Measurements of the leading neutron
production in√polarized pp collision at s=200 GeV

UNI VE

FO

KYOTO

U

ND

JAPAN

97

KY O

O

ITY
RS

T

Manabu Togawa

8
ED 1

A Dissertation Submitted in Partial Fulfillment of
the Requirement for the Degree ofDoctor of Science
Department of Physics
Kyoto University
January, 2008

Abstract
In the 2001−2002 run period at Relativistic Heavy Ion Collider (RHIC), polarized protons√
were accelerated to 100 GeV for the first time, with collisions at s=200 GeV. We performed
a pilot experiment at IP12 to search for non-zero single transverse asymmetry, AN , in π 0
production at very forward angles based on the observation of large asymmetries in large xF√in s = 19.4 GeV at Fermilab. The experiment was primary designed to detect photons fromπ 0 s but was also capable to detect neutrons in the forward angle, ±2.8 mrad. The detector forneutral particles was placed downstream of the RHIC DX dipole magnets, which bent out aproton beam and other charged particles.

Observed asymmetries for photons and π 0 s were consistent with zero within the errors,
whereas a significant asymmetry was observed for neutrons. The value of the asymmetry is
AN = (−0.090 ± 0.006 ± 0.009) × (1.0+0.52−0.25 ), where the errors are statistical, systematic and thescale error is from the beam polarization uncertainty. This discovery enabled to monitor the
beam polarization at the collision points at RHIC.
For further investigation about this large asymmetry in the leading neutron production andits origin, we have measured the xF dependence of AN and cross section with the PHENIX√
detector in the similar kinematics region at s=200 GeV. The observed asymmetries are consistent with the IP12 result within the errors and show almost no xF dependence. The measured
cross sections are consistent with the xF scaling claimed by the ISR experiment which measured√
the neutron production in pp collisions at s=30.6–62.7 GeV. These cross sections for large xF
neutron production, as well as those in ep collisions by ZEUS, are mainly reproduced by a pionexchange. Therefore, the observed large asymmetry for the neutron production is considered tocome from the interference between a spin-flip amplitude due to the pion exchange and non-flip
amplitudes from other Reggeon exchange.

Contents
1 Introduction

I

5

1.1

AN for inclusive productions in the forward kinematics . . . . . . . . . . . . . . .

6

1.2

Leading neutron production . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

7

1.2.1

Definition of kinematics for the leading neutron production . . . . . . . .

7

1.2.2

ISR experiment at CERN . . . . . . . . . . . . . . . . . . . . . . . . . . .

9

1.2.3

H1 and ZEUS experiments at HERA . . . . . . . . . . . . . . . . . . . . . 11

1.2.4

Regge theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

1.2.5

Meson-Cloud Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

IP12 experiment17

2 IP12 experiment

18

2.1

Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

2.2

RHIC-AGS complex for the Polarized Proton Collider . . . . . . . . . . . . . . . 19

2.3

2.2.1

Polarized ion source . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

2.2.2

Depolarization resonance . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

2.2.3

Partial snakes in AGS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

2.2.4

Siberian snake and Spin rotator in RHIC . . . . . . . . . . . . . . . . . . 24

2.2.5

pC- and pp-CNI polarimeters . . . . . . . . . . . . . . . . . . . . . . . . . 25

Experimental setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
2.3.1

Hodoscopes for beam-beam counters . . . . . . . . . . . . . . . . . . . . . 28

2.3.2

East detector : Electromagnetic Calorimeter . . . . . . . . . . . . . . . . 28

2.3.3

West detector : Hadron Calorimeter . . . . . . . . . . . . . . . . . . . . . 30

2.3.4

Data taking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

3 Analysis

36

3.1

Selection of the collision event . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

3.2

Particle identification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

3.3

π 0 mass spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

3.4

Asymmetry calculation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

3.5

AN decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
1

2

CONTENTS

3.6

Neutron asymmetry measured with the HCal . . . . . . . . . . . . . . . . . . . . 45

3.7

Multiplicity of hodoscopes associated with neutron sample . . . . . . . . . . . . . 46

4 Results of the IP12 experiment

48

II

49

PHENIX experiment

5 PHENIX experiment

50

5.1

Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

5.2

Experimental setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
5.2.1

Zero Degree Calorimeter (Shower Max Detector, Charge veto counter) . . 51

5.2.2

Beam Beam Counters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

5.2.3

PHENIX data acquisition system . . . . . . . . . . . . . . . . . . . . . . . 56

5.2.4

Data taking for the leading neutron sample . . . . . . . . . . . . . . . . . 56

6 Simulation study
6.1

6.2

Study of the ZDC and SMD responses . . . . . . . . . . . . . . . . . . . . . . . . 59
6.1.1

Energy linearity and resolution for the ZDC . . . . . . . . . . . . . . . . . 61

6.1.2

Energy response in the detector edge . . . . . . . . . . . . . . . . . . . . . 61

6.1.3

Position resolution in the center region . . . . . . . . . . . . . . . . . . . . 62

6.1.4

Position response in the detector edge . . . . . . . . . . . . . . . . . . . . 66

6.1.5

Systematics for the position measurement . . . . . . . . . . . . . . . . . . 67

Study of the neutron identification by Pythia event generator . . . . . . . . . . 69
6.2.1

Event structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

6.2.2

Neutron identification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

6.2.3

Background contamination after the neutron identification . . . . . . . . . 77

7 Analysis for the cross section
7.1

59

80

Cross section calculation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
7.1.1

Cross section at the PHENIX experiment . . . . . . . . . . . . . . . . . . 80

7.1.2

Cross section at the ISR experiment . . . . . . . . . . . . . . . . . . . . . 80

7.2

Energy distribution in the very forward kinematics . . . . . . . . . . . . . . . . . 80

7.3

Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
7.3.1

Energy gain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

7.3.2

Neutron yield . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

7.4

Study of the beam gas background . . . . . . . . . . . . . . . . . . . . . . . . . . 82

7.5

Relative efficiency to the SMD cut between the real and simulation data . . . . . 82

7.6

Energy Unfolding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
7.6.1

Unfolding method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

7.6.2

Unfolding the data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

3

CONTENTS

7.7

Estimation of the efficiency for the experimental cut . . . . . . . . . . . . . . . . 87

7.8

Result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

7.9

Systematic errors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
7.9.1

Scaling errors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

7.9.2

Systematics for the energy linearity . . . . . . . . . . . . . . . . . . . . . . 90

7.9.3

Systematics for the determination of the beam axis on the ZDC geometry

8 Analysis for the single transverse spin asymmetry

91
94

8.1

The study of the beam axis on the detector geometry

8.2

Azimuthal angle dependence of AN . . . . . . . . . . . . . . . . . . . . . . . . . . 98

8.3

. . . . . . . . . . . . . . . 94

8.2.1

Asymmetry calculation

8.2.2

Preparation of the simulation sets for each trigger . . . . . . . . . . . . . 98

8.2.3

Estimation of the smearing effect . . . . . . . . . . . . . . . . . . . . . . . 100

8.2.4

Systematic errors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

xF dependence of AN. . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

8.3.1

Asymmetry calculation. . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

8.3.2

Estimation of the smearing effect . . . . . . . . . . . . . . . . . . . . . . . 111

8.3.3

Estimation of the mean xF in each bin . . . . . . . . . . . . . . . . . . . . 111

8.3.4

Systematic errors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114

9 Results of the PHENIX experiment

118

9.1

Cross section . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118

9.2

Azimuthal angle dependence of AN . . . . . . . . . . . . . . . . . . . . . . . . . . 119

9.3

xF dependence of AN. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119

10 Discussion

124

11 Conclusion

126

A Definition of the single transverse spin asymmetry130

A.1 Necessity to be finite AN. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130

A.2 Square root formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
A.2.1 Physics asymmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132
A.2.2 Detector asymmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132
A.2.3 Luminosity asymmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
A.3 Luminosity formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
A.4 Background contamination
B Local Polarimeter. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134
135

B.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
B.2 Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137

4

CONTENTS

B.3 Measurement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
B.3.1 ATNransverse . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
B.3.2 ALongitudinaland ALongitudinal
LR
UD
B.4 Estimation of PL /P. . . . . . . . . . . . . . . . . . . . . . . . 138. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138

B.5 Summary table . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142
C Uncertainty in bunch shuffling143

C.1 Uncertainty from the bunch number . . . . . . . . . . . . . . . . . . . . . . . . . 143
C.2 Residual asymmetry in bunch shuffling . . . . . . . . . . . . . . . . . . . . . . . . 144
C.2.1 Expected Ashuf f le values for φ-dependence asymmetry . . . . . . . . . . . 147
D Charged particles at BBC associated with leading neutron

148

D.1 Multiplicity distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148
D.2 Single transverse spin asymmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . 148
E Leading neutron asymmetries at various center of mass energies

150

F Polarization of inclusive Λ0 production

152

Chapter 1

Introduction
The property of the nucleon which comprises the real world is one of the most important interestsin physics. In the 1970s, experiments of electron-proton Deep Inelastic Scattering (DIS) foundthe intrinsic structure that we currently consider as elementary particles namely quarks andgluons (partons) in the nucleon. DIS efforts have allowed to extract important information on
the partonic structure of the nucleon to a high accuracy. The Quantum Chromo Dynamics(QCD) parton model has been shown to be very reliable in the perturbative region.
On the other hand, small momentum transfer reactions which can not be described by the
perturbative QCD are still less explored. They can improve our knowledge on the inner hadronic
reaction. The prominent effects are the consequence of the strong correlation between sea-quarkand sea-antiquark with the quantum number of the meson [1], for example, a pion exchange
via p → nπ and p → ∆π. The measurements of the neutron in the very forward kinematics,

which is called a leading neutron, in ISR at CERN [2, 3] and, H1 and ZEUS at HERA [4]-[9]are successfully described by a model based on an One Pion Exchange (OPE) in Regge calculus[10]-[15] with the absorptive correction [14]. Moreover, there is a meson-cloud model which deals
with the quantum number of the meson as a hadronic structure. This model also describe theleading neutron production in the ISR experiment by the nπ state in the proton (a pion-cloud)[16].
Such works are especially important for nuclear physics, as one wants to know what amountof the nucleon-nucleon interaction is describable in terms of meson exchange forces. Therefore,
detailed investigations for the leading neutron are necessary.
Leading neutron from polarized pp collision has one more interesting behavior, single transverse spin asymmetry (AN ) which is defined asAN ≡dσ ↑0 − dσ ↓0,
dσ ↑0 + dσ ↓0(1.1)

where superscripts of ↑ and ↓ indicate spin up and down of proton beams, respectively. “0”denotes the unpolarized state. We discovered finite AN for leading neutron productions at a
12 o’clock interaction point (IP12) experiment at Relativistic Heavy Ion Collider (RHIC) andreport on this thesis.
56

CHAPTER 1. INTRODUCTION

The IP12 experiment was initially motivated to search for sizable AN of production particles√
in very forward region, 0±2.8 mrad, in polarized pp collision at s=200 GeV for monitoring thebeam polarization at the collision point in RHIC. The AN for π 0 production was one candidate√
based on the observation of large asymmetries in large xF at s = 19.4 GeV (section 1.1).
Thus, the detector was primary designed to detect photons from π 0 s, but was also capable todetect neutrons. We did not observe finite AN for photons and π 0 s, whereas we discovered large
AN for leading neutron productions.
For further investigation, we have measured the xF dependence of AN and cross sectionof leading neutrons with the PHENIX detector in similar kinematics and energy regions. The
observed asymmetries are consistent with the IP12 result within the errors and show almost noxF dependence. The measured cross sections are consistent with the xF scaling claimed by the
ISR experiment. These cross sections for large xF neutron production, as well as those in ep
collisions, are mainly reproduced by the OPE model in the Regge framework. Therefore, theobserved large asymmetry for neutron productions is considered to come from the interference
between a spin-flip amplitude due to the pion exchange and non-flip amplitudes from otherReggeon exchange (Appendix A.1). Remarkably, the OPE model successfully describe one ofpolarized phenomena of inclusive hadron production; a polarization of inclusive Λ productionin pp collision (Appendix F). Moreover, the meson-cloud model also describes the cross sectionof leading neutron productions. (section 1.2)
Before introducing the measurements of leading neutrons in RHIC, extensive experimental
and theoretical reviews are described.

1.1

AN for inclusive productions in the forward kinematics

AN for inclusive productions, for example in pp or DIS, has been thought to be originated froma transversity distribution, δq, of polarized Parton Distribution Function (PDF). δq is definedas,δq ≡ q↑↑ (xB ) − q↑↓ (xB ),↑(↓)

where q↑(1.2)

is the probability of (anti-)parallel spin state for the parton to the nucleon. xB is a

Bjorken-x which is defined as a fraction of longitudinal momentum of the parton to that of thenucleon. According to the expectation from the leading-twist pQCD, AN should be suppressed√
as αs mq / s, where mq is the quark mass [17]. Therefore, the experimental observation oflarge AN at high center of mass energy of 19.4 GeV in pp collision at Fermilab, shown in next√
section, was initially a surprise and this asymmetry remains to the RHIC energy at s=200
GeV [18, 19, 20].
Recently three different mechanisms, Sivers effect1 [21], Collins effect [23] and Higher-twist
effect [24, 25], have been studied extensively as the possible origin of AN in high energy collision.
1

The Sivers PDF has suggested the relation of orbital angular momentum of partons in the nucleon [22].

7

CHAPTER 1. INTRODUCTION

However, these theories would not be applicable to our measurements since kinematics is in nonperturbative region; pT coverage is approximately below 0.2 GeV/c due to the acceptance limit.
We do not discuss them in this thesis.
AN of pion productions at FNAL E704
The experiment E704 was carried out at Fermi National Accelerator Laboratory (FNAL) for
the study of polarization effects using 200 GeV/c (anti-)proton beam incident on a 1.0 m√
liquid hydrogen target ( s=19.4 GeV). Significant AN were observed in this experiment for the
fragmentation region of polarized proton beams [26]–[29].
Figure 1.1 shows AN for inclusive pion productions vs. xF in the region of 0.7 < pT < 2.0
GeV/c, where xF is Feynman-x which is defined as a fraction of the incident proton momentum
carried by the pion in initial direction of the proton. The observed AN are strikingly large,
increasing with xF and reaching a magnitude of 40% for charged pions around xF =0.8. Thereπ−
is a clear sign dependence of the pion charge, with Aπ+
N >0 and AN <0, and magnitudes andthe xF dependence are approximately same. The π 0 asymmetry is positive but magnitude isabout a half of charged pions.
Pion asymmetries have not only the xF but also pT dependence. Figure 1.2 includes AN
values in lower pT region, 0.2 < pT < 0.7 GeV/c, in case of π ± . The magnitudes are smallerin the low pT region and it indicates that AN increases dramatically in pT above 0.7 GeV/c.
0

Figure 1.3 shows the pT dependence of AπN in the region of 0.5 < xF < 0.8. It also indicatessame behavior as π ± , increasing with pT .
Our experiment was initially motivated by this π 0 asymmetry; we needed to find significant
AN to monitor spin directions of proton beams at the PHENIX interaction point (Local Po0

larimeter; Appendix B). Even if the extrapolation of AπN to pT below 0.2 GeV/c, which is our
experimental coverage, seems to be small or zero, AN for photons decayed from π 0 with higher
pT has a possibility to be measured.

1.2

Leading neutron production

1.2.1

Definition of kinematics for the leading neutron production

Kinematics of the leading neutron production for pa → nX reaction based on the Reggeon(meson) exchange model is shown in Figure 1.4. xF and pT of the leading neutron are definedas,xF= pL /pL(max) = En cos θn /Ep ∼ En /Ep ,

(1.3)

pT

= En sin θn ∼ xF Ep θn .

(1.4)

From these variables, the momentum transfer squared from the proton can be described as,p2t ∼ − T − t0
xF

where, t0 ≡1 − xF(m2n − xF m2p ).
xF

In ep collision, we can evaluate −Q2 by measuring the recoil electron.

(1.5)

8

CHAPTER 1. INTRODUCTION

AN

Figure 1.1: AN for inclusive π + , π − and π 0
productions from polarized pp scattering in√
s=19.4 GeV at FNAL E704. They are plotted as a function of xF in 0.7< pT <2.0 GeV/c.
[27]

Figure 1.2: AN vs. xF for inclusive π + and√π − from polarized pp scattering in s=19.4
GeV in two pT ranges, 0.2<pT <0.7 GeV/c and0.7<pT <2.0 GeV/c. [27]

0.2

0.15

Figure 1.3: AN vs. pT for inclusive
π 0 productions from polarized pp scat√tering in s=19.4 GeV. The data isshown for 0.5< xF <0.8. [26]

0.1

0.05

0
0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2
2.2
pT (GeV)9

CHAPTER 1. INTRODUCTION

X

-Q2

a

s’

R
p
(mp,Ep)tθnN(mn,En)

Figure 1.4: A schematic diagram of the leading neutron production, pa → nX, on the Reggeon
exchange model showing with Lorentz invariant variables s′ , Q2 and t. a is proton or positronfor pp or e+ p reactions. R indicates a Regge trajectory with isospin odd such as π, ρ, a2 andPomeron-π in the Regge theory. In case of the pion exchange model, R means π.

1.2.2

ISR experiment at CERN

The ISR experiments at CERN measured leading neutrons in pp collisions at√
s=30.6, 44.9, 52.8

and 62.7 GeV [2, 3]. Neutrons were detected in a Sampling Total Absorption Counter (STAC)
with the energy resolutions of 26% and 19% for 20 GeV and 30 GeV neutrons, respectively.
They were placed for covering 20±4, 66±11 and 119±11 mrad (finite angle) and 0±1 mrad
(zero degree) for the measurement of very forward kinematics.
Invariant cross sections as a function of xF for various pT , which can be evaluated by equation
1.4 using the combination of three finite angles (θn ) and four proton energies (Ep ), are shown in
Figure 1.5. Only in the zero degree measurement (pT ≈0 GeV/c), large cross section is shownaround xF =0.8. Invariant cross sections at pT ≈0 GeV/c for various center of mass energies areplotted in Figure 1.6 and they are well scaled by xF , not by the center of mass energy. These
interesting behaviors are well described by the pion exchange model (section 1.2.4).
There is one more evidence of the pion exchange based on the Regge phenomenology as
Figure 1.7. Invariant cross sections are plotted as a function of (1 − xF ) for xF ≥0.88 and fitted

by straight line. In the Regge theory, cross section is described with a Regge trajectory α(t) as
πE

d3 σ
d2 σ
aR ′
∼
−
∝ (1 − xF )1−2α(t) σtot
(s , t),
dp3
dtdxF

(1.6)

aR (s′ , t) is total cross section for a and R scattering in Figure 1.4. From the slope
where σtot

in Figure 1.7, α(t) can be extracted. As an average all over the energies, it is obtained asα(0)=0.11±0.15 and is consistent with the intercept of the pion trajectory, απ (0) = 0. 2
2

In case of Pomeron trajectory, αP (t = 0)=1.08.

10

CHAPTER 1. INTRODUCTION

Figure 1.5: Invariant cross sections of neutron productions vs. xF for various pT in pp collisions
at ISR. The lines are hand drawn to guide to eye. [2]30s=30.6 GeV
s=44.9 GeV
s=52.8 GeV

25

3

2

3

E d σ/dp (mb/GeV /c )Neutron spectra at zero-angle from ISR experiments

s=62.7 GeV

3

20
15
10
5
0
0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9
1
xF = pL/p max

Figure 1.6: Invariant cross sections of leading neutrons vs. xF for various center of mass energies in pp Figure 1.7: Invariant cross sections of leadcollisions at ISR. [2]ing neutrons as a function of s′ /s ∼ 1 − xF
at the momentum transfer squared t ≈ 0.
[2]

11

CHAPTER 1. INTRODUCTION

1.2.3

H1 and ZEUS experiments at HERA

H1 and ZEUS experiments at HERA measured leading neutrons in ep collision at

√

s∼300

GeV [4]∼[9]. We show a part of ZEUS results in this section. At the ZEUS experiment, aForward Neutron Calorimeter (FNC) detected leading neutrons with the energy resolution ofpσ/E = 65%/ E(GeV), covering 0±0.8 mrad.

Result of the cross section is plotted in Figure 1.8 with the ISR pp result. Neutron production

from ep scattering also has a peak structure around xF =0.8 and is consistent with the curve
which is calculated by the OPE model for ISR results scaled by 0.41. Moreover, they measured
the D meson production associated with the leading neutron. The charm production can begenerated via a photon-gluon fusion in ep collision and is sensitive to the gluon in a pion if
the pion is exchanged. Result is shown in Figure 1.9. The data can be reproduced by the
RAPGAP[30]+OPE model only.
According to the results, they also tried to extract a structure function of pion, F2π (xπ , Q2 )where xπ = xB /(1 − xF ) in the framework of the OPE model [5]. F2π (xπ , Q2 ) is expected to beevaluated by the measurement of a neutron-tagged structure function, F2LN (xB , Q2 , xF ), withthe following relationship,
F2π (xπ , Q2 ) = Γ(Q2 , xF )F2LN (xB , Q2 , xF ),(1.7)

where Γ(Q2 , xF ) is the inverse of the pion flux factor integrated over the measured t region
and corrected for the t-averaged absorption effect. Obtained F2π are shown in Figure 1.10
with the curve for the F2 (xB , Q2 ) of the proton scaled by 0.36 and the calculation from GRVparameterization [31]. It has approximately the same xB and Q2 dependence as F2 of the proton.
The data provide new constraints on the shape of the pion structure function for xπ <10−2 .

1.2.4

Regge theory

In 1935, Yukawa predicted that there must be a particle, now known as the pion, which would
carry the strong interaction [32]. However, we now know that although the pion exchange is
an important component of the static force in high energy scattering, exchange of a very large
number of particles is involved. Regge theory [33] provides a simple quantitative description of
the combined effect of all these particle exchanges.
Regge theory is the framework based on the analytic property of scattering amplitudes
and it has been studied before the establishment of QCD. Unlike QCD which is starting fromthe first principle, Lagrangian, it is difficult to obtain a comprehensive assessment for theparticle interaction. However, its technique is extremely useful even now, especially in the
non-perturbation frame.
Cross section for inclusive scattering in the region of small momentum transfer
The cross section for an inclusive reaction, ab → cX, in the region of small momentum transfer

is described by a diagram of Figure 1.11. Rb scattering term, total cross section Rb, can be

CHAPTER 1. INTRODUCTION

12

Figure 1.8: Differential cross sections normalFigure 1.9: Differential cross sections for
γp→nX
ized by σtotof the leading neutron vs. xFneutron-tagged D ∗± vs. neutron xF from ep colat pT ≈0 GeV/c. Closed and open circles show
lisions at ZEUS. D ∗± was detected in the Cenep → nX at ZEUS and pp → nX at ISR, retral Tracking Detector (CTD) covering |η| <1.5.
spectively. The curve is the expectation of the
Only RAPGAP+OPE model can reproduce theOPE model for ISR data scaled by 0.41. [5]data. [7]Figure 1.10: F2π as a function of xπ
for the pion in bins of Q2 determined
for 0.64 < xF < 0.82. The pion flux
used to determine F2π is the effective
OPE flux (EF) used in hadron-hadron
reaction. Only statistical uncertainties are shown. The dotted lines are
the F2 (xB , Q2 ) of the proton, scaled
by 0.361. The solid curves are F2π
obtained from the GRV parameterization. [5]CHAPTER 1. INTRODUCTION

13

replaced as Figure 1.12 by an optical theorem with new Reggeon Rk (Triple-Regge diagram).
A term Figure 1.12 contributes to the cross section, d2 σ/dtdxF , asi(φ(α1 (t))−φ(α2 (t)))
bR ′gacRi (t)gacRj (t)Gij(1 − xF )1−α1 (t)−α2 (t) σtot(s ),k (t)e(1.8)

where, gacRi (t) and gacRj (t) are the couplings of Reggeon Ri and Rj to the hadron ac, respectively. Gij
k (t) is the triple-Reggeon vertex and φ(α(t)) is the phase arising from the signaturefactor, 1 ± e−iπα(t) . All contributions of Reggeon for Ri,j,k should be considered to obtain thebR (s′ ), convenient database provided by Particle Data Group (PDG) [34]. If s′
result. For the σtotbR (s′ ) can be described by the Pomeron exchange (R =Pomeron).
is bigger, σtotk

The leading neutron production in pp scattering, pp → nX, has been obtained by exchangesof π, ρ and a2 mesons. In case of the pion exchange, the cross section is described as follows[10].
2dσpp→nX−t
1 2gpnπpπ ′=(1 − xF )1−2απ (t) [F (xF , t)]2 σtot(s ),dtdxF π
4π 4π (t − m2π )2(1.9)

2 /4π = 27.5 [35] is the pnπ coupling constant and α (t) is approximately α′ t with α′ =where gpnππ

1 GeV−2 for the pion trajectory. F (xF , t) is the form factor included the effect of the final state
re-scattering of the neutron. For the hadronic charge exchange experiments, a good descriptionof most of the pn → pX data [36] is obtained using Bishari flux [37], which corresponds toequation 1.9 with F (xF , t)=1.

Equation 1.9 can be represented by a Regge factorization with the pion flux factor, fπ/p (xF , t),simply 3 .
dσpp→nXpπ ′= fπ/p (xF , t)σtot(s ).
dtdxF π(1.10)

Invariant cross section is obtained as Ed3 σ/dp3 ≈ −(1/π) · d2 σ/dtdxF (Equation 1.6) andcalculation results are plotted with ISR data in Figure 1.14. Here, cross sections were calculated

by not only the pion exchange but also ρ, a2 exchanges. Two step process, p → ∆ → n were also

considered. Regge theory well describes the leading neutron production and shows remarkable
component is from the pion exchange. As a result for the leading neutron production in highs′ , triple-Regge diagram ππP as Figure 1.13 is considered to be dominant.
3

Recently novel theoretical tools including leading neutron production based on standard pQCD have beendeveloped with a non-perturbative distribution (fracture function) which allow to absorb collinear singularities
at leading order in the QCD coupling constant.

14

CHAPTER 1. INTRODUCTION

c

2
a

c

a

R

a
R

=

b

c

X

R

b

b

Figure 1.11: Left : Amplitude for ab → cX via Reggeon exchange. It is same as right diagram.

c

c

a

a
Ri

b

n
p

p
π

Rj
Rk

n

b

t=0

Figure 1.12: Triple Regge diagram for inclusive
scattering amplitude. It is same as Figure 1.11
using an optical theorem for Rb scattering with
new Reggeon Rk .

p

π
P

p

t=0

Figure 1.13: Dominant process for the leafing
neutron production in high s′ .

Figure 1.14: Invariant cross sections for inclusive neutron productions, pp → nX, as a functionof xF for p2T =0 (GeV/c)2 . The long dashed curve shows the contribution from the pion exchange.
The dotted curve is the ρ,a2 -exchange contribution, the dashed curve shows the contribution
from the two-step process p → ∆ → n and the dot-dashed line is sum of them. The solid curve
represents the sum of all components. The experimental data are from ISR. [10, 15]15

CHAPTER 1. INTRODUCTION

1.2.5

Meson-Cloud Model

The meson-cloud model deals with the proton as a bare baryon surrounded by a virtual meson
cloud, such as p → nπ and p → ∆π states4 . This model gives a best description for the recent
interesting result from the Drell-Yan experiment at FNAL E866, there is an excess of the d¯quark relative to the ū quark in the nucleon sea [41], via the d¯ quark in the π + of a pioncloud (p → nπ + ). D’Alesio and Pirner applied to the meson-cloud model to leading neutron
productions in the ISR experiment and successfully describe by the pion-cloud model [16].
Cross section for the leading neutron production
A schematic view of the leading neutron production based on the pion-cloud model is shownin Figure 1.15. The pion is stripped from one proton and scattering with other proton. The
remaining neutron runs forward on the proton axis roughly. This process is basically same as
the pion exchange shown as Figure 1.4. Thus, the differential cross section is given by the pion
flux factor times the total p-π cross section in the same way of the Regge calculation as, 5dσpp→nXpπ ′= Fnπ (xF , pT )σtot(s ),2
dxF dpT π(1.11)

where the flux factor Fnπ (xF , pT ) gives the probability for the splitting of a proton to theneutron-pion system and is calculated based on the meson-cloud model.
On the meson-cloud framework, proton is described as,
(
)
XXZ
√
2
λλ′
|p ↑i =
S |p0 ↑i +dxF dpT φBM (xF , pT ) |B, M ; xF , pT iλλ′ BM=√S(|p0 ↑i +

×

"r

XZ

′

dxF dp2T φλλN π (xF , pT )λλ′

1|p, π 0 ; xF , pT i +3r2|n, π + ; xF , pT i + · · ·
3

#)

,

(1.12)′

where, φλλBM (xF , pT ) is the probability amplitude to find a baryon B with longitudinal momentum fraction xF , transverse momentum pT and helicity λ and a meson M with longitudinal
momentum fraction (1 − xF ), transverse momentum −pT and helicity λ′ inside a proton with√spin up. S is the renormalization constant, which is fixed by hp|pi = 1 and gives the amplitude

for the bare proton. The pion flux in equation 1.11 is calculated as,
2
2 X λλ′φN π (xF , pT ) .
Fnπ (xF , pT ) = π
3′(1.13)

λλThe light-cone, covariant Regge and Skyrme-type model approaches [42, 43] were performed to
calculate the pion flux factor. In addition, if an impact parameter b in Figure 1.15 is small, the4

A similar fluctuation for intrinsic strange and charm in proton, such as p → K + Λ and DΛc , can be possible.
However they are less strong because they are higher excitation states. [38, 39]
5
We use dp2T = −xF dt from equation 1.5. Negative sign is ignored in this discussion.

CHAPTER 1. INTRODUCTION

16

projectile proton and the neutron are re-scattering which is called the absorption effect. The
absorption correction is also discussed in [16].
Invariant cross section is obtained as Ed3 σ/dp3 ∼ −(xF /π) · d2 σ/dxF dp2T and shown in

Figure 1.16. Calculated lines are scaled by a factor of 1.2 for the normalization of other contri-

butions, for example ρ exchange and ∆ → n, estimated to be 20%. Data points show the cross√sections in pp collision at s = 53 GeV from the ISR experiment. All models describe well the
cross section of the leading neutron production.

Figure 1.15: A schematic view of the Figure 1.16: Invariant cross sections for the inclusive
inclusive neutron production based neutron production in pp collision as a function of xF
on the pion-cloud model. If an im- at pT =0 (GeV/c) calculated by three different models
pact parameter b is small, the pro- based on the pion-cloud model. Data points show the√jectile proton and the neutron are re- cross sections at s = 53 GeV from the ISR experiment.
scattering (absorption effect). [16][16]

In this thesis, we report measurements of leading neutron productions in polarized pp col√lision at s=200 GeV from the IP12 experiment (Part I; Chapter 2–4) and the PHENIX
experiment (Part II; Chapter 5–9). For the IP12 experiment, the experimental overview is
described in Chapter 2. The analysis for the asymmetry calculation is discussed in Chapter
3 and the results are presented in Chapter 4. For the PHENIX experiment, the experimental
overview is described in Chapter 5. The simulation study for the leading neutron detection are
described in Chapter 6. The analyses for the cross section and the asymmetries are discussed
in Chapter 7 and 8, respectively. Chapter 9 shows the results of the PHENIX experiment. We
present the discussion in Chapter 10 and the conclusion in Chapter 11.

Part I

IP12 experiment

17

Chapter 2

IP12 experiment2.1

Overview

Extracting the spin state of partons in the nucleon is one of main physics objects at RHIC
which has the capability of colliding polarized protons [44] 1 . Most of spin related analyses forinelastic scattering will extract the spin information of partons though the spin asymmetries,
which are defined as differences of cross sections with the different spin combinations of beams
as,
+0

−0

++

+−

• Longitudinally polarized combinations : AL ≡ − σσ+0 −σ, ALL ≡ σσ++ −σ
+σ+−
+σ−0
↑0

↓0

↑↑

↑↓

−σ
2
• Transversely polarized combinations : AN ≡ σσ↑0 +σ
AN N ≡ σσ↑↑ −σ
↓0 ,
+σ↑↓

where superscripts +(−) indicates the helicity state of beams.
A layout of the RHIC facility is shown in Figure 2.1. In the RHIC ring, proton beams, which
are called Blue and Yellow beams, preserve their polarizations (section 2.2.1) by a Siberian snake(section 2.2.4) as the vertically transverse direction and polarizations have been measured by
the pC-CNI polarimeters (section 2.2.5). We can study the physics which are related to thetransverse spin by colliding them directly (Transverse run period). For the study of the parton
helicity distribution, which will appear as ALL , it is necessary to rotate spin directions of protons
to the longitudinal. The spin rotator magnets (section 2.2.4), which have been installed upand down-streams of the collision points at PHENIX and STAR, enable us to take longitudinalbeams by rotating protons (Longitudinal run period).
For the confirmation of the longitudinal polarization at collision point, we need to develop
a new polarimeter which can evaluate the spin direction by a production process with a sizable
asymmetry in pp collision (Local Polarimeter : details at Appendix B). To establish the local
polarimeter system, we searched for a single transverse spin asymmetry, AN , at 12 o’clockinteraction point (IP12 experiment) in 2001-2002. Neutral particle detection in the very forward
kinematics, ±2.8 mrad, was chosen due to accessibility with the existing PHENIX detectors [46].
1

Other is the searching for Quark Gluon Plasma (QGP) state by colliding heavy ions as RHIC’s name indicates[45].
2
In some literature, ANN is denoted as AT T .

18

CHAPTER 2. IP12 EXPERIMENT

19

The IP12 experiment was originally designed by a ElectroMagnetic Calorimeter (EMCal)√
to detect the photon from π 0 s based on the observation of a large pion asymmetry at s=19.4
GeV (section 1.1). The detector consisted of a scintillation counter as a charged particle veto,a preshower EMCal, a 60 (5×12)-element array of PbWO4 EMCal and the two scintillation
counters with an iron block in between to distinguish neutrons. The energy resolution wasp∆E/E ≃ 10%/ E(GeV) and the position resolution was about 1 mm. We also installed a

Hadron Calorimeter (HCal) to measure only neutrons in the opposite side. The setup included
a photon veto, consisted of a 5 cm thick lead block followed by a scintillator, the HCal made oftungsten plates with optical fiber readout planes and five PbWO4 crystals for the reconstructionof neutron position. For the neutron measurement with the HCal, the energy resolution was
estimated to be 40–50% and the position resolution evaluated to be 3–4 cm studied by thesimulation which was prepared based on the test beam result.
For the data taking, pp collision was triggered by the scintillator hodoscopes located ±1.85

m from the collision point covering |η| = 2.2 to 3.9 and we also required some energy depositin each calorimeter. Average polarization for the Blue and Yellow beams were 12 and 17 %,respectively.

2.2

RHIC-AGS complex for the Polarized Proton Collider

The study of high energy polarized proton beams has been a long term program at BNL with thedevelopment of polarized beams in a Booster and an Alternating Gradient Synchrotron (AGS)rings for the fixed target experiments. The capability of polarized proton beams has been
extended to the RHIC machine. The first collision at RHIC was in 2000 and the performancehas been improved every year. The main components used for the acceleration of proton beams
at RHIC are shown in Figure 2.1.
We will describe the overview of the RHIC as a polarized proton collider. More details arefound in references [47, 48]. Polarized proton injection uses an optically-pumped polarized H
ion source (section 2.2.1). It produces 500 µA in a single 300–400 µs pulse, which corresponds
to 9–12×1011 polarized H. The polarization have reached 80% or more at the source. There
are several steps from polarized H pulse to a bunched polarized proton beam; the LINAC, theBooster, the AGS then the RHIC.
A pulse of polarized H ions are accelerated to 200 MeV kinetic energy in the 200 MHz
LINAC. The pulse of H ions is strip-injected and captured into a single bunch as polarized
protons in the Booster. The single bunch of polarized proton is accelerated in the Booster to
1.5 GeV kinetic energy and then transferred to the AGS, where it is accelerated to 24.3 GeV(RHIC 100 GeV run parameter). Then, the polarized protons are transferred to the RHIC. The
AGS to RHIC transfer line has been designed to transport proton beams in the energy range,from 20.6 GeV to 28.3 GeV. Each of RHIC rings can be filled with up to 120 polarized proton
bunches from the AGS, in which case the time between bunch crossing at Interaction Points

20

CHAPTER 2. IP12 EXPERIMENT

IP12 experiment(2001-2002)
Hydrogen gas jet target system
RHIC pp CNI polarimeter

RHIC pC CNI polarimeter

Siberian Snake
Spin rotator

PHENIXBlue ring
Yellow ringSTAR
Polarized Ion
Source

AGS-to RHIC Transfer Line
BOOSTERAGS quasi elastic polarimeter
AGS
Warm Snake

200 MeV polarimeter

AGS pC CNI polarimeter

RF dipole
Cold snake

Figure 2.1: A layout of the RHIC facility. Polarized protons are accelerated from the source
through a LINAC, a Booster synchrotron, and the AGS before being injected to the RHIC rings.
Several of the components used to maintain polarization throughout the acceleration stages areshown. Location of polarimeters are also noted. The places of IP12 and PHENIX experiments
are 12 o’clock and 8 o’clock points as a clock.

Figure 2.2: The pattern of the polarization signs of the bunches in the two counter-rotating
beams in RHIC. It is typically desirable to collide equal numbers of (++,+−,−+,−−) bunches
at each experiment, where +− represents a bunch in one beam with polarization up colliding
with a bunch in the other beam with polarization down in transverse pp mode.

CHAPTER 2. IP12 EXPERIMENT

21(IPs) is 106 nsec.
In the RHIC operation, for example 2005 year run, rings were filled with 111 polarized
proton bunches and 9 abort bunches to estimate a beam gas background. The same pairs ofbunches interact at each intersection region but spin directions of pairs are different in eachinteraction. It is typically desirable to collide equal numbers of (++,+−,−+,−−) bunches at
each experiment, where +− represents a bunch in one beam with polarization up colliding witha bunch in the other beam with polarization down. One solution which would satisfy all theintersections is to load the bunches of one ring (+ +−−, etc.), and load the bunches of the otherring with (+ − +−, etc.) as shown in Figure 2.2. The polarization sign of each bunch is decided

at the exit of H ion source where polarization direction is aligned in transverse. This is very
useful for the asymmetry analysis since any common systematic errors, for example detector
asymmetry, are canceled out by using those spin combinations (see Appendix A). After fillingof both rings is complete, the beams are accelerated to the flat-top energy.
Polarized proton beams have been accelerated, stored and collided in the RHIC rings at
center-of-mass energies of 62.4, 200 and 410 GeV so far. The acceleration of polarized beamin circular acceleration is complicated by the presence of numerous depolarization resonances.
During acceleration, the polarization may be lost when the spin precession frequency passes
through depolarizing resonances (section 2.2.2). Therefore the polarization is maintained by
the use of two partial Siberian snakes in the AGS (section 2.2.3) and two full Siberian snakes[49]–[51] in each RHIC ring (section 2.2.4). Besides constant polarized beam deliveries tothe experiments (PHENIX, STAR, etc.), the beam-development has also been continued. In
FY2006, polarization and intensity for the average store in RHIC reached 60% and 1.5×1011
protons/bunch, respectively. Polarized protons were first accelerated to the highest beam energy
of 205 GeV in RHIC with a significant polarization measured at top energy in FY2005 [52] andfurther high-energy beam commissioning has been continued in FY2006 towards the maximum
beam energy of 250 GeV.
The brief history of the RHIC facility, which is focused on the polarized proton beam
acceleration, is summarized in Table 2.1.

2.2.1

Polarized ion source

The polarized H− has been generated by Optically Pumped Polarized Ion Source (OPPIS)[53] was constructed at TRIUMF from the KEK OPPIS source (Figure 2.3). H+ ions are
extracted from the 29 GHz Electric Cycrotron Resonance (ECR) proton source and enter therubidium cell, which has polarized rubidium vapor by optically pumped using 795 nm laser andits polarization is kept by 2.5 T solenoid magnet. H+ capture a polarized electron by H+ -Rb
interaction and become hydrogen atom. Then, the hydrogen pass through a Sona region where
the polarization is transferred from electron spin to H nucleus spin by the Sona transition [54].
H capture un-polarized electron from sodium vapor and become polarized H− ions. It achieved
85% polarization for 10×1011 protons per pulse at FY2006.

CHAPTER 2. IP12 EXPERIMENT

FY2001(2001–2)
FY2002(2001–2)
FY2003(2002–3)
FY2004(2003–4)
FY2005(2004–5)
FY2006(2005–6)

Polarized proton acceleration was commissioned in one RHIC ring (Blue ring)with two snakes and one pC polarimeter.
All snakes and pC polarimeters in both RHIC rings were commissioned andoperational. First transverse-spin polarized proton collisions were achieved.
Spin rotators at RHIC and pC polarimeter at AGS were commissioned andoperational. First longitudinal-spin polarized proton collisions were achieved.
AGS warm snake was commissioned. Hydrogen gas jet target wascommissioned and operated for the measurement of absolute polarization.
The calibration for pC polarimeters (in the blue-ring and yellow-ring) werecompleted.
AGS cold snake was commissioned.

Table 2.1: Summary of the history focused on the polarized proton acceleration in RHIC.

Figure 2.3: A schematic view of BNL OPPIS.

22

23

CHAPTER 2. IP12 EXPERIMENT

2.2.2

Depolarization resonanceBean polarization during acceleration can be compromised by depolarization mechanisms driven
by magnetic field. The evolution of the spin direction of polarized protons in external magnetic
field is governed by the Thomas-BMT equation [55],
dP~
=−
dt



e
γm

h

i
~ ⊥ + (1 + G)B~ k ) × P~ ,
Gγ B

(2.1)

where the polarization vector P~ is expressed in the frame that moves with the particle. γ is~⊥
Lorentz factor, γ = E/m, and G=1.793 is the proton anomalous magnetic momentum. B~ k are perpendicular and parallel magnetic fields to the beam direction, respectively. Thisand B
simple precession equation is very similar to the Lorentz force equation which governs theevolution of the orbital motion in an external magnetic field,


e
d~v
~ ⊥ × ~v .
=−
B
dt
γm

(2.2)

From comparing these two equations, it can readily be seen that, in a pure vertical field, thespin precessed Gγ times orbital revolution; Gγ, which is called spin tune, νsp , gives the numberof spin precession frequency. The Thomas-BMT equation also shows that at low energies (γ∼1)~ k can be quite effective to the spin motion, but at high energies, transverse
longitudinal fields B~ ⊥ is dominated.
fields B
There are two main types of depolarizing resonances.
• Imperfection resonances : It is driven by magnet errors and misalignment. It arises when
νsp = Gγ = n, where n is an integer. Therefore, these are separated 523 MeV steps.

• Intrinsic resonances : It is driven by the focusing fields and arises when νsp = Gγ =kP ± Qy , where k is an integer, Qy is a vertical betatron tune and P is a superperiodicity.

The superperiodicity is the number of repeated section of bending and focusing magnets. The
betatron tune indicates a betatron frequency around ring. For example, P =12 and Qy ∼8.8 at

the Brookhaven AGS. For most of the time during the acceleration cycle, stable spin direction
coincides with the main vertical magnetic field. Close to a 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 calculated
analytically and is given by the Froissart-Stora equation [56],2
Pf = 2e−π|ǫ| /2α − 1 Pi ,(2.3)

where Pi and Pf are the polarizations before and after crossing resonances. ǫ is the resonancestrength and α is the rate of resonance crossing, α = d(Gγ)/dθ. When the beam is slowly(α ≪ |ǫ|2 ), accelerated through the resonance, the spin vector will adiabatically follow the stablespin direction resulting in spin flip. However, for a faster acceleration rate partial depolarization

24

CHAPTER 2. IP12 EXPERIMENTor partial spin flip will occur. Traditionally, the intrinsic resonances are overcome by using
a betatron tune jump, which effectively makes α large, and the imperfection resonances are
overcome with the harmonic corrections of the vertical orbit to reduce the resonance strength.
At high energy, these traditional methods become difficult and tedious because the strengthof imperfection resonances generally increase linearly with the beam energy. The calculated
strength of imperfection and intrinsic resonances for AGS and RHIC are shown in Figure 2.4–
2.6. To avoid these resonances, proton beams have been accelerated with rotating spin direction
by snake magnets; partial snakes in AGS (section 2.2.3) and Siberian snake in RHIC (section
2.2.4).

2.2.3

Partial snakes in AGS

We used a warm and a cold snakes to preserve beam polarizations in the AGS acceleration.
The warm snake is composed of a normal helical dipole magnet and has been used with a 5%snake strength (s=0.05, corresponds to 9 degrees) of original Siberian snake (section 2.2.4). As
a result, spin tune obeys,cos(πνsp ) = coss
2cos(πGγ),(2.4)

which does not satisfy the imperfection resonance conditions. However, it can not overcome
strong intrinsic resonances. It has been treated by the spin resonance excitation using a RF
dipole [57]. A stronger Siberian snake could also be more effective to overcome strong intrinsic
resonances in AGS. A superconducting helical dipole magnet as a 15% partial Siberian snake,the cold snake, is developed and in use.

2.2.4

Siberian snake and Spin rotator in RHIC

To preserve beam polarization in RHIC acceleration, we used Siberian snake which is composedof four superconducting helical dipole magnets, producing a central field of up to 4 T. It rotates
the spin direction by 180 degrees without a net orbital distortion in each cycle. Spin motion
image is shown in Figure 2.7 (Left). As a result, spin tune is change from νsp = Gγ to
1/2, independent of the beam energy. Therefore, neither imperfection nor intrinsic resonanceconditions can ever be met as long as the betatron tune is different from a half-integer. With one
or two Snakes all depolarizing resonances should be avoided. However, if the spin disturbancefrom small horizontal fields is adding up sufficiently between the Snakes, depolarization canstill occur. This is most pronounced when the spin rotations from all the focusing fields add
up coherently which is the case at the strongest intrinsic resonances. At RHIC, two Snakes canstill cope with the strongest intrinsic resonance. It was successfully demonstrated to preserve
the beam polarization at 60% in FY2006.
The Spin rotators, which is basically same composition as Siberian snake, placed aroundthe experiments to rotate the spin direction by 90 degrees again without generating net orbit
distortions. Spin motion image is shown in Figure 2.7 (Right). Spin rotators are required at the25

CHAPTER 2. IP12 EXPERIMENT

intersection points of PHENIX and STAR to allow for the measurements of spin effects withlongitudinally polarized protons; The spin rotators rotate the spin direction from the verticaldirection into the horizontal plane on one side of the interaction region to provide longitudinalpolarization and restore it to the vertical direction on the other side.
Design and construction of the snakes and rotators is discussed in [58, 59].

2.2.5

pC- and pp-CNI polarimeters

The single transverse spin asymmetry with the beam polarization P is expressed by,
AN =

1
ǫN
P

(2.5)

where ǫN is called raw asymmetry which is calculated by the production yields (section A.2.1
and A.3). We can extract P by measuring ǫN of the physics process which has known AN .
The proton-carbon Coulomb Nuclear Interference (pC-CNI) polarimeter, which takes ad2
vantage of sizable analyzing power, ApC
N ∼ 0.01 at −t∼0.01–0.02 GeV , in the elastic scatteringof polarized protons with carbon nuclei, serves as a fast feedback tool to tune up the beam
acceleration. ApC
N which originates from the interference between electromagnetic force andhadronic force was initially measured by the AGS experiment E950 [60].
The pC-CNI polarimeters are installed in the AGS and RHIC rings. They employed theultra-thin carbon ribbon target (3.5 g/cm2 thick and 5 µm wide typically), which have beendeveloped at IUCF. Polarimeters regularly collected one million events of recoil carbons from theelastic scattering process within 10 seconds. The pC polarimeter measures beam polarization
every a few hours in a store and also measures bunch-by-bunch polarization.
pC
The accuracy of the pC-CNI polarimeter was limited by the uncertainty of ApC
N . The AN was

measured at proton beam momentum in 22 GeV/c and we extrapolates it to 100 GeV/c using
a theoretical calculation [61]. The uncertainty of ApC
N directly affects the beam polarization
uncertainty. For example, (1.00±0.32) scaling uncertainty was applied to 2001-2002 run.
Absolute beam polarization has been measured by a proton-proton (pp)-CNI polarimeterwith a well calibrated polarized proton target; since it utilizes pp elastic scattering for beamand target protons, we can extract the beam polarization in high accuracy from the well known
target polarization. The accuracy of absolute beam polarization, ∆Pb /Pb has reached to 0.06
[62, 63].

2.3

Experimental setup

A plan view of the IP12 experiment is shown in Figure 2.8 and schematic views of each component are shown in Figures 2.9 and 2.10. Detectors placed in very forward directions, centeredon a production angle of 0◦ , behind the RHIC DX dipole magnets, which removed produced
charged particles.

26

CHAPTER 2. IP12 EXPERIMENT

Figure 2.4: The calculated strength of intrinsic spin resonance through the AGS
acceleration as a function of Gγ ∼1.9×
beam energy in GeV.

Figure 2.5: The calculated strength of intrinsic Figure 2.6: The calculated strength of imperspin resonance through the RHIC acceleration fection spin resonance through the RHIC acceleration as a function of proton energy.
as a function of γ = E/m.

Figure 2.7: Spin motion images through Siberian snake (Left) and spin rotator (Right).

27

CHAPTER 2. IP12 EXPERIMENT

West
East
EM Calorimeter
Hadronic Calorimeter(EMCal)(HCal)
Figure 2.8: A plan view ofCollision-Trigger
Preshower
N2
the IP12 experiment, not to
Hodoscopes
Postshower γ
CVN1scale. Shown are the prin-DX
Yellow BeamDX
Blue Beam

Collision Point

PbWO4
Plastic Scintillator

ciple components of the experiment. Schematic viewsof the east and west components are shown in Figure
2.9 and 2.10, respectively.

Lead
Iron

Figure 2.9: A schematic
view of the east detector;
the EMCal.

Figure 2.10: A schematic
view of the west detector;
the HCal.

CHAPTER 2. IP12 EXPERIMENT

2.3.1

28

Hodoscopes for beam-beam counters

The collision-trigger hodoscopes (beam-beam counters) were located ±1.85 m from the collision

point. The hodoscopes were formed with four sets of rectangular scintillators, with full azimuthal
coverage, and pseudo rapidity acceptance |η| = 2.2 to 3.9 in the vertical and horizontal directions(Figure 2.11). The time resolution of the hodoscopes gave a vertex resolution of 23 cm. Detailsof the selection of collision event are discussed in section 3.2.

Figure 2.11: A schematic view of hodoscopes. Sensitive areas are shown in the left. A pair wasplaced ±1.85 m up- and down-stream of collision point and covered 2.2< |η| <3.9 in horizontal
and vertical rapidity regions.

2.3.2

East detector : Electromagnetic Calorimeter

The east detector, facing the Blue beam, consisted of a scintillation counter as the charged particle veto (dimensions 10.5 cm × 25 cm × 0.6 cm, CV in Figure 2.8), a preshower electromagnetic

calorimeter, and a 60-element array of PbWO4 crystal electromagnetic calorimeter (EMCal).

These were followed by two scintillation counters (dimensions 10 cm × 24 cm × 0.6 cm, N1 andN2 in Figure 2.8) with a 2.8 cm thick iron block between them to observe hadron shower.

The preshower and the EMCal arrays were built from PbWO4 crystals of 2.0×2.0× 20.0 cm3 ,coupled to 3/4-inch photomultiplier tubes (HPK R4125) via silicone cookies and filters of 1/10
light reduction. The preshower consisted of five crystals forming a horizontal hodoscope, 10 cm(horizontal) × 20 cm (vertical) × 2 cm (deep). The EMCal was a 5 (horizontal) × 12 (vertical)
crystal array, 20 cm (22 radiation length, X0 ) deep.

A schematic view is shown in Figure 2.9. Picture of the EMCal and its coordinate aredisplayed in Figure 2.12 and 2.13.

CHAPTER 2. IP12 EXPERIMENT

29

Test experiment for the EMCal at SLAC
The preshower and the EMCal were calibrated by the electron beam at Final Focus Test Beam
(FFTB) Facility at SLAC. The FFTB electron beam was bunched as a few psec in 10 Hz andone bunch has a capability to have several electrons. We requested 10 GeV per electron at
the experiment. The EMCal was located on the plate which can be moved less than 1 mm
accuracy by electric operation. We corrected the gain tower by tower and evaluated the energy
and position resolutions with this experiment.
Figure 2.14 shows the observed energy spectrum which was a sum of all tower energies. We
confirmed the EMCal has a good energy linearity up to 100 GeV. In Figure 2.15, the energy
resolution was evaluated over the inverse square root energy and it is described as,
∆E
0.11
≈p
.
E
E(GeV)(2.6)

The reason why the resolution was poor than the typical calorimeter made of PbWO4 is the filter
for light reduction to keep the energy linearity. For the measurement of two photons decayedfrom π 0 separately, we employed a clustering algorithm; gathering towers which have energy
deposit above threshold. In this test, we also evaluated the effect for the energy measurementof the clustering algorithm with 0.1 GeV threshold. Figure 2.16 shows the cluster energies as afunction of the incident position for 30, 50 and 70 GeV incident energies. Typically the cluster
energy was reduced to 97% from the incident energy. If incident position was in the edge region
on the EMCal, for example y=10–12 cm, the cluster energy was more reduced due to the shower
leakage. These effects should be corrected based on this result.
The position is reconstructed by a centroid method with the logarithmic weights [64],
P




Ei
Wi xi
X= P
,
Wi = Max 0.0, ln(2.7)+ W0 ,
Wi
Esum

where, xi and Ei are the position and the deposit energy in each tower above threshold. Esumis the sum of Ei , which is defined as the cluster energy. W0 is a weight parameter and set to 4
in this analysis. The calculated position as a function of incident one is shown in Figure 2.17.
The position was well reconstructed by this method and we obtained position resolutions as
1.4 mm to 1.0 mm for 30 GeV to 70 GeV electrons, respectively. With the simulation study,
we consider it was not an intrinsic resolution, dominated by a beam spread (see next section).
In the edge region, the response of the position reconstruction was changed due to the shower
leakage as well as the energy response. It is corrected based on this result too.
Simulation by Geant3
At the IP12 experiment, the noise level was worse than the test experiment; it corresponds to
0.4 GeV, and we required 0.5 GeV threshold for the clustering algorithm. For the estimationof the effect to the clustering, for example the energy reduction and the energy resolution etc.,
we prepared a Monte Carlo simulation for the EMCal detector by Geant3 [65] and evaluatedCHAPTER 2. IP12 EXPERIMENT

30

its reliability by comparing with the result of SLAC experiment. Responses of the energy
and position measurements from the simulation are plotted in Figure 2.16 and 2.17 with theexperimental data. They agree well except for the position resolution; that for the simulation is
0.5 mm. It is because of the beam spread of electron beam which was mentioned as a few mmfrom the accelerator and we concern it was about 1.3 mm by this measurement. The clustering
response with the 0.5 GeV threshold was studied by the simulation.
For the neutron measurement, we did not test by known hadron beam. The responses of theenergy and position measurements for the neutron were estimated by the simulation, GHEISHA
was used for hadron reactions [65], and the energy response is shown in Figure 2.18. The cluster
energy fraction for neutrons is Ecluster : En ∼ 1 : 3 with a ∼ 100% energy resolution. As theneutron energy, we were using the cluster energy which were calibrated for photons since it is
impossible to do the energy correction. The position resolution was estimated to be 5 mm.

2.3.3West detector : Hadron Calorimeter

Actually we observed finite AN in the neutron sample detected with the EMCal during run
period. To confirm the neutron asymmetry, we installed a hadron calorimeter (HCal) facing
the Yellow beam, in west side as shown in Figure 2.8. The setup included a photon veto (γ̄in Figure 2.8) which consisted of a 5 cm thick lead block followed by a scintillator, the HCal,
and the postshower which was formed by five PbWO4 crystals as a horizontal hodoscope3 , toidentify left and right production of the neutron. A schematic view is shown in Figure 2.10.
The HCal was a prototype of the ZDC which have been installed in the RHIC collision
points (Details are described in section 5.2.1). The HCal had a transverse dimension of 10 cm
× 10 cm, and 2 interaction lengths, λI . The measured energy with the HCal was calibrated

by the cosmic-ray data taking after the IP12 experiment and its resolution was expected to be
40–50% for the energy above 20 GeV. The position was reconstructed by the postshower using
the centroid method and its resolution at the detector center was estimated to be 3–4 cm. Theenergy and position resolutions were estimated by the Geant3 simulation which well reproduce
the results of the test beam experiment [66]. The asymmetry was obtained by comparing theleft and right scattering only.

2.3.4

Data taking

For the data taking, we required the trigger as a coincidence of following conditions,• A collision event, with ≥1 Minimum Ionizing Particles (MIP) in each hodoscope, within
a 5 ns collision window (pp collision event).

• Minimum energy deposit in detectors, more than ∼5 GeV in the sum of EMCal or ∼20
GeV in the HCal.

3

Construction is same as the preshower in east detector.

CHAPTER 2. IP12 EXPERIMENT31

We also synchronized it with the RHIC clock which is the same frequency of the beam and sentfrom the RHIC accelerator. The energies and signal timings of all detectors were recorded by
Analog to Digital Converters (ADC; FERA LeCroy 4300B for the EMCal and LeCroy 2249W
for others) and Time to Digital Converters (TDC; LeCroy 2228A). The information of polarization directions (up, down or un-pol.) for both beams were provided by the RHIC accelerator
and sorted by Programmable Logic Unit (PLU; LeCroy 4508). They were recorded by the coincidence register (LeCroy 4448). A block diagram of the readout system for the IP12 experimentis shown in Figure 2.19. The trigger rate was 100–300 Hz, and the average live time of theexperiment was 75%. We collected 70 million and 8 million events for the EMCal and HCal
samples, respectively.
In this run period, average polarization for Blue and Yellow beams were (0.116±0.002±0.020)
and (0.169±0.002±0.020) respectively, errors show the statistical and systematic natures. In addition, the scaling error, (1.0±0.32), which was from the calibration of the polarimeter analyzing
power should be added (section 2.2.5). For the AN measurement, we added the systematic errors to the scaling errors by quadratic sum. Total scaling errors were (1.0±0.36) and (1.0±0.34)
for Blue and Yellow beams, respectively. Scaling errors should be affected to AN as 1.0+0.57−0.27
and 1.0+0.52−0.25 for Blue and Yellow beams based on a correction of 1/P (equation 3.5).

CHAPTER 2. IP12 EXPERIMENT

32

Figure 2.12: Pictures of the EMCal. Single PbWO4 crystal is mounted to the photomultiplieras bottom picture. The EMCal consists of 5 times 12 PbWO4 crystals as shown in the top left.

Figure 2.13: The definition of coordinates for
the EMCal.

CHAPTER 2. IP12 EXPERIMENT

33

Figure 2.14: The energy spectrum for the electron beam at SLAC.

Figure 2.15: The energy resolution of the EMCal for electrons as a function of the inverse squareroot energy.

CHAPTER 2. IP12 EXPERIMENT

34

Figure 2.16: The responses of cluster energies for various incident energies as a function ofincident y position. Closed and open circles indicate these from the test experiment and thesimulation, respectively. The responses was changed in the region of 10–12 cm where is theedge region on the EMCal (see Figure 2.13).

Figure 2.17: The reconstructed position vs. incident y position for 30 GeV electrons. Dashedlines are hand draws to guide the eyes. Error bars show the position resolutions which areabout 1.4 mm and 0.5 mm for the experimental and the simulation values, respectively. These
differences would be caused by the beam spread of electron and we concern it was about 1.3
mm by this measurement. The responses were changed in the edge region as well as the energy
response.

CHAPTER 2. IP12 EXPERIMENT

35

Figure 2.18: The cluster energy in the EMCal as a function of the incident energy in case ofneutrons estimated by simulation. The EMCal was calibrated by the electron beams. The
response was Ecluster : En ∼ 1 : 3 with a ∼ 100% energy resolution.

Figure 2.19: A block diagram of the read out system for the RHIC-IP12 experiment.

Chapter 3

Analysis
In this chapter, analysis procedure is introduced by the EMCal analysis. Analysis for the HCal
is described in section 3.6.

3.1

Selection of the collision event

The collision was identified by the z-vertex distribution which was reconstructed by the time
information of hodoscopes as follows,
E−HodoW−Hodoz−vertex = (Tave.
− Tave.
)c/2,

(3.1)

E−Hodo and T W−Hodo are average TDC values, which were truncated means after slewing
where Tave.
ave.

correction in this analysis, of the east and west hodoscopes, respectively. c is light speed. Figure
3.1 shows a z-vertex distribution and the collision vertex was seen around 0 cm with the widthof ∼50 cm. We found peaks at ±185 cm and they are called the beam gas events, a particle

shower events by the interaction between the beam and particles in the beam pipe. Basically,

the width of the beam gas event, ∼23 cm, is dominated by the vertex resolution of hodoscopes.
For the selection of the collision events with the beam gas as low as possible we used the

E−Hodo +information of not only the z-vertex but also a TDC sum, which was defined as, (Tave.
W−Hodo )c/2, and their correlation is shown in Figure 3.2. Since the RHIC clock was used as
Tave.

a common started TDC, the TDC sum of the beam gas should be smaller than the collision
event; the beam gas occurred earlier. We selected the collision events by a diamond cut in offline analysis as shown in Figure 3.2. Maximum contamination of the beam gas were evaluatedby mirror images of the diamond cut toward the beam gas peaks on the z-vertex, which are ±185cm, at their peaks in the TDC sum, which is 1300 cm. Summary of beam gas contamination
are listed in Table 3.1.
The beam gas contamination was studied by the data which were taken without the trigger
bias; most of the beam gas events were removed by the hardware trigger at physics data taking(Figure 2.19). As a conservative estimation, the beam gas contamination was increased withthe maximum beam intensity based on an assumption of a proportional relation between the

36

37

CHAPTER 3. ANALYSIS

contamination of beam gas and the beam intensity. Normalized values also shown in Table 3.1.
Finally, we conclude that the collision were identified with 3% beam gas contamination.
The total number of collision event : 620296
The number of event in the
diamond cut for beam gas
Blue beam gas
8977
Yellow beam gas
4951
Sum
13928

Fraction(%)
1.45±0.02
0.80±0.01
2.25±0.02

Normalized fraction to
maximum intensities (%)
2.41±0.026
0.86±0.012
3.27±0.028

Table 3.1: The contamination of the beam gas to the the collision event. Contamination was
estimated by the data without trigger bias which removed most of the beam gas events. The
fraction of the beam gas was increased with the maximum beam intensity conservatively basedon an assumption of a proportional relation between the beam gas contamination and the beam
intensity.

3.2

Particle identification

As mentioned in section 2.3, only neutral particles were expected to be detected in the acceptance because any charged particles would be swept away by the DX magnet. The KL0 was a
possible background for our measurement. At the ISR energies, the KL0 fraction to the neutron
in a similar kinematical region was estimated to be 3-4 % from observed charged kaon samples[2]. We have included no correction for the KL0 background in this analysis and considered thatdata samples are mixture of photons and neutrons.
The photon and neutron samples were identified according to their behaviors.
• Photon (γ-ID) : Most of them convert in the preshower (2.2 X0 ), and its shower is expected
to be stopped in the EMCal (22 X0 ). There would be no hit in the post scintillators(N1&N2).
• Neutron (n-ID) : Most of them pass through the preshower (0.11 λI ) without interaction.
It converts in the EMCal (1.1 λI ) and the generated shower hits post scintillators.

As a common cut, we required no charge veto activity to remove unexpected charged particle
event.
The purity of the photon and neutron samples were obtained by comparing the event fractions of the photon and neutron samples with and without the preshower activity by 15 MeV
threshold. Figure 3.3 shows the energy distributions deposited in the EMCal for (a) the photon
sample and (b) the neutron sample. The solid curves were selected the photon and neutron
samples using the CV, N1 and N2 except for the preshower, namely γ-PID and n-PID. We define the actual number of photons and neutrons in γ-PID as Nγγ−PID and Nnγ−PID , respectively,
and also define these for n-PID as Nγn−PID and Nnn−PID . In case of the photon sample, it iscategorized with (dashed curve) and without (dotted curve) preshower activity; the number

CHAPTER 3. ANALYSIS

38

Figure 3.1: The z-vertex reconstructed by the hodoscopes.

E−Hodo + T W−Hodo )c/2 vs. (T E−Hodo − T W−Hodo )c/2 = z-vertex. Three islands
Figure 3.2: (Tave.
ave.
ave.
ave.
from the left hand side indicate the yellow beam gas, collision and blue beam gas. The diamondcut, shown in this plane as “collision event cut”, was applied to the selection of the collision
event. Maximum contamination of the beam gas were evaluated by the mirror images of the
diamond cut shown in this Figure.

39

CHAPTER 3. ANALYSIS

of events with the preshower, N γ−PID w P re , and without the preshower, N γ−PID w/o P re areexpressed as follows.
N γ−ID = N γ−PID w P re = PγP re · Nγγ−PID + PnP re · Nnγ−PID ,N γ−PID w/o P re = (1 − PγP re ) · Nγγ−PID + (1 − PnP re ) · Nnγ−PID ,(3.2)

where PγP re and PnP re are the conversion probabilities in the preshower for photons and neutrons
which can be calculated by radiation length (0.9 cm) and interaction length (18 cm) for PbWO4 ,PγP re = 1 − exp [−2(cm)/0.9(cm)] ∼ 0.892,
PnP re = 1 − exp [−2(cm)/18(cm)] ∼ 0.105.

(3.3)

Same description can be used to the neutron identification by changing from γ-PID to nPID1 We can obtain actual number of photons and neutrons in each PID samples, Nγγ−PID ,Nnγ−PID , Nγn−PID and Nnn−PID , by those formulas. Finally, we obtained the actual number ofphotons in the photon sample and neutrons in the neutron sample, defined as PγP re · Nγγ−PID
and (1 − PnP re ) · Nnn−PID , in equation 3.2.

Possible systematics is from the different efficiencies of the N1 and N2 cut for the hadron

showers which are generated at the preshower and EMCal. Naively, the efficiency for theconversion at upstream of the EMCal seems to be 100% , and 0% in case of the conversion at
down stream. We assumed the efficiency for the conversion at the EMCal is 50% which is anaverage of up and downstream cases. Of course, it would be near 100% since the generated
shower was boosted toward the N1 and N2. However, we can not estimate it correctly, 50%
systematics was added conservatively. This systematics lies in only PnP re which are required theN1 and N2 cut2 .
As a result, purities for the photon and neutron samples in the deposit energy above 20
GeV were 0.82±0.07 and 0.994±0.005, respectively. Errors show systematic only; statistics are
negligibly small. The particle identifications and purities are summarized in Table 3.2.
Photon ID
Neutron ID

Energy cut (GeV)
>20
>20

CV
×
×

Pre Shower (MeV)
>15
<15

N1
×

N2
×

Purity0.82±0.07
0.994±0.005

Table 3.2: The particle identifications for the photon and neutron samples. The energy cut forneutrons was applied to the measured energy calibrated for photons.
and × indicate the hit
requirements of the scintillators as 1 MIP or more, and below 1 MIP, respectively. The purities
are defined as actual numbers of photons and neutrons in the photon and neutron samples,
respectively.

In this case, N n−ID = N n−PID w/o P re .
50% is not directly to PnP re . It should be added for relative values with and without the preshower activity,
such as (1−PnP re )/(PnP re ).
1

2

40

CHAPTER 3. ANALYSIS

3.3

π 0 mass spectrum

π 0 candidates were constructed for all pairs of clusters with the requirements of both cluster
energies above 15 GeV. Figure 3.4 shows the invariant mass spectra, Mγγ , in samples with andwithout the γ-ID. Clear π 0 peak was seen and its width be narrower with the γ-ID. The reason
why π 0 peak was shifted by 5 MeV with the γ-ID is the elimination of the events that thephoton energy was lost by the conversion in upstream materials due to the requirement of no
CV hit. Both mass shift and reduction in number of events are understood from the simulation
using the upstream geometry, in particular the conversion in a 2.1 cm thick stainless steel plate
just downstream of the DX magnet shown in Figure 2.8.
For the measurement of π 0 AN , we used a larger data sample, without the γ-ID. The π 0 peak
center for events without γ-ID was used for an absolute energy calibration and tower-by-tower
for the final calibration. We selected events within ±20 MeV of the peak center as π 0 sample
and the background fraction was estimated using a Gaussian plus polynomial fit, it is 21±3.8%.

3.4

Asymmetry calculation

For the measurement of a left-right asymmetry, it is necessary to eliminate the zero degree region
around the beam axis. The acceptance cut was applied as radii 5 – 40 mm from the centerof the detector (5<r<40 mm), or production angles 0.3 – 2.2 mrad. The acceptance area wasdivided into 20 pieces in a radial pattern (see Figure 3.5). For the asymmetry calculation, we
employed a square root formula which cancels other systematics, such as a detector asymmetry
until third order by taking average of the number of events in one piece for spin up (Nφ↑ ) and↓
opposite piece for spin down (Nφ+π); those measurement should be same physics region due toparity invariance. More details are discussed in Appendix A. The asymmetry was calculatedas,qǫN (φ) = q↓
Nφ↑ Nφ+π−↓Nφ↑ Nφ+π
+

q
q

↑
Nφ+π
Nφ↓
↑
Nφ+π
Nφ↓

.

(3.4)

It is smeared from the initial AN (φ) by a smearing effect; the wrong assignment of the position
due to the position resolution. After the correction of the polarization, P , and the smearing
effect, ǫsmearing, we can obtain initial AN (φ) as,φ
AN (φ) =11
ǫN (φ).
P ǫsmearing
φ

(3.5)

The asymmetry was calculated in counterclockwise starting at φ=π/2 to check a sine modulationin the azimuthal angle which is characteristic of single transverse spin asymmetry. AN is defined
as an amplitude of the sine modulation as,
AN (φ) = AN sin(φ − φ0 ),where φ0 allows a deviation of the polarization direction.

(3.6)

41

number of events

CHAPTER 3. ANALYSIS

4

10

(a)
3

10

2

10

10
1

number of events

0

10

20

30

40

50
60
70
80
90 100
EMCal deposited energy (GeV)

4

10

(b)
103
2

10

10
1
0

10

20

30

40

50
60
70
80
90 100
EMCal deposited energy (GeV)

arb. unit

Figure 3.3: Panel (a) shows the deposited energy distribution in the EMCal for events used to
select the photon sample required as no CV and no N1&N2. The solid curve displays all events,
the dashed curve displays events with preshower energy (this is photo sample), and the dottedcurve shows events with no preshower energy. Panel (b) shows the energy distribution for the
events used to select neutrons required as no CV and N1&N2 activities. The solid curve is forall events, the dashed curve is for events with no preshower energy (neutron sample), and thedotted curve is for events with preshower energy.
1
0.8

Figure 3.4: Invariant mass of pairs ofenergy clusters in the EMCal. We required energy above 15 GeV for bothclusters. 444K events with no additional selection requirement (Solid)
and that for 35K with photon identification (Dashed).

0.6
0.4
0.2
0
0

0.05

0.1

0.15

0.2
0.25
0.3
2
invariant mass (GeV/c )42

CHAPTER 3. ANALYSIS

In the RHIC operation, both of the Blue and Yellow beams have been polarized. It enable
us to measure the forward and backward asymmetries at the same time. Because the EMCal
was facing the Blue beam, the forward (xF >0) asymmetry was defined as using polarization
patterns of the Blue beam. Similarly the backward (xF <0) asymmetry was using those of the
Yellow beam only.
The ǫSmearing
was estimated by the simulation and it was 0.92 for the neutron measurementφwith the 5 mm position resolution. Error was applied 100%, ±0.08, conservatively since thatfor the hadron reaction was just estimated by the simulation. It should be better in case ofphotons, with the position resolution ≈1 mm. However, we can just separate their particles

statistically based on the study of section 3.2. In this analysis, same correction, ǫSmearing=
φ
0.92±0.08, was applied conservatively.
In this chapter, we show the ǫN only and the AN values, after the correction of the polarization and the smearing effect, are shown in next chapter (chapter 4). The asymmetry
calculations were performed for the neutron, photon and π 0 samples as shown in Figure 3.6.
We could not see significant asymmetries in the photon and π 0 samples, but obvious large negative asymmetry appeared in the forward neutron sample. Line shows the fit result of the sinemodulation (equation 3.6) and we obtained the φ0 as −0.148 ± 0.069. For the final values, we
fixed φ0 = −0.148 for the forward ǫN based on this result, and φ0 = 0 for the backward ǫN

assuming that the polarization direction of the Yellow beam was transverse completely. Afterthe φ0 was fixed, reduced χ2 for the forward neutron being 1.47. We attributed a large reduced χ2 to the scattered beam background observed higher rate in the EMCal tower near the
beam pipe. Whether or not the background has a spin dependence, it affects the azimuthal
dependence of equation 3.6. Its contribution to the systematic uncertainty was estimated by√
the reduced χ2 , where we have assigned a systematic uncertainty of 0.47×δAN . In case of
backward asymmetries, ǫN of all samples are consistent with zero. It indicates the forward finite
asymmetry is not fake from the measurement as well as the zero backward asymmetry.

24 cm

10 cm

φ= π/2φ=0

φ=-π/2

Figure 3.5: The acceptance definition for theazimuthal dependence of AN .

43

CHAPTER 3. ANALYSIS

2×10

Backward ∈N for neutron sample

χ /ndf = 13.25 / 8
p0 = -0.0096 ± 0.0006
p1 = -0.1484 ± 0.0689
2

-2

1.5
1

∈N

∈N

Forward ∈N for neutron sample

2

1
0.5

0

0

-0.5

-0.5

-1

-1

-1.5

-1.5

-0.5

0

0.5

1

∈N

Forward ∈N for photon sample
2×10

χ /ndf = 13.00 / 9
p0 = -0.0025 ± 0.0013
p1 = -0.1484 ± 0.0000

1.5
1

2

0

0

-0.5

-0.5

-1

-1

-1.5

-1.5

0

0.5

1

∈N

Forward ∈N for π0 sample
2

×10

2

1.5
1

2

0

0

-0.5

-0.5

-1

-1

-1.5

-1.5

-0.5

-1

-0.5

1.5
φ

0

0.5

1

1.5
φ

0

0.5

1

1.5
φ

χ /ndf = 7.56 / 9
p0 = 0.0009 ± 0.0025
p1 = 0.0000 ± 0.0000
2

-2

×10

1
0.5

-1

1

2

1.5

0.5

-2
-1.5

0.5

Backward ∈N for π0 sample

χ /ndf = 11.12 / 9
p0 = -0.0019 ± 0.0025
p1 = -0.1484 ± 0.0000

-2

-2
-1.5

1.5
φ

∈N

-0.5

0

χ /ndf = 6.62 / 9
p0 = -0.0023 ± 0.0013
p1 = 0.0000 ± 0.0000

-2

×10

1
0.5

-1

-0.5

1.5

0.5

-2
-1.5

-1

Backward ∈N for photon sample

2

-2

-2
-1.5

1.5
φ

∈N

-1

2

1.5

0.5

-2
-1.5

χ /ndf = 13.98 / 9
p0 = 0.0005 ± 0.0006
p1 = 0.0000 ± 0.0000

-2

×10

-2
-1.5

-1

-0.5

0

0.5

1

1.5
φ

Figure 3.6: The azimuthal angle dependence of ǫN for the neutron (Top), photon (Center) andπ 0 (Bottom) samples. Closed and open circles show the forward and backward asymmetries,
respectively. They were fitted by the sine curve, f (φ) = p0 × sin(φ − p1) as equation 3.6. p1
for the forward and backward were fixed as -0.1484 and 0.000 except for the forward neutron
sample.

44

CHAPTER 3. ANALYSIS

3.5

AN decomposition

The asymmetry of neutrons (AnN ) and photons (AγN ) were obtained by the AN of the neutronn−ID) and photon (Aγ−ID) identified samples through their relation of,
(AN
N
n−ID
= (1 − Pn ) · AγN + Pn · AnN ,
AN

Aγ−ID
= (1 − Pγ ) · AnN + Pγ · AγN ,
N

(3.7)

where Pn and Pγ are the purity of neutrons in the neutron sample and photons in the photon
sample, respectively (section 3.4). Equation 3.7 is represented as,
AnN

=

AγN

=

i
h
1
n−ID
,
− (1 − Pn )Aγ−ID
Pγ AN
NPn + Pγ − 1
i
h
1
n−ID
.
Pn Aγ−ID
−
(1
−
P
)A
γ
N
N
Pn + Pγ − 1(3.8)

Statistical errors can be calculated as,
δAnN

=

δAγN

=

q
1
n−ID 2) + (1 − Pn )2 (δAγ−ID)2 ,(Pγ )2 (δANN
Pn + Pγ − 1q1n−ID 2(Pn )2 (δAγ−ID) .
)2 + (1 − Pγ )2 (δANN
Pn + Pγ − 1(3.9)

Systematic errors for Pn and Pγ are delivered by,(δAnN )sys. =(δAγN )sys. =n−ID− Aγ−ID|q|ANN(Pγ )2 (δPn )2 + (1 − Pn )2 (δPγ )2 ,(Pn + Pγ − 1)2n−ID− Aγ−ID|q|ANN(Pn )2 (δPγ )2 + (1 − Pγ )2 (δPn )2 .
(Pn + Pγ − 1)2(3.10)

This decomposition can be applied to ǫN with the assumption of the same P and ǫsmearinginφequation 3.5. The ǫN for the neutron and photon after the decomposition are listed in Table
3.3.
In case of π 0 AN , the dilution from background should be corrected as follows,q
π 0 +bg 2
π 0 +bgbg2(δAN) + r 2 (δAbg
AN
− rAN
N)
π0
π0
AN =
, δAN =,1−r
1−r

(3.11)

0where r indicates the background fraction, defined as r ≡ N bg /(N π + N bg ). Detail of thedilution effect is explained in Appendix A. In this analysis, we used r=0.210±0.038 which wasestimated in section 3.3. The ǫN for the background π 0 were estimated by the sample of the
combinatorial background dominates in Figure 3.4, where (|Mγγ − 0.135|>0.020) with assumingthat it was same as the background ǫN in the π 0 sample (|Mγγ − 0.135|<0.020). They wereconsistent with zero as shown in Figure 3.7. Results after the dilution correction are also listed
in Table 3.3.

45

CHAPTER 3. ANALYSIS

∈N

3

×10

Backward ∈N for π0 bg sample

χ /ndf = 4.70 / 9
p0 = 0.0010 ± 0.0032
p1 = -0.1484 ± 0.0000
2

-2

2

3

∈N

Forward ∈N for π0 bg sample

1

0

0

-1

-1

-2

-2

-1

-0.5

0

0.5

1

1.5
φ

2

-2

2

1

-3
-1.5

χ /ndf = 10.62 / 9
p0 = 0.0008 ± 0.0033
p1 = 0.0000 ± 0.0000

×10

-3
-1.5

-1

-0.5

0

0.5

1

1.5
φ

Figure 3.7: The azimuthal angle dependence of ǫN for the π 0 background. The definition issame as the π 0 asymmetries in Figure 3.6 but sample was selected the out of π 0 mass range in
Figure 3.4, where |Mγγ − 0.135|>0.02.
ǫN (×10−3 )NeutronPhotonπ0Forward-9.655 ± 0.641 ± 0.044-0.974 ± 1.588 ± 0.742-2.614 ± 3.300 ± 0.000

Backward
0.528 ± 0.640 ± 0.018-2.982 ± 1.590 ± 0.300
0.869 ± 3.322 ± 0.000

Table 3.3: The ǫN for the neutron, photon and π 0 after the decomposition. The first and second
errors show statistics and systematics, respectively.

3.6

Neutron asymmetry measured with the HCal

To confirm the neutron asymmetry, we added the HCal facing the Yellow beam (section 2.3).
With the robust photon veto system, most of photons converts in the 5 cm lead block (8.9
X0 )3 and is removed by requiring the gamma veto activity. Sample is expected to be almost
neutrons.
Since the HCal can only reconstruct the x-position due to the structure of the postshower,
the raw asymmetry was calculated as left-right asymmetry,qq
NL↑ NR↓ − NR↑ NL↓q
ǫN = q,NL↑ NR↓ + NR↑ NL↓(3.12)

where L(R) indicates the acceptance of left (right) part which is defined as Figure 3.8. By

this acceptance definition, the ǫN is reduced according to the cosφ dependence as ǫN = AN ×R
R
cosφdS/ dS and it is estimated to be 0.50. The ǫN is also smeared by the position resolution,

3–4 cm for the HCal, and total smearing was 0.37 estimated by the simulation. We added 100%3

The conversion ratio is estimated to be 99.99%

CHAPTER 3. ANALYSIS

46

systematic error for the smearing effect due to the position resolution, it was 0.50-0.37=0.13,
in the same way of the EMCal analysis (section 3.4). Finally, we obtained the smearing effect,ǫSmearing, was 0.37±0.13. In this data taking period (nearly end of 2001-2002 run), polarizaHCal
tions were improved to be 0.160 and 0.181 for the Blue and Yellow beams, respectively. Samesystematic and scaling error which are mentioned in section 2.3.4 should be added.
In the EMCal analysis, scattered beam background was suggested by the large χ2 fromthe fit result of the neutron asymmetry (section 3.4). In the HCal side, it was estimated by
the comparison of events in the left and right acceptances. Figure 3.9 shows the reconstructed
position distribution of the sample for AN calculation and the background ratio was defined asbg
a differences of counts in the left and right, RHCal= (NL − NR )/(NL + NR )∼0.055. However,

the asymmetric distribution in the left and right can rise from other origins, for example, thebeam axis was shifted to the HCal center and/or the postshower calibration was insufficient.
We just added it as a systematic uncertainty.
Figure 3.10 shows the ǫN as a function of the measured energy with the HCal. Accord-

ing to the 40–50% energy resolution, events were distributed above 100 GeV broadly. They
were merged in 90 to 100 GeV bin in Figure 3.10 and we used sample in 30–100 GeV for theasymmetry calculation. Significant negative neutron asymmetry was observed in the forward
kinematics and not in the backward. Asymmetry does not show any dependence on the neutron energy within the energy resolution and we fitted them by constant to evaluate amplitudes
which are written in Figure 3.10. We obtain the AN after the correction of the polarization andthe smearing effect as equation 3.5. The results are shown in the next chapter.

3.7

Multiplicity of hodoscopes associated with neutron sample

In Figure 3.11 we present the uncorrected multiplicity observed by the hodoscope for the neutron
sample of the EMCal data with deposited energy above 20 GeV. The vertical and horizontal
slats for each hodoscope overlap over 67% of the area, so that a correct estimation of themultiplicity is (uncorrected multiplicity)/1.67.
As seen in the figure, the multiplicity for the forward beam counter, in the direction of theneutron, is low, hmultiplicity (forward)i∼2. The multiplicity in the backward beam counteris large, hmultiplicity (backward)i∼7. Therefore, we observe a clear separation of beam andtarget fragmentation multiplicity, with a large asymmetry for neutrons produced forward fromthe polarized beam, in the direction of low multiplicity. This pattern was confirmed with
the HCal data, where the forward and backward beam counters are reversed. The low forward
multiplicity is consistent with a simple exchange process producing the forward neutron, p → n.

47

CHAPTER 3. ANALYSIS

Y

unit : cm
Position distribution for AN calculation in HCal

LEFT

10

RIGHT
10000

X

8000

(0,0,-1800)

6000

4000

1

2000

HCal view
from collision point

0

-6

-4

-2

0

2

4

6
xpos (cm)Figure 3.8: Definition of a coordinate Figure 3.9: The reconstructed position distribution by
for the HCal and the acceptance for the postshower. Negative and positive x positions were
the AN calculation.
included in the left and right acceptances in Figure 3.8.

∈N

Neutron ∈N by HCal
1.5

×10

-2

Backward : p0 = 0.0016 ± 0.0011

1
0.5
0
-0.5
-1
-1.5

Forward : p0 = -0.0085 ± 0.0011
-2
0

10

20

30

40

50

60

70
80
90
100
Measured energy (GeV)

Figure 3.10: The energy
dependence of ǫN for theneutron productions detected with the HCal.
Closed and open circles
show the forward andbackward
asymmetries,
respectively. Events whichwere distributed above 100
GeV were merged in 90-100
GeV bin and we just say
about the horizontal axis
as the measured energy
due to the 40–50% energy
resolution.

×10

number of events

3

Solid line : Forward

1000

Dashed line : Backward
800

600

400

200

0
0

2

4

6

8
10
12
14
16
Uncorrected hodoscope multiplicity

Figure 3.11: Uncorrected
multiplicity observed by hodoscopes for the neutron
sample of the EMCal datawith deposited energy above
20 GeV. The solid line showsthat for the forward counter(east) and the dashed line
shows that for the backward
counter (west).

Chapter 4

Results of the IP12 experiment
AN values after corrections of the polarization and smearing effect are summarized in table 4.1.
The AN of photon and π 0 are consistent with zero within the statistics, whereas significant large
AN in the forward neutron production was observed in the EMCal, and it was confirmed by the
HCal. The magnitudes and signs are consistent, AN ≈ −10% with the scaling error ×1.00+0.52−0.25 .

It was unexpectedly large asymmetry in this low pT range; the range in pT for the neutrons

can only be estimated from the limits of the acceptance (equation 1.4), which give pT < 0.22
GeV/c, for the limits En < 100 GeV and θn < 2.2 mrad, since polarization phenomena which
has been observed so far seem to vanish in such low pT (cf. pion AN and Λ polarization, section
1.1).
The measurements of inclusive neutrons in the similar kinematics for unpolarized pp andep collisions have been studied and these cross sections are consistent with the OPE model
predictions (section 1.2). And the OPE model can suggest the finite neutron asymmetry by the
spin-flip amplitude due to the pion exchange. It is necessary to check the cross section at the
RHIC energy whether the OPE model is applicable or not.
As a further experiment, the PHENIX experiment has measured leading neutrons in similar
kinematics with the improved energy measurement. The PHENIX experiment enables us to
extract the cross section and the xF dependence of AN .
AN results from EMCal
ForwardNeutron -0.090 ± 0.006 ± 0.009
Photon -0.009 ± 0.015 ± 0.008
π0-0.024 ± 0.031 ± 0.002
AN results from HCal
Neutron -0.126 ± 0.017 ± 0.045

Backward
0.003 ± 0.004 ± 0.003-0.019 ± 0.010 ± 0.004
0.006 ± 0.021 ± 0.001
0.028 ± 0.019 ± 0.010

Table 4.1: The results of AN obtained at the IP12 experiment. Errors show the statistical andsystematic, respectively . Scaling errors from the beam polarization for the EMCal, 1.0+0.52
−0.25
+0.57
and 1.0−0.27
for the forward and backward AN , respectively, are not included. For the HCal,scaling errors are nearly same for the forward and backward AN , it is 1.0+0.52−0.26 .

48

Part IIPHENIX experiment49

Chapter 5

PHENIX experiment
We performed the leading neutron measurement at PHENIX as a further measurement of theIP12 experiment. Remarkably, the energy resolution was improved to ∼20% at the 100 GeV√
neutron and it is expected to extract the cross section and the xF dependence of AN at s=200
GeV. The data from 2005 run was used for the present thesis and one notable comment is thatthe beam polarization measurement was dramatically improved with the pp-CNI polarimeterto be ∆P/P ≈0.09. (cf. ∆P/P ≈0.35 at 2001-2002 run)

5.1

Overview

The RHIC-PHENIX experiment has measured leading neutrons by a Zero-Degree Calorimeter(ZDC) with a position-sensitive Shower-Max Detector (SMD) which cover ±2.8 mrad of theforward and backward directions [66]. One ZDC module has 1.7 λI and 51 X0 . It achieves

21% energy resolution for the 100 GeV neutron by locating three ZDCs in series. The SMD is
an x-y scintillator strip hodoscopes, 7 for the x-coordinate and 8 for the y-coordinate, inserted
between first and second ZDC modules. The neutron position can be obtained by calculating
the center of gravity of showers which are generated in the first ZDC module. The position
resolution depends on the neutron energy and it was estimated to be ∼1 mm for the 100 GeV
neutron, for example.

Since detectors are located in the same kinematics of the IP12 experiment, downstreamof the DX magnet, charged particles from collisions are expected to be removed. A forward
scintillation counter has been installed between the DX magnet and the ZDC to remove charged
particle backgrounds from other sources in the same way of the IP12.
We prepared two sets of triggers for the neutron measurement. One is the ZDC self trigger
and the other is a coincidence with charged particles. Charged particles are detected by a BeamBeam Counter (BBC) which are placed in up and downstream of the collision point coveringwith ±(3.0–3.9) and 2π in η and φ spaces, respectively. We collected 6.5 million and 17.6 million
events for the ZDC self trigger and that with a BBC coincidence, respectively.

50

51

CHAPTER 5. PHENIX EXPERIMENT

5.2

Experimental setup

A plan view of the experimental setup for the leading neutron measurement at PHENIX isshown in Figure 5.1. Schematic views of whole PHENIX detectors [46] are also shown in Figure
5.2.

1800 cm

SOUTH

NORTH

BBC
5 cm
Dx

Dx

Collision point
Blue Beam
Yellow BeamZDC (W-Cu alloy)
Charge veto counter (Plastic Scintillator)
SMD (Plastic Scintillator)
Figure 5.1: A plan view of the experimental setup at PHENIX, not to scale. Shown are theprinciple components for the leading neutron physics.

5.2.1

Zero Degree Calorimeter (Shower Max Detector, Charge veto counter)

The ZDC is a hadron calorimeter for the measurement of neutrons in very forward kinematics.
The response of the ZDC was tested by 100 and 160 GeV proton beams by a prototype ZDC[66]. The prototype was not same as ZDCs which have been installed in RHIC 1 . The prototype
is composed of tungsten plates with PMMA2 -based communication grade optical fibers. One
prototype has 2.0 λI and four modules were located in series at the test experiment. On thebasis of the test bench, present ZDC was designed.
Present ZDC consists of Cu-W alloy absorbers with PMMA-based communication grade
optical fibers and it corresponds to 1.7 λI for one module,. Figure 5.3 shows a design of
one ZDC module. A photomultiplier (Hamamatsu R329-2) collects Cherenkov lights from theoptical fibers in one module. Three ZDCs are located in series (5.1 λI ) at ±1800 cm away fromthe collision point within the small acceptance, covering ∼10 cm in the trasverse plane. It has

been calibrated by observing the one neutron from peripheral heavy ion collisions; the 100 GeV
neutron, diverged by less than 2 mrad from the beam axis, provides a single neutron peak asshown in Figure 5.6. It achieved 21% energy resolution for the 100 GeV neutron.

The position-sensitive SMD is an x-y scintillator strip hodoscopes inserted between first andsecond ZDC modules where is the maximum hadronic shower approximately. The x-coordinate(horizontal) is sampled by 7 scintillator strips of 15 mm width, while the y-coordinate (vertical)
1

In the RHIC ring, the ZDC is a global detector. They are installed in all interaction points and used as aluminosity monitor.
2
Polymethylmethacrylate (Index 1.49)

CHAPTER 5. PHENIX EXPERIMENT

Figure 5.2: Schematic views of whole PHENIX detectors.

52

CHAPTER 5. PHENIX EXPERIMENT

53

is sampled by 8 strips of 20 mm width, tilted by 45 degree (Figure 5.5). The active area inthe transverse plane is 105 mm × 110 mm (horizontal×vertical). Their signals are collected bya multi-channel photomultiplier (Hamamatsu H6568, M16) through the Wave Length Shifter(WLS) fibers. The neutron position can be reconstructed using the SMD scintillators whichhave energy deposits above the threshold as follows.
x, y =

SM D
PNmulti.

i

ene(i) · pos(i),SM D
PNmulti.
ene(i)i(5.1)

where ene(i) and pos(i) are the pulse hight leads to the energy deposit and the position for the
SM D
i-th scintillator, respectively. The number of scintillators above threshold is shown as Nmulti.

and it is defined as a SMD multiplicity. The response of position measurement is studied by
the Monte Carlo simulation in chapter 6. SMD scintillators were calibrated relatively by MIP
peaks of the cosmic-ray data taken at the beginning of 2003 run. Figure 5.7 shows a raw ADC
spectrum of the cosmic sample and we can see a pedestal and MIP peaks around 50 and 100channels, respectively. Actually the cosmic data was collected by usual CAMAC system which
was different gain to the PHENIX data acquisition system. A LED light was used for the gain
connection between them. We calibrated the MIP peak about 70 channel after the pedestal
subtraction and require a threshold as 40 channel for the centroid method. However there is alarge uncertainty for the gain connection by the LED system. It would affect the estimation of
the detector efficiency by the simulation in section 7.7. Fortunately, we can estimate the relative
SMD efficiency for the real and simulation data by selecting nearly pure neutron samples (section
7.5).
For the charged particle elimination, the 3.3 mm thick plastic scintillator is installed in frontof the ZDC. It covers 10 × 12 cm, it is almost same acceptance as the ZDC. It is calibrated formatching the MIP peak to 100 channel as shown in Figure 5.8. Typically we require the cut by
50 ch for the selection of charged events.
A whole ZDC system is shown in Figure 5.4. In this analysis, we used a SOUTH ZDC
detector which is facing Yellow beam only. Its coordinates are defined according to the PHENIX
coordinate system as shown in Figure 5.9.

5.2.2

Beam Beam Counters

The BBC consists of 64 one-inch mesh dynode photo multiplier tubes (Hamamatsu R6178) with
3 cm thick quartz radiators as a Cherenkov radiator (Figure 5.10). BBC elements are mounted
as the beehive structure around the beam pipe (Figure 5.11) and located ±144 cm away fromthe interaction point with the coverage of 3< |η| <3.9 and 2π in azimuthal angle.

The BBCs are used to measure the collision vertex point in a direction of the beam axis

using hit time informations in NORTH and SOUTH parts. The time resolution of the single
BBC element is 52 psec. They are used as the PHENIX minimum bias trigger; they can take√about 50% inelastic interaction for pp collision at s=200 GeV (Details are in section 5.2.4).

54

CHAPTER 5. PHENIX EXPERIMENT

Figure 5.3: A schematic design of the present ZDC module. Units are shown in mm.

Figure 5.4: A picture of whole ZDC system. Three ZDCs
are placed in series and the SMD is installed between firstand second ZDC modules. The scintillator as a charge
veto counter is placed in front of the ZDC.

Figure 5.5: A picture of
SMD; 7 scintillators for the
x-coordinate.

55

CHAPTER 5. PHENIX EXPERIMENT

htmpDsumN
Entries
123727
Mean
307.7
125.3
RMS
2
84.18 / 80
χ / ndfProb
0.353-35.73 ± 7.49
p0
1.179 ± 0.042
p1
99.71 ± 4.38p2
100 ± 1.0
p3
21.36 ± 1.31
p4

ZDC energy distibution in CuCu collision at sNN = 200 GeV.

350
300
250
200
150

Figure 5.6: The energy distribution
with the√ZDC in CuCu collision at SN N = 200 GeV.
We can see an one neutron peak clearly.

100
50
0
0

50

100

150

200

250

300

350

400 450 500
Energy (GeV)

Forward counter distribution

Cosmic-ray data in SMD. (CH2)

8000
7000

104

6000
3

10

5000
4000

102

3000
2000

10

1000
1
0

50

100

150

200

250

300

350

400

450 500
ADC value

0

100

200

300

400

500

600
700
1 MIP = 100 channel

Figure 5.7: The ADC distribution of the SMD Figure 5.8: The charge distribution of thefor the cosmic-ray data. The pedestal and MIP charge veto counter in pp collision. It is calipeaks are shown around 50 and 100 ch, respec- brated by the MIP peak to 100 ch.
tively.

YYφcollision point(0,0,0)+R

X(west)φ=0unit : cm

10

φ
X

θ

(0,0,-1800)

φ=0

Z(NORTH)
θ=0

PHENIX cordinate system

SOUTH ZDC view
from collision point

Figure 5.9: The coordinates for the SOUTH ZDC according to the PHENIX coordinate system.

CHAPTER 5. PHENIX EXPERIMENT

56

Figure 5.10: A single BBC consists of a one-inch Figure 5.11: A BBC array consists of 64 BBC
mesh dynode photo-multiplier tube with a 3 cm elements. They are installed ±144 cm away
thick quartz radiator.
from the collision point with the coverage of3< |η| <3.9 and 2π in azimuthal angle.

5.2.3

PHENIX data acquisition system

PHENIX detectors are designed to make measurements with high interaction rate; design interaction rates for pp and AuAu are 500 kHz and few kHz, respectively. PHENIX data acquisition(DAQ) system [68, 69] has a capability for such high rate through the pipelined and dead-timeless features implemented to the detector front ends.
The schematic diagram of the PHENIX DAQ is shown in Figure 5.12. Signals from detector
are converted to digital information and buffered in Front End Modules (FEM) which include
signal amplifiers, shapers and time decisions. Each subsystems such as BBC, ZDC etc. produce
a local level-1 trigger (LVL1) for taking physics interest events and send it to Global Level-1
trigger (GL1). Once a trigger is accepted at GL1, Global Timing Module (GTM) requests FEM
to send the data to Data Collection Modules (DCM). DCMs perform zero suppression, error
checking and data reformatting. Many parallel data streams from DCMs are sent to Sub-Event
Buffer (SEB). In the final stage, Assembly Trigger Processor (ATP) fetch the data from SEB
and concatenate the event fragments.
The recorded data are sent to the tape device in High Performance Storage System (HPSS)in the RHIC Computing Facility (RCF).

5.2.4

Data taking for the leading neutron sample

Following LVL1 triggers are used to collect the leading neutron sample.
1. Energy deposit in the NORTH or SOUTH ZDC above 5 GeV : ZDCN|S
2. One or more hits in 64 photo-multipliers of both NORTH and SOUTH BBC :BBCLL1(NoVtxCut)

CHAPTER 5. PHENIX EXPERIMENT

57

Figure 5.12: A block diagram of the PHENIX data acquisition system.

3. Same as 2, but requiring z-vertex being ± 30 cm (Figure 5.13) : BBCLL1(Defined as a PHENIX minimum bias trigger)

The ZDC self trigger, ZDCN|S, was used for both the cross section and the asymmetry
analyses. For the asymmetry analysis, a combination of 1 and 3, namely BBCLL1&(ZDCN|S),
was also studied 3 . Such coincidence study is very interesting since charged particles in the BBC
would be generated by a pion-proton scattering in the Regge framework (X in Figure 1.4). The
comparison of asymmetries for these two data samples would give us additional information
for the neutron production mechanism. Data collections were scaled with the factor of ∼400

and ∼25, correspond to ∼100 Hz and 200–300 Hz, for ZDCN|S and BBCLL1&(ZDCN|S),
respectively due to the DAQ bandwidth4.

In transversely polarized pp run at 2005, we collected 6.5 million and 17.6 million eventsfor ZDCN|S and BBCLL1& (ZDCN|S) triggers, respectively. The trigger counts of the BBCLL1(NoVtxCut) were used to the luminosity estimation in this analysis. By the van der
Meer/vernier scan analysis, the cross section of BBCLL1(NoVtx) events was 22.9±2.2 mb [70].
Integrated luminosity was calculated for the transverse run period as,
L=3

N of BBCLL1(NoVtxCut)
5536229087=∼ 242 (nb−1 ).
σBBC
22.9(mb)(5.2)

BBCLL1&(ZDCN|S) trigger has been used to the local polarimeter. Details are shown in Appendix B.
A full PHENIX DAQ bandwidth was achieved 5 kHz at 2005 year run. Most of the bandwidth has been used
to other physics triggers.
4CHAPTER 5. PHENIX EXPERIMENT

58

Vertex distribution ( RUN#178565 )
×10

3

1200

BBCLL1(NoVtxCut)

1000
BBCLL1Figure 5.13: z-vertex distributions determined by BBCs. The BBCLL1 has
been required vertex cut ±30 cm atonline.

800

600

400
200

0
-300

-200

-100

0

100
200
300
Vertex distribution (cm)

Chapter 6

Simulation study
An absolute scale for the energy measurement is determined by the 100 GeV single neutron
peak from the heavy ion collision. However, neutrons coming from pp collisions are below 100
GeV. In case of the SMD, it is not tested by any beams. A simulation study is important toevaluate the detector responses for the neutron measurement below 100 GeV and discussed inthis chapter. The performance for the neutron identification is also discussed.
The simulation was prepared by Geant3 with GHEISHA [65] as a hadron interaction(Figure 6.1) which was well reproduce the response of the prototype ZDC. As an event generator,
a single neutron event generator and Pythia (version 6.220) [71] were prepared. The single
event generator creates neutrons as a function of xF and pT . The xF distribution which wasused for the simulation input was determined as a differential cross section, dσ/dxF , in the
cross section analysis (chapter 7). The pT distribution is difficult to determine by the PHENIX
data alone since the position and energy resolutions are insufficient. The pT distribution fromthe ISR result, exp(−4.8±0.3 pT ), as shown in Figure 6.2 were used to the simulation input,
assuming pT scaling from the ISR to the PHENIX energies. To check a reliability of thisassumption, distributions of radius from the detector center, r, for the real and simulation data
were compared based on the relation of pT ∝ r as,
pT = En sinθn = En √rr 2 + d2r
∼ En ,
d(6.1)

where d is the distance from the collision point to the detector, corresponding to 1800 cm. The
comparison of r distributions with the integration of measured neutron energies 20–120 GeV isshown in Figure 6.3, and they agree well. It means the assumption of pT scaling is reasonable.
To check the relevance of the simulation, the energy resolution for the 100 GeV neutron wasstudied as shown in Figure 6.4. The energy resolution is about 22% and consistent with theobserved width of one neutron peak from the CuCu data as shown in Figure 5.6.

6.1

Study of the ZDC and SMD responses

The response of the energy measurement at the ZDC is compromised by 1) a non-linearity ofphoto electron yield to the neutron energy, and 2) a hadron shower leakage at the detector edge59

60

CHAPTER 6. SIMULATION STUDY

Figure 6.1: A plan view of the simulation setup drawn by Geant3. the ZDC, BBC, beam pipe
and all materials around the ZDC are included.

Figure 6.2: pT distributions ofneutron productions for 0.3<
xF <0.4 (left) and 0.4< xF <0.7
(right) from the ISR experiment.
The pT shape was obtained as ∝exp(−4.8±0.3 pT ). [3]Comparison of radii distributions btw real and sim

Real data

25000

Simulation
20000

Figure 6.3: r distributions for the real andsimulation data with the ISR pT shape.
Distributions are well agree within r<4 cm.
The ISR pT shape is expected to be applicable to our data.

15000

10000

pT shape for simulation : exp(-4.8 pT)

5000

Measured energy : 20~120 GeV
0
0

0.5

1

1.5

2

2.5

3

3.5

4
R (cm)

61

CHAPTER 6. SIMULATION STUDYZDC output with 100 GeV neutron by simulation

Entries = 10000
Mean = 98.984

180
160

RMS = 24.155

140

χ /ndf = 344.44 / 170

120

Constant = 153.39 ± 2.13

100

Mean = 100.19 ± 0.2580

Sigma = 21.78 ± 0.19

260

Figure 6.4: The measured energy distribution for 100 GeV neutrons estimated by the simulation and the widthis 21.78±0.19. It is consistent with thewith of one neutron peak in heavy ion
collision, 21.36±1.31 (Figure 5.6).

40
20
0
0

20

40

60

80

100

120

140 160 180 200 220
Measured energy (GeV)(edge effect). The SMD response will be changed by same reasons too. In this section, studiesof the ZDC and SMD responses by the simulation with the single neutron event generator are
discussed.

6.1.1

Energy linearity and resolution for the ZDCThe energy linearity and resolution were estimated by output responses for various incident
energies of neutrons which were from 20 to 100 GeV with 10 GeV intervals in the simulation.
Top of Figure 6.5 shows the mean of output energy as a function of the incident energy. Absolute
scale was determined by 0 and 100 GeV in the same way of the real data. As a result, ZDC
responses of energy measurements below 100 GeV have the non-linearity; output value is 15
GeV for the 20 GeV incident energy, for example. We applied a correction of the non-linearity
to the real data based on this result. As a systematic error for this correction, we decided touse the difference between the linear and non-linear cases, conservatively. It is estimated in theanalysis part, section 7.9.2, as a variation of the cross section results.
Bottom of Figure 6.5 shows the energy resolution as a function of the inverse square root
incident energy. The energy resolution for the 20–100 GeV neutron is described as,
64.64
∆E
=p
+ 15.13 (%).
E
E (GeV)

6.1.2

(6.2)

Energy response in the detector edge

The edge effect was studied by the prototype ZDC with the 100 GeV proton beam at CERN
as shown in Figure 6.6. Plots are the energy response and resolution as a function of incident
positions for x and y. Basically the energy response decreases in the edge area, but only in theedge of positive y, it increases. It is caused by the fibers in the top region for the connection to
the PMT (Figure 5.3); the leaked shower hits them directly. The simulation which was prepared
for the prototype reproduced this effect well.

62

CHAPTER 6. SIMULATION STUDY

output neutron energy

Energy linearity of simulation
100

80

60

40

20

0

0

20

40

60

80

100
input neutron energy

Energy resolution (%)

Energy resolution estimated by simulation

χ2/ndf = 61.23 / 7
P0 = 15.13 ± 0.27
P1 = 64.64 ± 1.96

30

28

26

24

22
0.1

0.12

0.14

0.16

0.18

0.2
0.22
-1/2
-1/2
Inverse square incident energy E (GeV )Figure 6.5: Top) The mean of output energy as a function of the incident neutron energy
estimated by the simulation. Solid line indicates a linear response. Bottom) The energy resolution as a function√ of the inverse square root incident energy. Solid line shows the fit result by∆E/E = [P0]/ E + [P1].

Figure 6.7 shows the study for the present ZDC estimated by the simulation. The response
and resolution in the edge area act a same behavior of the prototype result except for the edgeof positive y. It is because of the wide coverage of the absorber plates in positive y region forthe present ZDC as an upgrade to reduce the edge effect.
This difference of the edge effects in positive and negative y can be seen in the real datawith the SMD position cut. Figure 6.8 shows energy distributions measured at the ZDC fory >0 cm and y <0 cm by the SMD cut. Not only the different response but also the high energy
tail in y >0 cm are seen. We should not use the edge area to reduce the edge effect.
According to this study, 95–100% energy response is obtained by selecting r < 3 cm as an
acceptance cut.

6.1.3

Position resolution in the center region

The neutron position was calculated by the centroid method as equation 5.1. The position
resolutions were estimated by the simulation in this section. Figure 6.9 shows the output
position distributions and their resolutions as a function of the neutron incident energy for x
and y. Since distributions were not be reproduced by gaussian, the resolution was taken as
RMS. The position resolution was estimated to be around 1 cm for the neutron energy at 100
GeV. A reliability of this estimation will be discussed in section 6.1.5.

CHAPTER 6. SIMULATION STUDY

63

Figure 6.6: The energy response and resolution of the prototype ZDC as a function of theincident neutron position studied by the 100 GeV proton beam at CERN. Gray lines indicate
the responses estimated by the simulation and they well reproduced the real data.

64

CHAPTER 6. SIMULATION STUDY

110

150

Response (%)

Vertical scan ( Response )Response (%)

Horizontal scan ( Response )
105

140

100

130

95

120

90
110
85
100
80
90

75
70

80

65

70

60
-5

-4

-3

-2

-1

0

1

2

3

4
5
x (cm)

60
-5

-4

-3

-2

-1

0

1

60

20 GeV neutron input
30 GeV neutron input
40 GeV neutron input
50 GeV neutron input
60 GeV neutron input
70 GeV neutron input
80 GeV neutron input
90 GeV neutron input
100 GeV neutron input

55
50

45

35

30

30

25

25

20

20
-2

-1

0

1

3

4
5
y (cm)

45

35

-3

2

50

40

-4

4
5
y (cm)

55

40

-5

3

Resolution (%)

Vertical scan ( Resolution )60

Resolution (%)

Horizontal scan ( Resolution )

2

2

3

4
5
x (cm)

-5

-4

-3

-2

-1

0

1

Figure 6.7: The energy response and resolution of the present ZDC as a function of incident
neutron position for various incident energies estimated by the simulation.
ZDC energy distribution (with position cut)
104

Figure 6.8: Energy distributions measured with
the ZDC by the SMD position cut. Solid anddashed distributions indicate the SMD cut withy >0 and y <0, respectively. Not only the different response but also the high energy tail in
y >0 cm are seen.

103

102

10

1

0

50

100

150

200
250
300
Measured energy (GeV)65

CHAPTER 6. SIMULATION STUDY

hxpos
7471

Entries600

Mean-0.01356

RMS

0.8482500

χ / ndf
Position resolution of x-position
2

300

200

100

1
1.787 ± 0.04188

p1-0.01843 ± 0.001557

p2

9.193e-05 ± 1.275e-05

1.3
1.2
1.1
1
0.9
0.8

-4

-3

-2

-1

0

1

2
3
4
5
x position (cm)Measured y-position in incident y = 0 cm at 100 GeV

400

20

hypos
Entries

7516

Mean

0.1997

RMS

1.044

350

30

40

50 60 70 80 90 100
Incident neutron energy (GeV)χ / ndf
Position resolution of y-position
20.0009164 / 6
1

Prob

Position resolution by RMS (cm)

0
-5

p0

1.5
1.4

400

0.002502 / 6

Prob

Position resolution by RMS (cm)Measured x-position in incident x = 0 cm at 100 GeV

1.7

p0
p1

-0.0176 ± 0.000942

1.6

p2

8.503e-05 ± 7.714e-06

1.967 ± 0.02534

1.5

300
250

1.4

200
150

1.3
1.2

100
1.1
50
0
-5

1
-4

-3

-2

-1

0

1

2
3
4
5
y position (cm)

20

30

40

50 60 70 80 90 100
Incident neutron energy (GeV)

Figure 6.9: Left) Output position distributions of x (top) and y (bottom) for the 100 GeV
neutron injected in the ZDC center, (x,y)=(0,0) cm, evaluated by the simulation. The position
resolution was obtained as their RMS. Right) The position resolution as a function of the incident neutron energy for x (top) and y (bottom). They were well reproduced by the polynomial
2 fit.

66

CHAPTER 6. SIMULATION STUDY

6.1.4

Position response in the detector edge

In the edge area, the position measurement is also affected by the shower leakage. Figure 6.10
shows the output distribution of the x-position for various incident x-positions at the 100 GeV
neutron estimated by the simulation. Their peak and RMS values as a function of the incident
position for x and y are shown in Figure 6.11. If incident position was in the edge area, theoutput position was shifted to the detector center caused by the shower leakage. This effect does
not depend on the neutron energy. This misalignment caused by the edge effect is correctedbased on this result.
Incident x position = -4 cm

Incident x position = -3 cm

Incident x position = -2 cm

500
500
500
400
400

400

300
300
200

100

0
-6

-4

-2

0
2
4
6
Reconstructed x position (cm)

300

200

200

100

100

0
-6

Incident x position = -1 cm

-4

-2

0
2
4
6
Reconstructed x position (cm)0-6Incident x position = 0 cm

-4

-2

0
2
4
6
Reconstructed x position (cm)Incident x position = 1 cm

600
500

500
500

400

400
400

300

300
300

200

200

200

100

100

100

0
-6

-4

-2

0
2
4
6
Reconstructed x position (cm)0-6Incident x position = 2 cm

-4

-2

0
2
4
6
Reconstructed x position (cm)0-6Incident x position = 3 cm

-4

-2

0
2
4
6
Reconstructed x position (cm)Incident x position = 4 cm

600
450

500

400

500
400

350

400

300
300

250

300

200
200

200

150

100

100

0
-6

0
-6

100
50

-4

-2

0
2
4
6
Reconstructed x position (cm)

-4

-2

0
2
4
6
Reconstructed x position (cm)

0
-6

-4

-2

0
2
4
6
Reconstructed x position (cm)Figure 6.10: The output x-position distributions for 100 GeV neutrons with various incident
positions estimated by the simulation. The y-position is basically same behavior as the xposition. (see Figure 6.11)

67

CHAPTER 6. SIMULATION STUDY

5
4
3
2
1
0
-1
-2
-3
-4
-5
-5

-4

-3

-2

-1

0
1
2
3
4
5
Incident x position (cm)Vertical scan ( Position )Reconstructed y position (cm) [error is RMS]

Reconstructed x position (cm) [error is RMS]

Horizontal scan ( Position )

5
4
3
2
1
0
-1
-2
-3
-4
-5
-5

-4

-3

-2

-1

0
1
2
3
4
5
Incident y position (cm)Figure 6.11: The peak value of the output x- (left) and y-position distributions (right) as afunction of incident position for 100 GeV neutrons estimated by the simulation. Error bars
indicate the RMS of position distributions. Output was not on the gray line which displays alinear response.

6.1.5

Systematics for the position measurement

Responses of position measurements by the SMD were evaluated by the simulation in previous sections, 6.1.3 and 6.1.4. However, the hadron response of Geant3 simulation has large
uncertainty in general. 1 In this section, the reliability of the position measurement is discussed.
The reliability was studied by comparing hadron shower shapes of the real and simulation
data. Before the comparison, cut parameters for the neutron identification and the cross section
for the simulation input, dσ/dxF , were decided. The neutron identification and the analysis forthe cross section are described in section 6.2.2 and chapter 7, respectively.
For the discussion of the hadron shower shape, two variables were defined as follows,• Shower Shape (SS) : Figure 6.12 (left) shows the charge distribution in each SMD in one
event. For the definition of the SS, we used the charge in the i-th scintillator above the
P SM D
SM D isthreshold, Ci , and defined the charge sum, Csum , as Csum = Nmulti. Ci , where Nmulti.

the SMD multiplicity. And the charge fraction in each scintillator were defined as fi =Ci /Csum . The SS for one event was defined as Figure 6.12 (Right); the fi are plotted as afunction of hscintillator positioni minus houtput position calculated by centroid methodi,

which means the shower center is moved to zero. For the statistical evaluation, the SS
was calculated event by event and normalized by statistics. From the SS, we can see theprofile of the shower width.
• Peak Fraction (PF) : The PF is defined as the highest charge fraction among all scintil1

Not only Geant3 simulation but also any other simulations.

68

CHAPTER 6. SIMULATION STUDY

Hadron shower distribution in one event

chargeFraction of shower charge sum

Hadron shower distribution in one event
3500
3000
2500
2000
1500
1000
500

0.5

0.4

0.3

0.2

0.1

0
-6

-4

-2

0

2
4
6
scintillator position (cm)

0
-10

-8

-6

-4
-2
0
2
4
6
8
10
(scintillator position)-(measured position) (cm)Figure 6.12: A definition of the shower shape (SS) in case of the y position. Left) The SMD
charge vs. each SMD scintillator in one event. It is expected to be a hadron shower shapeat the SMD. The vertical dashed line shows the position calculated by the centroid method
and the horizontal dotted line is the threshold for the centroid method (equation 5.1). Right)
It is a modified picture of the left one. Horizontal axis indicates hscintillator positioni minus
houtput position calculated by the centroid methodi and vertical axis shows a charge fractionin each scintillator to the sum of charges.

lators; PF = Max (fi ). In case of Figure 6.12 (Right), the PF is the third scintillator fromthe right-hand edge and the PF value is about 0.52. The PF distribution is expected tobe the reference of a shower height.
These were calculated for x and y independently. We compared these distributions in each SMD
multiplicity since the hadron shower shape highly depend on the SMD multiplicity.
The comparison of SS and PF distributions between the real and simulation data are shownin Figure 6.13,6.15 for x and 6.14,6.16 for y, respectively. The shapes for y were reproduced by
the simulation well. However these for x were obviously different, especially that of the PF, ina high SMD multiplicity event. Maximum systematic error for the position measurement canbe discussed with the matching SS and PF shapes for x in case of the SMD multiplicity = 7,
which shows the worst agreement between the real and simulation data.
There was one more disagreement between the real and simulation data as shown in Figure
6.17, displays the SMD charge sum vs. each SMD charge for the real and the simulation in
top-left and top-right, respectively. In the real data, there was a correlation between them; if
the SMD charge sum was larger, the pedestal of SMD moves to higher. We are considering thereason is the cross talk possibly caused by the leakage of photo-electrons in the optical fibers andthe multi-anode readout of photomultiplier is also suspicious. It can be reproduced by adding
a base offset depending on the charge sum (Smearing by the flat method shown in later).
For the quantitative discussion, SMD charges in the simulation were smeared to match SS
and PF shapes to the real data by following 2 methods (Figure 6.18).
• Neighboring method : A fraction (fneighboring ) of charge in the i-th scintillator is shared

CHAPTER 6. SIMULATION STUDY

69

to neighbor two scintillators (In case of the edge scintillator, the charge is shared to just
one side). The fneighboring is the smearing parameter and this sharing is applied to all
scintillators. It is expected that shower shape is wider.
• Flat method : A fraction (ff lat ) of charge in the i-th scintillator is shared to all scintillators.

The ff lat is the smearing parameter and this sharing is applied to all scintillators2 . It is
expected to reproduce the cross talk which is seen in the correlation between the SMD
charge sum and each SMD charge. Shower shape is also expected to be wider.

Both smearing were applied so that the charge sum is conserved.
The behavior considering as the cross talk can be reproduced by the flat method. Bottomleft and bottom-right in Figure 6.17 show the correlations after the flat smearing with the ff lat= 0.05 and 0.10, respectively. The distribution with ff lat = 0.05 looks reasonable to the real
data and 0.10 is too much.
For the estimation of systematic error, the flat smearing with ff lat = 0.05 was applied to thesimulation first and the neighboring method was used to match SS and PF shapes. Figure 6.19
show SS and PF shapes for x at the SMD multiplicity = 7 after the smearing of the neighboring
and the flat methods with fneighboring = 0.28 and ff lat = 0.05, respectively. Smearing parameterfor the neighboring method was determined by minimizing the χ2 value of the histogram fit for
PF shapes between the real and simulation data. After the smearing for matching PF shapes,
the SS shape of the simulation also reproduces that of the real data. After the smearing, theposition resolution as a function of the incident energy was obtained as Figure 6.20. In theleft panel, closed and open circles indicate the position resolutions estimated by the simulation
without and with the smearing, respectively. In the right figure, the ratio of them is shown andthe position resolution increases 14.1% after the smearing. Finally, we conclude that responses ofthe position measurements estimated by the simulation are reliable within the 14.1% systematic
uncertainty for the position resolution.

6.2

Study of the neutron identification by Pythia event generator

The event structure in detector acceptance at pp collision was studied by Geant3 with Pythia
event generator. The neutron identification and its reliability are discussed in this section.

6.2.1

Event structureFirst, we studied what kind of particles are detected in the ZDC with the 5 GeV energy threshold
which was required for the neutron event trigger, ZDCN|S (Figure 6.21). In most of events,
about 92%, only one particle is detected with the ZDC in each pp collision. Figure 6.21 left
shows a tag-number of the particle which is detected with the ZDC. The tag-number is defined in2

It is same as that a fraction of charge sum is shared to all scintillators.

70

CHAPTER 6. SIMULATION STUDY

SMD multiplicity = 2

SMD multiplicity = 3

SMD multiplicity = 4
0.18

0.25
0.25

0.16
0.2

0.14

0.2

0.12
0.15

0.15

0.1
0.08

0.1

0.1

0.05

0.05

0.06
0.04
0.02

0
-10 -8

-6

-4

-2

0

2

4

6

8

10

0
-10

-8

-6

-4

-2

0

2

xpos (cm)

4

6

8

10

0
-10 -8

SMD multiplicity = 5

-4

-2

0

2

4

6

8

10

xpos (cm)

SMD multiplicity = 6

0.18

-6

xpos (cm)

SMD multiplicity = 7

0.16
0.14

0.16
0.14
0.14

0.12
0.12

0.12

0.1
0.1

0.1

0.08

0.08

0.08

0.06

0.06

0.06
0.04

0.04

0.02

0.02

0
-10 -8

-6

-4

-2

0

2

4

6

8

10

0
-10

0.04
0.02

-8

-6

-4

-2

0

2

xpos (cm)

4

6

8

10

0
-10 -8

-6

-4

-2

0

2

xpos (cm)

4

6

8

10

xpos (cm)Figure 6.13: The SS shapes for the real (solid line) and simulation (dashed line) data for xposition are plotted in each SMD multiplicity. Vertical scale is normalized by event number.
SMD multiplicity = 2

SMD multiplicity = 3

SMD multiplicity = 4

0.25

0.22

0.16

0.2

0.14
0.2

0.18

0.12

0.16
0.14

0.1

0.15

0.12

0.08

0.1

0.1

0.06

0.08
0.06

0.04
0.05

0.04

0.02

0.02
0
-10 -8

-6

-4

-2

0

2

4

6

8

10

0
-10

-8

-6

-4

-2

0

2

ypos (cm)

4

6

8

10

0
-10 -8

-6

-4

-2

0

2

ypos (cm)

SMD multiplicity = 5

4

6

8

10

ypos (cm)

SMD multiplicity = 6

SMD multiplicity = 7

0.16

0.14

0.14
0.14

0.12

0.12

0.12

0.1

0.1

0.1

0.08

0.08

0.08
0.06

0.06

0.06

0.04

0.04

0.04

0.02

0.02

0.02

0
-10 -8

-6

-4

-2

0

2

4

6

8

10

ypos (cm)

0
-10

-8

-6

-4

-2

0

2

4

6

8

10

ypos (cm)0-10 -8

-6

-4

-2

0

2

4

6

8

10

ypos (cm)

SMD multiplicity = 8
0.14
0.12
0.1
0.08
0.06
0.04
0.02
0
-10 -8

-6

-4

-2

0

2

4

6

8

10

ypos (cm)Figure 6.14: The SS shapes for the real (solid line) and simulation (dashed line) data for yposition are plotted in each SMD multiplicity. Vertical scale is normalized by event number.

71

CHAPTER 6. SIMULATION STUDY

SMD multiplicity = 2

SMD multiplicity = 3

SMD multiplicity = 4

1000
1400

600
800

1200

500

1000
400

600
800

300
400

600

200

400
200

100
0
0

200

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1

0
0

PFx

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1

0
0

PFx

SMD multiplicity = 5

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1PFx

SMD multiplicity = 6

SMD multiplicity = 7

3500

2500

5000
3000
2000

4000

2500
1500

2000

3000

1500
1000

2000
1000

500

1000

500
0
0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1

0
0

PFx

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1

0
0

PFx

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1

PFx

Figure 6.15: The PF distributions for the real (solid line) and simulation (dashed line) data for
x-position are plotted in each SMD multiplicity.
SMD multiplicity = 2

SMD multiplicity = 3

450

SMD multiplicity = 4
800

500

400

700
400

350

600

300
500

300

250

400
200
200

300

150

200

100

100
100

50
0
0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1

0
0

PFy

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1

0
0

PFy

SMD multiplicity = 5

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1PFy

SMD multiplicity = 6

SMD multiplicity = 7

2000
1000

2500

1800
1600

800

2000

1400
1200

600

1500
1000
800

400

1000

600
400

200

500

200
0
0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1

PFy

0
0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1

PFy

0
0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1PFy

SMD multiplicity = 8
3000
2500

2000
1500
1000
500
0
0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1

PFy

Figure 6.16: The PF distributions for the real (solid line) and simulation (dashed line) data fory-position are plotted in each SMD multiplicity.

CHAPTER 6. SIMULATION STUDY

72

Figure 6.17: The correlation between the SMD charge sum vs. SMD charge in one scintillator
for x position. The correlations for the real (top-left) and simulation (top-right) are plotted.
Here, we show the charge for the scintillator ID number 8 as a horizontal axis, for example.
In the real data, we can see the correlation between them; if the SMD charge sum is larger,
the pedestal of one SMD moves to higher. This behavior is not reproduced by the simulation.
The correlations after the flat smearing with ff lat =5% (bottom-left) and 10% (bottom-right) inthe simulation are shown. The correlation is reproduced reasonably by the flat smearing with
ff lat =5%.

73

CHAPTER 6. SIMULATION STUDY

Figure 6.18: Schematic views of the smearing procedures for the neighboring method (left) andflat method (right) in case of y position (The number of scintillators = 8). (Left : neighboring
method) A fraction, fneighboring , of the charge in i-th scintillator is shared to neighbor two
scintillators. (Right : flat method) A fraction, ff lat , of charge in the i-th scintillator is shared
to all scintillators. Figures just show the smearing for one scintillator, and the smearing wasapplied to all scintillators above threshold so that the charge sum is conserved.

SS for x after smearing : SMD multi. = 7

PF for x after smearing : SMD multi.=7

0.14
5000
0.12
4000

0.1
0.08

3000

0.06
2000
0.04
1000

0.02
0
-10 -8

-6

-4

-2

0

2

4

6
8 10
xpos (cm)00

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1PF

Figure 6.19: SS and PF shapes after the smearing for x position at the SMD multiplicity =7. Solid and dashed lines show the shapes for the real and simulation data, respectively. For
reproducing the real data, we determined the smearing parameters for the neighboring and flat
methods as 0.28 and 0.05, respectively. The smearing parameter of neighboring method was
decided by minimizing the χ2 value of the histogram fit for PF shapes between the real and thesimulation data. After the PF shape was matched, SS shape also reproduced that of the real
data.

74

CHAPTER 6. SIMULATION STUDY

Ratio of smearing to default with PosCut r<4cm
Ratio (%)

Position resolution by RMS(cm)Position resolutin with PosCut r<4cm

2

1.3

1.25

1.8
1.2

1.6
1.15

1.4

1.1

1.2

1.05

1
0.8
0

10

20

30

40 50 60 70 80 90 100
Incident neutron energy (GeV)

1
0

10

20

30

40 50 60 70 80 90 100
Incident neutron energy (GeV)

Figure 6.20: Left) The position resolution as a function of the incident neutron energy in r<4cm estimated by the simulation. Closed and open circles show the position resolutions without
and with the smearing to match the SS and PF shapes respectively, and the ratio is shown in
the right figure. It looks independent of the incident energy and the average value of 14.1% isconsidering as a maximum systematic error for the estimation of the position resolution.

Geant3 and summarized in Table 6.1. Main particles detected with the ZDC are the photon,
neutron and proton. Measured energy distributions for these three particles are plotted in
Figure 6.21 right 3 .
In the first expectation, only neutral particles, the photon and neutron, are detected with
the ZDC since there is the DX magnet. However the proton has a chance to deposit its energy
in the ZDC as follows. Figure 6.22 shows the event structures of protons for various incident
energies. A bent low energy proton makes a particle shower by hitting the DX magnet or beampipe, and particles from the shower can hit the ZDC. Such proton event can be removed by the
charge veto counter.
Physics processes for major three particles estimated by Pythia are summarized in Table
6.2. Most of the photon and neutron are generated by a diffractive and a gluon scattering
processes. 4 Protons are generated by the elastic and diffractive processes.

6.2.2

Neutron identification

As mentioned in the previous section, major backgrounds are photons and protons for theneutron measurement. They can be removed based on their behaviors as follows.
3

High energy tail above 100 GeV is caused by the edge effect; the hadron shower hits fibers in the top region.
Details are discussed in section 6.1.2.
4
In Pythia hard process, the leading neutron is generated mainly from a string fragmentation (∼65%) andthe decay from ∆0 , ∆+ , ∆− , Λ0 . the forward photon is decayed from the π 0 (∼91%) and η(∼7%).

75

CHAPTER 6. SIMULATION STUDYParticle number in GEANT3 tag

Energy distribution for single particle in ZDC

3

×10

450

all eventγ
neutronproton

5

10

400
104

350
300

3

10

250
200

2

10
150
100

10

50
0
0

5

10

15

20
25
Particle number

1

0

50

100

150

200

250
300
ZDC energy (GeV)√
Figure 6.21: The event structure which is detected with the ZDC in pp collision at s=200
GeV was studied by Geant3 with Pythia simulation. The ZDC threshold has set to 5 GeV
in the similar to the ZDCN|S trigger. Here, I show the event that one particle is detected with
the ZDC in each pp collision (about 92%). The particle tag-number in Geant3 (summarizedin Table 6.1) is plotted in the left. Major particles, the photon, neutron and proton are shownas the tag-number = 1, 13 and 14, respectively. Measured energy distributions for these threeparticles with the ZDC are plotted in the right figure.
Tag-number
1
2
3
5
6
8
9

Particle
γ
e+
e−
µ+
µ−
π+
π−

Tag-number
10
11
12
13
14
15
25

Particle
KL0
K+
K−
n
p
n̄
p̄

Table 6.1: The tag-numbers for particles in Geant3
Physics process
qq → qq
q q̄ → q q̄
q q̄ → gg
qg → qg
gg → q q̄
gg → gg
Elastic scattering
Single diffractive (XB)
Single diffractive (AX)Double diffractive
Low-pT scattering
Total

Neutron (µb)
35
<1
<1
268
9
352
0
<1
462
328
551
2006

γ (µb)
46
<1
<1
358
12
468
0
2
527
413
651
2479

Proton (µb)
14
<1
<1
95
3
114
446
387
107
92
137
1395

Table 6.2: Cross sections for major three particles detected with the ZDC in each physics
process studied by Geant3 with Pythia. Same energy threshold as ZDCN|S trigger, 5 GeV,
was required.

CHAPTER 6. SIMULATION STUDY

76

Figure 6.22: The event structure of the proton for various incident momenta. In case of the
momentum below 70 GeV/c, the particle shower is generated by hitting the DX magnet or
beam pipe and it can get into the ZDC acceptance.

CHAPTER 6. SIMULATION STUDY

77• The behavior for both photons and protons : The deposit energy in the ZDC is distributedin the lower energy part, less than 20 GeV, as shown in Figure 6.21.

• The behavior for photons : Photons stop in the first ZDC module which has 51 X0 . Thus,
photons can be removed by requiring some energy deposit in the SMD or second ZDC.

• The behavior for protons : Charged particles in the proton shower are expected to deposit
the energy in the charge veto counter. It can be removed by requiring charge veto cut.

The neutron energy above 20 GeV and the charge veto cut were required as a basic cut inthis analysis and most of proton events were removed by this cut. There are two methods toremove the photon as follows
1. Using the SMD :The requirement of SMD hits; more than one scintillator above threshold (the SMD
multiplicity ≥ 2) for both x and y → Neutron purity = 93.6±0.3%
2. Using the second ZDC module :The second ZDC energy deposit above 20 GeV → Neutron purity = 93.6±0.5%
Main backgrounds after the neutron identifications are the K 0 and proton. At the analyses of
the cross section and the asymmetry, the SMD cut was applied automatically since the position
information calculated by the SMD is necessary. The second ZDC cut was used to the estimationof the SMD cut efficiency which is discussed in section 7.5.
The purities which are described in this section were estimated for the neutron energy above
20 GeV. In the cross section and the asymmetry analyses, we required also the acceptance cut,r <X cm and/or more higher energy cut. In these cases, the purities would become better andare estimated in each analysis section.

6.2.3

Background contamination after the neutron identification

Background contamination after the neutron identification estimated by Geant3 with Pythia
and their reliabilities are discussed in this section.
K 0 contamination
At the ISR experiment, the K 0 contamination to the neutron measurement were estimated fromthe K ± measurements [2]. Conclusions were 10 % at xF =0.2 and less than 4 % at xF >0.4.
Table 6.3 lists the fraction of the K 0 and proton contamination after the neutron identification√
with the acceptance cuts, r<3 cm, in case of pp at s=200 GeV estimated by the simulation.
The fraction of the K 0 to the neutron in Pythia is consistent with the ISR result. It indicates
the K 0 contamination estimated by Pythia is reasonable. However there is no measurement
experimentally. Thus, we have included no correction for the K 0 contamination in this analysis
as well as the IP12 analysis (section 3.2).

CHAPTER 6. SIMULATION STUDY

78

Proton contamination
The proton background is very sensitive to the materials around the ZDC and the magnet tuningin the accelerator. Systematics for the proton contamination estimated by the simulation were
evaluated by a fraction of charged events in the charge veto counter.
Figure 6.23 shows the comparison of charge distributions in charge veto counters between
the real and simulation data normalized by their statistics. Noise was estimated by the pedestal
width of the real data and the width = 16.64 ch was incorporated into the simulation. For thisanalysis, photon events were removed by requiring the second ZDC module cut (section 6.2.2)
since a converted photon in upstream materials hits the charge veto counter. A fraction of
proton events can be evaluated as a fraction of charged candidates which is the events in one
MIP or more. Their fractions were 0.424 and 0.279 for the real and simulation, respectively.
Proton events for the real data were about 1.5 times higher than that of the simulation.
A threshold dependence of the selection of charged particle candidates was also studied.
Typically we applied 50 channel to the threshold and checked the ratio of the proton fractionin the real and simulation for ±10 channel (Table 6.4). Their fluctuation was less than 1% andthe threshold dependence is negligibly small. Therefore, factor 1.5 was reasonable value for thefraction of charged candidates between the real and simulation data. It would be including thesystematics of the proton generation in Pythia, the material tunes in Geant3 and the magnet
tuning in the accelerator.
Since it is just estimated by simulation, we have included no correction to the results, thisis added as a systematic error. According to the study for the structure of the proton event
in section 6.2.1, the proton event would be detected in the direction of beam bending which
is negative x for the SOUTH ZDC. This behavior was confirmed by the real data as shownin Figure 6.24 which is a scatter plot of the x position determined by the SMD vs. the charge
distribution in the charge veto counter. Most of charged candidates were distributed in negativex region. It indicates that the proton background was not related to the proton spin and we
can assume the proton background AN equals to zero. We added its systematic error by the
dilution method discussed in Appendix A.4 with Abg
N =0.
Multi-particles event
The event structure in pp collision has been studied as far as the one particle detection in one
event. From Pythia study, the event which has two or more particles into the ZDC acceptanceis about 8%, it depends on the acceptance cut. We have included this type of background into
the systematic error.

79

CHAPTER 6. SIMULATION STUDY

K 0 (%)
13.91±1.04
6.61±0.41
2.77±0.19
1.71±0.13
1.13±0.10
0.61±0.07
0.48±0.07
0.18±0.05
0.005±0.003

xF
0.1∼0.2
0.2∼0.3
0.3∼0.4
0.4∼0.5
0.5∼0.6
0.6∼0.7
0.7∼0.8
0.8∼0.9
0.9∼1.0

proton (%)
1.65±0.34
2.67±0.26
4.27±0.23
5.24±0.23
4.50±0.20
4.00±0.19
3.91±0.21
3.48±0.22
15.26±0.53

Table 6.3: The background fraction of the K 0
and proton after the neutron identification inr<3 cm estimated by Geant3 with Pythia in√
pp collision at s=200 GeV.

x-position vs. charge vetox position (cm)Comparison of CV shapes between real (heavy line) and sim (thin line)

Charged event

104

Real data (ZDCN|S trigger)
103

5
4

104

3
2
1

Simulation

103

0
102

-1
-2
-3

10

102

-4
0

500

1000

1500

2000
1 MIP = 100

-5
0

50

100

150

200

250

300 350 400 450 500
charge veto (1 MIP = 100 ch)

Figure 6.23: The comparison of charge distribu- Figure 6.24: The x position calculated by thetions in charge veto counters between the real SMD vs. the charge distribution in the charge
and simulation data with the second ZDC cut veto counter for the real data. Most of charged
events were distributed in negative x region(photon veto).
which is the direction of beam bending by the
DX magnet.

Charge veto counter cut
40
50
60

Real
0.461
0.424
0.408

Sim
0.302
0.279
0.270

Ratio (Real/Sim)
1.525
1.519
1.510

Table 6.4: Fractions of charged candidates in the charge veto counter after requiring the photon
veto. They were evaluated for three thresholds, 40, 50(default) and 60 ch, to check the thresholddependence. Ratios of them are also listed and they are threshold independent.

Chapter 7

Analysis for the cross section
7.1

Cross section calculation

7.1.1

Cross section at the PHENIX experiment

Since it is difficult to determine pT by the PHENIX data alone, the cross section was estimatedas a differential with respect to xF as,dσ
Nneutron 1
=
,
dxF
L
dxF

(7.1)

where, Nneutron is counts for the neutron sample after the correction of cut efficiencies (section
7.5, 7.7) and the energy unfolding (section 7.6).

7.1.2

Cross section at the ISR experimentInvariant cross sections measured at the ISR experiment were converted to differential crosssections for the comparison with the PHENIX data. The conversion formula to dσ/dxF is
described with the approximation in the forward kinematics as,
Z
d3 σ
d3 σ
1 xF d2 σ
dσ
2π
E 3 ≈
E 3 pT dpT ,→=d p
2π dxF pT dpT
dxF
xF Acc. d p(7.2)

where Acc. means the pT range for the PHENIX acceptance cut (for example, 0< pT <0.11xF
GeV/c for the r <2 cm cut).

7.2

Energy distribution in the very forward kinematics

For the cross section analysis, 65 million events taken by the ZDCN|S trigger were used. The
acceptance cut was decided as r<2 cm to choose the similar kinematics of the ISR experiment;
the detection angle is about 1 mrad. It is expected to be maximum pT ≈0.11 GeV/c for the

100 GeV neutron by this cut. We assumed a beam axis on the ZDC geometry was same as the

ZDC center in this analysis. However, it is a possible that the beam axis was shifted. The beamaxis search on the ZDC geometry is shown in section 8.1 and the effect for the cross section is
discussed in section 7.9.3.
80

CHAPTER 7. ANALYSIS FOR THE CROSS SECTION

81

Figure 7.1 shows the energy distribution measured with the ZDC after the neutron identification and the acceptance cut. A detailed discussion of the neutron identification are in section
6.2.2. The measured energy spectrum has a peak structure as shown in Figure 7.1.
ZDC energy distribution with neutron ID nad acceptance cut

800 ×10

3

700
600
500
400
300
200
100
0
0

20

40

60

80

100

120
140
ZDC energy (GeV)

Figure 7.1: The energy distribution measured with the ZDC after the neutron identificationand the acceptance cut (r<2 cm, corresponds to pT <0.11 GeV/c).

7.3

Stability

In this section, we discuss the stability checks we have made run-by-run for the neutron measurement.

7.3.1

Energy gain

If a gain of the ZDC was changed in the run period, it appears in a variation of peak positions.
The stability of the energy gain can be evaluated by the fluctuation through the run period asshown in Figure 7.2. The energy gain was stable.

7.3.2

Neutron yield

The ratio of the neutron yield to the BBCLL1(NoVtxCut) counts through the run period givethe stability of the cut condition for the neutron measurement. The ratio, RN , was defined as,
RN =N of counts with the neutron ID and the acceptance cut.
N of BBCLL1(NoVtxCut) counts(7.3)

Figure 7.3 shows the ratio through run period with a constant fit result. The neutron yield wasstable through the measurement.

82

CHAPTER 7. ANALYSIS FOR THE CROSS SECTION
Mean of measured peak in ZDC through RUN5

χ2 / ndf
Prob
p0

74
72

Neutron production ratio throuth RUN5

20.2 / 24
0.685668.03 ± 0.1741

70χ2 / ndf

0.011

17.75 / 24

Prob0.0105p0

0.81490.009273 ± 2.714e-05

0.01

68

0.009566
0.009
640.0085

62
×10

3

60

178.45178.5

178.55178.6
Run number

×10

3

0.008

178.45178.5

178.55

178.6
Run Number

Figure 7.2: The peak value of the energy distri- Figure 7.3: The ratio of the neutron yield tobution through the run period.
the BBCLL1(NoVtx) counts through the run
period.

7.4

Study of the beam gas backgroundAs one of the backgrounds, we need to consider a beam gas event which is the particle shower
event occurred by an interaction between the proton beam and particles in the beam pipe. It isremoved by the requirement of the collision vertex since most of beam gas events distribute inout of the collision time window. However, in case of the ZDC self trigger, we cannot distinguishthe collision event and beam gas event. The amount of the beam gas event is estimated in thissection.
As a RHIC operation in 2005 run, there were 9 abort bunches in each beam. In the PHENIX
interaction point, abort bunches of one beam cross to the filled bunch of other beam. In thesecrossing, only the beam gas event can occur, whereas the collision can never. We can estimate
the fraction of the beam gas background by the events in abort bunches assuming there is nodifferences of beam conditions in all bunches.
Figure 7.4 shows the event counts in each bunch crossing after the neutron identification andthe acceptance cut. Abort bunch crossings are found as dashed areas in Figure 7.4. The fractionof the beam gas background was estimated to be 0.0062±0.0004 in average. We conclude that
it was negligibly small.

7.5

Relative efficiency to the SMD cut between the real andsimulation data

The neutron hit position was calculated by the centroid method using the information of scintillator charges above the threshold (equation 5.1). In this analysis, same threshold, 40 ch, wasapplied to the real and simulation data and the efficiency of the acceptance cut using the SMD
was estimated by the simulation in section 7.7. However, the SMD cut efficiency between thereal and simulation data can be different since absolute calibration was not well known in the

83

CHAPTER 7. ANALYSIS FOR THE CROSS SECTIONCounts for each crossing after applying neutron ID and acceptance cut.
104

103

102

10
0

20

40

60

80

100

120
Crossing

Figure 7.4: The event counts in each bunch crossing after the neutron identification and theacceptance cuts. Dashed areas show the events in the abort bunch crossings that only beamgas events occur. (31–39 : abort bunches in the Yellow beam and 111–120 : abort bunches in
the Blue beam)

real data (section 5.2.1). These difference of efficiencies can be estimated using the nearly pureneutron sample by the neutron identification with the second ZDC cut (section 6.2.2). The
SMD cut efficiency for the neutron measurement was evaluated by a following formula,ǫSM D cut =N of (Neutron ID with the second ZDC cut)&(the SMD cut).
N of (Neutron ID with the second ZDC cut)

(7.4)

The efficiencies for the real and simulation are shown in Figure 7.5 as a function of the measured
energy. A relative efficiency as a ratio of the real and simulation data is shown in the leftbottom. It is nearly equal to one but it depends on the measured energy. The energy spectrum
was corrected based on this result before the energy unfolding in section 7.6.

7.6

Energy Unfolding

The measured neutron energy with the ZDC is smeared by the energy resolution. For the extraction of an initial energy distribution, it is necessary to unfold the measured energy distribution.
In this section, the energy unfolding [72] is discussed .

7.6.1

Unfolding method

We assume that the initial distribution x(E) is smeared to the measured distribution y(E ′ ) andthis smearing is described by a linear combination. Their relation can be given by the matrix

84

CHAPTER 7. ANALYSIS FOR THE CROSS SECTION

Real data : SMD cut efficiency for neutron-2

Sim data : SMD cut efficiency for neutron
-2

105

105

100

100
Efficiency

110 ×10

Efficiency

110 ×10

95
90
85

95
90
85

80

80

75

75

70

70

65

65

60
0

20

40

60

80

100 120 140
ZDC energy (GeV)

60
0

20

40

60

80

100 120 140
ZDC energy (GeV)

Relative SMD cut efficiency for neutron-2

Relative ratio of efficiency ( sim / real )110 ×10

105
100
95
90

Fit by polynomial 285

y = p0 + p1*x + p2*x^28075p0 = 0.957743 ± 0.014748

70

p1 = 0.001679 ± 0.000348

65

p2 = -0.000013 ± 0.000002

60
0

20

40

60

80

100 120 140
ZDC energy (GeV)

Figure 7.5: The SMD cut efficiencies for the real (Left) and simulation data (Right) in Top two
figures. Bottom figure shows the ratio of the efficiencies for the simulation to the real data.

85

CHAPTER 7. ANALYSIS FOR THE CROSS SECTION
A(E ′ , E) which is a transition matrix for the energy smearing.
y(E ′ ) = A(E ′ , E)x(E).

(7.5)

It can be represented as ~y = A~x. The extraction of the initial distribution is possible with aninversion of the matrix A.
~x = A−1 ~y,Vx = A−1 Vy (A−1 )T ,(7.6)

where Vx and Vy are the error vectors for ~x and ~y , respectively. 1
If the smearing effect is large, the result looks terrible and is too sensitive to a small changeof A. It can be discussed using an orthogonal decomposition. The matrix A is expressed by,
A = U DU −1 ,(7.7)

where U is a transform matrix and D is a diagonal matrix. The diagonal elements of the matrix
D are the eigenvalue λj of the matrix A. Equation 7.6 is represented as,
y = A~x = U DU −1 ~x,~
U −1 ~
y = DU −1 ~x→~c = D~b,(7.8)

where ~c = U −1 ~y and ~b = U −1 ~x are new vectors transformed from ~y and ~x, respectively. Eachof the coefficients bj and cj in ~c = D~b is transformed independently of any other coefficient byusing eigenvalue λj ,cj = λj · bj .

(7.9)

In order to perform the unfolding, the coefficients cj have been influenced by statisticalfluctuations of the elements of measured vector ~y . The bj which includes the information of
initial vector x is obtained by bj = cj /λj . The statistical fluctuation of the cj is delivered tothe bj as magnified values in case of small eigenvalue λj . It is the reason why the result is
unsatisfactory. Reasonable result can be obtained by cutting the cj which has a large statisticaluncertainty.

7.6.2

Unfolding the data

Figure 7.6 shows the scatter plot of the initial and measured neutron energies estimated by thesimulation. The transition matrix A was calculated by it. A binning is set to 20 GeV due tothe 20% energy resolution for the 100 GeV neutron.
First, the coefficients cj were calculated and shown in Figure 7.7. Three sets of the transition
matrix A, which have the same energy resolution but different initial energy distributions2 , wereIf ~x and ~y have different dimensions, A has zero-row or -column. In this case, we can not calculate A−1 sincea determinant is zero. We need to calculate A−1 as a pseudo-inverse matrix, A+ , which is a generalized inverse
matrix. It still satisfies equation 7.6.
2
Initial shapes are prepared to increase, be flat and decrease as a function of xF . These shapes are close to
the cross sections at pT ≈ 0.0 GeV/c, 0.2 GeV/c and 0.4 GeV/c in the ISR results (Figure 1.5).
1

86

CHAPTER 7. ANALYSIS FOR THE CROSS SECTION

prepared to check the statistical error propagation of the cj . Basically cj distributions should
be same since it depends on the energy resolution only, not on the initial distribution. However,if the cj has large statistical uncertainty, its value is easily fluctuated by a small change of A.
From Figure 7.7, the c4 has a large statistical uncertainty and it is delivered to the b4 magnified
by the smallest eigenvalue λ4 3 . The c4 should be removed for the reasonable unfolding. The
statistics was scaled for the reduction with the c4 cut so that the statistics was conserved.
Energy spectra before and after the unfolding are plotted in Figure 7.8. Horizontal axis ischanged to xF by equation 1.3.
cj values calculated by 3 sets of transition matrix A from simulation.
Incident energy (GeV)c

Relation between measured and incident energy estimated by simulation

100
90

7

10

80
70
60

106

50
40
30

105

20
10
0
0

20

40

60

80

0

100
120
140
160
Measured energy (GeV)

1

2

3

4

5

6
j

Figure 7.6: A relation between the initial and Figure 7.7: The values of the coefficients cj calthe measured neutron energies estimated by the culated by three sets of the transition parameter
simulation. z-axis was normalized as a sum of A which have the same energy resolution, buthorizontal values in each vertical bin. The tran- different initial distributions. The c at j=4 hassition matrix A was calculated by this relation. a large statistical uncertainty.
Energy unfolding
×10

6

22
20
18

Before unfolding

16

After unfolding

Figure 7.8: Energy distributions before and after the energy unfolding.
The unfolding was performed so thatstatistics was conserved.

14
12
10
8
6
4
2
0
0

3

0.2

0.4

0.6

0.8

1

1.2

1.4
xF

Typical eigenvalue, λj , for the A being smaller starting at one with increasing j. For example, {λ0 , λ1 , λ2 ,λ3 , λ4 }={1.00, 0.82, 0.59, 0.37, 0.12} in our case.

87

CHAPTER 7. ANALYSIS FOR THE CROSS SECTION

7.7

Estimation of the efficiency for the experimental cut

The efficiency of the experimental cut, including the neutron identification and the acceptancecut, for the unfolded xF distribution was estimated by the simulation with the single neutron
event generator. The efficiency was defined as follows.
ǫcut ef f =N of counts after the experimental cut with the detector resolution.
N of simulation inputs in the acceptance(7.10)In this analysis, the acceptance cut was performed by the radius, r, thus, the efficiency canbe visualized by looking at the pT distribution bin-by-bin based on the equation 1.4, pT =xF · Ep · θn ≈ 0.056 · xF · r GeV/c.

Figure 7.9 shows the simulated pT distributions (dashed line) in each xF bin starting at

0.2 with 0.2 interval. Maximum pT for the acceptance cut, r <2 cm, in each xF are given byax = 0.11 · x GeV/c shown as dot-dashed lines. Thus, the expected p distribution in the
pM
F
T
T

acceptance is below dot-dashed lines. However actual pT distributions with the experimental

cut were smeared due to the position resolution shown as solid lines. Ratio of their counts are
the efficiency for the experimental cut as equation 7.10 and listed in table 7.1. The errors were
derived from the uncertainty of the pT shape obtained by the ISR experiment, exp(−4.8±0.3pT ); the maximum variations of the efficiencies using exp(−4.5 pT ) and exp(−5.1 pT ) to thatusing exp(−4.8 pT ) are shown. These variations were added as a systematic error.
The solid curves also show the pT resolution derived from the position resolution and theresolution seems to be insufficient to determine the pT with present detectors. Mean values ofsimulated pT distributions in each energy region are also listed in table 7.1.
Neutron xF
0.2–0.4
0.4–0.6
0.6–0.8
0.8–1.0

Expected pT region
for r <2 cm (GeV/c)
0–0.033
0–0.056
0–0.078
0–0.100

Mean pT with theexperimental cut (GeV/c)
0.050
0.071
0.087
0.099

Efficiency (Systematics
derived from pT shape)
0.707 ± 0.060 (8.5%)
0.777 ± 0.037 (4.8%)
0.735 ± 0.017 (2.3%)
0.658 ± 0.018 (2.7%)

Table 7.1: The expected pT region, mean pT value and the efficiency for the experimental
cut estimated by the simulation (Figure 7.9). The errors for efficiencies are derived from theuncertainty of the pT shape obtained from the ISR experiment, exp(−4.8±0.3 pT ).

7.8

Result

The cross section was obtained after the correction of the energy unfolding and the cut efficiency
as shown in Figure 7.10. Error bars only show the statistical uncertainty derived from theunfolding. Before the physics discussion in chapter 9, systematic errors are discussed from next
sections.

88

CHAPTER 7. ANALYSIS FOR THE CROSS SECTION

pT distribution : neutron xF at 0.3 (0.2-0.4)

pT distribution : neutron xF at 0.5 (0.4-0.6)
40000

30000

35000
30000

25000

25000
20000
20000
15000
15000
10000

10000

5000
0
0

5000
0.05

0.1

0.15

0.2

0.25
pT

pT distribution : neutron xF at 0.7 (0.6-0.8)
50000

0
0

0.05

0.1

0.15

0.2

0.25
pT

pT distribution : neutron xF at 0.9 (0.8-1.0)
30000
25000

40000

20000
30000
15000
20000
10000
10000

0
0

5000

0.05

0.1

0.15

0.2

0.25
pT

0
0

0.05

0.1

0.15

0.2

0.25
pT

Figure 7.9: Simulated pT distributions using the pT shape measured at the ISR experiment.
Initial pT distributions are shown as dashed line in each xF region. Expected pT region for
the acceptance, r <2 cm, is below dot-dashed line which is the maximum pT calculated as≈ 0.11 · xF GeV/c. However actual pT distributions with the experimental cut, including theneutron identification and the acceptance cut, were smeared as solid lines due to the position
resolution. The ratio of a) Entries of pT distribution in the experimental cut (solid line) to b)
Entries in initial pT shape below the dot-dash line are the efficiency for the experimental cut.

89

CHAPTER 7. ANALYSIS FOR THE CROSS SECTION

dσ/dx (mb)

Cross section of forward neutron production
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

xF

Figure 7.10: The cross section after the correction of the cut efficiency and the energy unfolding.
Error bars only show the statistical uncertainties derived from the unfolding.

7.9

Systematic errors

Systematic errors for the cross section measurement are discussed in this section. Scaling errors
are summarized in section 7.9.1 and bin-by-bin correlated systematics are discussed in section
7.9.2 or later. Only the bin-by-bin systematics for the cut efficiency were evaluated in section
7.7. All errors were estimated as the ratio of the variation to the final cross section values.
Total errors were calculated by quadratic sum.

7.9.1

Scaling errors• The background estimation by the simulation with Pythia event generator. (section
6.2.3).

– Background contamination (in the measured neutron energy with the ZDC from 20
to 140 GeV for the acceptance cut of r < 2 cm)∗ BG from the proton : 3.6% (Original value is 2.4%. the proton contamination
estimated by the simulation should be 1.5 times as discussed in section 6.2.3)∗ Multi particles detection in each collision : 7%
• Systematics for the luminosity estimation by BBC counts– 9.7% (22.9±2.2 mb for the BBCLL1 trigger)• In total : 12.5%

90

CHAPTER 7. ANALYSIS FOR THE CROSS SECTION

7.9.2

Systematics for the energy linearity

In section 6.1.1, the ZDC energy response for the neutron below 100 GeV was estimated by
the simulation and it has the non-linearity. This non-linearity was corrected by the energy
unfolding because its effect is included in the transition matrix A (section 7.6). Since thehadron interaction was just estimated by the simulation, systematics should be considered. We
considered it as a variation of the cross section evaluated with a different matrix A with thelinear response.
Figure 7.11 shows differential cross sections calculated with the transition matrix A for the
non-linear (default) and linear energy responses. Table 7.2 lists the ratio of the variation to thefinal value which was calculated with the non-linear response in each xF bin. They are added
as a systematic error in each bin.

dσ/dx (mb)Differential cross section (dσ /dx) with non-linear and linear energy response for unfolding.

0.9

Energy response is non-linear (default)
0.8

Energy response is linear

0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

xF

Figure 7.11: Differential cross sections calculated by the transition matrix A with the non-linear(default) and the linear energy responses.

xF
0.2∼0.4
0.4∼0.6
0.6∼0.8
0.8∼1.0

Systematic error (%)
14.33
30.16
7.46
12.59

Table 7.2: Systematic errors for the energy response. Errors are shown as a ratio of the variation
to the final cross section value.

CHAPTER 7. ANALYSIS FOR THE CROSS SECTION

7.9.3

91

Systematics for the determination of the beam axis on the ZDC geometry

In this analysis, we assumed that the beam axis points to the ZDC center. Searching for thethe beam axis on the ZDC geometry was performed by the position scan of the asymmetry insection 8.1. Results of the beam axis searches for x and y were +0.46 cm and −1.10 cm fromthe ZDC center, respectively.

The shift of the beam axis also appear in the position distributions at high SMD multiplicity
event. Figure 7.12 shows the position distribution as a function of the SMD multiplicity for thereal and simulation data. In the low multiplicity case, the real and simulation data agree wellfor both x and y. But at the high multiplicity events, the center for x and y are shifted andtheir tendencies are consistent with the results of the beam axis searches.
However the beam axis shifts were only shown in pp collision as discussed in section 8.1. We
do not understand the reason why. In this analysis, we suffice them by the estimation of theeffect to the cross section measurement as systematic error. For this estimation, cross sections
were calculated in the different acceptances according to the result of the beam axis shift while
keeping r<2 cm and their variations were applied as systematic error. The results of crosssections are shown in Figure 7.13 and the ratio of the variations to the final values are listed inTable 7.3.

92

CHAPTER 7. ANALYSIS FOR THE CROSS SECTION

6

4

2

6

4

2

0

0

-2

-2

-4

-4

-6
0

1

2

3

4

-6
0

5
6
7
8
SMD multiplicity for x

SMD multiplicity dependence for x-position (Sim data)

6

4

2

-2

-4

-4

3

4

5
6
7
8
SMD multiplicity for x

4

5
6
7
8
9
SMD multiplicity for y

2

-2

2

3

4

0

1

2

6

0

-6
0

1

SMD multiplicity dependence for y-position (Sim data)

y position (cm)x position (cm)

SMD multiplicity dependence for y-position (Real data)

y position (cm)x position (cm)

SMD multiplicity dependence for x-position (Real data)

-6
0

1

2

3

4

5
6
7
8
9
SMD multiplicity for y

Figure 7.12: Position distributions for x (Left) and y (Right) as a function of the SMD multiplicity for the real (Top) and simulation data (Bottom). Asymmetric distribution for y at low
SMD multiplicity was due to the different edge effect for the top and bottom regions (section
6.1.2). Center positions of the real data are shifted at high SMD multiplicity for both x andy and tendencies are consistent with the results of the beam axis searches in section 8.1. The
beam axis shift can not be reproduced by the simulation, thus the shifts are not caused by thedetector geometry.

93

CHAPTER 7. ANALYSIS FOR THE CROSS SECTION

dσ/dx F (mb)Differential cross section (dσ /dx F) with various cut
0.7
Beam center = ( 0.00, 0.00 )Beam center = ( 0.46, 0.00 )0.6

Beam center = ( 0.00,-1.10 )Beam center = ( 0.46,-1.10 )

0.5

0.4

0.3

0.2

0.1
0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

xF

1

Figure 7.13: Cross sections were calculated in the different acceptances according to the resultof the beam axis search. The centers of acceptances were (x,y)=(0.00,0.00), (0.46,0.00), (0.00,1.10) and (0.46,-1.10) cm with keeping r <2 cm.

xF
0.2∼0.4
0.4∼0.6
0.6∼0.8
0.8∼1.0(x,y)=(0.46,0.00) (%)
0.32
1.03
0.34
0.46

(x,y)=(0.00,-1.10) (%)
0.38
16.04
0.69
4.34

(x,y)=(0.46,-1.10) (%)
1.03
26.27
2.04
6.94

Table 7.3: Differences of the cross section values by moving the acceptance center. Values were
calculated as the ratio of variations to the final cross section value in each xF bin. Maximum
differences shown as bold characters were added as a systematic error.

Chapter 8

Analysis for the single transverse
spin asymmetry
In this chapter, analyses for single spin asymmetries are shown. The definition of single transverse spin asymmetry (AN ) are written in Appendix A. The azimuthal angle and xF dependencies of AN are evaluated and shown in section 8.2 and 8.3, respectively.

8.1

The study of the beam axis on the detector geometry

The ZDC center was aligned to a beam axis at the beginning of 2003 run. Basically we assumed
that the beam axis was on the ZDC center in this analysis. Moreover we can obtain the beamaxis on the detector geometry by a position scan of the AN which is calculated by the luminosity
formula since AN should be zero kinematically at the beam axis (Appendix A). Figure 8.1 shows
schematic views of position scans. Scans went through x and y directions starting at negative
region and we called the “horizontal scan” and “vertical scan”, respectively. At the verticallypolarized pp collision, which is an usual setting in transverse run at RHIC, the AN should appear
in the horizontal scan only; the AN is expected to be zero at the beam axis and distributed
symmetric around it. The AN for vertical scan should be zero in all region at that time. At thehorizontally polarized pp collision, the AN is expected to appear in the vertical scan only. We
can extract beam axes for x and y by looking at intersections of zero-asymmetry in these scans.
Figure 8.2 and 8.3 show results of the position scans for the vertically and horizontally polarized pp collisions, respectively1 . We used BBCLL1&(ZDCN|S) data samples in this analysis.
The AN distributions for all scans give the consistent shapes as we expected. The beam axes
for x and y were obtained as +0.46±0.08 cm and −1.10±0.14 cm for x and y, respectively.

However this estimation has some systematics. It appears in the vertical scan for horizontally

polarized pp run in the forward kinematics as a systematic error for the relative luminosity, R;
The values of AN should be zero for all bins, but the values were systematically shifted to thenegative. Same R was used to the horizontal scan. Therefore, we cannot conclude the shift ofthe beam axis in x estimated by the AN scan alone.
1

Horizontally polarized pp collision was performed at 2006 run.

94

CHAPTER 8. ANALYSIS FOR THE SINGLE TRANSVERSE SPIN ASYMMETRY95

For the beam axis in y, its shift is so large. According to the RHIC complex, the fluctuationof the beam axis should be a few mm. For more discussions, we checked position distributionsin heavy ion data which was CuCu collision at 2005 run2 and the result is Figure 8.4. We canalso evaluate the beam axis by these distributions since 100 GeV neutrons from the heavy ion
peripheral collision are emitted within a few cm diffusion at the ZDC transverse geometry [2].
The beam axis for x and y were obtained as 0.28±0.01 cm and −0.07±0.01 cm, respectively.

There are inconsistencies between the pp and heavy ion data, especially in the y-axis. We do

not understand the reason why.
The beam axis shift that we observed should be considered as systematic errors for the
results. Systematics were defined as variations of the cross section and asymmetry results
obtained by moving the center of acceptance while keeping same cut region (for example, r<2
cm for the cross section analysis).

Horizontal scan
ZDC detector

Vertical scan
ZDC detector

-5cm

-5cm

5cm

5cm

Figure 8.1: Schematic views of the position scans of AN calculated by the luminosity formulafor searching beam axes in x (Left) and y (Right). The gray region, 1 × 10 and 10 × 1 cm forthe horizontal and vertical scans, respectively, is the acceptance of the AN calculation in eachbin in Figure 8.2 and 8.3.

2

It was performed just before pp run period.

CHAPTER 8. ANALYSIS FOR THE SINGLE TRANSVERSE SPIN ASYMMETRY

Positon scan of forward LR luminosity asymmetry in transvrese run

10 ×10-2

Horizontal scan
( Forward )8
6

Positon scan of backward LR luminosity asymmetry in transverse run∈N/pol.

∈N/pol.

-2
10 ×10

6
4

2

2

0

0

-2

-2

-4

-4

-6

-6

-8

-8
-3

-2

-1

0

1

2

3
4
5
xpos (cm)Positon scan of forward UD luminosity asymmetry in transverse run∈N/pol.

-2
10 ×10

-10
-5

-2

-1

0

1

2

3
4
5
xpos (cm)-2

Vertical scan
( Forward )

6

6
4

2

2

0

0

-2

-2

-4

-4

-6

-6

-8

-8
-3

-2

-1

0

1

2

3
4
5
ypos (cm)Vertical scan
( Backward )

8

4

-4

-3

10 ×10

8

-10
-5

-4

Positon scan of backward UD luminosity asymmetry in transverse run∈N/pol.

-4

Horizontal scan
( Backward )

8

4

-10
-5

96

-10
-5

-4

-3

-2

-1

0

1

2

3
4
5
ypos (cm)Figure 8.2: The position scan of ǫN calculated by the luminosity formula at the transversely
polarized pp run. The finite asymmetry is seen at the horizontal scan in the forward kinematics
and its intersection within ǫN =0 should be the beam axis in x. Based on this result, the beamaxis was obtained as +0.46±0.08 cm in x for the ZDC geometry.

CHAPTER 8. ANALYSIS FOR THE SINGLE TRANSVERSE SPIN ASYMMETRY

Positon scan of forward LR luminosity asymmetry in radial run

10 ×10-2

Horizontal scan
( Forward )8
6

Positon scan of backward LR luminosity asymmetry in radial run∈N/pol.

∈N/pol.

-2
10 ×10

6
4

2

2

0

0

-2

-2

-4

-4

-6

-6

-8

-8
-3

-2

-1

0

1

2

3
4
5
xpos (cm)Positon scan of forward UD luminosity asymmetry in radial run∈N/pol.

-2
10 ×10

-10
-5

-2

-1

0

1

2

3
4
5
xpos (cm)-2

Vertical scan
( Forward )

6

6
4

2

2

0

0

-2

-2

-4

-4

-6

-6

-8

-8
-3

-2

-1

0

1

2

3
4
5
ypos (cm)Vertical scan
( Backward )

8

4

-4

-3

10 ×10

8

-10
-5

-4

Positon scan of backward UD luminosity asymmetry in radial run∈N/pol.

-4

Horizontal scan
( Backward )

8

4

-10
-5

97

-10
-5

-4

-3

-2

-1

0

1

2

3
4
5
ypos (cm)Figure 8.3: The position scan of ǫN calculated by the luminosity formula at the horizontally
polarized pp run. The finite asymmetry is seen at the vertical scan in the forward kinematics
and its intersection within ǫN =0 should be the beam axis in y. Based on this result, the beamaxis was obtained as −1.10±0.14 cm in y for the ZDC geometry.

98

CHAPTER 8. ANALYSIS FOR THE SINGLE TRANSVERSE SPIN ASYMMETRY

hx

x-position from CuCu data
Entries
Mean
700

RMS

0.96652

χ / ndf
Prob
Constant

600
500
400

105.8 / 97
0.2544695 ± 4.1

Mean

0.2834 ± 0.0060

Sigma

0.8679 ± 0.0099

700

Entries

79420

Mean

0.009809

RMS

600

1.055

χ2 / ndfProb

500

0.3672

Constant

400

101.1 / 97
632.4 ± 3.9

Mean-0.07071 ± 0.00769

Sigma

0.9657 ± 0.0139

300

300
200

200

100

100

0
-5

hy

y-position from CuCu data
79420
0.2335

-4

-3

-2

-1

0

1

2

3
4
5
x-position (cm)

0
-5

-4

-3

-2

-1

0

1

2

3
4
5
y-position (cm)Figure 8.4: Position distributions for x and y for the CuCu data in 2005 run. Neutron energy
was selected to be above 300 GeV, corresponds to three neutrons or more from the peripheral
collision. Beam axes looks almost center in the ZDC geometry.

8.2

Azimuthal angle dependence of AN

8.2.1

Asymmetry calculation

The acceptance definition for the azimuthal angle (φ) dependent asymmetry is shown in Figure8.5. For the comparison with the previous measurement, the IP12 experiment (see part I),
same acceptance cut, 0.5<r<4.0 cm, was applied. The acceptance area was cut to 16 pieces ina radial pattern and asymmetry was calculated by the square root formula in the same way of
the IP12 experiment. (section 3.4)3 .
For this analysis, we used 6.5 million (3 fills) and 17.6 million (4 fills) events for the ZDCN|S
and BBCLL1&(ZDCN|S) trigger samples, respectively. The energy cut was required to select40–120 GeV in the measured energy with the ZDC. Before merging ǫN ’s of all fills, they weredivided by polarizations in each fill; the polarization can be fluctuated fill-by-fill due to a timedependence of the acceleration performance. The polarization in each fill is listed in Table 8.1.
Merged ǫN divided by the polarization are plotted in Figure 8.6. Finite asymmetries are seenin the forward kinematics for both trigger samples and they can be fitted by the sine curve(equation 3.6) well.

8.2.2

Preparation of the simulation sets for each trigger

Measured energy distributions for two trigger samples after the experimental cut were different
as shown in Figure 8.7. The neutron energy associated with charged particles was lower thanthat without the association. It is appropriate in the view of energy conservation.
We should prepare two sets of simulations for the estimations of the smearing parameters,ǫsmearing, which are correlated to the position resolution and the position resolution is correlated
φ3

The definition of φ is different with the IP12 experiment. The φ=0 at the PHENIX corresponds to theφ=−π/2 at the IP12. To perform the φ dependence, both experiments calculated AN in counterclockwise.

CHAPTER 8. ANALYSIS FOR THE SINGLE TRANSVERSE SPIN ASYMMETRY

99

6

φ=π/2

600

4
500
2
400

φ=π

0

φ=0

-2

300
200

-4

100

-6
-6

-4

-2

0

2

4
6
xpos (cm)Forward Neutron Asymmetry φ distribution ZDCN|S trigger-20

Forward Neutron Asymmetry φ distribution BBCLL1&(ZDCN|S) trigger-2

10 ×10∈N / pol.

Figure 8.5: The acceptance definition for theφ dependence of AN . The acceptance cut was
required 0.5<r<4.0 cm in the same way of theIP12 experiment. The acceptance was also divided to 16 pieces in a radial pattern and theasymmetry was calculated by the square root
formula starting at φ=0 to π in counterclockwise. In this figure, only even pieces are plotted.

χ /ndf = 8.09 / 6
p0 = -0.044 ± 0.006
p1 = 1.454 ± 0.137
2

8
6

10 ×10∈N / pol.

ypos (cm)

Acceptance definition for φ-dependence of AN

6

4

4

2

2

0

0

-2

-2

-4

-4

-6

-6

-8

-8

-10
0

-10
0

0.5

1

1.5

2

2.5

3
φ (rad)Backward Neutron Asymmetry φ distribution ZDCN|S trigger
-2

1

1.5

2

2.5

3
φ (rad)Backward Neutron Asymmetry φ distribution BBCLL1&(ZDCN|S) trigger

10 ×10∈N / pol.

∈N / pol.

0.5-2

10 ×10
8

8

6

6

4

4

2

2

0

0

-2

-2

-4

-4

-6

-6

-8

-8

-10
0

χ2/ndf = 3.03 / 6
p0 = -0.056 ± 0.003
p1 = 1.594 ± 0.047

8

0.5

1

1.5

2

2.5

3
φ (rad)

-10
0

0.5

1

1.5

2

2.5

3
φ (rad)Figure 8.6: The φ dependence of ǫN divided by the polarization. Top and bottom figures show
the asymmetries in the forward and backward kinematics, respectively. Left and right figures
show them for the ZDCN|S and BBCLL1&(ZDCN|S) samples, respectively. Finite asymmetries are seen in the forward kinematics and can be fitted by the sine curve well. Backward
asymmetries were consistent with zero for both samples.

CHAPTER 8. ANALYSIS FOR THE SINGLE TRANSVERSE SPIN ASYMMETRY
Polarization
Fill number
6931
7294
7295
7296

Blue beam (%)
43.06±2.94
47.63±2.69
47.70±2.4546.30±2.30

100

Yellow beam (%)
52.25±3.50
46.30±8.23
50.67±3.25
52.20±3.23

Table 8.1: The polarization value in each fill which was used for the asymmetry analysis.
Fill#6931 had the BBCLL1&(ZDCN|S) data sample only. In the present analysis, we used the
SOUTH ZDC only. Thus, asymmetries in the forward and backward kinematics were calculatedby the Yellow and Blue polarizations, respectively. Systematic uncertainties of 6.2% and 5.9%
for the Yellow and Blue beams are not included.

with the neutron energy (section 6.1.3). The energy distributions for the simulation inputs were
determined in the same way of the cross section analysis (chapter 7). After some iterations bymodifying the input shapes4 , we obtained the simulations which well reproduce measured energy
distributions above 40 GeV as shown in Figure 8.8. Same simulations were used for the analysisof the xF dependence (Chapter 8.3).
Measured energy distributions within r<2 cm
4500
4000
3500

Figure 8.7: Measured energy distributions after
the neutron identification and the acceptancecut for the ZDCN|S and BBCLL1&(ZDCN|S)
samples.

3000
2500
2000
1500
1000

BBCLL1&(ZDCN|S)

500

ZDCN|S

0
0

8.2.3

20

40

60

80
100
120
Measured energy (GeV)

Estimation of the smearing effectThe ǫN is smeared from the AN by the wrong assignment of left and right due to the position
resolution. It is called smearing effect and it was estimated by the simulation. From equation
3.5, the smearing parameter, ǫsmearing, can be evaluated by the simulation as,φ
AOutput
smearing= NInput ,
ǫφ
AN

(8.1)

where, AOutput
corresponds to the ǫN of the real data; it is including effects of the experimentalN
cut and the position resolution. As AInput, we generated neutrons with the sine modulatedN
AN (φ) as equation 3.6 with AN =−0.10. The output is shown in Figure 8.9 in case of the
4

In the cross section calculation, we cannot obtain the detail shape because we concluded it with only threepoints. The small iteration to reproduce the measured energy distribution is necessary.

CHAPTER 8. ANALYSIS FOR THE SINGLE TRANSVERSE SPIN ASYMMETRY

Measured energy distribution : ZDCN|S

Measured energy distribution : BBCLL1&(ZDCN|S)

16000

16000

14000

14000

12000

12000

10000

10000

8000

8000

6000

6000

Real data

4000

0

20

40

60

Real data

4000

Simulation

2000

101

Simulation

2000
80
100
120
Measured energy (GeV)

0

20

40

60

80
100
120
Measured energy (GeV)

Figure 8.8: Measured energy distributions for the real and simulation data. Input shapes forthe simulations were modified to reproduce the measured energy distributions above 40 GeV
for the ZDCN|S and BBCLL1&(ZDCN|S) trigger samples.

ZDCN|S. Smeared amplitude was obtained as −0.076 and their ratio, 0.76, is a correction

factor of the smearing effect. We also estimated that for the BBCLL1&(ZDCN|S) by sameprocedure and obtain smearing parameters as follows.
• ǫsmearing[ZDCN|S]φ: 0.760 ± 0.015• ǫsmearing[BBCLL1&(ZDCN|S)] : 0.746 ± 0.016
φThey were not so different.
Estimation of AOutput
for the φ-depedence asymmetry by MC with initial AN=-0.10N

10 ×10

ANOutput-2

χ2/ndf = 5.77 / 6
p0 = -0.0760 ± 0.0015
p1 = 1.587 ± 0.020

8
6
4
2
0
-2
-4
-6
-8

-10
0

0.5

1

1.5

2

2.5

3

φ

Figure 8.9: The output of the φ-dependent
AN from the simulation which was modified to
the ZDCN|S sample. As a simulation input,
AN (φ)=AN sin(φ − π/2) with AN =−0.10 wasused. The output amplitude was reduced to−0.076 and the ratio of them, 0.76, is a correction factor of the smearing effect.

CHAPTER 8. ANALYSIS FOR THE SINGLE TRANSVERSE SPIN ASYMMETRY

8.2.4

102

Systematic errors

Background contamination
The background contamination was studied by the simulation with Pythia event generator.
After the neutron identification and the acceptance cut, the neutron purities were,• Without BBC trigger, ZDCN|S : 0.975 ± 0.006• With BBC trigger, BBCLL1&(ZDCN|S): 0.977 ± 0.010
Main background contributions were the K 0 and proton. According to the discussion in section
6.2.3, we added the systematic error contributed from the proton only. They were 1.4% and1.0% for the ZDCN|S and BBCLL1& (ZDCN|S) triggers, respectively and should be increased
1.5 times. Increased values, 2.1% and 1.5% were added as systematic errors.
The background from the beam gas event was also discussed in the cross section study(section 7.4). It was 0.22% and negligible.
Systematics from the beam axis shift
The systematic error for the determination of the beam axis was discussed in this section.
According to the results of beam axis searches, AN (φ) were calculated with center positions
as (x,y) = (0.46, 0.00), (0.00, -1.10) and (0.46, -1.10) cm while keeping the acceptance cut,
0.5<r<4.0 cm. Results fitted by the sine curve are shown in Figure 8.10. We cannot distinguishthe best fit from χ2 values. Thus, maximum variations to final values, which were calculated by(x,y) = (0.00, 0.00) cm, are added as systematic errors. Table 8.2 lists asymmetry amplitudes
and variations to final values.
ZDCN|S
Acceptance center in (x,y) cm(0.00, 0.00)(0.46, 0.00)(0.00, -1.10)(0.46, -1.10)
BBCLL1&(ZDCN|S)
Acceptance center in (x,y) cm(0.00, 0.00)(0.46, 0.00)(0.00, -1.10)(0.46, -1.10)

ǫN / pol.
0.0443±0.0063
0.0448±0.0063
0.0414±0.0062
0.0439±0.0062

Variation to the final value (%)“This is the final value”
1.13
6.55
0.90

ǫN / pol.
0.0559±0.0028
0.0530±0.0028
0.0576±0.0028
0.0559±0.0028

Variation to the final value (%)“This is the final value”
5.19
3.04
∼0

Table 8.2: The ǫN calculated with various center positions of the acceptance. Variations to final
values are also shown and maximum variations are displayed as bold characters. The maximum
one is added as systematic errors.

103

CHAPTER 8. ANALYSIS FOR THE SINGLE TRANSVERSE SPIN ASYMMETRY

χ2 /ndf = 8.09 / 6
p0 = -0.0443 ± 0.0063
p1 = 1.454 ± 0.137------ (x,y) = (0.46,0.00) -----χ2 /ndf = 3.34 / 6
p0 = -0.0448 ± 0.0063
p1 = 1.512 ± 0.135------ (x,y) = (0.00,-1.10) -----χ2 /ndf = 3.66 / 6
p0 = -0.0414 ± 0.0062
p1 = 1.417 ± 0.145------ (x,y) = (0.46,-1.10) -----χ2 /ndf = 9.98 / 6
p0 = -0.0439 ± 0.0062
p1 = 1.461 ± 0.136

8
6
4
2
0

Forward φ-asymmetry for various centers : BBCLL1&ZDCN|S

10 ×10-2∈N / pol.

∈N / pol.

Forward φ-asymmetry for various centers : ZDCN|S-2------ (x,y) = (0.00,0.00) -----10 ×10

8
6
4
2
0

-2

-2

-4

-4

-6

-6

-8

-8

-10
0

0.5

1

1.5

2

2.5

3
φ

------ (x,y) = (0.00,0.00) -----χ2 /ndf = 3.03 / 6
p0 = -0.0559 ± 0.0028
p1 = 1.594 ± 0.047------ (x,y) = (0.46,0.00) -----χ2 /ndf = 2.51 / 6
p0 = -0.0530 ± 0.0028
p1 = 1.575 ± 0.049------ (x,y) = (0.00,-1.10) -----χ2 /ndf = 3.32 / 6
p0 = -0.0576 ± 0.0028
p1 = 1.587 ± 0.046------ (x,y) = (0.46,-1.10) -----χ2 /ndf = 6.55 / 6
p0 = -0.0559 ± 0.0028
p1 = 1.569 ± 0.047

-10
0

0.5

1

1.5

2

2.5

3
φ

Figure 8.10: The φ dependence of ǫN calculated with various center positions of the acceptanceas (x,y) = (0.00, 0.00), (0.46, 0.00), (0.00, -1.10) and (0.46, -1.10) cm are plotted as circle,
square, triangle-up and triangle-down points, respectively. The horizontal value in each point
was shifted to be easily viewable: only circle points show correct values.

Systematics for the correction of the smearing effectSince the smearing effect is caused by the position resolution, systematic error of the position
resolution, 14.1% (section 6.1.5), should be reflected to uncertainties for the result. It wasestimated by a variation of the asymmetry value calculated with the 14.1% increased position
resolution in the simulation. Figure 8.11 shows output asymmetries with the default position
resolution and the 14.1% increased one. The asymmetry was more smeared with 4.2% reduction.
The variation is assigned as a systematic error for the smearing correction.
Effect of the detector asymmetry
A detector asymmetry is defined as a difference of acceptance for left and right (Appendix A).
The detector asymmetry is basically canceled out using the square root formula until thirdorder. In this section, the cancellation of the detector asymmetry is shown.
Figure 8.12 shows detector asymmetries calculated by moving the x-center of the acceptance;
x-centers were −0.4, −0.2, 0.0, 0.2, 0.4 cm. The detector asymmetry increased 0.25 by moving

the x-center from −0.4 cm to 0.4 cm. However physics asymmetries which are also shown in

Figure 8.12 did not change within 0.003. This result indicates the square root formula wascorrectly working for the cancellation of the detector asymmetry.

CHAPTER 8. ANALYSIS FOR THE SINGLE TRANSVERSE SPIN ASYMMETRY

104

Estimation of the systematic error of smearing effect for phi-dependence

10 ×10

ANOutput-2--- Defalt position resolution --2
χ /ndf = 8.83 / 6
p0 = -0.0762 ± 0.0009
p1 = 1.549 ± 0.012--- 14.1% increased postion resolution --2
χ /ndf = 5.59 / 6
p0 = -0.0730 ± 0.0009
p1 = 1.551 ± 0.012

8
6
4

Figure 8.11: The φ dependence of AN withthe default and 14.1% increased position resolutions estimated by the simulation are plotted
as closed and open circles, respectively. As asimulation input, AN (φ)=AN sin(φ − π/2) with
AN =−0.10 was generated. The variation of amplitudes was 4.2%.

2
0
-2
-4
-6
-8
-10
0

0.5

1

1.5

2

2.5

3

Detector asymmetry for various x-centerφMeasured asymmetry for various x-center-2∈N

Detector asymmetry-2

30 ×10x = -0.4 cm

×10
3.4

x = -0.2 cm

20

3.2

x = 0.0 cm
x = 0.2 cm

103

x = 0.4 cm

2.8

0

2.6
-10
2.4
-20

-30
0

2.2

0.5

1

1.5

2

2.5

3
φ

2

-0.4

-0.2

0

0.2
0.4
x-center (cm)Figure 8.12: Left) Detector asymmetries as a function of φ for various x-centers ; x-center for-0.4, -0.2, 0.0, 0.2, 0.4 cm. Right) Physics asymmetries, ǫN , as function of the x-center.

CHAPTER 8. ANALYSIS FOR THE SINGLE TRANSVERSE SPIN ASYMMETRY

105

Bunch shufflingA technique called “bunch shuffling” can be utilized to check uncertainties in asymmetries dueto the variation of beam characteristics bunch by bunch. The basic idea is the comparison of theasymmetry fluctuation to the statistical uncertainty by random assignments of spin patterns
in each bunch, which is called a bunch shuffling. If there is no systematic uncertainty, thefluctuation of asymmetry with the bunch shuffling should be same as the statistical uncertainty.
If there is, the fluctuation being larger than the statistical one.
Asymmetry calculations by the bunch shuffling were performed fill by fill and the results
divided by polarizations were fitted through the run period in each shuffling. For example,
results for 5 shuffles are shown in Figure 8.13. The χ2 value of the constant fit is described ineach shuffle with the assumption of the same ǫsmearing
in all fills as,
φ
χ2 =

n
X
(AN (f ) − hAN (f )i)2f2
σstat.
(f )

,

(8.2)

where f is the fill number with n being the total number of fills. Therefore, χ2 distributions
lead to the comparison of the asymmetry fluctuation, shown as numerator, to the statisticaluncertainty, shown as denominator, through the run period. We can evaluate bunch-by-bunchsystematics whether the χ2 /ndf is unit or not.
In this analysis, the bunch shuffling was performed 30,000 times. Figure 8.14 and 8.15
show χ2 /ndf distributions in each φ region for the ZDCN|S and BBCLL1&(ZDCN|S) samples,
respectively. We found large χ2 /ndf values in the first and last bins. It is caused by the residualof the physics asymmetry which would affect as,v"u 2 #u 1∆l∆AResidual∼ AN t1+,
Nnbunchhli∆(χ2 /ndf ) ∼(∆AResidual
)2
N
,
2
σstat

where nbunch is total bunch number in all fill, nbunch =(8.3)
ff ill nbunch .

P

hli and ∆l denote themean value and the bunch-by-bunch fluctuation of luminosity. Detailed calculation of theresidual asymmetry are shown in Appendix C.2. In the data sets, nbunch for the ZDCN|S and
BBCLL1&(ZDCN|S) were 195 and 245, respectively. For example, the value of ∆(χ2 /ndf ) for
first and last bins were ≈0.42 with ǫN /pol.≈0.057 for the BBCLL1&(ZDCN|S) trigger sample5 .
On the other hand, the χ2 /ndf value for bin#4 was almost unit since the ǫN /pol. was nearlyzero in this bin (Figure 8.6); the residual asymmetry was negligible.
In addition, we should consider the fluctuation of χ2 /ndf value coming from the total bunch
number. The χ2 /ndf can fluctuate as,2∆(χ /ndf ) ∼
5

1± p

1
(nbunch − 1)

Values of the ∆(χ2 /ndf ) for all bins are listed in Appendix C.2.

!2

− 1.

(8.4)

CHAPTER 8. ANALYSIS FOR THE SINGLE TRANSVERSE SPIN ASYMMETRY

106

Details are shown in Appendix C.1. It means that we cannot ignore the fluctuation in case ofsmall nbunch . For our data samples, values of ∆(χ2 /ndf ) for the ZDCN|S and BBCLL1&(ZDCN|S)√
are about 0.15 and 0.13, respectively. Systematical fluctuations are 0.15·σstat = 0.39·σstat and√0.13 · σstat = 0.36 · σstat for ZDCN|S and BBCLL1&(ZDCN|S), respectively. The systematics
estimated by the bunch shuffling is limited by this fluctuation.

Values of χ2 /ndf for all bins can be understood with the fluctuation caused by nbunch andthe residual asymmetry. We conclude the fake asymmetry from the bunch-by-bunch is lessthan 0.39σstat and 0.36σstat for the ZDCN|S and BBCLL1&(ZDCN|S), respectively. We did
not include these uncertainties to the final systematics in AN .
Constant fits for bunch shuffling results∈N / pol.

-2

10 ×10

1st shiffling : χ2/ndf = 1.512
2nd shiffling : χ2/ndf = 0.658
3rd shiffling : χ2/ndf = 0.359
4th shiffling : χ2/ndf = 0.688
5th shiffling : χ2/ndf = 1.435

8
6
4
2
0
-2
-4
-6
-8
-10
7292.57293
6931

7293.57294

7294.57295

7295.57296 7296.5
Fill number

Figure 8.13: Asymmetries were calculated by
the bunch shuffling and shown as a function of
the fill. In this picture, the bunch shuffling was
performed five times, for example. Results werefitted by constant through all fills and fit lines
are shown as solid lines. Values χ2 /ndf of fit
results are also shown.

107

CHAPTER 8. ANALYSIS FOR THE SINGLE TRANSVERSE SPIN ASYMMETRY22

χ /ndf for Bin 1

2

χ /ndf for Bin 2

χ /ndf for Bin 3

300

Entries = 30000

300

Entries = 30000

300

Entries = 30000

250

Mean = 0.857

250

Mean = 0.895

250

Mean = 1.188

200

RMS = 0.842

200

RMS = 0.893

200

RMS = 1.163

150

150

100

100

50

0
0

1

2

3

4

φ = 0 ~ 1π
8

50

5

0
0

6

7

8
9
2
χ /NDF

2

1

2

3

4

50

5

0
0

6

7

8
9
2
χ /NDF

Entries = 30000

250

Mean = 0.882

200

RMS = 0.866

150

1

2

3

4

7

8
9
2
χ /NDF250

Mean = 1.032

200

RMS = 0.901

200

RMS = 1.011

150

100

1

2

3

4

φ = 4π ~ 5π
8
8

50

5

0
0

6

7

8
9
2
χ /NDF

φ = 5π ~ 6π
8
8
1

2

3

4

5

6

7

8
9
2
χ /NDF2

χ /ndf for Bin 7

χ /ndf for Bin 8300

Entries = 30000

300

Entries = 30000

250

Mean = 1.280

250

Mean = 1.348

200

RMS = 1.240

200

RMS = 1.296

150

150

100

100

50

0
0

6

Mean = 0.904

0
0

2

5

250

5

8
9
2
χ /NDF

4

Entries = 30000

50

7

3

Entries = 30000

φ = 3π ~ 4π
8
8
6

2

χ /ndf for Bin 6

100

50

1

300

150

100

φ = 2π ~ 3π
882

χ /ndf for Bin 5
300

0
0

100

φ = 1π ~ 2π
882

χ /ndf for Bin 4
300

150

1

2

3

4

φ = 6π ~ 7π
8
8

50

5

0
0

6

7

8
9
2
χ /NDF

φ = 7π ~ π
8
1

2

3

4

5

6

7

8
9
2
χ /NDF

Figure 8.14: Distributions of χ2 /ndf obtained from the bunch shuffling for the ZDCN|S sample.
Each histogram indicates each bin for the φ-dependent AN . Solid gray line is the expected
χ2 /ndf distribution for 3 fills (ndf = 2) so that the asymmetry fluctuation by the bunch shuffling
equal to the statistical uncertainty.

108

CHAPTER 8. ANALYSIS FOR THE SINGLE TRANSVERSE SPIN ASYMMETRY22

χ /ndf for Bin 1

2

χ /ndf for Bin 2

Entries = 30000

250

Mean = 1.321

200

χ /ndf for Bin 3

Entries = 30000

250

Mean = 1.297

200

RMS = 1.053

RMS = 1.008

RMS = 0.931

150

150

100

100

100

0
0

1

2

3

4

φ = 0 ~ 1π
8

50

5

0
0

6

7

8
9
2
χ /NDF

2

1

2

3

4

φ = 1π ~ 2π
8
8

50

5

0
0

6

7

8
9
2
χ /NDF2

χ /ndf for Bin 4

Entries = 30000
Mean = 0.820200

Entries = 30000
Mean = 0.992

200

RMS = 0.659

100

4

φ = 3π ~ 4π
8
8

50

5

0
0

6

7

8
9
2
χ /NDF

2

1

2

Mean = 1.388

200

3

4

8
9
2
χ /NDF

Entries = 30000
Mean = 1.156

φ = 4π ~ 5π
8
8

50

5

0
0

6

7

8
9
2
χ /NDF

φ = 5π ~ 6π
8
8
1

2

3

4

5

6

7

8
9
2
χ /NDF

Entries = 30000

250

Mean = 1.525

200

RMS = 1.106

RMS = 1.182

150

150

100

100

50

2

7

χ /ndf for Bin 8

Entries = 30000

1

6

2

χ /ndf for Bin 7
250

0
0

5

RMS = 0.934

100

3

4

RMS = 0.783

100

2

3

200

150

1

2

250

150

0
0

1

χ /ndf for Bin 6

250

150

50

φ = 2π ~ 3π
882

χ /ndf for Bin 5

250

Mean = 1.165

200

150

50

Entries = 30000

250

3

4

φ = 6π ~ 7π
8
8

50

5

0
0

6

7

8
9
2
χ /NDF

φ = 7π ~ π
8
1

2

3

4

5

6

7

8
9
2
χ /NDF

Figure 8.15:
Distributions of χ2 /ndf obtained from the bunch shuffling for the
BBCLL1&(ZDCN|S) sample. Each histogram indicates each bin for the φ-dependent AN . Solidgray line is the expected χ2 /ndf distribution for 4 fills (ndf = 3) so that the asymmetry fluctuation by the bunch shuffling equal to the statistical uncertainty.

CHAPTER 8. ANALYSIS FOR THE SINGLE TRANSVERSE SPIN ASYMMETRY

109

Summary of systematic errors
Systematic errors are listed in this section. Values are presented as variations to final values. If
there are differences between two trigger samples, the ZDCN|S value is written up in parentheses.
Total uncertainties were calculated by quadratic sum.
• Estimation from the measurement : total 5.2% (6.6%)– Beam axis shift : 5.2% (6.6%)– Beam gas background for the ZDCN|S trigger : (<1%)• Simulation uncertainty : total 7.4% (8.0%)– Background contamination : 1.5% (2.1%)– Smearing effect from the position resolution : 4.2%
– Multiple hits in one event : 5.9% (6.5%)• Total : 9.0% (10.4%)• Polarization : 6.2% for the Yellow beam

CHAPTER 8. ANALYSIS FOR THE SINGLE TRANSVERSE SPIN ASYMMETRY

8.3

xF dependence of AN

8.3.1

Asymmetry calculation

110

The acceptance definition for the xF dependence of AN is shown in Figure 8.16. It required
0.5<r<3.0 cm, which is not same as the azimuthal angle dependence because we should not use
the edge region to avoid the edge effect (section 6.1.2). The acceptance was also divided to theleft and right sectors by the azimuthal angle cut which was determined to maximize the figureof merit according to the cosφ dependence of AN ,Rφ
( −φ0 0 cosφ dφ)2
S2
=
,
N
2φ0

(8.5)

and obtained as φ0 =66.8◦ . With this definition, the effective AN of present analysis is expected
to be
f
Aef
N
=
AN

R φ0

−φ0 cosφ dφ

2φ0= 0.788.

(8.6)

φ0 =66.8◦

This effect is a part of the smearing effect which is estimated in section 8.3.2.
Asymmetry was calculated by the square root formula with the number of event in the left
sector (NL ) and the right sector (NR ) in each measured energy (E) as,qq
NL↑ (E)NR↓ (E) − NR↑ (E)NL↓ (E)q.
ǫN (E) = q
NL↑ (E)NR↓ (E) + NR↑ (E)NL↓ (E)(8.7)

After the correction of the polarization, P , and the smearing effect, ǫsmearing
which is discussedE
in next section, we obtain initial AN (E) as,AN (E) =11
ǫN (E).
smearing
P ǫE
(E)

(8.8)

We used same data samples as the φ dependence analysis (section 8.2.1). According to
the 20% energy resolution at the 100 GeV neutron, we chose 20 GeV binning in the measured
energy with the ZDC starting at 40 GeV and last bin was decided from 80 to 120 GeV due tothe resolution.
The measured energy dependence of ǫN divided by the polarization are plotted in Figure8.17. Finite asymmetry can be seen in the forward kinematics for both trigger samples and theylook flat. The energy unfolding is a straight forward to extract the initial xF from the measured
energy in the same way of the cross section analysis, however, the correlated errors from theunfolding are so terrible and illusive6 . For the estimation of the xF dependence, the meanvalue of xF in each bin was estimated by the simulations which were modified to reproduce themeasured energy distributions for each trigger samples (section 8.2.2).
6
In this case, the correlated error would be ∼2.5 times higher than the statistics error before the unfolding.
It means the unfolded error for the ZDCN|S sample would overlap with AN = 0 in all bins. But it is not true.
These errors are highly anti-correlated within the finite asymmetry which is seen obviously before the unfolding.
Such errors would invite the misunderstanding.

CHAPTER 8. ANALYSIS FOR THE SINGLE TRANSVERSE SPIN ASYMMETRY

111

ypos (cm)

Acceptance definition for xF-dependence of AN

6

700

4

600

Figure 8.16: The acceptance definition for thexF dependence of AN . It required 0.5<r<3.0
400 cm to avoid the edge effect. We divided theregion in the left and right sectors by the az◦◦
300 imuthal angle cut, −66.8 < φ < 66.8 , to maximize the figure of merit according to the cosφ
200 dependence of A (equation 8.5).
N
500

2

0

-2

-4

-6
-6

8.3.2

100

-4

-2

0

2

4
6
xpos (cm)0

Estimation of the smearing effectThe ǫN is smeared from AN due to the position resolution discussed in section 8.2.3. The smearing effect was estimated by the simulation with the same acceptance cut and, then, included
the smearing from the acceptance cut of equation 8.6. Figure 8.18 shows the output asymmetry
as a function of the measured energy obtained by the simulation which was modified to the
ZDCN|S sample (see section 8.2.2). As a simulation input, we generated neutrons with thesine modulated A(φ) with AN =−0.10 in all xF regions. The AOutput
were obtained as (40–60,N
60–80, 80–120(GeV)) = (−0.050, −0.054, −0.055). Their variations to the initial amplitude

as equation 8.1 are correction factors for the smearing effect. Same calculation was performedby the simulation which was modified to the BBCLL1& (ZDCN|S) sample, and we obtainedǫsmearing(E) in each trigger sample as follows.
E• ZDCN|S: (40–60, 60–80, 80–120(GeV)) = (0.50±0.03, 0.54±0.03, 0.55±0.03)• BBCLL1&(ZDCN|S) : (40–60, 60–80, 80–120(GeV)) = (0.50±0.02, 0.59±0.02, 0.55±0.02)

8.3.3

Estimation of the mean xF in each bin

After the correction of the smearing effect, we obtain the measured energy dependence of AN .
The mean values of xF in each bin were estimated in this section.
The mean xF values for the ZDCN|S and BBCLL1&(ZDCN|S) samples were estimated by
the simulations. The xF distribution in each energy bin is shown in Figure 8.19. Mean xF values for three bins, (40–60, 60–80, 80–120(GeV)), are hxF i= (0.57±0.13, 068±0.11, 0.78±0.09)

and (0.55±0.12, 0.66±0.10, 0.75±0.08) for the ZDCN|S and BBCLL1&(ZDCN|S) samples, respectively. Errors were obtained as RMS values.

CHAPTER 8. ANALYSIS FOR THE SINGLE TRANSVERSE SPIN ASYMMETRY

Forward Neutron LR Asymmetry for ZDCN|S

Forward Neutron LR Asymmetry for BBCLL1&(ZDCN|S)-2∈N / pol.

∈N / pol.

-2

8 ×10
6

8 ×10
6

4

4

2

2

0

0

-2

-2

-4

-4

-6

-6

-8
0

20

40

-8
0

60
80
100
120
Measured energy (GeV)

Backward Neutron LR Asymmetry for ZDCN|S

40

60
80
100
120
Measured energy (GeV)-2∈N / pol.

∈N / pol.

20

Backward Neutron LR Asymmetry for BBCLL1&(ZDCN|S)

-2

8 ×10
6

8 ×10
6

4

4

2

2

0

0

-2

-2

-4

-4

-6

-6

-8
0

112

20

40

60
80
100
120
Measured energy (GeV)

-8
0

20

40

60
80
100
120
Measured energy (GeV)

Figure 8.17: The measured energy dependent ǫN divided by polarizations. Top and bottom
show results in the forward and backward kinematics, respectively. Left and right figures showthese for the ZDCN|S and BBCLL1&(ZDCN|S) samples, respectively. Negative AN can be seenin the forward region.

113

CHAPTER 8. ANALYSIS FOR THE SINGLE TRANSVERSE SPIN ASYMMETRY

OutputEstimation of AN

for the energy depedence asymmetry by MC with initial AN=-0.10

ANOutput

-2

0 ×10
-1

Figure 8.18: The AOutput
as a funcNtion of the measured energy estimatedby the simulation which was modified
to the ZDCN|S sample. As an input,
we generated A(φ)=AN sin(φ − π/2)with AN =−0.10 in all xF region. The
asymmetry was smeared to 50∼55%
due to the position resolution and theacceptance cut.

-2
-3
-4
-5
-6
-7
-8
-9
-10
0

20

40

60

xF distribution for measured energy 40~60 (GeV) : ZDCN|S

80
100
120
Measured energy (GeV)

xF distribution for measured energy 60~80 (GeV) : ZDCN|SxF distribution for measured energy 80~120 (GeV) : ZDCN|S

450
400
350

Mean = 0.568500

Mean = 0.682

Mean = 0.776
600

RMS = 0.125

RMS = 0.109

RMS = 0.088

400

500

300
400

300

250
200

300
200

150

200

100

100

100

50
0
0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1xF

xF distribution for measured energy 40~60 (GeV) : BBCLL1&(ZDCN|S)00

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1xF

xF distribution for measured energy 60~80 (GeV) : BBCLL1&(ZDCN|S)

900

0
0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1xF

xF distribution for measured energy 80~120 (GeV) : BBCLL1&(ZDCN|S)

1000

800

Mean = 0.547

700

RMS = 0.121

Mean = 0.664
800

1000

RMS = 0.103

Mean = 0.749
RMS = 0.083

800
600
600

500

600

400
400

400

300
200

200

200

100
0
0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1
xF

0
0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1
xF

0
0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1xF

Figure 8.19: The simulated initial xF distribution in each measured energy bin for Left) 40–60,
Center) 60–80 and Right) 80–120 GeV. Top and bottom figures show these for the ZDCN|S and
BBCLL1&(ZDCN|S) samples, respectively.

CHAPTER 8. ANALYSIS FOR THE SINGLE TRANSVERSE SPIN ASYMMETRY

8.3.4

114

Systematic errors

The systematic errors for the xF dependence are evaluated in this section. The methods arebasically same as the φ-dependence analysis (section 8.2.4). We describe the results only.
Background contamination
After the neutron identification and the acceptance cut, the neutron purities were,• Without BBC trigger, ZDCN|S : 0.976 ± 0.007• With BBC trigger, BBCLL1&(ZDCN|S): 0.979 ± 0.014
The proton backgrounds were 1.3% and 0.9% for the ZDCN|S and BBCLL1& (ZDCN|S) samples, respectively and should be increased 1.5 times. Increased values, 2.0% and 1.4% wereadded as a systematic errors.
Systematics for the beam axis shift
According to results of beam axis searches, AN (E) were calculated with various center positions
as (x,y) = (0.46, 0.00), (0.00, -1.10) and (0.46, -1.10) cm while keeping the acceptance cut andresults are shown in Figure 8.20. Table 8.3 lists asymmetry values and their variations to final
values which were calculated by the center of (x,y) = (0.00, 0.00) cm. Maximum variations to
final values are added as a systematic errors.
Forward ene.-dep asymmetry for various centers : ZDCN|S

Forward ene.-dep asymmetry for various centers : BBCLL1&(ZDCN|S)-2∈N / pol.

∈N / pol.

-2

0 ×10
-1

0 ×10
-1

-2

-2

-3

-3

-4

-4

-5

-5

-6

-6

-7

-7

-8
0

20

40

60
80
100
120
Measured energy (GeV)

-8
0

20

40

60
80
100
120
Measured energy (GeV)

Figure 8.20: The measured energy dependence of ǫN divided by the polarization calculated withthe acceptance center as (x,y) = (0.00, 0.00), (0.46, 0.00), (0.00, -1.10) and (0.46, -1.10) cm areplotted as circle, square, triangle-up and triangle-down points, respectively. Horizontal value ineach point was shifted to be easily viewable; only circle points show correct values.

CHAPTER 8. ANALYSIS FOR THE SINGLE TRANSVERSE SPIN ASYMMETRY

Acceptance center (x,y) cm(0.00, 0.00)(0.46, 0.00)(0.00, -1.10)(0.46, -1.10)

ZDCN|S
40∼60 (GeV)-0.0359-0.0390 (8.6%)-0.0282 (21.6%)-0.0345 (4.0%)60∼80 (GeV)-0.0344-0.0274 (20.3%)-0.0292 (15.0%)-0.0265 (22.8%)80∼120 (GeV)-0.0355-0.0344 (3.2%)-0.0278 (21.7%)-0.0534 (50.4%)

Acceptance center (x,y) cm(0.00, 0.00)(0.46, 0.00)(0.00, -1.10)(0.46, -1.10)

BBCLL1&(ZDCN|S)40∼60 (GeV)
60∼80 (GeV)-0.0326-0.0366-0.0307 (5.9%)-0.0384 (5.0%)-0.0363 (11.2%) -0.0397 (8.4%)-0.0344 (5.4%)-0.0391 (6.7%)

80∼120 (GeV)-0.0330-0.0331 (0.1%)-0.0386 (16.8%)-0.0320 (3.0%)

115

Table 8.3: The ǫN divided by polarizations calculated with various centers of the acceptance ineach measured energy bin. The variations to final values, which were calculated by the center of(x,y)=(0,0) cm, are written in parentheses. Maximum variations are shown as bold charactersand added as a systematic error.

Systematics for the estimation of the smearing effectFigure 8.21 shows the AOutput
with the default position resolution and 14.1% increased one.
N
Asymmetry was more smeared with ∼6.5% reduction. It was applied to systematic errors in

AN for the smearing correction.

Estimation of the systematic error of smearing effect for energy dependence

ANOutput

-2

0 ×10
-1

Default position resolution-2

14.1% increased position resolution

-3
-4
-5
-6
-7
-8
-9
-10
0

20

40

60

80
100
120
Measured energy (GeV)

Figure 8.21: The measured energy dependence of AOutput
with the defaultN
position resolution (Circle points)
and 14.1% increased one (Triangle
points) estimated by the simulation.
As an input, we generated neutrons as A(φ)=AN sin(φ − π/2) with
AN =−0.10. By increasing the position resolution, the smearing effects
were also increased and variations
were ∼6.5%.

Bunch shuffling
Details of the bunch shuffling are written in the φ dependence analysis (section 8.2.4). For thexF dependence, the bunch shuffling was performed by an up-down asymmetry since it would
be zero in the vertically polarized beam. Therefore, we can ignore the residual asymmetry fromthe finite AN (Appendix C.2). The acceptance definition of the up-down asymmetry is shown

CHAPTER 8. ANALYSIS FOR THE SINGLE TRANSVERSE SPIN ASYMMETRY

116

in Figure 8.22. It is obtained by rotating the acceptance of the left-right asymmetry 90 degrees.
Figure 8.23 shows results of the bunch shuffling. They were performed in each measured
energy bin (40–60, 60–80 and 80–120 GeV). As discussed in section 8.2.4, the χ2 /ndf values canbe fluctuated as 1.00±0.15 and 1.00±0.13 for the ZDCN|S and BBCLL1&(ZDCN|S) samples,
respectively. Results of bunch shuffling consistent with the χ2 /ndf = 1 within its uncertainty
in all bins. We conclude the fake asymmetry from the bunch-by-bunch is less than 0.39σstatand 0.36σstat for the ZDCN|S and BBCLL1&(ZDCN|S), respectively. We did not include theseerrors to the final uncertainties in AN .

ypos (cm)

Acceptance definition for UP DOWN AN
6
700
4

600
500

2

Figure 8.22: The acceptance definition for the
up-down asymmetry for the xF dependence. It
is obtained by rotating the acceptance of the
300 left-right asymmetry 90 degrees.
400

0

-2
200
-4

-6
-6

100

-4

-2

0

2

4
6
xpos (cm)0CHAPTER 8. ANALYSIS FOR THE SINGLE TRANSVERSE SPIN ASYMMETRY

BS result for 40~60 GeV : BBCLL1&(ZDCN|S)

BS result for 60~80 GeV : BBCLL1&(ZDCN|S)

BS result for 80~120 GeV : BBCLL1&(ZDCN|S)

240

220

Entries = 30000

250

Entries = 30000

220
200

Mean = 0.889

200

Mean = 1.024

180
160

RMS = 0.704

RMS = 0.822

140

150

120

50

180

1

2

3

4

5

140

BS result for 40~60 GeV : ZDCN|S

RMS = 0.885

120

80

80

60

60

40

40

0
0

6
χ2/ndfMean = 1.108

160

20

20
0
0

Entries = 30000

200

100

100
100

117

1

2

3

4

5

0
0

6
χ2/ndf

BS result for 60~80 GeV : ZDCN|S

1

2

3

4

5

6
χ2/ndf

BS result for 80~120 GeV : ZDCN|S

300

Entries = 30000

300

Mean = 1.109

250

Entries = 30000

300

Entries = 30000

Mean = 1.045

250

Mean = 1.036

250

200

RMS = 1.045
150

100

50

0
0

1

2

3

4

5

6
2
χ /ndf200

RMS = 0.992

200

150

150

100

100

50

50

0
0

1

2

3

4

5

6
2
χ /ndf00

RMS = 0.984

1

2

3

4

5

6
2
χ /ndfFigure 8.23: Results of the bunch shuffling performed by the up-down asymmetries in eachmeasured energy bin, Left) 40–60, Center) 60–80 and Right) 80–120 GeV. Top and bottom
figures show these for the BBCLL1&(ZDCN|S) and ZDCN|S samples, respectively.

Summary of systematic errors
Systematic errors are presented as variations to final values. If there are differences between two
triggers, the value of the ZDCN|S is written in the parentheses. Total errors were calculated byquadratic sum. Correlated errors are shown as bold characters.
• Estimation from measurement– Center shift : 11.2, 8.4, 16.8 (21.6, 22.8, 50.4) %– Beam gas background for the ZDCN|S trigger : (<1%)• Simulation uncertainty : total 9.0% (9.5%)– Background estimation : 1.4% (2.0%)– Smearing effect from the position resolution : 6.5%
– Multi hits in one event : 6.1% (6.6%).
• Total scaling error : 9.0% (9.5%)• Polarizations : 6.2% and 5.9% for the positive and negative xF , respectively.

Chapter 9

Results of the PHENIX experiment
9.1

Cross section

The differential cross section, dσ/dxF , for the leading neutron production in pp collision at√
s=200 GeV was evaluated and listed in Table 9.1. We just show the result of xF above 0.4
since the data at xF = 0.2 would be affected by the energy cut off before the unfolding. The
expected integrated pT region in each bin is 0–0.11xF GeV/c as equation 1.3 with the acceptancecut of r<2 cm. We have no correction for xF bin in this analysis.
For the comparison with the ISR result, their differential cross sections with the sameacceptance, it means same pT region, were calculated from the invariant cross sections byequation 7.2. As a pT shape, we used exp(−4.8pT ) which were obtained from 0.3<xF <0.7
region in the ISR energies (Figure 6.2). The shape should be more steep at large xF by reading
their results of Figure 1.5, however it is difficult to take correct shape from the figure. For
the estimation of the differential cross section in the PHENIX data1 and the ISR, we usedexp(−4.8pT ) as pT shape in all xF regions2 . The cross section values for the ISR are listed inTable 9.2 and differential cross sections are plotted in Figure 9.1. ISR shapes are reproduced
by a combination of polynomial 3 and gaussian well.
The comparison is shown in Figure 9.2. The absolute normalization errors for the PHENIX
measurement, 12.5 %, is not included. The cross section is consistent with the ISR data. xF
scaling has no violation in higher center of mass energy. This result prefers the OPE model.
√
dσ/dxF (mb) : pp at s = 200 (GeV)xF
dσ/dxF (mb)
0.50
0.205 ± 0.009 ± 0.082
0.70
0.588 ± 0.014 ± 0.048
0.90
0.572 ± 0.011 ± 0.083
Table 9.1: The result of the differential cross section for the leading neutron production in pp√
collision at s=200 GeV. The first and second errors show the statistics after the unfolding
and systematics. The absolute normalization error, 12.5 %, is not included.
1

pT shape was used to the simulation input.
In the rough estimation, the pT shape at xF = 0.8 is exp(−6.6pT ) and it means the differential cross sections
using exp(−4.8pT ) was over estimated; the values around xF = 0.8 would be 10% reduction for both results.
2118

CHAPTER 9. RESULTS OF THE PHENIX EXPERIMENT

9.2

119

Azimuthal angle dependence of AN

The azimuthal angle dependence of AN for the leading neutron production are plotted in Figure 9.3 and listed in Table 9.3. Results are shown in the forward kinematics only; these in
the backward were consistent with zero as shown in Figure 8.6. The error bars are statistical only and the scaling errors from the measurement, 10.4% and 9.0% for the ZDCN|S and
BBCLL1&(ZDCN|S) samples, are not included. Lines are fit results of the sine curve definedas equation 3.6. Results well satisfy the characteristics of AN .
As a result, AN is larger when BBC hits are required in this acceptance (0.5<r<4 cm).
It may also consistent with the OPE picture; the asymmetry can be generated by the pionexchange which is spin-flip interaction and the BBC coincidentally detects the charged particles
which are generated by proton-pion scattering. 3
As a comparison with the IP12 result, we compared the AN of BBCLL1&(ZDCN|S) sample. Amplitudes of AN measured at the PHENIX and the IP12 are (−0.075±0.004±0.008)×(1.00 ± 0.09) and (−0.090±0.006±0.009)×(1.00+0.52−0.25 ), respectively. Errors indicate the statis-

tics, systematics and the scaling uncertainty from the polarization measurement. They are
consistent within the errors including the scaling errors. 4

9.3

xF dependence of AN

The xF dependence of AN for the leading neutron production are plotted in Figure 9.4 andlisted in Table 9.4. Positive and negative xF points were obtained from the forward and backward asymmetries, respectively. Statistical uncertainties are shown as error bars and correlated
systematic uncertainties are shown as brackets. Scaling errors from the measurement, 9.5% and9.0% for the ZDCN|S and BBCLL1&(ZDCN|S) samples respectively, are not included. Scaling errors of the polarization measurement, 6.2% and 5.9% for the positive and negative xF
respectively, are not included too.
We can see the significant negative AN in the positive xF region and there is no energy
dependence within the errors. On the basis of the OPE model, the asymmetry is expected toappear the interference of the pion exchange and the other Reggeon exchange which has spin
non-flip amplitude. This result is expected to evaluate the other Reggeon contribution over xF .
We are waiting for the theoretical calculation to discuss in details.

3

Actually the charged particles detected with the BBC associated with the leading neutron have interesting
property; it also has a finite single transverse spin asymmetry. Details are in Appendix D.
4However the center values are not so consistent. One concern is the different detection coverages for the
charged particles; 2.2<|η|<3.9 in the horizontal and vertical directions for the IP12 experiment, and 3.0<|η|<3.9
for the PHENIX experiment.

120

CHAPTER 9. RESULTS OF THE PHENIX EXPERIMENT
Ed3 σ/dp3 (mb · GeV−2 · c3 )√√s = 30.6 (GeV)
s=44.9 (GeV)
xF
0.150
13.7 ± 1.5
0.200
9.6 ± 1.6
12.8 ± 1.4
0.250
9.6 ± 1.4
12.2 ± 1.3
0.300
9.8 ± 1.3
11.9 ± 1.2
0.350
10.0 ± 1.3
11.9 ± 1.2
0.400
10.5 ± 1.3
12.3 ± 1.2
0.450
11.2 ± 1.3
12.7 ± 1.2
0.500
12.3 ± 1.2
13.6 ± 1.1
0.550
13.4 ± 1.2
14.8 ± 1.1
0.600
15.1 ± 1.2
16.4 ± 1.1
0.650
17.4 ± 1.3
18.2 ± 1.3
0.700
20.3 ± 1.3
21.0 ± 1.4
0.750
23.3 ± 1.6
25.5 ± 1.7
0.775
25.0 ± 1.7
27.8 ± 1.8
0.800
26.6 ± 1.8
29.1 ± 1.8
0.825
27.4 ± 1.8
29.2 ± 1.8
0.85026.7 ± 1.9
28.0 ± 1.8
0.875
23.8 ± 2.5
25.3 ± 2.1
0.90018.6 ± 3.0
20.9 ± 2.8
0.925
13.5 ± 3.0
14.9 ± 2.80.950
8.3 ± 3.08,3 ± 2.8
0.975
3.2 ± 3.0
1.8 ± 2.8dσ/dxF (mb)0.150
0.000 ± 0.000
0.076 ± 0.008
0.200
0.069 ± 0.012
0.092 ± 0.010
0.250
0.085 ± 0.012
0.108 ± 0.012
0.300
0.103 ± 0.014
0.125 ± 0.013
0.350
0.120 ± 0.016
0.143 ± 0.014
0.400
0.141 ± 0.018
0.166 ± 0.016
0.450
0.167 ± 0.019
0.189 ± 0.018
0.500
0.200 ± 0.020
0.221 ± 0.018
0.550
0.236 ± 0.021
0.260 ± 0.019
0.600
0.285 ± 0.023
0.309 ± 0.021
0.650
0.349 ± 0.026
0.365 ± 0.026
0.700
0.431 ± 0.028
0.446 ± 0.030
0.750
0.521 ± 0.036
0.571 ± 0.038
0.775
0.573 ± 0.039
0.637 ± 0.041
0.800
0.624 ± 0.042
0.683 ± 0.042
0.825
0.657 ± 0.043
0.701 ± 0.043
0.850
0.654 ± 0.047
0.686 ± 0.044
0.875
0.595 ± 0.063
0.633 ± 0.053
0.900
0.474 ± 0.077
0.533 ± 0.071
0.925
0.351 ± 0.078
0.387 ± 0.073
0.950
0.220 ± 0.079
0.220 ± 0.074
0.975
0.086 ± 0.081
0.048 ± 0.075√
s=52.8 (GeV)
10.9 ± 1.4
10.0 ± 1.3
9.9 ± 1.2
10.0 ± 1.2
10.4 ± 1.1
10.9 ± 1.1
11.4 ± 1.0
12.2 ± 1.0
13.3 ± 1.1
14.7 ± 1.1
17.0 ± 1.3
19.8 ± 1.4
23.6 ± 1.6
25.5 ± 1.6
26.7 ± 1.6
26.7 ± 1.6
25.7 ± 1.7
23.8 ± 1.8
19.3 ± 2.6
13.8 ± 2.6
8.3 ± 2.6
2.7 ± 2.6√
s = 62.7 (GeV)
9.5 ± 1.2
8.5 ± 1.1
8.2 ± 1.0
8.5 ± 0.9
8.7 ± 1.0
9.1 ± 1.0
9.8 ± 1.0
11.1 ± 1.0
12.5 ± 1.0
14.0 ± 1.0
15.7 ± 1.1
17.9 ± 1.2
21.6 ± 1.3
23.5 ± 1.6
25.2 ± 1.6
26.4 ± 1.6
26.6 ± 1.6
25.8 ± 1.7
23.4 ± 2.5
16.3 ± 2.8
9.1 ± 2.8
1.8 ± 2.80.060 ± 0.008
0.072 ± 0.009
0.088 ± 0.011
0.105 ± 0.013
0.125 ± 0.013
0.147 ± 0.015
0.170 ± 0.015
0.198 ± 0.016
0.234 ± 0.019
0.277 ± 0.021
0.341 ± 0.026
0.421 ± 0.030
0.528 ± 0.036
0.585 ± 0.037
0.626 ± 0.038
0.641 ± 0.038
0.630 ± 0.042
0.595 ± 0.045
0.492 ± 0.066
0.359 ± 0.068
0.220 ± 0.069
0.073 ± 0.070

0.052 ± 0.007
0.061 ± 0.008
0.073 ± 0.009
0.089 ± 0.009
0.104 ± 0.012
0.123 ± 0.013
0.146 ± 0.015
0.181 ± 0.016
0.220 ± 0.018
0.264 ± 0.019
0.315 ± 0.022
0.380 ± 0.025
0.483 ± 0.029
0.539 ± 0.037
0.591 ± 0.038
0.633 ± 0.038
0.652 ± 0.039
0.645 ± 0.043
0.597 ± 0.064
0.424 ± 0.073
0.241 ± 0.074
0.048 ± 0.075

Table 9.2: Top) Original values of invariant cross sections from the ISR. They were obtained
from Reference [2]. Bottom) The converted values as a differential cross section, dσ/dxF , fromthe invariant cross sections by equation 7.2. As a pT shape, we used exp(−4.8pT ) and theintegrated pT region was 0–0.11xF (GeV/c), corresponds to the acceptance cut of the PHENIX
measurement.

121

CHAPTER 9. RESULTS OF THE PHENIX EXPERIMENT

dσ/dx F (mb)

Neutron spectra at zero-angle from ISR experiments
0.9

s=30.6 GeV
s=44.9 GeV
s=52.8 GeV

0.8
0.7
0.6

s=62.7 GeV

0.5
0.4
0.3
0.2
0.1
0
0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8
0.9
1
xF = pL/p max

Figure 9.1: The differential cross sections converted from invariant cross sections from the ISR
by equation 7.2. The values are listed in Table 9.2. They can be fitted by a combination ofpolynomial 3 and gaussian well.

dσ/dx (mb)

Cross section of forward neutron production
0.9
s=30.6 GeV : ISR data0.8

s=44.9 GeV : ISR data0.7

s=52.8 GeV : ISR data
0.6
s=62.7 GeV : ISR data
0.5
s=200 GeV : PHENIX data
0.4
0.3
0.2
0.1
0
0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

xF

√
Figure 9.2: The cross section of the leading neutron production in pp collision at s=200 GeV
shown as circles. We have no correction for the xF bin. Statistical errors are buried in eachpoint and systematic errors are shown as brackets. The integrated pT region in each bin is 0–0.11xF GeV/c. Shapes of the ISR results, which are the fit results of Figure 9.1, are also shown.
Absolute normalization errors for the PHENIX and ISR are 12.5% and 20%, respectively.

122

CHAPTER 9. RESULTS OF THE PHENIX EXPERIMENT-2
10 ×108

Forward Neutron Asymmetry φ distribution BBCLL1&(ZDCN|S) trigger

10 ×10-2

AN

AN

Forward Neutron Asymmetry φ distribution ZDCN|S trigger

8

A N = 0.058 ± 0.008

A N = 0.075 ± 0.004

6

6

4

4

2

2

0

0

-2

-2

-4

-4

-6

-6

-8 6.2% normatization uncerainty
is not included.
-10
0
0.5
1
1.5
2

-8 6.2% normatization uncerainty
is not included.
-10
0
0.5
1
1.5
2

2.5

3
φ (rad)2.5

3
φ (rad)Figure 9.3: The azimuthal angle dependence of AN for the leading neutron productions in the
ZDCN|S (Left) and BBCLL1&(ZDCN|S) (Right) samples. The measured neutron energy was
integrated from 40 to 120 GeV. We applied 0.5<r<4 cm to the acceptance cut, or production angles 0.3 - 2.2 mrad in the same way of the IP12 experiment. The errors show statistics only. The
scaling errors from the measurement, 10.4 and 9.0%, for the ZDCN|S and BBCLL1&(ZDCN|S)
samples are not including. Scaling uncertainty from the beam polarization, 6.2%, is not includedtoo.

AN (×10−2 )Azimuthal angle
0 < φ < π/8
π/8 < φ < 2π/8
2π/8 < φ < 3π/8
3π/8 < φ < 4π/8
4π/8 < φ < 5π/8
5π/8 < φ < 6π/8
6π/8 < φ < 7π/8
7π/8 < φ < π

ZDCN|S3.28 ± 1.68
6.08 ± 1.63
2.43 ± 1.59
-0.58 ± 1.58
0.61 ± 1.58
-6.04 ± 1.60
-4.46 ± 1.64
-6.72 ± 1.69

BBCLL1&(ZDCN|S)
7.66 ± 0.76
5.86 ± 0.73
3.61 ± 0.70
2.22 ± 0.69
-0.80 ± 0.69
-4.23 ± 0.71
-6.32 ± 0.73
-7.59 ± 0.76

Table 9.3: The results of the azimuthal angle dependence of AN for the leading neutron pro√duction in pp collision at s=200 GeV. Errors show the statistics only. The scaling errors fromthe measurement, 10.4 and 9.0%, for the ZDCN|S and BBCLL1&(ZDCN|S) samples are notincluded. Scaling uncertainty from the beam polarization, 6.2%, is not included too.

123

CHAPTER 9. RESULTS OF THE PHENIX EXPERIMENTxF-dependence of leading neutron asymmertry for ZDCN|S
10 ×10

AN

-2

8
6
4
2
0
-2
-4
-6
6.2% and 5.9% normatization uncerainties
for positive and negative xF are not included.

-8
-10
-1

-0.8

-0.6

-0.4

-0.2

-0

0.2

0.4

0.6

0.8

<xF> 1

0.8<xF> 1

xF-dependence of leading neutron asymmertry for BBCLL1&(ZDCN|S)AN-2

10 ×10
8
6
4
2
0
-2
-4
-6
-8
-10
-1

6.2% and 5.9% normatization uncerainties
for positive and negative xF are not included.
-0.8

-0.6

-0.4

-0.2

-0

0.2

0.4

0.6

Figure 9.4: The xF dependence of AN for the leading neutron production in the ZDCN|S (Top)and BBCLL1&(ZDCN|S) (Bottom) samples. We applied 0.5<r<3.0 cm to the acceptancecut, or production angles 0.3–1.6 mrad; it is not same as the azimuthal angle dependence tominimize the edge effect (section 6.1). According to the estimation of the mean xF , section
8.3.3, xF values were determined. The scaling errors from the measurement, 9.5 and 9.0%, for
the ZDCN|S and BBCLL1&(ZDCN|S) triggers are not included. Scaling uncertainties from thepolarizations, 6.2% and 5.9% for positive and negative xF respectively, are not included too.
ZDCN|S< xF >-0.776-0.682-0.568
0.568
0.682
0.776

(×10−2 )AN
1.455 ± 2.133 ± 3.276
1.155 ± 2.171 ± 1.442-0.202 ± 2.565 ± 1.546-7.158 ± 2.365 ± 1.546-6.325 ± 1.999 ± 1.442-6.500 ± 1.962 ± 3.276

BBCLL1&(ZDCN|S)< xF >
AN (×10−2 )-0.749-0.251 ± 1.070 ± 1.002-0.6640.320 ± 0.911 ± 0.520-0.547-0.371 ± 1.065 ± 0.727
0.547-6.487 ± 0.939 ± 0.727
0.664-6.189 ± 0.804 ± 0.520
0.749-5.965 ± 0.945 ± 1.002

Table 9.4: The results of the xF dependence of AN for the leading neutron production in√pp collision at s=200 GeV. First and second uncertainties show statistics and systematics, respectively. Scaling errors from the measurement, 9.5 and 9.0%, for the ZDCN|S and
BBCLL1&(ZDCN|S) samples are not included. Scaling uncertainties from the beam polarizations, 6.2% and 5.9% for the positive and negative xF respectively, are not included too.

Chapter 10

Discussion
In this thesis, we reported the discovery of the single spin asymmetry for the leading neutron√
production in pp collision at s=200 GeV. The measurement of cross section for the leading
neutron was performed as a further investigation and it is consistent with the xF scaling claimedby the ISR experiment which is well described by the OPE model in Regge calculus (section
1.2.4). Therefore, the observed large asymmetry for the neutron production is considered tocome from the interference between a spin-flip amplitude due to the pion exchange and non-flip
amplitudes from other Reggeon exchange. So far our knowledge of Reggeon exchange components for the leading neutron is almost the pion. The asymmetry would have the sensitivity
to the contribution of other spin non-flip Reggeon exchange even if it is small amplitude. The
theoretical calculation is on going.
In addition, we want to discuss our results based on the meson-cloud model. This model gives
a best description for the recent interesting result from the Drell-Yan experiment at FNAL E866.
1 The meson-cloud model was applied to the leading neutron production in the ISR experimentand successfully described with the pion cloud (section 1.2.5). It is interesting whether themeson-cloud model can describe the observed asymmetry or not. If it can, the asymmetry
would be a new observable for the discussion of the hadronic structure of the nucleon basedon the meson-cloud model. Clarification of the hadronic structure of the nucleon is importantto the proton spin physics since such system can have an orbital angular momentum and weneed to consider this contribution if the hadron structure is present. We are waiting for thetheoretical calculation to discuss in details.
Future measurement in RHIC
We reported the xF dependence of cross section and asymmetry on this thesis and so far
our detector is insufficient for the measurement of pT . Detector improvements are desired tomeasure more precise xF and evaluate pT , but improvements are difficult due to the acceptancelimit. However polarized pp collision can be realized at various center of mass energies in RHIC.
1
In this model, Drell-Yan process is generated by the interaction between the d quark in one proton and the¯
d quark in the π + of p → nπ + state for other proton. At that time, neutron should be generated in very forward
kinematics as the leading neutron. The coincidence of these measurements is interesting.

124

CHAPTER 10. DISCUSSION

Already125√
s=410 GeV and 62 GeV runs were performed in 2005 and 2006 run, and significant

asymmetries of the leading neutron were shown at online (Appendix E). In the near future,√
RHIC plans to accelerate the polarized proton to s=500 GeV. We will be able to performthe cross section in such higher energy. For the asymmetry, we will study not only the xF
dependence but also pT dependence which can be expected to be evaluated by the comparisonof different center of mass energies with fixed xF and measured angle, θn (pT is approximately
written as pT ∼ xF Ep θn (GeV/c) ).

Chapter 11

Conclusion
In 2001−2002 run period at RHIC, transversely polarized protons were accelerated to 100 GeV√
for the first time, with collisions at s=200 GeV. The IP12 experiment searched for a production
process with a single transverse spin asymmetry, AN , based on a large pion asymmetry in√forward kinematics from polarized pp collision at s = 19.4 GeV. The experiment was designed
to detect photons decayed from π 0 , but was also capable to detect neutrons in the very froward
kinematics, ±2.8 mrad. Detector was placed downstream of the RHIC Dx magnets, which bentout a proton beam and produced charged particles.

Observed asymmetries for photon and π 0 are consistent with zero within the statistics,
whereas a significant asymmetry was observed for neutrons in forward kinematics, AN =(−0.090 ± 0.006 ± 0.009) × (1.0+0.52−0.25 ), where the first and the second errors in the parenthesis areof statistical and systematic, respectively. The scaling error originates the beam polarization

uncertainty. The asymmetry in backward was zero. The asymmetry was confirmed by thehadron calorimeter as AN = (−0.126 ± 0.017 ± 0.045) × (1.0+0.52−0.26 ). It was unexpectedly large

asymmetry in this low pT range, the pT range for the neutrons was limited by acceptance which
gives pT <0.22 GeV, since polarization phenomena which have been observed so far seems tovanish in such low pT (cf. pion AN and Λ polarization).
For further investigation about this large asymmetry in the leading neutron production andits origin, we have measured the xF dependence of AN and cross section with the PHENIX√
detector in the similar kinematics region at s=200 GeV. The observed asymmetries are con-sistent with the IP12 result within the error. We also performed the xF dependence of AN asmean xF through 0.5<xF <0.9 with the resolution of ∼0.1 in RMS. Asymmetry showed almost√
no xF dependence. The measured cross section at s=200 GeV has no violation of xF scaling
with the results from the ISR experiment. These cross sections for large xF neutron production,
as well as those in ep collisions by ZEUS, are mainly reproduced by a pion exchange. Therefore, the observed large asymmetry for the neutron production is considered to come from theinterference between a spin-flip amplitude due to the pion exchange and non-flip amplitudes
from other Reggeon exchange.
On the basis of OPE model, the discovered neutron AN would have the sensitivity to the126CHAPTER 11. CONCLUSION

127

contribution of other Reggeon exchange even if it is a small amplitude. It would be a newobservable for the discussion of hadronic structure of the proton based on the meson cloudmodel. More precise measurement is expected to give a new knowledge in forward physics.
√
In the near future, RHIC plans to accelerate the polarized proton to s=500 GeV. We will
be able to perform the measurements at various xF for the cross section in such higher energy.
For the asymmetry, we will study not only xF dependence but also pT dependence which is
expected to evaluated by the comparison of different center of mass energies with fixed xF andthe measured angle, θn (pT is approximately written as pT ∼ xF Ep θn (GeV/c)).

Acknowledgments
Fist of all, I would like to express my gratitude to Prof. Ken’ichi Imai, who has been mysupervisor. I appreciate him to give me interesting physics talk and continuous encouragement.
I am grateful to Prof. Naohito Saito who teach me wide range of physics interests. I havelearned attitudes to the physics research as a physicist from him.
I would like to express my respect to Dr. Yuji Goto who preserves discussion and support
at BNL. He has provided fruitful discussion for not only this thesis but also other physics. I
could not complete this thesis without his support.
I would like to thank all the collaborators of the IP12 experiment group; Dr. Yoshinori
Fukao, Dr. Alexander Bazilevsky, Dr. Leslie Bland, Dr. A. Bogdanov, Dr. Gerry Bunce, Dr.
Hideto En’yo, Dr. Brendan Fox, Dr. Yuji Goto, Dr. John Haggerty, Prof. Ken Imai, Dr.
William Lentz, Dr. D. von Lintig, Prof. Ming X. Liu, Dr. Yousef Makdisi, Dr. Muto Ryotaro,
Dr. S.B. Nurushev, E. Pascuzzi, Dr. Martin Purschke, Prof. Naohito Saito, Dr. Fuminori
Sakuma, Dr. Sean Stoll, Dr. Kiyoshi Tanida, Dr. Junji Tojo, Dr. Yasushi Watanabe and Dr.
Craig Woody.
I would like to thank all the PHENIX collaborators and the staff of Collider-Accelerator
Department at BNL. The experiment has not been done without their enormous efforts. I
am deeply grateful to Dr. Gerry Bunce who has been supervisor at RIKEN BNL ResearchCenter. With his support, I could polish the analysis and finish my thesis. I am oblige for Dr.
Sebastian White who is supervisor of the ZDC detector group. He gave me many advice andsuggestion about analysis using the ZDC detector. I appreciate to Dr. Alexei Denisov for histeaching and advice for the ZDC detector operation. I would like to thank Dr. Mickey Chiu.
He has gave many advices and helps the ZDC operation at experimental running time. I can
not complete the leading neutron analysis without support from ZDC group. I am grateful to
all the members of RIKEN BNL Research Center for their support during my stay at BNL. I
would like to thank Ms. Pam Esposito, Ms. Tammy Heinz, Ms. Jane Lysik, Ms. Taeko Ito,
Ms. Chiharu Shimoyamada-Kawamura, Mr. Jun Nakamura and Mr. Ryousuke Maruyama for
their administrative and secretarial support.
I am grateful to all members of Radiation Laboratory in RIKEN. I am deeply grateful to theChief Scientist, Dr. Hideto En’yo. I am very impressed his leadership. He gave me many advices,suggestions, support and encouragements as an scientist with broader perspective. I appreciate
to Dr. Yasuyuki Akiba. He gave me many techniques and procedures of the physics analysis. I

128

CHAPTER 11. CONCLUSION

129

would like to express special thanks to Dr. Satoshi Yokkaichi. He gives useful comments andsuggestion for my analysis. I would like to express my gratitude to Ms. Noriko Kiyama, Ms.
Michiko Muroi and Ms. Karen Sakuma for their secretarial support and hospitality.
I would like to thank all members of “Experimental Nuclear and Hadronic Physics (NH)”group in Kyoto University. I am deeply grateful to Dr. Kiyoshi Tanida who has been one of myadvisers. He helps me to finish thesis work with his wide knowledge. He also read throughout
my thesis and give worthful comments. I am oblige for Mr. Masayuki Niiyama. Discussions
with him have been very fruitful for me, especially for the introduction part of this thesis. I
would like to express special thanks to Mr. Kazuya Aoki. I have enjoyed discussion with him
about physics, analysis, etc. They are valiant time for me. I am grateful to Prof. Masanobu
Nakamura, Prof. Haruhiko Funahashi, Prof. Tetsuya Murakami and Prof. Masaru Yosoi, forvaluable discussions. I would like to express my thanks to Ms. Ayako Furumori-Nagaoka, Ms.
Ayumi Nakao, Ms. Sonoko Chiaki and Ms. Shiori Yamamoto for their secretarial support andhospitality at NH group.
I would like to address my sincere thanks to all graduate students, especially in the same
grade, Dr. Yoshinori Fukao, Dr. Koji Miwa, Mr. Terunao Nakura, Dr. Hiromi Okada, Mr.
Shinsuke Ota and Mr. Satoru Terashima for their friendships and encouragements. I would liketo also appreciate to Dr. Yoshinori Fukao and Dr. Hiromi Okada for their careful reading andextensive feedback for this thesis; the present version is greatly improved, thanks to beneficial
comments.
Finally, I have to appreciate to my parents, Mr. Toshio Togawa and Ms. Mitsuru Togawa,and my brother, Mr. Makoto Togawa for their continuous support and encouragement.

Appendix A

Definition of the single transverse
spin asymmetry
Spin asymmetry is defined as a difference of the cross sections with the different spin directionsof the scattered particles . In case of single transverse spin asymmetry, it can be defined as,
AN ≡dσ ↑ − dσ ↓.
dσ ↑ + dσ ↓(A.1)

Superscripts of ↑ and ↓ indicate the polarization directions are vertically transverse in up anddown. From the rotational and parity invariance, AN should be zero at the pT =0 GeV and thevalues in the detection at left and right toward the spin direction should be symmetric. Thus
AN is represented as,
AN =σL↑ − σL↓

↑
↓
↑
σR
− σR
σL↑ − σR
=
−
=
.
↑
↓
↑
σL↑ + σL↓σR
+ σR
σL↑ + σR

(A.2)

Subscripts for L and R mean left and right areas toward the polarization axis. Positive AN
is defined for the left detection as convention. By this description, AN is called “left-rightasymmetry” sometimes.
AN can be represented to the useful expressions using the experimental yields and beampolarization, P , by formulas in section A.2.

A.1

Necessity to be finite AN

The most convenient way to handle spin is to use the helicity formalism. Transverse spin state,|↑i and |↓i, can be described by helicity spin state, |+i and |−i, as
1|↑i = √ (|+i + i |−i) ,2

1|↓i = √ (|+i − i |−i) .
2

The AN for the inclusive production in Figure 1.11 can be represented as,PP22
dσ ↑ − dσ ↓X |hcX | T | ↑i| −X |hcX | T | ↓i|
=
AN =
P
P
2
2
dσ ↑ + dσ ↓X |hcX | T | ↑i| +X |hcX | T | ↓i|P
2 Im X − T † cX hcX | T | +i
= P
P
2
2.
X |hcX | T | +i| +X |hcX | T | −i|
130

(A.3)

(A.4)

APPENDIX A. DEFINITION OF THE SINGLE TRANSVERSE SPIN ASYMMETRY 131
Here, numerator is known as cross section. For the non-zero AN , the helicity flip interaction asa denominator is necessary.
OPE model is one of possible candidates for the origin of the leading neutron AN since pionexchange is the fully spin flip interaction.

A.2

Square root formula

Asymmetry can be calculated by the square root mean of experimental yields for the combinations of spin up and down in left and right detector regions; NL↑ ,NL↓ ,NR↑ and NR↓ . Before theexplanation of square root formula, we should define these variables as follows,
NL↑ = σL↑ L↑ ΩL

NL↓ = σL↓ L↓ ΩL↑ ↑NR↑ = σRL ΩR

↓ ↓NR↓ = σR
L ΩR

(A.5)

where,• σ ↑(↓) : Cross section for spin up (down).
• L↑(↓) : Luminosity for spin up (down).
• ΩL(R) : Acceptance for left (right) geometry.
Equation A.5 can be represented by these variables of a spin average and spin asymmetry which
are defined as,
Cross section for left and rightσL↑ + σL↓
,
σL ≡
2
↑
↓
σR
+ σR
,
σR ≡
2

σ0 ≡ σL = σR

AN L ≡AN R ≡σL↑ − σL↓

σL↑ + σL↓
↑
↓
σR
− σR

↑
↓
σR
+ σR
(Rotational symmetry).

(A.6)

Luminosity
L≡L↑ + L↓,2

AcceptanceǫL ≡L↑ − L↓.
L↑ + L↓(A.7)ΩL − ΩRΩL + ΩR
,
ǫΩ ≡
.
(A.8)
2
ΩL + ΩR
Basically AN L and AN R in equation A.6 are same by rotational symmetry, however we shouldΩ≡consider the systematic uncertainty if AN L 6= AN R due to the some reasons which cannot be

studied. AN and its systematic uncertainty, ǫA , can be defined as,
AN ≡AN L + AN R
,
2

ǫA ≡

AN L − AN R.
AN L + AN R

(A.9)

APPENDIX A. DEFINITION OF THE SINGLE TRANSVERSE SPIN ASYMMETRY 132
It is assumed that the values of ǫA is small, |ǫA | ≪1.
Finally, equation A.5 can be represented as,

NL↑ = σ0 LΩ(1 + ǫL )(1 + ǫΩ ) [1 + (1 + ǫP )(1 + ǫA )P AN ]NL↓ = σ0 LΩ(1 − ǫL )(1 + ǫΩ ) [1 − (1 − ǫP )(1 + ǫA )P AN ]NR↑ = σ0 LΩ(1 + ǫL )(1 − ǫΩ ) [1 − (1 + ǫP )(1 − ǫA )P AN ]NR↓ = σ0 LΩ(1 − ǫL )(1 − ǫΩ ) [1 + (1 − ǫP )(1 − ǫA )P AN ] .

A.2.1(A.10)

Physics asymmetry

Physics asymmetry, which is shown as ǫN on this thesis, is defined as,qq
NL↑ NR↓ − NL↓ NR↑qǫP hys ≡ q
NL↑ NR↓ + NL↓ NR↑∼ P AN 1 − 2ǫA ǫP (P AN ) + (ǫ2A + ǫ2P )(P AN )2 + O(ǫ5 )∼ P AN ,(A.11)

where N is the experimental yield and its sub(super)scripts are same as equation A.2. For
the last approximation, we assume that P AN , ǫA , ǫP , ǫL and ǫΩ are small (These five small
quantities are represented as ǫ). This approximation is also applied to a detector and a luminosity asymmetries which are explained in next sections. The contribution of the systematic
uncertainties to AN is canceled out until the third order of ǫ by using square root formula.
Statistical uncertainty is obtained as,qs
NL↑ NL↓ NR↑ NR↓
1
1
1
1+ ↓ + ↑ + ↓δǫP hys = q
2q↑
NL NL NR NR
NL↑ NR↓ + NL↓ NR↑∼

1√
f or NL↑ ∼ NL↓ ∼ NR↑ ∼ NR↓ ,N(A.12)

where N is total number of yield; N = NL↑ + NL↓ + NR↑ + NR↓ .

A.2.2

Detector asymmetry

Detector asymmetry is defined as,ǫDet ≡qq

NL↑ NL↓ −NL↑ NL↓ −qq

NR↑ NR↓NR↑ NR↓∼ ǫΩ + ǫP (P AN ) − ǫA (P AN )2 + O(ǫ4 )∼ ǫΩ .
The uncertainty is obtained by equation A.12 with changingq
NR↑ NR↓ .

(A.13)q

NL↑ NR↓ +q

NL↓ NR↑ toq

NL↑ NL↓ +APPENDIX A. DEFINITION OF THE SINGLE TRANSVERSE SPIN ASYMMETRY 133

A.2.3

Luminosity asymmetry

Luminosity asymmetry is defined as,qǫLumi ≡q

NL↑ NR↑ −NL↑ NR↑ −qq

NL↓ NR↓
NL↓ NR↓∼ ǫL + ǫA (P AN ) − ǫP (P AN )2 + O(ǫ4 )∼ ǫL .

(A.14)qqq
The uncertainty is obtained by equation A.12 with changing NL↑ NR↓ + NL↓ NR↑ to NL↑ NR↑ +q
NL↓ NR↓ .

A.3

Luminosity formula

Physics asymmetry also can be calculated by using beam luminosity in the fixed detector region.
In case of the usage of left region, equation A.2 is represented as,ǫP hys (L) ≡NL↑ − RNL↓,↓

NL↑ + RNL

R≡

L↑
,
L↓

(A.15)

where N and L are the experimental yield and integrated luminosity. Superscripts are same as
equation A.2. R is called Relative Luminosity, defined as the ratio of luminosities for spin up
and down. In this analysis, BBCLL1 count is used for the luminosity calculation. N ↑ and N ↓can be described as ,NL↑ ∼ σ0 L↑ ΩL [1 + (1 + ǫP )AN P ]NL↓ ∼ σ0 L↓ ΩL [1 − (1 − ǫP )AN P ] .

(A.16)

Equation A.15 can be same representation as equation A.11 by square root formula. In thisformula, there is no contribution of the detector asymmetry, but the factor of the luminosity
remains as R = (L↑ /L↓ ) = (1 + ǫL )/(1 − ǫL ).

Assuming that the ∆R is negligibly small, statistical uncertainty is obtained as,q
NL↑ NL↓ (NL↑ + NL↓ )δǫP hys (L) = 2R 2NL↑ + RNL↓∼

1√at NL↑ ∼ NL↓ and R = 1,
NL

(A.17)

where NL is total number of yield detected in left region; NL = NL↑ + NL↓ .
We can obtain the asymmetry in right detection, ǫP hys (R), by same way. By rotational andparity invariance, the signs of ǫP hys (L) and ǫP hys (R) are opposite, but amplitudes should besame. If there are some systematics due to the some reasons which cannot be studied, it would
appear as ǫP hys (L) 6= ǫP hys (R).

Since AN must be zero at the beam center (pT =0 GeV), the beam center can be extracted

by the scanning of AN values calculated by the luminosity formula (it is used in section 8.1).

APPENDIX A. DEFINITION OF THE SINGLE TRANSVERSE SPIN ASYMMETRY 134

A.4

Background contamination

We should concern the effect of background contamination to the asymmetry measurement.
With the assumption of the very small ǫL ,ǫΩ ,ǫA and ǫP in equation A.10, the number of yields
with background contamination can be described as,ih)NL↑ = σ0 LdΩ (1 − r)(1 + P ASN ) + r(1 + P Abg
Nhi
NL↓ = σ0 LdΩ (1 − r)(1 − P ASN ) + r(1 − P Abg)
NhiNR↑ = σ0 LdΩ (1 − r)(1 − P ASN ) + r(1 − P Abg)
NhiNR↓ = σ0 LdΩ (1 − r)(1 + P ASN ) + r(1 + P Abg),
N(A.18)

where ASN is the asymmetry of signal. r and Abg
N are defined as a fraction of backgroundcontamination and its asymmetry respectively. Measured asymmetry by square root formulais,ihǫP hys = P (1 − r)ASN + rAbg
N .

(A.19)

S+bg
We can extract ASN by measured asymmetry, AN= ǫP hys /P , and its uncertainty as,q
S+bg 2
2
S+bg
bg
(δAN
) + r 2 (δAbg
A
− rAN
N)
ASN = N
, δASN =.
(A.20)1−r
1−r

It is same result for luminosity formula.

Appendix B

Local Polarimeter
B.1

Introduction

Extracting the spin state of partons in the polarized nucleon is one of main purposes at RHIC
which is the first and unique polarized proton collider in the world. Especially, the study of theparton polarization has been performed by measuring ALL which is defined as a differences of
the cross sections for different helicity combination with the longitudinally polarized collisions ,ALL ≡σ ++ − σ +−,σ ++ + σ +−(B.1)

where +(−) indicates that helicity state of the proton beam.
In the RHIC ring, proton polarization are keeping as vertically transverse. Absolute andfill-by-fill polarizations have been measured by the pp- and pC-CNI polarimeters, respectively.
A spin rotators which are installed up and downstream of the collision points at PHENIX andSTAR enables us to take the longitudinal beams by rotating the proton spin from vertical tolongitudinal at upstream and restore to vertical at downstream.
If the polarization direction is diagonal with finite polar angle after the rotation, measured
ALL has transverse component as,
easuredP B P Y AM
≈ PTB PTY AT T + PLB PLY ALL ,
LL

(B.2)

where, AT T is defined in the same way of ALL for the transverse polarization as,
AT T ≡σ ↑↑ − σ ↑↓.
σ ↑↑ + σ ↑↓(B.3)

P B and P Y are beam polarizations for Blue and Yellow beams, respectively. Subscripts, T and
L, indicate transverse and longitudinal components of polarization as P 2 = PT2 + PL2 . For theeasured , it is necessary to know P , P and A
extraction of ALL from P B P Y AM
TL
T T . AT T should
LL

be measured by the transversely polarized pp collisions. PT and PL for both beams can beobtained by the measurement of leading neutron AN as follows.
AN is governed with the sine modulation to the transverse spin component. Thus, expectedshapes of the azimuthal angle dependence of measured AN for A(φ) = −0.10 × sin(φ + φ0 )
135

136

APPENDIX B. LOCAL POLARIMETER

with various spin directions of the proton are shown in Figure B.1. We obtained shape (a) withthe vertically polarized pp collision at the spin rotator OFF. After the spin rotator is ON, theproton spin is rotated to any direction and the azimuthal dependence of AN is changed basedon its spin direction at the collision point. Rotation in φ and θ angles affect to φ0 as shape (b)and amplitude as shape (c), respectively. As shown in shape (c), we can check the longitudinal
direction by measuring AN being zero after the spin rotator is ON1 . Event if the transverse
component is remaining shown as (d), its amount can be evaluated by comparing the amplitudewith shape (a). Details procedures are shown in next section. Since the extraction of PT and
PL are performed at the PHENIX collision point locally, it is called “Local Polarimeter”.
We have collected the data by BBCLL1&(ZDCN|S) trigger for the local polarimeter analysis
to know PT and PL in the PHENIX minimum bias samples (section 5.2.4). At the commissioning
period for the spin rotator tuning, we take the data with full PHENIX DAQ bandwidth2 . We
check the spin direction within 1 hour and feedback the result to the accelerator to minimize
the transverse component, PT . Through the physics run period, the data have been collectedwith 100–200 Hz. We have estimated the transverse and longitudinal components through therun by offline analysis.
YφMeasured AN

Expected shapes for azimulthal depedence on measured AN

10

Y

×10-2

θ
a) Vertically transverse

0

-10
0

Y

a)
b)
c)
d)
0.5

1.5

2

2.5

3
φ

Z (θ=0)b) Horizontally transverseY

φ1

X (φ=0)θZ (θ=0)5-5φ

X (φ=0)X (φ=0)θφ

X (φ=0)θZ (θ=0)c) Longitudinal

Z (θ=0)d) Finite θ angle

Figure B.1: Expected shapes for the azimuthal dependence of measured AN for A(φ) = −0.10×
sin(φ + φ0 ) in case of various spin direction of the proton beam. Collision point is (x,y,z) =(0,0,0) cm.

1
2

This measurement has no sensitivity for the helicity sign since amplitudes being zero in both signs.
Full DAQ bandwidth are ∼1, 2 and 5 kHz for 2003, 2004 and 2005 year runs respectively.

137

APPENDIX B. LOCAL POLARIMETER

B.2

Procedure

As we mentioned in introduction, The spin direction at the collision point is evaluated by thecomparison of the measured AN amplitude at longitudinal run period, ALongitudinal, and thatNat transverse run period, ATNransverse . We extract the longitudinal and transverse componentsof the polarization direction at longitudinal period as,s 2ALongitudinal
PT
PL
PT
,
= 1−
= NT ransverse .
P
P
P
AN

(B.4)

Since it is possible that the transverse component is diagonal after the spin rotation, comparison
should be performed by not only the amplitude but also the phase of azimuthal dependence.
However, it is difficult to obtain correct values from the fit results of sine modulation in case ofnearly zero asymmetry since the amplitude and phase are complicated; fit result is labile. For
the stable operation. The comparison has been performed with the LR asymmetry, ALongitudinal,LR. PT /P can be represented as,
and UD asymmetry, ALongitudinalUD!
qLongitudinal
APT
LR, κU D ≡= κ2LR + κ2U D , where, κLR ≡P
ATNransverse

ALongitudinal
UD
ATNransverse

!

.

(B.5)

Here, measured parameters κLR and κU D are obtained as the average amplitude through therun period and the relation of them is included the phase information.

B.3

Measurement

In this note, I introduce the measurement of 2005 year run. Acceptances for the measurementsof LR and UD asymmetries were defined as Figure B.2. ǫN has been calculated by the squareroot formula with neutron yields. For the local polarimeter analysis, we can ignore the smearing
effect from the position resolution by taking the ratio of ATNransverse and ALongitudinalassumingNthe smearing effect has no time dependence. Neutron identifications were applied by the SMD
cut and no requirement of the forward scintillator; sample may include proton shower events3 .
However the contamination is no matter for the local polarimeter work since it is canceled bytaking ratio as well as the smearing effect. Since polarization would be changed fill by fill, we
calculated ǫN divided by the polarization in each fill.

B.3.1

ATNransverse

ransverse and AT ransverse as function of measured energies are shown inResults of forward ATLR
UD

Figure B.3. In both beams, significant asymmetries can be seen in the LR and not in the UD.
ransverse
It indicates that both beams were completely polarized in vertical and we can use ATLR

results as ATNransverse in equation B.5. We obtained ATNransverse value as an average in themeasured energy from 20 to 120 GeV. Their values are 0.044±0.002 and 0.042 ±0.002 for theBlue and Yellow beams, respectively.
3

At that time, proton shower event was not well studied.

138

APPENDIX B. LOCAL POLARIMETER

Acceptance definiction for UD

6
90
4

80
70

2

ypos (cm)ypos (cm)

Acceptance definiction for LR

6
100
4
80
2

60
50

0

60
0

40
-2

30
20

-4

40

-2

20

-4

10
-6
-6

-4

-2

0

2

4
6
xpos (cm)

0

-6
-6

-4

-2

0

2

4
6
xpos (cm)0Figure B.2: Acceptance definitions for Left-Right and Up-Down asymmetries (ALR , AU D ).

B.3.2

ALongitudinaland ALongitudinal
LR
UD

Results of forward ǫLongitudinaland ǫLongitudinal
divided by polarizations as ALongitudinal
and
LR
UD
LR
were calculated fill by fill shown as Figure B.4. The values were calculated asALongitudinalUDan average in the measured energy from 20 to 120 GeV in the same way of ATNransverse . The
average values through 2005 year period were obtained by constant fit. We obtained κLR andκU D from equation B.5 (see Table B.1). Significant transverse components were remaining forboth beams.
For the confirmation of this measurement, backward ǫLongitudinaland ǫLongitudinal
dividedLRUDby polarizations were performed as Figure B.5. Values were completely zero. It is consistent
with the zero-asymmetry in the backward region (negative-xF ) and confirms the small amountof asymmetries observed in the forward were not fake asymmetries.

B.4

Estimation of PL /P

We measured κLR and κU D so far. As a next step, PT /P and PL /P are calculated by equation
B.4 and B.5 but we should be careful to treat a boundary condition of these equations; PT /Pand PL /P can not exceed 1 and lower than 0 from the relation of unit circle. From equation
B.5, we can easily imagine that this problem appears as an asymmetric error of PT /P whenκLR and κU D are nearly equal to zero with the same magnitude of statistical errors. Usual
gaussian error propagation is wrong at that case. Unfortunately, this situation is pretty match
to longitudinal run period.
For the estimation of PL /P distribution, we calculated PL /P by equation B.4 through B.5
with the random gaussian generation of κLR and κU D values based on their errors many times.
PL /P distributions for 2003 and 2005 year runs are shown in Figure B.6. In 2003 year run,
distribution is completely asymmetric due to the reason mentioned above. In this case, we
obtained the center value of PL /P as most probable value. Statistics errors were evaluated as

139

Forward LR asymmetry for BLUE beam

Forward LR asymmetry for YELLOW beam

∈N / pol.

∈N / pol.

APPENDIX B. LOCAL POLARIMETER

0.1

hbasymn
Entries
Mean

74.72

RMS

0.05

0.1

40

21.98 / 9Prob0.008931

0.05

0

-0.05

-0.05

-0.1

-0.1
60

21.23 / 9
0.01167-0.04232 ± 0.00174

80 100 120 140
Measured energy (GeV)

0

20

40

60

80 100 120 140
Measured energy (GeV)

Forward UD asymmetry for BLUE beam

Forward UD asymmetry for YELLOW beam

∈N / pol.

40

28.98

χ 2 / ndf

∈N / pol.

20

RMS

p0

0

0

74.65

Prob

-0.04414 ± 0.00185

p0

40

Mean

30.97

χ 2 / ndf

hyasyms
Entries

0.1

hbasyms
Entries
Mean

91.27

RMS

0.05

0.1

40

4.517 / 9

Prob

0.05

0

-0.05

-0.05

-0.1

-0.1
20

40

60

RMS

36.65

χ 2 / ndf

4.92 / 980 100 120 140
Measured energy (GeV)

00.84130.001096 ± 0.001438

p0

0

0

80.57

Prob

0.87420.0002119 ± 0.0015500

p0

40

Mean

34.73

2

χ / ndf

hyasymn
Entries

20

40

60

80 100 120 140
Measured energy (GeV)

UD
Figure B.3: Results of forward ǫLR
N and ǫN divided by polarizations for Blue and Yellow beams
at transverse run period. They are plotted as function of the measured energy.

140

APPENDIX B. LOCAL POLARIMETER∈N / pol.

0.04
0.03

χ2/ndf = 99.66 / 102
Constant = -0.00328 ± 0.00042∈N / pol.

Forward LR asymmetries for YELLOW beam

Forward LR asymmetries for BLUE beam

0.04
0.03

χ2/ndf = 85.74 / 102
Constant = -0.00089 ± 0.00047

0.02

0.02

0.01

0.01

0

0

-0.01

-0.01

-0.02

-0.02

-0.03

-0.03

-0.04

-0.04
69006950 700070507100 715072007250 73007350

69006950 700070507100 715072007250 73007350

fill#∈N / pol.

0.03

χ2/ndf = 110.06 / 102
Constant = 0.00295 ± 0.00041∈N / pol.

Forward UD asymmetries for YELLOW beam

Forward UD asymmetries for BLUE beam

0.04

fill#

0.04
0.03

χ2/ndf = 140.59 / 102
Constant = 0.00613 ± 0.00042

0.02

0.02

0.01

0.01

0

0

-0.01

-0.01

-0.02

-0.02

-0.03

-0.03

-0.04

-0.04
69006950 700070507100 715072007250 73007350

69006950 700070507100 715072007250 73007350

fill#

fill#

UD
Figure B.4: Results of forward ǫLR
N and ǫN divided by polarizations for Blue and Yellow beams
as function of fill number, through longitudinal run period. Energy region was used from 20 to
120 GeV in the measured energy in the same way of ATNransverse . Maximum and minimum forvertical axes are set to the values of ǫTNransverse divided by the polarization.

141

APPENDIX B. LOCAL POLARIMETER∈N / pol.

0.04
0.03

χ2/ndf = 123.77 / 102
Constant = -0.00013 ± 0.00046∈N / pol.

Backward LR asymmetries for YELLOW beam

Backward LR asymmetries for BLUE beam

0.04
0.03

χ2/ndf = 96.44 / 102
Constant = 0.00011 ± 0.00044

0.02

0.02

0.01

0.01

0

0

-0.01

-0.01

-0.02

-0.02

-0.03

-0.03

-0.04

-0.04
69006950 700070507100 715072007250 73007350

69006950 700070507100 715072007250 73007350

fill#∈N / pol.

0.03

χ2/ndf = 80.25 / 102
Constant = 0.00006 ± 0.00040∈N / pol.

Backward UD asymmetries for YELLOW beam

Backward UD asymmetries for BLUE beam

0.04

fill#

0.04
0.03

χ2/ndf = 90.92 / 102
Constant = 0.00030 ± 0.00042

0.02

0.02

0.01

0.01

0

0

-0.01

-0.01

-0.02

-0.02

-0.03

-0.03

-0.04

-0.04
69006950 700070507100 715072007250 73007350

69006950 700070507100 715072007250 73007350

fill#

fill#

U D divided by polarizations for Blue and Yellow
Figure B.5: Results of backward ǫLR
N and ǫNbeams as function of fill number, through longitudinal run period. Energy region was used from
20 to 120 GeV in the measured energy in the same way of ATNransverse . Maximum and minimum
for vertical axes are set to the values of ǫTNransverse divided by the polarization.

142

APPENDIX B. LOCAL POLARIMETER

the region including 68.3% counts from center value for positive and negative regions. In 2005
year run, the distribution looks gaussian with the high statistics measurement. It indicatesusual gaussian error propagation would work in case of 2005 year statistics. These values are
summarized in section B.5.
PL /P distribution for BLUE beam at RUN3

P L /P distribution for BLUE beam at RUN5

2500

2000

χ 2 / ndf

1800

Prob

1600
2000

298.6 / 77
02027 ± 8.6

Constant

1400
Mean

0.995 ± 0.000

Sigma

0.0009761 ± 0.0000034

1200

1500

1000
800

1000

600
400

500

200
0
0.95

0.96

0.97

0.98

0.99

1
PL /P

0
0.95

0.96

0.97

0.98

0.99

1
PL /P

Figure B.6: PL /P distributions for 2003 (Left) and 2005 year (Right) runs. In 2003 year
run, distribution is completely asymmetric and we obtained the center value of PL /P as mostprobable value (solid line). Statistics errors were evaluated as the region including 68.3% counts
from center value for positive and negative regions (dashed line). In 2005 year run, distribution
looks gaussian with the high statistics measurement.

B.5

Summary table

Transverse and longitudinal components of the beam polarization at longitudinal run period
have been extracted by the local polarimeter analysis. Table B.1 shows the summary of resultsfrom 2003 to 2005 year runs. Systematic errors have been evaluated by bunch shuffling (section
8.2.4) and they are basically small.
Directions of beam polarizations were almost longitudinal for all longitudinal run period.
Year
—
2003
2004
2005

κLR
0.027±0.069
0.084±0.047
0.074±0.010

BLUE beamκU D
0.116±0.067
0.042±0.046
0.067±0.010

PL /P
0.995+0.003−0.0130.997+0.002−0.0070.995±0.001

YELLOW beam
κLR
κU D
PL /P
0.213±0.066
0.077±0.050
0.975+0.012−0.019−0.008±0.056 −0.074±0.045
0.998+0.002−0.0080.021±0.011
0.145±0.012 0.989±0.002

Table B.1: Summary table of κLR , κU D and PL /P in longitudinal run period from 2003 to 2005
year runs. Errors show statistics only.

Appendix C

Uncertainty in bunch shuffling
C.1

Uncertainty from the bunch number

Assuming the same luminosities for all bunches (bunch number n), asymmetry can be calculatedas,A=σ↑ − σ↓N↑ − N↓∼,σ↑ + σ↓N↑ + N↓(C.1)

where N ↑ and N ↓ are the number of yield for spin ↑ and ↓, respectively. The uncertainty ofasymmetry can be calculated with the assumption of that yield Ni is distributed with the mean
µ and σ as,
2

(∆A)

=



4 (N ↑ )2 (∆N ↓ )2 + (N ↓ )2 (∆N ↑ )2=1n(N ↑ + N ↓ )

4

 2
σ
(∆µ)2
,
=
µ
µ2

(C.2)

where ∆µ is the uncertainty for the mean which is described as (∆µ)2 = σ 2 /n. The uncertaintyof error depends on the uncertainty of µ. With the approximation of the gaussian distribution
for Ni as µ0 and σ0 , µ0 ∼ σ02 ,∆AN ∼ √
where N is the total yield, N =

P

1
1
=√ ,
nµ0
N

(C.3)

n Ni . This is usually used as a statistics error.

By the way, the unbiased variance of mean value µ, Sn , is written by,
X
1
Sn2 =
(Ni − µ)2 .
n(n − 1) n

(C.4)

When the bunch number n is higher, Sn2 = (∆µ)2 |n→∞ .

It must estimate the uncertainty of Sn if n is small number and ∆µ → ∆µ±∆Sn in equation

C.2. The uncertainty of Sn is calculated as,(2Sn ∆Sn )2 = ∆(Sn2 )2 =∼X
4(Ni − µ)2 (∆Ni )2n2 (n − 1)2 n
4σ4 .
n(n − 1)2
143

(C.5)

144

APPENDIX C. UNCERTAINTY IN BUNCH SHUFFLING

Here, the approximation of gaussian distribution of Ni in each bunch, (Ni − µ)2 = (∆Ni )2 = σi2and σ = σ1 = σ2 = · · · = σn are used. And Sn2 in the left hand side can be written as∼ σ 2 /(n − 1). So, ∆Sn is,(∆Sn )2 ∼=σ2n(n − 1)
(∆µ)2
n−1

→

∆A can be represented as,


∆µ
∆AN →
µ

∆Sn
1±
∆µ



1
∼√
N

∆µ
.
∆Sn = √
n−1

(C.6)



(C.7)



1± √

1
n−1

.

It indicates A can be fluctuated as a relative value to ∆A which depends on bunch number n.
For example, n = 245 and 195 for BBCLL1&(ZDCN|S) and ZDCN|S triggers respectively.
It means the relative uncertainty to the errors for them are 6.4% and 7.2%. In case of bunch
shuffling, it is affected to the χ2 values as follows,n

1 X (Ni − µ)2
n−1
σi2
i
 2 
√
2
1 ± 1/ n − 1 σ
1
√
= 1±
∼
.
σ2
n−1

2
1
2
∆(χ /ndf ) ∼
1± √
− 1.
n−1
χ2 /ndf

=

(C.8)

+0.15
χ2 /ndf values for BBCLL1&(ZDCN|S) and ZDCN|S can be fluctuated 1+0.13−0.12 and 1−0.14 respec-tively.

C.2

Residual asymmetry in bunch shuffling

After the bunch shuffling, some bunches are unchanged their spin direction and others arechanged to opposite. Starting at the number of yield of initial polarization direction ↑, N↑ , thenumber of yields for un-changing and changing the spin direction are written as N↑↑ and N↑↓

respectively. Its description is also applied for luminosity as L↑↑ and L↓↑ . These variables havefollowing relations.
N↑ = N↑↑ + N↑↓

L↑ = L↑↑ + L↓↑N↓ = N↓↑ + N↓↓

L↓ = L↑↓ + L↓↓ .

(C.9)

Asymmetry after the bunch shuffling, Ashuf f le , can be calculated by these variables as follows.
Ashuf f le =σ↑ − σ↓σ↑ + σ↓N↑↑ +N↓↑=L↑↑ +L↑↓N↑↑ +N↓↑L↑↑ +L↑↓−+N↑↓ +N↓↓L↓↑ +L↓↓N↑↓ +N↓↓L↓↑ +L↓↓145

APPENDIX C. UNCERTAINTY IN BUNCH SHUFFLING

A
=

 ↑

L↑ −L↑↓

2+A
∼

= A

= A

L↑↑ +L↑↓

−

 ↑

↑ +

L↓↑ +L↓↓

L↑ −L↑↓

L↓↑ −L↓↓L↑↑ +L↓L↓↑ +L↓↓L↑↑ − L↑↓

1
A
2

L↓↑ −L↓↓L↑↑ + L↑↓−L↓↑ − L↓↓L↓↑ + L↓↓!

L↑↑ L↓↓ − L↓↑ L↑↓(L↑↑ + L↑↓ )(L↓↑ + L↓↓ )L↑↑ L↓ − L↑↓ L↑(L↑↑ + L↑↓ )(L↑ + L↓ ) + (L↑↑ + L↑↓ )2= ABlum .

(C.10)↑(↓)

From second to third equation a relation of N↑

↑(↓)

= CL↑

↑(↓)

(1 + A) and N↓

↑(↓)

= CL↓

(1 − A)

with the asymmetry value A are used. From third to fourth equation, approximation A ≪ 1 isapplied and equation C.9 is used from fourth to fifth.

This equation means the Ashuf f le is depends on the asymmetry with the fluctuation of
luminosity if asymmetry is finite value. The mean value of Ashuf f le should be zero since themean values of each luminosity is L/4 with the total integrated luminosity, L ≡ L↑↑ +L↓↑ +L↑↓ +L↓↓ .
The fluctuation affects to the results of the bunch shuffling and it is estimated in next section.
The fluctuation of Blum is written as follows.
v!
uu ∂Blum 2t∆Blum =(∆L↑↑ )2 +
↑
∂L↑

∂Blum
∂L↑↓

!2

(∆L↑↓ )2 .

(C.11)

In case of L↑↑ , differential part is,
i h
i
h
↑
↑
↑
↑
↑
↑
↑
↑ 2
−
(L
+
L
)
−
2(L
+
L
)
L
(L
+
L
)(L
+
L
)
+
(L
+
L
)
↑
↓
↓
↑
↓
↑
↓ (L↑ L↓ − L↓ L↑ )
↑
↓
↑
↓
∂Blum
(C.12).
=i2h∂L↑↑(L↑↑ + L↑↓ )(L↑ + L↓ ) + (L↑↑ + L↑↓ )2
The mean value of it would be 2/L at L↑↑ = L↑↓ = L↓↑ = L↓↓ = L/4 and L↑ = L↓ = L/2. And ∆Lpart is,L↑↑ = n↑↑ hli(∆L↑↑ )2 = (∆n↑↑ )2 hli2 + (n↑↑ )2 (∆hli)2= (∆n↑↑ )2 hli2 + (n↑↑ )(∆l)2 ,(C.13)

where l is the integrated luminosity for each bunch crossing. hli and ∆l correspond to mean

and fluctuation for l. n↑↑ is the number of bunches which are unchanged spin direction from ↑as same notation as L↑↑ . It has the relation of n = n↑↑ + n↑↓ + n↓↑ + n↓↓ . The mean value of n isrepresented with the approximation of n↑↑ = n↑↓ = n↓↑ = n↓↓ = n/4 as,hn↑↑ i ∼

n
4

146

APPENDIX C. UNCERTAINTY IN BUNCH SHUFFLING(∆n↑↑ )2=1(n↑↑ + n↓↑ )
2



1
1−
2



∼

n
.
8

(C.14)In second equation, binomial distribution is considered. Equation C.13 is represented by L =nhli as,(∆L↑↑ )2="
 2 #
L2
∆l
1+2
.
8n
hli

(C.15)

L↑↓ leads to be same results.
Equation C.11 is represented as,v "u 2 #u1∆lt∆Blum ∼
1+2
.
n
hli

(C.16)

The fluctuation of Blum times asymmetry as shown in equation C.10 would appear in the
result of the bunch shuffling. For the χ2 /ndf distribution, the fluctuation of the data, denom2+ (A∆Blum )2 . Thus, allowable χ2 /ndf value of
inator in equation 8.2, would change to σstat.

bunch shuffling result with finite asymmetry is,2
σstat.
+ (A∆Blum )2.
2
σstat.
(A∆Blum )2∆(χ2 /ndf ) = χ2 /ndf − 1 =.
2σstat.

χ2 /ndf (with f inite asymmetry) =(C.17)

147

APPENDIX C. UNCERTAINTY IN BUNCH SHUFFLING

C.2.1

Expected Ashuf f le values for φ-dependence asymmetry

Expected bunch shuffling results for our φ-dependence asymmetries are summarized as follows.
Bin#
0 (0 < φ < π/8)
1 (π/8 < φ < 2π/8)
2 (2π/8 < φ < 3π/8)
3 (3π/8 < φ < 4π/8)
4 (4π/8 < φ < 5π/8)
5 (5π/8 < φ < 6π/8)
6 (6π/8 < φ < 7π/8)
7 (7π/8 < φ < π)f
Aef
N
0.0571
0.04370.02690.0165-0.0060-0.0315-0.0472-0.0567

σstat.
0.00570.00540.00520.00520.00520.00530.00540.0057

AN Blum
0.00370.00290.00180.0011-0.0004-0.0021-0.0031-0.0037∆(χ2 /ndf )
0.430
0.275
0.113
0.044
0.005
0.153
0.320
0.422

Table C.1: Expected Ashuf f le values for BBCLL1&(ZDCN|S) trigger.
Bin#
0 (0 < φ < π/8)
1 (π/8 < φ < 2π/8)
2 (2π/8 < φ < 3π/8)
3 (3π/8 < φ < 4π/8)
4 (4π/8 < φ < 5π/8)
5 (5π/8 < φ < 6π/8)
6 (6π/8 < φ < 7π/8)
7 (7π/8 < φ < π)f
Aef
N
0.02490.04620.0185-0.00440.0047-0.0459-0.0339-0.0511

σstat.
0.01280.01240.01210.01200.01200.01210.01240.0128AN Blum
0.00180.00340.0013-0.00030.0003-0.0033-0.0025-0.0037∆(χ2 /ndf )
0.020
0.073
0.012
0.001
0.001
0.076
0.039
0.084

Table C.2: Expected Ashuf f le values for ZDCN|S trigger.

Appendix D

Charged particles at BBC associatedwith leading neutron
So far we investigated neutron measurement in forward kinematics. In addition, charged particles associated with the leading neutron indicates interesting behaviors.

D.1

Multiplicity distribution

The multiplicity distribution observed by the BBCs are shown in Figure D.1 In case of the sample
collected by requiring BBC self trigger, multiplicities of both BBCs are same distributions (Leftfigure). However their distributions are changed in the neutron tagged sample; multiplicity offorward BBC is lower than that of backward (Right figure). Same behavior appeared at IP12experiment (see section 3.7).
This result may be explained by OPE model. In the OPE framework, charged particles will
be generated by the proton-pion scattering. Since energy of pion is lower than that of proton,multiplicity of charged particles in the neutron direction may be lower.

D.2

Single transverse spin asymmetry

The polarized phenomena were also observed in the charged particles for the sample of the neutron association; we can observe significant left-right asymmetries, especially in same direction
as neutron.
• Forward BBC (same direction as neutron) : -4.50 ± 0.50 ± 0.22%• Backward BBC (opposite direction as neutron) : 2.28 ± 0.55 ± 0.10 %Without the neutron association, their asymmetries are consistent with 0.
In focus on the the asymmetry for same direction as neutron, its sign is negative as well as
neutron asymmetry. One consideration is that the information of polarization is delivered to theproton-pion scattering directly. In this case, charged particle from the scattering themselves canhave asymmetry, but we may not know its sign. It is necessary to discuss with the theoretical
calculation.
148

APPENDIX D. CHARGED PARTICLES AT BBC ASSOCIATED WITH LEADING NEUTRON149

BBC hit multiplicity (BBC trigger)

BBC hit multiplicity tagged with forward neutron (neutron trigger)
5

5

10

10

10

4

104

103

103

102

102

10

10

1

1
0

5

10

15

20

25

30
35
40
BBC hit multiplicity

0

5

10

15

20

25

30
35
40
BBC hit multiplicity

Figure D.1: The Multiplicity distributions of charged particles in BBC. Left) sample of the
BBC self trigger (require at least 1 hit for both BBCs). Right) sample of the neutron tagged
trigger by SOUTH ZDC. The solid line shows that for forward BBC in the neutron detection(SOUTH) and the dashed line shows that for backward BBC (NORTH).

Appendix E

Leading neutron asymmetries atvarious center of mass energies
In the RHIC, pp collisions at√
s = 62.4 and 410 GeV were already performed. At that time,asymmetry studies were also performed as shown Figure E.1. We can see the significant asym√
metries in the forward kinematics in both energies as well as s=200 GeV.
√√
In case of s=410 GeV, it was the test acceleration to the next pp phase, at s=500 GeV.
They were successfully accelerated with keeping polarization. Leading neutron production keeps√
its large asymmetry to such higher energies and we expect to see finite asymmetry at s=500
GeV. It means the local polarimeter work in Appendix B will be able to do with the same√procedure at s=500 GeV.
√
Unfortunately, we do not have proper polarization values from CNI polarimeter at s=410
GeV since it was test. It is not possible to extract physics statement from the data. We are√
waiting the next high energy polarized pp collision, at s = 500 GeV.

150

APPENDIX E. LEADING NEUTRON ASYMMETRIES AT VARIOUS CENTER OF MASS ENERGIES151

Forward raw asymmetry by sqrt formula for s=62 GeV

Forward raw asymmetry by sqrt formula for s=410 GeV-2-2

Measured asymmetry

Measured asymmetry

1.5 ×10

1

0.5

0

3 ×10

2

1

0

-0.5

-1

-1

-2

-1.5
0

0.5

1

1.5

2

2.5

-3
0

3
φ

Backward raw asymmetry by sqrt formula for s=62 GeV

1.5

2

2.5

3
φ

-2

Measured asymmetry

Measured asymmetry

1

Backward raw asymmetry by sqrt formula for s=410 GeV-2

1.5 ×10

1

0.5

0

3 ×10

2

1

0

-0.5

-1

-1

-2

-1.5
0

0.5

0.5

1

1.5

2

2.5

3
φ

-3
0

0.5

1

1.5

2

2.5

3
φ

Figure E.1: The azimuthal angle dependence of single transverse spin asymmetries for various
center of mass energies. Asymmetry is calculated by square root formula and no corrections√
such as polarization are applied. Left and right indicate s=62.4 and 410 GeV, respectively.
Top and bottom shows forward and backward asymmetries, respectively.

Appendix F

Polarization of inclusive Λ0
production
Inclusive Λ0 productions from pp and pA reactions are polarized in perpendicular to the production plane automatically. The polarization can be measured through the decay of Λ0 → p + π − ;decayed proton distribution in the rest frame is givin by,

dN/dΩ ∝ (1 + αP cosθ),(F.1)

where θ is the angle between proton momentum and Λ0 spin direction. α is 0.647±0.013 [73].
The magnitude of polarization, P , can be extracted by measuring θ dependence of proton
distribution.
Λ0 polarization was first discovered by pBe reaction in 1976 at Fermilab [74] and has been
observed in the range of 12 to 2000 GeV/c (equivalent proton energy on fixed target) [75]-[82]1 .
√We introduce results from the R608 experiment at ISR, pp at s=31–62 GeV, which is similar√√
to the RHIC experiment, pp at s=200 GeV. Figure F.1, F.2 and F.3 indicate s, pT andxF dependences of Λ0 polarizations in the range of 0.2<xF <0.8 and 0.4<pT <1.3 GeV/c. Λ0√polarization is negative, there is no obvious s dependence and grows with increasing xF . Λ0polarization also grows with increasing pT up to ∼1 GeV/c and being flat in pT above 1 GeV/c.

Some mechanisms, Lynde semi-classical fragmentation model [90], Recombination model[91] and Fragmentation from high-spin baryon resonance [92] are discussed. They can explain
polarization sign but not for large magnitude 2 . Recently, OPE model suggests the descriptionof Λ polarization [94]. In this model, it is assumed a diagram shown in Figure F.4, which
is binary reaction πp → KΛ and the total πp cross section connected by the exchange of an

off-shell Reggeized pion, dominates for the inclusive Λ production in the fragmentation region.
Λ polarization is related to the binary reaction πp → KΛ which was measured by [95]-[98] andit successfully describes not only polarization sign but also large amplitude.

1
Polarization of roughly the same magnitude has been observed in the production by protons of every otherhyperon [83]-[89].
2Apparently, the attempt of fragmentation from high-spin baryon resonance leads to a reasonable quantitative
understanding of experimental situation [93].

152APPENDIX F. POLARIZATION OF INCLUSIVE Λ0 PRODUCTION

153

It is very interesting suggestion for our leading neutron measurement; this model can also
explain the large cross section of leading neutron production at pp and ep collisions (section
1.2). We would consider discovered neutron AN with polarized phenomena of the exchangedpion.

Figure F.1: Λ0 polarization vs. center of mass energy for three xF -pT bins from pp collision.
Horizontal lines are drawn as average polarization values for each bins. [82]√Figure F.2: Λ0 polarization vs. pT for fixed xF from pp collision in s=62 GeV. As explainedin the text, the calibration point, (1.3±1.3)%, is calculated using pπ − pairs which do not arise
from Λ0 . [82]

APPENDIX F. POLARIZATION OF INCLUSIVE Λ0 PRODUCTION

154

Figure F.3: Λ0 polarization vs. xF in pT >0.96√
GeV/c from pp collision in s=62 GeV (Average pT is 1.14 GeV/c) [82]. The open circles
are the results from Fermilab in pT >0.95 GeV/c[80].

Figure F.4: Single-pion exchange diagram for the process pp → ΛKX. [94]

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