Measurement of Cross Section and Single Spin
Asymmetries of W ±-Boson Production in Polarized√
pp Collisions at s = 500 GeV

Kenichi Karatsu
2011Abstract
The cross section and non-zero parity-violating asymmetry of W and Z production in polarized pp√
collisions at s = 500 GeV have been measured for the first time with the PHENIX detector in the
RHIC (Relativistic Heavy Ion Collider) Year-2009 run (Run 9).
The data are from polarized pp → e± +X over a pseudo-rapidity range of |η| < 0.35, where the e± withtransverse momentum of pT > 30 GeV/c mainly come from W and Z decays. The results were obtainedusing the integrated luminosity of 8.56 pb−1 accumulated during the Run 9. Averaged beam polarization
was about 39 % for the selected runs. Large parity violating longitudinal single-spin asymmetries of
+
+0.09
e−
±
AeL = −0.88+0.27−0.12 and AL = 0.91−0.71 are observed for inclusive high transverse momentum e ’s. The
obtained W ± boson production cross sections for the e± channels are σ(pp → W + X) × BR(W + →−−e+ νe ) = 146.3 ± 21.0(stat.)+3.5→ e− νe ) =−10.5 (sys.) ± 23.3(norm.) pb, and σ(pp → W X) × BR(W
+7.4
34.2 ± 13.1(stat.)−8.8 (sys.) ± 5.6(norm.) pb.
This is the first measurement of the cross section of W production in pp collisions and the results are
consistent with NLO (Next-to-Leading Order) and NNLO (Next Next-to-Leading Order) perturbative
QCD (pQCD) calculations. The measurement of AL performed in this thesis establishes a new anddirect way to probe flavor-separated polarized quark and anti-quark distribution functions (polarizedPDFs) using parity-violating weak interactions. The results are also consistent with NLO pQCD based
predictions from polarized PDFs which are extracted from previous polarized (Semi-Inclusive) Deep
Inelastic Scattering (DIS) experiments. Although the uncertainties of the obtained results are quite large
due to limited statistics, the results indicate that RHIC luminosity and PHENIX detector upgrades inprogress will make it possible in the future to significantly reduce the uncertainties of AL and improve
our knowledge of flavor-separated quark and anti-quark spin contribution to the proton spin.

Contents
1 Introduction
1.1 Parton Distribution Function (PDF) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.1.1 Unpolarized PDF . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.1.2 Polarized PDF . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2 Current Experimental Knowledge on PDF . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2.1 Unpolarized PDF . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2.2 Polarized PDF . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.3 Hadronic Production of W -boson . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.3.1 Cross Section and Kinematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.3.2 Identification of W Event - Jacobian Peak . . . . . . . . . . . . . . . . . . . . . . .
1.3.3 Charge Ratio . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.3.4 Single Spin Asymmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.4 Organization of Thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3
6
6
8
9
9
10
14
14
17
20
23
28

2 RHIC PHENIX
2.1 RHIC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.1.1 Polarized Proton Source . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.1.2 Polarized Proton Accelerator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.1.3 Polarimeters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2 PHENIX . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2.1 Beam-Beam Counters (BBC) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2.2 Zero-Degree Calorimeters (ZDC) . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2.3 PHENIX Tracking System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2.4 Electro-Magnetic Calorimeters (EMCal) . . . . . . . . . . . . . . . . . . . . . . . .
2.2.5 Trigger . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2.6 The Data Acquisition System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

29
29
32
33
34
38
41
42
45
50
55
57

3 RHIC Run 9
3.1 Luminosity and Delivered Integrated Luminosity . . . . . . . . . . . . . . . . . . . . . . .
3.2 Beam Polarization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.3 Absolute BBC Cross Section . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

59
59
59
61

4 Analysis
4.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2 Calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2.1 Energy Reconstruction by EMCal . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2.2 EMCal Energy Calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2.3 EMCal Tof Calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2.4 Momentum Reconstruction by DC and PC . . . . . . . . . . . . . . . . . . . . . .
4.2.5 DC/PC Calibration for Charge Sign Determination . . . . . . . . . . . . . . . . . .
4.3 Cross Section . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.3.1 Integrated Luminosity at PHENIX . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.3.2 Event Selection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.3.3 Background Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

64
64
66
66
67
73
74
76
81
82
85
88

1

4.4

4.3.4 Trigger Efficiency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.3.5 Efficiency of Geometry and Analysis Cuts . . . . . . . . . . . . . . . . . . . . . . .
4.3.6 Energy Smearing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.3.7 Cross Section Calculation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Spin Asymmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.4.1 Isolation Cut . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.4.2 Calculation of Spin Asymmetry with Spin Fit Technique . . . . . . . . . . . . . . .
4.4.3 Single Spin Asymmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

95
96
99
100
102
103
106
110

5 Results and Discussion
113
5.1 Cross Section of W → eν Decay . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
5.1.1 Charge Ratio of W production . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
5.2 Comparison of AL to various Polarized PDFs . . . . . . . . . . . . . . . . . . . . . . . . . 118
5.3 Future Prospects of Single Spin Asymmetry of W Production . . . . . . . . . . . . . . . . 120
6 Conclusion

123

Acknowledgment

125

A Fragmentation Function

126

B Fast Monte Carlo (FastMC) Simulation

129

C Luminosity Corrections for Pile-up and Accidentals

131

D Tracking Performance134

E Contamination from Residual Transverse Polarization
138
E.1 Residual Transverse Polarization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138
E.2 Contamination to Measured AL . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
F Spin Fit Quality Check

141

Reference1482

Chapter 1

Introduction
The proton is one of fundamental particles which comprise the universe, and a variety of researches havebeen carried out to understand its properties. Our recognition is that every matter can be constructedfrom a single atomic nucleus with a distinct number of protons and neutrons, surrounded by a cloud ofelectrons. We never imagined the proton had substructure before 1950s.
The situation has changed in the late 1950s to 1960s with the construction of large particle accelerators.
A number of ”elementary” particles, called hadrons, were produced by experiments, and their propertiesare very similar to the nucleons (protons and neutrons). It suggests that the proton is only a part ofhadrons, and we began to suspect the proton also consists of more elementary components.
In 1964, Gell-Mann and Zweig independently proposed a scheme to categorize many observed hadrons[1], in which hadrons are composed of smaller particles; quarks. All hadrons are described as combinationof quarks, and its quantum numbers are calculated from those of quarks which form the hadron. Baryons
including the proton and the neutron consist of three quarks, and mesons, such as pions and kaons, consistof a pair of quark and anti-quark. This simple model (constituent quark model) has been developed basedon flavor-SU(3) symmetry of the light quarks; u, d and s quarks. For instance, the proton is describedby uud quarks. Although the flavor-SU(3) is not an exact symmetry, it is approximately satisfied due tosmall difference in mass of these three quarks. Because of this feature, the model successfully categorizes
hadrons, and also explains various physics observables of hadrons, such as mass and magnetic moment.
The constituent quark model seemed to achieve successful outcomes.
Since late 1960s to 1970s, the existence of structure within the proton was established by experimentsof electron-proton Deep Inelastic Scattering (DIS) through γ ∗ exchange. However, the proton structure
revealed by the DIS experiments is more complicate than the static constituent quarks. The results
show that about a half the momentum of the proton can be described by the charged components,
quarks, but neutral particles have to be introduced to explain another half of the momentum. The
neutral particles are gluons. In the proton, gluons are created by dynamic processes; quarks radiate or
absorb gluons, a gluon converts into a quark anti-quark pair, and a quark anti-quark pair annihilates
into gluons. Such dynamical quarks and gluons in the proton (or hadrons) are called ”partons” (partonmodel). The theoretical framework to explain the interactions between quarks and gluons was developed
3

based on color-SU(3) symmetry and Quantum Chromo Dynamics (QCD), a theory of strong interaction,
was established as a part of the Standard Model of the modern physics. In the parton model, partonsare categorized into three groups; ”valence quarks” which carry the quantum numbers of hadrons, ”seaquarks” which are quark and anti-quark pairs generated by pair creation in the hadron vacuum, and”gluons” which are the mediators of the strong force. It was considered that quantum numbers (such as
charge, spin, magnetic moment and mass) of the proton could be explained in terms of valence quark inthe model.
Although some properties of the proton are explained by valence quarks, it was revealed by polarized DIS experiments that quarks and anti-quarks in the proton carry only ∼ 25 % of the proton spin
[2, 3, 4, 5]. Combining with the data from β-decay of baryon octet [6], these results indicate that sea
quarks are negatively polarized to cancel out the polarization of valence quarks. This picture contradicts
to our expectation that quantum numbers can be explained by valence quarks, and various efforts tounderstand the proton spin structure are stimulated. Because the DIS measurements only involve γ ∗exchange, they only provide sensitivity to the combined contributions of quarks and anti-quarks summed
over all flavors, and cannot provide information on the flavor-separated polarized quark and anti-quark
contributions. In particular, the anti-quark polarizations cannot be determined by the DIS. As the polarized DIS results indicate sea quarks (anti-quarks) also contribute the proton spin, direct measurement of
individual polarized anti-quark distributions is an important task to clarify the overall picture of the proton spin structure. More generally, it leads to our deeper understanding of the vacuum of QCD because
the anti-quarks are dynamically created from the vacuum inside the proton.
Semi-Inclusive DIS (SIDIS) measurements [7, 8, 9] are one approach to probe a separated quark andanti-quark contribution to the proton spin. The method of SIDIS is almost same as DIS except that partof the final state hadrons are also identified in SIDIS. This method combines information from the proton
and the neutron (or deuteron) targets, and uses correlations in the fragmentation process between thetype of observed hadron and the flavor of its originating parton. The fragmentation process is quantified interms of fragmentation functions. However, the dependence on the details of the fragmentation process
limits the accuracy of this method, and we still have a vague picture of the individual polarizations¯
especially on the light anti-quarks (ū, d).
An alternative and more direct way is to utilize the weak interaction. The features of the weakinteraction are that it violates the parity conservation and couples to the weak charge which is highly
correlated with the flavor. The latter characteristic is useful to select quark flavor in a reaction. Relativistic Heavy Ion Collider (RHIC) is a polarized proton-proton (pp) collider located at Brookhaven National
Laboratory. At RHIC, direct sensitivity to the polarization of u, ū, d, and d¯ quarks in the proton becomes
available using the production of W bosons [10, 11]. W bosons only couple to the left-handed quarks(qL ) and right-handed anti-quarks (q̄R ), i.e. uL d¯R → W + and dL ūR → W − , so the asymmetry of theW yield from flipping the helicity of a polarized proton is sensitive to the flavor separated quark and4anti-quark contribution. The asymmetry is called single spin asymmetry and defined as:AWL =dσ+ − dσ−,
dσ+ + dσ−(1.1)

where subscripts of + and − on dσ indicate the helicity states of proton beam. As W production inpolarized pp collisions can observe the pure weak interaction, parity is maximally violated and large AW
L
is expected. This parity-violating single spin asymmetry of W production provides us direct accesses¯ d¯ 1 . The merits of this measurement are that the production
to individual ∆u/u, ∆ū/ū, ∆d/d and ∆d/of W bosons occur at a scale of W boson mass (MW ∼ 80 GeV/c2 ) where reliable perturbative-QCD
calculations are possible and it is free from uncertainties in fragmentation functions when detecting
leptons from W decays [12, 13].
In addition to the polarized sector of quark distribution functions, the W production at RHIC also
has potential to constrain unpolarized quark distribution functions. In pp collision, the charge ratio of¯
unpolarized W + and W − cross sections (R = dσ(W + → l+ )/dσ(W − → l− )) will directly probe the d/ūratio [14, 15, 16], while W production in pp̄ collision involves much larger valence components and has
sensitivity to d/u ratio [17, 18, 19]. Isospin dependence in Drell-Yan (DY) production of muon pairs in
pp, pd scattering [20, 21, 22, 23, 24], violation of the Gottfried Sum Rule (GSR) [25, 26], and SIDIS
measurements [27] have shown a strong breaking of SU(2) symmetry in the anti-quark sea, with the ratio¯ rising to 1.6 or higher. A distinct advantage of extracting the d/ū¯ ratio from W boson production ind/ū
pp collision is that no correction for the nuclear effect in the deuteron and no assumption on the validityof charge symmetry (i.e. up = dn , un = dp , ūp = d¯n etc.) is required. This is in contrast to the DY
experiments and the Gottfried-sum measurement, which require nuclear binding corrections on the effectin the deuteron and the assumption of charge symmetry to relate the neutron with the proton parton
distributions.
As described above, the production of W bosons at RHIC have rich aspects to probe both polarized andunpolarized anti-quark distributions, therefore provide us deeper understanding of the proton structure.
This probe is quite unique on the point that rich and complicated structure of the QCD can be directly
accessed by means of the well-understood weak interaction. Moreover, combining results from otheraccelerators such as Tevatron and LHC, W measurement at RHIC extends our knowledge of the W
production in wide kinematic range, therefore it will benefit new particle searches, e.g. Higgs boson or
SUSY (super symmetry) particle search utilizing W -boson production, which are now ongoing at LHC.
In this thesis, the first observations of W -boson production in polarized pp collisions was performed√
with the PHENIX detector for s = 500 GeV at RHIC. The parity-violating single spin asymmetry of
W production is also measured for the first time. The data are from polarized pp → e± + X over a
pseudo-rapidity range of |η| < 0.35, where the e± with transverse momentum of pT > 30 GeV/c mainly
come from W and Z decays. The results, obtained from the data accumulated during RHIC Year-2009
run, are reported.
1 Here, q and ∆q represent the unpolarized and polarized quark distribution in the proton, respectively

5In this chapter, the parton model and parton distribution function (PDF) is introduced in Sec. 1.1.
The current experimental knowledges about the PDF is shown in Sec. 1.2. In Sec. 1.3, the details of
W boson production is described with expected charge ratio and single spin asymmetries at RHIC. The
organization of this thesis follows in Sec. 1.4.

1.1

Parton Distribution Function (PDF)

1.1.1

Unpolarized PDF

The structure of the proton has been studied with DIS experiments. The detailed discussion on theformalism of DIS can be found in [28]. DIS is a high-energy inelastic scattering between leptons andnucleons. Here we take ep DIS as an example:e(E, ~k) + P (M, 0) → e(E 0 , k~0 ) + X,

(1.2)

where kµ = (E, ~k) is the four-vector of the incident electron, and kµ0 = (E 0 , k~0 ) represents that of thescattered electron. In a similar way, Pµ = (M, 0) shows the four-vector of the target proton, where M
denotes the proton mass, and the final state hadron(s) is written as X. Figure 1.1 displays the diagramof ep inelastic scattering.

Figure 1.1: Diagram of ep inelastic scattering. The Figure 1.2: Diagram of ep inelastic scattering. Ittarget proton is destructed by the scattering.
is assumed that the electron beam is scattered bya single parton in the proton. Interaction between
partons is ignored in this approximation.
Bjorken scaling variable x (also called ”Bjorken x”) is defined as:
x≡

Q2
−q 2
=
,
2P · q2P · q(1.3)

where Q2 is defined as Q2 = −q 2 so that Q2 > 0. It can be interpreted as the fractional momentum ofparton as described later. A Lorentz invariant variable ν is introduced as ν = P · q/M which is equal to6E − E 0 in the laboratory frame. Another Lorentz invariant variable y is defined as y = P · q/P · k and isequal to ν/E in the laboratory frame.
The exact calculation of cross section is not possible due to the lack of the knowledge on the structureof the proton, but the structure can be parametrized with structure functions since the form is restricted
by QCD. The unpolarized cross section is written as:
d2 σ
4πα2
=
[(1 − y)F2 (x, Q2 ) + xy 2 F1 (x, Q2 )],
dxdQ2
xQ4

(1.4)

where F1 and F2 are the structure functions.
The formalism introduced so far does not depend on the parton model. Here, the process is interpreted
in the parton model. In the parton model, the structure of the proton is specified by a set of ”parton
distribution functions” (PDFs).
It can be shown that the Bjorken x (Eq. 1.3) is interpreted as the fractional longitudinal momentumof the participating parton when the Q2 is high enough to neglect the proton and the parton masses 2 .
Let ξP be the momentum of the parton, so that ξ is the fractional momentum of the parton. Since theparton and the proton masses are neglected:(ξP + q)2

= 0

2ξP · q + q 2

= 0

ξ

=

−q 2
.
2P · q(1.5)

Therefore, ξ = x and the Bjorken x can be interpreted as the fractional momentum of parton.
In the parton model, the (unpolarized) PDF is represented with q(x), and q(x)dx is defined as theprobability that quark with flavor q and momentum fraction x to x + dx is observed in the scattering.
Similarly, PDF for the gluon in the proton is defined as g(x) and PDF for anti-quark is q̄(x). By definition,
sum of the parton momentum results in the proton momentum:#
Z 1 "Xx(q(x) + q̄(x)) + g(x) dx = 1.
0

(1.6)

q

In addition, as the quantum number of the proton is carried by uud constituent quarks, following equations
are satisfied:
Z 1

Z 1[u(x) − ū(x)]dx = 2,

0



¯
d(x) − d(x)dx = 1,0

Z 1[s(x) − s̄(x)] dx = 0.

(1.7)

0

Figure 1.2 shows the parton model interpretation of DIS, where a parton with momentum fractionof x interact with the virtual photon. The cross section can be calculated as an incoherent sum ofparton-photon scatterings, and the structure functions F1 and F2 are identified as:
F1 (x, Q2 ) =1
1X 2
F2 (x, Q2 ) =e q(x),
2x
2 q,q̄ q

2 Initial transverse momentum of the parton, which is called the intrinsic k

7

(1.8)

T , is also neglected in the parton model.

where eq is electric charge of the parton in unit of the electron charge. The relationship between F1and F2 is called ”Callan-Gross relationship”, and is a consequence of scattering from spin 1/2 partons[29]. The relation is confirmed by experiments at SLAC [30]. The equation also indicates that F1 (x, Q2 )and F2 (x, Q2 ) are independent of Q2 . The scaling behavior was derived for Q2 → ∞ at fixed x (DIS
limit), and is called ”Bjorken scaling”. In fact, the scaling is proved by measurements except for small
Q2 dependence. The violation of the Bjorken scaling does not indicate substructure of the parton, but it
is caused by higher order effect such as interaction between partons.

1.1.2

Polarized PDF

To describe spin-dependent reaction, unpolarized PDF needs to be extended to spin-dependent PDF.
h
Spin-dependent PDF is defined as qH(x), where H and h denote helicity of the proton and that of the+parton in the proton. In similar way to unpolarized PDF, q+(x)dx means the probability that in the

scattering one observes the quark with flavor q, momentum fraction x to x + dx and positive helicity inthe proton with positive helicity. It is same for the gluon. It is easy to derive equations:+−q+(x) = q−
(x),

−
+
q+
(x) = q−
(x),

(1.9)

by Parity invariance. Obviously, unpolarized PDF is obtained by combining spin-dependent PDFs as+−
q(x) = q+(x) + q+(x). Instead of directly discussing spin-dependent PDF, we often use the difference ofspin-dependent PDFs:+−∆q(x) ≡ q+(x) − q+(x).

(1.10)∆q(x) is referred to as polarized PDF in this thesis. Based on the fact that the proton spin is 1/2, sumof contributions from spin and orbital angular momentum of the parton in the proton should reconstruct
the proton spin:
1=
2#
Z 1" X1(∆q(x) + ∆q̄(x)) + ∆g(x) dx + L,
2 q
0

(1.11)

where L indicates contribution of the orbital angular momentum carried by quarks and gluons.
At polarized sector, it is also possible to relate structure functions to PDFs. The cross sections forpolarized DIS are parametrized as:
d2 ∆σk
dxdQ2

≡
=

→
→
d2 σ⇐
d2 σ ⇒
−
dxdQ2
dxdQ2
2
2
16πα y
y
y
2y
[(1
+
−
γ
)g1 (x, Q2 ) − γ 2 g2 (x, Q2 )],
4
Q
2
4
2

(1.12)

where ⇒ and ⇐ denote the nucleon helicity state, → and ← denote helicity state of the incident lepton,
γ = 4M 2 x2 /Q2 , and g1 and g2 are the polarized structure functions. The polarized structure functions
are found to be:
g1 (x, Q2 ) =1X 2
e ∆q(x),2 q,q̄ q

g2 (x, Q2 ) = 0,

(1.13)

in the zeroth order parton model. g2 (x) was measured by several groups [31, 32] and g2 = 0 holds
approximately.
8

1.2

Current Experimental Knowledge on PDF

It is difficult to theoretically calculate x dependence of PDFs from the first principle. PDFs have been
extracted from several types of experiments so far, including DIS, pp̄ collision, and Drell-Yan production in
hadron-hadron collisions. Since any particular experiment covers a limited range of x and Q2 , fixed by thecenter of mass energy, measurements from a variety of experiments are combined into ”global analyses”
that attempt to extract the distributions for all partons in the proton or the neutron simultaneously. To
extract PDFs from these huge amount of data points 3 , it is necessary to adopt a certain theoretical model
to fit the data. As described above, the Bjorken scaling is violated and PDFs depend on Q2 because ofhigher order effect such as interaction between partons. The Q2 evolution of PDF follows the DGLAP
equation obtained by QCD [33, 34, 35]. Therefore, in global analysis, model of PDF for the analysis isparameterized as a function of Bjorken x at fiducial Q2 , which is usually around a few GeV2 , and the Q2
dependence of the model is calculated based on the DGLAP equation. Though the parameterization isdifferent in analysis groups, the number of the parameters is roughly 10 to 20.
It is important to note about the charge (isospin) symmetry between the proton and the neutron.
The symmetry relates PDFs of the neutron to those of the proton:up (x) = dn (x),dp (x) = un (x),sp (x) = sn (x),(1.14)

where p and n at subscript of PDF denote the proton and the neutron, respectively. The similar relation
is obtained for anti-quark PDFs and polarized PDFs. To extract structure function of the neutron, DIS
experiment with deuteron target is utilized, assuming the deuteron as the proton and the neutron which
are bound very weakly and almost independent.

1.2.1

Unpolarized PDF

Since DIS experiments can measure only structure function of F2 which is sum of PDFs (Eq. 1.8), it ishard to divide F2 into PDFs of each flavor. To disentangle F2 , several kinds of experiments are utilized
in addition to DIS with the proton or the deuteron target. DIS with neutrino beam [36] is useful because
charged weak current (W boson coupling) selects quark flavor. The reaction with negatively charged
lepton detected in the final state selects s, d or ū quarks. Similarly, W boson production in collision of
proton and anti-proton (pp̄) [17, 18, 19] is used for the flavor separation of u and d quarks. PDF of c quark
is tagged by Semi-Inclusive DIS (SIDIS) with D meson detected in the final state [37, 38, 39, 40, 41].
Lepton pair production (Drell-Yan process) in pp or pd (proton-deuteron) collision [20, 21, 22, 23, 24]
is available to distinguish PDFs of ū and d¯ since lepton pair is produced by annihilation of the quark
and the anti-quark. Measurement of inclusive jet production in pp̄ collision [42, 43, 44, 45] contributes todetermination of gluon PDF, especially at higher x region (x >∼ 0.01). DIS measurements with wide Q2
range also contribute to constrain on gluon PDF through the Q2 evolution at small x region (x <∼ 0.1).
Table 1.1 summarizes a rough indication of particular PDFs that the various data constrain [46].
3 Number of data points for unpolarized PDF analysis is ∼ 2600 while it for polarized PDF analysis is ∼ 500.

9

Processl± {p, n} → l± X
l± n/p → l± X
pp → µ+ µ− X
pn/pp → µ+ µ− X
ν(ν̄)N → µ− (µ+ )X
νN → µ− µ+ X
ν̄N → µ+ µ− X
e± p → e± X
e+ p → ν̄Xe± p → e± cc̄X
e± p → jet + X
pp̄ → jet + X
pp̄ → (W ± → l± ν)Xpp̄ → (Z → l+ l− )X

Subprocessγ∗q → qγ ∗ d/u → d/uuū, dd¯ → γ ∗¯(ud)/(uū)
→ γ∗
∗
0
W q→q
W ∗s → c
W ∗ s̄ → c̄γ∗q → qW + {d, s} → {u, c}γ ∗ c → c, γ ∗ g → cc̄
γ ∗ g → q q̄gg, qg, qq → 2jets
ud → W, ūd¯ → Wuu, dd → Z

Partonsq, q̄, g
d/u
q̄
¯
d/ū
q, q̄
s
s̄
g, q, q̄
d, s
c, g
g
g, q
u, d, ū, d¯dx rangex >∼ 0.01
x >∼ 0.01
0.015 <∼ x <∼ 0.35
0.015 <∼ x <∼ 0.35
0.01 <∼ x <∼ 0.5
0.01 <∼ x <∼ 0.2
0.01 <∼ x <∼ 0.2
0.0001 <∼ x <∼ 0.1
x >∼ 0.01
0.0001 <∼ x <∼ 0.01
0.01 <∼ x <∼ 0.1
0.01 <∼ x <∼ 0.5
x >∼ 0.05
x >∼ 0.05

Table 1.1: The main processes included in the current global PDF analysis ordered in three groups: fixedtarget experiments, HERA and the Tevatron. An indication of their dominant partonic subprocesses, theprimary partons which are probed and the approximate range of x constrained by the data are shown[46].

There are several groups extracting PDF from the experimental data [47, 48, 49, 50, 46, 51]. Theirresults show reasonable agreement each other. Figure 1.3 shows the MSTW2008 results of PDFs at NLO
at Q2 = 10 and 104 GeV2 [46]. The PDFs of u and d have a peak around x ∼ 0.1 to 0.2, since they
are the valence quark flavors 4 . The rises in low x of u, d, ū and d¯ distributions, which is originated fromthe coupling to gluon evolution, cancel for valence distribution. The PDFs of s(= s̄) and c(= c̄) aresuppressed compared to other flavors (u, d) because of their larger mass.

1.2.2

Polarized PDF

There are some predictions about integral of PDF over x, which is called ”sum rule”. For example,
some trivial sum rules are Eq. 1.6 and 1.7, in addition to Eq. 1.11. The most fundamental sum rule forpolarized PDF is derived by Bjorken [52, 53] based on the charge (isospin) symmetry of the nucleon:


Z 1
gA
1 gA
αs (Q2 )pn+ ... ,
(g1 (x) − g1 (x))dx =1−= 1.2695 ± 0.0029,(1.15)
6
g
π
g
V
V
0
where g1p and g1n are polarized structure functions for the proton and the neutron, respectively (seeEq. 1.13 for g1 ). gV and gA are the vector and the axial-vector weak coupling constants of the neutron
β-decay and determined precisely by experiments [6]. The term of 1 − αs /π + ... in the right hand side
is from higher order correction of QCD. This Bjorken sum rule is well verified by measurements [4, 54].
By assuming flavor SU(3) symmetry for the spin 1/2 baryon octet, the β-decay of hyperons can be
described by two parameters, F and D [55, 56]. At the same time, certain combinations of integral of
the polarized PDF is related to F and D as follows:
Z 1


gA
¯
(∆u(x) + ∆ū(x)) − (∆d(x) + ∆d(x))
dx = F + D =,gV
0
4 The valence quarks are u − ū and d − d¯ to be exact.

10(1.16)

Figure 1.3: The MSTW 2008 NLO PDFs at Q2 = 10 and 104 GeV2 [46].

Z 1



¯
(∆u(x) + ∆ū(x)) + (∆d(x) + ∆d(x))− 2(∆s(x) + ∆s̄(x)) dx = 3F − D = 0.586 ± 0.031.

(1.17)

0Note that these linear combinations of PDF is independent of Q2 in up to NLO. Equation 1.16 is identicalto the Bjorken sum rule of Eq. 1.15. Because the charge symmetry is included in the flavor SU(3)symmetry, the Bjorken sum rule also appears in this model. The right-hand side of Eq. 1.17 is evaluatedusing measured gA /gV for hyperon decays (Λ → p, Ξ → Λ and Σ → n) [6], where F and D is fixed by
Eq. 1.16.
In global analysis of polarized PDFs, some constraints, which is generally tighter than the unpolarizedcase due to limited experimental precision, are often imposed depending on analysis groups. Whatis frequently utilized are sum rules of Eq. 1.15, 1.16 and 1.17. Symmetric sea quark distributions,¯ = ∆s̄(x) = ∆s(x), are also often assumed.
∆ū(x) = ∆d(x)Following pioneering polarized DIS experiments at SLAC [57, 58, 59, 60] to measure g1 , EMC (European Muon Collaboration) at CERN measured g1 in x range of 0.01 to 0.7 with polarized muon beamand extracted integral of g1 [2, 3]. When only three light-flavor quarks, u, d and s, are taken into account, integral of each polarized PDF of the quarks are extracted by assuming Eq. 1.16 and 1.17 basedon the flavor SU(3) symmetry. The EMC results were interpreted as quarks and anti-quarks carry only
∼10 % of the proton spin, even though the uncertainty is large. This conclusion is a contrast to that11

valence quarks carry quantum numbers of hadrons. Many polarized DIS and SIDIS experiments havebeen performed to confirm the EMC results (see references in [61, 62, 63, 64, 65, 66, 67, 68]), andthe contribution of the quarks and the anti-quarks to the proton spin obtained by the recent analyses[61, 62, 63, 64, 65, 66, 67, 68] is 20 − 35 %. However, compared with unpolarized experiments, precisionof the polarized experimental data is still limited and the measured range of Q2 and x is much smaller.
This means the uncertainties of polarized PDFs remain much larger than unpolarized PDFs.
Figure 1.4 shows the result of a global analysis including both inclusive DIS and SIDIS measurements[64]. While the sum of quark and anti-quark distributions, separately shown for u and d quarks, is well
¯ are unconstrained. The uncertainties of anti-quark distributions
known, anti-quark distributions (ū, d)
are caused by two different sets of fragmentation functions, which is used in SIDIS measurements, referred
to as KRE [69] and KKP [70]. It is known that KRE and KKP sets of fragmentation functions havesignificant differences for the gluon fragmentation function and also non-negligible differences for flavor
separation [64] (see Appendix A for the detail of fragmentation function). While the KRE fit favors
SU(3) symmetric sea, KKP fit finds the polarization of ū is opposite to d¯ and s̄ in this global analysis.
Figure 1.5 displays other result from a global analysis [67, 68]. The difference from the Fig 1.4 isthat this global analysis includes ALL (double-spin asymmetry) of π 0 and jet production in pp collision
at RHIC [71, 72] as well as more data points from SIDIS experiments. Since gluon polarization isconstrained by those ALL measurements, ∆g in the figure is rather smaller than Fig. 1.4 in the range of
0.05 < x < 0.2. Thanks to more data from SIDIS, anti-quark distributions are also much constrained
than Fig. 1.4. However, the uncertainty is still large because of the uncertainty of fragmentation function,and the result of this global analysis still provides an argument that further experimental work is needed¯
to improve the accuracy of anti-quark polarized PDFs (∆ū, ∆d).
Measurement of single spin asymmetry of W production in polarized pp collision is one of unique ways
which can put more constraints on anti-quark polarized PDFs. Determining flavor separated polarized
PDFs is very important task to reveal overall picture of the proton spin structure, and it leads to our
deeper understanding of QCD. The detail of W measurement with pp collider is described in the next
section.

12

Figure 1.4: Parton PDFs from global analysis, named DNS, at Q2 = 10 GeV2 along with uncertainty
bands corresponding to ∆χ2 = 1 and ∆χ2 /χ2 = 2 % [64].

Figure 1.5: Parton PDFs from global analysis, named DSSV, at Q2 = 10 GeV2 along with uncertainty
bands corresponding to ∆χ2 = 1 and ∆χ2 /χ2 = 2 % [67]. The result from DNS, GRSV global analysis[64, 62] are also indicated in the figure.
13

1.3

Hadronic Production of W -boson

Our goal is to explore the details behind the small total quark and anti-quark spin contribution to theproton spin, which is currently determined by (SI)DIS, using W boson production. Highlights of the W
measurement at RHIC are the sensitivity to ∆ū from W − asymmetries in the backward rapidity region
and the similarly large sensitivity to the ∆d¯ from W + asymmetry in the central and forward rapidity
region. In addition, it has been recognized that charge ratio of W ± boson production at RHIC provides
unique and clean test of an SU(2) symmetry breaking in the anti-quark sea by measuring the unpolarized¯ ratio in the protons.
d/ūThe basic aspects of W production at hadron collider, charge ratio and parity-violating single spin
asymmetries are explained in the following subsections.

1.3.1

Cross Section and Kinematics

W bosons in pp colliders are produced by hard scatters between the quarks which are inside the protons.
Protons consist of partons, which are quarks and gluons as discussed in previous section. Figure 1.6
shows a schematic diagram of the leading-order (LO) W production process with a hard scatter between
a quark and a anti-quark. These two quarks couple to a W , and the W subsequently decays into a
lepton and a neutrino. The other partons in the protons are spectators to the event, and they form the”underlying event”.

Figure 1.6: Schematic diagram of W production at pp collisions. The process is divided into two parts:
PDFs (qa , qb ) and partonic cross section (σ̂).

14In order to calculate the cross section for any process at a hadron collider, it is necessary to convolutethe partonic cross section with the PDFs within the proton. The differential cross section of W productionin pp collisions is written as:Zdσ(pp → W X) =dx1 dx2

Xqa (x1 , Q2 )qb (x2 , Q2 )dσ̂(ab → W ),(1.18)

a,b

where the sum is over all possible partons; dσ̂(ab → W ) is the cross section for parton a and parton b tocreate W boson. q(x, Q2 ) represents PDF, and x1 , x2 are the fractional momenta carried by the partons2in the colliding hadron pair. Q2 is chosen renormalization and factorization scale, and it is MW(MW :mass of W boson) in W production.
In this framework, W production is divided into two parts: PDFs and partonic (or subprocess)cross sections. The treatment is called the ”factorization theorem” [73]. Factorization theorem plays animportant role as it ensures that the partonic cross section, dσ̂(ab → W ), only depends on the parton
species a, b, and does not depend on the choice of the initial state hadron (proton), nor the choice of the
final state. The partonic cross sections can be calculated with perturbative QCD. All the non-perturbative
phenomena are carried by PDFs and the theorem ensures that PDFs are universal. The same PDFs canbe used for both pp collisions and ep DIS.
At LO, the differential cross section for W + production can be written as:
√
2π
x1 x2
dσ
+
¯ 2 ) + d(x¯ 1 )u(x2 )}(W ) = K
GF ()[cos2 θc {u(x1 )d(xdxF
3
x1 + x2+ sin2 θc {u(x1 )s̄(x2 ) + s̄(x1 )u(x2 )}],

(1.19)

where u(x), d(x), and s(x) represent the up, down, and strange quark distribution functions in the
hadrons, and xF = x1 − x2 . GF is Fermi coupling constant and θc is the Cabbibo angle. The factor K
takes into account the contributions from first-order QCD corrections:
K ∼1+

8π
αs (Q2 ).
9(1.20)

2
In Eq. 1.19, the u(x), d(x) and s(x) are all evaluated at Q2 = MW
, and αs ∼ 0.1158 and K ∼ 1.323 atthe scale. It indicates that QCD processes are relatively unimportant for W production. An analogous
expression for W − production cross section is given as:
√
dσ
2π
x1 x2
−
(W ) = K
GF ()[cos2 θc {ū(x1 )d(x2 ) + d(x1 )ū(x2 )}dxF
3
x1 + x2+ sin2 θc {ū(x1 )s(x2 ) + s(x1 )ū(x2 )}].

(1.21)

From momentum and energy conservation, we can derive relations between x values and the rapidity of
the W boson (yW ) as:

where

√

MW
x1 = √ eyW ,sMWx2 = √ e−yW ,s(1.22)s denotes the center mass of energy of the colliding protons.

In a hadron (pp) collider, W bosons are identified primarily from W → µν or eν leptonic decays. Thisis because W → q q¯0 hadronic decay is usually buried inside a large QCD background (pp → jets), as are15the τ s from the W → τ ν process. Approximately 10 % of the W bosons decay into an electron and aneutrino.
The LO W boson production mechanism results in the W boson being polarized by means of the V −A
structure of the weak interaction as shown in Fig. 1.7. The top panel shows the helicity configurationof the incoming quark and anti-quark. The lower panel displays the preferred direction of e± quoting
the scattering angle θ∗ in the W center-of-mass system measured with respect to the positive z axis.
The V − A structure means that the weak current couples only to left-handed u and d quarks (orto right-handed ū and d¯ quarks). For ultra-relativistic quarks, helicity 5 and chirality (handedness) areapproximately equivalent, and this results in full polarization of the produced W bosons. The W leptonic
decay process also couples only to left-handed e− and right-handed ν̄ (or right-handed e+ and left-handed
ν). The conservation of angular momentum favors a decay with the final state lepton at a small angle withrespect to the initial state quark direction (and a similar small angle between the initial state anti-quarkand final anti-lepton). As a result, the W → lν decay distribution has angle dependence of:dσW →l− ν+d(cos θ∗ )∝ (1 ± cos θ∗ )2 .

(1.23)

Figure 1.7: Helicity configuration of W + (Left) and W − (Right) production showing on top the helicity
configuration of the incoming quark and anti-quark. The lower panel displays the preferred direction ofe± quoting the scattering angle θ∗ in the W center-of-mass system measured with respect to the positivez axis.
It is necessary to relate the lepton kinematics to yW , so that one can assign the momentum fraction(x1 , x2 ) of the quark or anti-quark through Eq. 1.22. Only then would it be possible to translate themeasured single spin asymmetry or charge ratio into a determination of the quark or anti-quark PDFs.
The rapidity of the W (yW ) is related to the lepton rapidity in the W rest frame (yl∗ ) and in the labframe (yl ) by:yl = yW + yl∗ ,where yl∗ =
1
1 + cos θ∗ln.
2
1 − cos θ∗

5 Helicity is the projection of the spin onto the direction of momentum.

16(1.24)

The term of cos θ∗ can be determined from the transverse momentum (pT ) of the lepton at LO 6 by:
pT = p∗T =MWsin θ∗ .
2(1.25)In this reconstruction, however, the transverse momentum of the W boson (QT ) is ignored. In reality,
it has a QT , resulting from higher-order contributions such as q q¯0 → W ± g and qg → W ± q 0 , or fromtransverse momentum of initial-state partons. Even though the W mass sets a large Q2 scale so that the
coupling constant (αs ) is not large, the corrections certainly need to be known for a reliable theoreticalextraction of PDFs from data. Therefore, the Monte-Carlo like code ”CHE” (standing for ”Collisions at
High Energies”) [12], which implements the Next-to-Leading-Order (NLO) calculation, is used for reliablepredictions of single spin asymmetry of W production. The CHE provides access to the full kinematicsof the final state particles, allowing the computation of any infrared-safe observable in hadronic collisions
and the implementation of realistic experimental cuts such as lepton pT cut. It is worth noticing thatthe same code can compute the unpolarized, the single polarized and the double polarized cross sections.
Figure 1.8 shows the correlation between the averages of the momentum fractions, hx1,2 i, and yl
computed by CHE for W production (Left: W − , Right: W + ) [12]. A remarkably strong correlation isfound between hx1,2 i and yl in both cases. Large negative lepton rapidity corresponds to small (large)
momentum fractions x1 (x2 ). The opposite occurs for large positive rapidities. As a rough approximation,
one can parameterize these correlations by the simple formulas:MW
hx2 i ∼ √ e−yl /2 .
sMW
hx1 i ∼ √ eyl /2 ,s

(1.26)

Since RHIC experiments (PHENIX and STAR) cover the rapidity range of |yl | ∼< 2, one can expect
sensitivity to the polarized quark and anti-quark distributions in the region 0.05 < x < 0.4.

1.3.2

Identification of W Event - Jacobian PeakW → eν is the decay mode which we use to measure the W production in this thesis. The neutrino
passes through the detector without interacting. The electron, on the other hand, leaves a track in the
tracking chamber, and also deposits its energy in the calorimeters that surround the interaction region at
PHENIX. Usually W production is identified by isolated charged leptons with large pT and large missing
transverse energy, due to the undetected neutrino. However, we only measure pT of decay leptons in thisthesis because our detector (PHENIX) is not hermetic and measurement of missing pT is not available.
Since the angle dependence of W → e cross section is given by Eq. 1.23, the differential cross section
is written as:

dσW →e− ν+d(cos θ∗ )3
= σW →e− (1 ± cos θ∗ )2 .
+ 8

(1.27)The transverse momentum (pT ) of the e± at LO is shown in Eq. 1.25 and its squared value is:
p2T =

2
MW
M2
sin2 θ∗ = W (1 − cos2 θ∗ ).
4
4

6 Note that there is an irreducible uncertainty of the sign17

(1.28)

Figure 1.8: Averages of the momentum fractions x1 , x2 as functions of the lepton rapidity for W − (Left)and W + production (Right) at RHIC [12].

Therefore, the pT dependence of the cross section can be calculated as:
dσ
dp2T

=

2
dσ
2 cos θ ∗ d(cos θ ∗ )MW=

1 + cos2 θ∗3
σW →e 2
4
MW cos θ∗=2
3 σW →e (1 − 2p2T /MW
)
p
.
2
2
2
2 MW
1 − 4pT /MW(1.29)

Here, the term of cos θ∗ in numerator of the second line is ignored because pT is symmetric againstθ∗ ↔ π − θ∗ . As we can notice from the last line, the pT distribution has a singularity at pT = MW /2
and this is called ”Jacobian peak” which is the feature of two body decay. If W boson does not have a
QT (transverse momentum) and ΓW = 0 (ΓW : mass width of W boson), the pT distribution would havethe singularity. In practice, however, the observed distribution is with integration over the momentum
distributions of interacting partons and that results in smeared pT distribution with finite peak.
Figure 1.9 displays the Jacobian peak of W → eν decay (Left) and its real distribution from UA2
experiment (Right) [74]. The event of W → eν decay can be identified by measuring high pT e± s, and therequirement adopted in this thesis is pT > 30 GeV/c. Note that the PHENIX cannot distinguish betweene± s from W decays and them from Z decays due to its limited acceptance. Therefore, the contaminationby e± s from Z decays are unavoidable in this thesis.

18

Figure 1.9: (a): Ideal Jacobian peak of ud¯ → W + → e+ ν reaction. (b): Smeared Jacobian peak measured
at UA2 experiment [74].

191.3.3Charge Ratio

A variety of models have been proposed to describe the sea quarks in the nucleon. The earliest models
have assumed SU(3) flavor symmetry of sea quarks in the nucleon. However, it has already been shown by
neutrino-induced charm (c) production experiments that the s quark contribution to the nucleon is not aslarge as u and d sea quarks [75]. This asymmetry can be attributed to the large mass difference betweens quark and u and d quarks. For the light u and d quarks, it was still possible that their distributionsin the nucleon sea are symmetric. To test the flavor symmetry of the u and d quarks in the nucleon sea,
one can measure the Gottfried sum [76]:
Z 1
Z
1 2 1
SG =
[F2p (x) − F2n (x)]/xdx = +[ūp (x) − d¯p (x)]dx,
3 3 00(1.30)

where F2p and F2n are the proton and neutron structure functions measured in DIS experiments. The
second step in Eq. 1.30 follows from the assumption of the charge symmetry (Eq. 1.14: up (x) =dn (x), dp (x) = un (x), and ūp (x) = d¯n (x)). With the assumption of a symmetric sea, ūp = d¯p , SGequals 1/3 (Gottfried Sum Rule: GSR). The most accurate test of the GSR is the result from the New
Muon Collaboration (NMC), which measured F2p and F2n over the region of 0.004 ≤ x ≤ 0.8 with muon
DIS [25, 26]. They determined the SG to be 0.235 ± 0.026, significantly below 1/3. This result provided¯
the first strong evidence that ū(x) 6= d(x).
¯ ratios as a function of x were measured using other experiFollowing the NMC measurement, the d/ū
mental techniques. These new measurements include the NA51 and E866 experiments with the Drell-Yan¯ asymmetry is clearly(DY) process [21, 22, 23, 24] and the HERMES experiment with SIDIS [27]. The d/ūestablished from these experiments.
¯
Various theoretical models, such as meson-cloud model etc., have been proposed to explain the d/ū¯
asymmetry, which are reviewed in [77, 78]. While these models can describe the general trend of the d/ū¯ data at large x (x > 0.2), where the E866 data
asymmetry, they all have difficulties explaining the d/ūsuggest a rapid fall-off of this ratio. Therefore, it is very important to have new measurements sensitive to¯ ratios at x > 0.2 in order to better determine mechanisms which generate the flavor asymmetric
the d/ū
nucleon sea. The upcoming Fermilab E906 Drell-Yan experiment plans to extend the measurement tolarger x region [79].
¯ asymmetry is available.
With the pp colliders at RHIC, an independent technique to study the d/ū¯ asymmetry can
By measuring the ratio of W + versus W − production in unpolarized pp collision, the d/ūbe determined with some distinct advantages over the existing methods [14, 15, 16]. The advantages are:¯
• Not necessary to assume the charge symmetry: All existing experimental results of d/ū
asymmetry depend on the comparison of the scattering cross sections of hydrogen and deuterium
targets. It is pointed out that the violation of the GSR and the results of DY experiments can becaused by charge symmetry violation as well as by flavor asymmetry of the nucleon sea [80, 81]. A
comparison between W production in pp collision with the DY experiments would disentangle theflavor asymmetry from the charge symmetry violation effects.
20• Free from any nuclear effect: The nuclear modification of parton distributions should be taken
into account for DIS and DY process involving deuterium targets [82, 83, 84, 85]. The nuclear
shadowing effect for deuteron at small x could lead to a 4 % to 10 % decrease in the evaluationof the SG by the NMC [82, 85]. Moreover, the nucleon Fermi motion at large x also affects theextraction of neutron structure function and would cause additional uncertainty in the evaluationof the SG [83]. The nuclear effects and the associated uncertainty are absent in W production in
pp production.
¯ flavor asymmetry at a Q2 scale of ∼• Large Q2 scale: The W production is sensitive to d/ū6400 GeV2 /c2 , significantly larger than all existing measurements. This offers the opportunity to
examine the QCD evolution of the sea-quark flavor asymmetry.
¯ ratio is the ratio of the xF distributions for W + and W −An observable directly related to the d/ū
production. When ignoring the smaller contribution from the strange quarks, the ratio can be derived
from Eq. 1.19 and 1.21:
R(xF ) =¯ 2 ) + d(x¯ 1 )u(x2 )dσ/dxF (W + )u(x1 )d(x=.
−dσ/dxF (W )ū(x1 )d(x2 ) + d(x1 )ū(x2 )(1.31)

For pp collision, it is evident that R(xF ) is symmetric with respect to xF = 0 (R(xF ) = R(−xF )). At¯ 1 )u(x2 ) and ū(x1 )d(x2 ) are negligible, thus we have:large xF (i.e. x1  x2 ), the terms of d(xR(xF  0) ∼¯ 2)
u(x1 ) d(x.
d(x1 ) ū(x2 )(1.32)¯u(x) d(x).
d(x) ū(x)(1.33)At xF = 0, where x1 = x2 = x, one obtains:
R(xF = 0) =As the u(x)/d(x) ratios are already well known by experiments at Tevatron [17, 18, 19], a measurement¯
of R(xF ) in pp collision gives an accurate determination of the ratio d(x)/ū(x)at x of ∼ 0.05 to 0.16 (seeFig. 1.8).
It is not the xF distributions of the W which are measured in practice but the charged leptons fromthe decay of the W bosons. The measured lepton ratio is defined as:
R(yl ) =dσ/dyl (W + → l+ ).
dσ/dyl (W − → l− )(1.34)The differential cross section dσ/dyl is obtained by convoluting the q q¯0 → W cross section for each xF
with the relevant W → lν decay distribution (Eq. 1.23).
Figure 1.10 shows the predicted lepton ratios R(yl ) calculated for four different PDFs [86]. The first
PDF used here is MRS S0’ [87]. It assumes symmetric ū and d¯ distributions. The other three PDFs allow
flavor asymmetry in nucleon sea. New experimental data from DY measurement by E866 Collaboration
is included in the global analysis performed by CTEQ6 [49], GJR08 [50] and MSTW2008 [46]. Therefore
R(yl ) for those three PDF are similar at yl = 0 (xF ∼ 0, x1 ∼ x2 ∼ 0.16) and are significantly higher
21

than the MRS S0’ case. Considering the acceptance of PHENIX experiment, which covers |η| < 0.35 incentral rapidity, Fig. 1.10 shows that a measurement of R(yl ) at PHENIX central rapidity region is able
to distinguish flavor symmetric and flavor asymmetric nucleon sea.

Figure1.10: Prediction of the charge ratio R(yl ) as a function of lepton rapidity for pp collision at√
s = 500 GeV using various PDFs [86].

221.3.4Single Spin Asymmetry

W production in polarized pp collisions at RHIC can observe the pure weak interaction and generate large
parity-violating single spin asymmetries (AL ). AL of W production will provide the first direct (flavor
separated) measurement of the polarized PDF of quark and anti-quark especially for u and d quarks¯ Past measurements in (SI)DIS were carried out with significant
in the proton (∆u, ∆ū, ∆d, and ∆d).
experimental precision only for the u quark polarized PDF (∆u). The W measurements can complete
the knowledge of quark and anti-quark polarizations.
The standard model production of W bosons proceeds through a pure V − A interaction. Thus, thehelicity of the incoming quark and anti-quark is fixed. In addition, the W boson couples to a weak chargethat correlates directly to flavors. The production of W bosons in pp collisions is dominated by u, d, ū¯ with some contamination from s, c, s̄, and c̄, mostly through quark mixing. Therefore, W bosonand d,
production is an ideal tool to study the spin-flavor structure of the nucleon. The LO production of a W +boson, ud¯ → W + , with polarized pp collision is illustrated in Fig. 1.11.
The parity-violating single spin asymmetry is defined by:AL =dσ+ − dσ−,
dσ+ + dσ−(1.35)

where subscripts of + and − on dσ indicate the helicity states of proton beam. At RHIC, one candetermine this asymmetry from either polarized beam by summing over the helicity states of the otherbeam.
For example, if the production of the W + proceeded only through the diagram in Fig. 1.11 (a), thesingle spin asymmetry would directly equal the longitudinal polarization asymmetry of the u quark in the
proton. When ignoring the smaller contribution from the strange quarks, the AL can be derived using
Eq. 1.19 as:+

AW
L (xF ) =−¯¯u−∆u(x1 )+ (x1 )d(x2 ) − u− (x1 )d(x2 )−−¯ 2 ) + u (x1 )d(x¯ 2 ) = − u(x1 ) .
u+ (x1 )d(x−(1.36)The second step in this equation follows from Eq. 1.9. Similarly, for Fig. 1.11 (b) alone,
¯+
¯ 1)
d¯+
+
∆d(x
+ (x1 )u(x2 ) − d− (x1 )u(x2 )
AW
(x
)
=
= ¯
.
F
L
+
+
¯
¯
d(x1 )
d+ (x1 )u(x2 ) + d− (x1 )u(x2 )(1.37)

In general, the asymmetry is a superposition of the two cases:+AW
L (xF ) =¯ 2 ) + ∆d(x¯ 1 )u(x2 )−∆u(x1 )d(x.
¯¯u(x1 )d(x2 ) + d(x1 )u(x2 )(1.38)

The asymmetry for W − can be obtained by interchanging u and d.
By identifying the rapidity of the W (yW ) 7 relative to the polarized proton, we can obtain direct
measures at LO of the quark and anti-quark polarizations, which is separated by quark flavor: as thequark (anti-quark) distribution is large (small) in large xF limit (i.e. x1  x2 ), the asymmetry will bedominated by the quark distribution and give direct access to −∆u(x1 )/u(x1 ). Likewise, for small xF
7 (x , x ) can be derived by y1
2
W

through Eq. 1.22.

23

Figure 1.11: Production of W + bosons in p~p collisions [10]. (a) ∆u is probed. (b) ∆d d is probed. The
proton on the top in each panel is taken to be the polarized proton, with either positive (left) or negative(right) helicity.

¯ 2 ) is negligible and the asymmetry is given by ∆d(x¯ 1 )/d(x¯ 1 ). The
limit (x1  x2 ), the term of u(x1 )d(xasymmetry of W − production is the direct probe for ∆d/d and ∆ū/ū with similar calculation.
This picture is valid for the predominant production of W bosons at QT ∼ 0. The experimental
difficulty is that the W boson is observed through its leptonic decay W → lν, and only the charged
lepton is observed. The four momenta of the W boson cannot be determined from the momenta of
its decay products. Therefore, the rapidity of the W boson cannot be measured directly. While it iscertainly valuable to determine the asymmetry, AL (xF ), it has been shown that the directly measurable
asymmetries in terms of the observed lepton, AL (yl ), is a viable alternative to the asymmetry of AL (xF )[13].
Experimentally, AL (yl ) is measured by the difference of leptons from W boson decays between positive
and negative helicity of protons in a particular rapidity region (N + (l), N − (l)), divided by the sum, and24

normalized by the beam polarization (P ):AL (yl ) =1 N + (l) − N − (l)·.
P N + (l) + N − (l)(1.39)

One can expect that, at least to some extent, the relation between momentum fractions (x1 , x2 ) and
rapidity of the W boson (yW ) will be inherited by the lepton. The correlation is already shown in
Fig. 1.8, and it indicates the combinations of PDFs predominantly probed by AL (yl ) will vary withyl . However, the underlying structure of the weak interactions also enters here. For W − production,neglecting all partonic processes but the dominant ūd → W − → e− ν one, the asymmetry at LO is given
by:

R
−

AeL (yl ) =whereR⊗(x1 ,x2 )⊗(x1 ,x2 )R[∆ū(x1 )d(x2 )(1 − cos θ∗ )2 − ∆d(x1 )ū(x2 )(1 + cos θ∗ )2 ]⊗(x1 ,x2 )[ū(x1 )d(x2 )(1 − cos θ∗ )2 + d(x1 )ū(x2 )(1 + cos θ∗ )2 ],(1.40)

denotes an appropriate convolution over momentum fractions, and θ∗ is the polar angleof the lepton in the W rest frame, with θ∗ > 0 in the forward direction of the polarized proton. Notethat θ∗ itself depends on the momentum fractions and the rapidity of lepton (see Eq. 1.24). At large
negative yl , one has x2  x1 and θ∗ ∼ π. In this case, the first terms in the numerator and denominatorof Eq. 1.40 strongly dominate, because the combination of parton distributions, ∆ū(x1 )d(x2 ), and theangular factor, (1 − cos θ∗ )2 , dominate over the second term. Therefore, the asymmetry provides a clean
probe of ∆ū(x1 )/ū(x1 ) at medium x1 values (x1 ∼ 0.05). With similar reason, the second terms in the
numerator and denominator of Eq. 1.40 dominate at forward rapidity of yl  0, and one can access to∆d(x1 )/d(x1 ) at relatively high x1 (x1 ∼ 0.4).
For the W + production channel, the single spin asymmetry is written as:R¯ 1 )u(x2 )(1 + cos θ∗ )2 − ∆u(x1 )d(x¯ 2 )(1 − cos θ∗ )2 ][∆d(x⊗(x ,x )e+
AL (yl ) = R 1 2¯ 1 )u(x2 )(1 + cos θ∗ )2 + u(x1 )d(x¯ 2 )(1 − cos θ∗ )2 ] .
[d(x(1.41)⊗(x1 ,x2 )Here, the distinction of the two contributions by considering large negative or positive lepton rapidities¯ 1 )u(x2 ) will dominate(yl ) is less clear than the case of W − . For example, the partonic combination ∆d(xat negative yl (yl  0), but θ∗ ∼ π at the same time so that the angular factor (1+cos θ∗ )2 become small.
¯ 2 ) is suppressed by the angular factor at positive yl
Likewise, the dominant combination of ∆u(x1 )d(xbecause θ∗ ∼ 0. Therefore, both terms in Eq. 1.41 will compete for all yl of interest.
Figure 1.12 shows AL (yl ) at RHIC calculated using the CHE MC program [12]. This framework allows
the prediction of the AL for various polarized PDFs of quark and anti-quark, and different curves in the
figures correspond to the different sets of polarized PDFs of DSSV [67, 68], GRSV [62] and DNS [64]:the ”standard” and ”valence” sets from GRSV (”GRSV std” and ”GRSV val”) have SU(2) symmetricand broken sea distributions, respectively. The ”DNS kre” and ”DNS kkp” sets correspond to fits tothe same data of DIS and SIDIS, but obtained using different sets of fragmentation functions [69, 70] toanalyze the SIDIS data. In the figure, a clear discrimination to the underlying PDF is observed in the
backward region (yl < 0) of W − production, which is caused by the uncertainty of ∆ū. Moreover, thiskinematic region allows to connect the detected lepton back to the W production kinematics (x1 , x2 ), and25

the sea quark polarization. The asymmetry becomes large and positive at large yl , which reflects the factthat ∆d(x) remains negative at high x for all sets of polarized PDFs considered here (see Fig. 1.4). For
W + production, the sensitivity is similar in the forward and central region (yl ≥ 0), which is originated¯ Overall, the asymmetry is negative because of the contribution from ∆u infrom the uncertainty of ∆d.
Eq. 1.41, which is known to be positive from (SI)DIS measurements.
In addition to the contribution from W exchange, the CHE code can also compute the backgroundarising from Z-boson and/or photon exchange at the same accuracy in perturbative QCD. The contribution from photon exchange, q q̄ → γ ∗ g followed by γ ∗ → l+ l− , may generate large contributions at NLO
when the high-transverse momentum photon splits almost collinearly into the lepton pair, producing
high-pT leptons with a very low invariant mass. A proper treatment of this background would require theaddition of a fragmentation contribution which is based on parton-to-dilepton fragmentation functions[88].
Figure 1.13 displays the asymmetries calculated with including leptons produced by Z/γ exchange[12]. As expected, the inclusion of ”background” leptons results in a reduction of the asymmetry due tothe increase of the unpolarized cross section. Because the PHENIX detector cannot distinguish electrons
from W /Z decay, this figure shows the prediction of AL for the PHENIX. Even though the asymmetryis smaller than the asymmetry of pure W sample, it still has a large band in backward region of negative
charge and in central and forward region of positive charge. The distinct AL corresponds to the different
polarized PDFs, which all describe the (SI)DIS data, due to the differences in the sea quark polarizations.
As the acceptance of the PHENIX covers |η| < 0.35 in central rapidity, −2.2 < η < −1.1 and 1.1 < η < 2.4
in forward rapidities, the measurement of AL of high pT leptons, which mainly come from W /Z decays,
at PHENIX will significantly improve the knowledge of the quark polarizations, and enable the first direct
determination of the anti-quark polarizations when it is used as an input to a global analysis together withthe (SI)DIS data. It will be possible to determine, first at a lower luminosity, whether the anti-quarks
are polarized, and at an increased luminosity, whether there is a symmetry between the polarizations ofū and d¯ in the proton or not.

26

−

+

−

+

Figure 1.12: Rapidity dependence of the NLO single spin asymmetries AeL for electrons and Ae+ for
positrons at RHIC, for the various sets of polarized parton distribution functions. Only leptons produced
by W ± boson exchange are considered [12].

Figure 1.13: Rapidity dependence of the NLO single spin asymmetries AeL for electrons and Ae+ for
positrons at RHIC, for the various sets of polarized parton distribution functions. Leptons produced by
W ± and Z boson exchange are considered [12].

27

1.4

Organization of Thesis

As discussed in the previous section, measurement of parity-violating single spin asymmetry of W boson
production in polarized pp collision at RHIC is crucial for further understanding of proton spin structure.
The measurement of cross section and charge ratio of W production in spin-averaged collisions also
provides a sensitive test of current unpolarized PDFs, as well as it is an important confirmation of
the theoretical understanding of the W production. For these purposes, production of W boson in pp√
collisions at s = 500 GeV at RHIC Year-2009 run has been studied via electrons from their leptonic
decays in the central rapidity region of the PHENIX detector (|η| < 0.35) at RHIC. The Year-2009 run√
was the first time to have pp collisions with s of 500 GeV at RHIC, and also the first opportunity tomeasure W bosons.
In analysis procedure, firstly, the pT distribution of inclusive electron-candidate particles is measured.
Then the background from photons and charged hadrons are estimated using both real data and MC
simulation. The remnant includes electrons from both W and Z decay, because the PHENIX cannot
distinguish them due to its limited acceptance. The cross section and charge ratio of W → eν is extrapolated from the measured distribution based on NLO (Next-to-Leading Order) and NNLO (Next
Next-to-Leading Order) perturbative QCD calculations. Finally, single spin asymmetry of W + Z production is evaluated and compared to theory predictions. The results obtained in this thesis is the firstmeasurement of parity-violating single spin asymmetry of W + Z production as well as it is the firstobservation of W + Z boson production in pp collision.
The organization of this thesis is as follows: Chapter 2 introduce the RHIC accelerator complex and
the PHENIX detectors. In Chapter 3, the conditions of RHIC Year-2009 run are summarized. Then, in
Chapter 4, the analysis of measurement of the electrons from W and Z decays is explained. The analysis
to evaluate the single spin asymmetry of W + Z production is also described here. In Chapter 5, resultsof the measurement and the comparison to theoretical calculations are shown. The future prospects ofmeasurements of charge ratio and single spin asymmetry of W production with expected constraint on
PDFs are also presented in this chapter. Chapter 6 is the conclusion of this thesis.

28

Chapter 2

RHIC PHENIX
This thesis is based on the data which were taken at the RHIC (Relativistic Heavy Ion Collider) with the
PHENIX detector at BNL (Brookhaven National Laboratory) in the United States during the Year-2009
run (called Run 9). RHIC Run 9 was the first time of the polarized pp collision at CMS (Center of Mass√
System) energy of s = 500 GeV. The detail of Run 9 is described in Chapter 3.
The data analyzed in this thesis are collected with the PHENIX detector using its two central arm
spectrometers. Each spectrometer covers |η| < 0.35 in pseudo-rapidity and ∆φ = π/2 in azimuth in anearly back-to-back configuration. The capability of charged particle tracking and electron identificationis necessary to measure single electrons from W or Z decay. The arms include drift chambers (DC), padchambers (PC) for electron tracking, an electromagnetic calorimeter (EMCal) for energy measurement,
and trigger electronics (ERTLL1). Beam-beam counters (BBCs), positioned at pseudo-rapidity 3.1 <|η| < 3.9, measure the position of the collision vertex along the beam (zvertex ) and provide the interaction
trigger (BBCLL1). In this chapter, the RHIC and the details of PHENIX detectors are presented.

2.1

RHIC

RHIC can accelerate and collide polarized proton beams for the first time in the world, which provides
us unique opportunity to study the spin property of proton through strong and weak interactions. The
spin structure of proton has been studied with DIS (Deep Inelastic Scattering) experiment where theinteractions are mediated by virtual photons. Gluons interact at leading order in pp collisions while
gluons only participate at higher order in DIS. Therefore, pp collisions are a good probe for the gluon
spin contribution to the proton. The W -boson production in polarized pp collisions provide informationon the flavor separation of the quark spin contribution. RHIC can accelerate polarized protons up to an√energy of 250 GeV which results in collisions at s = 500 GeV with design luminosity of 2×1032 cm−2 s−1 .
Figure 2.1 displays a schematic of RHIC accelerator complex. The polarized proton beam is produced
at optically-pumped polarized ion source (called OPPIS, see Sec. 2.1.1) 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. The protons are accelerated by a LINAC (Linear Accelerator) to a kinetic energy of29

200 MeV. It is injected into a Booster, and is accelerated up to 1.5 GeV. Then it is transferred to AGS(Alternating Gradient Synchrotron) and accelerated up to 24.3 GeV. It is injected into two independentrings at RHIC, via the AGS-to-RHIC transfer line and is accelerated up to 250 GeV. Each beam travels
in opposite direction and collides at the IPs (Interaction Points). Two independent beams are called theBlue (clockwise) and the Yellow (counter-clockwise) beams. RHIC has six IPs and they are referred toas IP12, IP2, IP4, IP6, IP8, and IP10 as shown in Fig. 2.1. Once RHIC is filled with beams, the beams
are kept circulating in the rings to provide collisions at the IPs. When the luminosity becomes too low,the beams are dumped and refilled. The sequence from injection to dump of the beam is called a fill.
One fill typically lasts ∼8 hours.
The beam in RHIC has bunch structure and each ring contains 120 bunches of polarized proton beam,with a time interval of 106 nsec. One of the remarkable features of RHIC is that the spin pattern of thebunches is arbitrary by freely flipping the spin of the protons at OPPIS. Figure 2.2 shows an example of aspin pattern assignment. The Blue beam has a spin pattern ” + + − −” while the Yellow beam has a spin
pattern ” + − + −”. In this way, all possible spin combinations (++, + −, − +, − −) can be obtained.
This feature greatly reduced systematic uncertainty which comes from time dependence of the detector
responses. The structure of the sequence of filled and unfilled bunches help to confirm the bunch IDs
which are sent from the accelerator control system to the experiments. The exact bunch identification is
crucial for precise calculation of the spin asymmetries.

30

Figure 2.1: The accelerator complex of RHIC for the polarized proton collision.

Figure 2.2: Example of the spin pattern of the beams. This spin pattern allows us to detect particles
from the collision with any helicity combinations at the same time.

31

2.1.1

Polarized Proton Source

The polarized proton beam is produced at optically-pumped polarized ion source (OPPIS) [89]. The
OPPIS technique for polarized H ion beam production was developed in the early 80’s at KEK, INR
Moscow, LAMPF and TRIUMF. Figure 2.3 displays a schematic drawing of the OPPIS, and its polarizing
scheme is illustrated in Fig. 2.4. The source of angular momentum is high-power lasers. Rb atoms
are optically pumped by titanium-sapphire lasers and electron-spin-polarized Rb atoms are produced.
H+ atoms are created with 29 GHz Electric Cyclotron Resonance (ECR) proton source. While they
pass through the Rb vapor cell, the polarized electrons are transferred from Rb to H atoms, and H+
atoms becomes electron-polarized H0 atoms. To prevent depolarization in the charge-exchange collisions,the optically pumped cell is situated inside the strong (2.5 Tesla) superconducting solenoid. Then thepolarization is transfered from electron to the H nucleus by the Sona transition [90]. Finally, electrons
are attached by the Na-jet ionizer cell and H− ions are produced.

Figure 2.3: A schematic drawing of the OPPIS.

Figure 2.4: The polarizing scheme of the OPPIS.

32

2.1.2

Polarized Proton Accelerator

The spin of the beam is oriented to the vertical direction, which is the stable direction during theacceleration in the normal synchrotron accelerator because the magnetic field to bend the beam is vertical.
However, there are many depolarizing resonances at certain beam energies, depending on parameters of
the accelerator. The ”Siberian Snake” magnets play important role to overcome the depolarization andmaintain the beam polarization [91]. Two Snakes are placed in each ring of RHIC and the location of one
Snake is opposite side of the other (3 o’clock and 9 o’clock) to avoid additional depolarizing resonancecaused by installing the single Snake. Each Snake in RHIC consists of four superconducting helical dipole
magnets of identical structure. The strength and the direction of the field are different between four
magnets (−4 to +4 Tesla), while they are independent of beam energy. By reversing the spin direction of
the beam without an orbit distortion at the entrance and the exit of the Snake, the resonance conditions
are shifted and the depolarization is avoided. Similarly, one other type of the Snake magnet is placed in
AGS. Unlike the Snake in RHIC, the AGS Snake is just one normal-conducting helical dipole magnet androtates the direction of the beam spin by only 9 degrees, which is enough to maintain the polarization in
the lower-energy synchrotron, such as AGS, with weaker depolarizing resonances. This partial rotation
forces to induce full spin flip at the beam energy on the depolarizing resonances, while it keeps spin
direction unperturbed except for such a depolarizing conditions.
It is necessary to collide beams with polarized longitudinally to measure single spin asymmetry of W
production process. To realize this, spin rotators are located before and after the interaction region of
PHENIX in each ring. The former rotates the beam spin from the vertical to the longitudinal direction, thelatter restores it to the vertical direction. Figure 2.5 shows a schematic drawing of the RHIC beams aroundthe PHENIX interaction point. It is also possible to collide beams with radial transverse polarization.
One spin rotator consists of four helical dipole magnets, whose structure is same as that for the Snakemagnet except two of them have alternate handedness of helical magnet. They are powered independently
to generate appropriate field needed for the spin rotation, which depends on the beam energy (−3 to+3 Tesla at around 100 GeV).

Figure 2.5: A schematic drawing of the RHIC beam near the PHENIX interaction point, when thelongitudinally polarized collisions are required.

33

2.1.3

Polarimeters

Three polarimeters are used to measure and monitor the beam polarization. Two in RHIC and one at
PHENIX experimental hall. Two types of polarimeters utilized in RHIC are proton-carbon polarimeter(pC polarimeter) [92], and polarized hydrogen gas jet target polarimeter (H-jet polarimeter) [93]. At
PHENIX experiment, the orientation of the beam polarization is monitored by PHENIX local polarimeter[94]. These three types of polarimeters utilize a sizable single transverse spin asymmetry AN . AN
is defined for a reaction between transversely-polarized beam and unpolarized beam (or target). It isdefined as:
AN =σlef t − σright,σlef t + σright(2.1)

where σlef t (right) is the cross section that the outgoing particle goes left (right) side when the polarization
is upward in view of the polarized beam. The measured raw asymmetry εN is εN = P AN , where P denotes
the beam polarization. Thus once the physics asymmetry AN is known, the beam polarization can becalculated as P = εN /AN . The three types of polarimeters are described in this subsection.
Proton-Carbon Polarimeter (pC Polarimeter)
The proton-carbon polarimeter (pC polarimeter) utilizes AN in the elastic scattering between polarized
proton beams and carbon target (ApC
N ) at very forward region, with four-momentum transfer of −t =(0.01 − 0.02) (GeV/c)2 . The size of ApC
N in the measured kinematic region is about 1.4 %. Due tothe small scattering angle of protons, recoil carbons are detected instead of the scattered protons. The
target should be thin for recoil carbon with small energy of 0.1 to 1 MeV to escape the target, andnot to influence on the beam. However, it is required to achieve high statistics at the same time. The
requirements are satisfied by using ultra-thin carbon ribbon target of 3 to 5 µg/cm2 with a width of
10 µm.
Figure 2.6 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 are detected by the silicondetectors placed on both sides of the target. The pC polarimeter collects ∼ 4 × 106 events per one
measurements which is typically one minute. It corresponds to a statistical uncertainty of 4 % whichis smaller compared to the systematic uncertainty (see Sec. 3.2). The pC polarimeter confirms that thebunch by bunch polarization variation is within the uncertainty of the measurements. The ApC
N was notknown at the RHIC energy and cannot be measured with the pC polarimeter system. Therefore, thepC polarimeter only provides relative variation of polarization for each fill. The H-jet target polarimeter
was utilized to obtain the absolute normalization of the polarization. The polarization from the H-jet
polarimeter were used to normalize the pC polarimeter results.
Polarized Hydrogen Gas Jet Target Polarimeter (H-jet Polarimeter)
The polarized hydrogen gas jet polarimeter (H-jet polarimeter) utilizes AN in pp elastic scattering (AppN ).
Since both beam and target are polarized, App
N can be calculated for either beam or target polarization,
34

Figure 2.6: The experimental setup of the pC polarimeter. Left: beam view of the detectors. The beamruns into the paper and hit the carbon ribbon target in the center of the beam pipe. Right: side view ofthe detectors. The beam runs from left to right. The recoil carbon is detected with the Si detectors.

by averaging target or beam polarization. The relation between measured asymmetries (εbeam , εtarget )and physics asymmetry (AppN ) is:App
N = εbeam /Pbeam = εtarget /Ptarget .

(2.2)

Here, Pbeam (Ptarget ) denotes the polarization of beam (target). Ptarget is measured by the Breit-Rabi
polarimeter. Therefore, App
N and Pbeam can be obtained from measured asymmetries. One of the beautiful
aspects in this measurement is that the physics asymmetry and the beam polarization are obtained withthe same experimental setup, which reduces systematic uncertainty. Figure 2.7 illustrates the pp elastic
scattering process (Left) and the experimental setup of the H-jet polarimeter (Right). The measured
kinematic range is −t ∼ (0.001 − 0.02) (GeV/c)2 where the asymmetry is large. The hydrogen gas jettarget crosses the RHIC beam from top to bottom at a speed of 1.6 × 103 m/sec. The density of the gas
jet target is ∼ 102 H atoms/sec. The target spin direction is vertical, and is reversed in every 10 minutes.
The recoil particle is detected with the silicon detectors which are placed on both sides of the targets.
2
Figure 2.8 displays the measured App
N as a function of −t in the range of 0.001 < |t| < 0.032 GeV/c√
at s = 13.7 GeV [93]. The App
N is about 4 % in the measured kinematic range, and the statisticaluncertainty is about 2.5 % for a single beam for the whole run. This is not enough for measurement ofpolarization variation for each fill. The pC polarimeter is used for this purpose instead, which provides
polarization with statistical uncertainty of the same level within one minute of measurement. The absolutebeam polarization obtained with H-jet target is used to calibrate the pC measurement.

35

Figure 2.7: Left: The schematics of the pp elastic scattering process. The recoil proton is observed while
the forward proton is not. Right: The experimental setup of the H-jet polarimeter. The beam runs
through the jet target. The recoil particle is detected with the Si detectors.

√
Figure 2.8: AN as a function of −t at s = 13.7 GeV. The error bars on the data points represent
statistical uncertainties. The lower band represents the total systematic uncertainties. The black line
shows the prediction for AN with the electromagnetic spin-flip, and the blue line is a fit to the data
allowing for a hadronic spin-flip contribution to AN [93].

36PHENIX Local Polarimeter
For the study of the sea quark polarization, longitudinally polarized proton collisions are necessary while
the stable polarization direction of the beam in RHIC ring is transverse as explained in Sec. 2.1.2. To
obtain longitudinally polarized proton collisions, the polarization direction is rotated from transverse
direction to longitudinal direction just before the interaction point (IP). Therefore, the polarization
orientation should be monitored at the PHENIX IP. It is done by utilizing single spin asymmetry AN
of forward neutron production in polarized pp collisions. AN vanishes when the beam polarization is
in longitudinal direction, and AN is non-zero for transversely polarized collisions (the size of AN is∼ 10 %). In the longitudinally polarized proton collisions, the residual transverse component of the beampolarization (PT ) can be determined by comparing the size of AN measured with spin rotator on andoff. Neutrons are detected with PHENIX Zero-Degree Calorimeters (ZDCs) and Shower Max Detectors(SMDs), which details are described in Sec. 2.2.2. Since the extraction of PT and PL are performed at
the PHENIX IP locally 1 , it is called ”Local Polarimeter”.
At the commissioning period for the spin rotator tuning, the data for the local polarimeter analysis iscollected with full PHENIX DAQ bandwidth 2 to know PT and PL . The spin direction is checked within
one hour and the result is provided to the Collider Accelerator Department as a feedback in order tominimize the transverse component (PT ) 3 . The local polarimeter is also used to monitor the polarization
direction during the longitudinal run period. The data for the local polarimeter analysis is collected with
100 − 200 Hz through the physics run period, and the transverse and longitudinal components throughthe run is estimated by offline analysis.

1P

T and PL show transverse and longitudinal components of polarization as P
2 Full DAQ bandwidth is ∼ 5 kHz (see Sec. 2.2.6).

2 = P 2 + P 2.
TL

3 A new method of local polarimeter utilizing scaler information from SMDs has been developed during Run 9 at PHENIX.
It enables us to obtain enough data for the feedback within five minutes [95].

37

2.2

PHENIXPHENIX [96] is one of the largest experiments at RHIC, located at the 8 o’clock intersection point (IP8).
PHENIX was designed to measure photons, leptons, and hadrons with excellent particle identification
capability and to deal with both high-multiplicity heavy-ion collisions and high event-rate pp collisions.
Figure 2.9 shows the definition of global coordinate system used in the PHENIX experiment, which
defines the geometrical center of the interaction point as the origin (0, 0, 0). Taking the beam-line as az-axis (North is positive z direction), the direction to West arm is defined as x-axis, and upward is definedas y-axis. The azimuthal angle φ is measured counter-clockwise relative to the positive x direction, andthe negative x direction is φ = π rad. The polar angle θ is defined as the angle relative to z-axis. Using
the polar angle θ, the pseudo-rapidity variable is expressed as:θ.
η = − ln tan
2

(2.3)

Figure 2.9: The PHENIX coordinate system.
The PHENIX experiment is composed of many sets of detectors. A collection of detectors with thesame type is called a subsystem. The subsystems can be categorized into three groups: global detectors,
two central arms and two muon arms. The global detectors consist of several subsystems and measure
the collision information. Two central arms, East and West arms, cover the pseudo-rapidity range of|η| < 0.35 and half in azimuthal angle (∆φ = π/2 × 2). They are designed to detect photons, electrons
and hadrons. Two muon arms, north and south arms, cover 1.2 < η < 2.4 and −2.2 < η < −1.2
respectively with a full azimuthal coverage. They are designed to detect muons. PHENIX has threemagnets: central magnet and two muon magnets. These magnets provide magnetic fields for momentum
measurement of charged particles.
Figure 2.10 displays a schematic view of the PHENIX detector setup. The global detectors are Beam
Beam Counters (BBCs), Zero Degree Calorimeters (ZDCs) which can be found in the lower panel. Shower
38

Max Detector (SMDs) are also placed inside ZDCs and are not displayed in the figure. These are usedin this thesis and is described in Sec. 2.2.1 and 2.2.2. The upper panel is a beam view of the central
arm detectors. The proton beams run perpendicular to the paper, at the center of the detectors. The
beam pipe is surrounded by the central arm detectors: Hadron Blind Detectors (HBD), Drift Chambers(DC), Pad Chambers (PC1, 2, and 3), Ring Imaging Cerenkov detectors (RICH), Aerogel Cherenkov
detectors, Time Expansion Chamber (TEC), and Electromagnetic Calorimeters (EMCal). The central
arm detectors used in this thesis are DC, PC1 and EMCal. The DC and PC1 are utilized to measure
the momentum of electrons from W decays. The details of the PHENIX tracking system at central arms
which include DC and PC are introduced in Sec. 2.2.3. The EMCal is used for the energy measurement
and the most important detector in this thesis. The details of the EMCal are described in Sec. 2.2.4. The
lower panel of the figure is a side view of the setup. The proton beams collide at the interaction point(IP), which is the center of the panel. The collision vertex distributes along with the z-axis approximately
in a Gaussian shape, with its center at z ∼ 0 and with a width of σ ∼ 60 cm. The pole piece of the central
magnet, which is called nose cone, surrounds the beam pipe for |z| > 41 cm and it limits the acceptanceof the central arms. The muon arm detectors are also shown in this panel: Muon Tracker (MuTr) andMuon Identifier (MuID), which are utilized to detect muons. The acceptance and purpose of PHENIX
subsystems which are used in this thesis are summarized in Table 2.1.
As the event rate at RHIC is too high to take and record every collision events at PHENIX, thetrigger which efficiently selects events of interest is necessary. Such trigger system is implemented to the
PHENIX, and explained in Sec. 2.2.5. At the end of this section, the PHENIX data acquisition (DAQ)
system is summarized (Sec. 2.2.6).
subsystemBBCη±(3.1 - 3.9)

φ
2π

ZDC

±2 mrad2π

DC
PC
EMCal (PbSc, PbGl)

±0.35
±0.35
±0.35

π/2 × 2π/2 × 2π/2 × 2

purposeprimary vertex detection, luminosity,
time-zero, level-1 trigger
primary vertex detection, luminosity,
level-1 trigger
charged particle detection, tracking
pattern recognition, tracking
photon and electron detection,level-1 trigger

Table 2.1: PHENIX subsystems and their acceptance and purpose.

39

Figure 2.10: The PHENIX detector. The upper panel shows a beam view of the PHENIX central arm
detectors. The lower panel shows a side view of the PHENIX global and muon arm detectors.

40

2.2.1Beam-Beam Counters (BBC)The Beam-Beam Counters (BBC) provide the information of collision point and start signal for timeof flight measurement [97]. There are two arrays of quartz Cherenkov detectors in PHENIX along thebeam line. They locate at 144 cm from the interaction point and surround the beam axis with the
10 cm of inner diameter and 30 cm of outer diameter. The pseudo-rapidity coverage is 3.0 < |η| < 3.9.
Each counter consists of 64 one-inch diameter mesh-dynode (15 step) Photo Multiplier Tubes (PMT:
Hamamatsu R6178) equipped with 3 cm quartz on the head of PMT as a Cherenkov radiator. They aresensitive to charged particles with β greater than 0.7, and BBC detects the particles such as π ± fromcollisions. The pictures of BBC are displayed in Fig. 2.11.
The start signal for timing measurement (T0 ) and the beam-beam collision point along beam axis
(zvertex ) are provided by using the average arriving time of particles between North (TN ) and SouthBBC (TS ) and their difference. They are calculated from:TN + TSL− ,2cc(TS − TN ),zvertex =
2
T0 =

(2.4)
(2.5)

where L is the half of the distance between the two BBCs (144 cm), and c is the speed of light. Whenthe collision vertex position is outside of the BBCs (|z| > 144 cm), the vertex position is reconstructed
as z = ±144 cm, where ± correspond to on either side (z > 144 or z < −144 cm) of the collision vertex.
With slewing correction, ∼ 2 cm of the position resolution is achieved in offline analysis for pp
collisions.

Figure 2.11: Pictures of BBC. Left top: The element of BBC. Left bottom: A BBC array comprising 64
BBC elements. Right: BBC after installation. The collision point is the left side of the picture, and BBC
is attached behind the central magnet.

41

2.2.2

Zero-Degree Calorimeters (ZDC)

The Zero-Degree Calorimeters (ZDCs) [98], together with Shower Max Detectors (SMDs) were equipped
to detect neutrons at very forward angle of less than 2.8 mrad. They are placed at 18 m away fromthe IP, and sit behind the DX dipole magnet as shown in Fig. 2.12. The DX magnets serve to sweep
away most of the charged particles, and neutral particles with long life, which are mainly neutrons andphotons, hit the ZDCs. Since protons which experience elastic or diffractive scatterings at the IP may
hit the beam pipes and induce showers, the resulting charged particles may hit the ZDCs. A scintillation
counter are placed in front of each ZDC for charged particle veto.

Figure 2.12: The ZDC location.Top: top view of the ZDC location.Bottom: beam view of the ZDC
location.Charged fragments are bended by the Dipole magnets towards the outside of the acceptance of
ZDC.
The ZDC consists of three ZDC modules each of which has 1.7 interaction length or 51 radiation
length of absorber. The three ZDC modules are placed in series to comprise a ZDC. Figure 2.13 shows themechanical design of a module of the ZDC. A module consists of 27 layers of tungsten absorber plates andPMMA (polymethylmethacrylate) optical fibers. A neutron generates a hadronic shower in the tungsten
plates, and charged particles in the shower emit Cerenkov radiation. The radiation is detected throughthe optical fibers with a PMT (Hamamatsu R329-02). The size of a tungsten plate is 10 cm wide, 18.7 cm
phigh and 0.5 cm thick. The energy resolution of ZDC was obtained to be δE/E ' 218/ E (GeV) %42

from the test beam results. Neutrons can be separated from photons with the energy deposit in the 2nd
module of the ZDC, since the electromagnetic showers from photons cannot penetrate the 1st modulewith 51 radiation length and do not reach the 2nd module. Taking the correlation between North andSouth ZDC also provides the background rejection due to single-beam interaction with the residual gas
in the beam pipe.
The SMD consists of scintillator hodoscopes: 7 scintillator strips with a width of 15 mm in the verticaldirection (to provide x-coordinate), and 8 strips with a width of 20 mm in the horizontal direction (toprovide y-coordinate). The SMD is placed between the 1st and the 2nd ZDC modules where the neutroninduced shower reaches its maximum. The hits of the showers at the SMD were weight-averaged toprovide the shower position. The position resolution of ∼ 1 mm is achieved for neutrons with an energyof 100 GeV.
The vertex position is reconstructed from the hit timing in the two ZDCs as in the case of BBCs, andthe position resolution is obtained to be ∼ 30 cm in online and ∼ 10 cm in offline.
The coincidence of hits in the two ZDCs defines ZDC trigger. The ZDC trigger serve as an independent
luminosity measure as well as the BBC trigger (BBCLL1), which is explained in Sec. 2.2.5. There are
two types of ZDC trigger with different vertex position cuts. One is ZDC narrow trigger which is definedwith the vertex position cut of |z| < 30 cm, and the other is ZDC wide trigger which is defined with thevertex position cut of |z| < 150 cm. Since the width of the vertex position variation is ∼ 60 cm, most of
the collisions are covered by the ZDC wide trigger. The ZDC trigger information is used in vernier scanfor estimating z dependence of the BBC efficiency (Sec. 3.3).

43

Figure 2.13: ZDC Mechanical Design. The mechanical design of ZDC. Dimensions shown are in millimeter. Top: one layer of the PMMA fibers. Bottom left: The side view of one module. Bottom right: The
beam view of one module.

442.2.3

PHENIX Tracking System

The tracking devices of PHENIX central arms reconstruct momenta of charged particles from theirbending angle in the magnetic field. The tracking system consists of the central magnet, DC (Drift
Chamber) and PC (Pad Chamber). In this subsection, the details about the tracking system of the
PHENIX central arm are described.
Central Magnet
PHENIX Central Magnet provides the axial field ofR

B · dl = 0.78 T·m at θ = π/2 rad. It is used todetermine the momentum of charged particles using magnetic bending. The Central Magnet is energizedby two, inner and outer, pairs of concentric coils, which can be operated separately, with the samepolarities or opposite polarities. In Run 9 period, the same polarity operation was chosen to maximallybend high pT electrons from W decay, so that the charge sign of e± can be separated. The operation iscalled as ”++ mode” according to the polarity of the magnetic field. The magnetic field produced by themagnets is shown in Fig. 2.14. The pole faces of the magnet, which is called nose cone, are positioned
at ±45 cm in z direction covering the rapidity range of ±0.35. The nose cone also serves as the hadron
absorbers for the muon spectrometers.
Drift Chambers (DC)The Drift Chambers (DC) are used to measure the charged particle trajectories in r-φ plane and toprovide the high resolution momentum determination.
The DC is located in the East and West arm at a radial distance of 2.02 < R < 2.46 m, respectively.
One of them is the mirror copy of each other, and each DC covers 90 degrees in azimuth and 1.8 m along
the z-direction which corresponds to |η| < 0.35 (see Fig. 2.10). They are filled with the gas mixture of
50 % Argon and 50 % Ethane.
The schematic drawing of the DC frame is shown in Fig. 2.15. Each DC consists of 20 sectors, each ofwhich covers 4.5 degrees in azimuth. In each sector, there are six types of wire modules stacked radially.
They are called X1, U1, V1, X2, U2, V2 plane. The sketch of a sector and the layout of wire position are
shown in Fig. 2.16.
The wires in X1 and X2 plane run in parallel to the beam axis in order to perform the track measurements in r-φ plane. They are followed by two sets of small angle U, V wire planes. U1, V1, U2, and V2
wires have stereo angle of about 6 degrees relative to the X wires in order to measure the z-coordinateof the track. The X wire modules contain 12 sense (anode) planes and 4 cathode planes. Both U and V
wire modules contain 4 sense (anode) planes and 4 cathode planes. They form the cells with a 2 ∼ 2.5 cm
drift space in φ direction. In this scheme, 40 drift cells are located at different radii in the DC frame.
The DC system contains roughly 6,500 anode wires (∼ (12 × 2 + 4 × 4) × 4 × 40). Each wires are
separated into two halves by the Kapton support at z = 0, and the signals are independently extracted.
Thus, the number of total readout channel is 13,000. The anode wires are separated by Potential (P) wires

45

PH ENIX

-4.0

-2.0

0.0

2.0

4.0

Z (m)

Magnetic field lines for the two Central Magnet coils in combined (++) mode
Figure 2.14: The magnetic field lines of the Central Magnet and Muon Magnets shown on a verticalcutaway drawing of the PHENIX magnets. The beams travel along the r = 0-axis in this figure andcollide at r = z = 0. Arrows indicate the field direction.

and surrounded by Gate (G) and Back (B) wires. The P wires form a strong electric field and separate
sensitive regions of individual anode wires. The G wires limit the track sample length to roughly 3 mm
and terminate the unwanted drift line. The B wire has a rather low potential and terminates most of
the drift lines from side. With a 50-50 mixture of argon-ethane gas, the stable drift velocity plateau at
5.3 cm/µsec is achieved for the field gradation from 800 V cm up to 1.4 kV cm. Therefore, the maximum
drift time in a cell is approximately 470 nsec (= 2.5/5.3 µsec). The position resolution was measured tobe 1.1 mrad along the φ-direction.

46

Figure 2.15: The construction of a DC frame.

Figure 2.16: Left: The layout of wire position within one sector of the DC and inside the anode plane.
Right: A schematic diagram of the stereo wire orientation.
47

Pad Chambers (PC)The Pad Chambers (PC) is used to determine the space points which are used for momentum determination in the z-direction (pz ). The PCs are the multi-wire proportional chambers with cathode readoutthat form three separate layers (PC1, PC2 and PC3) in the central arms. The PC1 layer is innermost
chamber located between DC and RICH, occupying 2.47 m through 2.52 m in radial distance from theinteraction point, the PC2 layer is placed behind RICH occupying 4.15 m through 4.21 m, and the PC3
is located in front of EMCal occupying 4.91 m through 4.98 m. The PC1 is essential for determination of
the three dimensional momentum by providing the z-coordinate at the exit of the DC. The combinationof the DC and the PC1 information provides the direction vector through RICH. The PC2 and the PC3
are needed to resolve ambiguities in outer detectors where about 30 % of the particle striking the EMCal
are produced by either secondary interactions or the particle decays outside the aperture of the DC andthe PC1.
Each detector contains a single plane of wires inside a gas volume bounded by two cathode planes asshown in Fig. 2.17. The gas was chosen to be the mixture of 50 % Argon and 50 % Ethane at atmosphericpressure. One cathode is finely segmented into an array of pixels, and a cell area of 8.4 × 8.4 mm2 wasadopted from the requirement of good position resolution in z-direction and a low occupancy even in thehigh track multiplicities. The position resolution was measured to be 1.7 mm for the PC1 along the wire(z-direction).
When a charged particle starts an avalanche on an anode wire, the charge induced on a number ofpixels is read out through specially designed readout electronics. Figure 2.18 shows the schematic diagramof the PC. In order to reduce the amount of electric and other noise, each cell of the PC contains threepixels and an avalanche must be sensed by all three pixels to form a valid hit in the cell (Fig. 2.18 (a)).
Because of huge electronic channels with this arrangement, the interleaved pixels are gathered together
as shown in Fig. 2.18 (b). Nine pixels are connected to a group and to a common readout channel, suchthat the three pixels in a cell are always connected to different but neighbor channels and each cell isdefined by its unique channel triplet (Fig. 2.18 (c), (d)). This solution saves a factor of nine in readout
channels compared to readout of every pixel.

48

Figure 2.17: The vertical cut through a PC.

Figure 2.18: The schematic diagram of the PC. (a) Each cell is divided into three pixels. (b) Nine pixels
are electrically connected to form one pad. (c) Pixels are connected to provide pads shifted each other.
(d) The layers of the pads are virtually produced. The signals from three overlapped pads identify the
hit in the corresponding cell.

49

2.2.4

Electro-Magnetic Calorimeters (EMCal)

The primary role of the Electro-Magnetic Calorimeters (EMCal) is to provide a measurements of energy
and hit position of both photons and electrons [99, 100]. The EMCal covers central rapidity (|η| < 0.35)
and half in azimuthal angle (∆φ = π/2 × 2). Two kinds of EMCal are installed in PHENIX as shown in
Fig. 2.10. One is Lead-Scintillator calorimeter (PbSc) and the other is Lead-Glass calorimeter (PbGl).
PHENIX has 8 sectors of EMCal, 6 of them are PbSc type, and other two are PbGl type. The PbGl
occupies the lower two sectors of East Arm (named E1, E2 sector in this thesis) and the PbSc occupies
other six sectors (named W0 - W3, E2, E3 sector). The EMCal surface is placed at 510 cm (PbSc) and550 cm (PbGl) in radial distance from the interaction point. The total numbers of towers are 15552 for
PbSc and 9216 for PbGl, and the other basic parameters of EMCal are summarized in Table 2.2. A
detailed description of each type of the EMcal is explained in this subsection.
parameters
Radiation length (X0 )Moliere radius
Nuclear interaction length (λI )Total η coverage
Total φ coverage
Size of sector surface
Number of towers in one sector
Tower cross section
Tower η × φ coverage
Tower depth

PbSc
2.1 cm
∼ 3.0 cm
44 cm
0.7
π/2 + π/44(wide)×2(high) m2
72(wide)×36(high)
5.2 × 5.2 cm20.011 × 0.011
37.5 cm (18 X0 , 0.85 λI )PbGl
2.8 cm
3.7 cm
38 cm
0.7
π/4
4(wide)×2(high) m2
96(wide)×48(high)
4.0 × 4.0 cm2
0.008 × 0.008
40.0 cm (14 X0 , 1.05 λI )Table 2.2: Basic parameters of two types of PHENIX EMCal.

Lead-Scintillator Calorimeter (PbSc)
The PbSc is a sampling calorimeter which is composed of 65 lead tiles and 66 scintillator tiles, stacked inalternate way. The thickness of lead tile (scintillator tile) is 1.5 mm (4.0 mm). The scintillator is madeof polystyrene (the bulk material), 1.5 % of p-Terphenyl (the primary fluorescent material) and 0.01 %of POPOP (wavelength shifting material).
A PbSc module consists of four towers (2 × 2), which are optically isolated and read out individually.
Figure 2.19 shows the internal view of the module. A module has 64 holes for the read-out fibers to passthrough, with 1.2 mm diameter and 9.27 mm spacing. The read-out fibers are made of wave length shifter(0.5 % POPOP) which pass through the entire module from the back side to the front side then returnto the back side after following smooth curves. The both edges of the fibers are gathered and attached to
1 inch diameter PMT (FEU115M, MELS, Russia). The attenuation length of the fiber is approximately
1 m, which affects the linearity of the energy measurement.
The four edges of the scintillator tiles are coated by aluminum to reflect the scintillation light except
one corner at the center. For calibration, a fiber is inserted in the center of the module and provides laser

50light into four towers from the corners. Figure 2.20 shows a schematic drawing of the laser calibration
system [99]. The laser light is split in three steps and delivered into 3888 modules (= 15552/4); its
amplitude is monitored with a PMT and photo diodes in all the light splitters. This calibration system
is to normalize the gain change of the PMT due to the operation conditions. The gain of the amplifier
for the photo diodes is also monitored by test pulses.
The energy resolutions were evaluated as follows from the test experiments using electron beams:
∆E
8.1 %
=p
⊕ 2.1 %,
E
E(GeV)(2.6)

where E is energy and ⊕ means quadratic sum. The 8.1 % in the first term of Eq. 2.6 is close to the
expected resolution from sampling as predicted by GEANT simulation. The main contributors to theconstant term are intrinsic non-uniformity, in particular tower boundaries, hot spots at fiber positions
and shower depth fluctuations. There is also a loss in the calorimeter response when a particle hits thecorner of the towers (shower leakage). Shower depth fluctuations are responsible for the variations in theamount of the light seen, and in the energy leakage via the front and the back surface of the calorimeter.
The position resolution depends on both energy and the incident angle of the beam (θ: a polar angle
orthogonal to the front surface of the module) due to the fluctuation of the shower depth. It is expressedas:
5.7 mm,∆x = ∆x0 ⊕ L sin θ, ∆x0 = 1.55 mm ⊕ p
E(GeV)(2.7)

where L is ∼ X0 (= 2.1 cm). These results are reproduced by GEANT simulation.
At the same test experiments, the response to the proton and charged pions with momentum of a few
GeV/c was studied. Only a part of the total energy is deposited in PbSc because of the small nuclear
interaction length. MIP (minimum ionizing particle) energy for the charged pions penetrating the tower
was measured to be 270 MeV.

51

Figure 2.19: The module of PbSc. One module consists of 4 towers. The fiber for the laser system isinserted at the center of the module and along the tower axis.

Figure 2.20: Schematic drawing of laser calibration system for PbSc.

52

Lead-glass Calorimeters (PbGl)
The PbGl is a Cherenkov calorimeter with 1.648 of index of refraction. The modules, which were previously used in WA98 [101] experiment at CERN, was re-used at PHENIX.
Each PbGl sector comprises 192 super-modules (SM) in an array of 16 (wide) by 12 (high). Figure 2.21
shows a PbGl SM which consists of 24 Lead-Glass towers (6 × 4). The towers within the SM are opticallyisolated with aluminized Mylar foils and each tower is attached to an PMT (FEU84). Steel sheets of
0.5 mm thickness are used to house the entire towers and PMTs.
The PbGl LED calibration system are installed to monitor the gain drift. Three LEDs with different
wave length are placed on the front of every SMs. The mirror foil on the top surface have a hole for each
PbGl tower for LED light to enter. A polystyrene reflective dome covers the LED system.
The intrinsic performance of PbGl was also evaluated with the test experiments using electron beams.
The measured energy resolutions is well described as:∆E5.9 %⊕ 0.8 %,=p
E
E(GeV)(2.8)

and the measured position resolution of PbGl is:
8.4 mm∆x = 0.2 mm ⊕ p.
E(GeV)(2.9)

The response of PbGl to hadrons is different from that of PbSc since PbGl is based on Cerenkov
detection. The deposit energy is suppressed for hadrons due to its Cerenkov threshold (106 MeV/c forcharged pions). The energy deposit of charged pions was evaluated at the test experiments, and it was∼ 460 MeV for 1 GeV/c, and ∼ 540 MeV for 4 GeV/c.

53

Figure 2.21: The schematic view of the super-module (SM) of PbGl including the LED system for thecalibration. The SM consists of 24 towers.

54

2.2.5

Trigger

The Level-1 trigger (LVL1) have been designed to record a significant fraction of the physics events ofinterest. The LVL1 trigger system is a pipelined and dead-time less. The triggers consist of two types:
one is the Local Level-1 (LL1) system, and the other is the Global Level-1 (GL1) system. The LL1
system communicates directly with the detectors such as BBC and EMCal. The input data from thesedetectors are processed by the LL1 algorithms at each RHIC beam crossing, and the trigger informationis transferred to GL1. The GL1 receives and combines the LL1 data to make a trigger decision. A
particular LL1 is often combined with other LL1 at GL1 4 .
The BBC LL1 trigger (BBCLL1) and EMCal LL1 trigger (ERTLL1) are described in this section.
These triggers were utilized to select the physics events of interest in this thesis.
BBC LL1 Trigger (BBCLL1)
The main trigger for events in PHENIX relies on a coincidence between the two BBC modules. The timing
information of BBC is used to determine the position of the collision point. The vertex position cut of|z| < 30 cm can be applied in the BBCLL1 trigger using position information obtained online, and its
position resolution is ∼ 5 cm. The vertex cut roughly matches to the acceptance of the PHENIX central
arms. Two types of BBCLL1 triggers with and without the vertex position cut are implemented. In thisthesis, the BBCLL1 trigger with the 30 cm vertex cut is simply referred to as the ”BBCLL1(Vtxcut)”,
while we explicitly write the BBCLL1 trigger without vertex cut as the ”BBCLL1(noVtx)”.
The BBCLL1 is defined as minimum-bias (MB) trigger-condition for the data in pp collisions, andit also serves as a luminosity monitor: the number of BBCLL1 is used in the integrated luminosity andvernier scan measurement.
EMCal LL1 Trigger (ERTLL1)
The EMCal provides signals of the high energy electron and photon events. Basic idea of ERTLL1 is toselect events with a deposit energy on the EMCal over a certain threshold. As the shower generated by aparticle hit on EMCal usually spreads over several towers, it is necessary for the efficient trigger to sum
the energy deposits on the neighboring towers. When a particle hits the corner of the tower, the typicalenergy deposit in one tower is ∼ 20 % of total energy, while it is ∼ 80 % with a hit at the center.
The conceptual diagram of the actual procedure to select the events is displayed in Fig. 2.22. Theenergies of 2 × 2 towers are summed at first by on an ASIC chip. The information of the 6 × 6 ASIC
chip (12 × 12 towers) are read out by one FEM (Front-end Electronics Module). At this stage, towers
do not overlap, and hence 36 energy sums of 2 × 2 towers are produced per one FEM. To eliminate theinefficiency in the case of a particle hit at the border of the 2 × 2 block, the energy sum of 4 × 4 towers
is formed from four 2 × 2 energy-sum, allowing the overlap of 2 × 2 regions. The communication among
ASIC chips is relayed over FEMs and the trigger organization is seamless over the EMCal sector. In
4 For example, ERTLL1 is combined with BBCLL1 to require the vertex cut of |z| < 30 cm.

55

total, 36 signals of 4 × 4 energy sum is obtained for one FEM. This enables us to collect more than 90 %of the energy of a particle hit in at least one of the 4 × 4 energy sum. Each 4 × 4 energy sum, as wellas the 2 × 2 energy sum, is compared to a certain threshold for a trigger decision. To avoid the effect of
noisy towers, which always provide a large signal due to the electrical noise, the trigger can be masked
with a unit of the FEM.
In the experiment, three thresholds are applied for the 4 × 4 energy sum. The values of the thresholds
are summarized in Table 2.3 with that for the 2 × 2 energy sum. The triggers are named ”ERT4x4a”,”ERT4x4b”, ”ERT4x4c” and ”ERT2x2”, respectively. Because the gain setting of EMCal is different√
between s = 200 GeV and 500 GeV pp collision, the energy thresholds are different. The gain is
reduced by roughly factor of 2 at 500 GeV run, so that the dynamic range of EMCal covers high pTelectrons from W decay (∼ 40 GeV/c).

Figure 2.22: The method to produce the signal for the EMCal trigger. The region of 12 × 12 towers
displayed in the figure are processed by one FEM. Signals of 2 × 2 energy sum are relayed across theboundary of 12 × 12 region.
√s
200 GeV
500 GeV

ERT4x4a2.1 GeV
4.2 GeV

ERT4x4b
2.8 GeV
5.6 GeV

ERT4x4c
1.4 GeV
2.8 GeV

ERT2x2
0.8 GeV
1.6 GeV

Table 2.3: The energy thresholds for the ERTLL1. The threshold is compared to the energy sum of the
4 × 4 or 2 × 2 towers in ERTLL1.

56

2.2.6

The Data Acquisition System

The PHENIX data acquisition (DAQ) system processes the signals from each detector, produces thetrigger decision, and stores the triggered data. The typical data logging rate of PHENIX was ∼ 1 kHz for
AuAu collisions and ∼ 5 kHz for pp collisions. The zero-suppressed event sizes are 160 kbytes for AuAu
and 40 kbytes for pp, respectively.
The block diagram of the data acquisition flow is shown in Fig. 2.23. The DAQ system employs
the concept of granule and partition. A granule is smallest unit, which consists of individual timing
control and data collection for each detector. A partition is the combination of granules, that share busy
signals and accept signals. This configuration makes it possible to run the DAQ in desired combinationof detectors.
Since the RHIC beams have the bunch structures, the PHENIX data stream has to be synchronizedwith the RHIC beams. The RHIC beam clock of 9.4 MHz is provided by the accelerator control system,
and is transfered to the Master Timing Module (MTM). The MTM distributes the clock into the GranuleTiming Modules (GTM) which are equipped into all detectors and the Global Level-1 (GL1). The GTM
delivers the clock, the control commands (Mode Bits) and event accept signal to FEMs of each detector,
and is capable of a fine delay tuning of the clock phase in ∼ 50 ps step in order to compensate the timing
difference among FEMs. The GL1 produces the first Level-1 (LVL1) trigger decision, combining LL1(Local Level-1) signals from detector components. The event trigger issued by GL1 is sent to the FEMsof all detectors.
FEMs manipulate FEE (Front-End Electronics) to process the raw signals of the PHENIX detectors.
At the same time, LL1 trigger signals are generated. FEE has buffering scheme and store the processedsignals for up to 40 bunch crossings for read-out by request from the event trigger.
When a event trigger is issued by the GL1, FEMs send processed signals into Data Collecting Modules(DCMs). The DCM has capability to receive 100 Gbytes of uncompressed event data per second at thehighest trigger rate. It provides data buffering, zero suppression, error checking, and data formatting.
The information from the DCMs are collected by the Partitioner and then sent to the Sub-Event Buffers(SEBs). The SEB is the front-end of Event Builder (EvB) and communicate with each granule. The
SEBs then transfer the data on request to a set of the Assembly and Trigger Processors (ATPs) under
the control of the Event-Builder Controller (EBC). Here, granule-by-granule information is rearranged,
and assembled as an event.
The event data are sent to the online monitoring system for online-level quality insurance, and sent
to the DAQ Linux machines to record it in hard drives. The stored data is used for generation of trigger
decision by the second level (LVL2) software trigger. The data storage is finally provided by HPSS-basedtape storage system with maximum transfer rate of 20 Mbytes/s. Combining the buffering to local disk,
the maximum data logging rate become ∼60 Mbytes/s.

57

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

Figure 2.23: PHENIX DAQ system.

58

SEB
SEB
SEB
SEB
ATM Switch
ATP

Linux Farm
~600 CPU's

tapestorage
1.2PB

20MB/s
250TB/yrChapter 3

RHIC Run 9
3.1

Luminosity and Delivered Integrated Luminosity

The data for this thesis were taken by means of the PHENIX detector with polarized proton beam of RHIC
at BNL during the Year-2009 run (Run 9). The duration of the Run 9 polarized pp run was about 6 weeks(first 2 weeks were for commissioning: March 5th to March 17th, and following 4 weeks were for physics
running: March 17th to April 13th) with the beam energy of 250 GeV, which corresponds to a center√of-mass energy ( s) of 500 GeV, and with the beam spin polarized longitudinally at the collision pointof PHENIX. The data taken in the period of physics running was used in the analysis. RHIC achieved
the maximum luminosity of 8.5 × 1031 cm−2 sec−1 and the average luminosity of 5.5 × 1031 cm−2 sec−1 .
Delivered integrated luminosity at PHENIX through 6 weeks was about 60 pb−1 . Figure 3.1 displays thedelivered integrated luminosity as a function of time [102].
PHENIX accumulated the integrated luminosity of about 17.2 pb−1 . After various quality assurance
selections for polarized beam, total amount of data used in this thesis is 8.56 pb−1 (see Sec. 4.3.1).

3.2

Beam Polarization

The beam polarization was measured at pC polarimeter normalized by the physics asymmetry (AN ) whichis calibrated by H-jet polarimeter (see Sec. 2.1.3). The luminosity weighted polarization is 0.38 ± 0.03
for Blue beam, 0.40 ± 0.05 for Yellow beam, and the averaged polarization ((PB + PY )/2) is 0.39 ± 0.04.
The uncertainties related to the beam polarization are summarized in Table 3.1 [103]. The systematic
uncertainty of the H-jet measurements includes an uncertainty on molecular-hydrogen contaminationin the polarized atomic hydrogen jet, which is represented as ”H-jet (molecular)” in the table. Other
backgrounds to the H-jet measurement (shown as ”H-jet (other sys.)” in the table) was estimated fromthe asymmetry in ”non-signal” strips, and variation in the results due to applying different cuts. In
Run 9, the pC measurement suffered from systematic problems related to high event rates due to smaller√√transverse beam size at s = 500 GeV and on thicker targets, compared to previous s = 200 GeV runs.
This uncertainty is shown as ”pC (time/rate dependence)” in the table. There is also an uncertainty fromnon-flat profile of the beam polarization. Due to the non-flat polarization profile, the beam polarization
59

Figure 3.1: Projected minimum and maximum integrated luminosities for polarized proton collisions at
250 GeV beam energy, with achieved luminosity at Run 9.

was seen differently by the H-jet polarimeter, the pC polarimeter (in fixed target measurement case), andwhen colliding beams at the interaction points (IPs) of RHIC experiments. The jet target at the H-jet
polarimeter was much wider than the beam width, such the polarization profile was weighted with thebeam intensity profile in the average beam polarization. The polarization measurement in pC polarimeter
depended on the positioning of the carbon target, and whether a horizontal or vertical target is used for themeasurements (beam profiles are different for the horizontal and the vertical targets). For colliding beams,
the polarization profile is weighted by a product of two beam intensity profiles in the transverse plane for
evaluation of average beam polarization. These difference between average beam polarizations are taken
into account when normalizing the pC measurements to the H-jet absolute polarization measurement,
and when providing the polarization values for RHIC experiments 1 . The uncertainty due to this reason
is represented as ”pC (pol. profile)” in Table 3.1 which reflects the uncertainty in the average over all fills
polarization profile. As a result, relative uncertainties of ∆P/P is 8.3 % (12.1 %) for the Blue (Yellow)
beam.
For a product of two beam polarizations PB · PY (used in double spin asymmetry measurements), therelative uncertainty ∆(PB · PY )/(PB · PY ) is 18.5 %. The uncertainty is 9.2 % for the average between
two beam polarization (PB + PY )/2 (used in single spin asymmetry measurements, when data from twopolarized beams are combined). These uncertainty was calculated assuming that the H-Jet systematic un1 Scanning a carbon ribbon target across the beam allows to measure intensity and polarization profiles in both verticaland horizontal directions in transverse plane.

60certainties as well as time/rate dependence uncertainty of the pC measurement are maximally correlated
between Blue and Yellow beams.
uncertainty
H-jet (stat.)H-jet (molecular)
H-jet (other sys.)pC (time/rate dependence)pC (pol. profile)
totalBlue beam
2.5 %
2.0 %
3.0 %
5.0 %
5.0 %
8.3 %

Yellow beam
2.7 %
2.0 %
3.0 %
10.0 %
5.0 %
12.1 %

Table 3.1: Uncertainties in the beam polarization measurement.

3.3

Absolute BBC Cross Section

The absolute integrated luminosity is required for the cross section measurement. BBC was utilized as
a luminosity counter at PHENIX and therefore, the calibration of the collision cross section seen by the
BBC is necessary. The calibration of BBC is defined as BBC cross section (σBBC ):LBBC =

RBBC
,
σBBC

(3.1)

where RBBC is the event rate measured by BBC and LBBC is the effective luminosity delivered to the
PHENIX.
The BBCLL1(Vtxcut) trigger requires at least one hit on both North and South components of thedetector and it requires an online vertex cut of |z| < 30 cm. To calculate the absolute calibration ofBBC, number of events for each bunch crossing from BBCLL1(Vtxcut) triggered data is counted and theluminosity of each bunch crossing is calculated by:
Lbunchmachine =fbeam NB NY,
2πσx σy(3.2)

where fbeam is the frequency of each bunch crossing (∼ 78 kHz), NB and NY are number of protons(∼ 1011 ) in Blue and Yellow beams respectively. σx and σy are the horizontal and vertical widths of theoverlap profiles of Gaussian beams.
To measure the σx and σy , the technique of Vernier Scan (or Van der Meer scan) [104] is used. The
method (suggested by Simon Van der Meer) scans one beam across the other in small steps to measuretransverse profile (horizontal and vertical). The triggered event rate for each step was measured. The
rate as a function of the beam-step position (”beam-step position vs. rate” plot) was plotted. Global
level-1 (GL1) scaler (scaler for each crossing) kept the track of the event counts for BBCLL1(Vtxcut)
trigger. This scaler information was utilized to calculate the event rate for each step. A Gaussian fit to
the plot gives the resultant width (σx , σy ). The beam intensity information of NB , NY is obtained fromthe Wall Current Monitor (WCM), and Lmachine can be calculated for each bunch crossing by Eq. 3.2.

61

It needs to evaluate the acceptance (εvertex ) of the BBCLL1(Vtxcut) trigger (out of the total machine
luminosity) in order to calculate LBBC from Lmachine . It was obtained by comparing BBCLL1(Vtxcut)
with a wide unrestricted trigger BBCLL1(noVtx) (see Sec. 2.2.5). Using the machine luminosity, thetrigger acceptance and event rate at the maximal overlap point (Rmax ), which was obtained from the”beam-step position vs. rate” plot, the BBC calibration for each bunch crossing was calculated from:
bunch
σBBC
=

bunch
Rmax
.
bunch
Lmachine · εvertex(3.3)

By averaging this value over all good non-empty bunches, σBBC was evaluated.
Experimental Procedures
The data of vernier scans were taken near the end of a fill, before dumping a given fill and most often
before the polarization measurement. The main feature of the procedure was that one of the beams
was moved in steps in transverse directions while keeping the other steady. The step sizes were chosen
to be of the order of or smaller than expected width of the Gaussian profile of the beam in transverse
directions, typically a few hundred microns. Beams were moved in steps and stayed for typically 30 to 40
seconds at each step to enable PHENIX to acquire enough events. The beam positions were recorded byseveral different Beam Position Monitors (BPMs) at different positions along the beam ring. Intensityof circulating beams were monitored by two different monitors: the Wall Current Monitor (WCM) andthe Direct Current Current Transformer (DCCT). The DCCT measures the average current induced
in a solenoidal coil around the beam pipe in a rather long time period (1 sec), and this large sample
time allows the DCCT to measure the total number of ions in RHIC to ∼ 0.2 % accuracy [105]. One
drawback of the DCCT was that, because of its large integration time, it also included debunched beam
in its measurement. To remedy this, the DCCT was used with the WCM. The WCM measured induced
voltage in an RLC circuit in a particular section of the beam pipe. The WCM samples at a very fast
rate (∼ 0.25 nsec) so as to give the bunch profile in beam (z) direction and was insensitive to debunched
beam/beam-gas. For the number of protons in beam, the WCM data, which is calibrated at the beginningof the fill when there is none or very little debunched beam, was utilized.
ResultIn the evaluation of σBBC , corrections due to following effects were applied [106]:• The loss of intensity (NB , NY ) due to collisions and scraping during the scan.
• The hour glass effect due to beam focusing (beam transverse profile (σx , σy ) depends on z-position
along beam axis) and beam crossing angle (beam may collide not head-on).
• The scale factor for the beam-step size (or σx , σy ) to match to the data from BPMs.
• The loss of counts due to multiple collision effect.
• The correction for εvertex , because the BBC has a z-depending efficiency of detecting events.
62

The first, second and third items affect to Lmachine calculation, the third item affects to Rmax measurement and the forth item affects to εvertex .
The first effect was corrected by seeing the NB · NY value as a function of time. This is a small
correction (∼ 1 % at most).
To correct for the second effect, the z-vertex profile of events at the different beam-step position was
utilized. As the beam centers moved further away, overlap area becomes less and less in the near the
focal point (z = 0) and relatively higher on away-sides along the beam direction. This produces adouble hump structure of the z-vertex profile. The crossing angle between the beams makes this doublehump structure askew. The vertex distribution was compared to the simulation which took into accountthe beam focusing parameter (β ∗ ) and crossing angle to extract these parameters 2 . The simulation
determined β ∗ and crossing angles to ∼ 10 % precision.
The fourth effect was corrected by the method described in Appendix C.
The final effect is corrected using the ZDC information. The ZDC is another luminosity counter located
far from the nominal collision center and is free of any z-dependence. The z-depending efficiency wasevaluated by comparing events that fired both the BBC and ZDC triggers with only ZDC triggered events
as a function of z-vertex measured by the ZDC. This z-dependence is taken into account in the calculationof εvertex .
Applying those corrections, the final BBC cross section σBBC value was obtained as σBBC = 32.51± 3.01(sys.) ± 1.19(stat.) mb., with averaging over all vernier scan runs, The uncertainties related theσBBC measurement are summarized in Table 3.2 [106]. The uncertainty is varied among different vernierscan runs as indicated in the table.
uncertainty
εBBC
NB · NYσx , σyσx , σy
Rmax
total
total

value
2.0 %
0.4 to 1.1 %1.5 to 3.3 %7.6 %3.1 to 4.5 %
9.3 %
10 %

comment
from the fit
from DCCT and WCM measurementfrom the fit
correction of hour glass effect and crossing angle
from the fit and correction of multiple collision effecttotal systematic uncertainties
stat. ⊕ sys.

Table 3.2: Uncertainties in the σBBC measurement.

2 The beam profiles in the simulation is Gaussian for both transverse and longitudinal direction.

63

Chapter 4

Analysis
4.1

Overview

The PHENIX is not hermetic, unlike most of the hadron collider detectors. It means that missingmomentum technique for the final states of charged-lepton and neutrino cannot be used to identify the
W and reconstruct its momentum. Instead, the strategy adopted is to detect the charged decay lepton
to determine its transverse momentum pT l and rapidity ηl . The relevant process therefore becomes the
single-inclusive reaction pp → lX. This chapter describes the data analysis to extract differential crosssection (dσ/dy) and single spin asymmetry (AL ) of high pT (30 < pT < 50 GeV/c) single e± over a√
pseudo-rapidity range of |η| < 0.35, in pp collisions at s = 500 GeV. At this energy, such high pTelectrons/positrons are mainly produced from W or Z decays. Since the PHENIX cannot distinguishbetween e± s from W decays and them from Z decays, dσ/dy and AL measured must include bothcontributions 1 .
The data taken with the ERT4x4b trigger 2 is used in this analysis. The brief analysis procedure is
as follows:
1. Calibrations of EMCal and DC/PC1 are carried out. (Sec. 4.2)
2. pT spectrum of inclusive electron-candidate is obtained with several analysis cuts. (Event Selection,
Sec. 4.3.2)
3. pT spectrum of single electrons from W and Z decays is obtained by statistically subtracting
background. The contribution of background is estimated using both real data and MC simulation.
(Sec. 4.3.3)
4. The differential cross section (dσ/dy) is evaluated. (Sec. 4.3.7)
5. The single spin asymmetry (AL ) of W + Z production is calculated. (Sec. 4.4)
Figure 4.1 displays the flow chart of the procedure of this analysis.
1 The treatment of the Z fraction in the cross section calculation is described in Sec. 5.1.
2 The 4 × 4 tiled EMCal trigger with energy threshold of ∼ 6 GeV (see Sec. 2.2.5 for more detail).

64

Figure 4.1: The flow chart of analysis procedure in this thesis.

65

The EMCal and DC/PC1 are key detectors for this analysis. With a summary of energy reconstruction
by the EMCal and momentum reconstruction by the DC, the calibration for these detectors are describedin Sec. 4.2. In order to obtain the inclusive spectrum of high pT electrons, various analysis cuts areapplied and estimation of background is essential to extract dσ/dy. They are explained in Sec. 4.3 with
the integrated luminosity used in the calculation of dσ/dy. The trigger efficiency and the efficiencies ofgeometry/analysis cut are also described in this section. In addition to the cuts for the dσ/dy calculation,
an isolation cut is also applied to evaluate AL in order to reduce hadronic background. To fully utilize theavailable statistics of high pT e± sample, a maximum likelihood technique is used in ”spin fit” procedureby which AL is calculated. The detail of AL extraction is represented in Sec. 4.4.

4.2

Calibration

4.2.1

Energy Reconstruction by EMCal

The detail of the energy reconstruction in PHENIX can be found in [107]. From EMCal, the following
information about an incident particle is extracted for the analysis;• Total energy,• Hit position,• Time of flight (Tof) information.
The energy deposit of a particle hit at EMCal spreads among several towers. To extract the information of the incident particle, it is necessary to find a group of towers which have energy deposit of theparticle. At first, a certain threshold is applied to select a hit tower. The threshold was 10 MeV for PbSc
and 14 MeV for PbGl. The higher threshold for PbGl is due to the larger noise in PbGl. The neighboring
hit towers around the selected hit tower were grouped into a cluster. If there were more than one maxima
of energy deposit in a cluster, they were split into clusters so that a cluster had only one maximum.
In PbGl, all energies in the cluster towers are summed and total energy is corrected for the incident
angle dependence which was parameterized from the test beam results [100]. In PbSc, so-called ”core”
tower technique is utilized to obtain the total energy deposit of a cluster. The sum is performed only
for core towers, instead of summing all tower energy in a cluster. Core towers are defined as those inwhich the incident particle is estimated to deposit the energy more than 2 % of the total energy. The
estimation is based on the electromagnetic shower profile which was also parameterized from the test
beam experiments. The energy sum of core towers is about 90 % of the total. The total energy isthen extracted with the impact angle dependence correction as in the case of PbGl. The impact angle
dependence is less than 4 %. At the same time, the energy losses caused by the attenuation in the fibers
and shower leakage, are also corrected. The ”core” tower technique was introduced to cope with the highoccupancy environment in heavy ion collision experiment. This degrades the energy resolution slightly66

(the constant term of the energy resolution in Eq. 2.6 is increased by 2 % to 3 %) but the achieved
resolution is good enough for the analysis.
The hit position of the incident particle is obtained by the center of gravity method. The positionsof towers in a cluster are weight-averaged by tower energies, and the dependence of the impact angle ofthe particle is also corrected.
The Tof of a cluster is defined by the Tof of the hit tower with maximum energy deposit. As the start
time of Tof varies in each tower, offsets need to be adjusted among different towers in offline analysis.

4.2.2

EMCal Energy Calibration

The EMCal is a crucial detector for high pT electron measurement. The value of pT is calculated by
E × sin θ (E: energy of EMCal cluster, i.e. pT in this thesis represents ET ) as the energy resolution isbetter than the momentum resolution in high pT region (pT >∼ 10 GeV/c). Therefore, energy calibrationof EMCal to assure reliable energy measurement in high energy region is very important for this analysis.
The energy calibration of EMCal is described in this subsection.
The time dependence of the energy gain is corrected tower by tower using the laser calibration systemfor both PbSc and PbGl. This semi-online calibration corrects only relative time-drift of each tower
separately. Therefore, the gain non-uniformity between towers remains. See Sec. 2.2.4 about the laser
calibration system. In the offline analysis, additional energy calibration is carried out utilizing themeasured π 0 and η peak, which is the physics signal and therefore most reliable. The offline calibration isdivided into 3 steps: tower-by-tower gain correction, run-by-run global gain correction (time dependence)and calibration for the absolute energy scale.
Tower-by-tower Gain Correction
The tower-by-tower energy calibration utilizes the peak position of π 0 in the invariant mass spectrum of
the cluster pairs. The deviation of the π 0 peak in the tower-by-tower spectrum from the nominal positionis assumed to reflect the gain shift. The mass spectrum for a certain tower (the target tower) consists ofthe cluster pairs which include the cluster located at the target tower. Figure 4.2 displays the concept
schematically. In this step, it is assumed that the shift of the π 0 peak for the target tower is caused by
the gain shift of the target tower and the effects from other towers are averaged and negligible. However,to achieve better calibration, this step is iterated several times 3 . The calibration is tried for all towers
in EMCal, including even towers with problem in the electrical circuit. Then, towers which cannot becalibrated are included in the masked towers (see Sec. 4.3.2). The reason of the failure in the calibration
is mainly due to the malfunctional electrical circuit.
Run-by-run Global Gain Correction
The time dependence of the EMCal energy gain is initially corrected using the laser system for both PbSc
and PbGl. This correction is executed semi-online. The residual time dependence is calibrated using the
3 Five times of the iteration processes have been carried out for Run 9 data.

67

Figure 4.2: The concept of the tower by tower gain correction. π 0 s whose decay photon hit the target
tower are collected to make the tower-by-tower mass spectrum.

measured π 0 peak. π 0 s are reconstructed sector-by-sector in every run 4 and the spectra are fitted to the
combination of Gaussian and polynomial. The shift of the π 0 peak from the normal position is assumed
to be the shift of the energy gain and additional factor is applied to the gain in the analysis. In thisstep, the observed π 0 peak is set to a certain value near 0.135 GeV/c2 . Figure 4.3 shows the comparisonof the π 0 peak position and the width before (black points) and after (red points) the run-by-run gain
correction as a function of the run number.
Calibration of Absolute Energy Scale
Since the run-by-run global gain correction is performed at the π 0 pT range of 4 to 10 GeV/c, the energy(pT ) dependence of energy scale (non-linearity) still remains 5 . The energy dependence of lower pT region
is caused by the energy threshold for the EMCal tower, ∼10 MeV, and it causes that the energy of thereconstructed clusters is lower than the incident energy. The energy dependence of higher pT region
is caused by the depth of Electro-Magnetic (EM) shower. The shower depth originated by a high pTphoton/electron reaches closer to the EMCal photo-tube and larger amount of light is measured than theshower originated by a lower pT (i.e. pT ∼ 4 GeV/c) photon/electron. This effect causes that the energyof the reconstructed clusters is higher than the incident energy. There is also a shower leakage effect forevents of high pT photon/electron due to the limited length of a EMCal tower. This effect causes thatreconstructed energy is lower than the incident energy. Therefore, the energy dependence in higher pT
4 The π 0 p range of 4 to 10 GeV/c is selected to earn enough statistics.
T
5 When selecting π 0 s with 4 ≤ p < 10 GeV/c, the mass spectrum is dominated by π 0 s with pTTenergy of lower/higher than 4 GeV/c region is not completely corrected yet.

68

∼ 4 GeV/c, and theFigure 4.3: π 0 peak position (top) and the width (bottom) before (black points) and after (red points)
the run-by-run gain correction. (Left side: PbSc, Right side: PbGl)

region includes these two effects.
The absolute energy scale is calibrated by comparing the pT dependence of π 0 peak to the fast Monte
Carlo simulation (FastMC: more detail is given in Appendix B). The depth of EM shower is taken into
account in the FastMC, therefore the behavior of energy scale in higher pT region can be evaluated in thismanner. The energy scale is tuned in the FastMC in term of an additional attenuation length (λadd ) 6 .
In the tuning of FastMC, λadd = 350 cm (150 cm) for the additional attenuation length of PbSc (PbGl)
were used.
The energy resolution also affects to the pT dependence of the width of π 0 , which is extracted fromthe data. Accordingly, the energy resolution in the simulation must be also tuned to fit the real data.
By the tuning of FastMC, 8.1 % for the energy dependent term and 5.0 % for the constant term of theenergy resolution in Eq. 2.6 were used.
Figure 4.4 (4.5) shows the comparison between before and after the the absolute energy scale correctionin the position and the width of the π 0 (η) peak. The shift of π 0 mass peak position in PbSc aroundpT ∼ 12 GeV/c (Fig. 4.4) is due to unfolding of overlapped photon showers in the EMCal (merging effect).
6 λ represents the attenuation length in the fibers. The energy is corrected by multiplying E −X0 /λ (E: energy without
correction, X0 : radiation length). Additional attenuation length is also included by multiplying E −X0 /λadd , and it mean
the effective attenuation length become smaller (λef f = λλadd /(λ + λadd )).

69

This effect is not included in MC and is not corrected. On the other hand, the effect is not observed in η
mass peak position which is flat well even in high pT region (Fig. 4.5) because the η mass is heavier thanπ 0 and the merging effect would be seen in much higher pT region 7 . The contribution to make the width
wider at low pT is dominated by the energy resolution of EMCal, while that at high pT is by the position
resolution. The difference between PbSc and PbGl is from the method of the clustering. Because only
the core towers are used to extract the cluster energy in PbSc, the energy loss due to the tower-by-tower
threshold, which is usually caused by the non-core towers, is automatically corrected.
In the real data, π 0 peak position becomes additionally lower compared to FastMC due to the unusual
π 0 s. The invariant mass of these unusual π 0 s has lower peak position than usual π 0 s reconstructed by thepure photons (see Appendix B). The size of the shift due to this effect is −1 ± 1 MeV/c2 [108]. Because
FastMC cannot simulate this effect, the peak position of π 0 is simply lowered by 1 MeV/c2 . The errorof this effect, 1 MeV/c2 , is included in the uncertainty of the energy scale calibration, and is only 0.7 %.
In addition, the slight difference of the π 0 peak position between sectors of EMCal is observed and is
expected to be from the mis-alignment of EMCal. The error from the mis-alignment is conservatively
evaluated to be 2 % in the energy scale. The systematic uncertainty of the energy scale is calculated by
the quadratic sum to be 2.1 %. Note that these uncertainties of the energy scale is for π 0 s with pT rangeof 4 to 10 GeV/c.
In this analysis, an additional uncertainty due to the extrapolation to high energy around 40 GeV
needs to be considered. From the test beam results [100], the linearity was confirmed to 2 %. By adding
them in quadrature, the uncertainty of absolute energy scale is 2.9 %.
The extracted energy resolution (8.1 % for the energy dependent term and 5.0 % for the constantterm) from the FastMC is used when including energy smearing effect in the theory calculation (e.g.
Sec. 4.3.6)

7 The inner angle between 2 photons (θ), the mass of decay particle (m) and the p of decay particle have a relation of:Tθ ∝ m/pT at lab. frame, when |θ|  1.

70Figure 4.4: π 0 peak position (top) and width (bottom) as a function of π 0 pT . Black (Red) points
represents before (after) the absolute energy correction. (Left side: PbSc, Right side: PbGl) The dashedblue lines in top figures indicates the mass value of π 0 (135 MeV/c2 ). The sift in high pT region of π 0
peak position is caused by the merging effect (2 photons from high pT π 0 are measured as 1 photons
because of finite position resolution of EMCal).

Figure 4.5: η peak position (top) and width (bottom) as a function of η pT . Black (Red) points represents
before (after) the absolute energy correction. (Left side: PbSc, Right side: PbGl) The dashed blue lines
in top figures indicates the mass value of η (548 MeV/c2 ).
71

Dynamic Range Check
The dynamic range of the ADC is also important for this analysis since the decay electron from W has
quite high pT (∼ 40 GeV/c), and the ADC must cover that energy range. Figure 4.6 shows the central
tower energy of clusters (denoted as ”Ecent ”) as a function of ADC (left: without applying Tof cut, right:
with applying Tof cut). See Sec. 4.2.3 for the Tof cut. Although some events around 60 GeV of Ecent seemto be saturated events in the left figure, those events are rejected with Tof cut (right figure) since such ahigh energy events are mainly caused by cosmic backgrounds. The Ecent value bears about 80 % of thewhole cluster energy. Since the target energy range for this analysis is 30 < (cluster energy) < 50 GeV
which corresponding to the Ecent range of around 20 to 40 GeV, the right figure shows that energy
dynamic range is large enough to cover the target energy region.

Figure 4.6: Left: ADC vs Ecent , Right: ADC vs Ecent with Tof cut (−10 < Tof < 20 ns).

72

4.2.3

EMCal Tof Calibration

Tof (Time of Flight) of EMCal is a used to remove cosmic ray backgrounds in this analysis. The Tof
calibration is done by adjusting the offset for each tower. The offset value for each tower is measured in
1 ≤ Ecent ≤ 2 GeV region and subtracted from Tof value.
Figure 4.7 shows the Tof distribution after subtracting offset values. Although event-by-event origin of
the time measured by BBC (t0 ) are subtracted in the offset calculation, they are not subtracted in Fig. 4.7
because they can be mis-reconstructed in the multi-collision event and not reliable when calculating eventby-event Tof value 8 . That results in the non-zero mean values and quite broad distribution (∼ 2 nsec).
This broad distribution is used in this analysis since ∼ 2 nsec resolution is enough to reject cosmic ray
background, and required Tof cut is −10 < Tof < 20 nsec. Note that any slewing correction is notapplied to the Tof value either, since such a good timing resolution is not necessary to reject only cosmic
backgrounds.

Figure 4.7: ToF distribution without subtraction of the t0 values fitted with Gaussian distribution (Left:
PbSc, Right: PbGl). The non-zero mean value is the result of not subtracting t0 values, and asymmetricshape is due to without applying slewing correction.

8 See Sec. 4.3.1 about multi-collision effect.

73

4.2.4

Momentum Reconstruction by DC and PC

The detailed explanation about the track reconstruction technique in PHENIX can be found in [109, 110].
The essential parts are briefly summarized in this subsection.

Figure 4.8: A particle trajectory and the parameters used in the track reconstruction. Left: beam viewof PHENIX central arm, Right: side view of PHENIX central arm.
Figure 4.8 shows the definition of track parameters in order to describe a charged particle trajectory
traveling through the axial magnetic field in PHENIX up to PC1. The track parameters are categorized
as follows:• Measured variables- α: The angle between the projection of trajectory in the (x, y) plane and the radial direction, at
the intersection point of the trajectory with the circle of reference radius RDC = 2.2 m.
- zpad : The intersection point of the trajectory with PC1 surface radius RP C1 = 2.45 m.
- β: Considering the plane which includes the z-axis and zpad , β is defined as the angle between
the projection of trajectory onto that plane and the z-axis.
- φDC : The φ-angle of intersection point of particle trajectory with the circle of radius with RDC .
• Variables to be reconstructed- θv : The angle between the initial direction of the particle trajectory and z-axis.
- φv : The initial azimuthal angle of the particle trajectory.
- pT : The transverse momentum.

74

The intersection of particle trajectory with various detector planes is uniquely determined by threevariables: θv , φv and pT . They are reconstructed from the measured variables: α, β, zpad , φDC andvertex information. The collision vertex is assumed to be (0, 0) in (x, y) plane in PHENIX. The z positionof the vertex is determined by the timing information of BBC as described in Sec. 2.2.1. The Hough
Transform technique is used for the track reconstruction [111]. In case of PHENIX, track reconstruction
started from finding the hits in X1 and X2 wires.
1. Project the DC hits onto (x, y) plane at z = 0.
2. Perform the Hough Transform using all possible X1−X2 hit combinations taking α and φDC as theparameters in Hough Space. For example if there are 6 hits in X1-wires and 6 hits in X2-wires,
total 36 combination are taken.
3. Associate X-wire hits with the reconstructed track.
4. Perform a Hough Transform of the UV-wires and associate the resulting UV hits with the reconstructed track in order to obtain the z information.
5. Associate PC1 hit.
The α measured in the DC is closely related to the field integral along the track trajectory. The
transverse momentum, pT (GeV/c) and the α-angle (mrad) have the following approximate relation:
pT ∼

K
,
α

where K ∼ 0.10 (rad GeV/c) is the effective field integral, expressed as:
Z
e
K=
lBdl.
R

(4.1)

(4.2)

Here, e is the elementary charge in the hybrid unit (e = 0.2998 GeV/c T−1 m−1 ) and R is the DC
reference radius.
However, due to the small non-uniformity of the focusing magnetic field along the flight path ofcharged particles, an accurate analytical expression for the momentum of the particles is not possible.
Therefore, the non-linear grid interpolation technique is used [112]. A four-dimensional field integral gridis constructed within the entire radial extent of the central arm for momentum determination based on
the DC hits. The parameters of the field integral are momentum p, polar angle at the vertex, θv , zveretx ,and radial distance r at which the field integral f (p, r, θv , zvertex ) is calculated.
The field integral grid is generated explicitly by swimming particles through the magnetic field map
9and numerically integrating to obtain f (p, r, θv , zvertex ) for each grid point. An iterative procedure isused to determine the momentum for reconstructed tracks, using Eq. 4.1 as an initial value. The initial
estimate of θv is given by the zvertex and DC hit position.
9 The magnetic field map is from a survey measurement.

75

The momentum resolution depends on the intrinsic angular resolution of the DC and the multiple
scattering of a charged particle as it travels up to DC due to the intervening material. As a result, themomentum resolution is about ∆p/p ∼ p(GeV/c) × 1 % and the reconstruction efficiency is above 99 %for a single track.

4.2.5

DC/PC Calibration for Charge Sign Determination

The typical bending angles of high pT electrons (30 < pT < 50 GeV/c), which are the target of thisanalysis, are quite small (∼ 2.5 mrad from Eq. 4.1) and the observed charge signs are occasionally
mis-identified due to the angle resolution of the DC (∼ 1.1 mrad). Other sources of the charge sign
mis-identification in such a high pT region are the beam position, and tracks incident to the anode wire
region of the DC.
The beam offset correction, the treatment of the anode wire region and the charge contamination due
to the DC resolution are discussed in this subsection.
Beam Offset
The actual beam position may be slightly different than exactly at the center of PHENIX central arms.
Since the carriage of PHENIX central arms were moved occasionally for a maintenance, an additional
offset effect was observed. Because tracking algorithm of PHENIX assumes the beam position as (x, y) =(0, 0), the beam position correction is important for the charge sign determination as well as momentum
measurement. For a single track, the angle resolution of the DC is about 1.1 mrad, which corresponds to
2.4 mm (= 220 cm × 1.1 mrad) at the beam position.
The DC measures the momentum of a track by measuring α, which is the angle the track projection
makes with respect to a perfectly straight track projection at a reference radius of 220 cm (see Sec. 4.2.4).
So if there is no magnetic field, the tracks do not bend and α should be equal to zero. Any deviationfrom α = 0 in zero field run data indicates that the beam position is off from (x, y) = (0, 0), and this canbe corrected by the apparent α.
Doing some trigonometry (see Fig. 4.9), it can be shown that the beam offset in the x and y direction(dx and dy respectively) are related to α and φDC . Thus, the plot of α vs. φDC gives us a way to extract
dx and dy. The relation between α, φDC , dx, and dy is:
α=

dx
dy
sin φDC −cos φDC .
RDC
RDC

(4.3)

Here, RDC denotes the reference radius at the DC (220 cm). This function is fit to the plot of α vs.
φDC , and the beam position (dx, dy) are extracted. Two figures in the left side of Fig. 4.10 display an
example of the fit. The fit was carried out in west and east arm separately because the location of arms
are not exactly symmetric relative to the beam position. Two figures in the right side of Fig. 4.10 show
the α distribution after the beam position correction. These figures represents that the deviation fromα = 0 is very small and the resolution is less than 1.1 mrad after the correction.

76

Several zero field runs are available through Run 9 to follow the beam position shift. Figure 4.11
shows the beam position (dx, dy) for each run. Although there is a jump around run 278000, which is
caused by the carriage movement, they are within a range of ±300 µm. Compared with other resolution(DC, PC1, EMCal), those shift is small enough.
In this analysis, a correction of α is applied by αrecal = α − αof f set (run, φ) in which αof f set is
calculated following Eq. 4.3. The correction is also checked using mass peak position of J/ψ.

Figure 4.9: Schematic drawing of trigonometry to extract beam position.

77

Figure 4.10: An example of the fit to extract the beam position from a zero field run. Left two figures
show the fit example and Right two figures show α distribution after the beam position correction. (Top:West arm, Bottom: East arm)

Figure 4.11: Obtained fit parameters from zero field runs. A jump around run 278000 is caused by the
carriage movement.
78

Anode Wire Region
As pT goes up, the particle go through the DC plane perpendicularly. The left right ambiguity cannot be
solved when high pT track is close to the anode wire due to its small bending angle and the DC design(Fig. 4.12). As a result, more than one track are reconstructed even if only one particle goes throughthat region (ghost track).

Figure 4.12: DC wire design
The effect is checked using zero field runs. Figure 4.13 displays the number of tracks as a functionof φDC . When requiring a pair of tracks with the opening angle less than 2 mrad 10 , periodical peaks
appear (Red distribution). The interval of the peaks corresponds to that of the anode wires (∼ 0.02 radfrom Fig. 4.12). These peaks are produced by the ghost tracks.
Since ghost tracks cause the charge mis-identification and they are not the real tracks, the anode wire
regions are removed with φDC mask. In total, 15 % of the DC acceptance is masked to avoid the anodewire region.

10 This can be calculated from the anode wire region. The region is 4 mm from Fig. 4.12 and that corresponds to ∼2 mradby the DC radius of 220 cm.

79

Figure 4.13: Black line shows the number of tracks without any cut as a function of φDC . When requiring
pair track, periodical peaks appear. (Red line) These peaks are corresponding to the anode wire region.
(Left: West arm, Right: East arm, zoomed in to clearly display intervals of pair track peaks)

Charge Contamination
In the high pT region, charge mis-identification is unavoidable due to the limited DC resolution in thismeasurement. Taking the ratio of the bending angle at 40 GeV/c (2.5 mrad) to the resolution of the DC(1.1 mrad), the probability of a positron to be mis-identified as a negative charge (i.e. as electron) is
1.1 % (= 2.3 σ). The charge contamination of electrons to positrons is also the same fraction. Note that
this number for the 40 GeV/c track should be taken as the maximum charge contamination because thee± s from W decays mainly have pT of 30 to 40 GeV/c.

80

4.3

Cross Section

The relationship of dσ/dy to the experimental observables is the following:Z
X (N meas − N bg )
d2 σ
dσ
1
1
1
= dpT
= ·
·
·
,
dy
dpT dyL εtrig · ·εreco · εsmear ∆y 30<p <50
∆pT

(4.4)

T

where:• L denotes the integrated luminosity.
• N meas means the number of inclusive electron-candidate in each pT bin, andP

N meas /∆pT is thenumber of counts measured in signal region of 30 < pT < 50 GeV/c.
• N bg represents the number of background in each pT bin, andP

N bg /∆pT stands for the total

number of background in 30 < pT < 50 GeV/c region.
• εtrig is the ERT4x4b trigger efficiency.
• εreco denotes the efficiency of geometry and analysis cuts.
• εsmear means the effect of energy smearing.
• ∆y is the rapidity bin width and is set to ∆y = 1 here (PHENIX acceptance is included in εreco ).
The details of the evaluation of every factors are described in this section. First of all, the integrated
luminosity used in this analysis (L) is represented in Sec. 4.3.1. Then the pT spectrum of inclusiveelectron-candidate (N meas ) is measured, and the total number of counts in 30 < pT < 50 GeV/c region
is extracted in Sec. 4.3.2. Here, several geometry and electron ID cuts are utilized to measure N meas .
They are also explained in this subsection. The most essential part of this analysis is the estimation of
background (N bg ). It is estimated using the real data and MC simulation and the detail is describedin Sec 4.3.3. After that, efficiency of trigger (εtrig ), geometry and analysis cuts (εreco ) and the effect ofenergy smearing (εsmear ) is evaluated in Sec. 4.3.4, Sec. 4.3.5, and Sec. 4.3.6, respectively. Finally, The
calculation of dσ/dy at 30 < pT < 50 GeV/c region is described in Sec. 4.3.7

81

4.3.1

Integrated Luminosity at PHENIX

The integrated luminosity is calculated from number of Minimum Bias (MB) trigger (N M B ) of the
PHENIX. The MB trigger of the PHENIX is BBCLL1 as described in Sec. 2.2.5, and BBCLL1(noVtx)
trigger is used as MB trigger in this analysis.
√
In RHIC Run 9 at s = 500 GeV energy, there is non-negligible probability of multiple collision√(multi-collision) event. The expected event rate at s = 500 GeV is ∼ 3.3 MHz (= σpp (500 GeV) × L)
11

, and this event rate means there is a collision almost in every beam crossing (beam crossing rate at

RHIC is 9.4 MHz (106 nsec), but this includes empty bunches). Therefore, that results in non-negligible
multi-collision event probability (∼ 10 %). Because the BBC is not capable to detect multi-collision
event, the N M B measured by the BBC underestimates the real rate. The correction to get ”true” rate
was not only determined by Poisson probability, but also depends on the efficiency of the BBC (the detail
how to correct multi-collision effect is described in Appendix C).
N M B is calculated as 5.58 × 1011 after the correction of multi-collision effect. The uncertainty of thecorrection of multi-collision effect is 2.1 % which comes from single side efficiency of the BBC (denotedas ”kN , kS ” in Appendix C). To get the integrated luminosity, this N M B is multiplied by the BBC crosssection (σBBC ) measured by the vernier scan (Sec. 3.3).
An additional factor which should be taken into account is the reduction of the total integrated
luminosity caused by the event vertex cut. It depends on the vertex width. Figure 4.14 shows the fractionof the events with 30 cm vertex cut (f vtx (±30cm)) from the BBC trigger sample. It shows a slight run
by run variation which is caused by the trigger rate dependence. Figure 4.15 shows the f vtx (±30cm) as
a function of the trigger rate defined as N M B /(# of Beam Crossings). Here, the (# of Beam Crossings)
is calculated from the run time duration multiplied by the time of 1 beam crossing (106 nsec) with notcounting empty bunches. The trigger rate dependence is caused by the multi-collision. Experimentally,
the BBC is not capable of handling the multi-collision event, and that leads to the mis-reconstructedvertex position. The f vtx (±30cm) obtained from luminosity weighted mean of Fig. 4.14 is 0.499, and f vtx
with other vertex cuts are also listed in Table 4.1. The uncertainty of the fraction is evaluated from its
trigger rate dependence. It is estimated by the inclination of the fit function in Fig. 4.15. As the trigger
rate spread over the range of 0.05 to 0.25, the corresponding width of f vtx is p1 × (0.25 − 0.05) = 0.040.
Thus, we assigned the uncertainty of f vtx (±30cm) as 4.0 % (= (0.040/2)/0.499).
vertex cut (cm)
40
30
20

fraction
0.620
0.499
0.352

Table 4.1: Fraction of events with vertex cut (f vtx ).

11 σ

√

s = 500 GeV and its value is ∼ 61 mb [113]. L is the luminosity
pp (500 GeV) is the total collision cross section atof RHIC in Run 9, and it is 5.5 × 1031 cm−2 sec−1 in average.

82

Figure 4.14: The fraction of events with a vertex cut Figure 4.15: The f vtx (±30cm) as a function of theto ones with no vertex cut (f vtx (±30cm)).
trigger rate, fitted with a polynomial function (n =1). The extrapolated point of (trigger rate) = 0 isused to evaluate the uncertainty of the fraction.

Vertex Reconstruction from EMCal and PC1 Hit Position
Since the multi-collision event causes the BBC to mis-reconstruct the z-vertex position (zvertex ), thezvertex reconstructed from EMCal cluster and PC1 hit position was used in this analysis for the vertex
cut. Figure 4.16 shows the difference between the EMCal−PC1 based zvertex and the BBC based zvertex .
The distribution is fit with ”Gaussian + constant” and the width is 2.23 ± 0.03 cm (p2 parameter). Thiswidth can be explained by the position resolution of EMCal (1 ∼ 2 cm), PC1 (∼ 1.7 mm) and BBC(∼ 2 cm). The tail part contains both mis-reconstructed BBC vertex and mis-association of EMCal andtrack (DC/PC1 information) 12 .
Figure 4.17 shows the distribution of EMCal-PC1 based zvertex . The cut at around 40 cm is due tothe nose cone of PHENIX central magnet (Sec. 2.2.3). In the analysis, the EMCal-PC1 based zvertex isused in the vertex cut and the requirement is |zvertex | < 30 cm.
The difference between the BBC based zvertex and the EMCal-PC1 based zvertex causes the change
in the f vtx (±30cm). Since the width of the difference is 2.2 cm, the fraction of events with 32.2 cm
and 27.8 cm vertex cut is calculated. These fractions are different from the f vtx (±30cm) only by 6.0 %.
Including this difference, the uncertainty of the fraction becomes 7.2 %.
Summary of Integrated Luminosity
The total integrated luminosity that were analyzed for this analysis is calculated via:L=X

f vtx (±30cm)NiM B Ci (RiBBC )/σBBC .

(4.5)

i=run

The values with the i subindex are calculated run by run. The factors in this calculation are describedin Table 4.2. The calculated total integrated luminosity with 30 cm vertex cut is 8.56 pb−1 .
12 The mis-association is caused by the fact that only φ matching is required when associating a EMCal cluster to a
DC/PC1 position. This loose association is used in this analysis because the BBC based zvertex is mis-reconstructed in
multi-collision event and the association of z-direction is not reliable. The detail of the association is described in Sec. 4.3.2.

83

Figure 4.16: ∆z cm distribution between the BBCvertex and the reconstructed vertex. The distribution is fit with Gaussian + constant and the width is
2.2 cm (p2 parameter). This width can be explainedby the position resolution of EMCal, PC1 and BBC.

Figure 4.17: zvertex distribution reconstructed from
EMCal and PC1 hit position. The cut at around
40 cm is due to the natural cut of the nose cone of
PHENIX central arms.

The uncertainty is from correction of multi-collision (2.1 %), f vtx (7.2 %) and σBBC (10 %). Adding
them in quadratic, the uncertainty of integrated luminosity is 13 % in total.
notation
f vtx (±30cm)

value
0.499

uncertainty
7.2 %

NiM B

The number of MB (BBCLL1(noVtx)) trigger.

Ci (RiBBC )σBBCexplanation
The fraction of events which are within the vertex
cut.

32.51 mb

2.1 %The correction factor for multi-collisions. Ci is afunction of the trigger rate (RiBBC ). More detailed
explanation is given in Appendix C.

10 %The cross section seen by BBCLL1(Vtxcut) as determined by vernier scans (Sec. 3.3).

Table 4.2: explanations of factors in Eq. 4.5

84

4.3.2

Event Selection

In order to evaluate N meas in Eq. 4.4, several cuts are applied in this analysis. They are:• Track Quality Cut: require hits on X1 and X2 plane of the DC and hit on the PC1.
• DC Mask: Mask anode wire neighbors 13 .
• Vertex Cut: |zvertex | < 30 cm 14 .
• EMCal Tof Cut: −10 < Tof < 20 nsec.
• EMCal Mask: Mask problematic FEMs of ERT4x4b trigger and EMCal towers.
• DC-EMCal Matching: |∆φ| < 0.01 rad.
• E/p Cut: Reject low momentum accidental match with the cut of E/p < 2.
The detail of EMCal mask, DC-EMCal matching and E/p cut is described in this subsection. Note thatthe performance of the DC/PC1 is not so stable during Run 9 (see Appendix D for tracking performance)
while the EMCal performance throughout Run 9 period was quite stable (e.g. see Fig. 4.3). Because atechnique to handle the run-by-run acceptance change of the DC/PC1 is employed when calculating theefficiency of geometry and analysis cuts (εreco ) in Sec. 4.3.5, no run is rejected in this analysis to earn as
many of e± s from W decays as possible.
The value of pT is calculated by E × sin θ (E: energy of EMCal cluster, i.e. pT in this thesis represents
ET ) as the energy resolution is better than the momentum resolution in high pT region (pT >∼ 10 GeV/c).
Figure 4.18 is the raw pT spectrum of inclusive e± candidates with applying all cuts listed above, andTable 4.3 shows the total counts in the figure at 30 < pT < 50 GeV/c region. These numbers are used in
the calculation of dσ/dy.
charge+−±total counts
62
17
79

Table 4.3: Total counts in a pT range of 30 to 50 GeV/c.

EMCal Hot/Dead Tower Mask
A hot tower is defined as a tower sending signals at significantly higher frequency than other towers,
which is known to be electronics noise. Distributions of the number of hits of each tower is created foreach EMCal sector 15 and towers which have multiplicity greater by 15 σ from the average are identified
13 The junction of North and South side of the DC/PC1 is also masked due to its inefficiency. ±3 cm from the junction
is rejected in this analysis.
14 EMCal−PC1 based z
vertex is used for vertex cut (Sec. 4.3.1).
15 The cut of cluster energy is more than 10 GeV is applied to make this distribution.

85

Figure 4.18: pT spectrum of inclusive electron-candidate with applying all cuts. Left: positive charge,
Right: negative charge

as hot tower. This threshold is quite large, but it can reject extremely noisy towers, and can keep the
EMCal acceptance as large as possible in order to observe rare W events. When there is a hot tower,
towers around that particular tower are also masked to avoid mis-reconstructed energy caused by the hot
tower. Towers for which the calibration constants cannot be determined are also masked. These towers
remain uncalibrated since most of them are dead towers. The edge towers are also masked to avoid theshower leakage effects. Two towers from each edge are assigned as the edge towers. In total, 4145 towers
are masked in this analysis. (PbSc: 2820 (18 %), PbGl: 1325 (14 %), most of them are edge towers)
In addition to hot/dead towers, noisy FEMs of ERT4x4b trigger are also masked (see Sec. 2.2.5 forFEM). Three FEMs in PbSc (2.8%) and one SM (1.6 %) in PbGl are masked because their trigger rateis rather higher than other FEMs.
Removing hot/dead towers and bad FEMs, roughly 79 % of PbSc and 84 % of PbGl are used in thisanalysis. By weighting on the number of towers of PbSc and PbGl, averaged active area of EMCal isabout 81 %. The hot/dead towers and bad FEMs are taken into account when calculating εreco (detection
efficiency and geometrical acceptance) in Sec. 4.3.5.
DC-EMCal Matching
Due to non-negligible fraction of multi-collision events, the zvertex measured by the BBC is not reliable
for the track association. Therefore, only φ value is used for the association in this analysis. If theinternal φ angle (|∆φ|) between a particular track and a EMCal cluster is less than 0.01 rad, these hits
are tagged as originated from the same particle.
Figure 4.19 shows the ∆φ distribution for clusters with energy more than 10 GeV. The distribution isfitted with the combination of Gaussian and constant. The width of the peak is ∼ 3 mrad (p2 parameter)
and this value can be explained by the φ resolution of the EMCal (∼ 3 mrad 16 ) and the DC (1.1 mrad).
16 This value is calculated from the position resolution of ∼ 1.5 cm at 500 cm away from collision point86

The tail part of the distribution is corresponding to the random track association and these events areused in the estimation of accidental track match (Sec. 4.3.3). The red histogram in the figure shows the∆φ distribution with E/p cut and it can be noticed the tail part (accidental match) is well reduced by
the cut.
This association technique allows one track to match multiple EMCal clusters. If there multiple
candidates in |∆φ| < 0.01 rad, the one with the highest momentum was selected.

Figure 4.19: ∆φ distribution of track matching for the cluster more than 10 GeV. Black: without an E/pcut. Red: with E/p < 2 cut.

E/p Cut
Since the electron deposits all of its energy in EMCal and the mass is small compared to its momentum,
the ratio of the momentum to the energy detected at EMCal (E/p) is around one in case of the electron track. On the other hand, the E/p values for charged hadrons become small because the energy
deposit by hadrons at EMCal is smaller than they actually carry. Therefore, the E/p cut can reject
the charged hadron track. The cut also rejects the accidental matching in this analysis as the effect isshown in Fig. 4.19. Because the momentum resolution becomes worse in high pT region, the width ofE/p distribution gets wider. In order not to drop W candidate events at high pT region, very loose cutof E/p < 2 is adopted 17 .
17 This cut does not reject events with small E/p values and it means the cut does not work for rejecting charged hadron
events. However, ERT4x4b trigger has a energy threshold of ∼ 8 GeV (Sec. 4.3.4) and it works like a cut for small E/pevents.

87

4.3.3Background Estimation

Because the PHENIX is not hermetic, missing-momentum technique cannot be used to detect W s. Therefore the observed process is only the single-inclusive reaction pp → lX, and there are several backgrounds
for the measurement:• Accidental track match: γs observed at EMCal accidentally match to charged tracks from jet
fragments.
• π 0 conversion: A fake high pT track would be observed when one of γs from π 0 decay is converted
to electrons before the DC and another γ is detected at the EMCal.
• Charged hadron: There is a possibility that high pT charged hadron deposit all energy in EMCal,
and observed like high pT electron.
• Heavy flavor decays: e± s from D/B meson decays contribute to background.
• Z/γ decays: Because mass of Z boson is quite close to it of W boson, e± s from Z boson decays
have similar momentum to e± s from W boson decays. However, it is hard to reject e± s from Z
decays by tagging both electron and positron with the limited PHENIX acceptance. Therefore,
they are treated as a part of W signal in the signal extraction. The fraction of Z decay contribution
is estimated using NLO calculation, and subtracted in calculation of the W → e cross section which
is described in Sec. 5.1.
• W → τ → e decays: The branching ratio of τ to e is about 20%. The contribution of thisdecay mode was evaluated in event generator (PYTHIA) and found to be negligible in the range of
30 < pT < 50 GeV/c with current statistics.
The main components of background are from accidental track match, π 0 conversion and charged hadron.
These components are estimated using the real data and MC simulation.
The background estimation is based on the pT distribution of all EMCal clusters (N EM Cal ) in which
the dominant component is γs from π 0 decays. The EMCal cluster distribution is produced without any
vertex cut though the vertex cut of the PHENIX nose cone (40 cm) is applied naturally. The procedureof the background estimation is following:
1. The probabilities of accidental track match (Pacc ) and π 0 conversion (Pπ0 ) are evaluated by real
data and MC simulation.
Then, the pT distributions of accidental track match and π 0 conversion are obtained by multiplying
those probabilities to N EM Cal ((Pacc + Pπ0 ) × N EM Cal ). Note that N EM Cal is made with the
40 cm vertex cut, therefore the probabilities derived here need to be multiplied by the additional
factor of vertex cut (0.80 = f vtx (±30cm)/f vtx (±40cm), see Sec. 4.3.1) in order to compare themwith Fig. 4.18.

88

2. The shape (pT dependence) of charged hadron background (N hadron ) is evaluated using MC simulation. The obtained shape is then scaled so that the low pT region (12 < pT < 20 GeV/c) of
Fig. 4.18 can be explained by the charged hadron ((scale) × N hadron ).
3. The combination of these components are used for the background evaluation:
N bg= f vtx (±30cm)/f vtx (±40cm) × (Pacc + Pπ0 ) × N EM Cal + (scale) × N hadron= 0.80 × (Pacc + Pπ0 ) × N EM Cal + (scale) × N hadron .

(4.6)

The detail of each part is described below.
All EMCal Clusters: N EM Cal
N EM Cal is the pT distribution of EMCal clusters with subtracting cosmic components. The cosmic
components is estimated using the EMCal Tof value (Sec. 4.2.3), with requiring ”out” of timing (−20 <Tof < −10 nsec or 20 < Tof < 40 nsec) for the EMCal clusters. Here, the Tof window is selected so thatthe window size matches to the size of Tof window which is used in this analysis (−10 < Tof < 20 nsec).
Then the distribution of N EM Cal is obtained by statistically subtracting ”out” of timing from ”in” oftiming distribution. The main component of N EM Cal is γs from π 0 decays.
In the background estimation, N EM Cal extracted here is fitted with a power law function to getsmooth spectrum. In the following discussion, the obtained fit function is used as N EM Cal .
Accidental Track Match: Pacc
The probability of accidental track match is extracted from the tail of ∆φ distribution which is usedin DC-EMCal matching (Fig. 4.19). The pT distribution of EMCal clusters with track association in
0.01 < |∆φ| < 0.02 window is produced and compared to the N EM Cal distribution. The obtained ratio
is 0.68 ± 0.02 % without any pT dependence, and this is used as Pacc . Note that this number naturally
includes the overall efficiency of geometry and analysis cuts (Sec. 4.3.5) since the pT distribution withtrack association in 0.01 < |∆φ| < 0.02 is made with all cuts, which is described in Sec. 4.3.2, except forthe 30 cm vertex cut and DC-EMCal matching cut 18 .
π 0 Conversion: Pπ0
Events which have their origin in π 0 conversion contribute to the background. Because the opening angleof two γs from high pT π 0 decay can be quite small, a fake high pT track would be observed when one of
γs is converted to electrons before the DC and another γ is detected at the EMCal.
The probability of such events is evaluated using single particle MC simulation. Single π 0 events arethrown to PISA with flat pT distribution (20 < pT < 70 GeV/c). The output distribution is weighted
by the cross section from theory of the NLO perturbative QCD (pQCD) calculation [114] according to
18 P
acc is not affected by the change of DC-EMCal matching cut because the width of ∆φ window does not change between|∆φ| < 0.01 and 0.01 < |∆φ| < 0.02.

89the input pT of π 0 . Then the fraction of EMCal clusters which is associated with the DC/PC1 tracks
are studied. As a result, obtained Pπ0 is 1.07 ± 0.06 % without any pT dependence. In addition to all
cuts described in Sec. 4.3.2, φ-dependent efficiency (see Sec. 4.3.5) is included in this study to include
the acceptance change of the DC/PC1 during Run 9 period.
The obtained Pπ0 can be checked with a simple calculation. The conversion probability of γ calculated
from the materials before the DC is 2.6 %, and the number which corresponds to Pπ0 is 0.96 % =2.6 × 0.152/0.35 × 0.85 (0.152/0.35: the efficiency of geometry and analysis cuts from Sec. 4.3.5, 0.85: DC
wire region cut from Sec. 4.2.5). This number is consistent with the value from MC simulation within
uncertainty 19 .
Uncertainty of Accidental Track Match and π 0 Conversion
It needs careful treatment when adding Pacc and Pπ0 , because these probabilities might have an overlap.
As discussed in above, Pπ0 is 1.07 % from single π 0 MC, and it is 0.68 % for Pacc . If all the accidental
track match is caused by the π 0 conversion, the added probability would be 1.07 %, and the value
would be 1.75 % if two values are fully uncorrelated. Therefore, the factor multiplied to N EM Cal (i.e.
0.80 × (Pacc + Pπ0 ) ) is: 0.86 % as the minimum, 1.40 % as the maximum value. In this analysis, 1.00 %is set as a default value for simplicity.
The uncertainties of Pacc , Pπ0 and the vertex cut factor (0.80 = f vtx (±30cm)/f vtx (±40cm)) are
2.9 %, 5.6 % and 2.0 % of each value 20 . They are ignored here because it is small enough comparing to
the uncertainty of Pacc + Pπ0 assigned above.
Charged Hadron: N hadron
It is difficult to evaluate the absolute value of N hadron from simulation, so the MC simulation is onlyused for extracting the relative shape (i.e. pT dependence) of N hadron . Single π ± are generated with flatpT distribution (0 < pT < 100 GeV/c) and thrown to PISA. The output distribution is weighted by thecross section from theory of the NLO pQCD calculation according to the input pT of π ± .
In the background estimation, only the shape of the obtained distributions is used. After subtracting
components of accidental track match and π 0 conversion (0.80 × (Pacc + Pπ0 ) × N EM Cal ) from Fig. 4.18,
the shape of charged hadron is scaled so that the low pT region (12 < pT < 20 GeV/c) of the figure canbe explained by the charged hadron.
Uncertainty of Charged Hadron
The uncertainty of charged hadron background is estimated by changing the normalized range. The
default normalized range is set as 12 < pT < 20 GeV/c, and it was changed to 16 < pT < 20 GeV/c.
The difference of the scale factor is 17 % and it is included as the systematic uncertainty.
19 The uncertainty of efficiency of geometry and analysis cuts: 9.8 % (Sec. 4.3.5).
20 The uncertainty of the vertex cut factor is calculated from f vtx (±30cm) = 0.499, f vtx (±40cm) = 0.620 and theuncertainty of 7.2 % for f vtx (see Sec. 4.3.1).

90Note that the theoretical uncertainties from NLO pQCD calculations are ignored here. It might affectto the pT dependence of N hadron , but is small compared to to other sources of systematic uncertainty(e.g. uncertainty of Pacc + Pπ0 ).
Estimated BackgroundThe background is estimated following Eq. 4.6. Figure 4.20 displays the result of background estimation(Left: positive charge, Righ: negative charge). As the figure shows, the dominant background around 30
to 40 GeV/c is from components of accidental track match plus π 0 conversion (0.8×(Pacc +Pπ0 )×N EM Cal :Red distribution in the figure).

Figure 4.20: Background estimation. (Left: positive charge, Right: negative charge) The Red distribution
is components of accidental track match plus π 0 conversion (0.8 × (Pacc + Pπ0 ) × N EM Cal ) and the Greenis charged hadron ((scale) × N hadron ).

91

Contribution from Heavy Flavor Decays
The contribution of e± s from heavy flavor decays is ignored in the above discussion. Since the π ± spectra
folded by the EMCal response (N hadron ) is used to normalize the low pT region, only the difference in the
shape between heavy flavor contribution and π ± contribution is the source of systematics. The FONLL(Fixed-Order plus Next-to-Leading-Log) calculation 21 is used to evaluate the difference as PHENIX√
single electron measurement at s = 200 GeV agrees with the calculation within uncertainty [119].
Figure 4.21 shows the background estimation including heavy flavor contribution (thick Black distribution). The Red distribution is the components of accidental track match plus π 0 conversion (0.80 ×(Pacc + Pπ0 ) × N EM Cal ), the Black shows charged hadron ((scale) × N hadron ), and the Blue displays
heavy flavor contribution in the figure. It can be noticed that the FONLL spectra shape is similar tothe shape of π ± and the contribution of heavy flavor is about 10 % of the whole EMCal clusters withtrack (Black points with error bar). The difference between two procedures of with/without heavy flavor
component is very small, as the process of normalization of the low pT region with charged hadron shapecompensates the difference.
Figure 4.22 shows the difference of two total background spectra (with/without heavy flavor) divided
by the one without heavy flavor. The difference is less than 1 % at pT > 10 GeV/c, and it can be safely
stated that the background estimation described above covers heavy flavor contribution naturally.

Figure 4.21: Background estimation with heavy
flavor component (thick Black distribution). The
Red distribution is components of accidental track
match + π 0 conversion, the Black shows charged
hadron, and the Blue displays heavy flavor contribution (FONLL calculation).

Figure 4.22: The fractional difference of backgroundestimation between with and without heavy flavor contribution. (”N bg (w.o. h.f.)” denotes theestimated background without heavy flavor, and”N bg (w. h.f.)” denotes it with heavy flavor component) The difference is very small.

21 The theory based on pQCD calculation about heavy flavor production [115, 116, 117, 118].

92

Uncertainty of Background Spectrum (Shape Uncertainty)
It is better to confirm the background spectrum using a different approach. The problem is that it is hardto get a pure background sample at PHENIX. The best spectrum representing the QCD background is
obtained by requiring anti-isolation cut in negative particle samples. The isolation cut is explained inSec. 4.4.1, and anti-isolation cut selects events which is rejected by the isolation cut.
The data points with statistical error bars in Fig. 4.23 show the sample of the QCD backgroundobtained by anti-isolation cut in negative charge events. The Red (Blue) histogram in the figure is theestimated background for positive (negative) charge particles.
The spectrum of the obtained QCD background (distribution of negative charge with anti-isolationcut) is compared with the distribution of estimated background. Figure 4.24 displays the ratio of the QCD
background over the estimated background for positive charge. The shape uncertainty is determined by
the figure as an angle range which is extracted by fitting with linear function with setting the averagevalue at 12−20 GeV/c (or 16−20 GeV/c) as an anchor point. The extracted range (it is 0.0054 ± 0.0078
in the figure) is consistent to 0, and ±1 σ range is used in the background band calculation.

Figure 4.23: The pT distribution of negative parti- Figure 4.24: the ratio of the QCD background overcle with applying ant-isolation cut (points with error the estimated background for positive charge. Thebar) which represents QCD background spectrum. shape uncertainty is determined by this figure.
he Red (Blue) histogram is the estimated backgroundfor positive (negative) charge particles.

93Summary of Background Estimation
Considering the uncertainty of Pacc + Pπ0 (default, min, max), the normalization range of charged hadron(12 < pT < 20, 16 < pT < 20 GeV/c) and the shape uncertainty (default, ±1 σ), 18 (= 3 × 2 × 3)
background distributions are made in total, and the background band is evaluated by taking the maximum
and minimum value at each pT bin among 18 distributions.
Figure 4.25 displays the evaluated background band (Magenta band) with the whole EMCal clusters
with track (Black distribution) (Left: positive, Right: negative charge). The background band at theregion of pT > 30 GeV/c is dominated by the shape uncertainty.
The number of background in the range of 30 < pT < 50 GeV/c is summarized in Table 4.4, and this104

Counts per 2 GeV bin

Counts per 2 GeV binnumber is used in dσ/dy calculation.

Positron candidates3

Raw counts

10

Estimated BG
102

10Electron candidates
3

10

102

10

1
10

104

1
20

30

40

50

60
70
pT (GeV/c)

10

20

30

40

50

60
70
pT (GeV/c)

Figure 4.25: The evaluated background band (Magenta) with the whole EMCal clusters with track (Black
distribution) (Left: positive, Right: negative charge).

charge+−±total number
62
17
79

background (min, max)
7.0 (5.7, 11.0)
6.4 (4.1, 9.1)
13.3 (9.9, 19.9)

Table 4.4: The number of events and background in the range of 30 < pT < 50 GeV/c.

94

4.3.4

Trigger Efficiency

The threshold and efficiency of ERT4x4b trigger is checked using number of π 0 from MB (BBCLL1)
triggered events. Red points in Fig. 4.26 show the turn-on curves of ERT4x4b as a function of π 0 pT for
both PbSc and PbGl. Since statistics is not enough to cover high pT region in MB triggered sample, thetrigger threshold is also checked using ERT2x2 triggered events in the following manner:( ERT4x4b× εERT2x2 |plateau ,/NπERT2x2ε= NπERT4x4b00/NπM0 B ,εERT2x2 = NπERT2x2
0

(4.7)

where εERT4x4b (εERT2x2 ) represents the efficiency of ERT4x4b (ERT2x2) trigger. The plateau of εERT2x2(εERT2x2 |plateau ) can be evaluated in MB triggered sample because the threshold of ERT2x2 is lower than
ERT4x4b (see Sec. 2.2.5), and statistics of ERT2x2 triggered data is larger than ERT4x4b triggered data.
Blue points in Fig. 4.26 displays the turn-on curves of ERT4x4b calculated in this manner. Note thatthe efficiency value at plateau may exceed 1.0 because of the indirect efficiency calculation. The turn-on
curves reach plateau level at 12 GeV and the plateau values are 0.98 ± 0.03 for PbSc and 1.02 ± 0.04 for
PbGl. These values are consistent with the expected values with taking masked FEMs into account (seeSec. 4.3.2) and it means the trigger efficiency of each FEMs is 100 % at pT > 12 GeV/c region.

Figure 4.26: ERT4x4b turn on curve (Left: PbSc, Right: PbGl). The red data points are from MB. The
blue one is based on clusters fired ERT2x2 multiplied by ERT2x2 efficiency.

954.3.5

Efficiency of Geometry and Analysis Cuts

The efficiency of geometry and the analysis cuts (εreco ) is evaluated using single particle MC simulation.
During Run 9 period, run-by-run fluctuation caused by the pile up and the acceptance change of the
DC/PC1 is quite large (see Appendix D for tracking performance during Run 9). In order to include thisfluctuation in the simulation, the φ-dependent efficiency of DC/PC1 is derived.
DC/PC1 Efficiency
The DC/PC1 efficiency including run dependence is extracted comparing the real data with single particle
MC. Since the DC and PC dead map input into the simulation is from run 276870, this run is taken as
the fiducial run. The procedure is:
Step 1 Compare φ distribution of the fiducial run with the simulation to determine the scale factor.
(Fig. 4.27)
Step 2 Make φ distribution accumulating whole available runs from Run 9 (276 runs). In this procedure,run-by-run rate correction was applied and the φ distributions of each run are averaged with a weightof luminosity.
Step 3 Using the scale factor obtained at Step 1, the accumulated φ distribution obtained in Step 2 isnormalized and compared with the simulation. (Fig. 4.28) Take a ratio for each bin and assigned
as an efficiency of that particular φ range.
The detail of each steps are described below.
Step 1:The φ distributions are generated from MB triggered sample to compare them with the simu-

lation. Applied cut to make histograms are:
• Vertex cut: |zvertex | < 30 cm 22• Require hits on X1 and X2 plane of the DC, and hit on the PC1• pT > 2.0 GeV/c cut for real data (pT = 10, 20, 30, 40, 50 GeV/c for simulation)• Dead map of DC/PC1 in the simulation is taken from run 276870 (fiducial run 23 )• Normalization of hists: by number of MB (for real data) or number of input (for simulation)
Because of a few statistics in real data, it is impossible to set exactly the same pT cut between the real
and the simulation data. Actually, there is a pT dependence in acceptance value for pT < 2 GeV/c.
However, it is constant above pT > 2.0 GeV/c region [119] and the difference of the pT cut applied here
does not affect to the acceptance calculation. The cuts are kept as simple as possible to be independent
22 z
vertex used here is reconstructed by the BBC. As described in Sec. 4.3.1, it may be mis-reconstructed, but it does notaffect to the evaluated value of εreco here.
23 The fiducial run is randomly chosen from runs in which rather larger area of DC/PC1 were active.

96

from the EMCal effect, and any electron cut that might be strongly coupled to EMCal information is notapplied. Overall εreco value including EMCal acceptance is evaluated after extracting DC/PC1 efficiency
and that is described later.
The scale factor to match the scale of the data and the simulation is calculated using the top histogramin Fig. 4.27 (North + South side of the DC/PC1). The same scale factor is used in the middle (Northside of the DC/PC1) and bottom (South side of the DC/PC1) histograms of Fig. 4.27 and agreement
between real data and simulation looks good. The systematic uncertainty of the scale factor is estimatedfrom the ratio of the real data and simulation in the middle and bottom hists of the figure.

Figure 4.27: Comparison of the φ distribution of the fiducial run (Red hist) to the simulation (Blackhist). The red shaded areas indicate the out of acceptance of PHENIX. Top: North+South side of the
DC/PC1 (the scale factor to match the scale of the data and the simulation is calculated from this figure),
Middle: North side, Bottom: South side, Blue lines indicate the rough boarder of EMCal Sectors.

Step 2:In order to accumulate φ histograms of whole runs in Run 9, the rate correction and luminosity

weighting is necessary.
The run-by-run rate dependence should be corrected when averaging φ histograms from different runs.
The rate dependence can be explained by the pile up effect of the DC which is studied in Appendix D.
The correction factor is calculated based on that study 24 .
After correcting the run-by-run rate dependence, φ histograms of whole runs are summed up withluminosity weighting.

24 The gate width of the DC (denoted as ”NCLK” in the Appendix D) which causes the pile up is fixed at the value of

3.1.

97

Step 3:Using the scale factor obtained at Step 1, the accumulated histogram is normalized to compare

to the simulation. Figure 4.28 shows the comparison of accumulated histogram to the simulation. The
ratio of the accumulated histogram and the simulation is assigned as the efficiency of that particular φ
range. The obtained φ-dependent efficiency is included when calculating the efficiency of the geometry
and analysis cuts (εreco ) using single particle MC simulation. Note that the efficiency value may exceed
1.0, however the value is included in the MC simulation without modification because the purpose of thisφ depending efficiency is to reproduce the φ distribution of the real data in the MC simulation.

Figure 4.28: Comparison of the accumulated φ distribution (Red hist) to the simulation (Black hist).
The Black histogram is exactly same as it in Fig. 4.27 except for the number of bins. The accumulated
hist is scaled with the factor obtained in Step 1. The red shaded areas indicate the out of acceptance of
PHENIX. Top: North+South side of the DC/PC1, Middle: North side, Bottom: South sied, Blue lines
indicate the rough boarder of EMCal Sectors.

Efficiency Calculation
The efficiency of geometry and the analysis cuts (εreco ) has been calculated using single electron/positron
MC. 50K events (pT = 10, 20, 30, 40, 50 GeV/c, 10K events for each pT , |zvertex | < 45 cm) for eachparticle were generated and thrown to PISA, which is the GEANT based MC simulation including the
response of whole PHENIX detector. Number of events in output of PISA was counted to calculate theefficiency. In this process, all cuts, which are described in Sec. 4.3.2, except for the anode wire region cut
are applied and φ-dependent DC/PC1 efficiency evaluated above was also taken into account.
The bias from the chosen fiducial run at Step 1 of φ-dependent efficiency evaluation is estimatedby several MC simulation studies including different fiducial runs. The efficiency of geometry and theanalysis cuts (εreco ) are calculated for each simulation.
98

There are three types of systematic uncertainty in this calculation. First one is the uncertainty ofthe scale factor to match the simulation and the real data 25 . Second one is the uncertainty of the rate
correction 26 . This uncertainty is evalated by comparing the efficiency value of ”with” and ”without” therate correction The difference can be taken as the maximum uncertainty since the gate width of the DC
is certainly more than 1 beam clock and the rate dependence should be corrected. The last one is thevariation of εreco among different MC data set 27 .
Combining all data, εreco = 0.152, and the uncertainty from the scale factor is 7.0 %. The differenceof ”with” and ”without” rate correction is evaluated as 6.6 %. The standard deviation of the efficiency
values in different MC data set is only 1.7 %, and therefore it can be stated that there is no charge or
pT dependence in εreco . As a result, 9.8 % systematic uncertainty is assigned for the efficiency value byadding them in quadratic.
Since the active EMCal fraction is ∼ 81 % (see Sec. 4.3.2) and the maximum acceptance value is 0.35
in PHENIX, it means the efficiency of tracking part is ∼ 54 % (= 0.152/0.35/0.81) at Run 9. Considering
the anode wire region cut, this number should be multiplied by a factor of 0.85 since the cut masks 15 %of the acceptance.

4.3.6

Energy Smearing

The energy smearing effect (εsmear ) should be taken into account due to the limited energy resolutionof EMCal. It is evaluated with NLO based event generator (CHE: see Sec. 1.3) including the energy
resolution of EMCal, which is extracted from the FastMC 28 .
The obtained cross section values (dσ/dy) in 30 < pT < 50 GeV/c range for with/without smearing
effect are compared to evaluate the energy smearing effect (1/εsmear ), and the obtained value is 1.01.

25 Introduced at Step 1 of the evaluation of φ-dependent DC/PC1 efficiency.
26 Applied at Step 2 of the evaluation of φ-dependent DC/PC1 efficiency.
27 2(positron/electron) × 5(different p ) × 5(different fiducial runs) MC simulation are executed in total
T√
28 used EMCal resolution is: ∆E/E = 0.081/ E ⊕ 0.05, see Sec 4.2.2

99

4.3.7

Cross Section Calculation

The calculation of dσ/dy at 30 < pT < 50 GeV/c region is represented in this subsection. The relationshipof dσ/dy to the experimental observables is:Z
X (N meas − N bg )
dσ
d2 σ
1
1
1
·
.
= dpT
= ·
·
dy
dpT dyL εtrig · ·εreco · εsmear ∆y 30<p <50
∆pT

(4.8)

T

Each factor is evaluated in previous subsections, and summarized in Table 4.5 with its uncertainty.
factorL

value
8.56 pb−1

uncertainty
13 %comment and Section
Sec. 4.3.1

N measSummarized in Table 4.6, Sec. 4.3.2

N bg

Summarized in Table 4.6, Sec. 4.3.3

εtrig

100 %

Sec. 4.3.4

εreco

0.152 × 0.85

1/εsmear

1.01

Sec. 4.3.6

∆y

1.00

PHENIX acceptance is included in εreco

9.8 %Ideal value is 0.35, Sec. 4.3.5

Table 4.5: The summary of each factor in Eq. 4.8.

The uncertainty of energy smearing (1/εsmear ) is ignored here, because it is estimated based on
the NLO calculation and the theoretical uncertainty is small compared to other sources of systematic
uncertainty (e.g. N bg ). The uncertainty of L and εreco is the scale uncertainties, and its value is 16 % in
total by quadratic sum. The uncertainty of N bg affects to the absolute value of the cross section.
Other uncertainties which might affect to the absolute value of the cross section are the uncertainty
of absolute energy scale and the charge contamination caused by the DC resolution. The uncertainty ofabsolute energy scale is 2.9 % (see Sec. 4.2.2), and this is ignored in this analysis as it is small comparedto the uncertainty of background estimation. The probability of charge mis-identification is 1.1 % (seeSec. 4.2.5). In term of count, this probability is corresponding to 0.19 (= 17 × 0.011) for positive and0.68 (= 62 × 0.018) for negative charged particle. As the contamination from charge mis-identification issmall compared to the uncertainty of background estimation, it is also ignored in this analysis.
Background counts are subtracted from number of measured events. Figure 4.29 shows dN/dpT after
subtracting the background (Left: positive, Right: negative charge). In this figure, the NLO calculation(CHE) with energy smearing is overdrawn (Red distribution). The yellow bands in the figures show therange of the different background assumptions and the horizontal bars indicate the default data points.
The larger bands close to the edge of normalization (pT ∼ 20 GeV/c) is mainly caused by the differenceof the normalization range of charged hadron (12 < pT < 20 or 16 < pT < 20 GeV/c). The statisticaluncertainties on each point show 68 %CL calculated with TRolke function in ROOT 29 . When the value
29 This function computes confidence intervals for the rate of a Poisson process in the presence of uncertain backgroundand efficiency [120].

100

is small, a difference from the simple estimation with Poisson statistics (=√S + N ) can be noticed.

The number of counts in the signal region are summarized in Table 4.6. The systematic uncertainties
from different background estimation are also listed in the table. Note that these numbers are not only
including W signal but also including Z/γ signal.
The cross section (dσ/dy) at 30 < pT < 50 GeV/c is then calculated. The numbers in Table 4.6 are
multiplied/divided by the factor listed in Table 4.5. The result is shown in Table 4.7 which also includes
the systematic uncertainty from the scale factor (16 %). For example, the cross section of positive charge

dN/dp (GeV/c) -1

7

1.5

Positron signal

5Electron signal

T

6

T

dN/dp (GeV/c) -1

is 50.2 = (62 − 7.0)/8.56/0.152/0.85 × 1.01). These values are compared to theory calculation in Sec. 5.1.

4

1

0.5

3
2

0

1
0
-1

20

-0.5
30

40

50

60

70

20

30

40

50

60

p T (GeV/c)

70

p T (GeV/c)

Figure 4.29: dN/dpT of W /Z decay electrons after background subtraction (Left: positive, Right: negative). The red distribution represents the NLO calculation (CHE) of W + Z with energy smearing.

charge+−±total number
62
17
79

background (min, max)
7.0 (5.7, 11.0)
6.4 (4.1, 9.1)
13.3 (9.9, 19.9)

signal counts (W and Z)
55.0 ± 7.9(stat.) +1.3
−4.0 (sys.)
10.6 ± 4.1(stat.) +2.3
−2.7 (sys.)
65.7 ±8.9(stat.) +3.4
−6.6 (sys.)

Table 4.6: The number of events in the range of 30 < pT < 50 GeV/c.

charge+−±dσ/dy (pb) at 30 < pT < 50 GeV/c
50.2 ± 7.2(stat.) +1.2
−3.6 (sys.) ± 8.0 (normalization)
9.7 ± 3.7(stat.) +2.1
−2.5 (sys.) ± 1.6 (normalization)
60.0 ± 8.1(stat.) +3.1
−6.0 (sys.) ± 9.6 (normalization)

Table 4.7: dσ/dy of electrons form W and Z decay at 30 < peT < 50 GeV/c.

101

4.4

Spin Asymmetry

The single spin asymmetry (AL ) is calculated by following formula:
εL

=

AL

=

N+ − N− · R,
N+ + N− · R
1· εL ,
P(4.9)(4.10)

where εL is called single ”raw” asymmetry, N is the electron yield from W and Z decay, R is the relative
luminosity and P is the beam polarization. R is defined using luminosity (L) as R = L+ /L− . The signof + (−) put on N or L represents the helicity states of proton beam. As RHIC provides 4 helicity statesof proton beams ((PB PY ) = (++), (+−), (−+), (−−): B (Y ) denotes Blue (Yellow) beam), one beampolarization is summed up relative to another beam polarization when calculating N+ , N− , L+ and L−
30. Since two (Blue and Yellow) polarized beams are available at RHIC, AL can be measured for bothbeam polarizations, and a particular observed event can be used in both AL measurements. However,
AL is not calculated with this manner in this analysis. The uncertainty should be assigned with Poisson
statistics due to its small statistics and it is difficult when using Eq. 4.9 and 4.10.
In order to compensate the small statistics, a technique called ”spin fit” is engaged in this analysis.
This technique is based on the following equations:
σ++ = N++ /L++ = (1 + AL PB + AL PY + ALL PB PY ) · σ,
σ+− = N+− /L+− = (1 + AL PB − AL PY − ALL PB PY ) · σ,σ−+ = N−+ /L−+ = (1 − AL PB + AL PY − ALL PB PY ) · σ,
σ−− = N−− /L−− = (1 − AL PB − AL PY + ALL PB PY ) · σ,

(4.11)

where σ represents the cross section of electron from W and Z decay without efficiency correction and
ALL is the double spin asymmetry. The values of σ, AL and ALL is obtained by fitting measured N
and L with Eq. 4.11. Substituting AL to εL with the relation of AL P = εL (P : averaged polarization,
P = (PB + PY )/2), Eq. 4.11 can be written as:σ++ = N++ /L++ = (1 + 2εL + εLL ) · σ,
σ+− = N+− /L+− = (1 + (PB − PY )/P · εL − εLL ) · σ,σ−+ = N−+ /L−+ = (1 + (−PB + PY )/P · εL − εLL ) · σ,
σ−− = N−− /L−− = (1 − 2εL + εLL ) · σ.

(4.12)

Using this formula, values of σ, εL and εLL = ALL PB PY (double ”raw” asymmetry) is obtained. Contrast
to the separated measurement of AL for Blue/Yellow beam polarizations, this technique naturally uses
all available statistics. In addition, the maximum likelihood method can be used in the fitting process
and the uncertainty of small statistics can be treated properly.
The procedure of AL calculation is fully described in this section 31 . First of all, a cut called isolation
cut is explained in Sec. 4.4.1. Since the contribution of background to the AL value is a serious problem,the fraction of background should be suppressed as lower as possible. Therefore, the isolation cut isemployed to improve the signal purity in the AL measurement. This cut is not used in the cross section
30 For example, N = N+++ + N+− , N− = N−+ + N−+ etc.
31 The study of contamination from a residual transverse component of the beam polarization is described in Appendix E.

102measurement as we can reliably subtract the background component with proper background estimationin the cross section calculation. Secondly, the calculation of single raw asymmetry (εL ) with spin fit
technique is presented in Sec. 4.4.2. Finally, AL value is evaluated in Sec. 4.4.3. The treatment ofsystematic uncertainty from the dilution factor of hadron background and beam polarization is also
explained in this subsection.

4.4.1

Isolation Cut

As the signature of e± s from W or Z decays is that they are observed as isolated events, an electron
isolation cut, which requires no jet activity in the vicinity of the electron, is applied to improve the signal
purity in the spin asymmetry analysis in addition to various cuts which are applied in the cross section
measurement (Sec. 4.3.2).
The reason why the isolation cut is not applied in the cross section measurement is originated in
the difficulty of its efficiency estimation. When estimating the efficiency of the isolation cut, a complete
run-by-run trace of the EMCal and DC acceptance change must be included in the simulation, but it isdifficult for Run 9 analysis due to the frequent acceptance change of the DC (see Appendix D). However,the background contribution to the cross section can be reliably subtracted as long as we properly estimate
the background. By these reasons, the isolation cut is not employed in the cross section measurement.
On the other hand, the contribution of background to the AL value is a serious problem because itis hard to estimate 32 and leaves quite large uncertainty in the AL measurement. However, there is afeature in the AL measurement that any cut can be applied as long as the cut is spin independent. This isbecause the cut efficiency does not affect to the spin asymmetry (AL or ALL ) values in Eq. 4.11 and 4.12
though it does affect to the σ value. Therefore, the isolation cut, which requires the sum of EMCal cluster
energy and track momentum in the cone radius of 0.5 rad from the electron candidate to be less than
2 GeV (Fig. 4.30), is applied in the spin asymmetry measurement to suppress the background fraction as
low as possible. The detail of the selection of EMCal cluster energy and track momentum which is usedin the summation of cone energy is summarized in Table 4.8.
In addition to the isolation cut, the zero degree track cut (|α| > 1 mrad) is also applied to reject
e± candidates which track angle is parallel to the wires of DC, so that the charge contamination can be
negligible. It corresponds to 3.2 σ or 0.07 % of the opposite charge sample 33 .
Figure 4.31 displays both spectra with (Red) and without (Black) the isolation cut (Left: positive,
Right: negative). The Magenta bands is the QCD background estimated in Sec. 4.3.3 scaled by aconstant factor which is evaluated by taking a ratio of the with/without isolation cut spectrum at 12 <pT < 20 GeV/c region (0.224 for positive, 0.258 for negative). Numbers of signal and background counts
in 30 < pT < 50 GeV/c region are summarized in Table 4.9.
32 The expected A value of the hadron background is 0, but there may be a contribution from hadronic decays of W andL
Z bosons and they are difficult to estimate accurately.
33 As the resolution of the DC is 1.1 mrad and α is about 2.5 mrad for 40 GeV/c track (see Sec. 4.2.5), |α| > 1 mrad
cut corresponding to (2.5 + 1.0)/1.1 = 3.2 σ away from the opposite charge, and this reduces the probability of charge
contamination.

103

EMCal cluster

DC/PC1 track• minimum energy cut:(cluster energy) > 0.15 GeV• charge veto: not requiring track match• require hits on X1 and X2 planeof the DC and hit on PC1• minimum momentum cut:(momentum) > 0.2 GeV/c

104

Counts per 2 GeV bin

Counts per 2 GeV bin

Figure 4.30: Schematical representationof the isolation cut. After calculating asummation of EMCal cluster energy for
γs (red circles) and track momentum for Table 4.8: The selection of EMCal cluster and DC/PC1 track
charged particles (blue lines) which are ob- for the summation of cone energy.
served inside a cone radius of 0.5 rad froma high pT electron event, require the summation is less than 2 GeV in order to selectisolated events.

Positron candidates
Before Isolation Cut
After Isolation Cut
Estimated BG

3

10

102

104

Electron candidates
3

10

102

10

10

1

1

10

20

30

40

50

60
70
pT (GeV/c)

10

20

30

40

50

60
70
pT (GeV/c)

Figure 4.31: The pT spectra with (Red) and without (Black) the isolation cut (Left: positive, Right:
negative). The Magenta band shows the estimated background.

charge+ (without isolation cut)(with isolation cut)
− (without isolation cut)(with isolation cut)

total
62
42
17
13

estimated background (min, max)
7.0 (5.7, 11.0)
1.6 (1.3, 2.5)
6.4 (4.1, 9.1)
1.7 (1.1, 2.4)

signal (W and Z)
55.0 ± 7.9(stat.)+1.3−4.0 (sys.)
40.4 ± 6.5(stat.)+0.3−0.9 (sys.)
10.6 ± 4.1(stat.)+2.3−2.7 (sys.)
11.4 ± 3.6(stat.)+0.6−0.7 (sys.)

Table 4.9: Number of events in the signal region (30 < pT < 50 GeV/c)

104

Check of the Isolation Cut
The signal fraction of with/without the isolation cut is expected to be approximately same for positive
and negative. Equation 4.13 shows the definition of the fraction (S denotes the number of signal), where
N represents the number of measured count, N iso and x are the number of count which pass the isolation
cut, N iso and y represent the number of count which is rejected by the isolation cut 34 , B and z is thenumber of estimated background and a is the ratio of with/without isolation cut at 12 < pT < 20 GeV/cregion (ratio for background: 0.224 for positive, 0.258 for negative). Here, x, y and z are independent
values. The uncertainty of this fraction is derived from Eq. 4.14.

r=

S iso
N iso − B iso
N iso − aB
x − az
=
=
=
,
iso
iso
S
N −B
(x + y) − z(N + N ) − B(4.13)

s
∆r

=

(
r

=

r

∂r
∂r
∂r
∆x)2 + ( ∆y)2 + ( ∆z)2
∂x
∂y
∂z
((

1
1
1
a
1
− )∆x)2 + ((− )∆y)2 + ((− iso + )∆z)2 .
S iso
S
S
S
S

(4.14)

Each term in the equations is shown in Table 4.10 35 , and the fraction is calculated as r+ = 0.73±0.08,
r− = 1.08 ± 0.29 by substituting these values. They are consistent within uncertainties.
The pT dependence of the isolation cut is studied based on PYTHIA MC. While there is no pT
dependence for e± s from W or Z decays, a slight pT dependence is observed for hadron backgrounds 36 .
The study also shows the pT dependence is absorbed in the estimated background band and does notaffect to the result.
charge

S

S iso

a

+
−

55.0
10.6

40.4
11.4

0.224
0.258

∆x
√
42
=
6.48
√
13 = 3.61

∆y
√
62
−
42
=
4.47
√
17 − 13 = 2.00

∆z
4.0
2.7

Table 4.10: Components to calculate isolation cut ratio.

34 The relation is N = N iso + N iso = x + y
35 The uncertainty of the estimated background (∆z) is assigned as 4.0 for positive and 2.7 for negative, so that they canfully cover the background bands.
36 The rejection for hadron backgrounds becomes larger in higher p region (p >∼ 30 GeV/c) than lower p region.
T
T
T

105

4.4.2

Calculation of Spin Asymmetry with Spin Fit Technique

Luminosity of Four Spin States
Number of BBCLL1 (MB trigger for the PHENIX) counts are sorted into four spin states, and used asluminosity (L++ , L+− , L−+ and L−− ). Although these numbers are not converted to the luminosity
dimension with some correction factor (e.g. σBBC , rate correction etc.), it only affects to the scale factor(i.e. σ in Eq. 4.11 and 4.12) and does not matter for spin asymmetry calculation.
Spin Fit with Maximum Likelihood Method
Number of events with applying the isolation cut (Red histograms in Fig 4.31) in the signal region(30 < pT < 50 GeV/c) are sorted into four spin states. As the beam polarization 37 and luminosity offour spin states have been already known, there remains 3 parameters (σ, εL , εLL ) in Eq. 4.12. These
values can be evaluated by fitting. However, the fit must be carried out with likelihood method in thiscase due to the small statistics.
The likelihood function can be written as:
L=Yf (Ndata ; µexpected ) = f (N++ ; µ++ ) · f (N+− ; µ+− ) · f (N−+ ; µ−+ ) · f (N−− ; µ−− ), (4.15)

4 spin states

where f (N ; µ) denotes a Poisson distribution with mean value of µ, and Ni is the number of events ineach four spin state (i stands for a spin state). Using luminosity of four spin states (Li ), µi is written as:µ++ = L++ · σ++ ,
µ+− = L+− · σ+− ,(4.16)µ−+ = L−+ · σ−+ ,µ−− = L−− · σ−− .
The explicit forms of σi are represented in Eq. 4.12, and they include 3 parameters of (σ, εL , εLL ). In
the maximum likelihood method, the most probable values of each parameter are determined so that thevalue of L become the maximum.
Figure 4.32 displays the result of (σ, εL , εLL ) scan for positive particles. The obtained probability
distribution is sliced to 3 planes ((εL , εLL ), (σ, εL and (σ, εLL )) at the point where the likelihood function
takes a maximum value (the left row of the figure show the image). The right row of the figure display
the probability distribution in each 3 planes (Top: (εL , εLL ), Middle: (σ, εL ), Bottom: (σ, εLL )). The
boundaries observed in the probability distributions come from the requirement of σi ≥ 0 in Eq. 4.12.
Figure 4.33 shows how the uncertainty is obtained. The uncertainties of εL and εLL are calculated in
the following manner:
1. Project the probability distribution to the εL or εLL axis.
2. Normalize the projected distribution.
3. Get the boundary where the integrated probability become 0.68.
37 P

B = 0.38,

PY = 0.40 and the averaged value (P ) is 0.39 with uncertainty of 9.2 % (see Sec. 3.2).

106

The shaded red region in Fig. 4.33 displays the assigned uncertainties. Note that the most probable vales
is changed between before and after projection because there is a correlation between εL and εLL (seeTop Right distribution of Fig. 4.32). We take the most probable value of the before projection as thecenter value of εL (εLL ) and the projected distribution is only used to evaluate uncertainty.
The results of spin fit with maximum likelihood method are summarized in Table 4.11. It includes εLand εLL for signal (30 < pT < 50 GeV/c) and background (12 < pT < 20 GeV/c) region. Table 4.12 shows
single raw asymmetries calculated from Eq. 4.9. The assigned uncertainties in this table are calculated√
in statistical way (1/ 2N ), and displayed here only to give rough idea how large the uncertainties are.
They are consistent with the results of the maximum likelihood method within uncertainty, therefore thespin fit seems reliable. The quality check of the spin fit technique is also done with a toy MC which isrepresented in Appendix F.

107

Figure 4.32: The result of (σ, εL , εLL ) scan for positive particles (30 < pT < 50 GeV/c). The left row
of the figure show the schematical drawing of projection to 3 planes ((εL , εLL ), (σ, εL and (σ, εLL )).
The right row of the figure display the probability distribution in each 3 planes (Top: (εL , εLL ), Middle:(σ, εL , Bottom: (σ, εLL )).

108

Figure 4.33: Left: Projection to the εL axis for positive particles (30 < pT < 50 GeV/c). Right:
Projection to the εLL axis for positive particles (30 < pT < 50 GeV/c). Blue (Red) lines display themost probable value before (after) projection, and the Red shaded regions show the 68 % of the total
integration.

charge+pT range (GeV/c)
12−20: background30−50: signal−

12−20: background30−50: signal

εL−0.02 ± 0.04
−0.33+0.11−0.09[−0.42, − 0.22]−0.03 ± 0.04
0.31+0.18−0.24[0.07, 0.49]

εLL
0.07 ± 0.06
−0.04+0.18−0.12[−0.16, 0.14]
0.03 ± 0.06
0.55+0.22−0.21[0.34, 0.77]

Table 4.11: Summary of asymmetries calculated from maximum likelihood method. The uncertainties
indicate the range in which 68 % of the total integration is included.

charge+−pT range (GeV/c)
12−20
30−50
12−20
30−50

counts
318
42
302
13

Blue Beam (εBL)−0.07 ± 0.04
−0.19 ± 0.11
−0.06 ± 0.04
0.23 ± 0.20

Yellow Beam (εYL )0.03 ± 0.04
−0.47 ± 0.11
0.00 ± 0.04
0.39 ± 0.20

maximum likelihood−0.02 ± 0.04
−0.33+0.11−0.09−0.03 ± 0.04
0.31+0.18−0.24

Table 4.12: Summary of raw asymmetries calculated from Eq. 4.9.

109

4.4.3Single Spin Asymmetry

Single spin asymmetry (AL ) is calculated from single raw asymmetry (εL ). In the calculation, the beampolarization and a dilution factor from background must be taken into account. Their uncertainties
are treated in statistical way in this analysis to include them into the probability distribution, whichis extracted from likelihood function at the previous section. This treatment enable us to calculate aconfidence level of AL in the range of −1 ≤ AL ≤ 1 (i.e. in the physics boundary). The procedure toinclude those factors (polarization and dilution factor) when converting εL into AL is described in thissubsection.
Dilution Factor
The background works as a dilution factor to the AL calculation. Since the background contamination
cannot completely removed, measured AL is written as:
Ameas
L

=
=
=

S+B
S+B
σ+
− σ−
S+B
S+B
σ+
+ σ−
S
S
σ+
− σ−
S+B
S+B
σ+ + σ−
S
S
σ+
+ σ−
+Z
AW
·
,
L
S+B
S+B
σ+
+ σ−

(4.17)

where σiS+B represents the cross section including both signal and background, and σiS is the cross sectionof signal (e± s from W /Z decay) 38 . Here, we assume that background does not have asymmetry (i.e.
+Z
BG
BG
σ+
= σ−
). Using Eq. 4.17, AW
can be extracted from the measured AL by:
L
+Z
AW
L

=
=

S+B
S+B
σ+
+ σ−
S + σS
σ+
−
N
Ameas
· .
L
S

Ameas
·
L

(4.18)In the last line, N means the number of observed events and S is the number of signal in observed N .
From this relationship, the dilution factor from background contamination is defined as D = N/S.
The number of signal and background with the isolation cut is presented in Table. 4.9. It shows thesystematic uncertainty of the estimated background is less than 1.0 for positive and negative charge.
However, the uncertainty of ±1.0 is conservatively assigned here. Compared with the statistical uncertainty, this uncertainty is very small and it is not necessary to examine it in detail. The dilution factor
and its uncertainty are calculated from:
D=

N iso
N iso
x
=
=
,
S iso
N iso − B isox − az(4.19)

s
∆D

=
=

∂D
∂D
∆x)2 + (∆(az))2
∂x
∂(az)
r
1
1
1
D (( iso − iso )∆x)2 + ( iso ∆(az))2 .
NSS(

38 The index of i denotes the helicity state of proton beam.

110(4.20)

Here, the notation of N iso , S iso , B iso , x and az is same with Eq. 4.13 and 4.14. Used numbers for the
calculation are also same as listed in Table 4.9 and 4.10, but ∆(az) = 1.0 is assigned here as mentioned
above. The results of the calculation is summarized in Table 4.13.
Because of the zero degree track cut (see Sec. 4.4.1), the charge contamination is 0.07 % of the opposite
charge and negligible.
charge+−total(N iso = x)42
13background(B iso = az)
1.6 ± 1.0
1.7 ± 1.0

signal of W + Z(S iso )40.4 ± 6.5(stat.) ± 1.0(sys.)
11.4 ± 3.6(stat.) ± 1.0(sys.)

dilution factor(D)
1.04 ± 0.03
1.14 ± 0.11

Table 4.13: The dilution factors from background.

Treatment of Systematic Uncertainties
The shape of the probability distribution obtained in the spin fit procedure is originated from statistical+Z
fluctuation. In order to set a confidence level on the physics asymmetry (AW), it needs to consider notLonly the statistical fluctuation but also uncertainties of dilution (D) and polarization (P ). The values of+Z
P/D and its uncertainties are listed in Table 4.14. The physics asymmetry (AW) is calculated by theLformula of:+Z
AW= εL /(P/D),
L(4.21)and the uncertainty of P/D is merged into statistical treatments in this analysis.
By smearing the probability distribution with the uncertainty of P/D 39 , the effect of the uncertaintyof P/D is included. Figure 4.34 shows the distributions with (Red) and without (Black) uncertainty of+Z
P/D. The horizontal scale in the figures is converted into the AW
space, and it can be noticed thatL

the distributions get slightly wider.
chargeP/D+

0.375 = 0.39/1.04−

0.342 = 0.39/1.14

uncertaintyp2
9.6 % (=
0.092 + (0.03/1.04)2 )pp
13 % (= (∆P/P )2 + (∆D/D)2 = 0.0922 + (0.11/1.14)2 )p(∆P/P )2 + (∆D/D)2 =Table 4.14: The values of P/D and its uncertainties.

The renormalization in the range of −1 ≤ AL ≤ 1 (indicated as Blue lines in Fig. 4.34) has been+Z
done when extracting the confidence level of AW. For example, the integration window is extendedLstarting from the most probable value, and the values where the integration become 68 % (95 %) of total
integration at −1 ≤ AL ≤ 1 window are assigned as the range of 68 % (95 %) confidence level. If one
side of the integration window reaches 1 or −1, only the other side of the window is extended.
The numbers of 68 % and 95 % confidence levels are shown in Table 4.15. This table also includes εLof the background region (12 < pT < 20 GeV/c).
39 For example, the smearing width for the positive charge is: |A × ∆P/D| = |(−0.33/0.375) × 0.096| = 0.084.
L111

Figure 4.34: The likelihood functions with (Red) and without (Black) P/D uncertainty (Left: positive,
Right: negative charge). The horizontal scale is converted in AL space, and Blue lines indicate thephysical boundaries (−1 ≤ AL ≤ 1)

charge+−pT range (GeV/c)
12−20: background30−50: signal
12−20: background30−50: signal

εL−0.02 ± 0.04
−0.33+0.11−0.09−0.03 ± 0.04
0.31+0.18
−0.24

+Z
AW
L
−0.88
0.91

AL 68 %CL[−1, − 0.61][0.20, 1]

AL 95 %CL[−1, − 0.21][−0.51, 1]

Table 4.15: Longitudinal single spin asymmetries (AL ) with various confidence levels.

112

Chapter 5

Results and Discussion
The obtained results of dσ/dy and AL are shown again here in Table 5.1 and 5.2. These results arecompared to theoretical calculations or the results from previous experiments in this chapter.
As mentioned in Chapter 4, these results include electrons from Z decays as well as W decays because
the PHENIX detector cannot distinguish them. Thus, the contamination from Z decays should be taken
into account in the calculation of the cross section of W → eν decay. In addition to the Z contribution, thefraction of W decays that were outside of the detector acceptance also must be corrected. The subtractionof Z contribution and evaluation of the cross section of W → eν decay is discussed in Sec. 5.1. The charge
ratio of W production and its brief future prospects are also described in that section. The measured
single spin asymmetry can be directly compared to theoretical calculations because the Z contribution+Z
is already taken into account (see Sec. 1.3). The comparison of AW
to the theoretical calculations isLdescribed in Sec. 5.2 followed by the future prospects of spin asymmetry measurement of W production
at RHIC in Sec. 5.3.
charge+−±dσ/dy (pb)50.2 ± 7.2(stat.) +1.2
−3.6 (sys.) ± 8.0 (normalization)+2.1
9.7 ± 3.7(stat.) −2.5 (sys.) ± 1.6 (normalization)
60.0 ± 8.1(stat.) +3.1
−6.0 (sys.) ± 9.6 (normalization)

Table 5.1: dσ/dy of electrons from W and Z decay at 30 < peT < 50 GeV/c.

charge
+
−

+Z
AW
L
−0.88
0.91

AL 68 %CL[−1, − 0.61][0.20, 1]

AL 95 %CL[−1, − 0.21][−0.51, 1]

Table 5.2: AL with various confidence level at 30 < peT < 50 GeV/c.

113

5.1

Cross Section of W → eν Decay

The cross section of W → eν decay (σ(W ) · BR(W → eν)) is estimated from the differential cross section(dσ/dy). The relationship of the cross section to the dσ/dy is:dσ(W ) · BR(W → eν) =dy
σ(W ) · BR(W → eν) =dσ· (1 − fz ),dydσ
1 − fz
1 − fz· (∆y∆φ/2π) ·= σ(W + Z)|PHENIX ·
,
dy
sW
sW

(5.1)
(5.2)

where the term of ∆y∆φ denotes the PHENIX acceptance (∆y = 0.35 × 2, ∆φ = π), fz represents the Z
contribution (fz = Z/(W + Z)) and sW is the fraction of the cross section within |y| < 0.35 in rapidity
1, 30 < pT < 50 GeV/c and |∆φ| < π.
The values of fz and sW are estimated from NLO (Next-to-Leading-Order) and NNLO (Next-Next-

to-Leading-Order) perturbative-QCD calculations [12, 121], and listed in Table 5.3 with input PDFs
[46, 122]. By taking the average among different calculations, fz is 6.72, 24.4, 10.5 %; sW is estimatedto be 11.2, 7.50, 10.2 % for W + , W − and W ± , respectively. The theoretical uncertainties from NLO
and NNLO calculations and varied parton-distribution functions (PDFs) are negligible compared to othersources of systematic uncertainty from the data.
The cross section of W → eν is summarized in Table 5.4, and compared with the Spp̄S, Tevatron and
LHC data [123, 124, 74, 125, 126] in Fig. 5.1. The statistical and systematic uncertainties are added inquadrature. Our results are the first measurement of the W cross section in pp collisions. The curves in
the figure show theoretical calculations [121]. The difference of theoretical calculation between pp̄ and√pp collisions at lower s is originated from the difference of involved quarks. In pp̄ collisions, quarks
which couple to W -boson are mainly valence quarks, while the dominant process of W production in pp√
collisions is the coupling of the valence and sea quarks. However, in much higher s, the momentum
fractions (x) of the involved quarks become smaller and the coupling of sea quarks dominates the processof W production in both pp and pp̄ collisions. Therefore, the difference of W cross section between pp̄
√√and pp collisions disappears in higher s collisions as can be noticed in the figure. In such a high s
collisions, heavy flavors, such as s, c and b quarks, also contribute to the W production, while they are√very small and ignored in lower s collisions. It is noticeable that all of the measured cross sections
are consistent with the theoretical calculation. This means that the established NLO and NNLO QCD
calculations and our knowledge of unpolarized PDF in wide x range (from ∼ 0.16 at RHIC to ∼ 0.01 at
LHC 2 ) are quite reliable.

1 Because the exact PHENIX acceptance is |η| < 0.35, it corresponding to |y| < 0.35 for electrons due to their small mass
value.
2 The Bjorken x is calculated as ∼ M /√s at central rapidity region.
W

114

PDF

chargeCHE (NLO)

MRST2002

NLO

MSTW2008

NNLO

MSTW2008

+
−
±
+
−
±
+
−
±

dσ/dy (pb) of W + Z(30 < peT < 50 GeV/c)
43.2
11.3
54.5
46.5
12.8
59.3
46.9
13.5
60.4

fz

sW

0.07010.267
0.111
0.06470.235
0.101
0.0669
0.231
0.104

0.113
0.07440.103
0.111
0.07550.101

Table 5.3: fz and sW values estimated from NLO and NNLO calculations.

charge+−±cross section of W → eν (pb)146.3 ± 21.0(stat.) +3.5
−10.5 (sys.) ± 23.3(norm.)
34.2 ± 13.1(stat.) +7.4
−8.8 (sys.) ± 5.6(norm.)
184.0 ± 24.9(stat.) +9.5
−18.4 (sys.) ± 29.5(norm.)σ ⋅ BR (pb)Table 5.4: Cross section of W → eν decay.

104

103
pp → W ±PHENIX W+PHENIX W
PHENIX W±
UA1
UA2
D0
CDF
+
ATLAS WATLAS W
ATLAS W±

pp → W±102

pp → W
pp → W+

300 400

2000

1000

3000

s (GeV)
Figure 5.1: Inclusive cross sections for W leptonic decay channel of this thesis and other measurements[123, 124, 74, 125, 126]. The curves are theory calculations [121].

115

5.1.1

Charge Ratio of W production¯ flavor asymmetry at
The W production at RHIC offers the unique opportunity of extracting the d/ūvery high Q2 without the complications associated with the charge symmetry breaking effect and nuclear
binding effect. As discussed in Sec. 1.3.3, the charge ratio of W + and W − production in central rapidity¯ with distinguishing flavor symmetric and asymmetric sea
region at PHENIX is a good probe for d/ūquark distributions.
Using Eq. 5.1, the evaluated charge ratio with obtained dσ/dy at PHENIX central rapidity is:
R = 6.4 ± 2.6(stat.) ± 1.7(sys.).

(5.3)

Here, the systematic uncertainty is calculated from larger value of the background estimation in Table 5.1
for simplicity, and assumed there is no correlation between positive and negative charge. Due to the large
uncertainty from the small statistics of negative charge, the obtained value cannot distinguish flavor¯
symmetric and asymmetric anti-quark sea 3 . However, the result also indicate that we can probe d/ū
asymmetry directly with more statistics, and the demonstration is shown below.
Future Prospect
Figure 5.2 displays the predicted charge ratios R(yl ) calculated for various PDFs, with the projected
sensitivities [86]. The statistical uncertainties of the projections are estimated for recorded luminosityof 300 pb−1 at RHIC [127]. The acceptance is for the PHENIX detector, which is |η| < 0.35 in central
rapidities, −2.2 < η < −1.1 and 1.1 < η < 2.4 in forward rapidities. The figure has clearly demonstrated
that a measurement of R(yl ) at PHENIX with more statistics is able to distinguish SU(2) flavor symmetric(assumed in MRS S0’) and asymmetric (assumed in other three PDFs) sea quark distributions.
¯ measurement, all of the theoretical models have difficulties to explain d/ū¯In the aspect of the d/ū
asymmetry at large x (x > 0.2), and E906 Drell-Yan experiment at Fermilab [79] plans to extend the¯ at large x region as
measurement in wide x range (0.1 ∼< x ∼< 0.4) to resolve the behavior of d/ūdescribed in Sec. 1.3.3. The future R measurement at RHIC can also provide complementary measurement¯ at x range of ∼ 0.05 to 0.16 without the complications which come from the charge symmetryof d/ūbreaking effect and nuclear binding effect.

3 Note that the value of f depends on assumed PDF and the central value will vary with different PDF, but it is notztaken into account here because of rather large uncertainty.

116

Figure5.2: Prediction of the charge ratio R(yl ) as a function of lepton rapidity for pp collision at√
s = 500 GeV using various PDFs. The projected sensitivities for a run with recorded luminosity of
300 pb−1 for the PHENIX detector are also shown [86].

117

5.2

Comparison of AL to various Polarized PDFs

Measured single spin asymmetry is directly compared to various polarized PDFs. The polarized PDFs
are extracted from fits of DIS and Semi-Inclusive DIS data [64, 67, 68, 62], and the theoretical AL values
are calculated using NLO with different polarized PDFs [12]. Table 5.5 shows the range of the theoretical
AL s of e± from W and Z decay in rapidity of |y| < 0.35 with 30 < pT < 50 GeV/c range. The measured
AL of 68 % confidence level and the value from DSSV PDF [67, 68] is also listed in the table.
Figure 5.3 displays the comparison of measured AL to the calculations. The discrimination power tothe choice of the underlying PDF is smaller comparing to Fig. 1.13 because of the different electron pT cut.
Nevertheless, there is still a band in AL of negative charge which comes from the uncertainty of ∆ū and∆d. Using the likelihood function obtained in Sec. 4.4.3, the confidence level at the theoretical AL s areevaluated. The experimental results are consistent with the theoretical calculations at 5−15 % confidence+−

level for AeL and at 18−36 % for AeL . An independent measurement of the single spin asymmetries of
W boson production at RHIC has been reported by the STAR collaboration [128], and it also shows theconsistent results with our measurement as well as theoretical predictions.
√
The observed asymmetries are sensitive to the polarized quark distributions at x ∼ MW / s ' 0.16.
Although the uncertainties of the obtained results are quite large due to limited statistics, this is the firstmeasurement of the single spin asymmetry of W + Z production and the results directly demonstrate theparity violating coupling between W bosons and light quarks. That means our analysis has established a
new and direct way to probe the spin structure of the proton using the weak interactions. It is expectedthat our knowledge of flavor-separated polarized PDFs will be improved in near future with the upgradesof RHIC luminosity and PHENIX detector. The future prospects of the AL measurement is described in
the next section.
charge+−measured AL(68 % CL)−0.88 +0.27−0.12(−1 ≤ AL ≤ −0.61)
0.91+0.09−0.71(0.20 ≤ AL ≤ 1)

AL from DSSV(|y| < 0.35, 30 < peT < 50)−0.35

AL from various PDFs(|y| < 0.35, 30 < peT < 50)−0.37 to −0.21

0.05−0.06 to 0.26

Table 5.5: AL values from theory calculation with various polarized PDFs.

118

AL

AL
1

1
+

W +Z
0.5

-

0

pT > 30 GeV/c

0.5|y|<0.35

0

0

-0.5

-0.5

-1

-1

-2

-1.5

-1

0

W +Z

-0.5

0

0.5

1

1.5

2

-2

y

pT > 30 GeV/c|y|<0.35

DSSV
DNS kkp
DNS kreGRSV std
GRSV val

-1.5

-1

-0.5

0

0.5

1

1.5

2

y

Figure 5.3: AL for electrons and positrons from W and Z decays. The error bars represent 68 % CL.
The theoretical curves are calculated using NLO with different polarized PDFs [12].

119

5.3

Future Prospects of Single Spin Asymmetry of W Production

RHIC luminosity and PHENIX detector upgrades are ongoing in order to reduce the uncertainties of
AL and extend the AL measurement to forward/backward rapidity. The upgrades lead to our better
understanding of the flavor separated polarized PDFs. Figure 5.4 displays the current constraints on thepolarized anti-quark distributions from the DSSV result [67, 68]. The uncertainties of the polarized PDFs
are shown with Yellow bands which mainly come from the uncertainty of the fragmentation functions
assumed in SIDIS measurements. The impact of the future AL measurement on anti-quark distribution,
which is estimated by one of global analysis groups, is presented in this section [12].

Figure 5.4: ∆u(x, Q2 = 10 GeV2 ) and ∆d(x, Q2 = 10 GeV2 ) as obtained in the DSSV global analysis[67, 68]. The bands correspond to changes of 2 % in the total χ2 of the fit.
The study is based on NLO calculation with PDFs from the DSSV global analysis. A pseudo”RHIC data set” was produced from the NLO calculation, using the central value of DSSV PDFs, by+−computing single spin asymmetries of AeL and AeL

4. Then the calculated asymmetries are randomly

shifted, assuming a Gaussian distribution of their statistical uncertainties. The statistical uncertainties√
are estimated using the formula of ∆AL = 1/(P Lσ), where σ is the cross section of W production. The
assumed polarization (P ) is 60 %, and the integrated luminosity of L = 200 pb−1 and 800 pb−1 were
produced. The rapidity coverage which is exactly same as the PHENIX and STAR detectors is taken into
account in this study. A new global fit for the polarized PDFs is then performed, for which the simulated
data set is included, and the impact of future W data from RHIC is estimated.
The result of this study is shown in Fig. 5.5, including their ∆χ2 /χ2 = 2 % uncertainties [12]. Note
4 Asymmetries are evaluated including both W and Z/γ boson exchange contributions

120

that this global fit only include the detector acceptance which is currently used for W measurement 5 . By
comparing the figure to Fig. 5.4, we can notice that there is a slight modification of the central distributions
due to the statistical fluctuation, however a clear reduction can be observed in their uncertainty bands.
This effect is very clear for x > 0.1 as expected, considering the rapidity coverage of the pseudo-”RHIC
data set” added to the global fit 6 . The decrease in the uncertainty band is more noticeable in ∆ūdistribution, which confirms the larger sensitivity of the e− asymmetries. At x ∼ 0.01 values, there isalmost no change in the distributions and uncertainties, because the single-spin asymmetries at RHIC
are rather insensitive to such small x values.
The impact of extending rapidity coverage by |η| < 0.35 and 1 < |η| < 2 for PHENIX and |η| < 2 forSTAR is also studied with psedo-”RHIC data set”. The extended rapidity for the PHENIX and STAR iscorresponding to the ongoing forward upgrade project of both experiments. Figure 5.6 shows the resultof the global fit with the case of larger luminosity and rapidity coverage [12]. The impact of RHIC data
can be noticed down to x of ∼ 0.05, thanks to the extended coverage in η.
The study described in this section does not include experimental issues (such as efficiencies for lepton
detection, correction of background subtraction and other systematic uncertainties, etc.), and that would
decrease the impact of the real data in the global fit. Nevertheless, the study shows that future RHIC W
measurement can definitely improve our knowledge of flavor separated polarized PDFs, and contributeto our deeper understanding of the proton spin structure.

5 |η| < 0.35 for the PHENIX and |η| < 1 for the STAR.

√ ±y /2
6 Using the lepton rapidity (y ), the Bjorken x is: hxl, and its value at yl ∼ 0 is x ∼ 0.16
1,2 i ∼ (MW / s)el121

Figure 5.5: ∆u(x, Q2 = 10 GeV2 ) and ∆d(x, Q2 = 10 GeV2 ) as obtained from a global analysis performed
by including the simulation data generated with the luminosity of 200 pb−1 and the current rapidity
coverage of RHIC detectors [12]. The bands correspond to changes of 2 % in the total χ2 of the fit.

Figure 5.6: ∆u(x, Q2 = 10 GeV2 ) and ∆d(x, Q2 = 10 GeV2 ) as obtained from a global analysis performed
by including the simulation data generated with the luminosity of 800 pb−1 and extended rapidity coverage of RHIC detectors [12]. The extended rapidity coverage is corresponding to the ongoing forward
upgrade project of PHENIX and STAR. The bands correspond to changes of 2 % in the total χ2 of the
fit.

122

Chapter 6

Conclusion
In summary, the first measurements of cross section and single spin asymmetry (AL ) of W and Z pro√duction in polarized pp collisions at s = 500 GeV were reported in this thesis. Contrast to previous(SI)DIS measurements, the W boson production at pp collisions is free from any fragmentation processby measuring its decay lepton, therefore provides us more direct probe for both polarized and unpolarizedanti-quark (sea quark) distributions, and it leads to our deeper understanding of the proton structure. W
bosons couple only the left-handed quarks and right-handed anti-quarks (uL d¯R → W + and dL ūR → W − ),so the single spin asymmetry of the W yield is sensitive to the flavor separated ∆q and, especially, ∆q̄. In
addition, the measurement of cross section and charge ratio of W production in spin-averaged collisions¯ asymmetry, as well as it isalso provides a sensitive test of the current unpolarized PDFs by probing d/ūan important confirmation of the theoretical understanding of the W production.
The experiment was performed with the PHENIX detector with the polarized proton beams provided
by RHIC at BNL. The data used in this thesis were taken during Year-2009 run. The analyzed data sample
has the integrated luminosity of 8.57 pb−1 , with average polarization of 39 %. W and Z production wasmeasured through inclusive pp → e± + X production over a pseudo-rapidity range of |η| < 0.35, where
the e± with transverse momentum of pT > 30 GeV/c mainly come from W and Z decays. The electrons
were detected by the PHENIX DC, PC1 and EMCal. The high pT photon/electron trigger which isconstructed by the signals from EMCal was used to collect the data sample.
The obtained W ± boson production cross sections for the e± channels are σ(pp → W + X)×BR(W + →−−e+ νe ) = 146.3 ± 21.0(stat.)+3.5→ e+ νe ) =−10.5 (sys.) ± 23.3(norm.) pb, and σ(pp → W X) × BR(W34.2 ± 13.1(stat.)+7.4−8.8 (sys.) ± 5.6(norm.) pb. This is the first measurement of W cross section in pp
collisions. The results are consistent with NLO and NNLO perturbative QCD (pQCD) calculations, andvalidity of the theoretical understanding of the W production is confirmed.
+

−

+0.09
e
The single spin asymmetries of AeL = −0.88+0.27−0.12 and AL = 0.91−0.71 are also measured for inclusive

high transverse momentum e± s (30 < pT < 50 GeV/c) for the first time. The non-zero AL s show thedirect demonstration of the parity-violating coupling of W to light quarks, and the results are consistent
with NLO pQCD based predictions from polarized PDFs which are extracted from previous polarized(SI)DIS results.
123

Acknowledgment
First of all, I would like to acknowledge Prof. T. Nagae, who is the leader of the experimental Nuclear andHadronic physics (NH) group, for his essential advice and comments about the experiment and physics.
He has read my thesis carefully and given me useful suggestions. I am deeply grateful to Prof. N. Saito,
who has been my supervisor. His abundant knowledge and precise suggestion always guided me. He hashelped me not only with his advice but also with support to concentrate on the research. Without his
help, I could not finish this work. I would like to express my sincere thanks to Dr. T. Murakami forhis kind supports for a great deal of encouragements. I could learn many techniques of the experiments
and analysis from him and could enjoy the research with his support. I also wish to thank Dr. T. Mibeand Dr. M. Niiyama, who have also read my thesis thoroughly and given me useful comments, for theirencouragement for the completion of this work.
I would like to express my deep thanks to Dr. K. Okada for his many thoughtful comments andsuggestions. This work is benefited especially from his idea and suggestions. I am grateful to themembers of W working group: Dr. M. Chiu, Dr. A. Bazilevsky, Dr. J. Haggerty, Dr. D. Kawall and
Dr. E. Kistenev. Discussion with them was beneficial and I was able to polish the analysis working
with them. I also would like to thank the members of the internal review committee (IRC120): Dr. M
GrossePerdekamp, Dr. A. Deshpande and Dr. B Sahlmueller.
I would like to acknowledge the Radiation Laboratory (Rad. Lab.) at RIKEN and RIKEN BNL
Research Center (RBRC) for the research activity as well as financial supports. Especially, Dr. H. En’yo,
who is the chief scientist of the Rad. Lab., often advised me about the research plan. I also would liketo express my gratitude to the members of RBRC: Dr. Y. Akiba, Dr. Y. Goto and Dr. I. Nakagawa.
I would like to thank the administrators of the PHENIX Computing Center in Japan (CCJ): Dr. T.
Ichihara, Dr. Y. Watanabe, Dr. S. Yokkaichi, Dr. T. Nakamura and Dr. Y. Ikeda. The analysis wasdone on CCJ and it was very effective with their supports. I am deeply grateful to all the collaboratorsof the PHENIX experiment, and the staff of Collider Accelerator Department at BNL. Without them theexperiment would not be possible.
I would like to express my thanks to the past and present staff of the NH group: Prof. K. Imai, Dr.
T. Kawabata, Dr. Kiyoshi Tanida, and Dr. H. Fujioka. Especially, to Prof. K. Imai who is the former
leader of the NH group. His deep understanding of nature and passion for physics always stimulated mywork. I am also grateful to the past and present member of our group: Dr. Y. Fukao, Dr. M. Togawa,124Dr. H. Iinuma, Dr. K. Aoki, Dr. J. Zenihiro, Dr. Y. Seki, Mr. K. Shoji and Mr. S. Dairaku for theirhelp, interesting discussions and the friendship at Kyoto and BNL.
I wish to express my appreciation to Mr. A. Tokiyasu and Mr. T. Hiraiwa. They are in the same
grade in the NH group and inspired interests in physics each other. I also would like to express special
thank to Mr. K. R. Nakamura. I always enjoyed working and discussing with him during my stay at
BNL. I wish to thank Dr. K. Kondo, Dr. J. Tamura, Dr. Y. Aramaki, Dr. Y. Yamaguchi, Dr. T.
Kanesue, Mr. K. Kijima and Mr. H. Asano for their help and the friendship. I could really enjoy mydaily life at BNL with them.
I would like to thank the NH secretary, Ms. S. Yamamoto, and Rad. Lab./RBRC secretaries, Ms.
N. Kiyama, Ms. K. Sakuma, Ms. K. Suzuki, Ms. E. Adachi, Ms. T. Ito, and Ms. S. Foster. I could
concentrate on the research with their support.
I wish to express my deepest gratitude to my parents, Hiroshi and Ikuko, and my brothers, Johji andYugo, for their support and encouragements of my continuing this work. I could not continue and finish
this work without their strong support.
Every last but not least, I am most grateful to my wife, Miki, for all her understanding and kind
supports. I would like to dedicate this thesis to her.

125

Appendix A

Fragmentation Function
In DIS or hard pp collision, a single parton with large momentum is knocked out from the proton. Since
the quark or the gluon cannot exist by itself, such a parton is hadronized and generates a ”jet”, which
consists of many hadrons with momentum almost parallel to that of the initial parton.

Figure A.1: Left: Diagram of SIDIS. In SIDIS experiment, one or more hadrons at the final state aredetected. Right: Diagram of hard pp collision. Inclusive measurement of hadrons is often performed. In
both SIDIS and pp collision, the reaction is divided into parts of PDF, FF and partonic cross section(factorization theorem).
In ep scattering, the reaction is divided into two parts; PDF and partonic cross section (factorization
theorem). While the cross section between the parton and the electron (partonic cross section) can becalculated precisely using perturbative QED, it is difficult to theoretically calculate PDF. When hadronat the final state is measured in DIS, so-called Semi-Inclusive DIS (SIDIS) measurement, it is necessary toformalize the hadronization process. The hadronization from the scattered parton is ”soft” QCD process,where pQCD does not work due to low energy scale, and the process is described by ”fragmentation
function” (FF). FF is represented as Dhf (z), where f and h indicate the parent parton and the hadron
after the fragmentation, respectively, and z is energy fraction of the hadron to the parton. Dhf (z)dz is126

defined as the probability density that the hadron h with momentum of zp to (z + dz)p is generated in
the jet originating in the parton f with momentum of p. For example, the differential cross section of
SIDIS with pion (π) measured in the final state (Left of Fig. A.1) is written as:
dσ ep→πX =

XZ 1
q,q̄

0

Z 1
dx
0

0

dzq(x)dσ̂ ep→q X Dqπ0 (z),(A.1)

where dσ̂ indicates the partonic cross section. The summation is performed for possible combinations of
the initial and the final parton (and anti-parton). When the hadron at the final state has spin, polarized
FF is defined in analogy with polarized PDF.
In the same way, reaction of pp collision can also be factorized as displayed in the right side of Fig. A.1.
Inclusive measurement of pion in pp collisions is divided into three parts; PDF, FF and partonic crosssection. Corresponding to the initial two protons, two PDFs appear. Unlike ep scattering, gluons in the
proton directly participate in the reaction in pp collisions. The cross section between partons is calculated
by pQCD instead of QED 1 . Then, the scattered parton is hadronized following FF. The process is written
as:
dσ

pp→πX

=

X Z

0

dx1 dx2 dzf1 (x1 )f2 (x2 )dσ̂ f1 f2 →f X Dfπ0 (z),(A.2)

f1 ,f2 ,f 0

where f indicates type of the parton and includes both the quark and the gluon.
It is important to note that, in ep scattering, this picture is valid at large Q2 where the virtual photon
interacts with a single parton in the proton. Similarly, large Q2 is required in pp scattering and it isconfirmed by the hadron production with high transverse momentum (pT ) 2 . Here, pT is defined as themomentum component perpendicular to momentum of the initial protons in center-of-mass system. The
large Q2 results in small αs , which assures validity of pQCD.
One of the remarkable features of the factorization theorem is the universality of PDF and FF. Thanks
to this feature, PDF or FF obtained by a certain experiment is available for other experiments and it
results in versatility and predictability of the theory. Based on the feature, FF is generally measured
by e+ e− collision because the reaction of e+ e− → q q̄ is precisely calculable by QED and there is noother theoretical uncertainty except for FF. In e+ e− collision with center-of-mass system of the incident
electrons, FF is related to cross section in the leading order as follows:
P
+ −
0
2 h
Eh
1
dσ e e →hX
2Ehq,q̄ eq Dq (z)P=.
,
z= √ =
2e+ e− →Z
dz
Eq
e
s
σtot
q,q̄ q(A.3)

Here, σtot is the total cross section of e+ e− collision. The summation is executed over possible quark√
flavors, while contribution from gluon does not appear in the leading order. s, Eh and Eq are center-ofmass energy, energy of the hadron and energy of the parton, respectively. Although only FFs of quarks
appear and the cross section has scaling on z in this equation, FF actually depends on Q2 with higher
order QCD, like PDF, and evolves following DGLAP equation for FF [129, 130, 131].
1 QED also contributes the process, but it is small due to smaller coupling constant of the electromagnetic interaction
than the strong interaction.
2 Empirically, boundary for the validity of the factorization theorem is around 1 GeV/c in p .
T127

To determine FF, several experiments of e+ e− collision and analyses to unify the experimental data
are performed (see [69, 70, 132, 133, 134] and references in them). However, from Eq. A.3, it can benoticed that it is difficult to distinguish FFs of each quark flavor because the right-hand side of theequation is always sum of FFs. For this reason, symmetric properties and phenomenological assumptions
are often adopted to extract FF from measured data. For instance, charge conjugation results in:
Dfh = Dfh̄¯,(A.4)

and additional isospin symmetry derives for positive pion:+++Duπ = Ddπ¯ ,+Ddπ = Dūπ .

In addition to these ”reliable” conditions, some constraints are imposed, such as Ddπ

(A.5)
+

= Dsπ

+

= Ds̄π+based on symmetry for non-valence quarks [69, 134]. Flavor-separated FFs are often obtained utilizing
experimental data where two hadrons are tagged at the final state [133]. For example, pion measured
with D meson tagged lead to pion FF from c quark, Dcπ . It is also hard to extract gluon FF in simple
measurement with one hadron tagged. To determine gluon FF, method with three jets tagged is applied.
One of the three jets originates in the gluon radiated by the quark which is the source of other jets.
After the determination of FFs, PDFs of quarks are extracted by SIDIS measurements in whichobtained FFs are used. To measure gluon PDF, pp collision is effective because gluon appears in thereaction at the leading order, and PDFs/FFs obtained above are valid in analyzing pp collisions.
In the same way as the unpolarized case, cross section of polarized DIS and pp collision can be also
factorized, and the universality of polarized PDF and FF is also available. Polarized FFs and PDF are
obtained by polarized e+ e− collision and (SI)DIS, respectively. However, compared with unpolarized
experiments, precision of the polarized experimental data is limited and rather larger uncertainties ofpolarized FFs and PDFs still remain as described in Sec. 1.2.2.

128

Appendix B

Fast Monte Carlo (FastMC)
Simulation
FastMC is a simple numerical simulation unlike GEANT and utilizes the measured cross section of π 0or η as the particle generator. The produced π 0 s decay into two photons isotropically in the rest frameof π 0 . In the generation of π 0 , the spread of the event vertex is simulated based on Gaussian with awidth of 30 cm, which is a little narrower than the measured width but the difference does not change
the results. FastMC simulates the photons from π 0 with the following setup:• Energy and position resolution
The initial energy and position is fluctuated following Gaussian distribution. The resolutions areevaluated by the test beam. In this simulation, the additional factor is applied to fit the real data.
• Electromagnetic shower profile
The profile of the electromagnetic shower is simulated and tower-by-tower energy threshold (∼10 MeV)
is applied based on the profile. Selection of the core towers in the PbSc clustering is also simulated.
See Sec. 4.2.1 about the clustering.
• Warn and edge towers
Masked region of EMCal due to the warn or edge towers is simulated. See Sec. 4.3.2 about thewarn and edge towers.
• Trigger efficiency of ERT
The trigger efficiency of ERT is applied for the single photon. See Sec. 2.2.5 about the ERT.
Because the Tof cut is difficult to simulate, they are not applied in the comparison between the data andFastMC.
The absolute energy scale is calibrated by comparing the pT dependence of π 0 /η peak to the FastMC
simulation. The energy scale is tuned in the FastMC with additional attenuation length (λadd ).
As well as the energy scale, the resolutions of the energy and position are also reflected in the pT
dependence of the measured width of π 0 /η. Therefore, the energy and position resolution in the FastMC
must be also tuned to fit the real data.
129

In the real data, the π 0 peak position becomes lower due to the unusual π 0 s. One of the sources ofsuch π 0 is the case that photons from the decay of π 0 convert into the electron positron pairs. π 0 is wellreconstructed using such e+ e− pairs, when they do not separate each other to form the single clusters.
π 0 from the decay of other hadrons, such as Ks0 and η, is another source. One other is π 0 generated
in the secondary interaction with the beam pipe or detectors. The invariant mass of two photon pair,Mγγ are calculated as:
2
Mγγ

=
=

2E1 E2 (1 − cos θ)θ
4E1 E2 sin2 ,
2

(B.1)

where E1 and E2 represent energy of two photon clusters and θ is the opening angle between the clusters. Because we assume photons detected in the EMCal are originated from the collision vertex whenreconstructing Mγγ , the reconstructed mass of the unusual π 0 s have lower peak position than the usual
π 0 s (see Fig. B.1).
The size of the shift due to these effect is evaluated by means of the GEANT simulation, and is−1 ± 1 MeV/c2 [108]. Because the FastMC cannot simulate this effect, the peak position of π 0 s is simply
lowered by 1 MeV/c2 in the FastMC.

Figure B.1: Left: A schematic drawing of (usual) π 0 mass reconstruction. Right: A schematic drawingof unusual π 0 mass reconstruction.

130

Appendix C

Luminosity Corrections for Pile-up
and Accidentals
The formulae of the rate corrections for the multi-collision effect is derived in this appendix.
The distribution for the number of collisions in a crossing follows a Poisson distribution:
P (n; µ) =∞X
e−µ µn
n=0

n!

,

(C.1)

where µ is the true rate of collisions per crossing which produce hits in the luminosity detectors. Once
one knows µ, the luminosity can be easily obtained as:true
L = RBBC/σBBC = µBBC /σBBC = µ/σpp,total ,(C.2)

where the BBC is used as the example luminosity detector. BBC is the BBC efficiency and σBBC isthe cross section seen by the BBC. The total collision cross section σpp,total can be taken from the PDG
parameterization [113]. σ(s) = 35.45 + 0.308 log2 (s/28.94) + 42.53(1/s)0.458 − 33.34(1/s)0.545 which gives√
60.859 mb for s = 500 GeV.
A collision can produce one of only four outcomes: only the North BBC is hit, only the South BBC
is hit, both are hit, or neither detector is hit. The luminosity correction must be applied for the mismeasurement of luminosity when using the BBC trigger rate as a luminosity monitor. For example, themis-measurements occur when more than one collision in the crossing produces events where both BBCs
are hit. The BBC only counts once and it undercounts in this case. There is also the possibility that
one can have a crossing where one collision produces a hit in the North BBC, and another produces a hit
in the South BBC. In this case, the BBC coincidence counts up when it should not, producing an overcounting of the coincidence rate.
For the following discussion, we define the following notation:• µ = true rate of all collisions which produce hits in the BBC per crossing. Note that µ ∈ [0, 1].
• B,N (z) = Probability given that there is a collision for having a hit only in the North BBC.
• B,S (z) = Probability given that there is a collision for having a hit only in the South BBC.
131

• 0 (z) = Probability given that there is a collision for having no hits in either BBC.
• BB (z) = Probability given that there is a collision for having hits in both BBCs, i.e. there is a
trigger.
• RBBN is the rate of BBC North triggers, including both singles and doubles.
• RBBS is the rate of BBC South triggers, including both singles and doubles.
• RBBC is the rate of BBC coincidence triggers.
• RBN is the rate of BBC north triggers, ”excluding” the contribution from BB events.
• RBS is the rate of BBC south triggers, ”excluding” the contribution from BB events.
Note that for a two arm detector, there are only the above four possibilities B,N , B,S , 0 , BB , andthus there is the condition of:
1 = BB (z) + B,N (z) + B,S (z) + 0 (z).

(C.3)

true
true
The true rates are given by RBBC
= µBB , RBN= µB,N , and so on. For completeness, there is a
true
true
true
relation of RBN= RBBN− RBBC, i.e., a single sided detector counts both singles and coincidence

events.
To calculate the distortion to the rates from pileup events and accidental coincidences from single hits,
we need to work out all the combinatoric probabilities of the four event types for crossings with n collisions.
Then, we need to select out the combinations of collisions which would count as a BBC coincidence, andsum them over the distribution of collisions per crossing given by Eq. C.1. Alternatively, we can calculate
the probability P (0) for the various combinations of collisions which do not trigger the BBC, and get theprobability of the BBC firing directly from 1 − P (0). In practice, this way is much easier. The probabilityof no coincidence in the BBC consists of collisions in which any combination of events produces hits only
in one arm. Summing up all these probabilities, one gets:
RBBC = 1 − P (0) = 1 −∞X
e−µ µn
n=0

n!

n
X

!
n−m
n−m
(n Cm m+n Cm m) − n0
B,N 0B,S 0m=0= 1 − e−µ(BB +B,N ) − e−µ(BB +B,S ) + e−µ(BB +B,N +B,S ) .

(C.4)

The second line of Eq. C.4 follows from the binomial theorem and the Taylor series expansion for e. The values can be extracted from beam clock triggered events and the vernier scan. Note that effects from
non-collisions, such as beam gas or beam scrape is ignored here. At low rates µ, the measured coincidence
rate reduces to:
RBBC = 1 − e−µBB + µ2 B,N B,S .

(C.5)

The 1 − e−µBB term corrects for the under counting due to multiple BBC coincidence events occurringin the same crossing, while the µ2 B,N B,S term accounts for the single hits accidentally forming acoincidence, which is exactly consistent with the prior expectations.
132

Although Eq. C.4 expresses the measured BBC coincidence rate (RBBC ) as a function of the true
singles and coincidence rates (µB,N , µB,S and µBB ), it is possible to reformulate the equation interms of only the measured singles rates (RBBN , RBBS ), coincidence rates (RBBC ), and the true BBC
coincidence rate (µBB ). Therefore, one can solve for the true BBC coincidence rate in terms of only
measured quantities. For instance, the inclusive rate in the North BBC (RBBN ) is calculated in thefollowing manner: Firstly the probabilities of ”no” having any hit in the north BBCs (PN (0)) is calculated,
and then take 1 − PN (0) as discussed above. PN (0) consists of all the combinations of collisions which
produce either no hit in either BBC or hits in only the south BBC, properly summed over the distributionof the number of collisions, and RBBN is written as:
RBBN = 1 − PN (0) = 1 −n∞Xe−µ µn Xn!

n−m(n Cm m)
B,S 0m=0
n=0−µ(1−0 −B,S )

=1−e

(C.6)

= 1 − e−µ(B,N +BB ) .

(C.7)

A similar equation exists for RBBS . Substituting into Eq. C.4:
RBBC = 1 − (1 − RBBN ) − (1 − RBBS ) + eµBB (1 − RBBN )(1 − RBBS ),which becomes:


true
RBBC
= ln

(1 − RBBN ) + (1 − RBBS ) − (1 − RBBC )(1 − RBBN )(1 − RBBS )

(C.8)


,

(C.9)

true
when solved for the true BBC coincidence rate (RBBC= µBB ). For the exclusive singles rate, theequation is:
true
true
RBN
= − ln(1 − RBBN ) − RBBC.

(C.10)

Therefore, we can get the true rates directly using Eq. C.9 and C.10 when measuring the singles andcoincidence rates of a detector.
trueEquation C.4 can be expressed in terms of RBBC(= µBB ) only by substituting in kN = B,N /BBand kS = B,S /BB as:
RBBC = 1 − e−µBB (1+kN ) − e−µBB (1+kS ) + e−µBB (1+kN +kS ) .

(C.11)

kN and kS can be determined when the cross sections seen by a single BBC versus the coincidence isevaluated, or one can empirically determine the ratios from data, such as beam clock triggered data orfrom scalers. For the 500 GeV pp case, we have determined that kN = kS = 0.282 ± 0.006. Then, therate correction factor (C(RBBC ): see Table 4.2) is calculated using Eq. C.11.

133

Appendix D

Tracking Performance
As a quality assurance of the DC and PC1 data set, number of tracks per MB (Minimum Bias: BBCLL1)
event (Ntrk /N M B ) was checked. Tracks with hits on X1 and X2 plane of the DC and on the PC1 withpT > 2.0 GeV/c are selected.
Figure D.1 shows Ntrk /N M B in each φDC bin as a function of run number (run number of thehorizontal axis is in arbitrary unit. Top panel: East arm, Bottom panel: West arm, Left side: Southside, Right side: North side). It can be noticed that some part of the DC/PC1 is unstable (e.g. 3.6 <φDC < 3.7) and some part of the DC/PC1 become dead (e.g. −0.2 < φDC < 0) during Run 9 period.

Figure D.1: Ntrk /N M B in each φDC bin as a function of run number. (positive charge, Top panel: Eastarm, Bottom panel: West arm, Left side: South side, Right side: North side). Blue boxes indicate thestable parts during Run 9 where is picked up to make Fig. D.2 and D.3.

134

Figure D.2 displays the Ntrk /N M B as a function of run number when choosing stable parts which areindicated with blue boxes in Fig. D.1 1 (Top panel: East arm, Bottom panel: West arm, Left side: Southside, Right side: North side). The Ntrk /N M B in this figure is slightly increased with time throughout
the Run 9 period. It turned out to be explained by multiple events in data holding period of the DC(pile up effect). Since the data holding time of DC is not one beam crossing clock, it results in counting
track events which occurred in the previous crossing at the recorded crossing in high trigger rate runs.
Figure D.3 shows the rate (= N M B /(# of Beam Crossings)) dependence of Ntrk /N M B and clear
correlation is observed. Expected rate dependence caused by the pile up effect are also drawn in the
figure. The function form of the expected rate dependence is:y = y0 ×x0
ln(1 − NCLK × x)
×,ln(1 − NCLK × x0 )
x

(D.1)

where x represents the rate (N M B /(# of Beam Crossings)), y is the Ntrk /N M B , (x0 , y0 ) stands for an
arbitrary point (in Fig. D.3, x0 is fixed at 0.1214 and y0 is a free parameter for fitting), and ”NCLK”
represents the data holding period of the DC (i.e. gate width, unit: beam crossing clock, this is also afree parameter for fitting). This function form is extracted by the following manner:
1. Take two points in Ntrk /N M B vs rate plane, for example, (x0 , y0 ) and (x, y).
2. Assuming Ntrk /N M B does not depend on the collision rate when N M B is calculated properly, thefollowing expression should have equality:
Ntrk= (const.),M
B
N
× (− ln(1 − NCLK × x)/(NCLK × x))(D.2)

where the term of (− ln(1−NCLK×x)/(NCLK×x)) is the correction factor of the multiple collisioneffect 2 .
3. Applying the relationship of Eq. D.2 to (x0 , y0 ) and (x, y), the function form of Eq. D.1 can beobtained.
The obtained NCLK from fitting is about 3.1 clocks that is consistent with the DC design (3 clocks).
This number is used in the acceptance calculation for rate correction (Sec. 4.3.5)
Figure D.4 shows the Ntrk /N M B as a function of run number (including unstable parts) and runby-run fluctuation become larger than Fig. D.2. The larger fluctuation are considered to come from therun-by-run acceptance change of the DC/PC1. Since a technique to handle this fluctuation is employedin acceptance calculation (see Sec. 4.3.5), no run is rejected and no φDC region is masked in this thesis.

1N
M B in this figure is calculated by integral over φ
DC bin of each part (i.e. East arm South side, East arm North
trk /Nside, West arm South side and West arm North side).
2 This correction is the first order approximation of the correction described in the Appendix C.

135

Figure D.2: Number of tracks per MB event (Ntrk /N M B ) at stable parts of the DC/PC1 as a function of
run number (positive combined, Left:South side, Right:North side, Top: East arm, Bottom: West arm).
The Ntrk /N M B is slightly increased with time throughout the Run 9 period due to the pile up effect.
The fluctuation observed in short time period is originated from the rate degeneration during a beam fill.

Figure D.3: Number of tracks per MB event at stable parts of DC/PC1 as a function of beam rate.
(positive charge, Left:South Side, Right:North Side)136

Figure D.4: number of tracks per MB event as a function of run number. (positive charge, Left:South
Side, Right:North Side) A large run dependence comes from the acceptance change is observed. For
example, large drop around run 277700 and run 278000 in East arm are corresponding to the acceptancelost and the decrease in West arm after run 279100 is mainly due to the dead area of −0.2 < φDC < 0(you can also notice these acceptance lost in Fig. D.1. The fluctuation observed in short time period is
originated from the rate degeneration during a beam fill.

137

Appendix E

Contamination from Residual
Transverse Polarization
E.1

Residual Transverse Polarization

As described in Sec. 2.1.3, the polarization direction is rotated from transverse to longitudinal just before
the interaction point (IP) and the presence of any residual transverse polarization should be monitored
at the PHENIX IP. The single transverse spin asymmetry (AN : Eq. 2.1 in Sec. 2.1.3) of forward neutron
production measured with ZDCs and SMDs is utilized for this purpose. The spin direction at thePHENIX IP is evaluated by the comparison of the measured AN amplitude at longitudinal run period,
ALongitudinal, and that at transverse run period, ATNransverse . The longitudinal and transverse componentsNof the polarization direction at longitudinal period is extracted by:s
2ALongitudinal
PT
PL
PT
= 1−
,
= NT ransverse .
P
P
P
AN

(E.1)

Since it is possible that the transverse component is diagonal after the spin rotation, comparison should be
performed by not only the amplitude but also the phase of azimuthal dependence. However, it is difficult
to obtain correct values from the fit results of sine modulation 1 in case of nearly zero asymmetry as fit
result is labile. For the stable operation, the comparison has been performed with the LR (Left-Right)
asymmetry, ALongitudinal, and UD (Up-Down) asymmetry, ALongitudinal. PT /P can be represented as:
LR
UD
!
!
q
LongitudinalLongitudinal
A
A
PT
UD
LR
= κ2LR + κ2U D , where κLR ≡, κU D ≡.
(E.2)
P
ATNransverse
ATNransverse
The phase information is included in the relation of them (e.g. κLR /κU D ).
During the Run 9 period, the LocalPol team at PHENIX has measured fill by fill residual transverse
polarization (PT /P ) with newly implemented local polarimeter scaler. The measured PT /P are 7 − 12 %in the blue beams and 3 − 5 % in the yellow beam [95]. These values are corresponding to more than
99 % of the relative longitudinal polarization (PL /P ). The uncertainty of the blue beam is larger due toa sign error in some of the RHIC beam position monitors, which caused errors in the orbit of the bluebeam.
1A
N is governed with the sine modulation to the transverse spin component.

138

E.2

Contamination to Measured AL

The residual transverse polarization, along with a nonzero AN for the electron/positron candidates inour sample, would create a left-right asymmetry, and that causes a contamination to the measured AL .
Considering the transverse polarization, the yield of electrons/positrons detected by PHENIX at theangle of (θ, φ) in polarized pp collisions is written as:
N (θ, φ) = σ(θ) ε(θ, φ) L [ 1

B
B
B
+ AB
L (θ)PL + AN (θ)PT cos(φ − φB )+ AYL (θ)PLY + AYN (θ)PTY cos(φ − φY )+ ALL (θ)PLB PLY + O(P~TB · P~TY ) + ... ].

(E.3)

Here, θ and φ represent the polar angle relative to z-axis and the azimuthal angle relative to the positive x
direction, respectively (see Fig. 2.9 in Sec. 2.2). The variables in this equation are described in Table E.1.
O(P~TB · P~TY ) in the equation shows higher order terms involving squared PT , and are ignored in thefollowing discussion because the magnitudes are 10−2 ∼ 10−3 lower than the longitudinal polarization.
notationσ(θ)
Lε(θ, φ)AjL (θ)AjN (θ)ALL (θ)PLj
PTj
φj

explanation
Unpolarized cross section at a scattering angle θ. (there is no φ dependence)
Integrated luminosity.
Electron/positron reconstruction efficiency
Longitudinal single spin asymmetry for the j (Blue or Yellow) beam andthe electron/positron measured at a scattering angle θ.
Transverse single spin asymmetry for the j beam. The azimuthal angle
dependence in the equation is taken from [135].
Longitudinal double spin asymmetry.
Longitudinal polarization of the j beam.
Transverse polarization of the j beam.
Azimuthal angle of the PTj relative to the vertical direction (i.e. positive ydirection).
Table E.1: explanations of variables in Eq. E.3.

The number of recorded electrons/positrons is given by the integral of Eq. E.3:Z
Ne =dθdφ N (θ, φ)
PHENIXB Y
fB P B + A
g
fY P Y + Ag
Bδ PB + A
Y δ PY + Ag= σ L ( 1+A
LL PL PL ),
L L
N B T
L L
N Y T

(E.4)

where:
Z
σ

=

f
AjL

=

g
AjN δj

=

g
A
LL

=

dθdφ σ(θ)ε(θ, φ),Z
1dθdφ AjL (θ)σ(θ)ε(θ, φ),σZ
1dθdφ AjN (θ)σ(θ)ε(θ, φ) cos(φ − φj )σZ
1dθdφ ALL (θ)σ(θ)ε(θ, φ).
σ
139



Z
Z
δj ∼ dφ ε(φ) cos(φ − φj ) / dφ ε(φ) ,Here, j denotes the beam (Blue or Yellow) and all integrals should be performed in the PHENIX acceptance (|η| < 0.35, ∆φ = π). Since Blue and Yellow beams are identical except for left-right reversal, thefollowing relation holds:
fB = AfY ≡ A ,
A
L
L
L

g
g
B = −AY ≡A ,
A
N
N
N

(E.5)

so that Eq. E.4 can be written as:
Ne = σ L ( 1 + AL PLB + AL PLY + AN δB PTB − AN δY PTY + ALL PLB PLY ),gwith one more substitution of ALL ≡ A
LL . Using this relation, Eq. 4.11 in Sec. 4.4 is modified as:σ++ = (1 + AL PLB + AL PLY + AN δB PTB − AN δY PTY + ALL PLB PLY ) · σ,






 σ+− = (1 + AL PLB − AL PLY + AN δB PTB + AN δY PTY − ALL PLB PLY ) · σ,
σ−+ = (1 − AL PLB + AL PLY − AN δB PTB − AN δY PTY − ALL PLB PLY ) · σ,





σ−− = (1 − AL PLB − AL PLY − AN δB PTB + AN δY PTY + ALL PLB PLY ) · σ.

(E.6)

(E.7)

One can see from the above equations that for a perfect PHENIX detector with the perfectly verticaltransverse polarization (i.e. φj = 0), the residual AN contributions are naturally suppressed by δj factors
as PHENIX covers symmetric regions around y-axis 2 .
However, in the real world, the effect on AL from a residual transverse polarization might not exactly
cancel because of (θ, φ) dependence of the acceptance from dead areas or other cuts, and also because
it is possible that there is a non-vertical transverse polarization (i.e. finite φj ). Using a toy MC whereparticles were distributed with cos(φ − φj ), we evaluated the effect which a non-zero vertical polarization
would have on the AL . At a phase of φj = 15◦ , the residual effect on AL is 5 % of AN (δj ∼ 0.05).
Putting realistic dead areas in the MC, the 5 % of the residual effect on AL is changed by only O(10 %).
Taking into account all effects, the AL could be modified by:
AN δj PTj ∼ 5 % × (12 %PL ) × AN = 0.006 × PL × AN .

(E.8)

With a 12 % of relative transverse component (PT /P ), the AN is still suppressed by over a factor of 100.
Even assuming that AN were 10 %, any contribution to AL from a nonzero AN and realistic residualtransverse polarization would be at ∼ 10−3 level, and could be ignored. Note that it is likely that the AN
is negligibly small since the AN at mid-rapidity has been measured to be zero up to pT of ∼ 10 GeV/c
at 200 GeV [136, 137], therefore the assumption of 10 % AN is a very conservative estimation.

2 In this case, δj ∼Rdφ cos φ /R

dφ = 0.

140Appendix F

Spin Fit Quality Check
Spin asymmetries in Sec. 4.4.2 are calculated from the maximum likelihood method. Although the results
are compared to the calculation from Eq. 4.9 and they are consistent with each other, it is better to confirm
the reliability of fitting process in different way.
As the likelihood function (Eq. 4.15) has three parameters (σ, εL , εLL ), and only four spin states(++, +−, −+, −−) is available in the fitting process, only one degree of freedom (DOF) remains. When
there are plenty of statistics, it can be checked whether the fit is reasonable or not by calculating chisquare value and comparing it to the chi-square distribution with one DOF. However, it is difficult for thisanalysis to be checked in that manner because the uncertainty does not follow the Gaussian distribution
due to the small statistics.
The easiest way to check the validity of the spin fit is to produce a probability distribution of Eq. 4.15.
Q
Once three parameters are determined, L = fi (i denotes four spin states) can be calculated. In orderQ
to make a probability distribution of L = fi , a toy MC is demonstrated in the following procedures:
1. Using σ data , εdata, εdataL
LL (evaluated values from the data) and Li (relative luminosity), generate a
set of pseudo four spin states (NiM C ) with TRandom3::Poisson() 1 .
CC
2. Fit NiM C with the maximum likelihood method to get σ M C , εMand εM
L
LL . Note that these values

are not exactly same as σ data , εdata
and εdata
L
LL , respectively.
CMCMC
3. Using σ M C , εM
, calculate L =L , εLL , Li and Ni

Q

fi .

Q
These procedures are repeated 500 times and a probability distribution of L = fi is generated.
Q
Figure F.1 shows the probability distribution of L = fi made by this toy MC (Left: positive, Right:
negative charge). In order to compare the probability distribution to chi-square distribution with one
DOF, L is converted in the following variable 2 :var. = −2 · (ln(L) − ln(A)),
p√
Q
C
A = 4 spin states 1/ 2πσi , σi = µM
i

(F.1)

1 TRandom3 is a random number generator implemented in ROOT.
2 The probability distribution is not expected to match to the chi-square distribution because of our small statistics. Thisvariable transformation is just to check how the probability distribution differs from the chi-square distribution.

141

(µi is calculated from Eq. 4.16). This formula is extracted by comparing chi-square to L =Qfi in thelarge statistical limit where fi (Poisson distribution) can be approximated to Gaussian distribution. The
Red lines drawn in Fig. F.1 are the probability value of the real data. Since they seem normal value inthe probability distribution, it can be concluded that the spin fit with the maximum likelihood method
is reasonable.

Figure F.1: Probability distribution obtained by repeating process (1-3) 500 times (Left: positive, Right:
negative charge). The Red line indicates the probability value of the real data and it looks normal value
in the distribution. The Black curve shows the chi-square distribution with one DOF. The probability
distribution is not so different from the chi-square distribution for positive charge. On the other hand, it
is slightly different for negative charge because of the small statistics.

142

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