping amplifier, and an externally controlled discriminator.

54

CHAPTER 2. EXPERIMENTAL SETUP
4.5 degr.

ϕ-direction
Wire orientations, top view

z-direction

V2 − planes

U2 − planes

r-direction

Termination wires

Back wire

V1 − planes~ 1800 mm

X2 − planes

Gate wires
U1 − planes
Potential wire

sense (anode)
X1 − planes

wire

anode planecathode plane

Sector, side view

U

X

V

ϕ-direction

Figure 2.14: (Left) The layout of wire position within one sector and inside the anodeplane. (Right) A schematic diagram, top view, of the stereo wire orientation.

The TMC chip can memorize bit information in time bins whose interval is 1/32 of the
input clock period. Using four times rate of the RHIC crossing clock as the input clock,the ASDs discriminator output is memorized with a single bin resolution of roughly
0.8 nsec in TMC.

2.2.5

Pad chamber (PC)The PCs are used to measure the space points which determine momentum in z direction(pz ) with the DC hit information. Figure 2.15 shows the PC mounted on the DC. This
position measurement is also important to determine the projection points on RICH,
EMCal, and HBD detectors. The PCs are the cathode readout multi-wire proportional
chambers located 2.47 < R < 2.52 m in radial distance from the collision point. EachPC also covers 90 degrees in azimuth and |η| < 0.35 in pseudorapidity as well as theDC.
Each detector contains a single plane of wires inside a gas volume bounded by two
cathode planes as shown in Fig. 2.16. The gas was chosen to be the mixture of 50%
Argon and 50% Ethane at atmospheric pressure. One cathode is finely segmented into
an array of pixels. A cell structure is made up with three pixels and has a dimension of
8.4 × 8.4 mm2 which satisfies the requirement of good position resolution in z direction
and a low occupancy even in the high track multiplicities. The position resolution of
the PC was measured to be 1.7 mm along the z direction.

2.2. PHENIX DETECTOR SYSTEM

55

Figure 2.15: PC mounted on DC.

Figure 2.16: Vertical cut through a chamber.

Anode wire
Field wire

8.4mm
Center pixel

1.5mm
2.7mm

Side pixel

8.2mm

Figure 2.17: (Left) The pixel geometry. (Right) A cell defined by three pixels locatedthe center of the figure.

When a charged particle starts an avalanche on an anode wire, charges are induced
on a number of pixels. Figure 2.17 shows the schematic diagram of the PC readout

56

CHAPTER 2. EXPERIMENTAL SETUP

pixels. In order to reduce the amount of electric and other noise, each cell of the PC
contains three pixels and an avalanche must be sensed by all three pixels to form avalid hit in the cell. Because of huge electronic channels with this arrangement, theinterleaved pixels are gathered together. Nine pixels are connected to a group and to acommon readout channel, such that the three pixels in a cell are always connected to
diﬀerent but neighbor channels and each cell is defined by its unique channel triplet.
This solution saves a factor of nine in the number of readout channels compared toreadout of the all pixels. The PC signals are read out by custom made integrated circuits
which contain charge amplifiers and discriminators. Many operational characteristicsof the circuits are remotely programmable via serial communication.

2.2.6

Ring-imaging Čerenkov (RICH) Detector

RICH

PC
DC

BBC

(a) Top view.

(b) Cutaway view.

Figure 2.18: (a) A top view and (b) a cutaway view of the RICH detector.

The RICH is a threshold-type gas Čerenkov counter and the primary detector used
to identify electrons in PHENIX and shown in Fig. 2.18(a). The RICH also provides anelectron trigger together with the EMCal. It is located in the radial region of 2.5-4.1 m.
The RICH covers pseudorapidity |η| < 0.35 and 2 × π/2 in azimuth.
Figure 2.18(b) contains a cutaway drawing of the RICH detector. The RICH contains 48 composite mirror panels, forming two intersecting spherical surfaces. The
spherical mirrors focus Čerenkov light onto two arrays of 1280 UV PMTs (Hamamatsu
H3171S), each located on either side of the RICH entrance window. The PMTs are
fitted with 2′′ diameter Winston cones and have magnetic shields that allow them tooperate at up to 100 G. The minimum thickness of radiator gas seen by any particle
is 87 cm, the maximum is about 150 cm. The RICH is filled with CO2 gas for theradiator. The radiator gas is maintained at a pressure of 0.5′′ of water above ambient.
The RICH has a Čerenkov threshold of γ = 35, which corresponds to p > 20 MeV/c
for electrons and p > 4.9 GeV/c for charged pions. The average number of hit PMTs
per electron track is ∼ 5, and the average number of photoelectrons detected is ∼ 10.
Below the pion threshold the e/π separation is ∼ 104 in p + p collisions.

2.2. PHENIX DETECTOR SYSTEM

57

The preamp output drives both a charge and a timing measurement channel. The
timing measurement uses a discriminator followed by a TAC while the charge measurement employs a CMOS charge-integrating amplifier followed by a variable-gain amplifier. Both of the above analog outputs are stored in Analog Memory Units (AMU). The
analog data are digitized with ADC boards when the level-1 trigger signal is received.

2.2.7

Electromagnetic calorimeter (EMCal)

Figure 2.19: Assembled EMCal west arm, which consists of 4 PbSc sectors.
The EMCal [89] plays an important role for the detection of electrons and photons.
The EMCal in a half central arm covers central rapidity of |η| < 0.35 and azimuthal
angle of ∆ϕ = π/2. The EMCal in west arm is shown in Fig. 2.19. EMCal is composedof two types of calorimeters, lead-scintillator (PbSc) and lead-glass (PbGl). The radial
distance from collision point is ∼5.1 m for PbSc and ∼5.4 m for PbGl. A box with alabel of PbSc or PbGl in Fig. 2.5 corresponds to a sector. PHENIX has eight sectors of
EMCal, six of them are PbSc and other two are PbGl. Both type covers pseudorapidity
range of |η| < 0.35.
A PbSc sector is composed of 3 × 6 super-modules, and the super-module consists

58

CHAPTER 2. EXPERIMENTAL SETUP

of 6 × 6 modules, each of which consists of 2 × 2 towers. The tower is the minimum
unit of individual read-out in EMCal. On the other hand, a PbGl sector is composedof 12 × 16 super-modules, each of which consists of 4 × 6 towers. The total numbers of
towers are 15552 for PbSc and 9216 for PbGl.
The energy, time of flight, and energy sum for electron and photon triggers arederived from EMCal signals. The energy and trigger are used in this analysis. The
analog signals from PMTs are processed by ASIC chips on EMCal front-end electronics.
An ASIC chip processes four adjacent towers forming 2×2 block, which is the minimum
unit for trigger decision. The ASIC divides the signals for TAC process and ADC
process. In the TAC process, the divided signal is transformed to a digital signal bya discriminator and sent to the TAC. In the ADC process, the signal is amplified byvariable gain amplifier (VGA) with a gain of ×4 - 12 with 5 bit resolution. The signal
is summed up with one from other three towers to provide the energy sum of 2 × 2
towers and sent to trigger boards. As mentioned in Sec. 2.3.1, the analyzed data in thispaper is obtained with the 2 × 2 energy sum trigger with RICH signal. One unit of thetrigger decision, namely a group of 2 × 2 towers, is called a trigger tile.

2.2.8

Hadron blind detector (HBD)

A challenging issue for the heavy flavor electron measurement is to reject the dominantbackground of electron pairs from γ conversions and Dalitz decays of π 0 and η mesons
which are mediated by virtual photons. The HBD aims to considerably reduce thephotonic electron pairs utilizing the distinctive feature of the e+ e− pairs, namely theirsmall opening angles.
The HBD is a position-sensitive Čerenkov detector operated with pure CF4 gas
as the radiator. It covers pseudorapidity |η| < 0.45 and 2 × 3π/4 in azimuth. The
coverage is larger than the acceptance of the other detectors in the central arm in orderto detect photonic electron pairs with only one track reconstructed in the central arm
and the other outside of the central arm acceptance. Figure 2.20 shows the top viewand exploded view of the HBD. The HBD has a 50 cm long radiator directly coupled
in a windowless configuration to a readout element consisting of a triple Gas Electron
Multiplier (GEM) stack, with a CsI photo-cathode evaporated on the top surface of
the interior GEM and pad readout at the exterior of the stack. The readout element in
one half HBD arm is divided into six sectors. The expected number of photoelectrons
for an electron track is about 20 and it is consistent with the measured number. Since
the HBD is placed close to the collision point, the material thickness should be small
to reduce conversions. The total thickness to pass through the HBD is 0.024 × X0 andthe thickness before the GEM pads is 0.007 × X0 , where X0 represents the radiation
length.
The hadron blindness of the HBD is achieved by operating the detector in the socalled reverse bias (RB) mode as opposed to the standard forward bias (FB) mode (seeFig. 2.21). In the RB mode, the mesh is set at a lower negative voltage with respect to
the GEM and consequently the ionization electrons deposited by a charged particle inthe drift region towards the mesh as shown in the right panel in Fig. 2.21. Consequently,
the signal produced by a charged particle results only from (i) the collection of ionization

2.2. PHENIX DETECTOR SYSTEM

59(a) HBD from top view.

(b) HBD exploded structure.

Figure 2.20: (a) Top view of the HBD showing the location of the HBD in the central
magnet. (b) Exploded view of one half HBD arm. CF4 gas is filled in the volume as
the Čerenkov light radiator.

60

CHAPTER 2. EXPERIMENTAL SETUP

Figure 2.21: Triple GEM stack operated in the forward bias mode (left) and in the
hadron-blind reverse bias mode (right). During the experiment, the HBD is operated
in the reverse bias mode.

(a) HBD west arm. (East arm is removed.)(b) HBD east arm.

Figure 2.22: Side views of installed HBD (a) west and (b) east arms.

charge from only a thin layer of ∼ 100 µm above the top GEM which is subject to theentire three-stage amplification, and (ii) the collection of ionization charge in the firsttransfer gap (between the top and the middle GEMs) which is subject to a two-stage
amplification only. The ionization electrons produced in the second transfer gap and in
the induction gap generate a negligible signal since they experience one and zero stages

2.2. PHENIX DETECTOR SYSTEM

61

of amplification, respectively. For a drift region and a transfer gap of 1.5 mm each anda total gas gain of 5 × 103 , the mean amplitude of hadron signal drops to ∼ 10% of its
value in the FB mode [93].
The readout pad plane consists of hexagonal pads with an area of 6.2 cm2 (hexagon
side length a = 1.55 cm) which is comparable to, but smaller than, the blob size which
has a maximum area of 9.9 cm2 . Therefore, the probability of a single-pad hit by anelectron entering the HBD is very small. On the other hand, a hadron traversing the
HBD will produce a signal predominantly localized in a single pad. This provides anadditional hadron rejection of the HBD.
The relatively large pad size also results in a low granularity thereby reducing thecost of the detector. In addition, since the signal produced by a heavy flavor electron is
distributed between two or three pads, one expects a primary charge of several photoelectrons per pad, allowing the operation of the detector at a relatively moderate gainof a few times 103 . This is a crucial advantage for stable operation of a UV photon
detector.
The HBD is located in a field free region that preserves the original direction of thee+ e− pair. The two spots images created by the electron pair which has a small opening
angle are merged each other. Therefore electron tracks can be identified as originatingfrom electron pairs from π 0 or η Dalitz decays or γ conversion pairs if the corresponding
hit in the HBD has an amplitude that is twice to a regular amplitude.
HBD readout electronics

Figure 2.23: Block diagram of HBD front-end module.
The charge signal from each readout pad of the GEMs is amplified by a custom
designed hybrid preamplifier (IO1195-1) developed by the Instrumentation Division at

62

CHAPTER 2. EXPERIMENTAL SETUP

BNL [95]. The gain was set to give an output of ±100 mV for an input signal of 16 fC(105 e), which corresponds to an average primary charge of 20 photoelectrons at a gas
gain of 5 × 103 . It has a peaking time of 70 nsec and a decay time of 1 µsec. Whenconnected to the GEM pad, the noise was measured to be ∼ 103 e, which is equivalent
to 0.2 photoelectrons (p.e.) at a gain of 5 × 103 . The preamp output is delivered to ashaper located in a HBD front-end electronics.
The overall layout for the HBD front-end electronics is shown in Fig. 2.23, andfurther details are given elsewhere [96]. The electronics receives the preamp signals
using a diﬀerential receiver, which provides some additional shaping, and then digitizes
them using a 65 MHz 8 channel 12 bit flash ADC. The gain of the shaper was set suchthat a preamp input charge of 16 fC produces a signal of 160 ADC channels (0.1 fC/ch).
The preamp signal was set in the range of ±1.5 V around the ADC common mode
voltage, which eﬀectively uses only half of the full dynamic range of the ADC. The
output of the ADC is serialized and sent to FPGA (ALTERA Straitx III 60), which
encodes the data to accomplish the communication with the PHENIX DAQ system.
HBD gain control during experiment

Relative gain

10

P/T dependence of gain

1

10-1

2.35

2.4

2.45

2.5

2.55

2.6

2.65

P/T (Torr/K)

Figure 2.24: P/T dependence of relative gas gain of HBD GEM filled with CF4 . The
gain is normalized to 1 at P/T = 2.55 Torr/K. The line represents a fit of the data
points with an exponential function.

The gas gain of the GEM pads filled with CF4 is sensitive to gas density. The gas
density can be determined by the ratio of the pressure, P , and the temperature, T ,of the HBD volume, namely P/T . Figure 2.24 shows the P/T dependence of the gas
gain [92]. A P/T change by 1% induces a ∼ 20% change in the gas gain.

2.3. PHENIX DATA ACQUISITION

63

Figure 2.25: (Left) P/T variation during 2010 experiments. This plot also shows theadjustment of HV corresponding to the P/T variation. (Right) Gas gain variation
during same period in one HBD sector as a result of the HV adjustment. The verticalaxis represents calibration constants for the HBD gain which will be mentioned inSec. 3.1.5. Blue and red points correspond to 4021 V and 4044 V HV configurations,
respectively. Due to the HV control, the variation of the gas gain is inhibited and kept
constant within ∼ 10%.

To avoid these large excursions of the gain, the operating HV was discretely adjusted
during this experiment to compensate the gain variations by monitoring P/T in HBD.
The left plot in Fig. 2.25 shows the P/T variation during 2010 experiment. About 2%
variation was observed during this experiment. This plot also shows the adjustmentof HV corresponding to the P/T variation. The right plot in Fig. 2.25 shows the gas
gain variation during same period as a result of the HV adjustment. Blue and red
points in the figure correspond to 4021 V and 4044 V HV periods, respectively. The
HV adjustment reduced the variation of the gas gain and kept it constant within ∼ 10%
deviations.

2.3

PHENIX Data Acquisition

2.3.1

Triggers

The RHIC beam bunches in the blue beam and the yellow beam cross each other in
9.4 MHz frequency (106 nsec period) at the PHENIX interaction region and the total
data volume from PHENIX detectors to be recorded for an event in this p + p 200 GeV
run was about 40 kB. Due to the high frequency of collisions and the data volume, theacquisition of the event data from the all collisions is not realistic. Therefore, we needa Level-1 trigger for the data selection in online level.
To issue the Level-1 trigger, several trigger signals from PHENIX detectors specified
to measure diﬀerent physics channels are combined. For examples, triggers from BBC
are for minimum-bias data sampling, triggers from ZDC are for forward neutron analysis, and triggers from a combination of EMCal and RHIC are for mid-rapidity electron
and photon analysis. These triggers are named as local Level-1 (LL1) triggers.

64

CHAPTER 2. EXPERIMENTAL SETUP

In this section, two main LL1 triggers used in this analysis, a minimum-bias trigger(MB trigger) and a EMCal/RICH trigger for electron (ERT-E trigger), are explained.
MB trigger
The least-biased trigger for detecting inclusive p + p inelastic collisions is the MB
trigger√which consists of a coincidence between signals from BBC-N and BBC-S. In thisp + p s = 200 GeV run, the MB trigger required one or more hit tubes for both of
BBC-N and BBC-S (each counter consists of 64 tubes).
The timing information of BBC-N and BBC-S can be utilized to determine theposition of the collision vertex in online level with a resolution of ∼ 5 cm. However thetiming cut was not applied because the rejection power of the MB trigger was enoughfor the electron analysis.
The MB trigger was utilized in the electron analysis in order to require the signals
on BBC-N and BBC-S modules and determine the vertex position (Sec. 3.1.1) and also
for the estimation of the trigger eﬃciency of the ERT-E trigger (Sec. 4.2.3).
ERT-E trigger
ERT-E trigger selects collisions with electron tracks in the mid-rapidity region. This
trigger is provided by signals from EMCal and RICH which are sensitive for electron
tracks [97]. Basic idea of the trigger is to find a EMCal energy deposit over a threshold
and also require a RICH hit signal in an acceptance associated with the position of the
EMCal hit.
Each EMCal readout electronics for the ERT-E trigger processes the signals from144 (12 × 12) EMCal towers. The readout electronics has 6 daughter cards which hold
6 ASIC chips (36 ASIC chips in total) and each ASIC chip processes the signals froma block of 2 × 2 EMCal towers. The ASIC sums up the energy deposit in the 2 × 2
towers. When the total energy is over a threshold, the ASIC issues a trigger bit (ERT2×2 signal) to a ERT-LL1 electronics. The threshold during this p + p 200 GeV run
corresponds to ∼ 0.8 GeV. All 2 × 2 towers in the both arms are numbered serially as
Fig. 2.26(a) shows.
RICH has 5120 PMTs (2 (arms) × 2 (north and south) × 80 (ϕ-direction) × 16
(z-direction)). The RICH readout electronics for the ERT-E trigger forms PMT tiles
including 5(ϕ)×4(z) PMTs. Those tiles are not overlapping, and therefore there are
2 × 16 × 4 = 128 tiles in one arm. Each PMT tile sums up all analog signals fromthe 5×4 PMTs and digitizes the analog sum. When the electronics find a hit in thedigitized signals, a trigger bit (ERT-RICH signal) is issued to the ERT-LL1 electronics.
All PMT tiles are also numbered serially as Fig. 2.26(b) shows.
The trigger bits from EMCal and RICH are sent to the ERT-LL1 electronics which
decides the LL1 triggers for electron and photon analyses. For the ERT-E trigger decision, the ERT-LL1 electronics requires coincidence of the trigger bits of ERT-2×2 and
ERT-RICH. The ERT-LL1 electronics also requires the acceptance matching between
the hit positions of ERT-2×2 and ERT-RICH. The matching check is done with a lookup table created by Monte Carlo simulations. The look-up table is shown in Fig. 2.27
where the horizontal axis and the horizontal axis correspond to the number of EMCal

2.3. PHENIX DATA ACQUISITION

65(a) EMCal 2 × 2 towers.

(b) RICH PMT tiles.

Figure 2.26: Numbering definitions of (a)EMCal 2 × 2 towers and (b)RICH PMT tiles.

2 × 2 towers and RICH PMT tiles defined in Fig. 2.26(a) and Fig. 2.26(b) respectively.
As an example, the 13th EMCal towers need to be compared with 16th, 17th, 18th,
19th, 20th, 24th, 25th, 26th, 32nd, 33rd and 34th RICH channels. If any one of the
EMCal channels has a corresponding RICH hit, the ERT-E trigger is issued.
The ERT-E trigger was utilized for the data selection of the electron analysis.

66

CHAPTER 2. EXPERIMENTAL SETUPRICH Tile

Lookup table, pt=0.9GeV/c

250

200

150

100

50

0
0

20

40

60

80

100

120

140

160
180
EMCal Tile

Figure 2.27: Look-up table of EMCal 2 × 2 tower and RICH PMT tiles matchingobtained by Monte Carlo simulation. The horizontal axis represents the number of the
EMCal 2 × 2 tower and the vertical axis represents the number of the RICH PMT tiles.
Because PbSc corresponds to the EMCal tiles from 0th to 107th and PbGl corresponds
to the EMCal tiles from 108th to 171st (see Fig. 2.26(a)), slope of the matching tablechanges between the two regions. This look-up table is utilized for the ERT-E trigger
decision.

2.3.2

Data acquisition system

The signals from the PHENIX detectors were taken with the PHENIX data acquisition system (DAQ) [98]. Figure 2.28 shows a block diagram of the PHENIX DAQ.
As mentioned in Sec. 2.3.1, signals from some of PHENIX detectors are utilized
for the local Level-1 (LL1) trigger decisions. The LL1 triggers are transmitted to a
global Level-1 (GL1) module which scales down the LL1 triggers and decides the Level1 trigger issuance considering combinations of the scaled down LL1 triggers and also
busy signals from DAQ. When the GL1 module accepts the LL1 trigger for an event, aLevel-1 trigger signal is issued to granule timing modules (GTM) and then distributed
to front-end modules (FEM) on the PHENIX detectors in the PHENIX interaction
region.
FEM is a readout electronics implemented for each detector. FEM processes the raw
signals from the detector with amplification and shaping and converts them into digital
data. Since the RHIC beams have the bunch structures and collide in a 9.4 MHz period,
FEM receives a 9.4 MHz RHIC beam-clock (BCLK) synchronized with the collisions
for the data processing. BCLK is provided by the accelerator and distributed to FEM
by GTM as well as the Level-1 trigger signal. The digitized data are buﬀered for up to2.3. PHENIX DATA ACQUISITION

67

GL1
Busy

GTM

So
ut
C
ZD

Data packets
over fiber~350000 ch
Shield Wall

Missing:• ARCNET Serial Interface to FEMs
• High and Low Voltage Control andassociated control systems• Alarm system

MTM

RHIC clocks

DCM

Data Packets
over Parallel
FIFO InterfaceD
ov atapa er Pa
ss NE ck
in V etg
Bu IS ss To
kenPartitioner
DCB

FEM

Clock

RHIC
Computing
Facility
End
User

Busy

Clock, Trigger,
Mode bits
over fiberhA(e S u.g b s.E y
as ste
tD m
C
)

ZD

C

N

L1

or
th

TriggerCountingrooma few miles later'trigger' is a placeholder
for a much more complex'Local Level 1' trigger system

Interaction region

GL1 Global Level 1 Trigger
GTM Granual Timing Module
MTM Master Timing Module
DCB Data Collection Board
DCM Data Collection Module
FEM Front End Module
SEB Sub Event Buffer
ATP Assembly Trigger Processor

OnlineMonitoring

SEB
SEB
SEB
SEB
ATM Switch
ATP

Linux Farm
~600 CPU's

tapestorage
1.2PB

20MB/s
250TB/yr

Figure 2.28: Block diagram of the PHENIX DAQ. PHENIX detectors are called as
subsystems in this figure. See the text for details.

40 BCLK periods (about 4 µsec) to allow FEM to wait the Level-1 trigger reception.
When FEM receives the Level-1 trigger, FEM transmits the data to a data collection
module (DCM) with a tag of a event number which is a universal counter for all FEM
system.
Each detector system also has its own DCMs as well as FEMs. DCM performs zero
suppression for the received data to compress the data volume and also error checking.
This procedure produces compressed data packets.
Compressed data packets from all detector system are sent to PHENIX event builders(EvB) in parallel for event assembly. EvB consists of sub-event buﬀers (SEB), an asynchronous transfer mode (ATM) switch and assembly/trigger processors (ATP). SEB
is a buﬀer to allow for the event assembly on ATP. To assemble data on ATP, data
packets from a large number of SEBs must converge on one ATP. This convergencecauses excesses of data volumes sent to ATP over an acceptable size. The ATM switch
solves the problem with ATM technology based on quality-of-service features. As theresult, the detector-by-detector data packets are assembled to the event-by-event data
packets on ATM.
The assembled data are transmitted to Linux machines to be recorded on hard drives
and tape storages (HPSS). The data are also transmitted to online monitoring system
to control data qualities in online level. End users can use the recorded data on thestorages for oﬄine analysis.
During this run, the trigger rate of ∼ 5 kHz and the DAQ band width of ∼ 200 MB/s

68
were achieved with this PHENIX DAQ.

CHAPTER 2. EXPERIMENTAL SETUP

Chapter 3
Electron Analysis
3.1

Conventional Techniques for Electron Analysis

For electron selection in the PHENIX central arms, some conventional techniques using
the detector signals from other than the HBD are already established and form thebasis for the electron analysis. Also in the electron measurement using the HBD, weutilized these techniques as well as the HBD response information. We explain theseconventional methods in the following sections.
In Sec. 3.1.1-3.1.5, we introduce the basic electron selection with the PHENIX detectors. Electrons sampled by these techniques largely include photonic and non-photonic
background. In the previous measurement, two methods, “cocktail method” and “converter method”, were used to estimate the background fraction [29]. Though our new
measurement with the HBD basically does not use these methods other than a part of
the “cocktail method” results of dielectron decays of light vector mesons as describedin Sec. 3.2.6, we briefly explain these estimation methods in Sec. 3.1.6 to make clear
the diﬀerence of the analysis techniques between the new and previous measurements.
The detailed description about these methods can also be found elsewhere [29].

3.1.1

Vertex position determination

In the PHENIX experiment, the collision vertex position is obtained by using the timing
diﬀerence between the hits in BBC-N and BBC-S. The vertex position, zvtx , and thecollision timing, T0 , are represented asc (TS − TN )2
TS + TN
L
=
− ,
2
c

zvtx =

(3.1)

T0

(3.2)

where TN and TS represent the measured hit timing in BBC-N and BBC-S and L
represents the half of the distance between the BBC-N and BBC-S and is equal to1440 mm, and c is the speed of light. The obtained vertex position zvtx is utilized forvertex cut and the track reconstruction with DC and PC information. The collision
timing T0 is also used for the measurement of the drift time in DC.
69

70

3.1.2

CHAPTER 3. ELECTRON ANALYSIS

Track reconstruction by DC and PC

Charged track reconstruction and momentum reconstruction

Figure 3.1: A particle trajectory and the parameters used in the DC track reconstructionin the beam view of PHENIX central arm.
Track reconstruction for charged particles is preformed with hit information of the
DC and PC, and vertex position zvtx [99, 100]. At the first step, reconstruction of the
projected track on the X-Y plane for the DC hits is performed employing a combinatorial Hough transform technique [101, 102]. In this technique, the drift chamber hits
in X1 and X2 planes are mapped pair-wise into ϕ-α space, where ϕ is the polar angle
at the intersection of the track with a reference radius near the mid-point of the DC(R = 220 cm) and α is the inclination of the track at that point. Figure 3.1 provides
a schematic illustration of these variables. Since the orientation of the magnetic field
in CM is parallel to the Z axis, the inverse of α is corresponding to the transverse
momentum of the particles.
After the reconstruction of the projected track in the X-Y plane, track reconstruction in the Z direction for the DC hits is first attempted by integrating informationfrom reconstructed PC clusters, which contain z information. If there is a unique PC
association, the three-dimensional track on the DC and PC is reconstructed by thez-coordinate of the PC cluster and vertex position zvtx . If there is no PC cluster association, or if there are multiple PC association solutions, the reconstruction in the Z
direction is performed by the DC stereo wires with a Hough transform again. If the3.1. CONVENTIONAL TECHNIQUES FOR ELECTRON ANALYSIS(a) Track reconstruction eﬃciency.

71(b) Momentum reconstruction resolution.

Figure 3.2: (a) The reconstruction eﬃciency in simulated central Au + Au collisions
as a function of the transverse momentum of the track. (b) Momentum resolution inmuon events as a function of momentum of the muons. [99, 100]

stereo wires find a unique associated track and determine the z-coordinate at the reference radius in the DC, the three-dimensional track on the DC is reconstructed by thedetermined z-coordinate on the DC and vertex position zvtx . The track reconstruction
also determines the polar angle of the track momentum at the reference radius in theDC, θ.
Due to non-uniformity of the focusing magnetic field along the flight path of thecharged particle traversing the CM, a field-integral grid which is a look-up table ofthe path-integrated field for the particle trajectory from the collision point to a radius
position r was employed for track and momentum reconstruction. The grid is definedon discrete inputs of zvtx , the magnitude of the momentum of the track at the collisionvertex p, and its polar angle θ0 . To interpolate the integrated field to any input values,
a fast Runge-Kutta method [103] is employed.
For each reconstructed track, the interpolated field integral and the trajectory determined by the field integral are calculated iteratively adjusting the input values, pand θ0 , and also the azimuthal angle of the track momentum at the collision vertex, ϕ0 ,so that the trajectory represents the all DC and PC hits associated to the reconstructed
track. For the first iteration, the initial θ0 is set to be measured θ and the initial p is
set to be an expected value from measured α. Using the iterative procedure, the p, θ0 ,and ϕ0 , namely the three-dimensional momentum vector, are reconstructed.
Figure 3.2(a) shows the reconstruction eﬃciency of above method in simulated central Au + Au collisions as a function of the transverse momentum of the track, andFig. 3.2(b) shows the momentum resolution in this reconstruction for simulated muon

72

CHAPTER 3. ELECTRON ANALYSIS

events with event vertices ranging |zvtx | < 40 cm, which will not decay as a function of
momentum of the muons [99, 100]. Multiple scattering dominates the resolution at low
momentum, while detector resolution determines the resolution at higher momentum.
The reconstructed track is extrapolated or interpolated to RICH, EMCal and HBD
to obtain the projected positions on the detectors. The projected position is used tosearch associated hit on the detector around the track.
DC alignment correction

Figure 3.3: Parameters of DC hit information and beam spot position in the DC alignment analysis.

To measure the momentum precisely, alignment of the wires in the DC was correctedby using charged tracks in zero magnetic filed obtained during this run. If there is nomagnetic filed, the tracks do not bend and α should be zero. Any deviation from zero
indicates that the cards do not set at correct positions which we suppose. The wrongalignment can be corrected by this measured deviation in α.
The first step in the alignment procedure is to find the beam oﬀset which is assumed to be at the original point of the DC coordinate in momentum reconstruction
as explained above. The actual position of the beam may be diﬀerent from the centerof the arms. Also, since the carriage gets moved out occasionally, this also relatively
adds to a beam oﬀset eﬀect. Figure 3.3 shows a cartoon of the geometry. The beamoﬀset in the x and y directions, dx and dy respectively, are related to dα and ϕ, where3.1. CONVENTIONAL TECHNIQUES FOR ELECTRON ANALYSIS

73

dα represents the measured α in the zero-field events. Thus plotting α vs. ϕ gives us away to extract dx and dy. The relation between these variables are
R sin ϕ − dy,R cos ϕ − dx

(3.3)

−dy cos ϕ + dx sin ϕ
R

(3.4)

tan(ϕ + dα) =orsin dα =assuming dx, dy ≪ R, where R is the middle radius of X1 and X2 in DC which is equal
to 220 cm. This function is fitted to the plot of α vs. ϕ to extract dx and dy.
The second step of the alignment is a correction for the rotation and tilt of anodecards, which the anode wires of the DC are attached to. Though the anode cards are
physically held in place with pins, they can shift slightly. The rotation of the card
induces non-zero α in zero-field events. Using the measured α’s at the X1 layer and theX2 layer, the rotation of the card can be corrected. The card tilt also causes z dependent
deviation of ϕ. This deviation is relatively corrected looking at ∆ϕ = ϕ2 − ϕ1 vs. z,
where ϕ1 (ϕ2 ) is the ϕ measured at the X1 (X2) layer. In these zero-field events, ϕ1 andϕ2 should be identical in whole z coordinate.

3.1.3Electron ID by RICH

The number of hit RICH PMTs in an annular region with an inner radius of 3.4 cm
and an outer radius of 8.4 cm around the track projection on the RICH is counted andwas required to be more than 1 hit PMTs in this measurement. With the requirement,
the RICH detector can provide a pion rejection factor better than 104 with an electroneﬃciency close to 100% in 200 GeV p + p collisions.

3.1.4

Electron ID by EMCal

EMCal cluster
The electromagnetic shower of an incident particle at EMCal spreads among several
towers. To extract the energy deposit, it is necessary to cluster the group of hit towers
which associate to the shower. The detailed description for the clustering method andenergy reconstruction of EMCal hits are already described elsewhere [104, 105]. The
total energy in the cluster towers is calculated as the energy deposit and corrected forthe incident angle dependence which is determined from beam tests [106]. The positionof the cluster is determined by the weighted average of tower energies, and also correctedwith the incident angle of the particle.
EMCal calibration
The detailed EMCal calibration procedure is also described elsewhere [107, 108, 105].
The time dependence of the tower-by-tower energy gain is corrected using the laser system for both PbSc ad PbGl, which is described in Sec. 2.2.7. The remaining long-term

74

CHAPTER 3. ELECTRON ANALYSIS

time dependence and gain non-uniformity among the towers is also calibrated using π 0
peak measurement by 2γ pair reconstruction in oﬄine analysis. This calibration procedure is divided into three steps: long-term time dependence correction for the global
towers, tower-by-tower dependence correction, and energy non-linearity correction.
The long-term time dependence of the EMCal gain is performed with run-by-run
π 0 peak reconstruction. The invariant mass spectra of reconstructed 2γ pairs are fitted
to the combination of Gaussian (signal) and polynomial (background). The correct
peak position must be larger than the real π 0 mass of 135.0 MeV/c2 [61], due to finite
energy resolution and steep pT shape of π 0 cross section [108]. Displacement of thepeak position from the correct position is calibrated tuning the global EMCal gain.
The tower-by-tower gain correction is also performed with same technique as above.
For each tower, 2γ pairs which create a cluster hit on that tower are sampled andinvariant mass spectrum for the pairs are created. Using the π 0 peak position, thetower gain is calibrated. Since the peak position is also aﬀected by the other tower
gains, this step is iterated several times for the whole towers.
After above calibrations, an energy non-linearity still remains due to the energy
threshold for the EMCal towers (∼ 10 MeV). For the energy non-linearity correction,a fast Monte Carlo simulation (FastMC), which is a simple numerical simulator of thedetector response, is used to determine the non-linearity as a function of the detectedcluster energy. For the simulation, π 0 → 2γ events are generated using the measured
π 0 cross section. The FastMC simulates the energy and position fluctuation and theshower profile to reproduce the energy deposit on the EMCal towers and compares itwith the real incident energy.

Electron identification by EMCal
For electrons, the energy deposit on EMCal, E, and the magnitude of the reconstructed momentum on DC and PC, p, must be identical due to their small mass.
Therefore the ratio, E/p, was required to be close to 1. Practically, the peak positionof electron E/p distribution is located at ∼ 0.98 for electrons of pT > 1.0 GeV/c. It
is because the depth of the shower from an electron is one radiation length (∼ 2 cm)
shallower than that from a γ which is used in EMCal energy calibration, and thus thelight from the electron is slightly more attenuated in the fibers than the light from theγ [104]. Since the energy resolution of the EMCal depends on the momentum of theelectron track, the cut boundaries were changed in diﬀerent momentum range. The
shower profile of the EMCal cluster was also required to be as the electromagnetic
shower.
The association with the DC/PC reconstructed track is required for the EMCal
cluster. The displacement between the projected position of the track and the EMCal
cluster position is decomposed in ϕ and z directions and required within 4.0 σ deviation
for the both directions, where σ represents RMS of Gaussian fitted to the displacement
distributions for these components.

3.1. CONVENTIONAL TECHNIQUES FOR ELECTRON ANALYSIS

75

Reverse bias
10

Hits/event

1
10-1
10-2
10-3
10-4

0

20

40

60

80

100

ADC counts
Figure 3.4: Pulse-height distribution in a HBD pad.

3.1.5

Electron ID by HBD

HBD gain calibration
The gain of each detector module is conveniently and precisely determined by exploiting the scintillation light produced by charged particles traversing the CF4 radiator andtranslated into the unit of number of detected photoelectrons (p.e.). The scintillation
signal is easily identified in the low amplitude part of the pulse-height distribution as illustrated in Fig. 3.4. A steep exponential distribution at very low amplitudes attributed
to scintillation photons can be seen in the figure. A longer tail at higher amplitudes
corresponds to ionization of the gas in the drift gap.
The mean value of the exponential distribution corresponds to the inverse slope ofthe distribution. Therefore the gain G of the detector is obtained from
S −1
G=
⟨m⟩

(3.5)

where S is the slope of the scintillation exponential shape at low amplitudes and ⟨m⟩is√ the average number of scintillation photons in a fired pad. In p + p collisions at
s = 200 GeV, ⟨m⟩ is enough close to 1 such that the gain is readily given by the
inverse slope of the exponential distribution, G = 1/S.1 A line in Fig. 3.4 represents
1

In Au+Au collisions, however, the inverse slope increases with the number of charged particles
traversing the detector since ⟨m⟩ increases with the number of the scintillation light.

76

CHAPTER 3. ELECTRON ANALYSIS

Matching, Δ Φ

Matching, ΔZ

40000
30000

Yield

Yield

30000
20000

20000

10000

10000

0

-0.1

0

0
-20

0.1

-10

0

ΔΦ (rad)20(b) z-direction.

20

2

15

1.5

σ (cm)σ (mrad)(a) ϕ-direction.

10

5

0

10

ΔZ (cm)

1

0.5

0

0.5

1

1.5
2
p (GeV/c)(c) σ in ϕ-direction.

2.5

3

0

0

0.5

1

1.5
2
p (GeV/c)

2.53(d) σ in z-direction.

Figure 3.5: (a, b) Matching of electron tracks in ϕ (a) and z (b) directions. The solidlines represents the fits to a Gaussian function. (c, d) Matching resolution of electron
tracks σϕ (c) and σz (d) as a function of momentum.

the determined slope by fitting. The typical calibration constant is ∼ 10 ADC/p.e..
HBD cluster
The HBD was not implemented in the previous PHENIX measurements and was firstused in this electron measurement. The electron identification using the HBD is discussed in this section.
In the 200 GeV p + p collisions, multiplicity in the HBD is small enough so thatclustering method for the HBD signals works for this analysis. Figure 3.6 is an event
display showing hit pads at a typical multiplicity. In the figure, a projection position ofa reconstructed electron track is also shown with a yellow-filled circle. The associated3.1. CONVENTIONAL TECHNIQUES FOR ELECTRON ANALYSIS

77

Charge [p.e.]
20.0
19.0
18.0
17.0
16.0
15.0
14.0
13.0
12.0
11.0
10.0
9.0
8.0
7.0
6.0
5.0
4.0
3.0
2.0
1.0

Figure 3.6: Event display of one HBD sector. Yellow filled circle represents the projection position of reconstructed electron track. Groups of pads surrounded with redlines are reconstructed HBD clusters. There is an associated cluster at the projection
position of the reconstructed track.

cluster can be found around this track.
The clustering is proceeded by the following procedure. The hit pads are separated
into two categories by their charge. One of the two is “seed pad” that has a signal
charge more than 3.0 p.e. and the another one is “non-seed pads” that has a signal
charge from 1.0 p.e. to 3.0 p.e.. For the first step, seed pads neighboring each otherare clusterized and create a “seed cluster”. For each seed cluster, all non-seed pads
neighboring this seed cluster are searched and added to the seed cluster to obtain the
final cluster for the analysis. This clustering is performed across the HBD sectors. The
position of the cluster is defined as the center of pad charge and the cluster charge isdefined as the sum of the all pad charges. Association of the cluster with the DC/PC
reconstructed track is required. The matching between the HBD cluster position andreconstructed track is shown in Fig. 3.5. The position of the cluster is required within
3.5 σ deviation in the both ϕ and z directions as well as the EMCal cluster. The clusters
created by this algorithm is also enclosed with red lines in Fig. 3.6.

3.1.6

Heavy flavor electron estimation in previous measurements

Electrons sampled by these techniques consist of heavy flavor electrons, photonic
background from Dalitz decays of light neutral mesons and γ conversions, and nonphotonic background from Ke3 decays, dielectron decays of light vector mesons, andDrell-Yan process. The yield of the background is to be known to estimate the yieldof the heavy flavor electrons. In the previous measurement, the background yield wasestimated with the “cocktail method” and “converter method” as follows (see also

78

CHAPTER 3. ELECTRON ANALYSIS

detailed descriptions elsewhere [28, 29]).
Cocktail method
A cocktail of electron spectra from background sources is calculated using a Monte
Carlo event generator of hadron decays and then subtracted from the inclusive electron
spectra. This technique requires the precise cross section spectra of these hadrons.
π 0 cross section spectrum have been well determined from charged and neutral π
measurements and π 0 is the largest component in the background. Cross section spectraof light mesons, η, ρ, ω, η ′ , and ϕ mesons, can be reproduced by mT scaling from theπ 0 cross section with an accuracy within ∼ 20% [78]. The normalization of the spectra
relative to the π 0 spectrum was obtained by the ratios of the mesons to the pions athigh pT region (pT > 5.0 GeV/c).
Since K meson has large decay lengths, electrons from Ke3 decays have diﬀerent
acceptance and diﬀerent detection eﬃciency from other electrons which come from thecollision point. Therefore, the estimation of the contribution from the Ke3 decays
needs a full GEANT simulation taking into account the exact electron selections. The
method of the Ke3 electron estimation is also used in the present analysis as describedin Sec. 3.2.6.
Heavy flavor electrons from heavy quarkonia decays (J/ψ and Υ), which are also
signal electrons in this analysis, have been also estimated using the cocktail method.
J/ψ cross section measured at PHENIX was utilized for the J/ψ→e+ +e− contribution,and Υ cross section calculated by NLO pQCD was utilized for the Υ→e+ + e− contribution. As the result, the J/ψ contribution to the total heavy flavor electrons was foundto be less than 2% in pT < 1.25 GeV/c and increased to ∼ 20% up to pT = 5.0 GeV/c,
whereas the Υ contribution was less than 1% in pT < 4.0 GeV/c and increased to ∼ 3%
up to pT = 5.0 GeV/c.
The Drell-Yan background have also been estimated by using LO pQCD calculated
cross section. This contribution was less than 0.5% in pT < 5.0 GeV/c and negligibly
small in this analysis.
Converter method
With the converter method, the yields of photonic and non-photonic electrons are
obtained by measuring the diﬀerence between inclusive electron yields with and withouta photon converter of precisely known thickness: a brass sheet of 1.680% radiation
length. These yields can be expressed as the following relations:
NeConv-out = Neph + Nenon-ph ,NeConv-in = Rph Neph + (1 − k)Nenon-ph .

(3.6)
(3.7)

Here NeConv-out (NeConv-in ) is the measured electron yield without (with) the converter.
Neph (Nenon-ph ) is the photonic (non-photonic) electron yield. k represents a small lossof Nenon-ph owing to the converter, which was evaluated to be ∼ 2.1%. Rph represents
how much the photonic electron yield is increased by the converter, which is calculated
to be ∼ 2.41 using material information around the collision point within accuracy of3.2. HEAVY FLAVOR ELECTRON ANALYSIS WITH HBD

79±2.0%. This value was also confirmed with a GEANT simulation including geometricaleﬀects.
The yields from non-photonic background can not be estimated only with the converter method. Though these contributions are smaller than the photonic electrons,
they can not be negligible especially in the low pT region. The results of the cocktail
method for Ke3 decays and light vector mesons were used for the estimation.

3.2

Heavy Flavor Electron Analysis with HBD

3.2.1

Overview

With the improved purity from the HBD, the double helicity asymmetry of the heavy
flavor electrons was obtained. In this section, we discuss how the heavy flavor electron
analysis and the purification of the heavy flavor electron sample using the HBD were
performed.

3.2.2

Data selection

Data used in this analysis were recorded by PHENIX from the end of April to thebeginning of July 2009. The data set was selected by the ERT-E trigger in a coincidencewith the MB trigger for the single electron analysis and by the MB trigger for thedetector calibration, where the ERT-E and MB triggers are described in Sec. 2.3.1.
The tracking eﬃciency of the PHENIX central detectors, especially due to HBD, issensitive to the track vertex position. Hence a collision vertex cut of |zvtx | < 20 cm wasapplied for the data selection.
After a vertex cut of |zvtx | < 20 cm and data quality cuts, an equivalent of 1.4 × 1011
MB events, corresponding to 6.1 pb−1 , sampled by the ERT-E trigger were analyzed.

3.2.3

Electron selection

The electron selection cuts for this analysis are named eID-Cut and listed in Table 3.1.
As described in Sec. 3.1.1, hit signals of RICH, EMCal, and HBD around the projected
positions of the reconstructed charged track and E/p ratio selection were required in
the eID-Cut.
In addition to these conventional electron selection, we also applied another selectionfor HBD cluster signal for further background rejection as follows. For the new selection
based on HBD, we introduced a new variable, qclus , ashbdcharge: qclus
Total charge of the associated HBD cluster calibrated in units of the number ofphotoelectrons (p.e.). The applied thresholds are diﬀerent for one sector due to
its lower eﬃciency compared with other sectors in this measurement.
Charged particles traversing the CF4 volume in the HBD produce also scintillation light,
which has no directivity and creates hits with only small charge on random locations inCHAPTER 3. ELECTRON ANALYSISCounts/MB events

80

4

×10-6

3.5
3
2.5
2
1.5
1
0.5
0
0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

E/p

2

Figure 3.7: E/p distributions for 0.5 GeV/c < pT < 1.0 GeV/c reconstructed charged
tracks with the eID-Cut except the E/p cut. The boundaries of the E/p cut for the
momentum region are shown by dashed lines in the plot.

the GEM pads. To remove the background hits from the scintillation light, a minimum
charge and a minimum cluster size were required for the HBD hit clusters. During thismeasurement, the eﬃciency for the Čerenkov light in one HBD sector was low comparedwith other sectors. Hence we applied a diﬀerent charge cut to that HBD sector for theelectron selection.
The E/p distribution for the selected tracks with these eID-Cut without the E/p
cut is shown in Fig. 3.7. The clear peak around E/p = 1 corresponds to electrons
and the continuous distribution around the peak consists of mainly electrons from Ke3
decays and mis-identified hadrons. The reason that the peak position is slightly smaller
than 1.0 is due to the diﬀerence of EMCal shower depth between electron and photon
as explained in Sec. 3.1.4. As the figure shows, the fraction of these background tracks
in the reconstructed electrons after applying eID-Cut was small. The fractions of theKe3 decays and the mis-identified hadrons are described in Sec. 4.1.1 and Sec. 4.1.2.
As mentioned in Sec. 2.2.8, we can remove the photonic electrons and purify theheavy flavor electrons on the basis of the associated HBD cluster charge. The selection
cut is represented as npe-Cut (non-photonic electron cut) and also shown in Table 3.1.

3.2.4

Detector stability

The stability of the DC, EMCal and HBD were checked with run dependence of multiplicities in the detectors. For the stability check, we used multiplicity calculated fromMB trigger dataset, which has no trigger bias.

3.2. HEAVY FLAVOR ELECTRON ANALYSIS WITH HBD

81

Table 3.1: Electron ID cuts and non-photonic electron cuts used in the electron analysis.
eID-Cut(electron ID cut)4.0σ matching between track and EMCal cluster# of hit tubes in RICH around track ≥ 2
3.5σ matching between track and HBD cluster
shower profile cut on EMCal
0.57 < E/p < 1.37 (0.5 GeV/c < pT < 1.0 GeV/c )0.60 < E/p < 1.32 (1.0 GeV/c < pT < 1.5 GeV/c )0.64 < E/p < 1.28 (1.5 GeV/c < pT < 5.0 GeV/c )# of hit pads in HBD cluster ≥ 2
qclus > 8.0 p.e.
( qclus > 4.0 p.e. for one low-gain HBD sector )

80
70
60
10-3
50
40
30

Sector Index in Z-coordinate

DC Board

npe-Cut(non-photonic electron cut)
8.0 < qclus < 28.0 p.e.
( 4.0 < qclus < 17.0 p.e. for one low-gain HBD sector)

90
80
70
10-3
60
50
40
30

20
20

10-4

10
10
0
0

10-4
100

200

300

400

500

0
0

100

200

300

400

Run Index(a) DC (East-South)500

Run Index(b) EMCal (Sect0)

Figure 3.8: (a) Run number dependence of DC track multiplicity as function of avariable corresponding to ϕ0 in half of one DC arm. (b) Run number dependence of
EMCal hit multiplicity as function of tower number projected in z-direction in a EMCal
sector.
Figures 3.8(a) and (b) display run dependence of multiplicity of reconstructed tracks
in a typical DC region and multiplicity of EMCal clusters in a typical EMCal region,
respectively. The horizontal axis in Fig. 3.8(a) is run number and the vertical axis
corresponds to track azimuth at vertex position ϕ0 , which is described in Sec. 3.1.2.
The color gradation represents the multiplicity of the reconstructed tracks in a MB
trigger event. Because we separated the all runs into four diﬀerent groups and required
diﬀerent fiducial-acceptance for these groups, ϕ0 distribution is changing during theexperiment and ϕ0 distribution in same fiducial-acceptance cut is stable. Each group

CHAPTER 3. ELECTRON ANALYSIS
10-2
90
80
70

10-3

60

HBD PadHBD Pad

82

10-2
90
80
70

10-3

60

50

50

40

40
-4

10-4

10
30

30

20

20

10
0
0

10
10-5
100

200

300

400

500

0
0

10-5
100

200

300

Run Index(a) Sect9 South

400

500

Run Index(b) Sect9 North

Figure 3.9: HBD pad-by-pad multiplicity as a function of run number in a typical HBD
sector.
with same fiducial-acceptance is called “run group”. For the calculation of the heavyflavor-electron cross section, the yields of heavy flavor electrons in the four run groups
are separately estimated, because each run group has a diﬀerent acceptance coverage.
The horizontal axis in Fig. 3.8(b) is run number and the vertical axis is EMCal sector
number projected in z-direction. The color gradation represents the multiplicity of thereconstructed clusters in a MB trigger event. We confirmed the fluctuations of these
multiplicities are consistent within statistical fluctuations.
Figure 3.9 shows the hit multiplicity in a typical HBD sector. Hit pads with HBD
cluster charge of q > 1.0 p.e. were selected for this study. The fluctuation of the HBD
multiplicity is large compared with the DC and EMCal due to the diﬃculty of the HBD
operation with a stable gain control.
Run dependence of the resulting electron yield after eID-Cut and npe-Cut per an
MB trigger event is shown in Fig. 3.10. Figure 3.10(a) is the result after applying
the cuts related with HBD (HBD signal association, qclus selection, and HBD cluster
size cut) and figure 3.10(b) is the one before applying them. The diﬀerent run groups
are distinguished by four colors, red, blue, green, and yellow. We confirmed that theyields per a MB trigger in each run group are consistent with each other during thisexperiment.

3.2.5

Non-photonic electron analysis with HBD

We categorize the HBD clusters into three types according to the sources of theclusters. A cluster created by a single spot of Čerenkov light from a non-photonic
electron as shown in Fig. 3.11(a) is defined as a single cluster. On the other hand,a cluster created by merging spots of Čerenkov light from a track pair of photonic
electrons as shown in Fig. 3.11(b) is defined as a merging cluster. However a portion of
the photonic electrons which has a large enough opening angle such that the two cluster>
0.1 rad) creates two separated single clusters as shown inare not merging (typically ∼Fig. 3.11(c). Therefore the single cluster is created by both of the non-photonic electron
and the photonic electron with a large opening angle.
We also define a third type of HBD cluster created by scintillation light, which wecall a scintillation cluster. Scintillation hits which accidentally have large hit charges

Electrons/MB(scaled:20cm)

3.2. HEAVY FLAVOR ELECTRON ANALYSIS WITH HBD

0.24

χ2 / ndf

×10-3

p0

83
250.7
142.3
56.91
84.11/ /268
123
47
71

4.192e-05
4.663e-05
5.375e-05 ± 1.646e-06
4.739e-05
1.078e-06
7.525e-07
9.756e-07

0.22
0.2
0.18
0.16
0.14
0.12
0.1
0.08
0.06
0.04
0.02
0

282

284

286

288

290

×103
292

Runnumber(a) with HBD cut
χ2 / ndf

×10

Electrons/MB(scaled:20cm)

-3

p0

265.3
152.9
87.73
87.46/ /268
123
47
71
0.0004865
0.0003876
0.0004076
0.0004131 ± 2.861e-06
2.264e-06
3.278e-06
4.901e-06

0.8
0.7
0.6
0.5
0.4
0.3
0.2
282

284

286

288

290

×103
292

Runnumber(b) without HBD cut

Figure 3.10: (a) Reconstructed electron yield per an MB trigger event after applying
eID-Cut and npe-Cut, and pT > 0.5 GeV/c per an MB trigger event. (b) Reconstructed
electron yield per an MB trigger event without HBD related cuts. Red, blue, green andyellow points represent to diﬀerent run groups.

and have neighboring hit pads can comprise clusters. Photonic electrons from γ conversions after the HBD GEM pads do not create Čerenkov light in the HBD gas volume.
Hence they basically do not have associated clusters in the HBD and they are rejected

84

CHAPTER 3. ELECTRON ANALYSIS(a) Non-photonic electrons.

(b) Photonic electrons (merging (c) Photonic electrons (separated
cluster).
clusters).

Figure 3.11: Responses of the HBD for (a) non-photonic electrons and (b, c) photonic
electrons. (b) Most of the photonic electron pair create merging clusters. (c) However
a portion of photonic electrons, which have a large opening angle, creates separated
clusters.

by the HBD hit requirement in the eID-Cut. However, a portion of these are accidentally associated with scintillation clusters, satisfy the eID-Cut, and survive in thereconstructed electron samples.
We estimated yields of these clusters from the distribution shape of the HBD cluster
charge. We also estimated the fraction of the small component of single clusters generated by photonic electrons which have the large opening angles as described. Then we
determined the non-photonic electron yield. Subtracting additional background electrons from Ke3 decays and e+ e− decays of light vector mesons, we obtained the yield of
the heavy flavor electron yield. In the following section, details of the above proceduresare described.
HBD cluster charge distributions
All clusters associated with the reconstructed electrons can be classified into the above
three types. The yield of the electrons associated with the single clusters must be
evaluated in order to estimate the yield of the heavy flavor electrons. The shapes of theqclus distributions for the three cluster types are quite diﬀerent since merging clusters
have basically double the charge of single clusters and the charge of scintillation clusters
is considerably smaller than the charge of the single cluster. Using the diﬀerence in the

HBD cluster charge distribution
single cluster

HBD cluster charge distribution
single cluster

0.05

merging cluster
scinti. cluster × 0.1

0.05

merging cluster
scinti. cluster × 0.1

0.04

0.04
0.03

0.03

0.02

0.02

0.01

0.01

0
0

85

dP/dqclus [1/p.e.]dP/dqclus [1/p.e.]

3.2. HEAVY FLAVOR ELECTRON ANALYSIS WITH HBD

10

20

30

40

50

60

70

(a) Sect9 South

80

90

100

qclus [p.e.]

0
0

10

20

30

40

50

60

70

80

90

100

qclus [p.e.](b) Sect9 North

Figure 3.12: HBD cluster charge (qclus ) probability distribution. The distributions arenormalized as integral = 1 for single cluster (blue) and merging cluster (red) and integral= 0.1 for scintillation cluster (yellow).
shapes, we estimate yields of these clusters as follows.
The qclus distributions of single and merging clusters were estimated by using lowmass unlike-sign electron pairs reconstructed with only the eID-Cut, which consist ofalmost photonic electron pairs. We defined the unlike-sign electron pairs whose twoelectrons were associated with two diﬀerent HBD clusters as separated electron pairs
and the pairs whose two electrons were associated to the same HBD cluster as merging
electron pairs. The qclus distribution of the single clusters were estimated by the qclus
distribution of the separated electron pairs and the qclus distribution of the merging
clusters were estimated by the qclus distribution of the merging electron pairs. These
distributions were normalized to each other with respect to their integrals.
The above estimations of qclus distributions for the single and merging clusters are
not identical to ideal definitions of these clusters. The ideal definition of the single
cluster is a cluster which consists of Čerenkov light only from one electron or positron
track, and one of the merging cluster is a cluster which consists of Čerenkov light fromboth of electron and positron tracks. Therefore, the qclus distributions of the above
estimation and the ideal definition are slightly diﬀerent. The diﬀerence is attributed
to a bias from the reconstruction of the electron pairs, which requires a minimum
momentum for the tracks. This eﬀect was estimated by using a GEANT3 simulationand found to modify the qclus distributions ∼±5%. The modification was applied to the
distributions.
The normalized distributions are denoted as fcs (qclus ) for the single clusters andfcm (qclus ) for the merging clusters. The qclus distribution for the scintillation clusters was
also estimated by the distribution of the hadron tracks reconstructed by the DC/PC
tracking and the RICH veto. The distribution was also normalized and is denoted as
fcsci (qclus ). These estimated distributions in a typical sector are plotted in Fig. 3.12.
The peak qclus of the merged clusters (red) is about twice as large as one of the single
clusters (blue). The distribution of the scintillation clusters (yellow) has small qclus
values compared with the distribution of the single clusters. The distribution functions
introduced above are summarized in Table 3.2 for the reference.

86

CHAPTER 3. ELECTRON ANALYSIS

Yield estimation of single clusters
The qclus distribution for the reconstructed electrons by applying eID-Cut is fitted witha superposition of the three normalized distribution
ns × fcs (qclus ) +nm × fcm (qclus ) +nsci × fcsci (qclus ),(3.8)

where ns , nm and nsci are the fitting parameters and represent the numbers of thereconstructed electrons associating to single clusters, merging clusters and scintillation
clusters after applying eID-Cut respectively. The variables are also summarized inTable 3.2. The fraction of non-photonic electrons and photonic electrons are diﬀerent
in diﬀerent pT region of the reconstructed electron sample. Therefore the fitting was
performed for each pT region and ns (pT ), nm (pT ) and nsci (pT ) for each pT region were
determined. In fitting the diﬀerent pT regions, pT independent normalized distributions,fcs (qclus ), fcm (qclus ) and fcsci (qclus ), were employed because the velocity of electrons in
pT region of interest is close enough to the speed of light in vacuum such that theyield of Čerenkov light from the electron is nearly pT independent. We also compared
the shapes of the distributions in diﬀerent pT regions to confirm that the eﬀect fromthe track curvature is small enough to be ignored even if at pT ∼ 0.5 GeV/c. On
the other hand, fcs (qclus ), fcm (qclus ) and fcsci (qclus ) for diﬀerent HBD sectors vary about
10% between HBD sectors with nominal gain, and fitting is performed for each sector
individually to account for this variation. The single low-gain sector exhibits a larger
variation and is fitted separately.
The qclus distribution for the reconstructed electrons with transverse momentum pTranging from 0.75 GeV/c to 1.00 GeV/c and the fitting result are shown in Fig. 3.13
for one HBD sector. The charge distribution of the reconstructed electrons can be
reproduced by the superposition of the 3 components excellently.
The total numbers of the reconstructed electrons after applying both of eID-Cut
and npe-Cut for the three cluster types, which are ∫represented as∫ ñs , ñm and ñsci ,
qmax
qmax
are calculated by applying the npe-Cut eﬃciencies of qmin
dqfcs (q), qmin
dqfcm (q) and∫ qmaxdqfcsci (q) to the fitting results, ns , nm and nsci , respectively. In the integrals, qmin
qmin
and qmax represent the HBD charge boundaries in the npe-Cut of 8 p.e. and 28 p.e.
(4 p.e. and 17 p.e. for the low-gain sector). The variables, ñ, are also summarized inTable 3.2 for the reference. Figure 3.14 shows the yield spectra from the calculation asfunctions of pT .
Yield estimation of separated photonic electrons
The estimated ñs is the sum of the numbers of non-photonic electrons and photonic
electrons which create the separated clusters on HBD. In the following description,
we denote the photonic electrons which create merging clusters as merging photonic
electrons (MPE) and those which create separated single clusters as separated photonic
electrons (SPE). In this section, the number of SPE is estimated to obtain the yield of
the non-photonic electrons.

3.2. HEAVY FLAVOR ELECTRON ANALYSIS WITH HBD

Table 3.2: Summary of variables used in hbdcharge analysis.

variable

description

fcs

Distribution for the single clusters
normalized with respect to the integral.

fcm

Distribution for the merging clusters
normalized with respect to the integral.

fcsci

Distribution for the scintillation clusters
normalized with respect to the integral.

ns

Number of single clusters
after applying eID-Cut.

nm

Number of merging clusters
after applying eID-Cut.

nsci

Number of scintillation clusters
after applying eID-Cut.

ñs

Number of single clusters
after applying eID-Cut and npe-Cut.

ñm

Number of merging clusters
after applying eID-Cut and npe-Cut.

ñsci

Number of scintillation clusters
after applying eID-Cut and npe-Cut.

87

CHAPTER 3. ELECTRON ANALYSISCounts88
4000

q dist. for reco. e
clus
single cluster : f sc
merged cluster : f mc
scinti. cluster : f sci
csuperposition

3500
3000
2500
2000
1500
1000
500
0
0

10

20

30

40

50

60

70

80

q

90

clus

100

[p.e.]

Yield

Figure 3.13: Charge distribution of HBD clusters associated with reconstructed electrons with a transverse momentum ranging from 0.75 GeV/c to 1.00 GeV/c (black),
and the charge distribution for each component, i.e., single clusters (ns fcs , blue), merging clusters (nm fcm , red), and scintillation clusters (nsci fcsci , green). The superpositionof these components is also shown (purple).

105

single cluster : ~nsmerged cluster : ~n104

scinti. cluster : ~nsci

m

103
102
10
1

1

2

3

4

5

p [GeV/c]T

Figure 3.14: Yield spectra of HBD clusters after applying eID-Cut and npe-Cut estimated from the HBD cluster charge fitting. The plot shows the spectrum for the single
clusters (ñs , blue), the spectrum for the merging clusters (ñm , red) and the spectrum
for the scintillation clusters (ñsci , green). The error bars represent fitting errors.

3.2. HEAVY FLAVOR ELECTRON ANALYSIS WITH HBD

89CountsFigure 3.15: A half of an annular region around the reconstructed electron track on the
HBD for the definition of hbdringcharge. The inner and outer radii of the annularregion are 7.0 cm and 8.0 cm respectively. The direction of the half region is determinedas the opposite side to the edge of the HBD sector to avoid ineﬃciency around the edge.

q

104

ring

dist. for reco. e

fitting result
spe

103

SPE component : f r

non-spenon-SPE component : f r

2

10

10

1
0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

q

ring

1.8

2

[p.e./cm2]

Figure 3.16: HBD charge distribution in the annular region for the reconstructed electrons with a transverse momentum ranging from 0.75 GeV/c to 1.00 GeV/c (black),
and the fitting result of the charge distribution for electrons with correlated charges(nspe frspe , green) and without correlated charges (nnon-spe frnon-spe , blue), and the superposition of the fitting results (red).

In the case where an electron track was reconstructed and it was SPE, the partnerelectron track of the pair will have created an additional HBD signal around the recon-

90

CHAPTER 3. ELECTRON ANALYSIS
Table 3.3: Summary of variables used in hbdringcharge analysis.

variable

description

frspe

Distribution for SPE
normalized with respect to the integral.

frnon-spe

Distribution for non-photonic electrons and MPE
normalized with respect to the integral.

nspe

Number of SPE
after applying the eID-Cut and npe-Cut.

nnon-spe

Number of electrons other than SPE
after applying the eID-Cut and npe-Cut.

structed track, as Fig. 3.11(c) shows. This property is utilized to estimate the numberof SPE. For this estimation, we defined a new value, qring , ashbdringcharge: qring
The total charge in the HBD pads centered on a half of an annular region with an
inner radius of 7.0 cm and an outer radius of 8.0 cm around the track projectionof HBD as shown in Fig. 3.15. Avoiding ineﬃciency around the edge of the HBD
sector, we employed a half of the annular region and the direction of the half region
is determined as the opposite side to the edge as the Fig. 3.15 shows. The qringvalue is normalized by the area of the half of the annular region in the definition.
The choice of 7.0 cm to 8.0 cm comes from three facts: (1) the distribution of distance
between separated clusters of SPE has a maximum around 7.0 cm, (2) few HBD clusters
have radii larger than 7.0 cm, and (3) larger area includes more scintillation backgroundand makes the ratio of the signal to the background worse.
Whereas the qring distributions for the non-photonic electrons and MPE consist of
signals only from scintillation light, the one for SPE consists of the correlated signals
around the tracks as well as signals from scintillation light. In the following description,
we denote the qring distribution for SPE as frspe (qring ) and one for the non-photonic
electrons and MPE as frnon-spe (qring ) as summarized in Table 3.3.
frnon-spe (qring ) can be estimated by hadron tracks and electron tracks with large qclus
values which consist almost entirely of MPE. Because hadrons as well as MPE do notcreate any correlated signals around their tracks, the qring distributions of the tracks
are created by only the scintillation light.
frspe (qring ) was estimated by using simulations. The dominant photonic electrons
come from the Dalitz decays of π 0 and η and γ from their decays which convert in ma-

3.2. HEAVY FLAVOR ELECTRON ANALYSIS WITH HBD

91

terials. We simulated the detector responses for the Dalitz decay and the γ conversion
events of the neutral mesons by a GEANT3 simulation [79] configured for the PHENIX
detector system. With the simulation, the qring distributions for the Dalitz decays andthe photon conversions from π 0 and η were determined. The qring distribution for thecorrelated signals was obtained as the summation of these distribution with known
branching ratios and a production ratio between π 0 and η which is determined by Tsallis distributions, which are the mT scaling spectra for hadron productions [78]. In thesimulation, the contribution from the scintillation light was not included. Considering
the contribution, frnon-spe (qring ), which is identical to the qring distribution from only the
scintillation light, was convoluted to the result to obtain frspe (qring ). Both distributions,frspe (qring ) and frnon-spe (qring ), were normalized with respect to the integrals.
The qring distribution for the reconstructed electrons by applying eID-Cut andnpe-Cut was fitted with the superposition of the qring distributions, frspe (qring ) andfrnon-spe (qring ), as
nspe × frspe (qring ) + nnon-spe × frnon-spe (qring ),(3.9)

where nspe and nnon-spe are fitting parameters and represent the numbers of SPE andother electrons in the qring distribution respectively as summarized in Table 3.3. As
well as the fitting for the qclus distribution, the fitting for the qring distribution was also
performed for each electron pT region and each HBD sector. Figure 3.16 shows a fitting
result in one HBD sector in the electron pT region from 0.75 GeV/c to 1.00 GeV/c.

3.2.6

Non-photonic background estimation

Yield estimation of heavy flavor electrons
Using the above fitting results of ñs and nspe , the yield of non-photonic electrons, N npe
was estimated as
N npe (pT ) = ñs (pT ) − nspe (pT ).

(3.10)The remaining background for the heavy flavor electrons in the non-photonic electron
sample comes from Ke3 decays and e+ e− decays of light vector mesons. Electrons fromthe Drell-Yan process are also background. However, the Drell-Yan contribution is
known to be less than 0.5% of total heavy flavor electrons in pT range of 0.0 < pT <5.0 GeV/c and can be ignored. Therefore, we can determine the yield of the heavy
flavor electrons from N npe by subtracting the components of the Ke3 electrons and theelectrons from light vector mesons, which are described in the following sections, asN HF e (pT ) = N npe (pT ) − N Ke3 (pT ) − N LVM (pT ),(3.11)

where N Ke3 (pT ) and N LVM (pT ) represent the electrons from the Ke3 decays and thelight vector meson decays, respectively. With this method, the heavy flavor electron
yield, N HF e (pT ), was obtained.

CHAPTER 3. ELECTRON ANALYSIS

Ke3 yield at ∫ dtL=1

92

K+
K
KL
KS

10-6
10-7
10-8
10-9

10-10
10-11
10-12
10-13
0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

p [GeV/c]T

Ke3 Cross Section [mb GeV -2c3]

Figure 3.17: Yield of electrons from Ke3 decays of ∫K + (blue), K − (red), KL (green) andKS (yellow) normalized as integrated luminosity, dtL, is equal to 1 mb after applying
eID-Cut and npe-Cut.

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

1

2

3

4

5

p [GeV/c]T

Figure 3.18: Estimated cross section of Ke3 in Run9 configuration. The red spectrum
represents after applying eID-Cut and npe-Cut and the blue spectrum represents after
applying same cuts without all HBD cuts.
Non-photonic background electrons from Ke3 decays
Electrons from Ke3 decays are main non-photonic background for the heavy flavor
electrons especially at low pT region. Then, the yield of the Ke3 electrons is to be
subtracted from the total non-photonic electron yield. In addition, because the Ke3

3.2. HEAVY FLAVOR ELECTRON ANALYSIS WITH HBD

93electrons have unknown background spin asymmetry of the K production, it introduces
systematic uncertainty in the spin asymmetry. Hence, it is essential to suppress the Ke3
electrons as much as possible for this spin asymmetry measurement.
Though the previous analysis already estimated the eﬀective cross section of theKe3 electron production, this cross section can not be used in this analysis because thenew electron selections based on HBD signals, qclus cut and cluster size cut, significantly
aﬀect the cross section. The HBD selections require only Ke3 decays occurring inside
the HBD gas volume. Therefore, the Ke3 electron cross section under the new selections
needs to be estimated.
We generated 8M + 8M K ± , 2M KL , and 2M KS events with an event generator.
For the K cross section in the generator, we employed a mT scaled cross section which
has been measured in PHENIX [78] and assumed that cross section spectra of all K
productions are identical. We set parameters in the mT scaled cross section as dσ/dy =4.23 mb and T = 125.4 MeV and n = 9.81, where the parameter notations are consistent
with the previous measurement [78]. Decays of K mesons were simulated by GEANT3.
Same electron selections as the single electron analysis, namely eID-Cut and npeCut, are employed. The estimated yields of these reconstructed∫ Ke3 are shown in
Fig. 3.17 as functions of peT . The yield spectra are normalized as dtL = 1 mb. The
blue spectrum corresponds to K + , the red to K − , the green to KL and the yellow to
KS .
The small statistics of KL decays come from long flight length of KL , cτ ∼ 15 m. The
statistical uncertainties on KL decays are too large to subtract Ke3 contribution fromthe non-photonic electron yield. Hence, Ke3 yields of KL decays was estimated from Ke3yield of K ± decays. The requirement of the E-p matching on eID-Cut constrains theflight length of the Ke3 decay to l < 60 cm, and the requirement of the association with
HBD signal constrains the flight length to l < 20 cm. The radius of the HBD volume(55 cm) is enough small compared with the decay lengths of K ± and KL , cτ =3.7m
and cτ =15m, to assume the flight length distribution of these Ke3 decays in the HBD
volume are flat. Ignoring the curvature of the tracks of K ± due to their charges, theyield of the Ke3 electrons from KL can be estimated with the yields from K ± . The
relation between these yields can be written as,
NLKe3 =

N+Ke3 + N−Ke3 BRL /cτL,2
BR± /cτ±(3.12)

Ke3
where N±,L
represent the yield of the Ke3 decays from K ± and KL and BR± represents
the branching ratio of K ± → e± νπ 0 and BRL represents the branching ratio of K L →e± νπ ∓ and cτ±,L represent the decay lengths of K ± and KL .
From the obtained yield of Ke3 electrons, the Ke3 electron cross section is determinedfrom the yield and the acceptance and reconstruction eﬃciency for the single electronsin |η| < 0.5, Aϵrec , which were estimated using a GEANT3 simulation. The detailed
Aϵrec estimation is described in a later section for cross section estimation (Sec. 4.2).
The spectrum of Aϵrec (pT , ∆η = 1) is shown in Fig. 4.6. Using Aϵrec , the Ke3 crosssection can be represented as,

E

1
1
N Ke3
d3 σ
|
=
,
Ke3
dp3
2πpT Aϵrec (∆η = 1) ∆pT(3.13)

94

CHAPTER 3. ELECTRON ANALYSIS

where N Ke3 ≡ N+Ke3 +N−Ke3 +NLKe3 +NSKe3 represents total yield of the Ke3 electrons inthe simulation and ∆pT represents the bin width of pT . The obtained Ke3 cross section
is shown in Fig. 3.18. The red spectrum is the cross section after applying eID-Cut andnpe-Cut and the blue spectrum is the cross section after applying the same cuts without
all HBD related cuts (HBD cluster association cuts, qclus cuts, and HBD cluster sizecut). Comparing the read and blue spectra, we found that the HBD reduced the Ke3
electrons by a factor of ∼0.4. It is a sizable contribution to measure the spin asymmetryof the heavy flavor electrons. For this single electron analysis, the red spectrum was
subtracted from the obtained non-photonic electron cross section spectrum.
Background electrons from light vector mesons
The large opening electron pairs from decays of light vector mesons V →e+ e− (where
V denotes a vector meson) are another non-photonic background of the heavy flavor
electrons. The fraction of this background in the heavy flavor electrons is small, lessthan 2%, over whole pT region in this measurement. The light vector mesons haveanother e+ e− decay mode of V →e+ e− + S (where S denotes a pseudoscalar meson),but the electron pair is photonic electron and has small opening angle as same as Dalitz
decay π 0 , η→e+ e− + γ and is dropped in the above HBD analysis.
The cross sections of the background electrons from the light vector mesons havebeen already estimated in the previously published result [28]. The cross sections wereused to estimate the background.

Max. Nh/Ne

Possible hadron background

0.025
0.02

0.015
0.01
0.005
0

-0.005
-0.01
0.6

0.8

1

1.2

1.4

1.6

1.8

2

2.2

p [GeV/c]T

Figure 3.19: Upper limit of hadron fraction in the inclusive electron events after applying eID-Cut and npe-Cut as a function of pT . I added sum of the center value and theRMS to the systematic uncertainty of the heavy flavor electron yield.
The RICH, EMCal, and HBD selections eﬀectively reject hadrons. However, small
fraction of hadrons can be included in the electron tracks. For the hadron estimation,

3.2. HEAVY FLAVOR ELECTRON ANALYSIS WITH HBD

95

we prepared two datasets, electron enhanced data (e-TrkSet) and hadron enhanced
data (h-TrkSet). The e-TrkSet consists of reconstructed tracks on the DC/PC with anassociated EMCal cluster which has electromagnetic shower profile, and the h-TrkSet
consists of reconstructed tracks with an associated EMCal cluster which does not havesuch profile. We applied electron selections (eID-Cut and npe-Cut without shower
profile selection: e-Cut) and hadron selections (h-Cut), which consists of track quality
selection and RICH veto. Then, four datasets were created by (e-TrkSet, h-TrkSet)×(eCut, h-Cut). The e-TrkSet after applying e-Cut (e-TrkSet&&e-Cut) is identical to thedataset after applying eID-Cut and npe-Cut. Since hadrons are dominant comparedwith electron tracks in charged tracks produced in pp collisions, the three datasets otherthan e-TrkSet&&e-Cut include a lot of hadron tracks.
To estimate the number of hadrons in e-TrkSet&&e-Cut, we used following relation,# of hadrons in (e-Cut&&e-TrkSet) =# of hadrons in (h-Cut&&e-TrkSet) ×)(# of hadrons in (e-Cut&&h-TrkSet).
# of hadrons in (h-Cut&&h-TrkSet)(3.14)The reconstructed tracks in h-TrkSet&&h-Cut consists almost of hadron tracks. Therefore, we can determine upper limit of the number of hadrons in e-TrkSet&&e-Cut as# of hadrons in (e-Cut&&e-TrkSet) <# of tracks in (h-Cut&&e-TrkSet) ×()# of tracks in (e-Cut&&h-TrkSet).
# of tracks in (h-Cut&&h-TrkSet)(3.15)The resulting upper limit of hadrons as a function of pT is shown in Fig. 3.19. We added
sum of the center value and the RMS to the systematic uncertainty of the heavy flavor
electron yield for pT < 1.25 GeV/c, and added same value as 1.00 < pT < 1.25 GeV/cregion for 1.25 < pT GeV/c.

Chapter 4
Results
4.1

Heavy Flavor Electron Yield

4.1.1

Systematic uncertainty on raw yield

The systematic uncertainties for the heavy flavor electron yield come from the fits for
qclus distribution and qring distribution, the estimations of Ke3 background, and themis-identified hadron background.
The most significant source in these contributions is the fitting uncertainty for the
qring distribution. We changed the inner and outer radii of the annular region into 6.0 cm
and 7.0 cm, and also into 8.0 cm and 9.0 cm, respectively, from the default radii of 7.0
cm and 8.0 cm. The uncertainty of the fitting was estimated from variations of obtained
nspe after these changes. The estimated uncertainties depend on the momentum anddecrease from about 16% of the heavy flavor electron yield at the low momentum region,
0.50 < pT < 1.00 GeV/c, to about 2% above 1.75 GeV/c.
The fitting uncertainty for the qclus distribution comes from the estimation of thebias in the charge distribution shape due to the electron pair reconstruction. The
systematic uncertainty from this eﬀect is estimated to be less than 2% by simulations.
In the low momentum region, 0.50 < pT < 1.00 GeV/c, uncertainties from theKe3 contribution and the hadron mis-reconstructions are also not negligible. The uncertainty from the Ke3 contribution comes almost from the uncertainty on the crosssection of K meson production used in the Ke3 estimation. The systematic error fromthe uncertainty is about 4% of the total heavy flavor electron yield in the low momentum region and decreases to less than 1% over 0.75 GeV/c. We also estimatedthe upper limits of the hadron contamination due to mis-reconstructions employing a
hadron-enhanced dataset. As a result, we determined the upper limits as 4% of thetotal heavy flavor electron yield in the low momentum region which decreases to lessthan 1% over 1.5 GeV/c. The upper limits are assigned as the systematic uncertainties
from hadron mis-reconstructions. Table 4.1 and Fig. 4.1 summarizes the systematic
uncertainties on the heavy flavor electron yield.
97

98

CHAPTER 4. RESULTS

Table 4.1: Relative systematic uncertainties given in percent on the heavy flavor electron
yield.

ΔNHFe/NHFe

sourceuncertainty
pT range (GeV/c)
hbdringcharge fitting
16%( 0.50 < pT < 0.75 )6% ∼ 4% ( 0.75 < pT < 1.75 )2%( 1.75 < pT )hbdcharge fitting
2%( 0.50 < pT < 0.75 )< 1%( 0.75 < pT )Ke34%
( 0.50 < pT < 0.75 )< 1%( 0.75 < pT )hadron mis-ID
4%( 0.50 < pT < 0.75 )< 1%( 0.75 < pT )0.18

Total Systematic Error Ratio

0.16

HBD Ring Charge Fitting

0.14

HBD Cluster Charge Fitting

0.12

Ke3 Yield
Hadron Mis-ID

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 4.1: Systematic uncertainties of the single electron yield for all sources. These
uncertainties are normalized by the single electron yield. The histograms represent
total uncertainty (yellow), uncertainty from the HBD ring charge fitting (red), fromthe HBD cluster charge fitting (blue), from Ke3 estimation (green) and from hadroncontamination estimation (purple).

4.1.2

Raw yield and signal purity of heavy flavor electron

From Eq. 3.10 and Eq. 3.11 and the discussion in the Sec. 4.1.1, the heavy flavor electron
yield spectrum with the systematic uncertainties is obtained. The spectrum is shownin Fig. 4.2. We also show the yield of inclusive reconstructed electrons after applying
the eID-Cut and npe-Cut and the estimated Ke3 contribution. The electrons from e+ e−
decays of the light vector mesons are not shown in Fig. 4.2, but they are less than 5%

Number of Electrons

4.1. HEAVY FLAVOR ELECTRON YIELD

99

total reconstructed electrons
105

heavy flavor electrons
Ke3 electrons

104

103

102
1

2

3

4

5

p [GeV/c]T

Figure 4.2: Heavy flavor electron yield spectrum. The black square points represent the
total number of the reconstructed electrons after applying the eID-Cut and npe-Cut.
The red circle points represent the estimated yield of the heavy flavor electrons. The
yellow bands represent the systematic uncertainties for the heavy flavor electron yield.
The green triangle points with dashed lines represent the estimated Ke3 contribution
with systematic uncertainties shown by light-blue bands.

compared with the heavy flavor electron yield in this pT range.
The ratio of the non-photonic electron yield to the photonic electron yield in thismeasurement,
N npe (pT )R(pT ) ≡ reco
Ne (pT ) − N npe (pT )(4.1)

where Nereco denotes the total number of reconstructed electrons after applying theeID-Cut and npe-Cut, is shown as the top plot in Fig. 4.3. In Eq. 4.1, we assumed
the fraction of mis-identified hadrons in the reconstructed electrons after the cuts is
negligible as shown in Fig. 3.7, and so the number of photonic electrons can be represented as Nereco (pT ) − N npe (pT ). The measured ratio in the previous measurement [28]is also shown in Fig. 4.3. The previous measurement employed another methods forthe background estimation, namely a cocktail method and a converter method. In the
cocktail method, a cocktail of electron spectra from various background sources wascalculated using a Monte Carlo event generator of hadron decays. On the other hand,in the converter method, an additional photon converter around the beam pipe wasintroduced for part of the experiment to estimate the photonic electron background.
The ratio is improved by a factor of about 2 or more in pT > 1.0 GeV/c comparedwith the previously measured result due to the rejection of photonic electrons with the
HBD.

100

CHAPTER 4. RESULTS

The signal purity is defined as the ratio of the yield of the heavy flavor electrons to
the reconstructed electrons after applying the eID-Cut and npe-Cut,
D(pT ) ≡N HF e (pT ).
Nereco (pT )(4.2)

The result is shown as the bottom plot in Fig. 4.3. We also show the result of thesignal purity in the previous measurement. Comparing with the previously measured
result, the signal purity is improved by a factor of about 1.5 around pT of 0.75 GeV/c
to 2.00 GeV/c.

4.2

Cross Section of Heavy Flavor Electron

The invariant cross section is calculated as,
E

d3 σ
1 1
1
N (∆pT , ∆y)=,3dp
2πpT L Aϵrec ϵtrig ∆pT ∆y(4.3)

where L denotes the integrated luminosity, A the acceptance, ϵrec the reconstruction
eﬃciency, ϵtrig the trigger eﬃciency, and N the estimated number of heavy flavor electrons.

4.2.1

Integrated luminosity

The luminosity, L, was calculated from the number of obtained MB triggers divided
by the cross section of MB trigger acceptance. For the latter cross section, a value of
23.0 mb with a systematic uncertainty of 9.6% was estimated from van-der-Merr scan
results [109] corrected by the relative changes in the BBC performance.

4.2.2

Detector acceptance and reconstruction eﬃciency

The combination of the acceptance and reconstruction eﬃciency, Aϵrec (pT ), was estimated with a GEANT3 simulation, which simulates the full detector responses. In
the acceptance calculation, it is important to check that the acceptance of each detector in the real data and the simulation agrees. Therefore, hit distributions of thereconstructed electrons in each detector with the MB dataset and those of the simulation dataset were compared. The acceptance and reconstruction eﬃciency can be
gen gendetermined to be Aϵrec = N rec (prec(pT ), where N rec (N gen ) is the number of
T )/N
gen
reconstructed (generated) electrons and prec
T (pT ) is reconstructed (generated) pT of
the electrons. The results are shown in Fig. 4.4 and Fig. 4.5. In Fig. 4.4, the black
histogram represents the data distribution and the red histogram represents the simulation distribution. Figure 4.5 shows the diﬀerence of the data and simulation (data− sim) divided by total hit counts in the simulation data (the integral of simulation
histogram). The total fractions of all positive diﬀerence (blue number) and all negative
diﬀerence (red number) are also shown separately. From these fractions, we assigned

Non-photonic/Photonic

4.2. CROSS SECTION OF HEAVY FLAVOR ELECTRON

101

10

1

2009 this result
2005 (cocktail method) PRL.97.252002

Signal Purity S/(S+BG)

2005 (converter method) PRL.97.252002

10-1

1

0.9
0.8
0.7
0.6
0.5
0.4
0.3
2009 this result

0.2

2005 (cocktail method) PRL.97.252002

0.1
0
0.5

2005 (converter method) PRL.97.252002

1

1.5

2

2.5

3

3.5

4

4.5

5

p [GeV/c]T

Figure 4.3: (top) Ratio between the yields of the non-photonic electrons and the photonic electrons in the reconstructed tracks. The red circles and the blue squares represent this analysis result and the previous result respectively. The error bars and bands
represent the statistic and the systematic errors. (bottom) Signal purity which is aratio of the yield of the heavy flavor electrons to the total reconstructed electrons.

a systematic uncertainty from the acceptance estimation is assigned to be 8% as aconservative value.
From the GEANT3 simulation, it was found that Aϵrec (pT ) has very small pT dependence at pT > 0.5 GeV/c and the value is 4.7%. The estimated Aϵrec (pT ) is shown

102

CHAPTER 4. RESULTS

220
160

200

140

180

120

160
140

100
120
80

100

60

80
60

40
40
20
0
-100

20
-80

-60

-40

-20

0

20

40

60

80

100

0
-100

-80

-60

-40

-20

0

20

40

60

DC z [cm]80

100

DC z [cm](a) z distribution of hits in east DC(b) z distribution of hits in west DC

600

500

500
400
400
300
300
200
200
100

100
0
-1

-0.5

0

0.5

1

1.5

2

2.5

3

3.5

4

DC φ [rad](c) ϕ distribution of hits in north-side DC

0
-1

-0.5

0

0.5

1

1.5

2

2.5

3

3.5

4

DC φ [rad](d) ϕ distribution of hits in south-side DC

Figure 4.4: (a - d) Comparison of DC hit distributions in data and GEANT3 simulation. Figures (a) and (b) correspond to z distributions of hits in East and West DC,
respectively, and figures (c) and (d) correspond to ϕ distributions of hits in North-side
and South-side DC, respectively.

in Fig. 4.6.
The applied electron cuts other than the cuts related to the HBD are identicalto the previous measurement. Therefore, we employed same criteria to assign thesystematic uncertainties from the eﬃciency of the electron identification as the previous
measurement. The assigned uncertainties on the DC, RICH and EMCal cut are thefollowing:
RICH eID cut
2%prob cut
0.5%
e/p cut
2%.
We also estimated systematic uncertainty from HBD cuts. There are two HBD cutsfor the electron identification, namely qclus cut and cluster size cut. We compared the4.2. CROSS SECTION OF HEAVY FLAVOR ELECTRON
pos:3.769% neg:5.443%

103

pos:4.722% neg:3.368%
0.003

0.002
0.002
0.001
0.001
0
0
-0.001-0.001-0.002
-100

-80

-60

-40

-20

0

20

40

60

80

100

zed [cm]

-100

-80

-60

-40

-20

0

20

40

60

80

100

zed [cm](a) z distribution of hits in east DC(b) z distribution of hits in west DC

pos:6.077% neg:4.367%

pos:5.213% neg:6.758%
0.004

0.004
0.002
0.002
0
0

-0.002-0.002-0.004-1-0.004-0.006
-0.5

0

0.5

1

1.5

2

2.5

3

3.5

4

DC φ [rad](c) ϕ distribution of hits in north-side DC

-1

-0.5

0

0.5

1

1.5

2

2.5

3

3.5

4

DC φ [rad](d) ϕ distribution of hits in south-side DC

Figure 4.5: (a - d) Diﬀerence of the data and simulation (data − sim) divided bytotal hit counts in the simulation data. Figures (a - d) correspond to Fig. 4.4(a - d),respectively. The positive (negative) region is filled with blue (red). The two numbers
at the top of the plots represent fractions of the integrated positive and negative areawith respect to the integrated area of the original histograms in Fig. 4.4.

qclus distribution and the cluster size distribution for non-photonic electrons between
data and simulation. Figure 4.7 shows the comparison of the data (black line) and thesimulation (red circles). The distributions of the data are estimated by using single
cluster hits in photonic electron pair events on the HBD.
In the cluster size distributions, this method can introduce a bias on the distributions, which enhances events with larger cluster size, because the single cluster can
include additional hit pads created by Čerenkov light or scintillation light from thecounterpart electron of the electron pair, which are accidentally not clusterized into theanother HBD cluster associated with the counterpart track. Therefore, we comparedalso cluster size distributions estimated from electron pairs associated to J/ψ→e+ + e−
decays. For this estimation, the electron pairs with pair mass 2.8 < Me+ e− < 3.3 GeV/c2
were selected from data. The electron pairs from J/ψ have large opening angle, andthen the HBD hits of a track do not aﬀect to the ones of the another track. Hence, thecluster size distribution from the J/ψ decays has no bias and suitable for the comparison.

CHAPTER 4. RESULTS
Acc.×Reco. Efficiency

104

0.05

0.04

0.03

e+
e(e++e-)/2

0.02

0.01

0

1

2

3

4

5

p [GeV/c]T

Figure 4.6: Aϵreco spectra as functions of electron pT after applying eID-cut and npeCut. The blue and red points are positron and electron, respectively. The black points
are the averaged Aϵreco for positrons and electrons.

The distributions of data and simulation are enough consistent each other so that
we can conservatively assign the systematic uncertainty from eﬃciency of the HBD hit
selection to be half value of percentage of rejected electron tracks by these selections.
Figure 4.8 shows HBD cluster charge distribution of the simulation data without the
charge cuts, and Fig. 4.9 shows HBD cluster size distribution of the simulation data
without the cluster size cut. The fractions of rejected events by the cuts are also shownin the plots. From the diﬀerences of comparisons and the fractions of the cuts, weassigned systematic uncertainties as:
HBD cluster charge cut
3%
HBD cluster size cut
3.5%.
As the result, the acceptance and reconstruction eﬃciency including systematic
uncertainties is obtained as Aϵrec (pT ) = 4.7% × (1 ± 8 × 10−2 (acc.) ± 6 × 10−2 (reco.)).

4.2. CROSS SECTION OF HEAVY FLAVOR ELECTRON

105

0.05
0.05
0.04
0.04
0.03

0.03

0.02

0.02

0.01

0.01

0
0

10

20

30

40

50

60

70

80

90

100

0
0

10

20

30

40

50

60

HBD cluster charge [p.e.](a) qclus distribution (Sect9 South)

80

90

100

(b) qclus distribution (Sect9 North)

0.5

0.5

Dalitz Cluster Size:3.190

Dalitz Cluster Size:3.255
0.4

0.4

J/psi Cluster Size:2.901

J/psi Cluster Size:3.013
0.3

0.3

Sim Cluster Size:2.868

Sim Cluster Size:2.822

0.2

0.2

0.1

0.1

0
0

70

HBD cluster charge [p.e.]

1

2

3

4

5

6

7

8

9

10

HBD cluster size

0
0

1

2

3

4

5

6

7

8

9

10

HBD cluster size(c) HBD cluster size distribution (Sect9 South) (d) HBD cluster size distribution (Sect9 North)Figure 4.7: Comparisons between (black lines) data and (red circles) GEANT3 simulation for (a, b) qclus distribution and (c, d) HBD cluster size distribution for non-photonic
electrons at a typical sector, respectively. The qclus distribution and the cluster size distribution from the data are estimated by using single cluster hits in photonic electron
pair events on the HBD. The green lines in the plots (c, d) represent the cluster size
distribution estimated by electron pairs from J/ψ (2.9 < Me+ e− < 3.3 GeV/c2 ) in data.

106

CHAPTER 4. RESULTS

0.05

Fraction of Cut Off = 2.58%

0.04

0.04

0.03

0.03

0.02

0.02

0.01

0.01

0
0

Fraction of Cut Off = 2.99%

0.05

10

20

30

40

50

60

70

80

90

100

HBD cluster charge [p.e.]

0
0

10

20

(a) Sect9 South

30

40

50

60

70

80

90

100

HBD cluster charge [p.e.](b) Sect9 North

Figure 4.8: qclus distributions from GEANT3 simulation without qclus cuts at a typical
HBD sector. The numbers shown at corners of the plots are the fractions of cut-oﬀ
electrons by the qclus cut of qclus > 8.0 p.e..

Fraction of Cut Off = 7.20%

0.45
0.4

0.4

0.35

0.35

0.3

0.3

0.25

0.25

0.2

0.2

0.15

0.15

0.1

0.1

0.05

0.05

0
0

1

2

3

4

5

6

7

8

9

Fraction of Cut Off = 7.29%

0.45

10

0
0

1

2

3

4

5

6

7

HBD cluster size(a) Sect9 South

8

9

10

HBD cluster size(b) Sect9 North

Figure 4.9: HBD cluster size distributions in GEANT3 simulation without HBD cluster
size cuts at a typical HBD sector. The numbers shown at corners of the plots are thefractions of cut-oﬀ electrons by the cluster size cut of size ≥ 2.

4.2. CROSS SECTION OF HEAVY FLAVOR ELECTRON

4.2.3

107

Trigger performance

Trigger Efficiency

The eﬃciency of the MB trigger for the hard scattering processes, including heavy flavor−2electron production, is ϵMB
trig = 79.5% × (1 ± 2.5 × 10 ). The eﬃciency of the ERT-E
e|MB
trigger for electrons under requiring the MB trigger, ϵtrig (pT ) ≡ ϵtrig (pT )/ϵMB
trig , can becalculated by the ratio of the number of the reconstructed electrons in the MB trigger
dataset with coincidence of the ERT-E trigger to the one without the coincidence. The
e|MB
eﬃciency ϵtrig is shown in Fig. 4.10 as a function of pT . Whereas we used the calculated
eﬃciency values for the momentum region of pT < 1.25 GeV/c, we assumed a saturated
eﬃciency for pT > 1.25 GeV/c and estimated the value with a fitting as shown in
Fig. 4.10. The fitting result is ϵplateau = 56.5% × (1 ± 3.6 × 10−2 ). The systematice|MB
uncertainty of ϵtrig for pT < 1.25 GeV/c was assigned to be ∆ϵ/ϵ ∼ 4%, which wasestimated from the statistical uncertainty in the eﬃciency calculation, and 3.6% for
pT > 1.25 GeV/c from the fitting uncertainty. The total trigger eﬃciency ϵtrig (pT ) cane|MBbe calculated with the above two eﬃciencies as ϵtrig (pT ) = ϵMBtrig × ϵtrig (pT ).
0.8
0.7
0.6
0.5
0.4
0.3
0.2

χ2/ndf = 7.04/7

0.1

Plateau Eff. = 56.5 % × (1 ± 3.6 ×10-2)

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

p [GeV/c]T

Figure 4.10: Eﬃciency of the electron trigger for reconstructed electrons under thecondition that the MB trigger was issued. The red line represents the fitting resultwith the constant function and the green band represents the fitting uncertainty.

4.2.4

Cross section of heavy flavor electron

Table 4.3 summarizes the systematic uncertainties on the cross section of the heavy
flavor production other than the uncertainties from the heavy flavor electron yieldalready shown in Table 4.1. All systematic uncertainties listed in Table 4.3 are globallye|MBcorrelated over whole pT region (pT > 1.25 GeV/c for the uncertainties on ϵtrig ).

108

CHAPTER 4. RESULTS

Table 4.2: Data table for the cross section result corresponding to Fig. 4.11.
3
pT [GeV/c] E ddpσ3 [mb×GeV−2 c3 ] stat. error[mb×GeV−2 c3 ] syst. error[mb×GeV−2 c3 ]0.612
2.12×10−3
0.04×10−3
0.47×10−3
0.864
7.93×10−4
0.09×10−4
1.11×10−4
−4
−4
1.115
2.78×100.03×10
0.37×10−4
1.366
1.09×10−4
0.02×10−4
0.13×10−4
1.617
4.77×10−5
0.08×10−5
0.58×10−5
−5
−5
1.867
2.34×10
0.05×10
0.27×10−5
2.118
1.15×10−5
0.04×10−5
0.13×10−5
−6
−6
2.369
6.05×10
0.20×100.68×10−6
2.619
3.28×10−6
0.19×10−6
0.37×10−6
2.869
1.82×10−6
0.11×10−6
0.20×10−6
−6
−6
3.120
1.08×100.07×10
0.12×10−6
3.370
6.20×10−7
0.41×10−7
0.69×10−7
−7
−7
3.620
4.07×10
0.26×100.45×10−7
3.870
2.42×10−7
0.19×10−7
0.27×10−7
4.121
1.59×10−7
0.15×10−7
0.18×10−7
−7
−7
4.371
1.07×10
0.11×10
0.12×10−7
4.621
8.02×10−8
1.11×10−8
0.89×10−8
4.871
5.38×10−8
0.71×10−8
0.60×10−8
The measured cross section of heavy flavor electrons is shown in Fig. 4.11 andtabulated in Table 4.2. A correction for bin width [110] is applied to the pT value ofeach point. The figure also shows the previously published result [28]. The new result
excellently agrees with the previous result within the uncertainties.
Note that in this paper we employed a new analysis method with the HBD whereas
the previous measurement employed diﬀerent methods, the cocktail method and theconverter method. The consistency of the measured cross sections between thesemeasurements means that we succeeded to confirm the reliability of the new analysis method. In addition, we also confirmed that the additional photonic backgrounds
generated in the HBD material are enough removed to measure the heavy flavor electrons.
The electron cross section from J/ψ → e+ + e− decays estimated by the cocktail
method [29] and a fixed order next-to-leading log (FONLL) pQCD calculation of theheavy flavor contributions to the electron spectrum [111] are also shown in Fig. 4.11.
The J/ψ contribution to the heavy flavor electrons is less than 2% in pT < 1.25 GeV/cand increase to ∼20% until pT = 5.0 GeV/c. The FONLL pQCD calculation showsthat the heavy flavor electrons in the low momentum region are dominated by charm
quark decays, and the contribution from bottom quarks in pT < 1.25 GeV/c is less than5%.

4.2. CROSS SECTION OF HEAVY FLAVOR ELECTRON

109

Table 4.3: Relative systematic uncertainties on the cross section due to uncertainties
in the total sampled luminosity, trigger eﬃciencies, and detector acceptance. Thesesystematic uncertainties are globally correlated in all pT regions except pT < 1.25 GeV/ce|MB
for the uncertainties on ϵtrig .
sourceMB trig. cross sect.
acceptance A
reco. eﬃciency ϵrec
MB trig. eﬃciency ϵMB
trig
e|MB
e trig. eﬃciency ϵtrig

uncertainty pT range (GeV/c)
9.6%
8%
6%
2.5%
∼ 4%
0.50 < pT < 1.25
3.6%
1.25 < pT

CHAPTER 4. RESULTS

Ed3σ/dp3 [mb×GeV-2c3]

110

pp→(e +e )/2+X+

10-2

s=200GeV (|ηe|<0.35)-

2009 this result
2005 (PRL.97.252002)
J/ ψ →e++e- (cocktail method)

10-3

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

10-4
10-5

10-6

d σ/FONLL - 1

10-7
2.5
2
1.5
1
0.5
0
-0.5
-1

1

2

3

4

5
p [GeV/c]T

Figure 4.11: (top) Invariant diﬀerential cross sections of electrons from heavy-flavor
decays. The red and the blue circles represent this analysis result and the previous
result, respectively. The error bars and bands represent the statistical and systematic
errors. The scaling uncertainty from the van-der-Merr scan is not included in thesystematic error bands because the same uncertainty must be considered for both theresults of 2009 and 2005. The purple dashed dotted line is electron cross section fromJ/ψ → e+ + e− decays estimated from the cocktail method [29]. The solid and dashedcurves are the FONLL calculations. (bottom) Diﬀerence of the ratio of the data andthe FONLL calculation from 1. The upper and lower curve shows the theoretical upperand lower limit of the FONLL calculation.

4.3. SPIN ASYMMETRY OF HEAVY FLAVOR ELECTRON

4.3

Spin Asymmetry of Heavy Flavor Electron

4.3.1

Beam polarization111

For determination of spin asymmetry, beam polarization during the run is to be known.
Fill-by-fill beam polarization is provided by the H-jet polarimeter and pC polarimeter√
located at IP12 as described in Appendix B. The measured beam polarization at s =200 GeV collisions is shown in Fig. 4.12 [112]. During this experiment, the average
beam polarizations for the Blue and Yellow beams are PB ∼ 56% and PY ∼ 57%,
respectively.
Run9 sqrt(s)=200 GeV: Polarization in collision vs Fill #0.8Blue beam

0.7
0.6
0.5
0.4
0.3
0.2
0.1

Global normalization uncertainty 4.7%010600

10700

10800

10900

11000

Run9 sqrt(s)=200 GeV: Polarization in collision vs Fill #0.8

Yellow beam

0.7
0.6
0.5
0.4
0.3
0.2
0.1
0

Global normalization uncertainty 4.7%

10600

10700

10800

10900

11000

Figure√4.12: Fill-by-fill beam polarization for the Blue and Yellow beams during the2009 s = 200 GeV run. The top (bottom) plot corresponds to the Blue (Yellow)
beam. The horizontal axis represents the fill number.
The uncertainty from the beam polarization measurement propagates into the uncertainty of resulting spin asymmetry. In this double-spin asymmetry measurement, theuncertainty on the product of the two polarization, PB PY , propagates. This uncertainty

112

CHAPTER 4. RESULTS

comes from uncertainty on the normalization between the H-jet and pC measurement,time dependent inconsistency between the polarization results of the two pC polarimeters, and uncertainty on spatial profile of beam polarization. The estimated uncertainty
is∆(PB PY )= 8.8%.
(4.4)
PB PY

4.3.2

Spin asymmetry calculation

Since parity is conserved in QCD processes, thereby finite single spin asymmetries of
the QCD processes are disallowed. Therefore, using Eq. 1.39 we express the expected
electron yields for each beam-helicity combination as
exp
N++(N0 , ALL ) = N0 (1 + |PB PY |ALL )exp
N−− (N0 , ALL ) = N0 (1 + |PB PY |ALL )/r−−exp
N+−(N0 , ALL ) = N0 (1 − |PB PY |ALL )/r+−exp
N−+(N0 , ALL ) = N0 (1 − |PB PY |ALL )/r−+ ,

(4.5)

exp
where N±±(N0 , ALL ) denote the expected yields for collisions between the blue beamhelicity (±) and the yellow beam-helicity (±) and N0 is the expected yield in collisionsof unpolarized beams under the same integrated luminosity as the ++ beam-helicity
exp(N0 , ALL ) are used for fitting functions to estimate ALL as describedcombination. N±±
below. PB and PY represent the polarization of the beams. The relative luminosity isdefined as the ratio of the luminosity in the beam-helicity combinations,( r++ ≡ 1 )
L++
r−− ≡
L−−
L++
r+− ≡
L+−
L++
r−+ ≡
,
L−+

(4.6)
(4.7)
(4.8)
(4.9)

where L±± represent the integrated luminosity in the beam-helicity combinations shownby the subscript. The relative luminosity is determined by the ratios of MB trigger
counts in the four beam-helicity combinations. The number of the MB trigger counts
is enough larger than the number of the detected electrons such that the statisticaluncertainties of the r±± are negligibly small compared with the statistical uncertainties
of the electron yield.
As discussed in Sec. 1.3.3, the following eﬀect can create a bunch-dependent eﬃciency which produces an eﬀective spin asymmetry:bunch-by-bunch diﬀerence of length of beam bunch
The length of beam bunch along the beam axis varies bunch-by-bunch, and theGaussian width of the track vertex distribution varies from ∼ 56 cm to ∼ 57 cm
in the present analysis. In addition, reconstruction eﬃciencies of tracks from
diﬀerent zvtx are also diﬀerent. Therefore, these eﬀects create a bunch-dependenceof the reconstruction eﬃciency.

4.3. SPIN ASYMMETRY OF HEAVY FLAVOR ELECTRON

113

readout electronics specificDue to a readout feature of the EMCal front-end electronics, the thresholds for
EMCal signals in even-number and odd-number beam crossings are not exactly
same. This feature can create bias on the yield of electrons in a specific beamhelicity combination, which is critical for the spin asymmetry measurement.
The first eﬀect is estimated by using a toy Monte Carlo calculation. The zvtx dependenceof the reconstruction eﬃciency obtained with a GEANT3 simulation was used for the
calculation. As the result, the eﬀect was found to be Aeﬀective∼ 5×10−5 and negligibly
LL
small in this analysis. To take account of the second eﬀect, the even crossing data andthe odd crossing data were analyzed separately. The two separated results were mergedwith weighted mean as explained in the following description.
The double-spin asymmetry for inclusive electrons after applying eID-Cut and npeCut, which include not only the heavy flavor electrons (S) but also the backgroundelectrons (BG), is determined by simultaneously fitting the yields of electrons in eachexpof the four beam-helicity combinations with the expected values N±±(N0 , ALL ) from
Eq. 4.5, where ALL and N0 are free parameters. To perform the fit, a log likelihoodexp
method assuming Poisson distributions with expected values of N±±(N0 , ALL ) (Eq. 4.5)
was employed.
The fit was performed for electron yields in each fill to obtain the fill-by-fill doublespin asymmetry. Due to the EMCal readout feature, the fitting was performed for theeven crossing data and odd crossing data separately, and the results were merged toobtain the final ALL value. Examples of the raw yields and the fitting result at a specific fill are shown in Fig. 4.13(a). The horizontal axis of the plots corresponds to fourbeam-helicity combinations, ++, −−, +−, and −+ from left to right, and the verticalaxis corresponds to measured N±± r±± at each beam-helicity combination. The plotcorresponds to electrons with transverse momentum ranging 0.75 < pT < 1.00 GeV/c.
exp(N0 , ALL )r±± according to
The red dashed lines represent the fitting results of N±±
Eq. 4.5, and therefore the results for ++ and −−, and for +− and −+ are same. To
confirm the fitting quality for all √fills, reduced-χ2 (χ2 /N DF ) was calculated by setting statistical uncertainty to be N±± r±± . The fill-by-fill reduced-χ2 for electrons
with 0.75 < pT < 1.00 GeV/c is plotted in Fig. 4.13(b). The vertical axis is the
reduced-χ2 and the horizontal axis is fill number. In this figure, the red line represents
χ2 /N DF = 1. It is confirmed there is no significant fill dependence in reduced-χ2 .
To check the reduced-χ2 distribution, the points were projected the vertical axis. Figure 4.13(c) shows the reduced-χ2 distributions, which are obtained by the projectionsof Fig. 4.13(b). The distributions should be consistent with the χ2 distribution of
N DF = 2 (∝ exp(−x/2)), because the number of the input parameters is four (N±± )and the number of the fitting parameters is two (N0 and ALL ). The exp(−x/2) distributions fitted to the obtained reduced-χ2 distributions is also plotted. The resulting ALLof electrons with 0.75 < pT < 1.00 GeV/c is plotted in Fig. 4.13(d). It is confirmed that
all asymmetries in diﬀerent fills are consistent with each other within their statisticaluncertainties. The double-spin asymmetries from the whole even or odd crossing datawith fitting to the fill-by-fill asymmetry result were obtained as shown in Fig. 4.13(d)as a red line. The resulting double-spin asymmetries of even and odd crossing data
are listed in Table 4.4. The asymmetries of even and odd crossing data are consistent

114

CHAPTER 4. RESULTS

Reduced Chi-Square

Num. of Electrons×Relative Luminosity

A LL :0.0495± 0.0423 PB :0.602 PY :0.587
1040
1020
1000

7
6
5
4

980
3
960

2

940

1

++

0

+

+

10650 10700 10750 10800 10850 10900 10950 11000

beam-helicity combination [0:++ 1:-- 2:+- 3:-+]fill number(b) Reduced-χ2 vs. fill numberχ2 / ndf
p0
7

27.6 / 29
4.634 ± 0.546

ALL(a) Electron yield at a specific fill

χ 2 / ndf
ALL

105.6 / 80
-0.00528 ± 0.005562

0.4

6

0.2
5

0

4
3

-0.2

2

-0.4
1
0
0

-0.6
1

2

3

4

5

6

(c) Reduced-χ

7
8
9
10
Reduced Chi-Square210650 10700 10750 10800 10850 10900 10950 11000

fill number(d) Fill-by-fill ALL result

Figure 4.13: (a) An example of the electron yields for even crossing data at a specific fill.
The four horizontal bins correspond to four beam-helicity combinations of ++, −−, +−,and −+ from left to right, respectively. The vertical axis is the electron yield normalized
by the relative luminosity in each beam-helicity combination, N±± r±± . The red lines
represent the ALL fitting results according to Eq. 4.5. (b) Reduced-χ2 for the ALL fitting
as a function of fill number (even crossing). The red line represents χ2 /N DF = 1. (c)Projection of the reduced-χ2 values for the ALL fitting. (even crossing) The red curve inthe plot shows the χ2 /N DF distribution where N DF = 2 (∝ exp(−x/2)). The dashedvertical line represents the position of χ2 /N DF = 1. (d) Fill-by-fill asymmetry resultsfor even crossing data (points) and pol0 fitting (red line) for the fill-by-fill asymmetry
results. The horizontal dashed line represents AS+BG= 0. All these four plots are
LL
results of electrons with transverse momentum ranging 0.75 < pT < 1.00 GeV/c.
each other. The final double-spin asymmetry for inclusive electrons, AS+BG(pT ), wasLL
calculated as the weighted mean of these even-crossing and odd-crossing asymmetries.

4.3.3

Background spin asymmetry

The double-spin asymmetry in the heavy flavor electron production, AHFe
LL , was determined from1
1 − D(pT ) BG
AHFe
AS+BG
(pT ) −ALL (pT )(4.10)LL (pT ) =LL
D(pT )
D(pT )

4.3. SPIN ASYMMETRY OF HEAVY FLAVOR ELECTRON

115

Table 4.4: Table of the obtained double-spin asymmetries of inclusive electrons fromeven and odd crossing data.
S+BG
even AS+BG(×10−2 ) odd ALL(×10−2 )LL
1.0 ± 0.6
−0.1 ± 0.7−0.5 ± 0.6−0.3 ± 0.6
1.0 ± 0.8
−0.5 ± 0.8
0.5 ± 1.1
1.5 ± 1.1
2.1 ± 1.6
−3.0 ± 1.7−1.6 ± 2.3−0.0 ± 2.3
2.1 ± 3.1
4.1 ± 3.2
−1.9 ± 4.3−6.2 ± 4.3−6.6 ± 5.5−3.3 ± 5.7−1.6 ± 7.1
11.4 ± 7.3

0

AπLL

pT region (GeV/c)
0.50 < pT < 0.75
0.75 < pT < 1.00
1.00 < pT < 1.25
1.25 < pT < 1.50
1.50 < pT < 1.75
1.75 < pT < 2.00
2.00 < pT < 2.25
2.25 < pT < 2.50
2.50 < pT < 2.75
2.75 < pT < 3.00

0.02
0

-0.02
-0.04
-0.06
-0.08
-0.1
-0.12
2

4

6

8

10 π0

p [GeV/c]T√
Figure 4.14: Measured π 0 double-spin asymmetry with same 2009 experiment of s =0
200 GeV pp collision. The red curve is a cubic function fitted to AπLL . The two blue0
curves are also cubic functions fitted to shifted AπLL by the uncertainties to plus andminus directions.
where ABG
LL represents the spin asymmetries for the background electron production, andD represents the signal purity defined in Eq. 4.2 and shown in Fig. 4.3. As previously
discussed, most of the background electrons come from Dalitz decays of the π 0 and η, orfrom conversions of photons from decays of those hadrons. The production mechanism
for the π 0 and η is expected to be very similar up to ∼ 10 GeV/c [11, 14]. Therefore, we
assume identical spectra for double-spin asymmetries of π 0 production and η production,0and estimated ABG
LL from only the π double-spin asymmetry measured in this PHENIX

116

CHAPTER 4. RESULTS

0

AπLL→e

0.006
0.005
0.004
0.003
0.002
0.001
0

-0.001-0.002
0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

pe [GeV/c]
T

Figure 4.15: Convolution performed with the π 0 spectrum and the AπLL (pπT ) for eachpeT region. The points are the result of the convolution with the red curve in Fig. 4.14
and the blue and the red lines in this plot are the results of the convolutions withthe upper and the lower blue curves in Fig. 4.14. The uncertainties of the backgroundasymmetry are defined with the diﬀerences between the blue points and the black points,ee∆√h (pT ), and the diﬀerences between the red points and the black points, ∆l (pT ), as|∆h (peT )∆l (peT )|.
run [73].
√
00
For AπLL (pπT ) input, a result in 2009 experiment of s = 200 GeV pp collision wasused, which is shown in Fig. 4.14 [73]. To obtain the spectrum, the data are fittedwith a cubic function. The fitting result is shown with the red curve in Fig. 4.14. Twocurves are upper and lower uncertainty of the spectrum shape estimated by the fitting0with shifted AπLL by the uncertainties to upper and lower directions. Expected ABG
LL
0from the AπLL spectrum is simulated using a toy Monte Carlo reproducing π 0 Dalitz
decay. The obtained asymmetry ABG
LL is shown in Fig. 4.15. The points in Fig. 4.15
are the result of the convolution with the red curve in Fig. 4.14 and the blue and thered points in Fig. 4.15 are the results of the convolutions with the upper and the lower−2−2
blue curves in Fig. 4.14. The resulting ABG< ABG
in
LL is −0.1 × 10
LL < 0.1 × 10
−2
BG
−2
0.5 < pT < 2.5 GeV/c and 0.1 × 10 < ALL < 0.2 × 10 in 2.5 < pT < 3.0 GeV/c,with uncertainties less than 0.2 × 10−2 . The background asymmetry is subtracted fromthe inclusive electron AS+BG
according to Eq. 4.10.
LL
Transverse polarization eﬀect
A transverse double-spin asymmetry AT T , which is defined by the same formula as
Eq. 1.2 for the transverse polarization, can contribute to ALL through the residualtransverse components of the beam polarization. The product of the transverse components of the beam polarization was measured to be ∼ 10−2 in this run. For π 0

4.3. SPIN ASYMMETRY OF HEAVY FLAVOR ELECTRON

117

production, the AT T is expected to be ∼ 10−4 based on an NLO QCD calculation [113].
If we assume the transverse asymmetries of π 0 and heavy flavor electrons are comparable, we arrive at the value of ALL ∼ 10−6 . This value is quite negligible compared withthe precision of the AS+BG
measurement of ∼ 10−3 .
LL

Helicity pattern dependence

HFe
ALL

4.3.4

Spin Pattern 1
Spin Pattern 2Spin Pattern 3
Spin Pattern 4

0.3
0.2
0.1
0
-0.1
-0.2
-0.3
0.5

1

1.5

2

2.5

3

p [GeV/c]T

Figure 4.16: Comparison of the resulting AHFe
LL from fills with each spin patterns. The
black, red, blue, and green points represent spin patterns of P 1, P 2, P 3, and P 4,
respectively.
As described in Sec. 2.1, there are four beam-helicity patterns in order to confirm no
beam-helicity-pattern dependence is present. The beam-helicity-pattern dependence isproduced by any correlations between raw yields in diﬀerent bunches. For the confirmation that there is no such correlations, we need to check consistency between AHFe
LL ’s
obtained from diﬀerent beam-helicity pattern fills to confirm that there is not any eﬀects
from previous bunch, which make the measured asymmetry values incorrect. The AHFe
LL
for the four patterns are shown in Fig. 4.16. The deviations between these asymmetries
are within their statistical uncertainties and we concluded there is no such eﬀect.

4.3.5

Spin asymmetry of heavy flavor electron

Systematic uncertainties on AHFe
LL are separated into scaling uncertainties and oﬀset uncertainties. The scaling uncertainties come from uncertainty in the beam polarization,
PB and PY , and the signal purity, D. The uncertainty from the beam polarization isestimated as ∆(PB PY )/PB PY = 8.8% which is globally correlated over the whole pTrange. The oﬀset uncertainties come from uncertainties in the relative luminosity, r,
and the background asymmetry, ABG
LL . The uncertainty in the relative luminosity is

CHAPTER 4. RESULTS
HFe
ALL

118
0.15

AHFe
LL p+p s=200GeV (|ηe|<0.35)
0.1
0.05
0

-0.05

ALL
syst. error from dilution factor-0.1

syst. error from rel. lum. and ABG
LL

HFe(not including 8.8% pol. scale error)
-0.15

0.5

1

1.5

2

2.5

3

p [GeV/c]T

Figure 4.17: Double-spin asymmetry of the heavy flavor electron production. The red
error bars represent scaling systematic uncertainties from the dilution factor and theblue error bands represent oﬀset systematic uncertainties from relative luminosity andthe background spin asymmetry.

globally correlated, and is estimated by comparing the relative luminosity measured by
the MB and ZDC triggers. This uncertainty is determined to be ∆r = 1.4 × 10−3 . The
systematic uncertainties are summarized in Table 4.5.
The result of the double-spin asymmetry of heavy flavor electrons is shown in
Fig. 4.17 and tabulated in Table 4.6. We show systematic uncertainties for scaling
and oﬀset separately in the figure. The measured asymmetry is consistent with zero.

4.3. SPIN ASYMMETRY OF HEAVY FLAVOR ELECTRON

119

Table 4.5: Systematic uncertainties and the types of the uncertainties. The scaling
uncertainty denotes an uncertainty on scaling of the raw asymmetry AS+BG
and theLL
oﬀset uncertainty denotes an uncertainty on the absolute value of the asymmetry. The“global” in this table means the uncertainties are globally correlated in all pT regions.
The scaling uncertainty is represented as the ratio of the uncertainty to the signal(∆S/S) given in percent and the oﬀset uncertainty is represented as the absolute valueof the uncertainty.
sourcesignal purity D
B PY )
polarization ( ∆(P)
PB PY
relative luminosity R
background asymm. ABG
LL

uncertainty
∼ 6%
8.8%
0.14 × 10−2
0.2 × 10−2 × 1−DDtypescaling
global scaling
global oﬀset
oﬀset

eTable 4.6: Data table for the AHF
result corresponding to Fig. 4.17.
LL
HF e
pT [GeV/c]
ALL
stat. error syst. error (oﬀset) syst. error (scale)−2
0.612
2.83×10
2.66×10−2
0.75×10−2
0.50×10−2
0.864-1.20×10−2
1.21×10−2
0.30×10−2
0.08×10−2
−2
−2
−2
1.115
0.76×101.30×10
0.21×100.04×10−2
1.366
2.08×10−2
1.63×10−2
0.18×10−2
0.10×10−2
1.617-0.69×10−2
2.18×10−2
0.17×10−2
0.03×10−2
−2
−2
−2
1.867
-1.39×10
2.68×10
0.16×100.03×10−2
2.118
4.82×10−2
3.46×10−2
0.16×10−2
0.09×10−2
−2
−2
−2
2.369
-5.91×10
4.40×10
0.16×10
0.11×10−2
2.619-6.97×10−2
5.47×10−2
0.16×10−2
0.13×10−2
2.869
6.43×10−2
7.07×10−2
0.16×10−2
0.12×10−2Chapter 5
Discussion
5.1

Constraint on ∆g(x)In this section, we discuss√ constraint of ∆g from the measured double-spin asymmetry.
In p+p collisions at s = 200 GeV, heavy flavor electrons with momentum ranging 0.50 < pT < 1.25 GeV/c are mainly produced by open charm events. Opencharm production is well described by pQCD calculations, which can be factorized
into charm quark production at the partonic level and fragmentation of the charm
quarks into charmed hadrons. Using polarized and unpolarized parton distributionfunctions (PDFs), the respective cross sections for charm quark production are estimated. pythia8 [76, 77] simulations are then used to model the fragmentation andsemi-leptonic decay processes. The spin asymmetry AHFe
LL is the ratio of the polarized
and unpolarized cross sections, and a comparison between the measured and calculated
values of AHFe
LL can thereby provide constraints on the gluon polarization ∆g.
For this discussion, we calculated the charm quark cross section in the partonic
level using an LO pQCD calculation [25]. In LO pQCD calculations, only gg → cc̄ andq q̄ → cc̄ are allowed for the open charm production. The charm quarks are primarily
created by the gg interaction in the unpolarized hard scattering, and the q q̄ contribution
is known to be just a few percent in this momentum region [24]. In addition, theanti-quark polarizations are known to be small from semi-inclusive DIS measurements
precisely enough that both DSSV [23] and GRSV [74] expect contribution of polarizedq q̄ cross section to the double-spin asymmetry of the heavy flavor electrons in |η| < 0.35
and pT < 3.0 GeV/c to be ∼ 10−4 [24], which is much smaller than the accuracy of thismeasurement. Therefore, in this analysis of ∆g, we ignore the q q̄ interaction and assume
the asymmetries are due only to the gg interaction. Under the assumption, the spin
asymmetry of the heavy flavor electrons is expected to be approximately proportional
to the square of polarized gluon distribution normalized by unpolarized distribution,|∆g/g(x, µ)|2 .
To calculate the cross section of the gg → cc̄ process, CTEQ6M [48] was employed
for the unpolarized PDF. For the polarized PDF, we assumed |∆g(x, µ)| = Cg(x, µ)
where C is a constant. The charm quark mass was assumed as mc = 1.4 GeV/c2 andthe factorization scale √in CTEQ6 and the renormalization scale were assumed to be
identical to µ = mcT ≡ pcT 2 + mc 2 . The resulting cross sections for a case of C = 0.5
121

122

CHAPTER 5. DISCUSSION

are shown in Fig. 5.1. Figure 5.1(a) shows the unpolarized cross section, σ pp→cc̄ (pcT , η c ),and Figure 5.1(b) shows the polarized cross section, ∆σ pp→cc̄ (pcT , η c ), and Figure 5.1(c)shows the asymmetry of the cross section, app→cc̄
(pcT , η c ) ≡ ∆σ pp→cc̄ /σ pp→cc̄ (pcT , η c ).
LL
The fragmentation and decay processes were simulated with pythia simulation.
We generated pp → cc̄ + X events and selected electrons from the charmed hadrons,
D+ , D0 , Ds , Λc and their antiparticles. We scaled the charm quark yield in pythia
with respect to the pQCD calculated unpolarized and polarized cross sections to obtain
unpolarized and polarized electron yields from charmed hadron decays under thesecross sections. We also applied a pseudorapidity cut of |η| < 0.35 for the electrons tomatch the acceptance of the PHENIX central arms. The shape of the expected spin
asymmetry AHFe
LL (pT ) is then determined from the simulated electron yields.
Figure 5.2 shows the distribution of the gluon Bjorken x contributing to heavy flavor
electron production in the momentum range 0.50 < pT < 1.25 GeV/c, from pythia.
Using the mean and the RMS of the distribution for 0.50 < pT < 1.25 GeV/c, we
determine the mean x for heavy flavor electron production to be ⟨log10 x⟩ = −1.6+0.5−0.4 .
HFe
We calculated expected ALL (pT ) by varying C = |∆g/g|. Figure 5.3(a) shows
several of these curves, along with the measured points. χ2 values are calculated foreach value of C, along with related uncertainties. By assuming that the systematic
uncertainties on the points are correlated and represent global shifts, we defined thequantity χˆ2 as()
nχ̂2 (C) ≡ −2 log (2π) 2 P̂ (C)∫P̂ (C) ≡ dpdqN( (p)N (q)×)
oﬀset − 1+qγ i scale f (x ;C)∏n(yi +pϵisyst
(
) i )
syst
i=1 N
ϵistat√()2)(
scale 2ϵisyst∆(PB PY )i scale,γsyst =+ PB PY
yi√
where N (X) denotes normal probability distribution, i.e. N (X) = 1/ 2π exp(−X 2 /2),
n is the number of the data points and equal to three, and for the i-th data point, xi is
oﬀset
scale
the pT value, yi is the ALL value, and ϵistat , ϵisyst
and ϵisyst
represent the statistical,
oﬀset systematic and scaling systematic uncertainties, respectively. f (pT ; C) denotes
the expected ALL (pT ) for the parameter of C = |∆g/g|. ∆(PB PY ) is an uncertainty
i scalefor polarization. If we set the systematic uncertainties, ϵoﬀset
syst and γsyst , to zero, thenewly defined χˆ2 is consistent with the conventional χ2 .
The resulting χ̂2 curve is shown in Fig. 5.3(b), plotted as a function of C 2 = |∆g/g|2
because the curvature becomes almost parabolic. The minimum of χ̂2 , χ̂2min , is locatedat |∆g/g|2 = 0.0 which is the boundary of |∆g/g|2 . ∆χ̂2 ≡ χ̂2 − χ̂2min = 1 and 9
were utilized to determine 1σ and 3σ uncertainties. With these criteria, we foundthe constraints on the gluon polarization are |∆g/g(⟨log10 x⟩ , µ)|2 < 3.0 × 10−2 (1σ)
and 10.0 × 10−2 (3σ). The constraints are consistent with theoretical expectations for∆g/g(x, µ) at ⟨log10 x⟩ = −1.6+0.5−0.4 and µ = 1.4 GeV which are ∼ −0.006 from DSSV, ∼0.016 from GRSV(std) and ∼ 0.019 from GRSV(val) using CTEQ6 for the unpolarized
PDF.
The eﬀects of the charm quark mass and scale factor in the cross section calculation

5.2. FUTURE PROSPECT FOR ∆G(X) MEASUREMENT

123

were also checked by varying the charm mass from mc = 1.3 GeV/c2 to 1.5 GeV/c2
and the scale to µ2 = 0.75mcT 2 and 1.5mcT 2 . Figure 5.3(b) also shows the resulting χ̂2
curves. Considering the variation of the crossing position at ∆χ̂2 = 1, the constraint
including the uncertainties from the charm mass and the scale can be represented as+0.5−2|∆g/g|2 < (3.0+0.4−0.2 (mass)−0.3 (scale)) × 10 (1σ). The square root of the constraint
2−2|∆g/g| < 3.0 × 10 is plotted in Fig. 5.4 with other constraint of ∆g/g from SIDIS
experiments. As Fig. 5.4 shows, this measurement covers the unexplored Bjorken x
region and gives comparable constraint on ∆g/g. It is worth to note that the present
constraint on ∆g/g(x) is not exact 1σ deviation since it is square root of 1σ deviationof |∆g/g(x)|2 .
The integral of the CTEQ6
unpolarized PDF in the sensitive x region of ⟨log10 x⟩ =∫ 0.08+0.5−1.6−0.4 and µ = 1.4 GeV is 0.01 dxg(x, µ) = 4.9. Hence the constraint on the integral∫ 0.08of the polarized PDF at 1σ corresponds to | 0.01 dx∆g(x, µ)| < 0.85. We can exclude>
large gluon polarization at the low Bjorken x region, ∆G[0.01, 0.08] ∼1.

5.2

Future Prospect for ∆g(x) Measurement

The constraint on ∆g(x, µ) obtained through this measurement is not enough to constrain ∆G(µ) compared with the total proton spin 1/2. However, this study highlights
the possibility for constraining ∆g(x, µ) more precisely in the future with higher statistics and higher beam polarizations.
In addition, there are other future planned channels to investigate ∆g(x, µ) in suchsmall Bjorken x region. They are ALL measurements of π 0 production at the PHENIX
mid-rapidity
region (|η| < 0.35) and the forward-rapidity region√(3.1 < |η| < 3.9)√
in s = 510 GeV p + p collisions. The ALL measurements at s = 510 GeV√ are
sensitive to ∆g(x, µ) at smaller Bjorken x by a factor of about 2/5 compared with s =200 GeV. Especially the measurement at the forward-rapidity region can approach torather small Bjorken x region. The sensitive Bjorken x regions for the measurements areshown in Fig. 1.18(c). These measurements will complement the heavy flavor electron
measurement each other to constrain ∆g(x, µ) at the small Bjorken x region.

124

CHAPTER 5. DISCUSSIONd2σ/dpTcdyc [a.u.]

40000
35000
30000
25000
20000
15000
10000
5000
0

-4

-3

-2

yc

-1

0

1

2

3

4

9
10

8

7

6

5

4

3

2

1

pTc [GeV/c](a) Unpolarized cross section
d2Δσ/dpTcdyc [a.u.]

8000
6000
4000
2000
0

4

-4

-3
c

y

-2

-1

0

1

2

3

4

9
10

8

7

3

2

1

5
6 p c [GeV/c]
T(b) Polarized cross section

aLL=dσ/dΔσ

0.2
0.1
0
-0.1
-0.2
-4

-3

yc

-2

-1

0

1

2

3

4

9
10

8

7

6

5

4

3

2

1

pTc [GeV/c](c) Asymmetry (aLL ≡dσ/d∆σ)Figure 5.1: (a) Unpolarized and (b) polarized cross section of the charm production
as functions of charm transverse momentum pcT and charm rapidity y c estimated byusing a LO pQCD calculation [25]. In the calculation, CTEQ6M [48] was employed
for the unpolarized parton distribution functions (PDF). For the polarized PDF, we
assumed |∆g(x, µ)| = 0.5|g(x, µ)|. The charm mass was assumed as mc = 1.4 GeV/c2
and the scale factors for the factorizationin CTEQ6 and the renormalization are as√cc2
pT + mc 2 . (c) Asymmetry of the cross section,
sumed identical to µ = mT ≡pp→cc̄ cc
pp→cc̄
pp→cc̄ caLL (pT , y ) ≡ ∆σ/σ
(pT , y c ).

10dN/d(Log x)

5.2. FUTURE PROSPECT FOR ∆G(X) MEASUREMENT

5 ×10

125

-3

+

pp→cc+X→(e +e-)/2+X
s=200GeV |ηe|<0.35
0.50<pe <1.25 GeV/c

4.5
4

T

3.5
3
2.5
2
1.5
1
0.5
0
-3

-2.5

-2

-1.5

-1

-0.5

0

Log10 xFigure 5.2: Bjorken x distribution of gluons contributing the heavy flavor electron
production with momentum ranging 0.50 < pT < 1.25 GeV/c obtained from pythia
simulation. The distribution is normalized with respect to the number of total generated
charmed hadrons.

CHAPTER 5. DISCUSSION

HFe
ALL

126

0.15

AHFe
LL p+p s=200GeV (|ηe|<0.35)
0.1
0.05
0

-0.05
-0.1
-0.15
0.4

|∆g/g|=0.00|∆g/g|=0.10|∆g/g|=0.20|∆g/g|=0.30|∆g/g|=0.40|∆g/g|=0.50

0.5

0.6

AHFe
LL
syst. error from dilution factor
syst. error from rel. lum. and ABG
LL(not including 8.8%% pol. scale error)
0.7

0.8

0.9

1

1.1

1.2

1.3

p [GeV/c]Tχ2(a) Expected AHFe
LL for several |∆g/g|.
16mc=1.4GeV/c2, µ 2/m 2T =1.0
mc=1.3GeV/c2, µ 2/m 2T =1.0

1412mc=1.5GeV/c2, µ 2/m 2T =1.0
mc=1.4GeV/c2, µ 2/m 2T =0.75

2

∆χ =9

mc=1.4GeV/c2, µ 2/m 2T =1.5

10
8
6
2

∆χ =1

4
2
0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

|∆g/g|2(b) χ̂2 curves.

Figure 5.3: (a) AHFe
LL for |∆g/g| = 0.00, 0.10, 0.20, 0.30, 0.40, 0.50 are shown as the solidline, the dashed line, the dotted line, the dashed dotted line, the long-dashed dotted line,
the dashed triplicate-dotted line respectively. They are plotted with the measured data
points and the notation for the error bars are same as Fig. 4.17. (b) χ̂2 curves calculated
from (a) as a function of |∆g/g|2 . The black solid line is the default configuration. The
blue curves are after changing the charm mass to 1.3 GeV/c2 (dashed line) and to 1.5
GeV/c2 (dotted line) and the red curves are after changing the scale µ2 to 0.75mcT 2(dashed dotted line) and 1.5mcT 2 (long-dashed dotted line).

∆g/g

5.2. FUTURE PROSPECT FOR ∆G(X) MEASUREMENT

0.8

127

New COMPASS, high p , Q2>1 (GeV/c)2, 2002-2006
T

0.6
0.4

COMPASS, high p , Q2<1 (GeV/c)2, 2002-2003
T

COMPASS, open charm, 2002-2007
SMC, high p , Q2>1 (GeV/c)2
T
HERMES, high p , all Q2
T

0.2
0
-0.2
-0.4
-0.6
-0.8
10-2

PHENIX, HF electron 2009
Q2=(1.4 GeV/c)2
DSSV fit, µ 2=3(GeV/c)2
LSS fit with ∆G>0, µ 2=2.5(GeV/c)2
LSS fit with ∆G changing sign, µ 2=2.5(GeV/c)2

10-1

xg

Figure 5.4: Constraint on ∆g/g(x) from this analysis, which is shown with green hexagonal star, on other constraints from SIDIS experiments shown in Fig. 1.13. The verticalerror bar in our result corresponds to |∆g/g(x)|2 < 3.0 × 10−2 (1σ), and therefore not1σ deviation in ∆g/g(x).

Chapter 6
Conclusion
The measurement of the gluon polarization in small Bjorken x is essential to determine
the total gluon polarization G(µ). Previously, achievable Bjorken x was limited up to<
the minimum of 2×10−2 ∼x due to a minimum pT required from pQCD calculations.
To extend the Bjorken x region, the double spin asymmetry of the heavy flavor electron production is an ideal probe because enough energy scale in this production is
guaranteed due to the large mass of the heavy quarks. In this work, we firstly measured
the double spin asymmetry in the PHENIX experiment at RHIC (BNL) using√s = 200 GeV p + p collisions, and obtained the constrain of the gluon polarization in
the small Bjorken x region from the measured asymmetry.
The measurement of the heavy flavor electron production suﬀered large backgroundof photonic electrons and Ke3 electrons. In this work, we succeeded to significantly
suppress the background with the new detector, HBD, and improved the signal purity<<
by a factor of ∼ 1.5 around 0.75 ∼pT ∼2.00 GeV/c. Especially, the HBD also eﬃciently
reduces the Ke3 electrons by a factor of ∼ 0.4, because the HBD requires the position of
the Ke3 decay to be close to the beam vertex, R ∼ 10 cm. The double spin asymmetryof the K production is not well known, and therefore the reduction of the Ke3 electrons
is important to reduce uncertainty on the measurement of the spin asymmetry of heavy
flavor electrons. We also succeeded to confirm the reliability of the new analysis method
using the HBD by comparing the obtained cross section of the heavy flavor electronwith the previous result.
We have reported on the first measurement of the longitudinal double-spin asymmetry of heavy flavor electrons, which are consistent with zero. Using this result, we
c 2−2
obtained a constraint of |∆g/g(log10 x = −1.6+0.5−0.4 , µ = mT )| < 3.0 × 10 (1σ). The
result covers unexplored Bjorken x region in other ∆g(x, µ) measurements and gives
comparable constraint compared with results from SIDIS experiments. This constraint
is consistent with the existing theoretical expectations with GRSV and DSSV.
With improved statistics and polarization, the helicity asymmetry of heavy flavor
electron production can provide more significant constraints on the gluon polarization,and complement other measurements of ∆G(µ). In addition, ALL measurements at
higher energy collisions at the PHENIX mid-rapidity and forward-rapidity regions will
also help to constraint ∆g(x) at small x region.

129

Acknowledgments
During I performed the analysis and wrote this PhD thesis, there were many persons
who provided generous support to me. I would like to express my thanks to all of themhere in this article. First of all, I would like to acknowledge Professor Dr. T. Nagae, whois the leader of our laboratory, and Lecturer Dr. T. Murakami for their careful supports
and suggestions to complete this thesis. I also greatly thanks to Professor Dr. N. Saito,
who has supervised and educated me since I enrolled in my graduate school. He taughtme the importance of the diligent attitude to research and the broad vision to physics.
I would deeply thanks to Vice-Chief Scientist Dr. Y. Akiba, who provided dedicatedsupports and a lot of suggestions in this analysis. He also advised me the importanceto take pride in my works. And I would also like to express my gratitude to Professor
Dr. K. Imai, who is the former leader of our laboratory. His strong passion towardphysics always encouraged me into research.
I would like to express my great thanks to also the persons who have worked on the
Muon Trigger Upgrade project together with me. Though the project is not directly
related to the present analysis, the time spent in the project was essential for me tobecome a researcher. Scientist Dr. I. Nakagawa and Associate Professor Dr. T. Mibe,
who always worked hard on the project, while they sometimes made our holidays fun
by holding parties and events. They taught me how we can make our research lives
interesting and worthwhile. Assistant Professor Dr. Y. Fukao also worked hard anddiscussed with me about physics. It was precious time for me to expand my knowledge
and think what we need for the future physics. I would also deeply thank to Mr. K. Shojiand Dr. K. Karatsu for telling me abundant knowledge about electronics and data
analysis. I was struck by their research attitudes to investigate any details. I express mygreat thanks to Dr. Y. Imazu, Mr. H. Oide, Ms. S. Park, Ms. M. Karatsu, Mr. Y. Ikeda
and Mr. K. Watanabe for making my research life in RIKEN and BNL glorious. They
worked on the Muon Trigger Upgrade project diligently and served as the biggest driving
force of the project. I thank Professor Dr. J. Lajoie and Dr. T. Kempel for their earnest
eﬀorts to complete the trigger electronics. And I would also like to express my deep
acknowledgment to Professor Dr. O. Sasaki and Professional Engineer Dr. M. Ikeno for
dedicated supports to product and develop our trigger electronics.
I also express great thanks to all the persons, who worked on the present analysis of the double spin asymmetry of the heavy flavor electron production. Scientist
Dr. K. Okada provided sound suggestions and comments on this work and knowledge
about the nucleon spin physics to me. I would acknowledge to Dr. K. Boyle, Associate
Professor Dr. K. Tanida, Scientist Dr. Y. Goto, and Dr. M. Durham for advising me a lot
in my analysis and the paper preparation. Professor Dr. T. Hemmick and Dr. I. J. Choi
131

132

CHAPTER 6. CONCLUSION

have kindly checked my publishing article. I thank Scientist Dr. M Stratmann for
detailed discussions about the constraint of the gluon polarization and preparation of
the calculation codes of the cross sections. I would also like to express deep thanks
to Professor Dr. B. Jacak and Scientist Dr. B. Johnson for devotedly managing the
publication of my article.
I would greatly acknowledge to Chief Scientist Dr. H. En’yo for supporting us bycreating comfortable environment for our research. I also deeply thanks to RIKEN
and RBRC oﬃce administrators, Ms. N. Kiyama, Ms. K. Suzuki, Ms. E. Adachi,
Mr. K. Mabuchi, and Ms. T. Ito, for managing my business trips and assisting myresearch life in RIKEN and BNL. Thanks to their helps, I could concentrate on thework. And I would like to thank to all the people who enjoyed the BNL life with me
and made it fun, Mr. S. Dairaku, Dr. T. Kojo, Scientist Dr. Y. Hidaka, Associated Professor Dr. K. Kondo, Associated Professor Dr. J. Tamura, Dr. T. Kanesue, Dr. Y. Yamaguchi, Dr. Y. Aramaki, Dr. K. Nakano, Dr. K. Aoki, Dr. Y. Ikeda, Mr. H. Asano,
Dr. T. Misumi, Mr. R. Akimoto, Mr. T. Kawanai, Mr. M. Nihashi, and many otherstudents from CNS, Hiroshima University, Tsukuba University, Rikkyo University, andTokyo Institute of Technology University. I would like to give a special thanks to Scientist Dr. Y. Watanabe, Scientist Dr. S. Yokkaichi, for kindly helping me especially on
computing troubles in the present analysis and cheering me up in RIKEN.
I am deeply grateful to all the members of my laboratory. I would express deeply
thanks to Associate Professor Dr. T. Kawabata for telling us the fun of nuclear physics
and making our laboratory more interesting. Assistant Professor Dr. M. Niiyama andAssistant Professor Dr. H. Fujioka have abundant knowledge about hadron physics,
and I learned a lot of things from them. I would also thank to Mr. A. Tokiyasu andMr. M. Moritsu for talking and counselling each other. The time spent in my laboratory
with them was precious days for me.
Last, I express grateful acknowledge to my both parents for their kind supports until
my graduation, for twenty-eight years. I owe them for all of my works, and dedicate
this thesis to them.

Appendix A
Deep Inelastic Scattering (DIS)
Experiment
The lepton-quark scattering in the DIS has same kinematics as electron-muon scattering, which is a scattering between diﬀerent Dirac particles. The diagram of theelectron-muon scattering is shown in Fig. A.1(a). Using the variables in the figure, thescattering amplitude in the process can be written ase4 e(L )µν (Lmu )µν ,q4∑∗≡[ū(k ′ , s′e )γµ u(k, se )] [ū(k ′ , s′e )γν u(k, se )]|Me-mu |2 =(Le )µν

(A.1)

s′e

(Lmu )µνe(A)= 2Le(S)µν + 2iLµν ,∑∗≡[ū(p′ , s′mu )γµ u(p, smu )] [ū(p′ , s′mu )γν u(p, smu )]

(A.2)

s′mu

= 2Lmu(S)+ 2iLmu(A)
,
µν
µν

(A.3)

where s and s′ represent initial and final spin states of the leptons, respectively. To study
the initial helicity dependence, the summation is taken only for the final spin states.
(A)(S)
Lµν and Lµν represent symmetric and asymmetric components for the commutationof µ and ν, and can be written as(S)Lµν(k, k ′ ) = kµ kν′ + kµ′ kν − gµν (k·k ′ − m2 ),
′
α
′ β
L(A)
µν (k, k , s) = ϵµναβ ms (k − k )

(A.4)
(A.5)

for the electron (interchange k to p, and k ′ to p′ for the muon), where m is mass ofthe electron (the muon). In the muon rest frame as shown in Fig. A.1(b), ignoring
the electron mass, the helicity-dependent and helicity-independent cross sections canbe represented as()
dσe-mu4α2 E ′2 E ′
Q2
2 θ
2 θ
(E, θ) =
sin
cos +
,
(A.6)
dΩ
Q4 E
2 2m2µ2
α2 E ′2 E + E ′ cos θd∆σe-mu(E, θ) =λe λmu ,dΩQ2 E 2
mµ
133

(A.7)

134

APPENDIX A. DEEP INELASTIC SCATTERING (DIS) EXPERIMENT

where λe and λmu represent incident helicities (±1) of electron and muon in the centerof-mass system, and mµ represent the muon mass.
k'ek'=(E',k')

k
γ*

μ
p

k=(E,k)eq=k–k'

θq=(ν,q)p'μp=(M,0)Time(a) Diagram of e-µ scattering.

(b) Kinematics in the muon rest frame.

Figure A.1: (a) Diagram of electron-muon scattering. (b) Kinematics in the muon rest
frame.
In the e-µ elastic scattering, the muon energy in the final state E ′ is determined fromE and θ as E ′ = E/(1 + (2E/mµ ) sin2 θ). However, when we consider lepton-parton
scattering, E ′ can not be determined only from the two variables due to the variationof the parton mass, xM . Therefore, diﬀerential form with respect to E ′ is better tointerpret the lepton-parton cross sections. The diﬀerential of E ′ requires to add a delta
function,
(
)
(
)
E
E
Q2
′
δ E −
=
δ ν−
(A.8)
E′
2mµ
1 + (2E/mµ ) sin2 θ
(
)
E 1
Q2
=
δ 1−
.
(A.9)
E′ ν
2mµ ν
Then, the diﬀerential form of the cross sections (Eq. A.6 and A.7) are given as()()
d2 σe-mu
4α2 E ′2
Q2
1
Q2
′
2 θ
2 θ
(E, θ, E ) =
cos +
sin
δ 1−
,(A.10)dΩdE ′
Q4
2 2m2µ
2 ν
2mµ ν
(
)
d2 ∆σe-mu
α2 E ′ E + E ′ cos θ 1
Q2′(E, θ, E ) =δ 1−
λe λmu ,(A.11)dΩdE ′
Q2 E
mµ
ν
2mµ ν
Using above results, we investigate relations between DIS cross section and quark
PDF. The cross section of DIS as shown in Fig. 1.3(a) can be represented as
α2 E ′
d2 σDIS
′
(E,
θ,
E
)
=
Lµν W µν ,(A.12)
′
2
dΩdE
Q E
)
∫ NX (1 ∑ ∏d3 p′n(A.13)Wµν (p, q, s) ≡4πM X(2π)3 2En′
n=1
(
)
NX
∑
∑⟨
⟩
(2π)4 δ 4 p + q −p′np, s|Jµ† |X ⟨X|Jν |p, s⟩(A.14)
n=1

sn

135
where s is initial nucleon spin and Wµν is a hadronic tensor as a function of p, q ands, which describes hadronic interaction with virtual photon, corresponding to Lµν oflepton. Current conservation ∂µ J µ = 0 requires following conditions,qµ W µν = 0,qν W µν = 0.

(A.15)As the result, Wµν can be written as(S)(A)
Wµν (p, q, s) = Wµν(p, q) + iWµν(p, q, s),(A.16)
)
(
qµ qν
(S)
Wµν
(p, q) ≡−gµν + 2 W1 (ν, q 2 )
q
(
)(
)
p·q
p·q
1
pν − 2 qν W2 (ν, q 2 ),(A.17)+ 2 pµ − 2 qµ
M
q
q
{
}
)
1 (
(A)
α
β
β
2
β
2
Wµν (p, q, s) ≡ ϵµναβ q M s G1 (ν, q ) +(p·q) s − (s·q) p G2 (ν, q ) ,M(A.18)

where M denotes the nucleon mass. W1 , W2 , G1 , and G2 are called structure functions
which describe structure of the nucleon. Especially at large Q2 region, the structure
functions represent the parton distribution, and therefore become functions of Bjorken
x and almost Q2 independent1 . This phenomenon is called “Bjorken scaling”. For
convenience, following dimensionless structure functions are conventionally used,
F1 (x, Q2 ) = M W1 , F2 (x, Q2 ) = νW2 ,g1 (x, Q2 ) = M 2 νG1 , g2 (x, Q2 ) = M ν 2 G2 .

(A.19)

Using above structure functions, the helicity-dependent and helicity-independent
cross sections of the DIS in the nucleon rest frame can be represented as()
d2 σDIS14α2 E ′2 2ν
′
2 θ
2
2 θ
2
(E, θ, E ) =sin F1 (x, Q ) + cos F2 (x, Q )
,
′
4
dΩdE
Q
M
2
2
ν
(A.20)
(
)
′
2
2
′
2
d ∆σDIS
2α E E + E cos θ
Q
1
′
2
2
(E,
θ,
E
)
=
g
(x,
Q
)
−
g
(x,
Q
)
λN λl ,
1
2
dΩdE ′Q2 E
MMνν(A.21)
where λN and λl represent the initial helicities (±1) of the nucleon and lepton in thecenter-of-mass system.
The DIS cross sections can be deduced also from the e-µ scattering cross sections. To
translate the e-µ scattering result into the DIS cross section, we replace the muon mass
mµ to the parton mass xM , add the quark charge factor e2q ((2/3)2 for u, c, t and (1/3)2
for d, s, b), and sum up for all quarks and anti-quarks with a weight of the unpolarized
parton density fq (x)dx for helicity-independent cross section or the polarized parton
1

Strictly speaking, small log Q2 dependence still appears due to higher order pQCD eﬀect in PDF
evolution as described in Sec. 1.4

136

APPENDIX A. DEEP INELASTIC SCATTERING (DIS) EXPERIMENT

density ∆fq (x)dx for the helicity-dependent cross section. Equations A.10 and A.11
are translated to be∑ ∫ 1
d2 σDISd2 σe-mu
2
=
e
dxf
(x)
(mµ →xM )qqdΩdE ′dΩdE ′
0
q
)
(
)
(
∑ ∫ 1
4α2 E ′2
ν
1
Q2
2
2 θ
2 θ
=
eq
dxfq (x)x cos +
sin
δ x−
4
Q
2
M
2
ν
2M ν
0
q
(
(
(
)
))
∑
∑
4α2 E ′2
θ
ν
θ
1
=
cos2
e2q xfq (x) +sin2
e2q fq (x)
,
4
Q
2
M
2
ν
q
q
∑
d ∆σDIS
=
e2q
′
dΩdE
q
2

(A.22)∫ 1

2

dx∆fq (x)0d ∆σe-mu(mµ →xM )dΩdE ′)(
α2 E ′ E + E ′ cos θ 1
Q2
dx∆fq (x) 2
λN λl ,
=
δ x−
Q
E
M
ν
2M
ν
0
q
(
)
α2 E ′ E + E ′ cos θ ∑ 2
1
=
e
∆f
(x)
λN λl .
q
q
Q2 E
M
ν
q
∑

∫ 1

e2q

(A.23)
Comparing between Eq. A.20 and A.22, and between Eq. A.21 and A.23, the relations
between the structure functions and quark PDF are deduced as1∑ 2
e fq (x),2 q q∑
F2 (x, Q2 ) = x
e2q fq (x),F1 (x, Q2 ) =(A.24)(A.25)q1∑ 2
g1 (x, Q2 ) =e ∆fq (x),2 q q(A.26)

g2 (x, Q2 ) = 0.

(A.27)

Therefore, the unpolarized and polarized quark PDF can be studied performing theunpolarized and polarized DIS experiments. Though this naive approximation resultsin g2 = 0, the longitudinally polarized DIS experiment is insensitive to g2 because
the coeﬃcient of g2 compared with g1 is 2M x/(E + E ′ cos θ) from Eq. A.21 which issuppressed by an order of M/E. In contrast, the spin-dependent cross section with thenucleon target polarized perpendicular to the lepton spin is sensitive to g2 .

Appendix B
Polarization Measurement at RHIC
Three polarimeters are used to measure and monitor the beam polarization. Two ofthem are fast carbon ribbon polarimeter (pC polarimeter) [81] and polarized hydrogengas jet target polarimeter (H-jet polarimeter) [82, 83], installed in the RHIC ring, andanother is PHENIX local polarimeter [84], installed at the PHENIX experimental area.
These three types of polarimeters measure a sizable transverse single spin asymmetries for elastic scattering or specific particles production. The transverse single spin
asymmetry is defined as
AN ≡σL − σR,
σL + σR(B.1)

where σL (σR ) represents cross section of the target process with an outgoing particle
in left (right) side with respect to the transverse spin direction in p + p↑ collisions. The
measurement is done by calculating a raw spin asymmetry of measured yields of thetarget particle, ϵN . The raw asymmetry can be represented as
ϵN = P A N ,(B.2)

where P denotes the beam polarization. Therefore, once the physics asymmetry AN is
known, the beam polarization P can be calculated as P = ϵN /AN . And also, by tracing
ϵN , we can detect the fluctuation in the beam polarization during the experiment. The
three types of polarimeters are introduced in this section.
The absolute value of the polarization is measured by the pC polarimeter and Hjet polarimeter. An accuracy of the polarization measurement should be achieved to∆Pbeam /Pbeam < 5% at any beam energy up to 250 GeV. Then ideal interactions forpolarimetry should satisfy the following conditions:
1. well-known or measurable and non-zero analyzing power,
2. high event rate interaction (large cross-section and/or thicker target) to save datataking time,
3. similar kinematics for diﬀerent beam momenta for common detector set up.
137

138

APPENDIX B. POLARIZATION MEASUREMENT AT RHIC

The elastic scattering of the polarized proton beam oﬀ a nuclear target A (p↑ + A →p + A) in the Coulomb nuclear interference (CNI) region is an good process for thismeasurement. We choose proton and carbon for A. However only one measurement
cannot satisfy the all requirements. Therefore, these two polarimeter complement themeach other. The pC polarimeter, which satisfies item 2 and 3, serves as a semi-on-line
beam polarization monitor during the beam acceleration and provides fill-by-fill oﬄine
Pbeam results tracing the polarization changing. The H-jet polarimeter, which satisfies
item 1 and 3, serves as an accurate absolute calibration for the pC polarimeter.
Fast carbon ribbon polarimeter (pC polarimeter)

Figure B.1: Picture of pC polarimeter set up at IP12.
The pC polarimeter (Fig. B.1) is set up at the downstream position of IP12 for Blue
beam and at the upstream position of IP12 for Yellow beam as shown in Fig. 2.2. This
polarimeter measures the transverse single spin asymmetry in the elastic scattering between polarized proton and carbon target, ApC
N , at very forward region by detecting therecoil carbons. The four-momentum transfer in the region is −t = 0.01 − 0.02 (GeV/c)2 .
ApC
N is known to be about 1.4% [81]. The target should be thin to achieve enough transparency for the recoil carbon with small energy of 0.1 − 1.0 MeV and also for the protonbeam. On the other hand, it is also required to achieve high statistics. The requirements are satisfied by using ultra-thin carbon ribbon target of 3 − 5 µg/cm2 with awidth of 10 µm is used.
Figure B.2 displays the experimental setup of the pC polarimeter. The target ribbon
is inserted into the beam and taken out after the measurement. Slow recoil carbons aredetected by the silicon detectors placed on both sides of the target. The pC polarime-

139

Si detectors

Recoil Carbon

cm
15

Beam

Beam
C filament
Beam’s-eye view of detectors

Side View

Figure B.2: The experimental setup of the pC polarimeter. Left: beam view of thedetectors. The beam runs into the paper and hit the carbon ribbon target in the centerof the beam pipe. Right: side view of the detectors. The beam runs from left to right.
Recoil carbon is detected with the silicon detectors.
ter collects ∼ 4 × 106 events per one measurements which is typically one minute. It
corresponds to a statistical uncertainty of 4% which is smaller compared to the systematic uncertainty of 7.2% for Blue and 9.3% for Yellow beams. The pC polarimeter
confirms that the bunch by bunch polarization variation is within the uncertainty ofthe measurements. ApC
N was not known in this energy and cannot be measured withthe pC polarimeter system . In this respect, pC provides only relative variation ofpolarization for each fill. H-jet target polarimeter results were used to normalize thepC polarimeter results. the major source of the systematic uncertainty assigned forthe beam polarizations, is the uncertainty of the absolute scale obtained with H-jet
polarimeter measurement.
Polarized hydrogen gas jet polarimeter (H-jet polarimeter)
The H-jet polarimeter (Fig. B.3) is set up at IP12. This polarimeter measures thetransverse single spin asymmetry in p + p↑ elastic scattering, App
N . Since both beampp
and target are polarized, AN can be measured by two method, namely the spin asymmetry for the target polarization and the spin asymmetry for the beam polarization.
The relation between the two raw asymmetries of the yield (ϵtarget , ϵbeam ) and physics
asymmetry (AppN ) is
App
N =

ϵbeam
ϵtarget=,Ptarget
Pbeam

(B.3)

where Ptarget (Pbeam ) represents the polarization of target (beam). From this relation,
the beam polarization can be determined as
Pbeam = Ptarget

ϵbeam.
ϵtarget

(B.4)

140

APPENDIX B. POLARIZATION MEASUREMENT AT RHIC

(a)

(b)

Figure B.3: (a) H-jet polarimeter in IP12. (b) Schematic drawing of the H-jet polarimeter.
Si detector
left

Jet target
Forward particle(Not observed)

beam

Recoil particle(detected)(a)Recoilparticlebeam

Si detectorright

Jet target(b)

Figure B.4: (a) The schematic of the p + p elastic scattering process. The recoil proton
is observed while the forward proton is not. (b) The experimental setup of the H-jet
polarimeter. The beam penetrate the polarized jet target. The recoil proton is detectedwith the silicon detectors.
The advantage of the H-jet polarimeter is that we can cancel out the common factorsof systematic uncertainties in ϵtarget and ϵbeam . Ptarget is measured by a Breit-Rabi
polarimeter [83]. The target polarization measured in the 2004 commissioning run was

141
Ptarget = 0.924 ± 0.018.
Figure B.4(a) illustrates the p + p↑ elastic scattering process. the measured kinematic range is −t = 0.001 - 0.02 (GeV/c)2 where the analyzing power becomes large.
Figure B.4(b) shows the experimental setup of the H-jet polarimeter. The hydrogengas jet target crosses the RHIC beam from top to bottom at a speed of 1.6 × 103 m/sec.
The flow of the gas jet target is ∼ 102 H atoms/sec. The target spin direction is vertical, and is reversed every 10 minutes. the recoil particle is detected with the silicondetectors which are placed on both sides of the targets.
PHENIX local polarimeter
For the study of the gluon polarization, longitudinally polarized proton collisions arenecessary while the stable polarization direction of the beam in RHIC ring is verticaldirection. To obtain longitudinally polarized proton collisions, the polarization orientation is rotated from vertical to longitudinal by spin rotators. The orientation in the
PHENIX collision point is monitored by the PHENIX local polarimeter. This polarimeter measure single spin asymmetry of forward neutron production in p + p collisions at
PHENIX, AnN . AnN vanishes when the beam polarization is rotated into exactly longitudinal. Therefore the transverse component in the beam polarization can be measured
with the non-zero AnN result. This feature is used to setup the spin rotators before theexperiment and to monitor the polarization orientation and the remaining transverse
component during the experiment.
Neutrons are detected with Zero-Degree Calorimeters (ZDCs) and Shower Max Detectors (SMDs). Detailed description about these detectors and the neutron detection
is in Sec. 2.2.

Bibliography[1] J. Ashman et al. Phys. Lett. B, 206:364, 1988.
[2] J. Ashman et al. Nucl. Phys. B, 328:1, 1989.
[3] J. Ellis and R.L. Jaﬀe. Phys. Rev. D, 9:1444, 1974.
[4] J. Kodaira. Nucl. Phys. B, 165:129, 1980.
[5] A. Airapetian et al. Phys. Rev. D, 75:012007, 2007.
[6] V. Y. Alexakhin et al. Phys. Lett. B, 647:330, 2007.
[7] V. Y. Alexakhin et al. Phys. Lett. B, 647:8, 2007.
[8] M.G. Alekseev et al. Phys. Lett. B, 690:466, 2010.
[9] M.G. Alekseev et al. Phys. Lett. B, 693:227, 2010.
[10] A. Adare et al. Phys. Rev. Let., 103:012003, 2009.
[11] A. Adare et al. Phys. Rev. D, 76:051106, 2007.
[12] S. S. Adler et al. Phys. Rev. D, 73:091102, 2006.
[13] A. Adare et al. Phys. Rev. D, 79:012003, 2009.
[14] A. Adare et al. Phys. Rev. D, 83:032001, 2011.
[15] P. Djawotho et al. arXiv, 1106.5769, 2011.
[16] B.I. Abelev et al. Phys. Rev. Let., 100:232003, 2008.
[17] A. Adare et al. Phys. Rev. D, 84:012006, 2011.
[18] Adeva B. et al. Phys. Lett. B, 420:180, 1998.
[19] A. Airpetian et al. Phys. Rev. D, 71:012003, 2005.
[20] A. Airpetian et al. Phys. Lett. B, 666:446, 2008.
[21] M. Alekseev et al. Phys. Lett. B, 680:217, 2009.
[22] M. Alekseev et al. Phys. Lett. B, 693:227, 2010.
143

144

BIBLIOGRAPHY[23] D. de Florian et al. Phys. Rev. D, 80:034030, 2009.
[24] J. Riedl, A. Schafer, and M. Stratmann. Phys. Rev. D, 80:114020, 2009.
[25] M. Karliner and R. W. Robinett. Phys. Lett. B, 324:209, 1994.
[26] I. Bojak and M. Stratmann. Phys. Rev. D, 67:034010, 2003.
[27] I. Bojak. ArXiv:hep-ph, 0005120, 2000.
[28] S.S. Adler et al. Single electrons from heavy-flavor decays in p+p collisions at
sqrt(s)=200gev. Phys. Rev. Let., 96:032001, 2006.
[29] A. Adare et al. Phys. Rev. C, 84:044905, 2011.
[30] S.D. Bass. The Spin Structure of the Proton. World Scientific, 1989.
[31] S.D. Bass. Rev. Mod. Phys., 77:1257, 2005.
[32] G. Sterman et al. Rev. Mod. Phys., 67:157, 1995.
[33] M. Anselmino et al. Phys. Rep., 261:1, 1995.
[34] E.W. Hughes et al. Ann. Rev. Nucl. Part. Sci., 49:303, 1999.
[35] B.W. Fillipone et al. Adv. Nucl. Phys., 26:1, 2001.
[36] R.L. Jaﬀe et al. ArXiv/hep-ph, 9602236, 1996.
[37] B. Lampe et al. Phys. Rept., 332:1, 2000.
[38] V. Barone et al. Phys. Rep., 359:1, 2002.
[39] R.L. Jaﬀe and N. Saito. Phys. Lett. B, 382:165, 1996.
[40] F.E. Close. An Introduction to Quarks and Partons. Academic, New York, 1979.
[41] F.E. Close and R.G. Milner. Phys. Rev. D, 44, 1991.
[42] S.B. Libby and G. Sterman. Phys. Rev. D, 18:3252, 1978.
[43] J.C. Collins et al. Nucle. Phys. B, 261:104, 1985.
[44] J.C. Collins et al. Nucle. Phys. B, 394:169, 1993.
[45] J. Babcock et al. Nucle. Phys. B, 19:1483, 1979.
[46] G. Bunce et al. Annu. Rev. Nucl. Part. Sci., 50:525, 2000.
[47] G. Altarelli. Phys. Rept., 81:1, 1982.
[48] J. Pumplin et al. JHEP, 07:012, 2002.

BIBLIOGRAPHY[49] H.L. Lai et al. Phys. Rev., D82:074024, 2010.
[50] A.D. Martin et al. Eur. Phys. J., C63:189, 2009.
[51] R.D. Ball et al. Nucl. Phys., B849:296, 2011.
[52] R.D. Ball et al. arXiv, page 1107.2652, 2011.
[53] F.D. Aaron et al. JHEP, 1001:109, 2010.
[54] S. Alekhin et al. Phys. Rev., D81:014032, 2010.
[55] M. Gluck et al. Eur. Phys. J., C53:355, 2008.
[56] P. Jimenez-Delgado and E. Reya. Phys. Rev., D79:074023, 2009.
[57] R.L. Jaﬀe and A. Manohar. Nucl. Phys. B, 337:509, 1990.
[58] A. Chodos et al. Phys. Rev. D, 9:3471, 1974.
[59] P.N. Bogoliubov. Ann. Inst. H. Poincare, 8:163, 1967.
[60] R.L.. Jaﬀe. Proc. Topical Conf. Baryon Resonances, 1976.
[61] J. Beringer et al. Phys. Rev. D, 86:010001, 2012.
[62] E. Leader et al. Phys. Rev. D, 73:034023, 2006.
[63] G. Altarelli et al. Phys. Lett. B, 212:391, 1988.
[64] R.D. Carlitz et al. Phys. Lett. B, 214:229, 1988.
[65] D.L. Adams et al. Phys. Lett. B, 336:269, 1994.
[66] Airapetian A. et al. JHEP, 1008:130, 2010.
[67] Adeva B. et al. Phys. Rev. D, 70:012002, 2004.
[68] C. Adolph et al. arXiv, 1202.4064, 2012.
[69] M.G. Alekseev et al. Phys. Lett. B, 676:31, 2009.
[70] D. de Florian et al. Phys. Rev. Lett., 101:072001, 2008.
[71] E. Leader et al. Phys. Rev. D, 82:114018, 2010.
[72] L. Adamczyk et al. arXiv, 1205.2735, 2012.
[73] A. Manion et al. J. Phys. Conf. Ser., 295:012070, 2011.
[74] M. Glück et al. Phys. Rev. D, 63:094005, 2001.
[75] M. Hirai et al. Nucl. Phys. B, 813:106, 2009.

145

146

BIBLIOGRAPHY[76] T. Sjostrand et al. arXiv, 0710.3820, 2007.
[77] T. Sjostrand et al. JHEP, 0605:026, 2006.
[78] A. Adare et al. Phys. Rev. D, 83:052004, 2011.
[79] GEANT 3.2.1 Manual.
geant.pdf, 1994.

http://wwwasdoc.web.cern.ch/wwwasdoc/pdfdir/[80] A. N. Zelenski et al. Hyperfine Interaction, 127:475, 2000.
[81] O. Jinnouchi et al. arXiv:nucl-ex, 0412053, 2004.
[82] H. Okada et al. Phys. Lett. B, 638:450, 2006.
[83] H. Okada et al. arXiv, 0712.1389, 2007.
[84] Y. Fukao et al. Phys. Lett. B, 650:325, 2007.
[85] K. Adcox et al. Nucl. Instrum. Methods Phys. Res., Sect. A, 499:469, 2003.
[86] K. Adcox et al. Nucl. Instrum. Methods Phys. Res., Sect. A, 499:480, 2003.
[87] K. Adcox et al. Nucl. Instrum. Methods Phys. Res., Sect. A, 499:489, 2003.
[88] M. Aizawa et al. Nucl. Instrum. Methods Phys. Res., Sect. A, 499:508, 2003.
[89] L. Aphecetche et al. Nucl. Instrum. Methods Phys. Res., Sect. A, 499:521, 2003.
[90] H. Akikawa et al. Nucl. Instrum. Methods Phys. Res., Sect. A, 499:537, 2003.
[91] M. Allen et al. Nucl. Instrum. Methods Phys. Res., Sect. A, 499:549, 2003.
[92] W. Anderson et al. Design, construction, operation and performance of a hadron
blind detector for the phenix experiment. Nucl. Instr. and Meth. A, 646:35, 2011.
[93] Z. Fraenkel et al. Nucl. Instr. and Meth. A, 546:466, 2005.
[94] A. Kazlov et al. Nucl. Instrum. Methods Phys. Res., Sect. A, 523:345, 2004.
[95] BNL Instrumentation Division. http://www.inst.bnl.gov/.
[96] C. Y. Chi et al. Proceedings of the 2007 IEEE NSS/MIC Conference, 2007.
[97] K. Barish. Studying the Spin Structure of Nucleons with High-Energy Polarized Protons. http://www.phenix.bnl.gov/WWW/trigger/pp/c-arm/doc/
Ken001010/doe-proposal.ps. restricted to PHENIX collaborators.
[98] S.S. Adler et al. Nucl. Instrum. Methods Phys. Res., Sect. A, 499:560, 2003.
[99] J. T. Mitchell et al. Nucl. Instrum. Meth., A482:491, 2002.
[100] K. Chikanian et al. Nucl. Instrum. Meth., A371:480, 1996.

BIBLIOGRAPHY

147[101] D. Ben-Tzvi et al. Pattern Recognition Lett., 11:167, 1990.
[102] M. Ohlsson et al. Comp. Phys. Commun., 71:77, 1992.
[103] J. Myrheim and L. Bugge. Nucl. Instrum. Meth., 160:43, 1979.
[104] H. Torii. Ph. D. thesis, 2004.
[105] K. Karatsu. Ph. D. thesis, 2011.
[106] T. C. Awes et al. arXiv, nucl-ex/0202009, 2002.
[107] Y. Fukao. Ph. D. thesis, 2007.
[108] K. Aoki. Ph. D. thesis, 2009.
[109] S.S. Adler et al. Phys. Rev. Let., 91:241803, 2003.
[110] G.D. Laﬀerty and T.R. Wyatt. Nucl. Instrum. Methods Phys. Res., Sect. A,
355:541, 1995.
[111] M. Cacciari, P. Nason, and R. Vogt. Phys. Rev. Let., 95:122001, 2005.
[112] RHIC Polarization Measurement at 2009 pp collisions. http://www4.rcf.bnl.
gov/~cnipol/pubdocs/Run09Offline/.
[113] A. Mukherjee et al. Phys. Rev. D, 72:034011, 2005.