0

Double helicity asymmetry ofinclusive neutral pion productionin polarized pp collisions at √s =62.4 GeV

Kazuya Aoki

A dissertation submitted in partial fulfillment of
the requirements for the degree of Doctor of Sccience

Department of Physics
Kyoto University
2009Abstract
We present the results of√double helicity asymmetry ALL of inclusive π 0 production inpolarized pp collisions at s = 62.4 GeV.
Polarized lepton-nucleon deep-inelastic scatterings (DIS) have revealed that the quark
spin carries only ∼ 25% of the proton spin, which contradict our naive expectation. It
stimulated various effort towards the understanding of the proton spin. Despite of thewide effort, there remains large uncertainty especially on gluon spin contribution to theproton. The double helicity asymmetry ALL of π 0 production in polarized pp collisions
is sensitive to the gluon contribution to the proton since the production is dominatedby quark-gluon and gluon-gluon scatterings in the initial protons. The gluons interact
at leading order in these processes unlike the DIS where gluons only contribute throughhigher order.
The measurement was performed with the PHENIX detector at the Relativistic Heavy
Ion Collider (RHIC) at Brookhaven National Laboratory (BNL) in the United States in√
Run 2006. The polarized proton beams collided at s = 62.4 GeV at RHIC and theintegrated luminosity of analyzed data sample is 40 nb−1 with an average polarization of
48%. We have measured π 0 via two photon decay, π 0 → γγ, and the decay photons weredetected with electromagnetic calorimeters (EMCal) of the PHENIX detector.
The measured kinematic range for ALL of π 0 is pT = 1 − 4 GeV/c, over a pseudorapid√
ity range of |η| < 0.35. The probed Bjorken x roughly scales with xT = 2pT / s. Thus
the results probe the higher range of Bjorken x√ of the gluon, xg , with better statisticalprecision than our previous measurements at s = 200 GeV in Run 2005. The measurements are sensitive to the gluon polarization in the proton for 0.06 < xg < 0.4. The
measured ALL is consistent with zero within the uncertainties. The data do not support alarge gluon polarization scenario, such as GRSV-max. The results were included in recent
global analysis of polarized
PDFs by the DSSV group. An truncated integral of ∆g(x)R12
was obtained to be 0.001 ∆g(x)dx = 0.013+0.106−0.120 for ∆χ = 1. The uncertainties are muchsmaller than the range previously allowed by the analysis which only includes DIS data.
In addition to ALL , the cross sections of π 0 production were measured to confirm
the applicability of perturbative QCD (pQCD) framework which the argument of ∆gextraction is based on. The results are consistent with√the pQCD calculation. Single spin
asymmetry AL , which is expected to be negligible at s = 62.4 GeV was measured andwas consistent with zero within the uncertainties. Double transverse spin asymmetry AT T
was measured to obtain a systematic uncertainty which come from the residual transverse
components of the beam polarizations. The maximal possible AT T effect on ALL wasfound to be < 0.15 · δALL , where δALL denotes the statistical uncertainty of ALL .

Contents
1 Introduction

12 Physics
2.1 π 0 production in pp collisions . . . . . . . . . . . . . . . . . . . . . . . . .
2.2 Deep inelastic scattering formalism . . . . . . . . . . . . . . . . . . . . . .
2.3 DGLAP equations and factorization . . . . . . . . . . . . . . . . . . . . . .
2.4 Factorization theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.5 Sum rules for PDFs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.5.1 Sum rules for unpolarized PDFs . . . . . . . . . . . . . . . . . . . .
2.5.2 Sum rule for polarized PDFs . . . . . . . . . . . . . . . . . . . . . .
2.6 Unpolarized PDFs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.7 Polarized PDFs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.8 Probing ∆g in polarized pp collisions . . . . . . . . . . . . . . . . . . . . .
2.9 xT and probed x range . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4
4
6
9
13
14
14
14
15
16
20
21

3 Experimental Setup
3.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2 Relativistic Heavy Ion Collider (RHIC) . . . . . . . . . . . . . . . . . . . .
3.2.1 Polarized proton source . . . . . . . . . . . . . . . . . . . . . . . . .
3.2.2 Polarization of proton beams . . . . . . . . . . . . . . . . . . . . .
3.2.3 Polarimeters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.3 PHENIX overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.4 PHENIX global detectors . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.4.1 Beam Beam Counter (BBC) . . . . . . . . . . . . . . . . . . . . . .
3.4.2 Zero Degree Calorimeter (ZDC) and shower max detector (SMD) .
3.5 PHENIX central arms . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.5.1 Electromagnetic calorimeter (EMCal) . . . . . . . . . . . . . . . . .
3.5.2 EMCal front end electronics . . . . . . . . . . . . . . . . . . . . . .
3.6 PHENIX trigger system . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.7 PHENIX DAQ system . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

25
25
25
29
30
36
39
44
44
46
50
50
56
58
58

i

4 Analysis
60
4.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
4.2 Run selection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
4.3 Beam polarization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
4.4 Relative Luminosity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
4.4.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
4.4.2 Bunch selection criteria . . . . . . . . . . . . . . . . . . . . . . . . . 65
4.4.3 Vertex width . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
4.4.4 A method to estimate the uncertainty on the relative luminosity . . 67
4.4.5 Results of relative luminosity analysis . . . . . . . . . . . . . . . . . 71
4.4.6 Event overlap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
4.4.7 Single beam background . . . . . . . . . . . . . . . . . . . . . . . . 73
4.5 π 0 reconstruction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
4.5.1 High-pT photon trigger performance . . . . . . . . . . . . . . . . . . 74
4.5.2 Clustering algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . 77
4.5.3 Quality assurance of the EMCal towers . . . . . . . . . . . . . . . . 78
4.5.4 Energy calibration of EMCal . . . . . . . . . . . . . . . . . . . . . . 78
4.5.5 Absolute energy scale . . . . . . . . . . . . . . . . . . . . . . . . . . 80
4.5.6 Reconstruction of π 0 . . . . . . . . . . . . . . . . . . . . . . . . . . 83
4.5.7 EMCal stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
4.5.8 Discussion on the background peak . . . . . . . . . . . . . . . . . . 86
4.5.9 Cosmic ray event under π 0 signal window . . . . . . . . . . . . . . . 87
4.5.10 Vertex cut difference in π 0 and BBC trigger . . . . . . . . . . . . . 89
4.6 Spin asymmetries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
4.6.1 Calculation of the asymmetries . . . . . . . . . . . . . . . . . . . . 93
4.6.2 Background ratio in signal window . . . . . . . . . . . . . . . . . . 95
4.6.3 Average pT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
4.6.4 Asymmetries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
4.7 Systematic uncertainties . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
4.7.1 Beam polarizations . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
4.7.2 Beam polarization orientations . . . . . . . . . . . . . . . . . . . . . 101
4.7.3 Relative luminosity . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
4.7.4 Bunch shuffling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
4.7.5 Double collision effect . . . . . . . . . . . . . . . . . . . . . . . . . 102
4.7.6 Summary of the systematic uncertainties . . . . . . . . . . . . . . . 104
5 Results and discussions
105
5.1 The cross section results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
5.2 Results of the double helicity asymmetries . . . . . . . . . . . . . . . . . . 107
5.3 Global analysis of polarized PDFs . . . . . . . . . . . . . . . . . . . . . . . 108
5.3.1 AAC global analysis . . . . . . . . . . . . . . . . . . . . . . . . . . 108
ii

5.4

5.3.2 DSSV global analysis . . . . . . . . . . . . . . . . . . . . . . . . . . 111
Further study on the spin structure of the proton . . . . . . . . . . . . . . 114

6 Conclusion

116

A z-dependent BBC-efficiency
120
A.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120√
A.2 z-dependent BBC-efficiency at s = 62.4 GeV . . . . . . . . . . . . . . . . 120
A.2.1 Procedure √with WCM and vernier scan . . . . . . . . . . . . . . . . 120
A.2.2 Results at s = 62.4 GeV . . . . . . . . . . . . . . . . . . . . . . . 121
B EMC clustering algorithm
124
B.1 PbSc . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
B.2 PbGl . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
C Some information on pol-PDFs
127
C.1 Q2 evolutions of parton contributions to the proton spin . . . . . . . . . . 127
C.2 DSSV polarized PDFs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128
C.3 GRSV . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128
D Subprocess cross sections for gg → q q̄ and gg → gg

iii

130

Chapter 1
Introduction
The proton is a very fundamental particle since all materials in the universe including
ourselves are made of it. Thus it is natural desire as an intellectual being, to understandit. The proton turned out to be composed of quarks and gluons, which are the elementary
particles to our best knowledge at present. Therefore, proton’s fundamental properties(such as charge, spin, magnetic moment and mass) should be understood in terms of theircomponents.
The proton has spin 1/2. Polarized lepton-nucleon deep inelastic scattering (DIS)
experiments have revealed that only ∼25% of the proton spin is carried by quark andanti-quark spin [1, 2, 3, 4], which contradict our naive expectation that most of the proton
spin is carried by quarks. It is called “spin crisis.” One may complain it since the fact
does not necessarily mean a breaking of a fundamental conservation law. Indeed, othercomponents such as gluon spin and/or orbital angular momentum can carry the rest of the
proton spin. It is sometimes called “spin puzzle” instead. Regardless of what we call it,
it initiated various effort towards the understanding of the proton spin structure. Despite
the wide efforts, the spin structure is still not well understood. There still remains large
uncertainty, especially on gluon spin contribution to the proton, ∆g. The main subjectof this thesis is to put constraint on it.
The spin structure have been studied mainly with lepton-nucleon DIS, where themajor contribution comes from the electromagnetic interaction. The gluons, however,are the mediator of strong interaction and do not react to electromagnetic interaction.
Thus the gluons in the proton only participates at higher order processes in DIS, withthe help of quarks which react to both strong and electromagnetic interactions. Thatis the reason why we don’t have much information on the polarization of gluon. There
are other efforts to explore gluon polarization in DIS by identifying final state hadrons
and enhancing the gluon participating processes ( Semi-inclusive DIS ). The following
experiments have measured helicity asymmetry of SDIS events with two high transverse
momentum hadrons in the final state and extracted ∆g/g. The HERMES experiment
at HERA used 7.5 GeV polarized positron beam and a polarized hydrogen target [5, 6].
1

2

CHAPTER 1. INTRODUCTION

The Spin Muon Collaboration (SMC) at CERN used 190 GeV polarized muon beam
scattered on polarized proton and deuteron targets [7]. The COMPASS experiment at
CERN utilized 160 GeV polarized muon beam scattered on a polarized 6 LiD target [8].
The COMPASS experiment has also measured SDIS events with D ∗ mesons (open charm)
tagged [9]. These measurements probe x region of 0.06 ∼ 0.4. However, the analyses are
based on Monte Carlo simulations at leading order (LO). Next-to-leading order (NLO)
calculations are not complete, and they suffer from large theoretical uncertainties.
Hard scatterings in polarized pp collisions are ideal tools to explore ∆g since the gluons in the proton participate the interaction directly at the leading order. However, it istechnically difficult to maintain the polarizations of proton beams through the acceleration. The difficulty was overcome by the invention of Siberian snakes [10, 11]. Relativistic
Heavy Ion Collider (RHIC) was given the ability to accelerate and collide polarized protons by Siberian snakes and it has been providing us an unique opportunity to explore
gluon spin in the proton directly through strong interactions [12]. Double helicity asymmetry (ALL ) of inclusive π 0 production in polarized pp collisions is sensitive to ∆g since
π 0 production is dominated by gluon-gluon (gg) and quark-gluon (qg) scattering for thecurrently accessible range√of pT . Since 2002, the PHENIX experiment at RHIC has beenmeasuring ALL of π 0 at s = 200 GeV [13, 14, 15, 16] and has put constraint on ∆g.
However, there still remains large uncertainty on ∆g at large Bjorken√x.
In Run√ 2006, we performed polarized pp experiment not only at s = 200 GeV, butalso at s = 62.4 GeV. Probed Bjorken x roughly scales with the scaling variable xT ,√which is defined as x√T = 2pT / s. At fixed xT , the cross√section of π 0 is two orders ofmagnitude larger at s = 62.4 GeV compared to that at s = 200 GeV. Thus the lower
center of mass energy have an advantage to√ probe large x gluons.
Originally low energy experiment at s = 62.4 GeV was motivated by heavy ion√
experiment. A new state of dense matter is formed in Au-Au collisions at sN N =200 GeV at RHIC. Parton energy loss in the produced dense medium results in highpT leading hadron suppression. Measurements of high pT data at lower energies are ofgreat importance in identifying the energy range at which the suppression sets in. They
require solid measurements of the cross section in pp collisions as a baseline for medium
effects. PHENIX has measured particle production in Au+Au and Cu+Cu collisions at√
sN N = 62.4 GeV and discussed them with the results of pp collisions as a√baseline whichwere obtained at Intersecting Storage Rings (ISR) at CERN. The energy s = 62 GeV isthe highest collision energy available at the ISR, the world’s first (unpolarized) pp collider.
At √the ISR, inclusive neutral and charged pion cross sections were measured several times
at s ∼ 62 GeV, but they have large uncertainties and have a large variation [17]. Having
both heavy-ion and baseline pp measurements with the same experiment is advantageous
as it leads to a reduction of the systematic uncertainties√ and, thus, to a more precise
relative comparison of the data. The π 0 measurements at s = 62.4 GeV is advantageous
for both heavy-ion and spin physics.
Measurements of other final states in polarized pp collisions have also been performed.

3
PHENIX has obtained preliminary
results for ALL of charged pions (π + , π − ) [18], η [19],√and direct photon√[20, 21] at s = 200 GeV. The STAR experiment at RHIC has published jet ALL at s = 200 GeV [22] and presented preliminary results for charged pions
ALL [23]. These measurements are complementary to the results of π 0 ALL since they
have different systematics. Measurements of π 0 ALL have disadvantage in determining
the sign of ∆g since ∆g contributes to ALL as a quadratic function (since gg and gq are
the dominant subprocesses). The (∆g)2 contribution cancels in the difference of π + andπ − thus they are important complementary measurements. However, their drawback isthe statistics due to the fact that an efficient trigger is not available. Direct photon production is dominated by qg → qγ and is also sensitive to the sign however, the drawback
is the low statistics.
In Chap. 2, we introduce the theoretical framework. In the framework, the spin
structure of the proton is described in terms of the polarized parton distribution functions(PDFs). The current experimental knowledge on the unpolarized and polarized PDFs is
also described in the chapter. The double helicity asymmetry ALL which is the measured
quantity in this thesis, is defined. In Chap. 3, the experimental setup is explained. RHIC,
which is the collider that provides polarized pp collisions, is introduced. and the PHENIX
detector, which is used in this thesis to detect π 0 from pp collisions, is explained. The
detailed analysis procedure
is discussed in Chap. 4. In Chap. 5, the results of ALL of√0
π production at s = 62.4 GeV, together with the results of cross section, are shownand the impact of the results on the gluon polarization in the proton is discussed. We
summarize our conclusion in Chap. 6.
The gluon spin contribution to the proton is written in either the upper or the lower
cases, ∆G or ∆g in the literature. There is a tendency to express integral of ∆g as ∆G.
Some experimentalists use ∆G instead of ∆g to show that their data is not sensitive tothe functional form of ∆g. However, there is no strict rule about it. Mostly the lower case∆g is used and these are not strictly distinguished in this thesis. When x dependence isof interest, it is written as ∆g(x). The upper case is used for the labels of some PDF
models (such as ∆G = 0), and an integral of ∆g.

Chapter 2
Physics - theoretical framework andpresent status of spin structure
functions
The goal of the thesis is to obtain information on the spin structure of the proton throughπ 0 production in polarized pp collisions. The spin (and unpolarized) structure is expressedby the parton model, in terms of the parton distribution functions (PDFs).
The parton model interpretation of π 0 production in pp collision is explained in Sec. 2.1.
Although this thesis is based on pp collisions, the structure of the proton has been studied with deep inelastic scatterings (DIS) before our experiment started. In Sec. 2.2, theformalism for DIS experiment for unpolarized and polarized targets are introduced. An
improvement of the parton model given by DGLAP equations, and factorization is introduced in Sec. 2.3. The factorization theorem, which validates the framework is explainedin Sec. 2.4. Some useful sum rules for the 1st moment of PDFs are known and are
introduced in Sec. 2.5. Experimental knowledge (before our experiment started) on unpolarized and polarized PDFs are shown in Sec. 2.6 and Sec. 2.7. The double helicity
asymmetry ALL which is the measured quantity in this thesis, is discussed in Sec. 2.8.
A scaling variable xT is introduced and the probed kinematical range are explained inSec. 2.9.

2.1

π 0 production in pp collisions

π 0 production in pp collisions is understood by the parton model [24]. The proton isconsidered to be a collection of point-like particles called “partons” and the interactionis understood as the incoherent sum of the partons’. This is so-called an impulse approximation. The partons in the proton are identified as quarks and gluons. Figure 2.1
illustrates π 0 production in pp collisions. A parton “a” from a proton and a parton “b”
4

2.1. π 0 PRODUCTION IN P P COLLISIONS

5

from another proton interact and the final state parton “c” fragments into π 0 .

fbProton 2

π0

b
xb

c Dπ

0

σ^
xa
a

Proton 1

fa
FF

PDF
subprocess cross section(parton-parton interaction)

Figure 2.1: π 0 production in pp collisions. The process is divided into three parts: PDFs,
subprocess cross sections, and FFs.
The unpolarized differential cross section is written asdσ = Σa,b,c

Z

dxa

Z

dxb

Z

c
dzc fa (xa , µ2 )fb (xb , µ2 )[dσ̂ab(xa P, xb P, µ2 )Dcπ (z, µ2 )](2.1)

fa (xa , µ2 ) is the parton distribution function (PDF) of a parton species “a” to have amomentum fraction xa of the parent proton with momentum P , and fb (xb , µ2 ) is for acparton “b” with a momentum fraction xb . dσ̂ab(xa P, xb P, µ2 ) represents the cross section ofparton-parton interaction (partonic cross section). D(z, µ2 ) is the probability for a parton“c” to fragment into π 0 , with π 0 having a fraction z of the parton’s momentum. It iscalled the fragmentation function (FF). µ2 represents equally chosen renormalization andfactorization scale (µ2 = µ2R = µ2F ). In this framework, π 0 production is divided into threeparts: PDFs, subprocess (or partonic) cross sections, and FFs. PDFs and FFs cannot becalculated with perturbative QCD (pQCD) and have to be determined by experiments.
The subprocess cross sections can be calculated with pQCD. The factorization scale µ2F
was introduced to renormalize the unperturbative soft physics part into PDFs and FFs.
The treatment is called the factorization and is discussed in Sec. 2.3 and 2.4.
The polarized differential cross-section is written in similar way,d∆σ ≡ dσ++ − dσ+−

6

CHAPTER 2. PHYSICS= Σa,b,c

Z

dxa

Z

dxb

Z

c
dzc ∆fa (xa , µ2 )∆fb (xb , µ2 )[d∆σ̂ab(xa P, xb P, µ2 )Dcπ (z, µ2 )](2.2)

where σ++ (σ+− ) denote the cross section for same (opposite) helicity combination of theinitial protons, and ∆fa (xa , µ) is the spin-dependent PDF of the parton a which is definedas∆fa (xa , µ) = fa+ (xa , µ) − fa− (xa , µ).
(2.3)
fa+ (xa , µ) (fa− (xa , µ)) is the PDF for parton a to have the same (opposite) helicity as thecparent proton’s. ∆fb (xb , µ) is defined in the same way for parton b. ∆σ̂ab(xa P, xb P, µ2 )represents the polarized cross section of parton-parton interaction. which is defined as
c
c
c
∆σ̂ab
(xa P, xb P, µ2) = [σ̂ab(xa P, xb P, µ2)]++ − [σ̂ab(xa P, xb P, µ2 )]+− ,(2.4)

where the subscripts ++ and +− denote the spin states of the interacting partons.

2.2

Deep inelastic scattering formalism

The unpolarized and polarized structure of the proton has been studied with deep inelastic
scattering (DIS). Thus we introduce its formalism in this section, and discuss higher ordercorrections in the next section. Its derivation and detailed discussion on the formalism
can be found in [25, 26].
DIS is a high-energy inelastic scattering between leptons and nucleons. We take electron proton deep inelastic scattering as an example. Let k µ = (E, k) be the four-vectorof the incident electron, k ′µ = (E ′ , k′) be that of the scattered electron. In a similar way,
let P µ = (M, 0) be the four-vector of the target proton, where M denotes the proton
mass. The invariant mass of the final state hadron(s) is defined as W . The process canbe described as:e(E, k) + P (M, 0) → e(E ′ , k′) + X(W, PX ),(2.5)

and can be illustrated as Fig. 2.2a).
Bjorken scaling variable x is defined as
x≡

−q 2
Q2
=
,
2P · q2P · q(2.6)

where Q2 is defined as Q2 = −q 2 so that Q2 > 0. It can be interpreted as the fractional
momentum of parton as described later. A Lorentz invariant variable ν is introduced asν = PM·q which is equal to the energy loss of the lepton (E − E ′ ) in the laboratory frame.
Another Lorentz invariant variable y is defined as y = Pp·k·q and is equal to the fractional

2.2. DEEP INELASTIC SCATTERING FORMALISM

a)

7

b)

k’

k’
k

e
k
e-

γ*

q = k-k’-

q = k-k’γ*
xPP
NucleonP
NucleonW

W

Figure 2.2: a) Deep inelastic scattering (DIS). b) DIS interpreted by the parton model.

energy loss of the lepton ν/E in the laboratory frame. The introduced variables are
summarized in Table. 2.1
The exact calculation of cross section is not possible due to the lack of the knowledge
on the structure of the proton, but the structure can be parametrized with structure
functions since the form is restricted by the Quantum Chromodynamics (QCD).
With the help of the Lorentz invariance of the matrix element, parity conservation,and current conservation, the unpolarized cross section is written asid2 σ2πα2 h
2
2
2
Y
F
(x,
Q
)
−
y
F
(x,
Q
)
,
=
+
2
L
dxdQ2
xQ4

(2.7)

where Y+ = 1+(1−y)2, F2 and FL are the structure functions. Some use another structure
function F1 (FL ≡ F2 − 2xF1 ) to describe the cross section.
The formalism introduced so far does not depend on the parton model. Here weinterpret the process in the parton model, as in the case of pp collisions. It can beshown that the Bjorken x (defined in Eq. 2.6) is interpreted as the fractional longitudinal
momentum of the participating parton when the Q2 is high enough to neglect the proton
and the parton masses. (In the parton model, initial transverse momentum of the parton,
which is called the intrinsic kT , is also neglected in the PDF parametrization.) Let ξP be
the momentum of the parton, so that ξ is the fractional momentum of the parton. Since
the parton and the proton masses are neglected,(ξP + q)2 = 0

8

CHAPTER 2. PHYSICS
2ξP · q + q 2 = 0
ξ =

−q 2
.
2P · q(2.8)

e2i xqi (x)(2.9)

Therefore, ξ = x and the Bjorken x can be interpreted as the fractional momentum ofparton. Figure 2.2b) illustrates the parton model interpretation of DIS, where a partonwith momentum fraction of x interact with the virtual photon. Let qi (x) be the parton
distribution function for a parton i. Here qi (x) is used instead of fi (x) to indicate thatthe participating partons are quarks. The polarized quark distribution ∆qi (x) is definedin a similar way as Eq. 2.3. The cross section can be calculated as an incoherent sum ofparton-photon scatterings. Then the structure functions F2 and FL are identified as
F2 (x, Q2 ) =XiFL (x, Q2 ) = 0.

(2.10)

The first equation indicates that F2 (x, Q2 ) is independent of Q2 . The scaling behavior
was derived for Q2 → ∞ at fixed x (DIS limit) by Bjorken [27] and is called the Bjorken
scaling. The 2nd equation is the Callan-Gross relationship, F2 = 2xF1 , as a consequenceof scattering from spin 1/2 partons [28]. Early SLAC data showed that the relation
holds. [29]The cross sections for polarized DIS are parametrized as→→d2 ∆σk
d2 σ⇐
d2 σ⇒
≡
−
dxdQ2
dxdQ2 dxdQ2"#2y16πα2 y2y2
2y2
(1 + − γ=)g1 (x, Q ) − γ g2 (x, Q ) ,Q4242d2 σ⇑→
d2 σ⇓→
d2 ∆σ⊥≡−
dxdQ2
dxdQ2 dxdQ2s(2.11)(2.12)(2.13)y2 y
8α2y
g1 (x, Q2 ) + g2 (x, Q2 ) ,= − cos φ 4 γ 1 − y − γ 2
Q
4 2




(2.14)

where ⇒ and ⇐ (⇑ and ⇓) denote the nucleon helicity (transverse spin) state, → and ←denote the incident lepton’s helicity state, γ = 4m2 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 ∆qi (x)2 i i

g2 (x, Q2 ) = 0,(2.15)(2.16)

in the zeroth order parton model. g2 (x) was measured by several groups [30, 31] andg2 = 0 holds approximately.

2.3. DGLAP EQUATIONS AND FACTORIZATION

9variable description
Q2
Q2
= 2M
.
x
= 2P
·q
ν
Bjorken scaling variable.
ν= p·q(= E − E ′ ).
M
The energy loss of the lepton in the laboratory frame.
y= ν/E = p · q/p · k.
The fractional energy loss of the lepton.
q= k − k ′ . The four-momentum transfer.
Q2= −q 2 > 0. The four-momentum transfer squared.
Y+= 1 + (1 − y)2
Table 2.1: Variables commonly used in DIS description. Please note that x, ν, y, and Q2
are written in terms of Lorentz invariant variables, but some of them are interpreted in
the laboratory frame.
The introduced structure functions and their interpretation in the parton model aresummarized in Table. 2.2.
structure func. interpretation in the parton model.
P
F1 (x, Q2 ) = i e2i xqi (x)¯= 49 {xu(x) + xū(x)} + 91 {xd(x) + xd(x)}+ 91 {xs(x) + xs̄(x)}2
FL (x, Q ) = 0
P
g1 (x, Q2 ) = 12 h i e2i ∆qi (x)i¯= 12 94 {∆u(x) + ∆ū(x)} + 19 {∆d(x) + ∆d(x)}+ 19 {∆s(x) + ∆s̄(x)}g2 (x, Q2 ) = 0
Table 2.2: The structure functions in DIS and their interpretation in the parton model.
The quarks heavier than s are neglected.

2.3

DGLAP equations and factorization

An improvement of parton model beyond the zeroth order was developed by Dokshitzer,
Gribov, Lipatov, Altarelli and Parisi, collectively known as DGLAP. The improvement isrefereed to as the DGLAP evolutions or Q2 evolutions.
Figure 2.3a) shows a zeroth order process where a parton (quark) in the proton with
momentum xP directly absorb virtual photon. However, the parton may emit gluon before

10

CHAPTER 2. PHYSICS
a)

γ*

xP
P

b)
γ*

γ*
^
σ

xP

ξP
P

ξP

Figure 2.3: a) Zeroth order parton model. b) Improved parton model.

(or after) absorbing the virtual photon, as in Fig. 2.3b). In Fig. 2.3b), ξP describes theinitial quark momentum, and it evolves to the quark with momentum xP . A variable z
is defined as z = Q2 /(2ξP · q), in the analog of Bjorken x. z is the fractional momentumof the evolved parton compared to the initial parton, xP = z · ξP .
The cross section is obtained by integrating over all possible gluon emissions. However,the cross section of such process is divergent in the limit of collinear gluon emission,
where the angle between the quark and the emitted gluon reaches zero. The divergenceis regulated by introducing an arbitrary cut-off κ2 . The cross section becomes to have alarge log term ln(Q2 /κ2 ). It is divided into two parts, by introducing a factorization scaleµF , as ln(Q2 /κ2 ) → ln(Q2 /µ2F ) + ln(µ2F /κ2 ). The term ln(µ2F /κ2 ) are put into PDFs andthe arbitrary cut-off κ2 is hidden under PDFs. The other term which is free from thearbitrary cut-off, is put into the subprocess cross sections. The treatment is called thefactorization.
After the factorization, PDF (and partonic cross sections) becomes dependent on afactorization scale µ2F . The final cross section should not depend on µ2F , but the scale2.3. DGLAP EQUATIONS AND FACTORIZATIONq

Pqq (z)

11

z

z

q

Pgq (z)
1-z
q

q

1-z
z

z

Pqg (z)Pgg (z)
1-z

1-z
Figure 2.4: The splitting functions.

should be larger than the arbitrary cut-off and should not be too large to make a termln(Q2 /µ2F ) moderate size which appear in partonic cross section. The factorization scale isusually chosen as µ2F = Q2 for deep inelastic scatterings, and µ2F = p2T for π 0 production inpp collisions. Although the factorization scale µ2F and the renormalization scale µ2R don’t
have to be equal, both are often chosen as µ2F = µ2R . The obtained cross-section should
not depend on the choice of the scale, but it does due to the truncation in perturbative
expansion. The dependence on the scale is often used as a guide for the theoreticaluncertainty.
The renormalized PDF cannot be calculated perturbatively as it includes the nonperturbative part. However, once PDFs are known for a certain scale, their evolutions
with ln µ2 can be calculated as∂∂ ln µ2qi (x, µ2 )g(x, µ2)!

=

αs (µ2 ) X 1 dξ
2π
x ξ
j
Z

Pqi qj ( xξ , αs (µ2 )) Pqi g ( xξ , αs (µ2 ))Pgqj ( xξ , αs (µ2 )) Pgg ( xξ , αs (µ2 ))!

qj (ξ, µ2)g(ξ, µ2)!

,(2.17)

where Pqq (z) is the probability distribution for q(ξ) → q(zξ)g((1 − z)ξ) splitting, Pqg (z)for g(ξ) → q(zξ)g((1 − z)ξ) splitting, which are called splitting functions. Other two
splitting functions are defined in similar way. The four types of the splitting functions in
Eq. 2.17 are summarized graphically in Fig. 2.4.
The splitting functions are expanded as power series in αs (µ2 ),αs (1)(0)
P (z) + · · ·Pqi qj (z, µ2 ) = δij Pqq(z) +2π qi qj
αs (1)(0)
Pqg (z, µ2 ) = Pqg(z) + Pqg(z) + · · ·2π(2.18)(2.19)

12

CHAPTER 2. PHYSICS

Pgq and Pgg are expanded in similar way as Eq. 2.19. The δij in the leading term of Eq. 2.18
indicates that one needs higher order term beyond LO to change the quark flavor. Becauseof charge conjugation and the flavor independence of the QCD Lagrangian, Pqg and Pgq
are independent of quark flavor and the same for q and q̄. The Pqi qj satisfy Pqiqj = Pq¯iq¯jand Pqiq¯j = Pq¯i qj and the leading order term vanishes unless qi = qj .
A non-singlet combination is the difference of qi and qj . Among the non-singlets, anuseful combination isqi− (x, µ2 ) = qi (x, µ2 ) − q¯i (x, µ2 ),(2.20)
which is the valence quark distribution for flavor i. The evolution of non-singlet distribution does not involve the gluon density since it cancels. The splitting functions are known
to NNLO accuracies.
Similar equations are obtained for polarized case and can be written as:∂∂ ln µ2∆q(x, µ2 )∆g(x, µ2 )!

αs (µ2 ) Z 1 dξ
=
2π
x ξ
!
∆Pqq ( xξ , αs (µ2 )) ∆Pqg ( xξ , αs (µ2 ))∆Pgq ( xξ , αs (µ2 )) ∆Pgg ( xξ , αs (µ2 ))∆q(ξ, µ2)∆g(ξ, µ2)!

(2.21)The splitting functions have been calculated at NLO accuracy [32, 33]. Non singlet
combinations ∆q−∆q̄, and ∆qi +∆q¯i −∆qj −∆q¯j does not couple to gluon in evolution. In
LO, quark helicity is conserved before and after the splitting thus ∆Pqq = Pqq , ∆Pqg = Pqg ,∆Pgq = Pgq . The 1st moment ∆Σ does not depend on µ2 at all at LO.
Beyond the leading order
The subprocess cross sections and DGLAP evolutions are expanded in power series of thestrong coupling constant αs . The leading order partonic processes are 2 → 2 reactions
thus the perturbative expansion starts at O(αs2 ). Next to leading order (NLO) calculations
include terms at O(αs3 ). In this section, αs2 is factored out and omitted as in the literature,since the factor αs2 is common for the process of strong interactions.
When the initial partons have just enough energy to produce a high-transverse momentum parton (which subsequently fragments into the observed hadron), the phase space
available for gluon bremsstrahlung vanishes, resulting in large logarithmic corrections to
the partonic cross section. We define√x̂T = 2p̂T / ŝ,(2.22)
where p̂T is the transverse momentum of scattered parton which fragments into observed
hadron, ŝ is the center of mass energy of the partonic processes. If x̂T reaches unity, leadinglarge contributions arise as αsk ln2k (1 − x̂2T ) at the kth order in perturbative expansion.

2.4. FACTORIZATION THEOREM

13

Such terms can be taken into account to all orders in αs by threshold resummation [34]. It
is called leading log (LL) calculations. One step higher order correction is called the next
to leading log (NLL) calculations which includes terms appear in the form αsk ln2k−1 (1−x̂2T )at kth order perturbative expansion, to all orders. The relation between the fixed order
perturbative expansions (LO, NLO, NNLO, ...) and the resummations (LL, NLL, NNLL,...) is illustrated in Fig. 2.5.
The log terms become significant when xˆT → 1. Therefore, they are significant in
fixed target experiments, while they are small at collider energies. In fact,√calculations
show that the NLL corrections are necessary for fixed target experiment at s ∼ 20√GeV
to describe the measured cross section [35], while NLL corrections are smaller at s =200 GeV [36].
The unpolarized
and polarized cross sections for inclusive hadron production in pp√
collisions at s = 62.4 GeV, which is this thesis is based on, are also calculated at both
NLO and NLL accuracies [36]. They are compared with the results obtained at PHENIX
experiment and discussed in Sec. 5.1.

LO

1

NLO

αsL2

αsL

αs

NNLO

αs2L4

αs2L3

αs2L2

αs2L

+…

αs3L6

αs3L5

αs3L4

αs3L3

+…

αs4L8

αs4L7

αs4L6

αs4L5

+…

αskL2k

αskL2k-1 αskL2k-2 αskL2k-3 +…

LL

NLL

NkLO

+…

NNLL

Figure 2.5: Fixed order perturbative expansion (LO, NLO, NNLO, ...) and resummation(LL, NLL, NNLL, ...). L represents the large log term.

2.4

Factorization theorem

We introduced the framework to deal with particle production in pp collisions, and ep
deep inelastic scatterings. π 0 production in pp collisions is written as an convolutionof PDFs, cross section of parton parton interactions, and FFs (Eq. 2.1). Factorization

14

CHAPTER 2. PHYSICS

theorem [37] plays an important role in this framework as it ensures that the cross section
cof parton-parton interaction, dσ̂ab(xa P, xb P, µ2), only depend on the parton species a, b
and c, and not depend on the choice of the initial state hadron (proton in this case), nor
the choice of the final state hadron (π 0 in this case). All the non perturbative phenomena
are carried by PDFs and FFs and the theorem ensures that PDFs and FFs are universal.
The same PDFs can be used for both pp collisions and ep deep inelastic scatterings. The
FFs are the same for pp collisions and other processes such as ee collisions. In fact, mostof the information on FFs are derived from hadron production in ee collisions.
The proofs of the factorizations are very technical and difficult tasks and rigorous
proofs are only given for a few processes such as DIS and Drell-Yan. However, we assume
that it holds for inclusive hadron production in pp collisions, which is the process that
this thesis is based on. And the cross sections calculated in the formalism agree withexperimental data as will be seen later.

2.5

Sum rules for PDFs

There are useful sum rules for the 1st moment of unpolarized and polarized PDFs. They
are summarized in this section.

2.5.1

Sum rules for unpolarized PDFs

Since QCD Lagrangian conserves fermion number and flavor, the following sum rules must
be obeyed by the parton densities for valence quarks in the proton:
Z 1

dx[u(x, µ ) − ū(x, µ )] ≡Z 10

¯ µ2 )] ≡dx[d(x, µ2 ) − d(x,Z 10dx[s(x, µ2 ) − s̄(x, µ2 )] ≡Z 10dx[c(x, µ2 ) − c̄(x, µ2 )] ≡

0

2

2

Z 1

dx uv (x, µ2 ) = 2(2.23)

dx dv (x, µ2 ) = 1(2.24)

dx sv (x, µ2 ) = 0(2.25)

Z 1

dx cv (x, µ2 ) = 0,(2.26)

2

2

0

Z 1
0

Z 1
0

0

and overall momentum conservation gives
Z 1
0

2.5.2

dx x

"

X

2

#

(qi (x, µ ) + q¯i (x, µ ) + g(x, µ ) = 1,i(2.27)

Sum rule for polarized PDFs

Total angular momentum conservation gives the important spin sum rule for the proton:11= ∆Σ(µ2 ) + ∆G(µ2 ) + Lz (µ2 ),(2.28)
22

2.6. UNPOLARIZED PDFS

15

where
∆Σ(µ2 ) =2

∆G(µ ) =

Z 1

dx

Z 1

dx∆g(x, µ2 ),

0

0

X



∆qi (x, µ2 ) + ∆q̄i (x, µ2 ) ,i(2.29)(2.30)and Lz (µ2 ) is the orbital angular momentum of quarks and gluons. ∆Σ(µ2 ) does not
evolve with µ2 at LO while ∆G(µ2 ) depends logarithmically on µ2 and the evolution of∆G(µ2 ) is compensated by that of Lz (µ2 ). A toy model calculation of the evolutions of
the first moments is presented in Appendix C.1.

2.6

Unpolarized PDFs

Unpolarized PDFs are extensively studied by deep inelastic scattering(DIS). The strategy
for the determination of PDFs is the following. Assume a reasonable functional forms for
PDFs at a certain input scale µ2 , then calculate structure functions and search for bestparameters to describe the data.
There are many groups such as CTEQ [38], MRST [39], and GRV [40] working on PDFs
with slightly different assumptions and functional forms. Parametrization is often donefor valence and anti-quarks, rather than quarks and anti-quarks since valence distributions
are non-singlets and are independent of gluon distribution.
We introduce the MRST2002 PDF set since the polarized and unpolarized cross sectionof π 0 production which this thesis is based on, are calculated with it [36]. PDFs areparametrized at input scale of µ2 = 1 (GeV/c)2 . Each of the valence quarks and the total
sea quark contribution are parametrized in the form
xq(x) = A(1 − x)η (1 + εx0.5 + γx)xδ(2.31)

where A is determined by the sum rule for the number of valence quarks of each type
(Eq. 2.24-2.26). The gluon distribution is parametrized asxg(x) = Ag (1 − x)ηg (1 + εg x0.5 + γg x)xδ − A− (1 − x)η− x−δ−(2.32)

where Ag is determined by the momentum sum rule (Eq. 2.27). The combination ū(x) −¯ is parametrized asd(x)¯x(ū(x) − d(x))= A(1 − x)η (1 + γx + δx2 )xδ

(2.33)

The functional form xδ (1 − x)η is assumed according the behavior in the limits at x →0, 1, suggested by Regge theory and the constituent counting rules[41], respectively. The
major ingredients for the determination of unpolarized PDFs are the structure functions

16

CHAPTER 2. PHYSICS

F2 (x, Q2 ) for the proton and the neutron (deuteron) targets with wide range of x andQ2 . The structure function F2 (x, Q2 ) for proton targets is shown in Fig. 2.6. Although
Bjorken scaling predicts that the structure function is independent of Q2 , the scaling
violation is visible. The scaling violation comes from the running coupling constant andQ2 evolutions. F2 for the proton and the neutron can be used for flavor separation,and
its Q2 dependence for gluons. Neutrino induced DIS and W boson charge asymmetry at
Tevatron help flavor separations.
Figure 2.7 shows the MRST2002 results of the PDF at NLO at Q2 = 4 (GeV/c)2 . u
and d distributions have a peak around x ∼ 0.1 − 0.2, since they are the valence quark
flavors. (The valence quarks are u − ū and d − d¯ to be exact.) The rise in low x cancels forvalence distribution, which come from the coupling to gluon evolution. s(= s̄) and c(= c̄)quark distributions are suppressed compared to other flavors because of their large mass.

2.7

Polarized PDFs

Polarized PDFs are also investigated by many groups such as AAC [43, 44], GRSV [45],LSS [46], and so forth. Here we show AAC03 results of polarized PDFs which is theresults obtained before our experiment started. AAC PDFs are obtained separately forthe valence u-quarks ∆uv (x), the valence d-quarks ∆dv (x), the sea quarks ∆q̄(x) (withthe assumption of flavor symmetric sea), and the gluons ∆g(x). The functional form for
PDFs in the analysis is:∆f (x) = [δxν − κ(xν − xµ )]f (x),(2.34)

where δ, ν, κ, and µ are free parameters, and f (x) is the corresponding unpolarized
PDF. The input scale is Q0 = 1 GeV/c. Although it is known that the unpolarized sea
quark distributions are different for different flavor, there are not enough information for
flavor separation in polarized PDFs. Thus flavor symmetric sea polarization is assumed.
The positivity condition |∆σ| ≤ σ does not necessarily mean |∆f (x)| ≤ f (x) at NLO
and higher. However, this condition is assumed since it avoids unphysical cross section|∆σ| > σ due to the lack of experimental data.
Under the assumption of flavor SU(3) symmetry, the β decays of spin 1/2 octet baryons
can be described by two parameters F , and D [47]. And certain combinations of polarized
PDFs are related to these parameters as [48, 49, 3],
a3 ≡
=
a8 ≡
=

∆U − ∆D
F + D = 1.269 ± 0.003∆U + ∆D − 2∆S
3F − D = 0.586 ± 0.031,(2.35)(2.36)¯ andwhere R∆U, ∆D, and ∆S are ∆U ≡ 0 dx(∆u + ∆ū), ∆D ≡ 0 dx(∆d + ∆d),1
∆S ≡ 0 dx(∆s + ∆s̄). These relations are utilized to constraint the parameters of PDFs.
R1

R1

17

10 9

2

i

F2(x,Q ) * 2 x

2.7. POLARIZED PDFS

10

8

10

7

H1
ZEUS
BCDMS
E665
NMC
SLAC

10 6
10 5
10 4
10 3
10 2

10

1

10

10

10

-1

-2

-3

10

-1

1

10

10

2

10

3

10

4

10

5

2

10
2

Q (GeV )Figure 2.6: The structure function F2 from DIS experiments.[42]6CHAPTER 2. PHYSICS

xf(x)

18

1.0
0.9
MRST2002
Q2 = 4(GeV/c) 2

0.8
0.7
0.6

u
d
ubar
dbar
s (= sbar)c (= cbar)
gluon

0.5
0.4
0.3
0.2
0.1
0

-3

-2

10

-1

10

10

x

Figure 2.7: The MRST2002 PDF sets at Q2 = 4 (GeV/c)2 [39]. The combinations of acolor (blue, black, red, or green) and a line type (solid, dashed-dotted, dotted, or dashed)represent parton species. The three lines of each parton type shows the central value ofthe parton density and its uncertainties.

Eq. 2.35 is equivalent to the famous Bjorken sum rule [50]1 gA
g1p(x) − g1n (x)dx = | |
6 gV
0

Z 1

αs (Q2 )
1−
,
π

"

#

(2.37)

where g1p (x) and g1n (x) are the polarized structure functions for the proton and the neutron
respectively. gV and gA are the vector and axial-vector coupling constant respectively, andthey are related to F and D as F + D = |gA /gV |.
Major ingredients for polarized PDFs are the polarized structure functions g1 (x, Q2 )for proton and deuteron targets. Fig. 2.8 shows the measured structure functions.
Figure 2.9 displays the AAC03 polarized PDF set. PDFs from other groups are also
overlaid. ∆uv and ∆dv have relatively small uncertainty and fair agreement was reachedamong the PDF sets. But the gluon polarized PDF have large uncertainties, and needs

2.7. POLARIZED PDFS

19

x g1p

further experimental information. The 1st moment of AAC03 results are summarized inTable 2.3. The large uncertainty is also seen in the 1st moment of the gluon polarization.
The quark spin contribution is 12 ∆Σ ∼ 0.1 and the rest of the proton spin might becarried by the gluon spin ∆G, but the uncertainty is too large to conclude it. The aim of
the thesis is to provide new data towards the determination of the polarized PDF of thegluons.

0.08
EMC
E142
E143
SMC
HERMES
E154
E155
JLab E99-117
COMPASS
CLAS

0.06
0.04
0.02

x g1d

0
-0.02
0.04
0.02

x g1n

0
-0.02
0.02

0

-0.02
10

-4

10

-3

10

-2

10

-1

1

x

Figure 2.8: The structure function g1 from DIS experiments.[51]20

CHAPTER 2. PHYSICS

0.5

2

2

2

Q = 1 GeV

AAC03
GRSV

0.4

0.2

BB

1

0.3

LSS

xDu (x)v0

xDg(x)

0.1
0
0.001

0.01

0.1

-1
0.001

1

0

0.01

0.1

1

0.01
0

xDdv(x)
-0.1

-0.01

AAC03
GRSV

-0.02

BB

-0.03

LSS
-0.2
0.001

0.01

0.1

1

xDq(x)

-0.04
0.001

Q2 = 1 GeV2
0.01

0.1

1

x

x

Figure 2.9: The AAC03 polarized PDF sets at Q2 = 1 (GeV/c)2 [44]. The AAC03 results
are compared with the results of other groups: GRSV [45], BB [52], LSS [46, 53, 54]. The
green bands are the uncertainties for the polarized PDFs obtained by AAC03. The statistical and systematic uncertainties in the experimental results are added in quadratureand the theoretical uncertainties are not included.

2.8

Probing ∆g in polarized pp collisions

0
Figure√ 2.10 displays the relative contribution of gg, qg, and qq scatterings to π production
at s = 62.4 GeV. As in the figure, the dominant subprocesses are qg and gg scatterings
for the measured pT range (1–4 GeV/c). Thus the π 0 production in pp collision is sensitive
to the gluon distribution.
The polarized cross section ∆σ is directly connected to the polarized PDFs as defined in Eq. 2.2. Experimentally, instead of directly measured ∆σ, the double helicity
asymmetry ALL is obtained which is defined as∆σ,σ
since experimental efficiency and normalization of cross section cancels.
ALL =(2.38)

2.9. XT AND PROBED X RANGE

21

∆Σ
∆G
∆Q̄
0.213 ± 0.138 0.499 ± 1.266 −0.062 ± 0.023
Table 2.3: 1st moment of the polarized PDFs in AAC03.
The double helicity asymmetry for subprocess is defined in similar way asâLL =∆σ̂.
σ̂

(2.39)

The âLL for various subprocesses are displayed in Fig. 2.11. Since the π 0 is detected in
midrapidity in PHENIX, the measured range roughly corresponds to cos θ ∼ 0. âLL for
qg and gg is positive except for gg → q q̄. However, the contribution of gg → q q̄ is smaller
than gg → gg by about three orders of magnitude as explained in Appendix D and overall
subprocess asymmetry is positive.

2.9

xT and probed x range√√
xT is defined as xT = 2pT / s, where s is the center of mass energy of pp collisions,and pT is the transverse momentum of π 0 . xT is interpreted as the fractional momentum(x) of initial parton in the proton when the two partons with the same x collide, thescattering angle is 90 degrees, and the fragmentation is neglected. Figure 2.12 illustrates
a scattering under the conditions mentioned above. The fragmentation process and precise
consideration of kinematics “smear” the rough argument above. x and xT are no longer
equal, and rather broad range of x contributes for a certain pT (or xT ) of π 0 as in Fig. 2.13
for example. However, the average x roughly scales with xT and x ∼ 1.7xT .
√Figure 2.14 displays the cross section for three different center of mass energy s
versus xT calculated √at NLO accuracy. The cross section √is two orders of magnitudelarger at fixed xT at s = 62.4 GeV compared to that at s = 200 GeV. Thus lower
center of mass energy can reach higher xT with the√ same integrated luminosity. We have0
measured and presented the results of π A√LL at s = 200 GeV based on the data taken
in Run 2005 [15]. The measurements at s = 62.4 GeV, which this thesis is based on,
are able to improve the accuracy
at high xT with smaller integrated luminosity compared√to the measurements at s = 200 GeV.

CHAPTER 2. PHYSICS

a^LL

22

1

C

A

0.75
1

B

0.5

0.9
0.8

D

0.25

0.7

qq+qq

0.6

0

0.5
0.4

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

-0.25

qg

0.3

-0.5

0.2

gg

0.1
0
0

-0.75
2

4

6

8

10

12
14
pT (GeV/c)

E
-1
-0.8

-0.4

0

0.4

0.8

cosθ
Figure 2.10: Relative contribution of√ processes to the production of π 0 at s =62.4 GeV.

Figure 2.11: The double helicity asymmetries for subprocesses âLL . θ is the scattering angle at the center of mass system.

2.9. XT AND PROBED X RANGE

23

p

T

partonx s/2

Proton 1

s/2

partonx s/2

Proton 2

s/2

p

T

√
Figure 2.12: A collision with two partons have equal momentum x s/2, when the scattering angle is 90 degrees. Fragmentation process is neglected. The pT of scattered particle√
is pT = x s/2 and xT = x in this case.

a.u.

pT =1.0-2.0 GeV/c
Average: -9.977E-01
RMS : 3.448E-01

log10(x)0Figure√ 2.13: The x range contributes to π production for pT = 1−2 GeV/c in pp collisions
at s = 62.4 GeV, calculated in NLO pQCD [55].

24

CHAPTER 2. PHYSICS

50
0

Ge

20
0

V

Ge
V

eV
G

[pb/GeV]

1010

.4
62

d
dpT d

10 8
10 6
10 4
10 2
10 0

measured range
62.4 GeV
200 GeV
0.01

0.02

0.05

0.1

0.2

xT 0.5Figure 2.14: The cross sectionof π 0 production in pp collisions versus xT for three different√
center of mass energies s √= 62.4 GeV, 200 GeV, and 500 GeV. The red arrow indicatesthe measured xT range√ at s = 62.4 GeV in this thesis. The blue arrow describes themeasured range at s = 200 GeV in Run 2005.

Chapter 3
Experimental Setup
3.1

Overview

This thesis is based on the data which were taken at the Relativistic Heavy Ion Collider(RHIC) with the PHENIX detector at Brookhaven National Laboratory (BNL) in the
UnitedStates during the Run 2006. During Run 2006, a polarized pp experiment at√
s = 200 GeV√was performed for about 13 weeks (Mar.5 – Jun.5). And a polarized pp
experiment at s = 62.4 GeV followed and was for about two weeks (Jun.6 – Jun.20).
√
The data taken√at s = 62.4 GeV was used in this thesis. The first one third of
the experiment at s = 62.4 GeV was with transversely polarized proton beams. It was
followed by the experiment with longitudinally polarized proton beams. The integrated
luminosity used in this thesis is 40 pb−1 for longitudinal runs with average polarizationof 48%.
In this chapter, RHIC is briefly introduced in Sec. 3.2. Descriptions of the experimental
setup of the PHENIX detector follows in Sec. 3.3.

3.2

Relativistic Heavy Ion Collider (RHIC)

Relativistic Heavy Ion Collider (RHIC) provides high energy heavy ion collisions andpolarized pp collisions. One of the major goals of the heavy ion experiment is to investigate
a new state of matter which is referred to as Quark Gluon Plasma (QGP). RHIC canaccelerate ions as heavy as Au up to an energy of 100 GeV per nucleon, which results in√heavy ion collisions at sN N = 200 GeV. RHIC can also accelerate and collide polarized
proton beams 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 deep inelastic scatterings (DIS) where the interactions are
mediated by virtual photons. Gluons interact at leading order in pp collisions while gluons
only participate at higher order in deep inelastic scatterings (DIS). Therefore, pp collisions25

26

CHAPTER 3. EXPERIMENTAL SETUP

Figure 3.1: RHIC accelerator complex.

are a good probe for the gluon spin contribution to the proton. The production of W
in polarized pp collisions provide information on the flavor separation of the quark spin
contribution. RHIC canof 250 GeV which√ accelerate polarized protons up to an energy 32
results in collisions at s = 500 GeV with design luminosity of 2 × 10 cm−2 s−1 .
Figure 3.1 shows an aerial view of RHIC accelerator complex and Fig. 3.2 displays
its schematic. The polarized proton beam is produced at optically-pumped polarized ion
source (OPPIS) [56] 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
pulse is accelerated by Linear Accelerator (LINAC) to a kinetic energy of 200 MeV.
It is injected into Booster, and is accelerated up to 1.5 GeV. Then it is transferred toAlternating Gradient Synchrotron (AGS) and accelerated up to 24.3 GeV. It is injected
into two independent rings at RHIC, via AGS-to-RHIC transfer line. Each beam travels
in opposite direction and collides at the interaction points (IPs). Two independent beams
are called the Blue (clockwise) and the Yellow (anti-clockwise) beams. RHIC has six
IPs and they are referred to as IP12, IP2, IP4, IP6, IP8, and IP10 as in the case of aclock. Once RHIC was filled with beams, the beams are kept circulating in the rings toprovide collisions at the IPs. When the luminosity becomes too low, beams are dumped
and refilled. The sequence from injection to dump of the beam is called a fill. One filltypically 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. Each bunch is filled with predetermined3.2. RELATIVISTIC HEAVY ION COLLIDER (RHIC)

H-jet
polarimeter

Blue

Beam

w
Yello

Beam

27

RHIC
pC polarimeters

IP12

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

Booster

Polarimeter
200MeV
Polarimeter

AGS

RF dipole
Cold snake

Warm snake

AGS pC Polarimeter

Figure 3.2: RHIC accelerator complex.

28

CHAPTER 3. EXPERIMENTAL SETUP

a)

b)



4

3 2

4 3

1



1

Yellow beam

2

 

4

1





3

1
Blue beam
1

c)

2

2

Yellow beam
3

2

3

4
Blue beam

4

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

Figure 3.3: A spin pattern of RHIC polarized proton beams. An arrow represents the
direction of a beam. A box corresponds to a bunch and the sign (+ and −) in the box
denotes the predetermined spin state of the proton beams in the bunch. The colors ofarrows and boxes represent the Blue and Yellow beams. a) Blue bunch 1 collides at Yellow
bunch 1 and provide helicity same collisions. b) One beam clock after a). Blue bunch 2
and Yellow bunch 2 collide and helicity opposite collisions occur. c) The resulting spincombination of the collisions. The spin pattern provides all possible spin combination ofcollisions.

3.2. RELATIVISTIC HEAVY ION COLLIDER (RHIC)

29

polarization sign. Figure 3.3 shows an example of a spin pattern assignment. The Blue
beam has a spin pattern “+ + −−” while the Yellow beam has a spin pattern “+ − +−”.
In Fig. 3.3a the Blue bunch 1 and the Yellow bunch 1 collide and collisions with thesame helicity are obtained. One clock after Fig. 3.3a, the Blue bunch 2 and the Yellow
bunch 2 collide and collisions with the opposite helicity are obtained as in Fig. 3.3b. In
this way, we obtain all possible spin combinations at the same time as in Fig. 3.3c. Thisfeature greatly reduced systematic uncertainty which comes from time dependence of thedetector responses. In Run 2006, 111 bunches out of 120 bunches are filled in each ring.
(A bunch out of 111 for each beam is used for tune measurements and is not used forphysics measurement.) The 1st bunch of the Blue beam collides at the 81st bunch of the
Yellow beam at PHENIX IP, which results in 18 non-colliding bunches. (Blue (Yellow)
unfilled bunches collides Yellow (Blue) filled bunches). The non-colliding bunches can beutilized to measure single beam background. And 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.

3.2.1

Polarized proton source

Figure 3.4: A schematic drawing of the RHIC OPPIS.

The polarized proton beam is produced at optically-pumped polarized ion source (OPPIS) [56]. The OPPIS technique for polarized H ion beam production was developed

30

CHAPTER 3. EXPERIMENTAL SETUP
Laser(796nm)+Rb

Rb
Protonsource

H+

H0

Sona
transition

H0

Na ionizer
cell

H-

Figure 3.5: The OPPIS scheme. The labels inside circles denote the names of the atoms,
which include the number of electrons, whether electrons are explicitly drawn or not. For
example, the circle with the label H0 with an electron around it does not mean H nuclei
has two electrons.

in the early 80’s at KEK, INR Moscow, LAMPF and TRIUMF. Figure 3.4 displays aschematic drawing of the OPPIS, and its polarizing scheme is illustrated in Fig. 3.5. The
source of angular momentum is high-power lasers. Rb atoms are optically pumped bytitanium-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 transfered from Rb to H
atoms, and H+ atoms becomes electron-polarized H0 atoms. To prevent depolarizationin the charge-exchange collisions, the optically pumped cell is situated inside the strong(2.5 Tesla) superconducting solenoid. Then the polarization is transfered from electron
to the H nucleus by the Sona transition [57]. Finally, electrons are attached by the Na-jet
ionizer cell and H− ions are produced.

3.2.2

Polarization of proton beams

The proton looses its polarization during the acceleration unless special actions are taken.
The proton precesses when it feels magnetic fields. The spin precession of the proton isgoverned by the Thomas-BMT equation[58, 59] in the laboratory frame as:dS
e=−[(1 + Gγ)B⊥ + (1 + G)Bk ] × S,
dt
γm

(3.1)

where S is the spin vector of the proton, B⊥ and Bk are the transverse and the longitudinalcomponents of the external magnetic fields respectively, and G = (g − 2)/2 = 1.7928473 is
the anomalous magnetic moment of the proton, and γ = E/m is the Lorentz factor. Sincea planar circular accelerator only has the vertical guide fields (B⊥ 6= 0, and Bk = 0), theproton spin vector precesses with the angular frequency of Gγωc , where ωc = eB⊥ /γm isthe cyclotron frequency. (A turn which corresponds to the proton’s motion was subtracted

3.2. RELATIVISTIC HEAVY ION COLLIDER (RHIC)

31

from the term 1+Gγ.) Thus the precession is Gγ times faster than the orbital motion, inthe existence of the magnetic field. The number of spin precession per one evolution isreferred to as the spin tune, νs . In an ideal planar circular accelerator, νs = Gγ.
The polarization of the proton beams are maximal at the source, and there is no repolarizing mechanism. Instead, there are many depolarizing resonances at certain beamenergies. Depolarizing resonance condition is satisfied when the spin precession frequency(or the spin tune) equals the frequency of an encounter between spin-perturbing magnetic
fields and the beam. Two main types of depolarizing resonances are: the imperfection
resonances, and the intrinsic resonances. The imperfection resonances arise from themagnetic field error and misalignment, and occur whenνs (= Gγ) = n,(3.2)

where n is an integer. Thus the imperfection resonances exist at every step of ∆γ = 1/G,
thus ∆E = mc2 /G = 523 MeV in acceleration. The intrinsic resonances arise fromhorizontal components of the focusing fields, and occur whenνs (= Gγ) = kP ± νy ,(3.3)

where k is an integer, P is the super-periodicity, which is defined as the number of identical
periods of the accelerator components, and νy is the vertical betatron tune, which is thenumber of the betatron oscillations per revolution. (At the AGS, P = 12 and νy ∼ 8.8.)
The stable spin direction is the precession axis. Thus in the absence of a spin resonance,
the stable spin direction is the same as the magnetic field of the accelerator. Close to aspin resonance, the stable spin direction is perturbed away from the vertical direction by
the resonance driving fields.
When a polarized beam is accelerated through an isolated resonance, the final polarization can be described by the Froissart-Stora formula[60]:π|ε|2Pf= 2e− 2α − 1,
Pi

(3.4)

where Pi and Pf are the polarizations before and after crossing the resonance respectively,and α = dGγ/dθ is the acceleration rate per radian of the orbit angle. Thus for avoiding
depolarization during a resonance crossing, |ε|2 ≪ 2α/π (results in Pf /Pi = 1) or |ε|2 ≫2α/π (results in Pf /Pi = −1) are required. When the beam is slowly accelerated or
the resonance is strong enough (in the latter case) the spin vector adiabatically follows
the stable spin direction, resulting in a complete spin flip without polarization losses.
Traditionally the intrinsic resonances are overcome by using a betatron tune jump, which
effectively make the resonance stronger, and the imperfection resonances are overcome
with the harmonic corrections of the vertical orbit to reduce the resonance strength ε butthese methods becomes difficult at high energy.

32

CHAPTER 3. EXPERIMENTAL SETUP

The invention of the Siberian Snake [10, 11], which generates a 180 degrees spinrotation about a horizontal axis, gave a solution to the problem. Utilizing two Siberiansnakes, the stable spin direction remains unperturbed at all times as long as the spinrotation from the Siberian Snake is much larger than the spin rotation due to the resonancedriving fields. The Imperfection and intrinsic resonances are both overcome by Siberian
Snakes. Such a spin rotator is constructed by helical dipole magnets. The spin motion
with snakes are explained in detail later in this section. At lower energy synchrotrons
such as AGS with weaker depolarizing resonances, a partial snake, which rotates the spinless than 180 degrees, are sufficient to preserve the spin direction unperturbed at theimperfection resonances [61]. A 5.9 % 1 warm snake (with normal conducting magnets),
together with 10% cold snake (with superconducting magnets) are utilized to overcome
the depolarization resonances at AGS.
Definition of axes in the particle rest framezx
Direction of
the beamyFigure 3.6: The definition of axes in the particle rest frame.

The definition of axis in the particle rest frame used in the next two subsections, isexplained in Fig. 3.6. The x and y axes is in the accelerator plane. The y axis is along
the direction of the beam, and the x axis is perpendicular to the y axis and faces inwards.
The z axis is in the transverse direction.
Spin motion with single snake
Figure 3.7 illustrates the spin motion with a single Siberian snake configuration [62]. For
simplicity, the accelerator is drawn as a combination of two bending arcs and two straight1

The amount of the rotation induced by snakes is expressed as the fraction of 180 degrees. 5.9% snake
rotates the spin by 10 degrees (180 × 0.059).

3.2. RELATIVISTIC HEAVY ION COLLIDER (RHIC)

33

b)

a)

z

ke
snape-1
ty

z

y

z

C

Gγ /2
mod 2π

x

y

B

1

x

A

y

2

y

y

C

x

Gγ /2
mod 2π

z

x

1
y

A

2

Gγ /2 mod 2π

c)

z

ke
snape-1
ty

z
B
Gγ /2
mod 2π

B

x

z

x

z

ke
snape-1
ty

y
C

x

y

z

x
x
y

A

1,2

Figure 3.7: The spin motion with a single snake configuration. This is a slightly modified
version of the drawing in [62]. The red arrows show the spin of the beam. The purple
dotted arrows show the spin of the beam after one revolution. The green arrows represent
the rotation axis of the Siberian snake.

34

CHAPTER 3. EXPERIMENTAL SETUP

sections. The spin precesses Gγ/2 times around z axis in each bending arc (Gγ times per
one turn). A Siberian snake which rotates the spin around y axis is installed in one of
the straight sections. This type of snake is called “Siberian snake type-1”. A local axis in
the particle rest frame is drawn for each point (A, B, and C). The red arrows represent
the spin of the beam, the purple dotted arrow represent the spin direction after one
revolution. The green arrow shows the rotation axis of the Siberian snake. Figure 3.7a)shows the z component of spin motion at point A. From A to B, the spin stays in the samez direction. The spin is rotated by the snake from z to -z direction between B and C.
Then the spin stays the same direction −z from C to A and back to the original position
with the opposite spin. Therefore, the spin flips every turn. Figure 3.7b) illustrates the xcomponent of spin motion at point A. From A to B, the spin precesses Gγ/2 times aroundz axis. The snake rotates the spin around y axis by π and the spin precesses Gγ/2 times
around z axis between C and A. Therefore, the x component of spin at A flips every turn.
Figure 3.7c) shows the y component of the spin at point A. In contrast to the case of z
and x components, the component y remains the same direction. In total, the spin tuneis ν = 12 , which is independent of energy Gγ. (The spin tune is Gγ in the absence of asnake.) Therefore, spin perturbing kick cancels every turn and does not accumulate. (Forexample, suppose the spin is in the y direction at point A as in Fig. 3.7c) and suffers froma spin perturbing kick around x axis at point A. The perturbing kick rotates the spin
and the spin becomes (0,cos χ,sin χ). Then the x and z component flips after one turn
due to the snake, and the spin becomes (0,cos χ,-sin χ) = (0,cos(−χ),sin(−χ)). Again theperturbing kick rotates the spin in the same direction, the spin goes back to the original
direction y.) This feature is very powerful in dealing with imperfection resonances. It
also works for intrinsic resonances unless the betatron tune is close to a half integer.
The stable spin direction is y direction at point A. At other points however, the stablespin direction depends on position and energy (Gγ). For example, if we consider a point
half way between A and B, the angle between the stable spin direction and the y axis isGγ/4 mod 2π. RHIC has 6 interaction points. Therefore, the dependence of stable spin
direction on energy and the position cause difficulty in manipulating the spin direction ateach interaction point.
Spin motion with two snakes
Figure 3.8 illustrates the spin motion with two Siberian snakes installed. As in the case of
Fig. 3.7 (single snake configuration), the accelerator consists of two bending arcs and two
straight sections. One of the straight sections is equipped with snake type-1. The otherstraight section is equipped with snake type-2 which rotates the spin by π around x axis.
Figure 3.8a), b), c) show the z, x, and y components of the spin at point A. In contrast to
the case of the single snake configuration, The z component stays the same and the othertwo components flips every turn. The spin tune is ν = 21 and the stable spin direction is
the z direction at all times, independent of energy, and the position. (The stable direction

3.2. RELATIVISTIC HEAVY ION COLLIDER (RHIC)

35

b)

a)

z

ke
snape-1
ty

z

C

x

Gγ /2
mod 2πzkesnape-2z tykesnape-1
ty

z

x

y

B

y

z

x
y

A

x

ke
snape-2
z ty

1,2

Gγ /2
mod 2π

y

A*

c)

x

A

2

A*

x
z

ke
snape-2
z ty

x
y

y

y

ke
snape-1 C
ty
y

B

1

Gγ /2 mod 2π

z
z

z

x

x
y

x

Gγ /2
mod 2π

y

B

y

C

x
y

A

1

2

A*

Figure 3.8: The spin motion with two snakes configuration. The red arrows show the spinof the beam. The purple dotted arrows show the spin of the beam after one revolution.
The green arrows represent the rotation axis of the Siberian snake.

36

CHAPTER 3. EXPERIMENTAL SETUP

is opposite each other between the two arcs.) Spin perturbing kicks cancel every turn.
The snakes overcome both imperfection and intrinsic resonances unless the intrinsic
resonance conditions are half integer. Such configuration with two snakes are utilized in
RHIC to preserve the polarization of the proton.
Interaction points
The spin is rotated in front of the IPs by spin rotators, so that we get longitudinallypolarized collisions at the IPs. The spin direction of the outgoing beams from the IPs
are rotated again to the transverse direction. Fig. 3.9 shows a schematic drawing of the
RHIC beams around the PHENIX interaction point.
Two large experiments, PHENIX and STAR experiments are located at IP8 and IP6
respectively. There are also smaller experiments such as PHOBOS (IP10), BRAHMS and
PP2PP (share IP2). In Run 2006, PHENIX, STAR and BRAHMS were operated and theother experiments were shut down before that. In 2002, a large forward neutron asymmetry was discovered at IP12 test experiment and is utilized as PHENIX local polarimeter.
Since 2004, Gas-jet absolute polarimeter experiment is being operated at IP12 which isutilized to determine the beam polarization normalization.
transverse
spin rYELLOW beam(outgoing BLUE beam)

ota
t

or

longitudinal

PHENIX

s

a
rot
pin

tor

transverse

BLUE beam(outgoing YELLOW beam)interaction point

Figure 3.9: A schematic drawing of the RHIC beam near the PHENIX interaction point,
when the longitudinally polarized collisions are required.

3.2.3

Polarimeters

Three polarimeters are used to measure and monitor the beam polarization. Two in RHIC
and one at PHENIX experiment. Two types of polarimeters utilized in RHIC are fast
carbon ribbon polarimeter (pC polarimeter) [63], and hydrogen gas jet target polarimeter(H-jet polarimeter) [64]. At PHENIX experiment, the orientation of the beam polarization
is monitored by PHENIX local polarimeter [65]. These three types of polarimeters utilize
a sizable single transverse spin asymmetry AN . AN is defined for a reaction between

3.2. RELATIVISTIC HEAVY ION COLLIDER (RHIC)

37

transversely polarized beam and unpolarized beam (or target). It is defined as
AN =σlef t − σrightσlef t + σright(3.5)

where σlef t(right) is the cross-section that the outgoing particle goes left(right) side when
the polarization is upward in view of 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, beam polarization can be calculated as P = εN /AN . The three types ofpolarimeters are introduced in this section.
Fast carbon ribbon polarimeter (pC polarimeter)
The fast carbon ribbon polarimeter (pC polarimeter) utilizes AN in the elastic scattering
between polarized proton beams and carbon target (ApC
N ) at very forward region, with2
four-momentum transfer of −t = (0.01 − 0.02) (GeV/c) . The size of ApC
N in the measured
kinematic region is about 1.4%. Due to the small scattering angle of protons, recoilcarbons are detected instead of the scattered protons. The target should be thin forrecoil carbon with small energy of 0.1 − 1 MeV to escape the target, and not to influenceon 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 − 5 µg/cm2 witha width of 10 µm is utilized.

Si detectors

Recoil Carbon

cm
15

Beam

Beam
Carbon ribbon
Beam’s-eye view of detectors

Side View

Figure 3.10: The experimental setup of the pC polarimeter. Left: beam view of thedetectors. The beam runs into the paper and hit the carbon ribbon target in the centerof the beam pipe. Right: side view of the detectors. The beam runs from left to right.
Recoil carbon is detected with the Si detectors.

38

CHAPTER 3. EXPERIMENTAL SETUP

Figure 3.10 displays the experimental setup of the pC polarimeter. The target ribbon
is inserted into the beam and taken out after the measurement. Slow recoil carbons aredetected by the silicon detectors placed on both sides of the target. pC polarimeter collects∼ 4 × 106 events per one measurements which is typically one minute. It corresponds toa statistical uncertainty of 4% which is smaller compared to the systematic uncertaintyof 7.2% for Blue and 9.3% for Yellow beams. pC polarimeter confirms that the bunchby bunch polarization variation is within the uncertainty of the measurements. ApC
N was
not known in this energy and cannot be measured with the pC polarimeter system. In
this respect, pC provides only relative variation of polarization for each fill. H-jet target
polarimeter was utilized to obtain the absolute normalization of the polarization. The
H-jet polarimeter results were used to normalize pC polarimeter results. The major sourceof the systematic uncertainty assigned for the beam polarizations, is the uncertainty of
the absolute scale obtained with H-jet polarimeter measurement.
Polarized hydrogen gas jet polarimeter (H-jet polarimeter)
The polarized hydrogen gas jet polarimeter (H-jet polarimeter) utilizes AN in pp elasticpp
scattering (AppN ). Since both beam and target are polarized, AN can be calculated for
either beam or target polarization, by averaging target or beam polarization. The relation
between measured asymmetries (εbeam , εtarget ) and physics asymmetry (AppN ) isApp
N = εbeam /Pbeam = εtarget /Ptarget ,(3.6)

where Pbeam (Ptarget ) denotes the polarization of beam (target). Ptarget is measured byBreit-Rabi polarimeter. Thus App
N and Pbeam can be obtained from measured asymmetries.
One of the beautiful aspects in this measurement is that the physics asymmetry andthe beam polarization are obtained with the same experimental setup, which reduces
systematic uncertainty.
Figure 3.11 illustrates the pp elastic scattering process. The measured kinematicalrange is −t = (0.001 − 0.02) (GeV/c)2 where the asymmetry is large. Figure 3.12 showsthe experimental setup of the H-jet polarimeter. The hydrogen gas jet target crosses the
RHIC beam from top to bottom at a speed of 1.6 ×103 m/sec. The density of the gas jettarget is ∼ 102 H atoms/sec. The target spin direction is vertical, and is reversed every
10 minutes. The recoil particle is detected with the silicon detectors which are placed on
both sides of the targets. App
N is about 4% in the measured kinematical range and the
H-jet measurement provided the statistical uncertainty of 6% for a single beam for thewhole run. This is not enough for measurement of polarization variation for each fill. pC
polarimeter is used for this purpose instead, which provides polarization with statisticaluncertainty of the same level within one minute of measurement. The absolute beampolarization obtained with H-jet target is used to calibrate the pC measurement.

3.3. PHENIX OVERVIEW

39

Si detector
left

Jet target
Forward particle(Not observed)

beam

Recoil particle(detected)

Figure 3.11: The schematics of the pp elastic scattering process. The recoil proton isobserved while the forward proton is not.

Recoilparticlebeam

Si detectorright

Jet target

Figure 3.12: The experimental setup of the
H-jet polarimeter. The beam runs throughthe jet target. The recoil particle is detected with the silicon detectors.

PHENIX Local Polarimeter
For the study of the gluon polarization, longitudinally polarized proton collisions are necessary while the stable polarization direction of the beam in RHIC ring is transverse as
explained in Sec. 3.2.2. To obtain longitudinally polarized proton collisions, the polarization direction is rotated from transverse direction to longitudinal direction just before theinteraction point. Thus the polarization orientation should be monitored at 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. This feature is utilized to setup
spin rotator magnet currents during the commissioning period. It is also used to monitor
the polarization direction during the longitudinal run period.
Neutrons are detected with PHENIX Zero-Degree Calorimeters (ZDCs) and Shower
Max Detectors (SMDs), which are described in detail in Sec. 3.4.2

3.3

PHENIX overview

PHENIX [66] 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 3.13 is the definition of the coordinate system used in this thesis. z axis isalong with the beam pipe, pointing to north, and the collision point is at the origin of
the coordinate. y axis is in the vertical direction, pointing up. x axis is pointing west.

40

CHAPTER 3. EXPERIMENTAL SETUP

The azimuthal angle is referred to as φ and the zenith angle is referred to as θ. The
pseudorapidity η is defined as"θη = − ln tan
2

!#

.

(3.7)

y(Top)φθPHENx(West)z(North)

inte
ract
ion
poin
t

IX

Figure 3.13: The PHENIX coordinate system.
The PHENIX experiment is composed of many sets of detectors. A collection of
detectors with the same type is called a subsystem. The subsystems can be divided into
three groups: two central arms, two muon arms, and global detectors. Two central arms,
east and west arms, cover the pseudorapidity range of |η| < 0.35 and half in azimuthal
angle. 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 witha full azimuthal coverage. They are designed to detect muons. The global detectors
consist of several subsystems and measure the collision information. PHENIX has threemagnets: central magnet and two muon magnets. These magnets provide magnetic fields
for momentum measurement of charged particles.
Figure 3.14 displays a schematic of the PHENIX detector setup. The upper panel isa beam view of the central arm detectors. The proton beams run perpendicular to thepaper, 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 Čerenkov detectors (RICH), Aerogel Cherenkov detectors, Time
Expansion Chamber (TEC), and Electromagnetic Calorimeters (EMCal). EMCal, which3.3. PHENIX OVERVIEW

41

is represented by the green boxes, are used in this thesis and is described in Sec. 3.5.1. The
lower panel of Fig. 3.14 is a side view of the setup. The proton beams run as indicated by
the red arrows and collide at the interaction point (IP) indicated by a star. The central
arm detectors which are not drawn in the schematic, cover the IP under and over thepaper. 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 thecentral magnet surrounds the beam pipe for |z| > 41 cm and it limits the acceptance ofthe central arms. The green boxes represent the global detectors: Beam Beam Counters(BBCs), Zero Degree Calorimeters (ZDCs). Shower Max Detector (SMDs) are placed
inside ZDCs and are not displayed in the schematics. These are used in this thesis and is
described in Sec. 3.4.1 and Sec. 3.4.2.
PHENIX subsystems and their acceptances and purposes are summarized in Table 3.1.
Subsystem
Beam-Beam Countersη
±(3.1 – 3.9)

∆φ
2π

ZDC

Zero-Degree
Calorimeter± 2 mrad

2π

DC
PC
TEC

Drift Chambers
Pad Chambers
Time Expansion
Chamber
Ring Imaging
Čerenkov Counter
Time Of Flight
Lead-Scintillator cal.
Lead-Glass cal.
Muon Tracker
Muon Identifier± 0.35± 0.35± 0.35

π
×2
2
π
×2
2
π
2

± 0.35π×2
2± 0.35± 0.35± 0.35±(1.2 – 2.4)±(1.2 – 2.4)

π
4
π
+ π4
2
π
4

BBC

RICH
TOF
PbSc
PbGl
MuTr
MuID

2π
2π

PurposePrimary vertex detection.
Luminosity. Time-zero.
Provides level-1 trigger.
primary vertex detection.
Luminosity.
Provides level-1 trigger
Charged particle detection.
Pattern recognition, tracking.
Pattern recognition, tracking.
dE/dx.
Electron identification.
Provides level-1 trigger†
Charged hadron identification.
Photon and electron detection.
Provides level-1 trigger
Muon/hadron separation.
Provides level-1 trigger†

Table 3.1: PHENIX subsystems and their acceptance and purpose. †The trigger was not
used in the analysis.

42

CHAPTER 3. EXPERIMENTAL SETUP

PHENIX Detector
PC3
PC2

PbSc

EMCal

CentralMagnet

PC3
TEC

PbSc

EMCal
PbSc

PbSc
DC

Beam

DC

RICH

RICH

PbSc

PbGl

BB

MPC HBD
PC1

PbSc

PC1

PbGl

aerogelx

TOF

Beam View

West

East
t

ne

So

uth

y

ag

Central Magnet
Retern Yoke

M

uo

nM

ag

nM

uo

rt

No

ne

t

hM

MPC
BB
ZDC South

Proton Beam ZDC North

Proton Beam

MuID

MuID

IP HBD
MuTr

Central Magnet
Pole Piece

y
z

South

Side View

North

Figure 3.14: The PHENIX detector. The upper panel shows a beam view of the PHENIX
central arm detectors. The eight outermost boxes represent EMCal. The lower panel
shows a side view of the PHENIX global and muon arm detectors. BBCs and ZDCs are
represented by the green boxes.

3.3. PHENIX OVERVIEW

43

EM
C

H

RIC

CentralMagnetbeam
Figure 3.15: A picture of the PHENIX central arm detectors. The proton beam run as
indicated by the black arrow. The IP is covered by the Central magnet and is not visible.
The outer most blue structure is a support for EMCal.

44

CHAPTER 3. EXPERIMENTAL SETUP

3.4

PHENIX global detectors

3.4.1

Beam Beam Counter (BBC)Beam Beam Counters (BBCs) [67] are used to determine collision time and the collisionvertex position, and to provide BBC trigger. They are composed of two identical components and are placed along the beam pipe, symmetrically to the interaction point (IP).
Each component is located 144 cm away from the IP. They cover forward rapidity of
3.0 < |η| < 3.9. Since +z direction corresponds to north and −z direction to south, thedifferent components are referred to as BBCN and BBCS. Figure 3.16 displays a pictureof a BBC. The outer diameter of the BBC is 30 cm and the inner diameter is 10 cm withclearance of 1 cm between the BBC and the beam pipe. Each BBC consists of 64 hexagonal quartz Čerenkov radiators with a refractive index of ∼1.5, each of which is attached
to a one-inch Hamamatsu R6178 photomultiplier tube. Figure 3.17 shows a picture of apair of a 3 cm quartz radiator and a phototube. They are sensitive to charged particles
with β greater than 0.7.
The BBC readout electronics chain consists of discriminators, shaping amplifiers, timeto-voltage converters (TVC) and flash ADCs (FADC). The timing and pulse height information is digitized real time and is stored in Digital Memory Units (DMUs). The BBC hit
information is sent to Beam-Beam Local-Level-1 (BBLL1) board to provide BBC trigger.
The collision vertex position is determined by using the time difference of the hits in
two counters. Let TN and TS be the measured hit timing in BBCN and BBCS, respectively.
The time zero (T0 ) and the collision vertex position can be calculated as
TN + TS L−
2cc(TS − TN )
=
,
2

T0 =
zvertex

(3.8)
(3.9)

where L is the half of the length between the two BBCs (144 cm), and c is the speed oflight. Thus when the collision vertex position is outside the BBCs (|z| > 144 cm), thevertex position is reconstructed as z = ±144 cm, where ± correspond to on which side(z > 144 cm or z < −144 cm) the collision vertex is.
Vertex position cut is implemented in BBC trigger using position information obtained
online. The position resolution is estimated to be ∼ 5 cm online. The vertex position
cut of |z| < 30 cm was applied and it roughly matches to the central arms’ (midrapidity
detector) acceptance. Two types of BBC triggers with and without vertex position cut
were used during the run. In this thesis, the BBC trigger with 30 cm vertex cut is simply
referred to as the “BBC trigger”, while we explicitly write the BBC trigger without vertex
cut as the “BBC no-vertex-cut trigger.”
Offline slewing correction improves the resolution and the position resolution of ∼2 cm
was achieved offline for pp collisions.

3.4. PHENIX GLOBAL DETECTORS

45

PMT
quartzFigure 3.16: A picture of a BBC. It consistsof 64 pairs of of a quartz and a phototube.

CentralMagnetFigure 3.17: A pair of a quartz and a phototube used in the BBCs.

BBC

ipe
beam pFigure 3.18: BBC installed.

46

CHAPTER 3. EXPERIMENTAL SETUP

3.4.2

Zero Degree Calorimeter (ZDC) and shower max detector(SMD)

Distance from the beam axis [cm]Zero-Degree Calorimeters (ZDCs) [68], together with Shower Max Detectors (SMDs) were
equipped to detect neutrons at very forward angle of < 2.8 mrad. They are placed at 18 m
away from the IP. Figure 3.19 shows the location of the ZDCs. As in the case of BBCs,
the different components are called ZDCN and ZDCS. They are placed behind the DX
magnets thus most of the charged particles are swept away and neutral particles with long
life, which are mainly neutrons and photons, hit the ZDCs. Neutrons can be identified
as explained later. 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 3.20 is a picture of a ZDC (south). It is placed between two beam pipes as
explained.
DX
Dipole Magnet

50

DX
Dipole Magnet

A

Beam
0

Beam
Neutrons

ZDC
South

Interaction
Point

-50

A

-20
-15
-10
-5
0
5
10
Distance from the interaction point along the beam axis [m]

15

ZDC
North
20

cross-section (A-A)
Beam pipe

ZDC

Beam pipe

100

Neutrons

Beam

-10

Distance from the beam axis [cm]Figure 3.19: The ZDCs’ location and the expected trajectories of the beams and theneutral particles such as neutrons produced in the collisions.

A ZDC consists of three ZDC modules each of which has 1.7 interaction length or3.4. PHENIX GLOBAL DETECTORS

47

PMT

pip
e
be
am

ZD C

Forward
Scintilator

e

am

pip

be

Figure 3.20: A picture of ZDC. ZDC is placed between two beam pipes. In front of ZDC,
there is a forwards scintillator for charged particle veto.

51 radiation length. The three ZDC modules are placed in series to comprise a ZDC.
Figure 3.21 shows the mechanical design of a module of the ZDC. A module consists of 27
layers of tungsten absorber plates and PMMA (polymethylmethacrylate) optical fibers.
A neutron generates a hadronic shower in the tungsten plates, and charged particles inthe shower emit Čerenkov radiation. The radiation is detected through the optical fibers
with a phototube (Hamamatsu R329-02). The size of a tungsten plate is 10 cm wide,
18.7 cm high and 0.5 cm thick. The tungsten plates and the optical fibers are tilted by 45
degrees to roughly match the direction of the Čerenkov radiation. The energy resolutionof a ZDC is obtained to be 21 % for neutrons at an energy of 100 GeV. Neutrons can beseparated from photons with the energy deposit in the 2nd module of the ZDC, since theelectromagnetic showers from photons cannot penetrate the 1st module with 51 radiation
length and do not reach the 2nd module.
An SMD consists of scintillator hodoscopes: 7 scintillator strips with a width of 15 mm
in the vertical direction (to provide x-coordinate), and 8 strips with a width of 20 mm in
the horizontal direction (to provide y-coordinate). The SMD is placed between the 1st andthe 2nd ZDC modules where the neutron-induced shower reaches its maximum (showermax). The hits of the showers at the SMD were weight-averaged to provide the shower
position. The position resolution of ∼1 mm is achieved for neutrons with an energy of
100 GeV.
The vertex position is calculated from the hit timing in the two ZDCs as in the caseof BBCs. The vertex position resolution is obtained to be ∼ 30 cm online and ∼ 10 cm
offline.
The coincidence of hits in the two ZDCs defines ZDC trigger. The ZDC trigger serve

48

CHAPTER 3. EXPERIMENTAL SETUP

as an independent luminosity measure as well as the BBC trigger, which is explainedin Sec. 3.4.1. Two types of ZDC triggers with different vertex position cuts were usedduring the run. One is ZDC narrow trigger which is defined with the vertex position
cut of |z| < 30 cm, and the other is ZDC wide trigger which is defined with the vertex
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.
Comparison between BBCs and ZDCs can be used to estimate the uncertainty on
luminosity measure. The method of evaluation of luminosity uncertainty is described inSec. 4.4.

49

(3
50
-5
50
)

20
0

50

3.4. PHENIX GLOBAL DETECTORS

5

.
φ0

56

45

232

136

306

100

100

Figure 3.21: ZDC Mechanical Design.

50

CHAPTER 3. EXPERIMENTAL SETUP

3.5

PHENIX central arms

3.5.1

Electromagnetic calorimeter (EMCal)

The PHENIX Electro-Magnetic Calorimeter (EMCal) [69] plays an important role for thedetection of photons from π 0 → γγ decay. The EMCal covers central rapidity (|η| < 0.35)
and half in azimuthal angle. (∆φ = π2 × 2). The PHENIX EMCal is composed of
two types of calorimeters. Lead-Scintillator (PbSc) calorimeters, and Lead-Glass (PbGl)
calorimeters. The basic parameters for EMCal are summarized in Table 3.2. A box witha label of PbSc or PbGl in Fig. 3.14 corresponds to a sector. PHENIX has 8 sectorsof EMCal, 6 of them are PbSc type, and other two are PbGl type. Both type covers
pseudorapidity range of |η| < 0.35. PbSc covers π2 + π4 in azimuthal angle, and PbGl
covers π4 . The minimum unit of EMCal with individual read-out is called a tower.
A PbSc sector is composed of 3 × 6 super-modules. A PbSc super-module consistsof 6 × 6 modules, each of which consists of 2 × 2 towers. A PbGl sector is composed of
12 × 16 super-modules, each of which consists of 4 × 6 towers.
The total numbers of towers are 15552 for PbSc and 9216 for PbGl. A detailed
explanation for each type follows in this section.

general parameters
radiation length (X0 )Moliere radiuschannel(tower)
cross section
depthη coverage
φ coverage
super-module
number of channels
sectornumber of super-modules
whole system
number of sectors
number of channels
η coverage
φ coverage

PbSc

PbGl

21 mm
∼30 mm

29 mm
37 mm

52.5 mm × 52.5 mm
375 mm (18 X0 )0.011 [rad]0.011 [rad]40 mm × 40 mm
400 mm (14 X0 )0.008 [rad]0.008 [rad]144 (12 × 12)

24 (4 × 6)

18 (3 × 6)

192 (12 × 16)

6
15552
0.7
90 (deg) + 45 (deg)

2
9216
0.7
45 (deg)

Table 3.2: Basic parameters of two types of PHENIX EMCal.

3.5. PHENIX CENTRAL ARMS

51

Lead Scintillator Calorimeter (PbSc)
The PbSc is a sampling calorimeter which is composed of 65 lead tiles and 66 scintillator
tiles, stacked in alternate way. The thickness of lead tile (scintillator tile) is 1.5 mm(4.0 mm). The scintillator is made of polystyrene (the bulk material) and 1.5% of pTerphenyl (the primary fluorescent material) and 0.01% of POPOP (wavelength shiftingmaterial) are added.
A PbSc module consists of four towers, which are optically isolated and read out
individually. Figure 3.22 shows the internal view of the module. A module has 64 holes
for the read-out fibers to pass through, 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 throughthe entire module from the back side to the front side then return to the back side after
following smooth curves. The both edges of the fibers are gathered and attached to
FEU115M phototubes (1 inch diameter, multi-alkali photocathode, made by MELS in
Russia.) The attenuation length of the fiber is approximately 1 m, which affects thelinearity 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 themodule and provides laser light into four towers through the corners. Figure 3.23 showsa schematic of the laser calibration system [70]. The laser light is split in three steps
and delivered into 3888 modules. The laser amplitude is monitored with a phototube
and photo diodes in all the light splitters. The laser calibration system is to normalize
the calibration change, due to the operation conditions. The gain of the amplifier for thephoto diodes is monitored by test pulses.
The performance of PbSc was evaluated with the test experiments with AGS at BNLand Super Proton Synchrotron (SPS) at CERN. Figure 3.24 displays the energy resolution for electromagnetic showers as a function of incident energy, obtained by the test
experiments. It is well described by∆E
8.1%⊕ 2.1%=q
E
E(GeV)(3.10)The 8.1% in the first term of Eq. 3.10 is close to the expected resolution from samplingas predicted by GEANT. The main contributor to the constant term is intrinsic nonuniformity, in particular tower boundaries, hot spots at fiber positions and shower depth
fluctuations. There will be a loss in the calorimeter response when a particle hits thecorner of the towers. 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 ofthe calorimeter.
The position resolution depends on the incident angle of the beam due to the fluctu-

52

CHAPTER 3. EXPERIMENTAL SETUP

Front side

Layers of lead andscintillator tiles
Wavelength
shifting fibers
TowersBack side
Figure 3.22: A module of PbSc. A module is composed of four towers, which are read-out
individually.

3.5. PHENIX CENTRAL ARMS

Figure 3.23: PbSc calibration

53

54

CHAPTER 3. EXPERIMENTAL SETUP

ation of the shower depth. The resolution can be expressed as
5.7 mm,∆x = ∆x0 ⊕ L sin θ, ∆x0 = 1.55 mm ⊕ q
E(GeV)(3.11)%

where L is ∼ X0 (= 21 mm) and θ is the incident angle to the PbSc tower axis. These
results are reproduced by GEANT simulation [71]. The energy deposit for a minimum
ionizing particle (MIP) was measured to be 270 MeV.
14

BNL 0.5-5.0GeV/c e

12

CERN 10-80GeV/c e

101.2% + 6.2% / E(GeV)

8
2.1% Ҷ 8.1% / E(GeV)

6
4
2
0

1

10

10

2

GeV
Figure 3.24: The energy resolution of PbSc for electromagnetic shower as a function of
the incident energy. The vertical axis shows ∆E/E.

Lead Glass Calorimeter (PbGl)
The PbGl is a Čerenkov type calorimeter. The modules are previously used in WA98[72]experiment at CERN and re-used at PHENIX. PbGl occupies two sectors out of eight.
A lead glass element has 4.0 cm × 4.0 cm cross section and is 40 cm long. This element
is the minimum unit of PbGl and is called a tower. 4 × 6 towers are gathered to form asuper-module. A schematic of a super-module of PbGl is shown in Fig. 3.25. The towers
within the super-module are optically isolated with aluminized Mylar foils and each tower
is attached to an FEU-84 phototube. Steel sheets of 0.5 mm thickness are used to housethe 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3.5. PHENIX CENTRAL ARMS

55

super-module. 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 evaluated with the test experiments with AGS
at BNL and with SPS at CERN. Figure 3.26 shows the energy resolution for electromagnetic showers obtained by the test experiments. No significant dependence on incident
angle was observed. The obtained resolution is well described as
5.9 %∆E⊕ 0.8 %=qE
E (GeV)(3.12)The position resolution for electromagnetic showers are also measured and is describedas
8.4 mm∆x = q⊕ 0.2 mm
E (GeV)(3.13)

photomultiplierwith housing

steel plates
mirror foil

lead glass matrix withcarbon fiber/epoxy

LED boardreflective coverphotodiode withpreamplifier

Figure 3.25: PbGl.
The response of PbGl to hadrons is different from that of PbSc since PbGl is basedon Čerenkov detection. The deposit energy is suppressed for hadrons due to its Čerenkov
threshold (106 MeV/c for charged pions). The energy deposit is examined at the test

56

CHAPTER 3. EXPERIMENTAL SETUP

Energy Resolution (%)

Lead-glass Test Results
14

A2 Test Beam

0 degrees
5 degrees
10 degrees
15 degrees
20 degrees

12
10

X1 Test Beam

8

0 degrees6ǻ/E = 5.95/sqrt(E) + 0.76

4
2
0

0

2.5

5

7.5

10

12.5

15

17.5

20

22.5

25

Energy (GeV)

Figure 3.26: The energy resolution of PbGl for electromagnetic shower as a function of
the incident energy. The vertical axis shows ∆E/E.

experiments. The deposit energy of charged pions is ∼ 460 MeV for 1 GeV/c, and∼ 540 MeV for 4 GeV/c.

3.5.2

EMCal front end electronics

Signals from PMTs are processed in the front end electronics (FEE) and the energy, timeof flight, and energy sum for high-energy photon trigger are derived. The analog signals
are processed by ASIC (application specific integrated circuit) chips on FEE. One ASIC
chip processes four adjacent towers forming 2 × 2 block, which is the minimum unit fortrigger decision. Figure 3.27 shows a block diagram of the analog part of FEE. Thereinserted a passive integrator with 93 Ω resistor and 500 pF capacitor and bias voltage of+4 V is applied to allow negative pulse input from the phototube. The current profile
from the phototube is converted to a voltage profile at point A, whose rise time is lessthan 5 nsec. It is not sensitive to the base shift and the fast voltage pulse is used for timeof light measurement.
The charge is collected onto the capacitor so that the voltage at the point B in Fig. 3.27
follows the charge collected, that is the collected energy. The voltage profile has a risetime of ∼ 100 nsec and long decay time of the order of 10 µsec. Thus the profile is almost a
step function with the amplitude proportional to the deposit energy. All of the remaining3.5. PHENIX CENTRAL ARMS
+4V

57

RHIC Beam Clock

A

Start
Discriminator

93΃

Hold

Low Gain
Out

Gain

x16

DAC

GND

TAC
Out

Ramp
Generator

VGA x4 − x12

B
500pF

Stop

High GainOut
2x2 sumout

Signals fromother PMTs

Sum
three of
2x2 sum fromother chips.

4x4 sumoutEnergymeasurement

From PMT

Energy sum
for Trigger

Thresh

TOF measurement

DAC

Sum

Analog Stage ASIC Chip

Figure 3.27: A block diagram of the analog part of EMCal FEE. One ASIC chip handle
signals from four PMTs. Only a flow for a PMT is shown.

analog processing stages up to ADC conversion are carried out within an ASIC chip[73],as shown in the Fig. 3.27. The signal is amplified by variable gain amplifier (VGA) with again of ×4 – ×12 with 5 bit resolution. The gain of VGAs can be remotely controlled via
ARCNet [74]. ARCNet is a local area network protocol available worldwide and is utilized
for the slow control of PHENIX. This allows the readout electronics to compensate , towithin a few percent over its range, for gain variations among phototubes which share thesame high voltage supply. This feature is useful for the trigger circuit and maximizing
the ADC dynamic range for all channels. Then the signal is divided into three. One isfurther amplified by 16 and used for energy measurement and is called “high gain” (HG).
Another signal is used for energy measurement without further amplification and is called“low gain” (LG). The other signal is used for energy sum for the trigger decision. The
mechanism to have both HG and LG is to maintain good resolution for the wide energy
range of 20 MeV – 30 GeV. The transition energy of the valid range between LG and HG
is about 1 GeV. The signals from four PMTs in the same ASIC chip are summed andprovide the energy sum of 2 × 2 towers for trigger decision. The analyzed data in thisthesis is obtained with 2 × 2 energy sum trigger, which is referred to as high pT photon

58

CHAPTER 3. EXPERIMENTAL SETUP

trigger. One unit of the trigger decision, a group of 2 × 2 towers, is called a trigger tile.
When a photon hit the center of a tower, 80% of the energy is deposited at the tower.
Therefore, the size of the trigger tiles is enough to trigger electromagnetic showers. Two
trigger circuits were used for a single tile of 2 × 2 energy sum and alternated in each beamclock count. In other words, trigger circuits are different for even and odd bunches. The
slight difference in trigger threshold in two circuits may lead false spin asymmetries. Thus
the counts in even and odd bunches are treated separately as explained in Sec. 4.5.1.
The analog outputs of TAC, LG and HG voltages are kept in analog memory units(AMUs), which consist of an array of capacitors. When a trigger is issued, the stored
analog signals are read out and digitized, then sent to the PHENIX Data Collecting
Modules (DCMs), which is described in Sec. 3.7.
The analog signals can be stored for 62 beam clocks. It corresponds to 7 µsec andlonger enough than 4 µsec, which is needed for the trigger decision.

3.6

PHENIX trigger system

A PHENIX Level-1 trigger is to pre-select events with potential interests before highlevel processing of events, based on primitive information from the PHENIX detector
subsystems. The Level-1 is fully pipelined and the decision is made whenever 9.4 MHz
RHIC beam clock is received. A PHENIX Level-2 trigger is based on data which areprocessed after accepting Level-1 trigger, thus is data-driven.
We only introduce Level-1 trigger here since the data used in this thesis are fullycollected with Level-1 trigger and without Level-2 trigger.
The PHENIX Level-1 trigger consists of two types: Local Level-1 (LL1) triggers anda Global Level-1 (GL1) trigger.
A LL1 trigger communicates directly with a detector or a set of detectors for trigger
decisions. Then the LL1 trigger decisions are sent to the GL1 system and it combines thetrigger information to provide the final decision. In the GL1 system, prescale, and logicaloperations such as AND and OR of LL1 trigger decisions can be made.
Among the PHENIX LL1 triggers, the BBC, BBC no-vertex-cut, ZDCwide, highpT photon, and pZDC triggers are used in this analysis. The BBC, ZDC, and high-pT
photon triggers are already introduced in Sec. 3.4.1, 3.4.2, and 3.5.1 respectively. The
pZDC trigger is defined as a coincidence of a hit in either BBC and a hit in either ZDC.
The trigger is symbolically expressed as “(BBCN or BBCS) and (ZDCN or ZDCS)”.

3.7

PHENIX DAQ system

Figure 3.28 shows a schematic of the PHENIX Data Acquisition (DAQ) System. Since the
RHIC beams have the bunch structures, the PHENIX detector has to be synchronized3.7. PHENIX DAQ SYSTEM

59

with the RHIC beams. 9.4 MHz of RHIC clock is provided by the accelerator control
system, and is transfered to the Master Timing Module (MTM). MTM distributes theclock into Global Level-1 (GL1) and the Granule Timing Modules which are equipped
into all subsystems.
The clock and the trigger decision issued by GL1 is sent to the Front-end Electronics
Modules (FEMs). FEMs manipulate FEE to process the raw signals of the PHENIX
detectors. FEE has buffering scheme and store the processed signals for up to 40 bunch
crossings for read-out by request from the trigger decisions.
When a trigger decision is issued by GL1, FEMs send processed signals into Data
Collecting Modules (DCMs). Information is collected by Partitioner and then sent to
Sub-Event Buffers (SEBs). SEBs then transfer the data on request to a set of Assembly
and Trigger Processors (ATPs) under the control of the Event-Builder Controller (EBC).
Here detector-by-detector information is rearranged to event-by-event information.
The rearranged data are sent to online monitoring system to online level quality control, and sent to DAQ Linux machines to record it in hard drives. Then they are recorded
into tapes (HPSS). PHENIX DAQ data taking ability is about 5 kHz.

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 I S ss To
kenPartitioner
DCB

FEM

Clock

RHIC
Computing
Facility
End
User

Busy

Clock, Trigger,
Mode bits
over fiber

h

A
(e Su
.g bs
.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 3.28: PHENIX DAQ system.

SEB
SEB
SEB
SEB
ATM Switch
ATP

Linux Farm
~600 CPU's

tapestorage
1.2PB

20MB/s
250TB/yrChapter 4
Analysis
4.1

Overview

In this chapter, the analysis procedure to extract double helicity asymmetry ALL of π 0
is explained. ALL is defined as Eq. 2.38. It is calculated experimentally by the following
formula.
1L++N++ − RN+−
ALL =
, R=
,
(4.1)
|PB ||PY | N++ + RN+−
L+−
where PB(Y ) denotes the beam polarization in the Blue (Yellow) RHIC ring, N is the π 0yield, L is the luminosity and ++ (+−) represents the helicity states of the proton beams.
R is the relative luminosity between bunches with the same and opposite helicities. The
proton beams are transversely polarized in RHIC rings, and the spin is rotated fromtransverse to longitudinal direction by the spin rotators as explained in Sec. 3.2.2. Since
the magnet currents of the spin rotators may not be adjusted properly, the polarizations
may have residual transverse components at the IP. Therefore, in addition to (absolute)
beam polarizations, beam polarization orientations are needed.
The necessary components for ALL measurement are• beam polarizations• beam polarization orientations• relative luminosity• spin dependent π 0 yields.
They are explained in the following sections.
60

4.2. RUN SELECTION

4.2

61

Run selection

DAQ is usually stopped after one hour of data taking and restarted even if nothing wrong
is found in the detector or the accelerator. A period during which DAQ is started andstopped is called a run. It is for convenience in the analysis. For example, time dependenceof the detector performance is studied run-by-run. On the other hand, the uncertainties
of relative luminosity, which is discussed in Sec. 4.4, is studied fill-by-fill since it depends
on the characteristics of the beams which are correlated among the runs in the same fill.
Asymmetry calculations were performed run-by-run, or fill-by-fill, which is determinedwith the statistics and the trigger efficiency. See Sec. 4.5.1 for detail.
In Run 2006, data reconstruction was started during the data taking. For some runs
PHENIX calibration database was not ready and reconstruction failed. It was not due
to the detectors used in this analysis. Such runs were not available at that time andthe amount was about 10%. Since recovering such small amount of data would notchange the results and do not add significance of the√ results, such runs were not included.
In total, 98 runs were available in Run 2006 at s = 62.4 GeV. Runs without somenecessary information or with wrong detector or accelerator settings, were discarded andnot analyzed. A fill (fill 8035) was removed due to missing polarization information.
Another fill (fill 8059) was removed since the STAR magnets frequently tripped andrecovered, which was known to affect the polarization orientation at the PHENIX IP.
After these selections, 86 runs (18 fills) survived and were analyzed. Out of 18 fills, 7 fills
are from the transverse run period, and 11 fills are from the longitudinal run period.
Figure 4.1 displays the raw rate of BBC trigger as a function of run number. The
BBC trigger serves as a luminosity measure for the PHENIX and the BBC trigger detects
inelastic events with a cross section of√σBBC = 13.7 mb [75]. It corresponds to 40% of
inelastic scatterings in pp collisions at s = 62.4 GeV. The inset in Fig. 4.1 is a closeup
of a certain period of the raw trigger rate. A group of points corresponds to a fill and therate, or the luminosity, decreases with increasing run number. The beams were dumped
after luminosity decreased by a factor of 2 ∼ 3. The DAQ has data recording abilityof about 5 kHz and are able to take data with multiple triggers. Since the BBC trigger
efficiency for inelastic scatterings (which include π 0 production) is only 40%, the highpT photon trigger was utilized during the run to collect π 0 data used in this thesis. See
Sec. 4.5.1 for the trigger performance. A typical setting of trigger mixture at the beginningof a fill is with the photon trigger (∼500 Hz), muon trigger (∼500 Hz), local polarimeter
trigger (∼1.5 kHz), the BBC trigger (∼ 1.5 kHz) and others such as clock trigger. The
photon trigger and muon trigger are without prescale, while prescale for other triggers
were adjusted so that the total rate does not exceed the DAQ limit of 5 kHz. The typical
live time is more than 90% as displayed in Fig. 4.2.

62

CHAPTER 4. ANALYSIS

Trigger rate [Hz]Raw rate of the BBC trigger
6000

6000

5000
4000
3000

5000

2000
1000

4000

0

206400206440

3000
2000
1000
0

205600205800206000206200206400206600Run number

Live time

Rate[Hz]

Figure 4.1: The raw rate of the BBC trigger vs run number.

5000

1

0.9
0.8

4000

0.7

0.6

3000

0.5
0.4

2000

0.3
0.2

1000

0.1

0

205600205800206000 206200

206400206600Run number

0205600205800206000 206200

206400206600Run number

Figure 4.2: The rate of events recorded by Figure 4.3: The live time of DAQ vs run
the PHENIX DAQ vs run number.
number.

4.3. BEAM POLARIZATION

4.3

63

Beam polarization

Polarization of Blue beam

Polarization of Yellow beam

Beam polarization

Beam polarization

The beam polarization was measured with two types of polarimeters as described inSec. 3.2.3. The pC polarimeter is utilized to measure fill by fill variations of the beampolarization, through AN of proton-carbon elastic scatterings. Since no significant depolarization was observed during a fill, several pC polarimeter measurements in a fill are
averaged and used for the asymmetry calculations. The absolute beam polarization is
given by the H-jet polarimeter.
Figure 4.4 displays the beam polarizations, obtained with pC polarimeter and normalized by H-jet polarimeter results. Therefore, the vertical axis shows the absolutepolarization. Statistical uncertainties and fill-to-fill uncorrelated uncertainties are shown.
0.65
0.6
0.55
0.5

0.65
0.6
0.55
0.5

0.45

0.45

0.4

0.4

0.35

0.35

0.3
8010

8020

8030

8040

8050

8060

8070

Fill number

Blue Beam.

0.3
8010

8020

8030

8040

8050

8060

8070

Fill number

Yellow Beam.

Figure 4.4: Beam Polarization.(Left: Blue Beam. Right: Yellow Beam.) The horizontal
axis shows RHIC fill number, and the vertical axis shows the beam polarization. Statisticaluncertainty and fill-to-fill uncorrelated systematic uncertainties are combined. Fill-to-fillcorrelated systematic uncertainty is not shown.
The luminosity-weighted average of beam polarization over 11 longitudinally (7 transversely) polarized fills used in this analysis, was
hPB i = 0.48 ± 0.007[stat] ± 0.035[syst],
hPY i = 0.48 ± 0.006[stat] ± 0.045[syst],

(4.2)
(4.3)

for the longitudinal run period and
hPB i = 0.49 ± 0.008[stat] ± 0.035[syst],
hPY i = 0.49 ± 0.008[stat] ± 0.045[syst],

(4.4)
(4.5)

for the transverse run period with a systematic uncertainty of 7.2% for the Blue beam, and9.3% for the Yellow beam. The major systematic uncertainty is the statistical uncertainty

64

CHAPTER 4. ANALYSIS

of H-jet polarimeter measurements which are used to normalize√fill-by-fill pC polarimetermeasurements. H-jet measurement for the Yellow beam at s = 62.4 GeV was notreliable due√ to high background. Therefore, the calibration parameter for the yellow
beam at s = 62.4 GeV, AN (yellow, 62), was obtained as
AN (yellow, 62) = AN (yellow, 200) ×

AN (blue, 62)
AN (blue, 200)

(4.6)

Additional systematic uncertainties were assigned
for the Yellow beam according to the√observed variations and dead layer drift at s = 200 GeV. The average of the polarization
product over the whole run was
hPB · PY i = 0.23 (0.24)(4.7)

with a systematic uncertainty of 13.9%.
The remaining transverse-spin component (PT /P ) were measured with the PHENIX
local polarimeter, by utilizing the single spin asymmetry AN of forward neutron production. It measures the polarization in horizontal (x) and vertical direction (y) in the beam
view. Then the transverse polarization component is calculated as:
hPT /P i =qhPx /P i2 + hPy /P i2,(4.8)

and the longitudinal polarization is calculated as
hPL /P i =q

1 − hPT /P i2.

(4.9)

The results are summarized in Table 4.1. The transverse components of the beam polarizations are consistent with zero, thus the proton beams were purely longitudinallypolarized within the uncertainties.
Blue
Yellow

hPx /P i
hPy /P i
hPT /P i
hPL /P i−0.071 ± 0.135 0.080 ± 0.162 0.107 ± 0.151 1.00 − 0.034−0.039 ± 0.116 0.105 ± 0.119 0.112 ± 0.117 1.00 − 0.025

Table 4.1: Remaining transverse component and the longitudinal component of the beampolarizations.

4.4

Relative Luminosity

A relative luminosity R is the luminosity ratio between the same and opposite helicity
collisions. It is required for the asymmetry calculations as in Eq. 4.1. In this section,
relative luminosity measurement and a method to estimate its uncertainty are described.

4.4. RELATIVE LUMINOSITY

4.4.1

65

Overview

The relative luminosity is defined as R = LL+−
, where L++ and L+− are the luminosity++
for the same and opposite helicity collisions, respectively. Although any counts which
is proportional to luminosity can be used for the measurement of R, it is preferable forthe counts to satisfy the following conditions: high statistics for high accuracy, no spindependence not to bias asymmetry measurement, small background, and linearity.
In this analysis, BBC trigger live counts were used for the relative luminosity measurement, since it satisfies all those requirements. Unfortunately BBC trigger rate (∼ 10 kHz)was more than the DAQ data recording ability of 5 kHz and cannot be recorded without
prescale. Thus GL1P scaler was utilized which keeps track of the number of live trigger
counts for each bunch. It makes the statistical accuracy free from the DAQ data recording
ability. The uncertainty on ALL due to the uncertainty of R, δARL
LL , can be calculated as
δARL
LL =

2N++ N+−PB PY (N++ + N+− R)2



∆R
R



∼

1
2PB PY



∆R
,
R


(4.10)

where N++(+−) denotes the π 0 yields in the same (opposite) helicity collisions. PB andPY are the beam polarizations.
To evaluate ∆R, BBC trigger counts (used for relative luminosity in this analysis)
were compared to counts of triggers other than BBC trigger itself. The ZDCwide trigger
was chosen for the transverse run period and the pZDC trigger was chosen for the longitudinal run period. The ZDCwide trigger is based on ZDCs which is another independent
luminosity measure and can be used for comparison.
However, the ZDCwide trigger rate√
was as low as the BBC trigger × ∼ 0.008 at s = 62.4 GeV. Thus the accuracy wasstatistically limited and not satisfactory. To achieve better accuracy, the pZDC trigger
was introduced and available for longitudinal run period. The pZDC is defined with BBCsand ZDCs as in Sec. 3.6 and also serves as a luminosity measure. Therefore, pZDC canbe used for comparison. (Since the BBC trigger and the pZDC trigger uses BBCs incommon, their dependence is considered in Sec. 4.4.6.)
The difference between the BBC trigger and the ZDCwide trigger (the pZDC trigger
for longitudinal run period) is partially due to the difference of the vertex position cut
as explained later in this section. Thus direct comparison of these will overestimate
the relative luminosity uncertainties. Such an effect is corrected by utilizing the vertex
distribution width.
At first, the quality of the bunches are investigated and only selected bunches wereanalyzed further. The selection criteria is described in Sec. 4.4.2.

4.4.2

Bunch selection criteria

The bunch crossing IDs are zero-based integer, 0 to 119. Although RHIC has 120 bunches
in each RHIC beam, the last nine bunches in each beam were not filled as explained in

66

CHAPTER 4. ANALYSIS
hist_bbcz0_000
Entries
9456
Mean
2.869
58.21
RMS
χ2 / ndf
26.47 / 28
Prob0.5471506.7 ? 7.1
Constant
Mean
3.682 ? 0.679
58 ? 0.6
Sigma

500

400

300

200

100

0
-250

-200

-150

-100

-50

0

50

100

150

200

z vertex position [cm]Figure 4.5: BBC z distribution. (Fill 8031, crossing number 0) It was fitted with Gaussian
to obtain the vertex width. The BBCs are located at |z| ∼ 144 cm and any collisions
outside the range are reconstructed as |z| ∼ 144 cm.
Sec. 3.2. The first bunch in the Blue beam collides on the 81st bunch in the Yellow beamat the PHENIX IR, which results in 18 (9 × 2) non-colliding bunch crossings. Two setsof non-colliding bunch crossings are called the “abort gaps”. The crossing ID = 1 wasdiscarded due to the specification of DAQ. A bunch in each ring (ID = 20, 60) is reserved
for accelerator control, and removed from the analysis. Consequently ninety nine bunch
crossings survive the criteria. In addition, bunches with the BBC trigger counts 3 RMS
less than the average were also removed since bunch crossings with small populations
imply that they have different vertex distributions from others.

4.4.3

Vertex width

It was obtained by fitting the BBC z-vertex distributions to Gaussian in BBCLL1(noVertexCut) triggered events fill by fill. Figure 4.5 shows an example of the fit ina fill. The bunch-by-bunch vertex distributions are well reproduced by Gaussians. The
distributions have peaks at |z| ∼ 144 cm since they are the places where BBCs are locatedand any collisions outside two BBCs are reconstructed at |z| ∼ 144 cm. See Sec. 3.4.1 fordetail.

4.4. RELATIVE LUMINOSITY

4.4.4

67

A method to estimate the uncertainty on the relative luminosity

Here we introduce a ratio ri defined as:
ri =

Ai
,
Bi

(4.11)

where i is the crossing ID, and Ai (Bi ) is the number of counts of trigger A(B) in the
crossing ID i (i = 0 − 119). If the triggers A and B are perfect luminosity measures, thenri should be constant. The ratio ri is calculated for each fill, and is fitted tori = C[1 + ALL PB PY ]= C[1 + εLL · sgn(PB PY )](4.12)(4.13)

where C and εLL are the fitting parameters. (when it is done for transverse runs, LL isreplaced by T T .) A possible dependence of the triggers on spin is taken into account byintroducing the parameter εLL in the fit. εLL is interpreted as the (raw) double helicity
asymmetry of A compared to B. It is difference of the double spin asymmetries of A andB,
B
εLL ∼ εA
(4.14)
LL − εLL

B
when εA
LL , εLL ≪ 1. And the raw asymmetries are related to the double helicity asymmetries as ALL = εLL/(PB PY ). This method provides the difference of the asymmetries
for two processes. The BBC trigger, which is used for relative luminosity calculations,and the ZDCwide and pZDC are expected to be spin independent. It is confirmed by thecomparison between BBC and ZDCwide (or pZDC).
The uncertainty propagates to ALL of π 0 as PY1PB δεLL , where δεLL denotes the uncertainty on the fit parameter εLL . The spin dependence εLL is expected to be zero andwas confirmed in the analysis. The BBC trigger was chosen as trigger B throughout theanalysis since it is the trigger used for relative luminosity calculations. Its spin dependence was evaluated by comparing it to ZDCwide or pZDC. ZDCwide was used as thetrigger A for the transverse run period, and the pZDC trigger was used as the trigger A
for the longitudinal run period as explained in Sec. 4.4.1. Figure 4.7 shows the results of
the ratio r for a fill. (It is a fill from longitudinal run period. Thus pZDC trigger wasused as the trigger A.) The solid line describes the fit function (Eq. 4.13). It was bent atremoved bunches for a guide. It has a huge reduced χ2 of 4248/95. ri are different fordifferent bunches but it cannot be explained by the spin dependence εLL. It is considered
to be due to vertex width variations.
The BBC trigger has a vertex position cut of z = ±35 cm, while pZDC does not.
Thus even if both work as perfect luminosity measures, wider vertex distribution makes
the ratio ri larger. Figure 4.6 illustrates the effect of the vertex cut on luminosity ratio.
A Gaussian represents z vertex distribution in a bunch crossing. Two Gaussians in the

68

CHAPTER 4. ANALYSIS

figure are with the same mean value of 0 but have different widths of σ = 60 cm andσ = 40 cm, and normalized so that to have the integrals over the whole z are the same.
σ = 60 cm is reasonable for vertex distribution in real experiment but σ = 40 cm isextremely narrower than the real distributions but it was chosen so that to illustrate theeffect clearly. The resolution of vertex cut is neglected in this discussion for simplicity.
The dashed line represents the vertex cut implemented in the BBC trigger. Thus the BBC
trigger counts collisions inside the vertex cut, while the pZDC trigger counts the wholez area. Let Ni,BBC and Ni,pZDC be the trigger counts of BBC and pZDC for the crossing
i respectively, and Ni,inside , Ni,whole be the counts for crossing i inside the vertex cut andthe counts for the whole distributions respectively. Then the ratio ri = Ni,pZDC /Ni,BBC is
proportional to Ni,whole /Ni,inside .
ri =Ni,pZDCNi,whole∝.
Ni,BBC
Ni,inside(4.15)

Thus the ratio between the two luminosity measures NpZDC /NBBC is different due to thedifference in width of the z vertex distributions. However, the difference due to the width
has nothing to do with the accuracy of the relative luminosity. Let c be the vertex cut,
and w be the vertex width. The fraction of events inside the vertex cut is calculated as
Z z′
2′fr (z ) = √exp(−t2 /2)dt.
(4.16)
π 0
where z ′ = c/w. Taylor expansion of fr (z ′ ) at z’=0 isz ′3+ ···.
(4.17)3fr (z ′ ) can be approximated as fr (z ′ ) ∼ z ′ when z ′ is smaller than one. Since Ni,whole /Ni,inside ∝1/fr (z ′ ) and z ′ ∝ 1/w, Ni,whole /Ni,inside is linearly dependent on the vertex width w.
Figure 4.8 shows an example of the correlation between the ratio ri and the vertex
width. They have clear correlation as expected. The plots were fitted to a linear function:
fr (z ′ ) = z ′ +f (w) = p0 + p1 w,(4.18)

where w denotes vertex width, and p0 and p1 denote fit parameters. Figure 4.9 shows thedeviation of ri from the fit divided by its statistical error δri . The reduced χ2 of the fit
is greater than unity which indicates the existence of effects other than the vertex width
dependence.
Such unknown uncertainties are accounted byqenlarging the uncertainty byqχ2 /NDF. If the deviation from the fit is greater than 2.5× χ2 /NDF, the bunch crossing
was removed from the asymmetry calculations since it implies that it has different vertex
distributions from other crossings. In Fig. 4.9, a bunch with a deviation of ∼ 11 wasremoved (crossing ID = 75.) Figure 4.11 displays corrected ratio r ′ = r/f (w) and the fit
results. Please note that the reduced χ2 of the fit was greatly reduced from 4248/95(∼ 45)
to 802.2/94(∼ 8.5), but thereq still remain systematic uncertainty. It was accounted byenlarging the fit error with χ2 /NDF.

4.4. RELATIVE LUMINOSITY

69

vertex cut

σ = 40cm

σ = 60cm-200 -150

-100

-50

0

50

100 150 200
z position [cm]Figure 4.6: The effect of vertex cut on luminosity ratio.

tmp

r = (pZDCwide / BBCLL1)

before correction
Entries

120

Mean

59
32.44

RMS
χ2 / ndf
Prob

0.18
0.17

C
ell

bad bunches

4248 / 95
00.1463 ? 0.0000-9.461e-05 ? 2.865e-04

abort gaps
0.16

0.15
0.14

0.13
0

20

40

60

80

100

120

crossing number

Figure 4.7: ratio r = pZDC / BBCLL1 and the fit result.

70

CHAPTER 4. ANALYSIS

χ 2 / ndf

0.158

932.7 / 95

Prob

0.156

| deviation from fit / error|r = (pZDCwide / BBCLL1)|deviation/error|0

p0-0.0140.002784

p1

0.002557

4.441e-05

0.154
0.152
0.15
0.148
0.146
0.144

12
10

2.5 x χ 2 / NDF

8

6
4
2

0.142

61

62

63

64

0

65

61

62

63

64

vertex width [cm]Figure 4.8: ratio r vs vertex width.

120

Mean

59
32.44

RMS

C
ell

bad bunches

tmp2

1st correction applied
1.2

abort gaps

217

Mean

58.6
32.58

RMS
χ / ndf
Prob

1.15

C
ell

-9.461e-05 ? 2.865e-04

0.16

Entries24248 / 95
00.1463 ? 0.0000

r ‘= r / f(w)r = (pZDCwide / BBCLL1)

Entriesχ2 / ndf
Prob

0.18
0.17

Figure 4.9: Deviations from the fit divided
by statistical uncertainties.

tmp

before correction65

vertex width [cm]802.2 / 94
00.9996 ? 0.0003-0.0005062 ? 0.0002880

1.1
1.05

0.15

1

0.14

0.95
0.9

0.13
0

20

40

60

80

100

120

crossing number

Figure 4.10: ri (before vertex width correction)

0

20

40

60

80

100

120

crossing number

Figure 4.11: ri′ (after vertex width correction)

4.4. RELATIVE LUMINOSITY

4.4.5

71

Results of relative luminosity analysis

Figure 4.12 shows the reduced χ2 vs δεT T for the transverse run period, and Fig. 4.13
displays the reduced χ2 vs δεLL for the longitudinal run period. They look reasonable
and no fills were removed. Figure 4.14 (4.15) displays εT T (LL) vs fill number and was
fitted to constant. For conservative estimationof uncertainty, the fill-by-fill statisticalq2
uncertainty for εfill , δεfill , is enlarged by χ /NDF when χ2 /NDF > 1 as mentioned
in the previous subsection, to account possible systematic uncertainty as mentioned in
the previous subsection. The enlarged error is indicated by red error bar in Fig. 4.14
and Fig. 4.15. The uncertainty δεT T (transverse) was found to be 7.5 ×10−4 , and δεLL
(longitudinal) was found to be 3.3 ×10−4 . Since the average polarization is hPB ·PY i = 0.24
(2.3) for transverse (longitudinal), the effect on double spin asymmetries will be δAT T =3.1 × 10−3 for transverse run period, and δALL = 1.4 × 10−3 for longitudinal run period.
0000The statistical accuracy of AπT T and AπLL for π 0 are δAπT T ∼ 7 × 10−3 and δAπLL ∼ 5 × 10−3
respectively. Therefore, the systematic uncertainties from relative luminosity are small
compared to the statistical uncertainties of π 0 asymmetries.
gr_elle_chi2ndf

χ2 /NDF

gr_elle_chi2ndf

χ 2/NDF

1.8

1.6

10
9
8

1.4

7
1.2

6
5

1

4
3

0.8
0.00150.002

0.00250.003

0.0035δεLL

Figure 4.12: χ2 /NDF VS εT T (Transverse
run period. ZDCwide)

4.4.6

0.00020.00040.00060.00080.001

0.0012δεLL

Figure 4.13: χ2 /NDF VS εLL (Longitudinal run period. pZDC)

Event overlap

The BBC and pZDC triggers use the BBCs in common. To evaluate how much the pZDC
trigger is dependent on the BBC trigger in statistical point of view, event overlap wasinvestigated using real data. We define a notation “A in B” which describes the number oflive counts of trigger-A in trigger-B triggered events. For example, “ZDCwide in BBC novertex-cut” means ZDCwide live counts in the BBC no-vertex-cut√ triggered events. The
results are shown in Table 4.2. For comparison, event overlap√ at s = 200 GeV was also
investigated. (We used to use ZDCwide in the analysis at s = 200 GeV.) ZDCwide in

72

CHAPTER 4. ANALYSIS

χ2 / ndf
Prob
p0

εTT

gr_fill_ell
0.006

1.552 / 6
0.956
-0.0004453 ± 0.0007543

0.004

0.002

0

-0.002-0.004
8014

8016

8018

8020

8022

8024

8026

Fill Number

Figure 4.14: εT T vs fill.

χ2 / ndf
Prob
p0

εLL

gr_fill_ell

0.003

8.774 / 10
0.5537-9.775e-05 ± 0.0003259

0.002
0.001
0
-0.001-0.002-0.003-0.004
8030

8035

8040

8045

8050

8055

8060

Fill Number

Figure 4.15: εLL vs fill.

4.5. π 0 RECONSTRUCTION

73

√
√
at
s = 62.4 GeV
BBC no-vertex-cut at s = 200 GeV and pZDC in BBC-no-vertex-cut√
are 0.7%, 1.2% respectively.
√ BBCno-vertex-cut in ZDCwide at s = 200 GeV and BBCno-vertex-cut in pZDC at s = 62.4 GeV is 21% and 19%, respectively. The event overlap
is in similar level at both energies. The BBC trigger and the pZDC (and ZDC) trigger arealmost independent in statistical point of view and can be used to obtain the accuracy ofrelative luminosity.
Trigger
BBC no-vertex-cut
ZDCwideBBC no-vertex-cut
pZDC

in
in
in
in

ZDCwideBBC no-vertex-cut
pZDC
BBC no-vertex-cut

Table
4.2: Event Overlap. The results from√s = 200 GeV in Run 2005 are shown.

4.4.7

√

62.4GeV
1.3 × 10−2
3.8 × 10−5
1.9 × 10−1
1.2 × 10−2

200GeV
2.1 × 10−1
6.7 × 10−3
N/A
N/A

s = 62.4 GeV in Run 2006 and andSingle beam backgroundSingle beam background is estimated by comparing the BBC trigger counts in colliding
and non-colliding bunch crossings. There are two types of non-colliding bunch crossings:
Blue beam only bunch crossings (31 − 39), and Yellow beam only bunch crossings (111 −
119). Among the Yellow beam only crossings, crossing ID = 115 was removed since it isreserved for GL1P scaler reset and the trigger counts in the bunch crossing is not correct.
Figure 4.16 shows the ratio between the average counts in colliding and non-colliding
bunch crossings versus run sequence number. Open (closed) circle is from Blue (Yellow)
beam only bunch crossings. Single beam background is < 0.35% and it has negligible
effect on δALL .

4.5

π 0 reconstruction

Spin dependent (bunch crossing dependent) π 0 yields are necessary for asymmetry calculations. π 0 is detected via two photon decay, π 0 → γγ. The decay photons are detectedwith PHENIX EMCal, which is explained in Sec. 3.5.1. The high pT photon trigger
was utilized to collect data. The trigger performance including its rejection power andefficiency for π 0 , is discussed in Sec. 4.5.1.
Deposited energy of photon-induced electromagnetic showers in EMCal spreads among
several towers, which is the minimum unit of individual read-out in EMCal. These towers
with deposited energy are gathered to form a cluster to measure the photon energy. The
clustering algorithm used in the analysis is explained in Sec.4.5.2.

74

CHAPTER 4. ANALYSIS

Single beam background ratio

Single beam background0.00350.003
0.00250.002
0.00150.001
0.0005
0
205

205.2

205.4

205.6

205.8

206

206.2

206.4

10

3

Run Number

Figure 4.16: The ratio between the average BBC trigger counts in colliding and noncolliding bunch crossings. Open circle: Blue beam only bunch crossings. Closed circle:
Yellow beam only bunch crossings.

EMCal is composed of many towers and some towers may not work properly. The
quality assurance of the EMCal towers is explained in Sec. 4.5.3. tower-by-tower relative
gain variations were corrected online as explained in Sec. 3.5.1. In addition, offline energy
calibration was performed with the measured π 0 peak position as described in Sec. 4.5.4.
The applied cuts to identify π 0 are explained, and obtained two-photon invariant mass
spectra are shown in Sec. 4.5.6. The EMCal stability for the whole run is evaluatedin Sec. 4.5.7. The background in the obtained two-photon invariant mass spectra are
discussed in Sec. 4.5.8 and Sec. 4.5.9.

4.5.1

High-pT photon trigger performance

The analysis is based on data collected with the high pT -photon trigger. We review thetrigger performance in this section. The trigger decision is based on the energy sum of
2 × 2 towers. A collection of 2 × 2 towers are referred to as a trigger tile. See Sec. 3.5.2
about the trigger.
Rejection power
The performance of the trigger is monitored by the rejection power which is defined as
NBBC /Nphotontrig where NBBC is the number of the BBC triggered events, and Nphotontrig
is the number of the high-pT photon triggered events respectively. The rejection factor
is defined using the BBC trigger since the BBC trigger is the best luminosity measure4.5. π 0 RECONSTRUCTION

75

in PHENIX as√described in Sec. 4.4. The BBC trigger detects 40% of the inelastic pp
scatterings at s = 62.4 GeV. Figure 4.17 shows the obtained rejection power of thehigh-pT photon trigger. The rejection power was about ∼ 10 during the run. A cluster ofruns corresponds to a fill. (Between fills, about ∼ 100 of run numbers were consumed for
detector calibrations and diagnostics of DAQ.) The time dependence of rejection poweris mainly caused by electronics noise. Sometime we have very hot towers which fire thetrigger at every crossing. Such towers are masked for trigger decision and such runs with
hot trigger tiles were discarded.
Rejection power of the high p photon trigger
Rejection power

T

14
12
10
8
6
4
2
0
205.4

205.6

205.8

206

206.2

x 10
206.4
206.6
Run number

3

Figure 4.17: Rejection power of the high-pT photon trigger

Circuit swap of the high-pT photon trigger
As described in Sec. 3.5.2, the high-pT photon trigger utilizes two different circuits for a
trigger tile and alternated in each beam clock count. Therefore, signals from even and odd
bunch crossings are treated with two different circuits. Different circuits lead to different
thresholds, and false asymmetries. It is avoided by calculating asymmetries separately
for even and odd bunch crossings.
The synchronization of the circuit usage for even and odd bunch crossings is done at
the beginning of a run. Therefore, a swap of the circuits between even and odd bunch
crossing may take place between two consecutive runs in a fill. Figure 4.18 shows anexample of such swap. The ratio Nphotontrig /NBBC , which is the inverse of the rejection
power, is plotted for each bunch crossings for a sector. The inverse of the rejection power

76

CHAPTER 4. ANALYSIS

is different from that in Fig. 4.17 by a factor of 8 which is the number of sectors. The
ratio is different for even and odd bunch crossings due to the slightly different thresholdof the circuits. It swaps between even and odd bunch crossings between consecutive runs
within the same fill. The probability to have such swap was small, about once in fiveruns. Please note that the synchronization of trigger circuits are done at the beginningof data taking thus trigger circuit swap never occur during a run.
Therefore, asymmetry calculations were performed run-by-run basis. Unfortunately
there is not enough statistics to calculate background asymmetry for the highest pT bin(3 − 4 GeV/c) in run-by-run basis. Thus fill-by-fill calculations were performed for thehighest pT only. As explained later in this section, the trigger efficiency reaches plateau
at pT > 3 GeV/c. Therefore, it is safe to calculate fill-by-fill at the highest pT bin.
ERT / BBCLL1 W3 RUN 206392

FILL 8058

1.22e-02 r 4.19e-05 χ2=41.5/47
1.24e-02 r 4.23e-05 χ2=39.7/47 *3.97 σB:even
R:odd

.018
.016

FILL 8058

ERT / BBCLL1 W3 RUN 206393
0.02
.018

2

1.25e-02 r 4.45e-05 χ =46.2/47 *1.23e-02 r 4.42e-05 χ2=61.8/47
3.21 σB:even
R:odd

.016

.014

.014

0.012

0.012

0.01

0.01

.008

.008
.006

.006
0

20

40

60

80
100
120
Crossing Number

0

20

40

60

80

100

120

Crossing Number

Figure 4.18: An evidence of trigger circuit swap between even and odd bunch crossings
between runs in a same fill. The ratio between the high-pT photon triggered events andthe BBC triggered events versus crossing number for a sector. Black histograms are for theeven crossings and red histograms are for the odd crossings. They are fitted to constantsfor even and odd bunches separately and the fit results with reduced χ2 are shown in thehistograms. The larger fit results are indicated with * and are swapped between Run
206392 (Left) and Run 206393 (Right). These two runs are in the same fill.

The high-pT photon trigger efficiency for π 0
The high-pT trigger efficiency is defined as the fraction of clusters which fire the high-pT
photon trigger in BBC triggered events. Figure 4.19 displays the high-pT photon trigger
efficiency for π 0 ’s vs pT . They are ∼ 55% (∼ 25%) at pT =1 GeV/c , increase towards
higher pT , and reach plateau of ∼ 98% (∼ 88%) for PbSc (PbGl). Noisy towers in EMCal,

4.5. π 0 RECONSTRUCTION

77

Trigger efficiency for π0

Trigger efficiency for π0

mainly due to electronics noise, are removed from trigger decision. The plateau can be
explained by the fraction of the removed towers in the trigger decision.

1

0.8

0.6

0.4

0.2

1

0.8

0.6

0.4

0.2

PbSc
0
0

1

2

3

4

5

6

pT (GeV/c)

PbGl
0
0

1

2

3

4

5

6

p (GeV/c)T

Figure 4.19: The high-pT photon trigger efficiency for π 0 versus pT . Left: PbSc. Right:
PbGl. Both PbSc and PbGl reach plateau at high pT and it is explained by the fractionof masked trigger tile.

4.5.2

Clustering algorithm

From EMCal, the following information about an incident particle is extracted for the
analysis.
• Total energy• Hit position• Photon probability
The energy deposit of a particle hit at EMCal spreads among several towers. To
extract information of the incident particle, it is necessary to find a group of towers whichhave energy deposit of the particle.
At first, a certain threshold is applied to select towers. The threshold is 10 MeV for
PbSc and 14 MeV for PbGl. The higher threshold for PbGl is due to the larger noisein PbGl. The neighboring towers among the selected towers are grouped into clusters.
If there are more than two maxima of energy deposit in a cluster, they are splitted into
clusters so that a cluster has only one maximum.

78

CHAPTER 4. ANALYSIS

In PbGl, all energies in the cluster towers are summed and total energy is corrected
for the incident angle dependence. In PbSc, “core” tower technique is utilized to extractenergy deposit of clusters. The sum is performed only for core towers, instead of summing
all tower energy in a cluster. Core towers are defined as those in which the incident particle
is estimated to deposit the energy more than 2% of the total energy. The estimation isbased on the electromagnetic shower profile which was parametrized in the test beam
experiments. The energy sum of core tower is about 90% of the total. Then the totalenergy is extracted with the impact angle dependence correction. The impact angle
dependence is less than 4%. The energy losses caused by the attenuation in the fibers
and shower leakage, are corrected. The “core” tower technique was introduced to copewith the high occupancy environment in heavy ion collision experiment. Although it is
not necessary in pp collisions, it is favorable to use the same algorithm in both pp andheavy ion experiment to reduce unnecessary systematic uncertainty due to the differenceof clustering algorithms. This degrades the energy resolution slightly (the constant termof the energy resolution is increased from 2% to 3%) but the achieved resolution is goodenough for the analysis.
The hit position of the incident particle is obtained by the center of gravity method.
The positions of towers in a cluster are weight-averaged by tower energy. Then thedependence of the impact angle of the particle is corrected.
Photon probability is calculated based on χ2 test of the observed energy distribution
in a cluster with an ideal electromagnetic shower profile which is parametrized by the test
experiment.

4.5.3

Quality assurance of the EMCal towers

EMCal consists of ∼ 25000 towers and some of the towers may not work as expected.
Towers which do not work as expected, are masked and not used for the analysis. Such
towers can be categorized into three types: dead, noisy and not-calibrated towers. A deadtower is defined as the number of hits in the tower is zero or considerably smaller thanother towers. A noisy tower is defined as a tower sending signals greatly higher frequency
than other towers, which is due to electronics noise. Distributions of the number of hits
for each tower, is created for each sector and towers which have multiplicity greater by
15 σ from the average are labeled as noisy and removed. Since energy leakage may affectthe energy measurement, in addition to those towers described above, towers next to thosebad towers, and the edge towers are also removed. Table 4.5.3 summarizes the fractionof the masked towers.

4.5.4

Energy calibration of EMCal

The time dependence of the gain in EMCal is corrected tower-by-tower basis with thelaser calibration system in EMCal. The calibration data are analyzed automatically after4.5. π 0 RECONSTRUCTION

Edge towers
Bad towers
Neighbor towers
Total towers

79
PbSc
PbGl
1272 (8.2%) 568 (6.2%)
288 (1.9%) 420 (4.6%)2060 (13.2%) 2110 (22.9%)
15552
9216

Table 4.3: The fraction of the masked towers.

calibration data taking, and stored into the PHENIX calibration database. In addition tothe online calibration, offline energy calibration was performed by utilizing the measured
π 0 peak position. The offline energy calibration utilizes physics signal and provide mostreliable energy scale. The energy scale is corrected for π 0 peak to have the world averageof π 0 mass obtained by Particle Data Group (PDG) [76] at first. After the calibration,
the scale is further corrected for energy smearing
effect as explained in Sec. 4.5.5.
√
The amount of the collected data at s = 62.4 GeV was not enough for the offline
tower-by-tower
energy calibration. However, before short low energy pp√√ experiment at
s = 62.4 GeV (∼ 2 weeks), longer high energy pp experiment at s = 200 GeV(∼√ 10 weeks) was performed. Thus the calibrations were performed with the data at
s = 200 GeV instead.

π0
π0
π0

Figure 4.20: A method of tower-by-tower energy calibrations. The boxes represent EMCal
towers. The box surrounded by thick lines is the target tower of the calibration. One ofthe decay photon of each π 0 in the figure hits the target tower. These π 0 ’s are used forthe calibration of the target tower.

CHAPTER 4. ANALYSIS
peak width [GeV/c 2 ]

80
0.015
0.014
0.013
0.012
0.011
0.010
0.009
0.008
0.007
1

1.5

2

2.5

3

3.5

4

p [GeV/c]T

Figure 4.21: π 0 width vs pT before (Black) and after (Red) energy calibration.
The offline calibration procedure is as follows. At first, run-by-run variations arecorrected using π 0 mass peak position. Then tower-by-tower gain correction follows.
Figure 4.20 illustrates the method of the tower-by-tower calibration. The energy scale ofa target tower is corrected with the measured π 0 -mass peak for photon pairs, one of eachpair hits the target tower. Although the mass shift due to the other tower is averaged
and have smaller effect than the target tower, it is still not negligible. It is overcome byiterating the process several times. Figure 4.21 displays the width of π 0 before and after
the calibration.
At the time of clustering, non-linearity due to energy leakage and light attenuation in
the fibers were corrected as in Sec. 4.5.2. However, there are residual non-linearity mainly
due to a finite energy threshold for individual towers to cut electronic noise. Residual nonlinearity was corrected by utilizing pT -dependent π 0 -mass peak-position and was obtained
to be
Eorg
Ecorrected =for PbSc(4.19)
0.003 + (1 − 0.01/Eorg )Eorg
for PbGl.
(4.20)
Ecorrected =0.021 + (1 − 0.02/Eorg )√The √above corrections were obtained with the data at s = 200 GeV, and applied fordata at s = 62.4 GeV.

4.5.5

Absolute energy scale

The absolute energy scale is calibrated by comparing the observed and the simulated π 0mass. For the purpose, Fast Monte Carlo (FastMC) simulation was used. FastMC is a

4.5. π 0 RECONSTRUCTION

81

simple Monte Carlo simulation unlike GEANT. It produces π 0 according to the measured
cross section, makes it decay into two photons, and smears the energies and the positionsof the photons according to the detector resolutions. Electro-magnetic shower profiles
are simulated with the parameters obtained with the test experiments. The same towermasks as in the analysis was used and the trigger efficiency was applied.
Finite energy resolution cause π 0 mass peak shift from the PDG (world average) value
due to the steep pT dependence of π 0 cross section. A π 0 with a certain measured pT may
have lower or higher pT in reality due to finite energy resolution. The invariant mass of
two photon pair, Mγγ and transverse momentum, pT are calculated as2Mγγ= 2E1 E2 (1 − cos(θ))= 4E1 E2 sin2

!

θ
,
2

(4.21)

and
pT =
=

q

q

|E T,1 + E T,2 |2(4.22)|E T,1 |2 + |E T,2 |2 + 2|E T,1 ||E T,2 | cos φ(4.23)

where E1 and E2 (E T,1 and E T,2 ) represent energy (transverse energy) of two photon
clusters, θ is the opening angle between the clusters, φ is the angle between E T,1 andE T,2 . When reconstructed pT is higher than the real pT , reconstructed mass is likely to
be larger than the real π 0 mass at the same time and vice versa. Due to the steep pT
dependence, effect from lower pT is more than higher pT which results in π 0 mass shift tohigher side. In addition, π 0 mass peak position becomes lower due to the following effects.
One of the sources of the unusual π 0 is so called “albedo”. One of decay photons may
convert into electron-positron pair, but still be reconstructed as a single cluster when thepair is close enough. Therefore, the reconstructed mass is close to that of π 0 . Anothersource is π 0 from decay of other hadrons such as Ks0 and η. The effects were evaluatedwith GEANT and obtained to be −1 ±1 MeV/c2 . The uncertainty is translated into 0.7%
in energy scale and is accounted as systematic uncertainty. In the FastMC, the π 0 mass
was lowered by 1 MeV/c2 to take into account the effect. Finally, the mass√was evaluatedby the FastMC and found to be about 137 MeV and the energy scale at s = 62.4 GeV
was corrected accordingly.
To reproduce the measured π 0 peak width vs pT , the constant term of EMCal energy resolution was increased to 4% (6%) in PbSc and PbGl. It is mainly due to theimperfection of the calibration.
Figure 4.22 displays the positions and widths of the observed and simulated π 0 mass
peak. Energy asymmetry, α, is defined asα=|E1 − E2 |,E1 + E2(4.24)

82

CHAPTER 4. ANALYSIS

where E1 and E2 are the photon energies of a pair. Figure 4.23 shows the energy asymmetry, α, distributions for pT = 1.0−1.5 GeV/c and pT = 2.5−3.0 GeV/c. The non-linearity
corrections as described in Sec. 4.5.4 were already applied for the plots. A cut was appliedfor Energy asymmetry α < 0.8 as described in the next subsection. The energy asymmetry, α, is not flat for low pT due to the energy unbalance between two photons cased by
the high-pT photon trigger. FastMC simulation and data agree well.

2

Mγ γ (GeV/c )2

Mγ γ (GeV/c )Slight difference of the π 0 peak position between sectors of EMCal is observed and is
expected from misalignment of EMCal. The uncertainty was evaluated to be 1%. For
the final systematic uncertainty of the energy scale, 1% (misalignment) and 0.7% (massshift) was added in quadrature and 1.2% was assigned. This has negligible effect on ALL
since observed ALL is flat.

0.144
0.142
0.14

0.144
0.142
0.14

0.138

0.138

0.136

0.136

0.134

0.134

0.132

0.132

0.13
0

1

2

3

4

5

6

7

8

9

0.13
0

10

1

2

3

4

5

6

7

0.015

2

0.014
0.013
0.012
0.011

9

10

0.015
0.014
0.013
0.012
0.011

0.01

0.01

0.009

0.009

0.008

0.008

0.007
0

8

pt (GeV/c)σγ γ (GeV/c )2

σγ γ (GeV/c )pt (GeV/c)

1

2

3

4

5

6

7

8

9

10

pt (GeV/c)

0.007
0

1

2

3

4

5

6

7

8

9

10

pt (GeV/c)

Figure 4.22: π 0 peak position vs pT (Top) and π 0 peak width (Bottom) for PbSc (left)
and PbGl (right). Blue points are obtained with simulations, Red points are from realdata.

4.5. π 0 RECONSTRUCTION

a.u.

a.u.

83

0.1

0.06
0.08
0.05
0.06

0.04
0.03

0.04

0.02
0.02
0.01

pT =1.0 - 1.5 GeV/c
0.1

0.2

0.3

0.4

pT =1.0 - 1.5 GeV/c
0.5

0.6

0
0

0.7
0.8
0.9
1
Energy Asymmetry

0.07

a.u.

a.u.

0
0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8
0.9
1
Energy Asymmetry

0.5

0.6

0.7
0.8
0.9
1
Energy Asymmetry

0.06

0.06
0.05
0.05
0.04

0.04

0.03

0.03

0.02

0.02
0.01

0.01

pT =2.5 - 3.0 GeV/c

0
0

0.1

0.2

0.3

0.4

pT =2.5 - 3.0 GeV/c
0.5

0.6

0.7
0.8
0.9
1
Energy Asymmetry

0
0

0.1

0.2

0.3

0.4

Figure 4.23: Energy asymmetry α for pT = 1.0 − 1.5 GeV/c (Top) and pT = 2.5 −3.0 GeV/c. Left plots are for PbSc and right plots are for PbGl. Blue points are obtainedwith simulations, and red points are from real data. The decrease in high α for pT =1.0 − 1.5 GeV/c is due to the minimal energy cut for single photons.

4.5.6

Reconstruction of π 0

As explained in the beginning of this section, π 0 yields are required for the asymmetry
calculations. The π 0 mesons are detected via two photon decay and the photons weredetected with PHENIX EMCal.
In this analysis, invariant mass spectrum was obtained for any pairs of clusters in
EMCal in an event. A peak at π 0 mass in the invariant mass spectrum was identified
as π 0 . The following criteria were applied for clusters or cluster pairs to reduce thebackground.
• Minimal photon energy cut,
• Shower profile cut,
• Energy asymmetry cut (α),

84

CHAPTER 4. ANALYSIS• Trigger tile matching for the higher energy cluster.

Minimal photon energy cuts were applied to reduce combinatorial background fromvery low energy clusters. The threshold was 0.1 GeV (0.2 GeV) for PbSc (PbGl). Higher
threshold for PbGl is due to the larger noise in PbGl. Energy distribution in a cluster iscompared with shower profile which is parametrized in the test experiment. Based on theχ2 test, clusters which has probability of less than 0.02 are discarded. Energy asymmetryis expected to be flat where trigger bias is small, while combinatorial background tends
to have large energy asymmetry due to large number of low energy clusters. Energy
asymmetry cut |α| < 0.8 was applied to further reduce the background. The backgroundratios for various cut conditions are summarized in Table 4.4. The background ratios forthe cuts applied for the analysis is in column (a).
Two-photon invariant mass spectrum for each pT bins are shown in Fig.4.24. In
addition to a peak corresponds to π 0 , there exists another peak near ∼ 0 GeV/c2 . The
peak originates from cosmic ray and hadrons and is discussed in detail in Sec. 4.5.8.
pT
Background ratio(GeV/c) (a)(b)(c)
1.0 − 1.5 35% 37% 46%
1.5 − 2.0 17% 20% 31%
2.0 − 2.5 10% 13% 27%
2.5 − 3.0 5.5% 8.8% 36%
3.0 − 4.0 3.9% 8.7% 57%
Table 4.4: The background ratio for different cuts. (c) with minimal photon energy cut,and trigger tile matching. (b) with shower profile cut in addition to the cuts imposed on(c). (a) with energy asymmetry cut in addition to the cuts imposed on (b). (cuts used
for the final results.)

4.5.7

EMCal stability

EMCal stability was investigated with run-by-run variation of π 0 identification. Figure 4.25a) displays run-by-run variations of π 0 peak position. A gain drift can be seen
and the π 0 mass increased by ∼2 MeV towards the end of the run. However, the peaks arewithin ± ∼ 2 MeV and we use ±25 MeV mass window. Therefore no further run-by-run
correction was applied. Figure 4.25b) and c) show the run-by run variations of the width
and the background ratio of π 0 peak. They are stable for the runs and no runs were
excluded by the EMCal stability check.

4.5. π 0 RECONSTRUCTION

Diphoton invariant mass pt:1.00-1.50
240

×10

3

85

Diphoton invariant mass pt:1.50-2.00
80000

220

70000

200
180

60000

160
140

50000

120

40000

100
80

30000

60

20000

40
10000

20
0
0

0.1

0.2

0.3

0.4
0.5
0.6
0.7
Invariant mass [GeV/c2 ]Diphoton invariant mass pt:2.00-2.50

0
0

0.1

0.2

0.3

0.4
0.5
0.6
0.7
Invariant mass [GeV/c2 ]Diphoton invariant mass pt:2.50-3.00

24000
22000
6000

20000
18000

5000

16000
14000

4000

12000
3000

10000
8000

2000

6000
4000

1000

2000
0
0

0.1

0.2

0.3

0.4
0.5
0.6
0.7
Invariant mass [GeV/c2 ]

0
0

0.1

0.2

0.3

0.4
0.5
0.6
0.7
Invariant mass [GeV/c2 ]Diphoton invariant mass pt:3.00-4.00

2500

2000

1500

1000

500

0
0

0.1

0.2

0.3

0.4
0.5
0.6
0.7
Invariant mass [GeV/c2 ]Figure 4.24: Two-photon invariant mass spectra. Both PbSc and PbGl types were used
to obtain the distributions.

86

CHAPTER 4. ANALYSIS

π0 BG/(sig+BG)a)

π0 width (GeV/c 2 )π0 mass (GeV/c 2 )

0.162
0.152
0.142
0.132
0.122

0.112
205.2
0.38
0.36 c)
0.34
0.32
0.3
0.28
0.26
0.24
0.22
0.2
205.2

205.9

×10
206.6
run number

3

×10
206.6
run number

3

205.9

0.014

b)

0.01320.01240.01160.0108
0.01
205.2

×10
205.9
206.6
run number

3

Figure 4.25: Run-by-run variation of π 0 identification. The horizontal axis is the runsequence number for the three plots. a) mass of π 0 . The range of the vertical axis
corresponds to the mass window used to identify π 0 . b) width of π 0 . c) background ratio.
They are stable for the whole run.

4.5.8

Discussion on the background peak

As shown in Fig. 4.24 there exists a peak below π 0 mass and they move towards π 0 peaks
with increasing pT . The peak comes from hadrons and cosmic rays. Hadrons and cosmic
rays make wider shower than photons and sometimes make two maxima in a cluster. Suchclusters are divided into two by the clustering algorithm, which makes a pair of clusters
with the same distance (thus opening angle). Since the width of a tower corresponds toan angle of 0.011[rad], the angle between the splitted towers is expected to be 0.011[rad]×2 ∼ 1.3 degrees. Figure 4.26 shows the two-photon opening angle distribution. The
opening angle of π 0 decreases with increasing pT while the opening angle of cosmic or
hadron events stay at the same expected angle.
To support the statement above, Time of Flight(TOF) distributions are shown for two
selected pT bins in Fig. 4.27. The black line shows the TOF distribution inside the π 0 mass
window, and the red line shows TOF under the cosmic ray and hadron background peakaround ∼ 0 GeV/c2 There are two components. Please note that the resolution of TOF
is limited to ∼ 3 nsec since BBC time zero subtraction was not done. In addition, timeof flight quality assurance is not complete. The plot is to show for a rough qualitative
statement only. As in the Fig. 4.27, one is collision related and the other is collision

4.5. π 0 RECONSTRUCTION87

pi0 opening angle ( PbSc )0.07

pt:3.50-4.00

0.06

pt:3.00-3.50

cosmicevents+hadrons0.05

π0pt:2.50-3.00

0.04

pt:2.00-2.50
pt:1.50-2.00
pt:1.00-1.50
pt:0.75-1.00
pt:0.50-0.75

0.03
0.02
0.01
0
0

2

4

6

8

10

12

14

16
18
20
open angle [deg]Figure 4.26: Opening angle distribution for each pT bin. (PbSc)
unrelated. The one with collision is from hadrons and it has longer TOF than photons.
The other component which is independent of collisions comes from cosmic rays. (Thepeak at −60 nsec corresponds to the events with TOF measurement failure or out of therange.)
hist_tof1 Good Bunch pi0_pt3.00-4.00_sig_PbSc

hist_tof1 Good Bunch pi0_pt2.00-2.50_sig_PbSc
10

4

10

3

10

3

10 2
10 2
10
10

1

1
-80

-60

-40

-20

0

20

40

60
80
TOF [nsec]

-80

-60

-40

-20

0

20

40

60
80
TOF [nsec]

Figure 4.27: TOF distributions for photon pairs within the π 0 mass window(black), andfor cosmic and hadron peak. Left: pT = 2.0 − 2.5 GeV/c. Right: pT = 3.0 − 4.0 GeV/c.

4.5.9

Cosmic ray event under π 0 signal window

As discussed in Sec. 4.5.8, the peak near ∼0 GeV/c2 in a two-photon invariant mass spectrum comes from cosmic ray and hadron events. The hadron events are collision related

88

CHAPTER 4. ANALYSIS

while the cosmic ray events are collision un-related. To estimate the amount of cosmic ray
background under the π 0 peak, two-photon invariant mass spectra were obtained from thenon-colliding bunches and is displayed in Fig. 4.28. The black histograms show invariant
mass spectra from the colliding bunches, the red histograms are from non-colliding bunch
crossings, scaled by the number of bunches. (The number of colliding bunches / thenumber of non-colliding bunches) Background in the lowest pT (1.0 − 1.5GeV/c) is ∼0.1%
Background in the highest pT (3.0 − 4.0GeV/c) is ∼ 1% or less. Cosmic ray events arethought to have zero asymmetries thus the effect on the measured asymmetry is negligible.
At the highest pT , the peaks near 0 GeV/c in black and red histograms match well. Thusin the highest pT , cosmic ray events dominate in the peak. For pT = 1.0 − 1.5 GeV/c, thepeak near ∼ 0 GeV/c2 is larger in PbSc than that in PbGl, and hadron events dominate
since PbGl detects Čerenkov radiation and is less sensitive to hadron events than PbSc.
pT = 1.0 - 1.5 GeV/cpT = 3.0 - 4.0 GeV/c
350

50000

Cosmic ray event ratio
inside the window

PbSc

40000

0.0012  0.0001

30000

Cosmic ray event ratio
inside the window

300
250

0.0080  0.0025

200

Signal window

Signal window

150

20000

100
10000

0

50
0

0

0.05

0.1

0.15

0.2

0.25

0.3

0

0.05

0.1

0.15

0.2

0.25

0.3

Diphoton invariant mass [GeV/c2]

Diphoton invariant mass [GeV/c2]

2200

Cosmic ray event ratio
inside the window

2000
1800

PbGl

1600

0.0011  0.0003

1400
1200

Cosmic ray event ratio
inside the window
0.0081  0.0058

60
50
40

Signal window

1000

70

Signal window

30

800
600

20

400
10

200
0

0

0.05

0.1

0.15

0.2

0.25

0.3

Diphoton invariant mass [GeV/c2]

0

0

0.05

0.1

0.15

0.2

0.25

0.3

Diphoton invariant mass [GeV/c2]

Figure 4.28: Two-photon invariant mass spectrum for pT = 1.0 − 1.5 GeV/c and pT =3.0−4.0 GeV/c, in PbSc and PbGl. The black histograms are for the events in the colliding
bunch crossings, and the red ones are for that in the non-colliding bunch crossings, scaledby the number of bunch crossings.

4.5. π 0 RECONSTRUCTION

4.5.1089

Vertex cut difference in π 0 and BBC trigger

The high-pT photon trigger was used to collect π 0 sample, while relative luminosity is
calculated with the BBC trigger counts. The vertex position cuts are slightly different
between them. Therefore, the effect of the cut difference on the measured ALL is estimatedin this subsection.
The vertex cut in the BBC trigger is implemented by utilizing the time difference
between the hits in BBCN and BBCS as explained in Sec. 3.4.1. The high-pT photon
trigger does not have explicit vertex cut. Instead, the z axis is surrounded by the central
magnet for |z| > 41 cm, thus the measured π 0 has a vertex cut-off but slightly different
from the vertex cut in the BBC trigger.
We define a ratio r = Nπ0 +BG /NBBC , where Nπ0 +BG is the counts for the signal window
in two-photon invariant mass spectrum for pT > 1 GeV/c which is defined in Sec. 4.6.1,
and NBBC is the BBC trigger counts. The ratio r can be written (in case there is novertex width variations) as r = C(1 + εLL sgn(PB PY )) where εLL is the raw asymmetryof π 0 + BG and it relates to the double helicity asymmetry as ALL = εLL/(PB PY ). r
depends not only on spin, but also the vertex width as discussed in Sec. 4.4. The purposeof the thesis is to measure ALL that is the dependence of r on spin. In this subsection,
the dependence of r on the vertex width is estimated with a simple Monte Carlo (MC).
The obtained dependence is utilized to estimate the overall effect on ALL . Necessary
ingredients for the MC are the z (vertex position) dependence of π 0 detection efficiency
and the BBC trigger efficiency.
The z dependence of the BBC trigger efficiency is expressed as
fBBC (z) = εBBCno−vcut (z) · εBBCvcut (z).

(4.25)

where εBBCno−vcut (z) is the z dependent BBC detection efficiency without vertex cut andεBBCvcut (z) is the z dependent BBC vertex cut efficiency. εBBCno−vcut (z) is measured andwell reproduced by a Gaussian with sigma of 95 ± 10 cm as described in Appendix A.
εBBCvcut (z) is measured by comparing the counts in BBC no vertex cut trigger with andwithout the BBC trigger fired. The results are displayed in Fig. 4.29 with a fit function.
It can be well approximated by a combination of two error functions
2
Erf(x) = √
π

Z x
0

!

exp(−t2 )dt

z − p1z − p3εBBCvcut (z) = p0 Erf √/2 · 1 − Erf √2p2
2p2(4.26)!!

/2.

(4.27)

where p0 , p1 , p2 , and p3 are the fit parameters. p0 is the normalization, p1 and p3 are
the vertex cut edge on negative and positive z sides respectively. p2 is the resolution of
the vertex cut. For the MC, the obtained histogram is utilized instead of the fit function
since the χ2 is not good. But the results did not change anyway.

90

CHAPTER 4. ANALYSIS

vertex cut efficiency

ratio
EntriesMean
RMS
χ2 / ndf
Prob
p0
p1
p2
p3

1
0.8

1262950-0.6846
22.73
1930 / 196
00.994 0.000-36.5 0.0
8.107 0.015
35.16 0.02

0.6
0.4

0.2
0
-100

-80

-60

-40

-20

0

20

40

60

80

100

BBC z vertex [cm]Figure 4.29: BBC trigger vertex cut efficiency.
The z dependence of π 0 detection efficiency with BBC z vertex reconstructed, is expressed as(4.28)
fπ0 ,BBC (z) = επ0 (z) · εBBCno−vcut (z).
where επ0 (z) is the z dependent π 0 efficiency which includes the acceptance of the EMCal.
Figure 4.30 displays the obtained επ0 (z).
The π 0 counts for |z| > 50 cm is only (7.17 ± 0.095) × 10−3 for the same helicity
combinations and (7.23 ± 0.095) × 10−3 for the opposite helicity combinations. They are
consistent within the statistical uncertainty. The statistical uncertainty propagates to
ALL as 9.5 × 10−5 /hPB PY i ∼ 4.1 × 10−4 thus it is negligible compared to the assigned
systematic uncertainty from the relative luminosity. Therefore, the following discussion
only consider |z| < 50 cm. Correcting BBC efficiency mentioned above only increase theamount of π 0 counts for |z| > 50 cm about 20% in counts thus does not change the results.
The dependence of r on vertex width is estimated by utilizing a simple MC with
the information obtained above. Input is a Gaussian distribution which simulates theobserved vertex distribution in BBC no vertex triggered events. As shown in Fig. 4.5,
the observed vertex distribution is well reproduced by a Gaussian for |z| < 100 cm. The
probability of π 0 to be observed for a randomly obtained z position is calculated by
επ0 (z)/εBBCno−vcut (z). And the probability of BBC counts to be observed is calculated byεBBC (z). The absolute efficiency is not the source of systematic uncertainty. Therefore,
the probability is made to be 100% for εBBCno−vcut where the probability is the highest
and 80% for επ0 not to exceed 100% by εBBCno−vcut correction. Figure 4.31 shows thevertex width dependence of the ratio between π 0 counts and the BBC trigger counts(with vertex cut) obtained by the simple MC. The measured vertex width is within theefficiency

4.5. π 0 RECONSTRUCTION

91

0.00250.002

0.00150.001

0.0005

0
-100

-80

-60

-40

-20

0

20

40

60

80

100

BBC z vertex[cm]Figure 4.30: z vertex dependence of π 0 relative efficiency.
range (55 < σ < 65 cm) and the ratio is linearly dependent on the width. The dependence
is obtained to be ∼ 3.6 × 10−4 /cm. (3.2 × 10−4 /0.88).
The bunch by bunch width for each fill were weight-averaged to obtain the widths forhelicity same and opposite combination separately to estimate the overall effect on ALL .
Figure 4.32 displays the difference of width in the different helicity combinations. The
difference is consistent with zero and obtained to be 0.053 ± 0.049 for the longitudinal run
period. Taking one σ, the uncertainty propagates to ALL as ∼ 0.36/hPB PY i ∼ 1.6 × 10−4 .
Similar results were obtained for the background window. The uncertainty is one ordersmaller than the uncertainty from the relative luminosity. Therefore, the vertex cut
difference have negligible effect on the measured ALL .

92

CHAPTER 4. ANALYSIS

χ2 / ndf

13.6 / 10

r

Prob

0.890

0.1922p0

0.8685 ? 0.0005531

p1

0.0003176 ? 9.131e-06

0.889
0.888
0.887
0.886
56

58

60

62

64
66
vertex width [cm]Difference of vertex width [cm]Figure 4.31: a MC results of width dependence.

χ2 / ndf
8.46 / 10
Prob
0.584
p0
0.05295 ? 0.04913

0.8
0.6
0.4
0.2
-0

-0.2
-0.4
-0.6
-0.8
8030

8035

8040

8045

8050

8055

8060

Fill number

Figure 4.32: difference in vertex width.

4.6. SPIN ASYMMETRIES

93

4.6

Spin asymmetries

4.6.1

Calculation of the asymmetries

The goal of the thesis is to extract double helicity asymmetry ALL of π 0 . In addition to
ALL , double transverse spin asymmetries AT T , and single longitudinal spin asymmetries
AL were also obtained for a study on systematic uncertainties. AL is defined in Sec. 4.6.4.
We take ALL as an example to explain the procedure to extract asymmetries since theprocedures are the same for AT T and AL .
The double helicity asymmetry ALL is calculated as
1 N++ /L++ − N+− /L+−PB PY N++ /L++ + N+− /L+−
1 N++ − RN+−=
where
PB PY N++ + RN+−

(4.29)

ALL =

R=

L++
L+−

(4.30)PB (PY ) is the polarization of the Blue (Yellow) beam and N++ (N+− ) is the particle yieldin the same (opposite) helicity bunch crossings, and R is the relative luminosity between
bunches with the same and opposite helicities. π 0 is identified by constructing invariant
mass of two decay photons and the background under π 0 peak is indistinguishable fromthe signals. The amount of π 0 signals could be obtained in principle by a fit for different
polarization sign separately and ALL can be directly calculated for π 0 signals. However,
ALL must be calculated run-by-run or fill-by-fill as explained in Sec. 4.5.1. Therefore, themethod suffers from π 0 yield-extraction uncertainty. Instead, the asymmetry for π 0 and0background, AπLL+BG , was calculated and the contribution on the asymmetry from the0π0background, ABG
LL , was subtracted to obtain the physics asymmetries of π , ALL .
0
π +BG
ALL
were calculated using the yields in signal window, which is defined as themass range 137 ± 25 MeV/c2 . 1 The width corresponds to ∼ 2 σ of the π 0 peak. The
0background asymmetry ABG
LL under π peak cannot be directly measured. It is replacedby the background asymmetry of the side band 177 − 217 MeV/c2 , assuming that thebackground asymmetry under the π 0 peak is the same as that in the side band. Figure 4.33
illustrates the signal window and the background window used for asymmetry calculations.
The lower side band was not used in this analysis to avoid possible effect from the cosmic
ray and hadron background peak. See Sec. 4.5.8 for the cosmic ray and hadron backgroundpeak. It was confirmed that the asymmetries for the lower and higher side band areconsistent with zero.
The subtraction is performed by the following formula:
0

0
AπLL

1

π +BG
ALL
− rABG
LL
=
1−r

where

r=

NBG
,
Nπ0 +BG(4.31)The peak position of π 0 mass is ∼ 137 MeV/c2 (which is slightly higher than the world-average ofπ mass) due to the energy smearing effect as explained in Sec. 4.5.5
094

CHAPTER 4. ANALYSIS

π0

δALL =r

0

δAπLL+BG2+ r 2 (δABG
LL )

1−r

2

.

(4.32)The procedure to extract the background ratio r is explained later.
For a calculation of ALL , the data are required to have more than 10 counts for sameand opposite helicity bunch crossings for the statistical uncertainty of Poisson distribution
to be approximated by that of Gaussian distribution. This is just rejecting low statistics
runs and does not bias the measurement.

Signal window

Background window

0

0.05

0.1

0.15

0.2

0.25

0.3
2

Diphoton invarnant mass [GeV/c ]Figure 4.33: The ranges used for asymmetry calculations. The red area corresponds tothe signal window (signal + background), and the blue area describes the backgroundwindow.
The statistical uncertainties of ALL is calculated as:vu

1
2RN++ N+− ut δN++
δALL =PB PY (N++ + RN+− )2
N++

!2

δN+−
+
N+−

!2

δR
+
R

!2

.

(4.33)

The calculation of statistical uncertainty needs careful consideration since the numberof signals no longer obey the Poisson distribution. Let N sig be the number of signals,
N trig be the number of triggered events, and hki (hk 2 i) be the average number (squared)of signals in one event. N trig obeys the Poisson distribution and its
q statistical uncertainty
is approximated by that of Gaussian distribution as δN trig = Ntrig . The statisticaluncertainty of N sig is calculated asv
u 2√
u hk isigδN = t× N sig .

hki(4.34)

4.6. SPIN ASYMMETRIESr95

hk 2 iis referred to as the enhancement factor since the statistical uncertainty is thathkiof a Poisson distribution times the enhancement factor. They were obtained for each pT
bins and were summarized in Table 4.5.

pT [GeV/c]
1.0 − 1.5
1.5 − 2.0
2.0 − 2.5
2.5 − 3.0
3.0 − 4.0

q

hk 2 i/hki
Signal window Background window
1.09
1.06
1.04
1.03
1.02
1.03
1.02
1.02
1.02
1.02

Table 4.5: The enhancement factor for the statistical uncertainties for each pT bins.

4.6.2

Background ratio in signal window

Background ratio in signal window, which is defined in Sec. 4.6.1, is necessary to subtract the background asymmetry under π 0 mass peak. The background ratio is definedBG
as r = N N, where NBG and Nπ0 are the number of background and π 0 counts
π 0 +NBG
in the signal window. (It was introduced in Eq. 4.31.) To evaluate the backgroundratio, the invariant mass spectra were fitted with a combination of three functions:fcos (x),fcomb (x), and fsig (x). fcos (x) represents the tail of the cosmic ray and hadron
events near ∼ 0 GeV/c2 and exponential was assumed. fcomb (x) describes combinatorial
background and a quadratic function or a linear function was assumed. Quadratic functions were assumed for pT < 2.5 GeV/c since low pT spectra were not well reproducedby linear functions. fsig (x) represents π 0 mass peak and Gaussian was assumed. Figure 4.34 displays an example of the fit. The black curve shows the sum of the functions
while red, purple and blue curves describe the contribution from the functions separately.
The curves were drawn for the range used for the fit. The π 0 mass peak was not well
reproduced by a Gaussian shape which results in large reduced χ2 , ∼19 for the lowest
pT and ∼ 3 for the highest pT bin. Thus the amount of signal and background in thesignal window is simply obtained by counting the measured yields, while the amount of
the background is estimated by the fit function. The statistical uncertainties assigned
from the fit were enlarged by square root of the reduced χ2 of the fit and assigned assystematic uncertainties. The difference between the counts and integral of the fit function in the signal window is within the assigned systematic uncertainties. The fit range
was also varied and the results were compared but the difference was within the assigned
systematic uncertainties.

96

CHAPTER 4. ANALYSISThe obtained background ratio and the systematic uncertainties for each pT bin aresummarized in Table. 4.6.
35000
30000
25000
20000
15000
10000
5000
0
0

0.05

0.1

0.15
0.2
0.25
Diphoton invariant mass [GeV/c 2]

Figure 4.34: Background ratio estimate by a fit. The functions fcos (x),fcomb (x), andfsig (x) were used. See text for the definitions. The black curve shows the sum of thefunctions while red, purple and blue curves describe the contribution from the functions
separately. The curves were drawn for the range used for the fit.

pT [GeV/c]
1.0 − 1.5
1.5 − 2.0
2.0 − 2.5
2.5 − 3.0
3.0 − 4.0

BG ratio in signal window
0.350 ± 0.008
0.170 ± 0.010
0.100 ± 0.015
0.055 ± 0.002
0.039 ± 0.004

Table 4.6: Background (BG) ratios and their systematic uncertainties in signal window
for each pT bins.

4.6.3

Average pT

The calculated asymmetries are plotted at the average pT , hpT i, for each pT bin. As in
the case of the asymmetries, it requires background correction to obtain the average pTof π 0 . It is obtained with the following formula:0
hpTπ +BG i − rhpBG
T i
π0
hpT i =

1−r

where

r=

NBG
Nπ0 +BG(4.35)

4.6. SPIN ASYMMETRIES97

The obtained average pT for each pT bin is listed in Table.4.7. The statistical uncertainties
and the effect from the systematic uncertainty of background ratio obtained in Sec. 4.6.2
are negligible compared to the absolute energy scale uncertainty of 1.2%.
pT [GeV/c]
1.0-1.5
1.5-2.0
2.0-2.5
2.5-3.0
3.0-4.00hpπT i [GeV/c]
1.21
1.70
2.20
2.70
3.32

Table 4.7: The average pT for each pT bin.

4.6.4

Asymmetries

Double helicity asymmetries ALL
The double helicity asymmetry ALL of π 0 were obtained as explained in Sec. 4.6.1. The
asymmetries were calculated run by run for pT < 3 GeV/c and fill by fill for pT > 3 GeV/c
as explained in Sec. 4.5.1. Figure 4.35 2 shows run-by-run ALL results for even bunch
crossings, signal window, and pT = 1.5 − 2.0 GeV/c as an example. It was fitted toπ 0 +BG
a constant and ALL
was obtained. ABG
LL was also obtained in the same way and0subtracted using π purity measured as in Sec. 4.6.2. The results of ALL were calculated
separately for even and odd bunch crossings, and they are averaged to obtain the final
results. The final results of ALL for even and odd bunch crossings separately and the
combined results are displayed in Fig. 4.36.
Double transverse spin asymmetries AT T
Our purpose is to measure ALL but the measured ALL is affected by the double transverse
spin asymmetry AT T through the remaining transverse components of the beam polarizations as discussed in Sec. 4.7.2. Thus AT T was also obtained in this analysis. AT T shouldhave azimuthal angle dependence and is proportional to cos(2φ). Therefore, measured
asymmetry Ameas
should be corrected for angle dependence to obtain the physics asymTT
metry AT T . But as explained in Sec. 4.7.2, the correction is not necessary for this analysis
without angle-dependence
and the correction was not performed. The measured AmeasTTcorrection was displayed in Fig. 4.37 and listed in Table. 4.8. The gray band indicate thesystematic uncertainty from the relative luminosity as discussed in Sec. 4.4.
2

Between fills, ∼ 100 run numbers are consumed for detector calibrations and diagnostics of DAQ.
Therefore, gaps of ∼ 100 runs appear in the figure.

98

CHAPTER 4. ANALYSISALL

A LL for signal window for even bunch crossings pT = 1.5-2.0 GeV/c

0.2

χ 2/NDF = 52.1/63
A LL = -0.0040 + 0.0071

0.1
0
-0.1
-0.2
205400205600205800206000 206200

206400206600Run number

0

π +BG
Figure 4.35: Run-by-run results of ALL
for even bunch crossings at pT = 1.5 −2.0 GeV/c.

A LL
0.08
0.06
0.04
0.02
0
-0.02
-0.04

Even/odd combined
Even bunches-0.06

Odd bunches

-0.08
-0.1

1

1.5

2

0

2.5

3

3.5
4
p T [GeV/c]

Figure 4.36: The results of AπLL for even (red) and odd (blue) bunch crossings and the
combined results (black).

4.6. SPIN ASYMMETRIES

99

A TT
0.15
Even/odd combined
Even bunches

0.1

Odd bunches

0.05

0

-0.05

-0.1

1

1.5

2

2.5

3

3.5
4
p T [GeV/c]

0

Figure 4.37: Results of AπT T for even (red) and odd (blue) and the combined results(black). The gray band shows the systematic uncertainty from the relative luminosity.

pT bin1.0 − 1.5
1.5 − 2.0
2.0 − 2.5
2.5 − 3.0
3.0 − 4.0

hpT i
1.22
1.70
2.20
2.70
3.33

0

AπT T
4.0 × 10−4 ± 6.8 × 10−3
3.0 × 10−2 ± 9.1 × 10−3
9.4 × 10−3 ± 1.6 × 10−2
−1.6 × 10−2 ± 3.0 × 10−2
2.3 × 10−2 ± 5.0 × 10−20

Table 4.8: Results of AπT T .

100

CHAPTER 4. ANALYSIS

Single spin asymmetries AL
Single spin asymmetry AL is defined as
AL ≡ −

1 σ+ − σ−,P σ+ + σ−(4.36)

where P is the polarization of the polarized beam, and σ+ and σ− are the cross sections
for the helicity + and − collisions respectively. Non-zero single spin asymmetry requires
parity violation in physics
to be very√
√ process, that is the weak interactions. It is expected−5small (AL < 10 ) at s =200 GeV [77] and be smaller than that at s = 62.4 GeV.
It is experimentally calculated as
AL = −

1 N+ − RN−,
P N+ + RN−

where.

Rsingle =L+
L−(4.37)

N+ (N− ) is the particle yields in the collisions with helicity + (−) state in one beam andeither helicity state in the other beam. Rsingle is the relative luminosity for helicity + and− beam collisions. It was measured in a similar way as described in Sec. 4.6.1. Since bothBlue and Yellow beams are polarized, AL can be measured for each beam separately.
Figure 4.38 displays the results of AL for Blue and Yellow beams separately and they
are consistent with zero as expected.
A L Blue beam.

A L Yellow beam.

0.04

0.04
Even/odd combined

0.03

Even/odd combined

0.03

Even bunches

Even bunches

Odd bunches

0.02

Odd bunches

0.02

0.01

0.01

0

0

-0.01

-0.01

-0.02

-0.02

-0.03

1

1.5

2

2.5

3

3.5
4
p T [GeV/c]

-0.03

1

Blue beam.

1.5

2

2.5

3

3.5
4
p T [GeV/c]

Yellow beam.
π0

Figure 4.38: The results of AL for the Blue (left) and Yellow (right) beams. Red points
are for even, blue for odd, and black for the combined results.

4.7

Systematic uncertainties

Sources of systematic uncertainties on the analysis are discussed in this section. Bunch
shuffling technique was utilized to confirm the validity of the assigned uncertainties and4.7. SYSTEMATIC UNCERTAINTIES

101

is described in Sec. 4.7.4. The assigned systematic uncertainties are summarized inSec. 4.7.6.

4.7.1

Beam polarizations

The uncertainty of ALL due to the beam polarizations propagates as,
δApol
LL =

δ(PB PY )
ALL .
PB PY

(4.38)The uncertainty correlates over all pT , thus behave as a scale uncertainty, where the central
values and the uncertainties are changed by the same factor. The systematic uncertainty
of 13.9% was assigned for the beam polarizations. The major contributor to the systematic
uncertainty is the statistical uncertainty in the H-jet polarimeter measurement.

4.7.2

Beam polarization orientations

The beam polarization orientations were measured with the PHENIX local polarimeteras described in Sec. 3.2.3. The fraction of the transverse components are hPT /P iB =0.11 ± 0.15 for the Blue beam, hPT /P iY = 0.11 ± 0.12 for the Yellow beam. Thus thebeams are fully longitudinally polarized within the uncertainties and the longitudinalcomponents are hPL /P iB = 1.000 − 0.023 for the Blue beam, hPL /P iY = 1.000 − 0.022
for the Yellow beam.
The beam polarization orientations affect in two ways: The effect from the uncertaintyof the longitudinal components of the beams, and the effect from AT T through transverse
components of the beams.
The effect from the uncertainty of the longitudinal components of the beams
The ALL was calculated as the polarization is fully longitudinally polarized. The uncertainty of ALL from the uncertainty of longitudinal components of the beam polarizations,δAuncert.long.
, is calculated as
LLδAuncert.long.
LL

= δ

1

′
′
PB,L
PY,L

ALL
=
′
′
PB,L PY,L
P

P

1
v
u
u
t

!

Ameas
LL

′
δPB,L
′
PB,L

!2

(4.39)
′
δPY,L
+
′
PY,L

!2

,

(4.40)′′and PY,L= Y,L. Thus a systematic uncertainty of 3.2% (2.3% ⊕where PB,L
= B,L
P
P
2.4%) was assigned for the effect. The uncertainty is pT -correlated and behaves as a scale
uncertainty.

102

CHAPTER 4. ANALYSISThe effect from the AT T through the transverse components of the beams
In addition, the measured ALL is affected by AT T through the transverse components ofthe beams. The effect on ALL is:δALL =

PB,T PY,T meas
AT T .
PB PY(4.41)The effect was calculated with the measured AT T and the polarizations. The central values plus one σ were assigned as the systematic uncertainties and is summarizedin Table 4.9. The assigned uncertainties are smaller compared to the major systematic
uncertainty from relative luminosity (besides the scale uncertainty from the beam polarization).

4.7.3

Relative luminosity

The uncertainty of relative luminosity is assigned as δALL = 1.4 × 10−3 as described inSec. 4.4.

4.7.4

Bunch shuffling

Bunch shuffling is a technique to evaluate the validity of the uncertainty of the asymmetry
calculations. The asymmetry calculations were performed in the same way as the true
asymmetry calculations but with the polarization signs randomly assigned to the bunch
crossings. Such calculations were repeated and accumulated. Figure 4.39 displays an
example of the shuffled ALL variations. They are fitted to Gaussian to obtain σ for the
shuffled variations. They are compared with the uncertainty for the true asymmetries.
Figure 4.40 display the ratio between the σ of the shuffled ALL variations and thestatistical uncertainty of true ALL versus pT . The statistical uncertainties were well
reproduced by the shuffled results and no further systematic uncertainties were foundin this analysis.

4.7.5

Double collision effect

The BBC trigger cannot distinguish two collisions if they occur at exactly the same
bunch crossing. Double collision would make measured luminosity lower than the theactual luminosity. Thus it is important to estimate the double collision rate.
The total cross section (σtot ) and the elastic scattering cross section(σel ) of pp collisions√
at s = 62.4 GeV are σtot ∼ 43 mb, and σel ∼ 7.5 mb, respectively [78]. Thus theinelastic scattering cross section becomes σinel ∼ 35.5 mb. The cross section of events
which the BBCs detect is σBBC = 13.7 mb, which is about 40% of the inelastic scatteringcross sections. The rate of the BBC trigger without vertex cut was at most 15 kHz

4.7. SYSTEMATIC UNCERTAINTIES

103

Shuffled ALL pT =1.0-1.5 GeV/chist_all_0_2

2000

Entries

10000

Mean

2.14e-05

RMS

0.003914

χ2 / ndf

6.558 / 12

Prob

1600

0.88542037 r 25.0

Constant

1.792e-05 r 3.924e-05

Mean
Sigma

1200

0.003915 r 0.000028

800
400
0
-0.03

-0.02

-0.01

0

0.01

0.02

0.03

Shuffled A LL

Figure 4.39: The shuffled ALL variations for the signal window and pT = 1.0 − 1.5 GeV/cin even bunch crossings.

σshuffling / σAσshuffling / σA

LL

1.2

LL

1.2

1.15

even bunch crossings

1.15

even bunch crossings

1.1

odd bunch crossings

1.1

odd bunch crossings

1.05

1.05

1

1

0.95

0.95

0.9

0.9

0.85

0.85

0.8

1

1.5

2

2.5

3

3.5

p

4
T

0.8

1

1.5

2

2.5

3

3.5

p

4
T

Figure 4.40: The ratio of the σ of the shuffled ALL variations and the statistical uncertaintyof true ALL for π 0 +background (left), and background (right) windows. Black circles arefor the even bunch crossings, and Red ones are for the odd bunch crossings.

104

CHAPTER 4. ANALYSIS√
at s = 62.4 GeV in Run 2006. Thus the real collision rate (inelastic scattering rate)
is at most 15/0.4 ∼ 38 kHz. Since RHIC has 9.4MHz ( 1/106 nsec ) basket of beamwith ∼100 out of 120 bunches filled, the single collision probability is ∼ 0.49%. The
collision probability obeys the Poisson distribution with parameter λ = 0.0049. Thus theprobability to have k collisions in a bunch crossing, pk , is calculated to bepk =e−λ λk.
k!

(4.42)

Therefore, probability to have more than one collision per bunch is ∼ 0.0012% and is
negligible.

4.7.6

Summary of the systematic uncertainties

The sources of systematic uncertainties are;• Relative luminosity.
• AT T contamination through transverse component of the beam polarizations.
• Background ratio in the signal window.
• (Scale uncertainty) Polarizations of the beams• (Scale uncertainty) Longitudinal components of the polarizations.
The last two items were treated separately since they are scale uncertainties. The
uncertainty of 14% (13.9% ⊕ 3.2%) was assigned. The systematic uncertainties are summarized in Table. 4.9. The systematic uncertainty is about 28% of the statistical uncertainty for the lowest pT where the statistics are the largest. The dominant systematic
uncertainty originates from the relative luminosity.
pT(GeV/c)1.0 − 1.5
1.5 − 2.0
2.0 − 2.5
2.5 − 3.0
3.0 − 4.0

AT T effect
0.87
10
3.9
6.9
11

Rel. lumi.
(×10−4 )
14
14
14
14
14

BG ratio
1.8
3.6
3.9
0.3
13

total
(×10−4 )
14
18
15
16
22

Table 4.9: Systematic uncertainties for ALL . The scale uncertainty of 14% which comesfrom the beam polarizations was not listed in the table.

Chapter 5
Results and discussions
0In√ this chapter, we present and discuss the results of π ALL in polarized pp collisions at
s = 62.4 GeV.
√
Although lower s has advantage in probing high x range, it should not be too low. In
fact, NLO pQCD calculation fail to describe low-energy fixed-target experiments. Therefore, it is important to confirm pQCD applicability since the polarized gluon distributions
are extracted based on pQCD. Before presenting and discussing the results of ALL , whichis the main subject of the thesis, the results of the cross sections are shown and the applicability of pQCD is discussed in Sec. 5.1. Then the results of ALL are presented anddiscussed in Sec. 5.2. Two global analyses on polarized PDFs which included our ALL
results are discussed in Sec. 5.3.

5.1

The cross section results

Figure5.1 presents the inclusive mid-rapidity π 0 invariant production cross sections at√
s = 62.4 GeV versus pT , from pT = 0.5 GeV/c to pT = 7 GeV/c [75]. They were
measured with the same data set as the asymmetries. An overall normalization uncertainty of 11% due to the uncertainty in absolute normalization of the luminosity is
not shown. The analyzed data sample with 0.76 × 109 BBC triggers corresponded to
55 nb−1 integrated luminosity. The measurements fall within the large spread of ISR
data [79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90]. A summary of the situation can befound in [17].
The data are compared to NLO and NLL pQCD calculations at a theory scale µ =pT , where µ represents equally-chosen factorization, renormalization, and fragmentation
scales [36]. See Sec. 2.3 about the scale. The NLL corrections extend the NLO calculations to include the resummation of extra “threshold” logarithmic terms to all orders in
αs . The log terms become important in the perturbative expansion at not very high energies because the initial partons have just enough energy to produce the high pT parton
105

CHAPTER 5. RESULTS AND DISCUSSIONS

3

E d σ /dp3 (mb GeV-2 c 3 )

106
10
1
10

10 -2
10

-3

10 -4
10

-5

10

-6

10 -7

4
(Data-QCD)/QCD

pQCD µ = pT
MRST2002 PDF; fDSS FF
NLL
NLO

-1

0

a)
b) NLO 2

4

6

8

4

6

8
p (GeV/c)20c) NLL

4
2
0
0

2

T

√
Figure 5.1: (a) The neutral pion production cross section at s = 62.4 GeV as a functionof pT (circles) and the results of NLO (solid) and NLL (dashed) pQCD calculations forthe theory scale µ = pT . (b) The relative difference between the data and NLO pQCD
calculations for the three theory scales µ = pT /2 (upper line), pT (middle line) and 2pT(lower line); experimental uncertainties (excluding the 11% normalization uncertainty)
are shown for the µ = pT curve. (c) The same as b) but for NLL pQCD calculations.
that fragments into a final pion. See Sec. 2.3 about the higher order corrections. The
MRST2002 parton distribution functions [39] and the fDSS set of fragmentation functions [91], which are extracted in NLO, are used in both NLO and NLL calculations.
We have previously
seen that the data are well described by NLO pQCD with a scale of√µ = pT at s = 200 GeV [92, 93, 15]. In√ contrast, NLO calculations with the same scale0
underestimate the π cross section at s = 62.4 GeV. However, it does not necessarilymean that NLO fails at this energy, since NLO calculation agree with data within thetheoretical uncertainties.
At the same time, it is known that NLO calculations are not always successful at
describing low energy fixed target data [94], while NLL calculations have been successful [35]. The NLL calculations have a smaller scale dependence and describe our data

5.2. RESULTS OF THE DOUBLE HELICITY ASYMMETRIES

107

well with µ = pT . The scale dependence is expected to be smaller when the effect fromtruncation in the perturbative expansion is smaller. However, as noted in [36], subleading
perturbative corrections to the NLL calculation may be significant. Therefore, we decided
not to choose one from another, and we show comparisons to both NLO and NLL at ascale of µ = pT .
The data and the pQCD calculations agree well at this energy and the data can beinterpreted in pQCD framework.

5.2

Results of the double helicity asymmetries

Figure 5.2 presents the measured double helicity asymmetry in π 0 production versus pT .
0
A scale uncertainty of 14% in AπLL due to the uncertainty in beam polarizations is notshown. The other systematic uncertainties are negligible, as discussed in the previous
chapters and checked using a technique to randomize the sign of bunch polarization.
Figure 5.2 also shows a set of ALL curves from pQCD calculations that incorporates
different scenarios for gluon polarization within the GRSV parametrization of the polarized parton distribution functions [95, 45]. GRSV-std corresponds to the best fit topolarized-DIS data. The other three scenarios in Fig. 5.2 (GRSV-max, ∆G = 0, and∆G = −G) are based on the best fit, but use the functions ∆g(xg ) = g(xg ), 0, −g(xg )at the initial scale for parton evolution (Q2 = 0.4 GeV2 ), where g(xg ) is the unpolarized gluon distribution, and ∆g(xg ) is the difference between the distributions of gluons
with the same and opposite helicity to the parent proton. In Fig. 5.2, we compare ourasymmetry data with both NLO and NLL calculations. Although ALL is smaller in NLL
calculations compared to that in NLO calculations, the√ difference is smaller than that at
Fermilab
fixed-target energies [36]. Similar to our s = 200 GeV results [13, 15], our√s = 62.4 GeV ALL data do not support a large gluon polarization scenario, such as
GRSV-max.
Figure √5.3 presents the measured ALL versus xT in π 0 production overlaid with theresults at s = 200 GeV [15]. Clear statistical improvement can be seen at higher xT .
For the measured pT range 2–4 GeV/c, the range of xg in each bin is broad and spans
the range xg = 0.06 − 0.4, as calculated by NLO pQCD [55]. Thus our data set extends
the sensitive xg range of ∆G and also overlaps the previous measurements, providing
measurements with the same xg but at a different scale.

CHAPTER 5. RESULTS AND DISCUSSIONS

A LL

108

0.1

0.08

ax

V-m
GRS

0.06
0.04

∆G= -G

0.02

GRSV-std

0
∆G=0

-0.02

NLO
NLL

-0.04
1

1.5

2

2.5

3

3.5
4
pT (GeV/c)√
Figure 5.2: The double helicity asymmetry for neutral pion production at s = 62.4
GeV as a function of pT (GeV/c). Error bars are statistical uncertainties, with the 14%
overall polarization uncertainty not shown; other experimental systematic uncertainties
are negligible. Four GRSV theoretical calculations based on NLO pQCD (solid curves)and on NLL pQCD (dashed curves) are also shown for comparison with the data (see text
for details). Note that the ∆G = 0 curves for NLO and NLL overlap.

5.3

Global analysis of polarized PDFs

Our results are already included in some of the global analysis. We introduce AAC results
in section 5.3.1, and DSSV results in section 5.3.2.

5.3.1

AAC global analysis

Asymmetry Analysis Collaboration (AAC) included the preliminary version of the resultsin their analysis [96]. The polarized gluon PDF by the AAC group is shown in Fig. 5.4.
AAC provided positively and negatively polarized gluon solutions separately in [43]. The
existence of two solutions can be understood by combining Eq. 2.38 and Eq. 2.2, andA LL

5.3. GLOBAL ANALYSIS OF POLARIZED PDFS

109

0.1
GeV
2
6
x(
ma
V
S
GR

Run2006 62.4GeV

0.08

Run2005 200GeV

0.06
0.04
GR

0.02

ax
m
SV

e
0G
0
(2

V)

)

0GeV)
0
2
(
d
t
s
GRSV
V)
d(62Ge
t
s
V
S
GR

0
-0.02

p (200GeV)
T

1

-0.04

2

3

4
1

0

0.02

0.04

5
1.5

6

7
2

0.06

8
2.5

0.08

pT (62.4GeV)
3

0.1

3.5

0.12

xT
√
Figure 5.3: Double helicity asymmetries for neutral pion production at s = 62.4 GeV
and 200 GeV as functions of xT . Error bars√are statistical uncertainties, with the 14%(9.4%) overall polarization uncertainty for s = 62.4 GeV (200 GeV) data which are
not shown. Two GRSV theoretical calculations based on NLO pQCD are also shown forcomparison with the data (see text for details.)
rewriting it as
ALL ∼

1
qq qq
gg
gq
(∆g∆g · âgg
+ ∆g∆qâgq
LL σ̂
LL σ̂ + ∆q∆qâLL σ̂ ) ,
σ

(5.1)

where ∆g and ∆q are the polarized PDFs for gluons and quarks respectively, and âLL(σ̂) are the partonic ALL (cross sections) for processes indicated by the superscripts. The
fragmentation functions were omitted for simplicity. It is a quadratic equation in terms of∆g and it makes ALL not sensitive to the sign of ∆g. (According to pQCD
√ calculations,contribution of qg exceeds that from gg for pT > 2 GeV/c (3 GeV/c) at s = 62.4 GeV(200 GeV) and thus high pT ALL have some sensitivity to the sign.)
AAC is the first polarized PDF analysis group to add uncertainty on PDF results. Assignment of uncertainties on PDFs is not straight forward since some of the experimental
uncertainties are correlated. PDF analysis is based on various experimental observables

110

CHAPTER 5. RESULTS AND DISCUSSIONS
pT [GeV/c]
1.0 − 1.5
1.5 − 2.0
2.0 − 2.5
2.5 − 3.0
3.0 − 4.0

hpT i [GeV/c]
1.21
1.70
2.20
2.70
3.32

background ratio
0.35
0.17
0.10
0.056
0.041

0

AπLL
(−1.1 ± 5.1) × 10−3(−10. ± 6.7) × 10−3
( 0.7 ± 1.2) × 10−2(−0.6 ± 2.2) × 10−2(−0.6 ± 3.5) × 10−2

Table 5.1: Results for π 0 ALL .
Data sets
DIS only
DIS + 62.4 GeV
DIS + 200 GeV
DIS + 62.4 GeV + 200 GeV∆G
0.47 ± 1.1
0.26 ± 0.39
0.37 ± 0.40
0.26 ± 0.31

Table 5.2: AAC results of the first moment of the polarized gluon PDFs obtained withvarious data sets.
from different experimental groups and treating correlated uncertainties in mathematically correct way is very difficult. Therefore, the statistical and systematic uncertainties
are added in quadrature, and all uncertainties are treated as uncorrelated, which overestimates the uncertainties. This also applies for DSSV analysis described in the next section.
AAC utilized Hessian method to assign uncertainty. It determines the uncertainty fromthe dependence of χ2 near its global minimum based on a Taylor expansion and keeping
only the leading term. This assumes a quadratic form in the displacements of all parameters from their optimum values. On the other hand, Lagrange multiplier method,which is used by DSSV described in the next section, does not use the assumption. It
was recently realized that the Hessian method tends to produce slightly larger uncertainty
compared to the Lagrange multiplier method for polarized gluon distribution. Therefore,
the uncertainty by AAC and DSSV cannot be compared directly, but the uncertainties
by the same group can be directly compared and discussed.
We only show positively polarized gluon solution in Fig. 5.4 since a negatively polarized
gluon solution is not available. The black curve shows the polarized gluon PDF (xg(x))
and its uncertainty obtained with DIS data only. It has a huge uncertainty as in the
figure. The blue curve and√ the blue shaded area show xg(x) and its uncertainty when0
the results of π ALL at s = 62.4 GeV are added in addition to the DIS data sets.
Significant improvement in the uncertainty, and slight decrease in the central value canbe observed. The figure is one of the proofs of the fact that the polarized pp collisions
work as a powerful tool to investigate the gluon polarization inside the proton. For thex∆g(x)5.3. GLOBAL ANALYSIS OF POLARIZED PDFS

111

DIS only
62 GeV
62+200 GeV

1.2
0.6

Q2=1 GeV 2

0
-0.6

-3

10

-2

10

-1

10

x

1

Figure 5.4: AAC results of the polarized gluon PDF. The black curves show the central
value and the uncertainties of the pol-gluon PDF obtained with DIS data sets only. The
blue curve and the shaded area shows the PDF obtained with DIS and 62.4GeV data.
The red curve and the shaded area shows the results obtained with DIS, 62.4GeV and
200GeV data. The blue and the red lines are overlapped.
√
red curve and the red shaded area, the results of π 0 ALL at s = 200 GeV in Run 2005
are also added. The uncertainty slightly decreased but the central shape does not show√
significant change from the one with s = 62.4 GeV data. The first moments of gluon
polarizations with the various data sets are summarized in Table 5.2.

5.3.2

DSSV global analysis

The DSSV group included the preliminary version of our results√in their global analysis [97]. It also includes PHENIX π 0 ALL preliminary results at s = 200 GeV in Run
2006 [98], STAR jet ALL [22] preliminary results, and semi-inclusive DIS data which issensitive to the flavor decomposition of the quark spin. Some authors of the DSSV groupwere involved in GRSV, whose results are compared with our measurements in Sec. 5.2.
Their polarized gluon PDF is shown in Fig. 5.5. The green band represent the uncertaintyof the PDF at the level of ∆χ2 = 1, and the yellow band corresponds to that at the levelof ∆χ2 /χ2 = 2%. In unpolarized PDF analysis, it is customary to assign uncertainties

112

CHAPTER 5. RESULTS AND DISCUSSIONS

with ∆χ2 /χ2 = 2% or 5% to account for large χ2 which may come from theoretical uncertainties or unaccounted experimental uncertainties. Therefore, uncertainty correspondsto ∆χ2 /χ2 = 2% is also provided. (The 2% increase in χ2 roughly corresponds to 2.8σ inthis case since the χ2 of the fit is χ2 = 392.6 with DOF= 441.)
The first moment of the DSSV best fit, and the integrals truncated at xmin = 0.001
with uncertainties at the levels of ∆χ2 = 1 and ∆χ2 /χ2 = 2% in Table 5.3. Below
xmin = 0.001, there is no constraining data set. Therefore, the deviation between the fulland the truncated integral is from extrapolation to x = 0.
In Fig. 5.5, GRSV-max, GRSV-std and GRSV-min (∆G = −G), which are introducedin Sec. 5.2, are overplotted for x∆g(x). The central value and uncertainty of DSSV ∆g(x)is smaller than the GRSV-std, which is the best fit in GRSV. And the uncertainty is
considerably smaller compared to the allowed range of GRSV parametrization.

0.2

d
V-st
S
R
G

0.1

DNS

ax

DSSV ∆χ=1
DSSV ∆χχ =2%

GRSV-mx∆g 0.3

GRS(∆G V-min
= -G
)

0
-0.1
-0.2

GRSV max. ∆g
GRSV min. ∆g

10

-2

10

-1

x

Figure 5.5: DSSV polarized PDFs. The green bands correspond to ∆χ2 = 1 and theyellow bands correspond to ∆χ2 /χ2 = 2%.
Figure 5.6 displays ∆g/g obtained by the DSSV together with that extracted from
Semi-inclusive DIS (SDIS) experiments by SMC [7], HERMES [5, 6], and COMPASS [9].
The SDIS results are extracted at LO and are based on Monte Carlo simulations, while theDSSV results since the DSSV are extracted at NLO. Therefore, they cannot be directly

5.3. GLOBAL ANALYSIS OF POLARIZED PDFS

113

xmin = 0
xmin = 0.001
2best fit∆χ = 1∆χ2 /χ2 = 2%
∆u + ∆ū
0.813
0.793 +0.011
0.793 +0.028−0.012−0.034+0.035∆d + ∆d¯−0.458 −0.416 +0.011−0.416 −0.025−0.009+0.059∆ū
0.036
0.028 +0.021
0.028 −0.059−0.020∆d¯−0.115 −0.089 +0.029−0.089 +0.090−0.029−0.080+0.010∆s̄
−0.057 −0.006 −0.012 −0.006 +0.028−0.031+0.702∆g−0.0840.013 +0.106
0.013 −0.314−0.120∆Σ
0.242
0.366 +0.015
0.366 +0.042−0.018−0.062
Table 5.3: The first moments of DSSV PDFs,

∆g/g
0.5

R1

2
2
xmin ∆f (x) at Q = 10 GeV .

COMPASS 2-had, Q2<1 GeV2
COMPASS charm
HERMES (prel.)
SMC

0

-0.5

2

2

Q =1 GeV22
Q =10 GeV

10

-1

x

Figure 5.6: Comparison of ∆g/g obtained by the DSSV group with extracted ∆g/g fromphoton-gluon fusion by SMC, HERMES, and COMPASS.

114

CHAPTER 5. RESULTS AND DISCUSSIONS

compared with each other. NLO calculations for these processes are still not available
and these data are not included in the DSSV analysis.

5.4

Further study on the spin structure of the proton

PHENIXof π 0√ALL (Run 2003 [13], Run 2004 [14], Run 2005 [15,√ have published the results16]) at s = 200 GeV, and π 0 ALL at s = 62.4 GeV√[75] (Run 2006, which this thesis
is based on). Recently PHENIX published π 0 ALL at s = 200 GeV extracted from the√data collected in Run 2006 [98]. It was demonstrated that π 0 ALL at s = 62.4 GeV [75]alone gave significant constraint on ∆g. DSSV√ performed global analysis including this0data, together with the results of π ALL at s = 200 GeV [98], and jet ALL [22, 99]
provided by STAR, and obtained new set of PDFs with much smaller uncertainty thanthe previously allowed range of ∆g.
ALL of π 0 with higher statistics will add further constraint on ∆g thus accumulating
data is one of the important effort which should be made.
√ In Run 2009, we plan to collect−1
data with integrated luminosity of about 50 pb at s = 200 GeV which is three times
largerstatistics than that in Run 2006. In addition, we have a plan to take data at√
s = 62.4 GeV in Run 2010. These data are expected to reduce the uncertainty for thesensitive x range (0.02 < xg < 0.4). Figure 5.7 displays the uncertainty of ∆g obtainedby the DSSV group(Fig. 5.7 top) and expected uncertainties when RHIC Run 2009 data
are added (Fig. 5.7 bottom).
However, the large uncertainty of√1st moment comes from lower x where no measurement exists. Collecting data at s = 500 GeV will extend the lower limit of thex range by a factor of about 2.5. An order of smaller x range can be investigated via
ALL of J/ψ detected with the forward muon spectrometers of PHENIX. However, theproduction mechanism of J/ψ is still not well understood. The mechanism should bestudied to extract information on ∆g. Lower x range, down to a few times 10−4 , canbe investigated with a future facility called e-RHIC [100]. e-RHIC will be the world’s
first polarized electron-proton DIS collider which consists of the existing polarized proton
ring at RHIC and an electron ring which will be installed. Electron energy of 10 GeV
and proton energy of 250 GeV are expected. At low x, the logarithmic derivative of thestructure function g1 in Q2 is proportional to the negative of ∆g.
ALL of other final state observables such as π +,− [18, 19], and direct photon [20, 21]
are also important since those have different systematics compared to π 0 . The gg process
which hides the sign of ∆g is common in π + and π − . Therefore, the difference is moresensitive to the sign of ∆g than π 0 . Direct photon production is dominated by qg → qγ
process and is sensitive to the sign of ∆g. Luminosity growth of RHIC and the acceptanceextention by the PHENIX detector upgrade with Silicon Vertex Tracker (SVT) provide
information on x dependence of ∆g via two particle correlations such as γ+jet. (γ isdetected with EMCal and jet is detected with SVT.)

5.4. FURTHER STUDY ON THE SPIN STRUCTURE OF THE PROTON

0.3

115

x∆g (current data)

0.2
0.1
0
-0.1
-0.2
0.3

GRSV std.
DNS

x∆g (projected data)

DSSV

0.2
0.1
0
-0.1
-0.2

GRSV ∆g = g
GRSV ∆g = -g

10

-1

x

1

Figure 5.7: Upper panel: x∆g at µ2 = 10GeV2 √from the NLO global√ analysis by
DSSV [97], which includes RHIC Run 2005 data at s = 200 GeV and s = 62.4 GeV.
The green band corresponds to ∆χ2 = 1 and the yellow band√shows ∆χ2 /χ2 = 2%. Lower
panel: Expected uncertainty band with RHIC Run 2009 at s = 200 GeV (50 pb−1 witha polarization of 60%.)
It is still possible that ∆G explains the missing piece of proton spin but the missing
piece might not be fully explained by the gluons. The other candidate and only one left is
the orbital angular momentum (OAM) of partons. Sivers function, which is a correlation
between a parton’s transverse momentum and the proton spin vector, contains informationof OAM in the proton. The gluon Sivers function can be accessed via transverse single
spin asymmetries of pions and D meson production in pp collisions [101, 102].
The polarized sea quark distributions are also not well
√ determined. It can be studied
with W boson production in polarized pp collisions at s = 500 GeV. The integrated
luminosity of about 300 pb−1 is expected to be achieved by Run 2013 and ∼ 3000 W +events and ∼ 3000 W − events are expected to be collected.

Chapter 6
Conclusion
We have measured double helicity
asymmetry ALL of inclusive mid-rapidity π 0 production√in polarized pp collisions at s = 62.4 GeV. Its kinematical coverage was pT = 1−4 GeV/c
and |η| < 0.35. It is sensitive to gluon polarization in the proton since gluon interacts
at leading order in π 0 production in polarized pp collisions. In addition to ALL , single
helicity asymmetry AL , and double transverse spin asymmetry AT T have been measured.
We have also presented the unpolarized cross section, obtained from the same data set.
The experiment was performed with the PHENIX detector, with the polarized proton
beams provided by RHIC at BNL in the United States. The data used in this thesis
were taken from Jun. 6 to 20 (two weeks) during Run 2006. The analyzed data sample
has the integrated luminosity of 40 nb−1 , with average polarization of 48%. π 0 production was measured through its two photon decay mode. The photons were detected by
the PHENIX EMCal. High-pT photon trigger which is constructed by the signals from
EMCal was used to collect the data sample. The measured ALL are consistent with zero
within statistical uncertainties. Major source of the systematic uncertainty is the scale
uncertainty of 14% which comes from the beam polarizations. Double transverse spin
asymmetry AT T was also measured to obtain the systematic uncertainty from the residual transverse components of the beam polarizations. The maximal possible AT T effecton ALL was obtained to be < 0.15 · δALL in all pT bins, where δALL denotes the statisticaluncertainty of ALL . Single spin asymmetry AL , which is expected to be negligible atthis energy, was measured for diagnostics and is consistent with zero within the statistical uncertainties. Other systematic uncertainties are small compared to the statisticaluncertainty, which is confirmed by a technique to randomize the sign of bunch crossing
polarization.
Comparisons of unpolarized cross section results to NLO and NLL theoretical calculations were performed. NLO and NLL agree with the data within theoretical uncertainties.
Therefore, the data can be interpreted in pQCD framework to obtain information of ∆g.
The ALL results extend the sensitivity to the polarized gluon√ distribution in the proton to higher xg compared to the previous measurements at s = 200 GeV. AAC and
116

117
DSSV global analyses show that our data have strong constraint power on ∆g beyond the
traditional polarized DIS data. The data do not support a large gluon scenario, such as
GRSV-max. An NLO global pQCD analysis by DSSV group
included the present data
R1
in this thesis, and obtained an integral of ∆g(x) to be 0.001 ∆g(x)dx = 0.013+0.106−0.120 for+0.70222222∆χ = 1 and 0.013−0.314 for ∆χ /χ = 2% at µ = 10 GeV . The uncertainty was significantly reduced compared to what is obtained with polarized DIS data only. For more
precise determination of ∆g, it is important to accumulate more
√ data. In Run 2009, weplan to collect data with integrated luminosity of 25√pb−1 at s = 200 GeV with beampolarizations of 65%. We also plan to take data at s = 62.4 GeV in Run 2010. These
data are expected to reduce the uncertainty and provide us further information on thespin structure of the proton.

Acknowledgments
I would like to express profound gratitude to Dr. Ken’ichi Imai, who is the leader ofthe experimental Nuclear and Hadronic physics group (NH). His words were always fullof passion for physics and I felt deep understanding of nature inside him that always
stimulated my work. He has read my thesis carefully and given me useful suggestions. I
am deeply grateful to Dr. Naohito Saito, who has been my supervisor. I learned not only
physics but also scientific approach through discussions and every day conversations. He
has read my thesis thoroughly and the discussion with him was vital in completing thethesis.
I cannot thank enough Dr. Yuji Goto who was the tutor at BNL. I would like toexpress my gratitude to the members of π 0 working group: Dr. Alexander Bazilevsky
and Dr. Kieran Boyle. Discussion with them was beneficial and I was able to polish theanalysis working with them. I would like to thank Dr. Hideto Enyo who is the chief
scientist of Radiation Laboratory at RIKEN for his support. I also would like to thankthe members of RIKEN BNL Research Center (RBRC): Dr. Yasuyuki Akiba, Dr. Gerry
Bunce, Dr Abhay Deshpande, and Dr. Kensuke Okada. I would like to thank the PHENIX
collaborators, and the staff of Collider Accelerator Department at BNL. Without themthe experiment would not be possible.
I would like to thank the members of the paper preparation group (PPG087): Dr.
Ken Barish, Dr. Alexander Bazilevsky, Dr. Robert Bennett, and Dr. Yuji Goto. I also
would like to thank the members of the internal review committee (IRC087): Dr. Ed
Kinney, Dr. Vassili Papavassiliou, and Dr. Henner Buesching.
I always enjoyed working and discussing with Dr. Manabu Togawa during my stay at
BNL. I learned a lot about physics and PHENIX-related matters. I also learned how tocook miso soup from him. I would like to thank Dr. Yoshinori Fukao and Dr. Kenichi
Nakano. Discussion with them was helpful in writing the thesis. I would like to thank thestaff of the NH group: Dr. Tomofumi Nagae, Dr. Tetsuya Murakami, and Dr. Kiyoshi
Tanida. I also would like to thank colleagues from the NH group: Dr. Hiromi Iinuma,
Mr. Kohei Shoji, Mr. Seishi Dairaku, Mr. Kenichi Karatsu, and Mr. Katsuro Nakamura.
I would like to thank the administrators of the PHENIX Computing Center in Japan(CCJ): Dr. Yasushi Watanabe, Dr. Satoshi Yokkaichi, Dr. Soichiro Kametani, Dr.
Tomoaki Nakamura. The analysis was done on CCJ and it was very effective with their
118

119
supports.
I would like to thank Dr. Werner Vogelsang, Dr. Marco Stratmann, and Dr. Masanori
Hirai who has kindley provided theoretical calculations.

Appendix A
z-dependent BBC-efficiency
A.1

Overview

The procedure to obtain z(vertex position)-dependentBBC-efficiency is explained in this√section. The procedure used for the data at s = 62.4 GeV (which the thesis is basedon) is explained and the results are presented in this chapter.

√s = 62.4 GeV

A.2

z-dependent BBC-efficiency at

A.2.1

Procedure with WCM and vernier scan

The collision vertex distribution at PHENIX IP is calculated with the beam profiles of theBlue and the Yellow beams. Then it is compared with the measured vertex distribution
with BBC at PHENIX and BBC efficiency is found.
The collision vertex distribution at PHENIX IP is calculated as
V (z) ∝Z Z Z

dxdydtDB (x, y, z, t) · D Y (x, y, z, t),(A.1)

where D B and D Y are the beam densities for the Blue and the Yellow beams, respectively.
The beam densities can be written as
D B (x, y, z, t) = DxB (x) · DyB (y) · DzB (z − ct),

(A.2)

for the Blue beam, andD Y (x, y, z, t) = DxY (x) · DyY (y) · DzY (z + ct),

(A.3)

for the Yellow beam. Transverse beam profiles are measured by the van der Meer techniqueor the vernier scan. In a vernier scan, the transverse widths of the beam overlap σx , σy120

A.2. Z-DEPENDENT BBC-EFFICIENCY AT√

S = 62.4 GEV

121

are measured by sweeping one beam across the other in small steps while monitoring the
BBC trigger rate. Figure A.1 displays the horizontal and the vertical beam profiles. (Eachplot has four points at zero since a scan starts from zero and go back to zero when a scanin one direction was finished.) The obtained profiles are well reproduced by Gaussians,
thus Dx (x), and Dy (x) are written as(x − x0 )2,
Dx (x) =
exp −
2σx2
σx 2π
!
1
(y − y0 )2√ exp −
Dy (x) =
,
2σy2
σy 2π
!

1
√

(A.4)
(A.5)
(A.6)

where σx and σy are the widths of transverse beam profiles. The hour glass effect is taken
into account by changing the widths as
z2
2
2
σx(y)
(z) = σx(y)· 1 + ∗2 ,
β
!

(A.7)

BBC rate [Hz]BBC rate [Hz]where β ∗ is beam focusing parameter. The longitudinal beam profiles were measured with
Wall Current Monitors (WCMs) as in Fig. A.2.
20

15

20

15

10

10

5

5

0

-2000

-1000

0

1000

2000
Step [µm]

0

-2000

-1000

0

1000

2000
Step [µm]Figure A.1: Horizontal (Left) and vertical (right) beam profiles measured with the vernierscan. The horizontal axis shows the step size of the beam displacement in µm. The verticalaxis is the BBC trigger rate.

A.2.2

Results at√
s = 62.4 GeV

The vertex distribution was simulated according to Eq. A.1, using the information explained above. The BBC vertex resolution of σzBBC = 2 cm was used in the simultaion.

122

APPENDIX A. Z-DEPENDENT BBC-EFFICIENCY

0.8

0.8

0.7

0.7

0.6

0.6

0.5

0.5

0.4

0.4

0.3

0.3

0.2

0.2

0.1

0.1
0

0

0

2

4

6

8

10

12 z[m] 14

0

2

4

6

8

10

12 z[m] 14

Figure A.2: Longiudinal beam profile measured with WCM for the Blue (Left) and for
the Yellow (Right) beams.

The z-dependence of BBC efficiency is well represented as a Gaussian with a width ofBBC
σef
f . Figure A.3 displays the vertex distribution measured with BBC overlapped withthe simulated results. The three plots show how the distribution changes with different
BBC
BBC
σef
in the simultaions. The left plot was with σef= 85 cm, and the middle was
f
f
BBC
BBC
with σef f = 95 cm, and the right was with σef f = 105 cm. As shown in the figure,the distribution is sensitive to the BBC efficiency. The z-dependent BBC efficiency wasBBC2
obtained to be σef
f = 95 ± 10 cm (∆χ = 9).

A.2. Z-DEPENDENT BBC-EFFICIENCY AT

0.018
0.016

σBBC
eff =85cm

0.018
0.016

√

S = 62.4 GEV

σBBC
eff =95cm

0.018
0.016

0.014

0.014

0.014

0.012

0.012

0.012

0.01

0.01

0.01

0.008

0.008

0.008

0.006

0.006

0.006

0.004

0.004

0.004

0.002

0.002

0.002

0
-200 -150 -100 -50

050

100 150 200
Z (cm)

0-200 -150 -100 -50

050

100 150 200
Z (cm)123

σBBC
eff =105cm

0-200 -150 -100 -50

050

100 150 200
Z (cm)

Figure A.3: Comparison of the z vertex distributions measured with BBC (Black) and calBBC
BBC
BBC
culated with monte carlo (Red) for σef
f = 85 cm, σef f = 95 cm,σef f = 105 cm(right).
BBC
The calculation for σef f = 95 cm is most compatible with the measured distribution.

Appendix B
EMC clustering algorithm
B.1

PbSc

In PbSc, instead of summing all of the energies deposited in towers in a cluster, the sum isperformed only for the core towers as explained in Sec. 4.5.2. Then the core tower energycore
E core was corrected to obtain the cluster energy Ecorr. The correction was divided into
three pieces. One is the correction from core energy to cluster energy, which is dependent
on the incident angle of the photon. The function used for the correction is,core
Eangle−corr

=

a =
b =
c =

E core,a(1 − b sin4 (θ) × (1 − c · log(E core )))0.918,
1.35,
0.003,

(B.1)
(B.2)
(B.3)
(B.4)

where E core is the core energy of the target cluster, θ is the impact angle of the incident
core
photon, and Eangle−corris the corrected core energy. The parameters a, b and c wereobtained with the test experiments. Fig. B.1 displays the correction factor (C clus =
core
core
Ecorr
). (Although the plot is for E core = 1 GeV, the energy dependence is negligible′ /E
at the energies of interest.) The impact angle θ is less than 24 deg. The core energy is
about 90% of the cluster energy, and the angle dependence correction is less than 4%.
The correction factor for shower leakage (C leak ) and attenuation in the fibers (C att )are obtained as
C leak =

1
2−

q

core
1 + d · log2 (1 + Eangle−corr)d = 0.0033,
C att =1core
exp (log(Eangle−corr) · X0 /λ)

124

,

,

(B.5)
(B.6)
(B.7)

125

C clusB.1. PBSC

1.13

1.12

1.11

1.1

1.09
0

5

10

15

20

25

impact angle[deg]Figure B.1: The correction factor from core energy to cluster energy, including angle
dependence, C clus .

X0 = 2 cm,
λ = 120 cm

(B.8)
(B.9)

where X0 is the radiation length, and λ is the attenuation length in the fiber. Then thecore
corrected energy Ecorr
becomes

1.01(B.10)

C att

C leak

core
core
Ecorr
= C leak C att Eangle−corr
.

1.05
1.04

1.008

1.03
1.02

1.006

1.01
1

1.004

0.99
1.002

0.98
0.97

1
0

0.96
1

2

3

4

5

6

7

8

9

10

E(GeV)

1

2

3

4

5

6

7

8

9

10

E (GeV)

Figure B.2: The correction factors for shower leakage, C leak (Left), and for attenuationin the fibers, C att (Right).

126

APPENDIX B. EMC CLUSTERING ALGORITHM

PbGl

C angleC nonlin

B.2
1.08
1.07

1.1

1.06

1.08

1.05

1.06

1.04

1.04

1.03

1.02

1.02
1
1.01
0.98

1
0

5

10

15

20

25

1

impact angle (deg)

2

3

4

5

6

7

8

9

10

E(GeV)

Figure B.3: The correction factors for angle dependence, C angle (Left), and for nonlinearity, C nonlin (Right).

The correction factors for angle dependence, C angle and non linearlity, C nonlin arecalculated as1,a(exp(bθ) − 1) + 1
a = −0.012043,
b = 0.077908 · 180/π,

C angle =(B.11)(B.12)(B.13)

andC nonlin
a
b
c

=
=
=
=

a + b log(E) + c log2 (E),
1.0386,−0.041423,
0.006064.

(B.14)(B.15)(B.16)(B.17)

Then the cluster energy, E, is corrected as
E corr = C angle C nonlin E,
where E corr is the corrected energy of the cluster.

(B.18)

Appendix C
Some information on pol-PDFs
C.1

Q2 evolutions of parton contributions to the proton spin∆GLq∆Σ / 2Lg
Q 2 [GeV2]

Figure C.1: Toy calculation of the Q2 evolutions of the contributions to the proton spinat leading order. [103]Figure. C.1 displays toy calculation of the Q2 evolutions of the contributions to theproton spin at leading order [103]. ∆Σ is the sum of quark spin contributions, Lq and
Lg is the orbital angular momentum of quarks and gluons respectively, ∆G is the gluon
spin contribution. The calculation was performed assuming ∆Σ = 0.25, ∆G = Lq = 0.2,
Lg = −0.025 at an initial scale Q0 = 1 GeV. The rise of ∆G ∝ log Q2 is compensatedby an opposite evolution of Lg . At Q2 → 0, 21 ∆Σ + Lq and ∆G + Lg become roughly
127

128

APPENDIX C. SOME INFORMATION ON POL-PDFS

equal [104].

DSSV polarized PDFs

Truncated 1st moment [x:0.001 → 1]

C.2

0.5

(∆u+ ∆ u)/2

0.4
0.3

∆Σ/2

0.2

∆g

0.1

∆u/2
∆ s /2∆ d /2-0.1
0(∆d+ ∆d )/2

-0.2
-0.3
1

10

Q 2 [GeV 2 ]10 2

Figure C.2: Truncated 1st moments of polarized PDFs by the DSSV group vs Q2 . Thelines between the points for different Q2 are straight lines and are drawn only for a guide.
Some of the points at Q2 = 10 GeV2 are shifted for clear separations. The uncertainties
corresponds to ∆χ2 = 1 are only plotted for Q2 = 10 GeV2 .
1Figure C.2 displays truncated 1st moments of polarized PDFs ( 0.001
dxf (x)) by the
DSSV group.

R

C.3

GRSV
Q2 [GeV2 ]
1
5
10

∆u
0.861
0.859
0.859

∆d
-0.405-0.406-0.406∆q̄
−0.063−0.064−0.064

∆g
0.420
0.708
0.828

∆Σ
0.204
0.197
0.197

Table C.1: 1st moments of polarized PDFs by the GRSV group [45]. GRSV-std.

C.3. GRSV129

Q2 [GeV2 ]0.4
14

GRSV-max
1.256
1.894
2.795

GRSV-std
0.240
0.420
0.668

GRSV-0 GRSV-min
0-1.2560.070-1.7630.160-2.495

Table C.2: 1st moments of polarized gluon PDFs for vairous senarios by the GRSV
group [55].

Appendix D
Subprocess cross sections for gg → q q̄and gg → gg
The cross sections of parton parton interactions for center of mass system (CMS) at LO,
when they are purely strong interactions, are given by [105],παs2dσ̂
= 2 |M|2 ,
ŝ
dt̂

(D.1)

where ŝ, t̂, and û (which appears later) are the Mandelstam variables for the partonparton interactions, σ̂ is the cross section of interest, αs is the strong coupling constant,and |M|2 is the spin-averaged matrix element squared. The metrix elements for gg → q q̄and gg → gg are,gg → q q̄

: |M|2 =

9 8
·
64 3

4 û2 + t̂2 û2 + t̂2−
9 ût̂ŝ2

9ût̂ ûŝ ŝt̂
gg → gg : |M|2 =3− 2 − 2 − 2
2
ŝ
û
t̂

!

!

(D.2)
(D.3)

Figure D.1 displays the spin-averaged matrix element squared for these subprocesses
as a function of cos θ∗ , where θ∗ is the scattering angle at CMS. The cross sections for
gg → gg is larger than that for gg → q q̄ by about three orders of magnitude.

130

|M|2

131

10 5
10 4
10 3

gg → gg

10 2
101

gg → qq

10 -1
10

-2

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

cos θ*

1

Figure D.1: The spin-averaged matrix element squared |M 2 | for gg → q q̄ and gg → gg.

Bibliography[1] J. Ashman et al. (European Muon), Phys. Lett. B206, 364 (1988).
[2] J. Ashman et al. (European Muon), Nucl. Phys. B328, 1 (1989).
[3] A. Airapetian et al. (HERMES), Phys. Rev. D75, 012007 (2007), hep-ex/0609039.
[4] V. Y. Alexakhin et al. (COMPASS), Phys. Lett. B647, 8 (2007), hep-ex/0609038.
[5] A. Airapetian et al. (HERMES), Phys. Rev. Lett. 84, 2584 (2000), hep-ex/9907020.
[6] P. Liebing (HERMES), AIP Conf. Proc. 915, 331 (2007).
[7] B. Adeva et al. (Spin Muon (SMC)), Phys. Rev. D70, 012002 (2004),
hep-ex/0402010.
[8] E. S. Ageev et al. (COMPASS), Phys. Lett. B633, 25 (2006), hep-ex/0511028.
[9] M. Alekseev et al. (COMPASS) (2008), 0802.3023.
[10] Y. S. Derbenev et al., Part. Accel. 8, 115 (1978).
[11] S. Y. Lee, Spin Dynamics and Snakes in Synchrotorons (World Scientific, 1997).
[12] G. Bunce, N. Saito, J. Soffer, and W. Vogelsang, Ann. Rev. Nucl. Part. Sci. 50, 525(2000), hep-ph/0007218.
[13] S. S. Adler et al. (PHENIX), Phys. Rev. Lett. 93, 202002 (2004), hep-ex/0404027.
[14] S. S. Adler et al. (PHENIX), Phys. Rev. D73, 091102 (2006), hep-ex/0602004.
[15] A. Adare et al. (PHENIX), Phys. Rev. D76, 051106 (2007), 0704.3599.
[16] Y. Fukao (PHENIX), Ph.D thesis.
Double helicity asymmetry for π 0 production in√polarized p + p collisions at s = 200 GeV: Implications for the polarized gluon
distribution in the proton. (2007).
[17] D. G. d’Enterria, J. Phys. G31, S491 (2005), nucl-ex/0411049.
132

BIBLIOGRAPHY

133[18] M. Astrid, talk given at the 18th International Spin Physics Symposium (SPIN2008), University of Virginia, USA. (2008).
[19] F. Ellinghaus (PHENIX) (2008), arXiv:0808.4124.
[20] S. S. Adler et al. (PHENIX), Phys. Rev. Lett. 98, 012002 (2007), hep-ex/0609031.
[21] R. Bennett (PHENIX), Talk given at the 18th International Spin Physics Conference(SPIN2008), University of Virginia, USA (2008).
[22] B. I. Abelev et al. (STAR), Phys. Rev. Lett. 100, 232003 (2008), arXiv:0710.2048.
[23] A. Kocoloski (STAR), AIP Conf. Proc. 1149, 277 (2009), 0905.1033.
[24] F. Halzen and A. D. Martin, Quarks and Leptons (Wiley, 1984).
[25] R. Devenish and A. C-Sarkar, Deep Inelastic Scattering (Oxford university press,
2005).
[26] A. V. Manohar, hep-ph/9204208 (1992).
[27] J. D. Bjorken, Phys. Rev. 179, 1547 (1969).
[28] J. Callan, Curtis G. and D. J. Gross, Phys. Rev. Lett. 22, 156 (1969).
[29] B. Povh, K. Rith, C. Scholz, and F. Zetsche, Teilchen und Kerne (Springer, 1994).
[30] K. Abe et al. (E143), Phys. Rev. D58, 112003 (1998), hep-ph/9802357.
[31] P. L. Anthony et al. (E155), Phys. Lett. B458, 529 (1999), hep-ex/9901006.
[32] R. Mertig and W. L. van Neerven, Z. Phys. C70, 637 (1996), hep-ph/9506451.
[33] W. Vogelsang, Phys. Rev. D54, 2023 (1996), hep-ph/9512218.
[34] G. Sterman, Nucl. Phys. B281, 310 (1987).
[35] D. de Florian and W. Vogelsang, Phys. Rev. D71, 114004 (2005), hep-ph/0501258.
[36] D. de Florian, W. Vogelsang, and F. Wagner, Phys. Rev. D76, 094021 (2007),
arXiv:0708.3060.
[37] J. C. Collins et al., Factorization of Hard Process in QCD in Perturbative Quantum
Chromodynamics (World Scientific, 1989).
[38] J. Pumplin et al., JHEP 07, 012 (2002), hep-ph/0201195.

134

BIBLIOGRAPHY[39] A. D. Martin, R. G. Roberts, W. J. Stirling, and R. S. Thorne, Eur. Phys. J. C35,
325 (2004), hep-ph/0308087.
[40] M. Gluck, E. Reya, and A. Vogt, Eur. Phys. J. C5, 461 (1998), hep-ph/9806404.
[41] S. J. Brodsky and G. R. Farrar, Phys. Rev. Lett. 31, 1153 (1973).
[42] C. Amsler et al. (Particle Data Group), Phys. Lett. B667, 194 (2008).
[43] M. Hirai, S. Kumano, and N. Saito, Phys. Rev. D74, 014015 (2006),
hep-ph/0603213.
[44] M. Hirai, S. Kumano, and N. Saito (Asymmetry Analysis), Phys. Rev. D69, 054021(2004), hep-ph/0312112.
[45] M. Gluck, E. Reya, M. Stratmann, and W. Vogelsang, Phys. Rev. D63, 094005(2001), hep-ph/0011215.
[46] E. Leader, A. V. Sidorov, and D. B. Stamenov, Eur. Phys. J. C23, 479 (2002),
hep-ph/0111267.
[47] N. Cabibbo, Phys. Rev. Lett. 10, 531 (1963).
[48] F. E. Close and R. G. Roberts, Phys. Rev. Lett. 60, 1471 (1988).
[49] F. E. Close and R. G. Roberts, Phys. Lett. B316, 165 (1993), hep-ph/9306289.
[50] J. D. Bjorken, Phys. Rev. 148, 1467 (1966).
[51] C. Amsler et al. (Particle Data Group), Phys. Lett. B667, 200 (2008).
[52] J. Blumlein and H. Bottcher, Nucl. Phys. B636, 225 (2002), hep-ph/0203155.
[53] E. Leader, A. V. Sidorov, and D. B. Stamenov, Phys. Lett. B462, 189 (1999),
hep-ph/9905512.
[54] E. Leader, A. V. Sidorov, and D. B. Stamenov, Phys. Rev. D58, 114028 (1998),
hep-ph/9807251.
[55] W. Vogelsang, private communications.
[56] A. Zelenski et al., Hyperfine Interactions 127, 475 (2000).
[57] P. Sona, Energia Nucleare 14, 295 (1967).
[58] L. H. Thomas, Phil. Mag. 3, 1 (1927).

BIBLIOGRAPHY

135[59] V. Bargmann, L. Michel, and V. L. Telegdi, Phys. Rev. Lett. 2, 435 (1959).
[60] M. Froissart and R. Stora, Nucl. Instrum. Meth. 7, 297 (1960).
[61] T. Roser, AIP Conference Proceedings 187, 1442 (1988).
[62] B. W. Montague, Phys. Rep. 113, 1 (1984).
[63] O. Jinnouchi et al., RHIC/CAD Accelerator Physics Note 171 (2004).
[64] H. Okada et al., Phys. Lett. B638, 450 (2006), nucl-ex/0502022.
[65] Y. Fukao et al., Phys. Lett. B650, 325 (2007), hep-ex/0610030.
[66] K. Adcox et al. (PHENIX), Nucl. Instrum. Meth. A499, 469 (2003).
[67] M. Allen et al. (PHENIX), Nucl. Instrum. Meth. A499, 549 (2003).
[68] C. Adler et al., Nucl. Instrum. Meth. A470, 488 (2001), nucl-ex/0008005.
[69] L. Aphecetche et al. (PHENIX), Nucl. Instrum. Meth. A499, 521 (2003).
[70] G. David et al., IEEE Trans. Nucl. Sci. 45, 705 (1998).
[71] CERNLIB, Geant3.
[72] M. M. Aggarwal et al. (WA98), Phys. Rev. Lett. 81, 4087 (1998), nucl-ex/9806004.
[73] M. S. Emery et al., IEEE Trans. Nucl. Sci. 44, 374 (1997).
[74] S. S. Adler et al. (PHENIX), Nucl. Instrum. Meth. A499, 560 (2003).
[75] A. Adare et al. (PHENIX), Phys. Rev. D79, 012003 (2009), arXiv:0810.0701.
[76] C. Amsler et al. (Particle Data Group), Phys. Lett. B667, 587 (2008).
[77] S. Arnold, K. Goeke, A. Metz, P. Schweitzer, and W. Vogelsang, Eur. Phys. J. ST
162, 31 (2008).
[78] C. Amsler et al. (Particle Data Group), Phys. Lett. B667, 364 (2008).
[79] A. L. S. Angelis et al. (CERN-Columbia-Oxford-Rockefeller), Phys. Lett. B79, 505(1978).
[80] C. Kourkoumelis et al. (R-806), Phys. Lett. B84, 271 (1979).
[81] A. L. S. Angelis et al. (CMOR), Nucl. Phys. B327, 541 (1989).

136

BIBLIOGRAPHY[82] T. Akesson et al. (Axial Field Spectrometer), Sov. J. Nucl. Phys. 51, 836 (1990).
[83] C. Kourkoumelis et al. (R-806), Zeit. Phys. C5, 95 (1980).
[84] A. G. Clark et al. (CSZ), Phys. Lett. B74, 267 (1978).
[85] F. W. Busser et al. (CCRS), Nucl. Phys. B106, 1 (1976).
[86] K. Eggert et al. (ACHM), Nucl. Phys. B98, 49 (1975).
[87] F. W. Busser et al. (CCR), Phys. Lett. B46, 471 (1973).
[88] F. W. Busser et al. (CCRS), Phys. Lett. B55, 232 (1975).
[89] M. Banner et al. (Saclay), Nucl. Phys. B126, 61 (1977).
[90] B. Alper et al. (British-Scandinavian), Nucl. Phys. B100, 237 (1975).
[91] D. de Florian, R. Sassot, and M. Stratmann, Phys. Rev. D75, 114010 (2007),
hep-ph/0703242.
[92] H. Torii (PHENIX), √Ph.D thesis. Midrapidity Neutral-Pion Production in ProtonProton Collisions at s = 200 GeV (2004).
[93] S. S. Adler et al. (PHENIX), Phys. Rev. Lett. 91, 241803 (2003), hep-ex/0304038.
[94] P. Aurenche, M. Fontannaz, J. P. Guillet, B. A. Kniehl, and M. Werlen, Eur. Phys.
J. C13, 347 (2000), hep-ph/9910252.
[95] B. Jager, A. Schafer, M. Stratmann, and W. Vogelsang, Phys. Rev. D67, 054005(2003), hep-ph/0211007.
[96] M. Hirai, private communications.
[97] D. de Florian, R. Sassot, M. Stratmann, and W. Vogelsang, Phys. Rev. Lett. 101,
072001 (2008), arXiv:0804.0422.
[98] A. Adare et al. (PHENIX), Phys. Rev. Lett. 103, 012003 (2008), arXiv:0810.0694.
[99] D. Staszak (STAR), Nucl. Phys. A827, 210c (2009), arXiv:0902.1345.
[100] C. Aidala et al., A high luminosity, high energy electron-ion-collider (erhic white paper) http://www.phenix.bnl.gov/WWW/publish/abhay/Home of EIC/NSAC2007/
070424 EIC.pdf (2007).
[101] M. Anselmino, U. D’Alesio, S. Melis, and F. Murgia, Phys. Rev. D74, 094011(2006), hep-ph/0608211.

BIBLIOGRAPHY

137[102] M. Anselmino, M. Boglione, U. D’Alesio, E. Leader, and F. Murgia, Phys. Rev.
D70, 074025 (2004), hep-ph/0407100.
[103] M. Stratmann and W. Vogelsang, J. Phys. Conf. Ser. 69, 012035 (2007),
hep-ph/0702083.
[104] X.-D. Ji, J. Tang, and P. Hoodbhoy, Phys. Rev. Lett. 76, 740 (1996),
hep-ph/9510304.
[105] E. Leader, Spin in Particle Physics (Cambridge University Press, 2001).