CONTENTS · Lab. t Departmen of ysics Ph o oky T Institute of hnology ec T ebruary F 2001. Abstract...

Master's Thesis

Flavor Asymmetry of the polarized

light Sea Quarks extra ted from

deep-inelasti S attering at HERMES

Fumiko Sato

Shibata Lab.

Department of Physi s

Tokyo Institute of Te hnology

February 2001

Abstra t

The spin ontent in the nu leon is an interesting subje t of nu lear physi s, and

it has been found that the quark ontribution is less than 30 % in the nu leon spin

in re ent experiments. On the other hand from the unpolarized experiment, it was

found that the unpolarized quark distributions of �u and

�

d are not equal : there is

an asymmetry of �u and

�

d.

The main subje t of this thesis is the extra tion of the polarized quark avor

distribution, espe ially the polarized avor asymmetry extra tion. The avor asym-

metry of light sea quarks is expressed as ��u��

�

d, where ��u(�

�

d) is the polarized

�u(

�

d) quark distribution. In this analysis, a new method for extra tion of avor

asymmetry of light sea quarks is proposed. It is based on the Chiral Quark Soliton

Model (CQSM) predi tion. With the new method, the single quark distributions

an be extra ted as well as the avor asymmetry.

At HERMES experiment the produ ed parti les are measured in Deep-Inelasti

S attering as well as s attered lepton. The Ring Imaging Cherenkov Counter is

operated to identify parti les sin e 1998. The study of the dete tor eÆ ien y is also

dis ussed in this thesis. The HERMES RICH an identify pions, kaons and protons.

16 types of in lusive and semi-in lusive spin asymmetry be ame available. Using

these asymmetries, the quark distributions are extra ted and their moments with

the original and the new methods are presented.

CONTENTS i

Contents

1 Introdu tion 1

2 Deep-Inelasti S attering (DIS) 3

2.1 DIS Kinemati s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2.2 Cross Se tion and Nu leon Stru ture Fun tion . . . . . . . . . . . . . 7

2.2.1 Unpolarized Cross Se tion and Stru ture Fun tion . . . . . . . 8

2.2.2 Polarized Cross Se tion and Stru ture Fun tion . . . . . . . . 8

2.3 The Quark Parton Model . . . . . . . . . . . . . . . . . . . . . . . . 9

2.3.1 The unpolarized PDF . . . . . . . . . . . . . . . . . . . . . . 10

2.3.2 The polarized PDF . . . . . . . . . . . . . . . . . . . . . . . . 11

2.3.3 Measurement of g

1

(x) . . . . . . . . . . . . . . . . . . . . . . . 12

2.4 Sum Rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.4.1 Gottfried Sum Rule . . . . . . . . . . . . . . . . . . . . . . . . 13

2.4.2 Bjorken Sum Rule . . . . . . . . . . . . . . . . . . . . . . . . 14

2.4.3 Ellis-Ja�e Sum Rule . . . . . . . . . . . . . . . . . . . . . . . 15

2.5 Semi-In lusive Measurement . . . . . . . . . . . . . . . . . . . . . . . 15

2.5.1 Fragmentation Fun tions . . . . . . . . . . . . . . . . . . . . . 16

2.5.2 Fragmentation Models . . . . . . . . . . . . . . . . . . . . . . 17

2.5.3 Hadron Asymmetry . . . . . . . . . . . . . . . . . . . . . . . . 19

3 HERMES Experiment 21

3.1 Experimental Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

3.2 The HERA Storage Ring . . . . . . . . . . . . . . . . . . . . . . . . . 21

3.3 The Internal Gas Target . . . . . . . . . . . . . . . . . . . . . . . . . 23

3.4 The HERMES Spe trometer . . . . . . . . . . . . . . . . . . . . . . . 24

3.4.1 The Tra king System . . . . . . . . . . . . . . . . . . . . . . . 25

3.4.2 The Parti le Identi� ation System . . . . . . . . . . . . . . . . 27

4 Ring Imaging Cherenkov Counter 29

4.1 Dete tor Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

4.2 Study of RICH Property . . . . . . . . . . . . . . . . . . . . . . . . . 33

4.2.1 Aerogel Radiator . . . . . . . . . . . . . . . . . . . . . . . . . 33

4.2.2 Mirrors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

4.2.3 Photon Dete tor Plane . . . . . . . . . . . . . . . . . . . . . . 41

4.3 Monte Carlo Simulation for RICH . . . . . . . . . . . . . . . . . . . . 42

4.3.1 Development of MC for RICH . . . . . . . . . . . . . . . . . . 44

4.3.2 PID EÆ ien y after Modi� ation of MC . . . . . . . . . . . . 48

ii CONTENTS

5 Extra tion of Polarized Quark Distributions 57

5.1 The Purity Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

5.1.1 De�nition of Purity . . . . . . . . . . . . . . . . . . . . . . . . 57

5.2 Evaluation of Purities . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

5.3 Assumptions on Sea Quark Polarizations . . . . . . . . . . . . . . . . 70

5.3.1 Assumption of Polarization Symmetri Sea . . . . . . . . . . . 70

5.3.2 Assumption of Sea Quark from CQSM . . . . . . . . . . . . . 71

5.4 Interpretation of Asymmetries on the

3

He and D Targets . . . . . . . 76

5.4.1 Asymmetry on the

3

He Target . . . . . . . . . . . . . . . . . . 76

5.4.2 Asymmetry on the D Target . . . . . . . . . . . . . . . . . . . 77

5.4.3 Method to use Asymmetry on

3

He and D . . . . . . . . . . . . 77

5.5 The Set of Analyzed Asymmetries . . . . . . . . . . . . . . . . . . . . 82

5.5.1 1995 Asymmetries on the

3

He Target . . . . . . . . . . . . . . 82

5.5.2 1996 and 1997 Asymmetries on the Proton Target . . . . . . . 83

5.5.3 1998 Asymmetries on the D Target . . . . . . . . . . . . . . . 84

5.5.4 The Sets of Input Asymmetries . . . . . . . . . . . . . . . . . 85

6 Analysis and Results 87

6.1 Results for the Polarized Quark Distributions . . . . . . . . . . . . . 87

6.1.1 The Valen e De omposition . . . . . . . . . . . . . . . . . . . 90

6.1.2 The Flavor De omposition . . . . . . . . . . . . . . . . . . . . 93

6.1.3 Flavor Asymmetry of Light Sea Quarks . . . . . . . . . . . . . 96

6.1.4 Single Quark Distribution . . . . . . . . . . . . . . . . . . . . 99

6.1.5 Results for the di�erent Purities . . . . . . . . . . . . . . . . . 102

6.2 Moments of Polarized Quark Distributions . . . . . . . . . . . . . . . 104

6.2.1 The Valen e and Flavor De omposition . . . . . . . . . . . . . 105

6.2.2 Flavor Asymmetry and Single Quark Distribution . . . . . . . 106

6.2.3 Comparison of Results with Predi tions . . . . . . . . . . . . . 108

6.3 Flavor Asymmetry Extra tion with Another Method . . . . . . . . . 110

6.3.1 Another Method for Flavor Asymmetry Extra tion . . . . . . 110

6.3.2 Theoreti al Model of the Fragmentation Fun tions . . . . . . 111

6.3.3 Results and Comparison with the Purity Matrix Method . . . 114

6.4 Simple Estimation of Systemati Errors . . . . . . . . . . . . . . . . . 116

7 Con lusions and Summary 119

A Detail of MC Development 121

B Detail of Analysis 125

B.1 Covarian e Matrix V

A

. . . . . . . . . . . . . . . . . . . . . . . . . . 125

B.2 Singular Value De omposition . . . . . . . . . . . . . . . . . . . . . . 127

C Table of Results 129

C.1 De�nition of the Binning in x . . . . . . . . . . . . . . . . . . . . . . 129

C.2 Input Asymmetries . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130

C.2.1 1995 Asymmetries on the

3

He Target . . . . . . . . . . . . . . 130

C.2.2 1996 Asymmetries on the Proton Target . . . . . . . . . . . . 131

C.2.3 1997 Asymmetries on the Proton Target . . . . . . . . . . . . 132

CONTENTS iii

C.2.4 1998 Asymmetries on the D Target . . . . . . . . . . . . . . . 133

C.3 Correlation CoeÆ ients for Parti le Count Rates . . . . . . . . . . . . 135

C.4 Results of Quark Polarizations and Distributions . . . . . . . . . . . 138

C.4.1 Valen e De omposition . . . . . . . . . . . . . . . . . . . . . . 138

C.4.2 Flavor De omposition . . . . . . . . . . . . . . . . . . . . . . 140

C.4.3 Flavor Asymmetry of Light Sea Quarks . . . . . . . . . . . . . 142

C.4.4 Single Quark Distributions . . . . . . . . . . . . . . . . . . . . 144

C.5 Moments of Polarized Quark Distributions . . . . . . . . . . . . . . . 146

C.5.1 Valen e De omposition . . . . . . . . . . . . . . . . . . . . . . 146

C.5.2 Flavor De omposition . . . . . . . . . . . . . . . . . . . . . . 147

C.5.3 Flavor Asymmetry of Light Sea Quarks . . . . . . . . . . . . . 148

C.5.4 Single Quark Distributions . . . . . . . . . . . . . . . . . . . . 149

1

Chapter 1

Introdu tion

The spin stru ture of the nu leon be ame one of the fundamental questions in

physi s, when results from the EMC experiment [1℄ showed that only a fra tion

of the nu leon's spin is arried by its quarks. Before that experiment, it has been

predi ted, that spin of the nu leon, like its magneti moment, is almost ompletely

arried by the quarks. As a onsequen e of this dis overy, a more general approa h

for the spin ontents of a proton, taking quarks, gluons and orbital momenta into

a ount, has to be used:

S

z

=

1

2

=

1

2

�� +�g + L

z

; (1.1)

where �� = �u+��u+�d+�

�

d+�s+��s is the ontribution of quark spins, �g

is that of the gluon spin and L is that of the orbital angular moment.

Further experiments [2℄ on�rmed the EMC results and obtained : �u

�

=

0:8;�d

�

=

�0:4 and �s

�

=

�0:1. But the ontributions of light sea quarks (��u;�

�

d), gluons and

the orbital angular momenta are not lear yet. In most further studies to measure

the ontribution of quark spins, the ontributions of light sea quarks are assumed to

be symmetri as ��u = �

�

d = �s = ��s to ompensate for the la k of experimental

data. The ontributions of valen e quarks have been extra ted from the di�erent

experiments and their results are almost onsistent, however, the spin stru tures

of light sea quark are not made lear omparing the valen e quarks. In addition,

the asymmetry between the spin of the �u and the

�

d quarks have been not obtained

learly yet. In the unpolarized experiment at HERMES, a avor asymmetry of the

unpolarized sea quark densities (�u�

�

d 6= 0) are measured already [3℄, and this mea-

surement may be a hint that there might be an asymmetry for the polarized light

sea quark densities. This unpolarized avor asymmetry have been obtained in other

experiment [4℄ [5℄.

This polarized avor asymmetry of light sea quarks is theoreti ally predi ted by

the 1=N

-expansion of Quantum ChoromoDynami s (QCD) [6℄, where it is al ulated

in the large-N

limit by the Chiral Quark Soliton Model. This model predi ts a quite

large polarized sea quark avor asymmetry. From the theoreti al side, the estimation

of the avor asymmetry are a hieved in [7℄ [8℄. Additionally there are the extra tion

of avor asymmetry with the HERMES and SMC data in [9℄ [10℄ and it suggests the

existen e of the avor asymmetry, although it is not on�rmed be ause the statisti s

is not enough. It is very interesting and hallenging to study of the avor asymmetry

of light sea quarks, ��u��

�

d, for solving the spin stru ture of nu leon.

2 CHAPTER 1. INTRODUCTION

One of the main tasks of the HERMES Experiment at DESY is the determination

of the quark spin distributions with semi-in lusive measurements in deep-inelasti

s attering. Semi-in lusive measurement allows us to study the spin ontributions of

ea h quark in the nu leon by tagging the stru k quark with help of the identi� ation

of the leading hadron in the �nal state. The HERMES Experiment started in 1995

and has obtained many important basi results already. Sin e 1998 a Ring Imag-

ing Cherenkov Counter (RICH) is installed, and this is one of the most important

devi es to tag the stru k quark from the target. The HERMES RICH uses two

radiators, aerogel and C

4

F

10

gas. In ident photons make a dual ring image on the

photon dete tor plane, whi h allows us to dete t three types of hadrons (�, K, p) in

the momentum region of 2 � 15 GeV. This dual radiator RICH provides a urate

identi� ation of the dete ted hadrons. With this RICH data we an extra t the

ontributions of quark spins to the total spin of the nu leon.

3

Chapter 2

Deep-Inelasti S attering (DIS)

2.1 DIS Kinemati s

Deep-Inelasti S attering (DIS) is one of the most important tools to investigate

the internal stru ture of the nu leon. The DIS pro ess of the intera tion between a

nu leon N and a lepton l an be written as follows:

l +N ! l

0

+X; (2.1)

where l

0

is the s attered lepton, h is the dete ted hadron and X stands for other

undete ted parti les. The Feynman diagram of this pro ess is shown in Fig. 2.1.

The in oming lepton whi h hits the nu leon intera ts with the quarks by the

weak or ele tromagneti intera tion, by ex hanging a neutral ve tor boson Z

0

or a

virtual photon

�

. But in the energy region of HERMES the ontribution of the

Z

0

boson an be negle ted be ause the mass of the Z

0

boson is far beyond the

kinemati s of HERMES. So from here on, we onsider the ele tro-magneti pro ess

whi h the ex hange of one photon

�

only.

Figure 2.1: The diagram of DIS. The kinemati al valuables are expressed in

Tab.2.2. X expresses the produ ed parti les in the fragmentation

pro esses.

4 CHAPTER 2. DEEP-INELASTIC SCATTERING (DIS)

The in lusive DIS variables

k = (E;

!

k

) Four-momentum of the in oming lepton

k

0

= (E

0

;

!

k

0

) Four-momentum of the s attered lepton

s = (0;

^

!

s

)

^

!

s

=

1

m

(j

!

k

j;

^

!

k

E) ;

^

!

k

=

!

k

=j

!

k

j

Spin four-ve tor of the in ident lepton

s

0

= (0;

^

!

s

0

) Spin four-ve tor of the s attered lepton

P = (M;

!

0

) Four-momentum of the target nu leon

S = (0;

^

!

S

) ,

^

!

S

= (sin� os �; sin� sin�; os�)

Spin four-ve tor of the target nu leon

� S attering angle of the lepton in the lab system

� =

P � q

M

lab

= E � E

0

Energy transfer to the target

q = (�;

!

q

) Four-momentum transfer to the target

Q

2

lab

= 4EE

0

sin

2

�

2

Negative square of the four-momentum transfer

W

2

= (P + q)

2

lab

= M

2

+ 2M� �Q

2

Invariant of the hadron �nal state

x =

Q

2

2P � q

lab

=

Q

2

2M�

Bjorken s aling variable

y =

P � q

P � k

lab

=

Q

2

2M�

Fra tional energy transfer of the virtual photon

Table 2.1: De�nition of DIS kinemati al values. These variables of DIS are

shown in Fig.2.2 and Fig.2.3.

2.1. DIS KINEMATICS 5

Figure 2.2: De�nition of DIS kinemati variables. The angles �, �, � and �

are de�ned by the in oming lepton k, the s attered lepton k' and

the spin four-ve tor of target nu leon S.

Figure 2.3: De�nition of s attering angles in DIS. The angles whi h are

shown here are identi al with those in Fig.2.2.


We next de�ne the kinemati s of the DIS pro ess. The kinemati variables are

de�ned in Tab.2.1. In this thesis, the dis ussion are based on the laboratory (lab)

system, whi h is the spe ial ase where an in oming lepton with four-momentum

k = (E;

!

k

) is s attered by a �xed target with four-momentum P = (M; 0). M is

the rest mass of the target nu leon. All quantities are expressed in units of and �h

as � �h � 1 and skip all afterwards.

First the four-momentum transfer q between the in oming lepton and the target

nu leon is de�ned as:

q

2

= (k

0

� k)

2

lab

�

�4EE

0

2

sin

2

�

2

; (2.2)

and the negative square of the four-momentum transfer squared is de�ned:

Q

2

� �q

2

= �(k

0

� k)

2

: (2.3)

The invariant mass W of the hadron �nal state is obtained by the four-momentum

transfer of the ex hanged photon and the target four-momentum P of the initial

state:

W

2

= (P + q)

2

= M

2

+ 2Pq + q

2

= M

2

+ 2M� �Q

2

; (2.4)

where the Lorenz invariant variable � is de�ned as:

� =

P � q

M

lab

= E � E

0

: (2.5)

In the laboratory frame � gives the transfer energy from the in oming lepton to the

target by the virtual photon. Using this Q

2

the two dimensionless variables x; y are

given by:

x =

Q

2

2P � q

lab

=

Q

2

2M�

; (2.6)

y =

P � q

P � k

lab

=

�

E

: (2.7)

x is the variable of Bjorken s aling and y is the fra tional energy transfer and these

an be determined in the region of 0 to 1.

In the elasti s attering where W = M :

2M� �Q

2

= 0: (2.8)

On the other hand in the inelasti s attering where W is larger thanM the relation:

2M� �Q

2

> 0; (2.9)

is obtained. The Bjorken s aling variable x is a measure for the inelasti ity of the

s attering pro ess. In the elasti pro ess x is de�ned as:

x = 1: (2.10)

In the inelasti pro ess x takes the value from 0 to 1:

0 < x < 1: (2.11)

2.2. CROSS SECTION AND NUCLEON STRUCTURE FUNCTION 7

2.2 Cross Se tion and Nu leon Stru ture Fun -

tion

In DIS, the pro ess o urs in a large region of the square of four-momentum transfer

Q

2

and energy transfer �. The ross se tion of DIS an be written as:

d

2

�

dE

0

d

=

�

2

Mq

4

�

E

0

E

L

��

W

��

; (2.12)

where � is the �ne stru ture onstant, L

��

is the lepton tensor andW

��

is the hadron

tensor. The lepton tensor an be obtained from QED al ulation:

L

��

(k; s; k

0

; s

0

) = [�u(k

0

; s

0

)

�

u(k; s)℄[�u(k

0

; s

0

)

�

u(k; s)℄

�

; (2.13)

and we an split into symmetri (S) and anti-symmetri (A) part:

L

��

(k; s; k

0

; s

0

) = L

��

(S)

+ iL

��

(A)

; (2.14)

where

L

��

(S)

= 2

n

k

�

k

0�

+ k

0�

k

�

� g

��

(k � k

0

�m

2

)

o

; (2.15)

L

��

(A)

= 2 m �

��

s

�

(k � k

0

)

�

; (2.16)

where m is the lepton mass, s and s

0

are ovariant spin four-ve tor of the initial

and �nal lepton, respe tively. In Eq.2.15-Eq.2.16, the notations are used, whi h are

shown as:

�

0123

= �1; (2.17)

g

��

=

8

>

<

>

:

1 � = � = 0;

�1 � = � = 1; or 2 or 3;

0 � 6= �:

(2.18)

Like the lepton tensor, the hadron tensor is separated into symmetri (S) and

anti-symmetri (A) part:

W

��

(q;P; S) =W

��

(S)

(q;P ) + iW

��

(A)

(q;P; S): (2.19)

The hadron tensor re e ts the stru ture of hadrons. Although the lepton tensor

an be obtained exa tly from the al ulation, the hadron tensor annot be al ulated

be ause hadrons are not pointlike parti les. Therefore it an be parameterized by so

alled stru ture fun tions. In the following se tions, the ross se tion and stru ture

fun tion in the unpolarized and polarized ase will be dis ussed.


2.2.1 Unpolarized Cross Se tion and Stru ture Fun tion

The spin-independent part of the hadron tensor W

��

(S)

an be expressed by dimen-

sionless stru ture fun tions, F

1

(x;Q

2

) and F

2

(x;Q

2

):

W

��

(S)

(q;P ) = 2

�g

��

�

q

�

q

�

Q

2

F

1

(x;Q

2

)

!

+

P

�

+

P � q

Q

2

q

�

!

P

�

+

P � q

Q

2

q

�

!

F

2

(x;Q

2

)

P � q

: (2.20)

The stru ture fun tions F

1

(x;Q

2

) and F

2

(x;Q

2

) are Lorentz-invariant and re e t

the internal stru ture of the nu leon. With the symmetri parts of the lepton and

hadron tensors, the unpolarized di�erential ross se tion in the lab system (Eq.2.12)

is:

d

2

�

dxdQ

2

=

d�

d

!

�

1

EE

0

�

x

"

F

2

(x;Q

2

) +

2�

M

F

1

(x;Q

2

) tan

2

�

2

!#

: (2.21)

These two stru ture fun tions F

1

(x;Q

2

) and F

2

(x;Q

2

) are independent or have

very weak dependen e of the value of Q

2

for the �xed x. Therefore, F

1

(x;Q

2

) and

F

2

(x;Q

2

) an be rewritten in the limit of the large Q

2

:

F

1

(x;Q

2

) ! F

1

(x); (2.22)

F

2

(x;Q

2

) ! F

2

(x); (2.23)

where x =

Q

2

2M�

is the momentum fra tion whi h the parton arries.

The inelasti stru ture fun tions F

1

(x) and F

2

(x) depend only on the x variable,

whi h is alled the Bjorken s aling.

2.2.2 Polarized Cross Se tion and Stru ture Fun tion

In the polarized ase, the ross se tion in ludes two types stru ture fun tions. One

is the two spin-averaged stru ture fun tions F

1

(x;Q

2

) and F

2

(x;Q

2

) ( ontaining in

W

��

(S)

) like in the unpolarized ase, other is the two spin-dependent stru ture fun -

tions g

1

(x;Q

2

) and g

2

(x;Q

2

) ( ontaining in W

��

(A)

). W

��

(A)

is expressed by g

1

(x;Q

2

)

and g

2

(x;Q

2

):

W

��

(A)

(q;P; s) = �

��

�

q

�

�

"

S

�

g

1

(x;Q

2

) +

s

�

�

S � q

P � q

P

�

g

2

(x:Q

2

)

!#

: (2.24)

The spin-dependent stru ture fun tions an be obtained by measuring the dif-

feren es of the ross se tions in the opposite target spin states be ause the spin-

independent stru ture fun tions are an eled in the ross se tion di�eren es. The dif-

feren es of ross se tion has only the L

��

(A)

W

��(A)

term and are obtained by g

1

(x;Q

2

)

and g

2

(x;Q

2

) as follows:

X

s

0

"

d

2

�

ddE

0

(k; s; P;�S; k

0

; s

0

)�

d

2

�

ddE

0

(k; s; P; S; k

0

; s

0

)

#

=

�

2

2Mq

4

�

E

0

E

4L

��

(A)

W

��(A)

(2.25)

=

8��

2

Q

4

y

E

"

(E + E

0

os �)g

1

(x;Q

2

)�

Q

2

�

g

2

(x;Q

2

)

#

: (2.26)

2.3. THE QUARK PARTON MODEL 9

In the Bjorken limit, the spin-dependent stru ture fun tions g

1

(x) and g

2

(x)

whi h are dependent only on x are obtained:

g

1

(x;Q

2

)! g

1

(x); (2.27)

g

2

(x;Q

2

)! g

2

(x): (2.28)

2.3 The Quark Parton Model

The basis of the Quark Parton Model (QPM) is that the proton stru ture is ob-

tained from the longitudinal momentum of onstituents whi h are alled 'partons'.

The harged partons are the quarks and others are the gluons. We an easily treat

the deep-inelasti s attering in the Breit-Frame (Fig 2.4), where the transverse mo-

mentum and the rest mass of the partons an be negle ted.

If P is de�ned as the four-momentum ve tor of proton target and �P is the part

of the proton momentum whi h is arried by the parton the following relation is

obtained in deep-inelasti s attering by this parton and lepton:

(�P + q)

2

= m

2

q

: (2.29)

Negle ting the mass of the parton m

q

, � is given by:

� =

Q

2

2M�

�

2

1 +

q

1 +Q

2

=�

2

=

2x

1 +

q

q + 4(Mx)

2

=Q

2

: (2.30)

In ase of Q

2

>> M

2

, Eq.(2.30) simpli�es:

� �

x

1 + (Mx)

2

=Q

2

� x: (2.31)

In the Breit-Frame, � (Na htmann valuable) whi h expresses the ratio of the

longitudinal momentum of parton to the total momentum of proton is identi al

with the Bjorken x in the large-Q

2

limit. In this frame, we noti e that the photon

whi h has the four-momentum q in the lab-frame intera ts with the parton whi h

has the momentum fra tion x = Q

2

=2M�.

e+’

e+

γξ

ξ P

- P

Figure 2.4: DIS Pro ess in Breit-Frame. �P expresses the momentum whi h

is arried by the parton.


From the above onsideration, we an de�ne the stru ture of nu leon using the

parton momenta. The nu leon ontains two types of quarks, the valen e and the

sea quarks. The valen e quarks de ide the quantum number of the nu leon and the

sea quarks are pairs of quarks and anti-quarks whi h are produ ed virtually from

the gluons by the strong intera tion.

The stru ture fun tion F

2

(x) an be expressed by quark momentum distributions

in the QPM as:

F

2

(x) = x �

X

f

e

2

f

(q

v

(x) + q

s

(x))

f

; (2.32)

in whi h f denotes the quark avor, q

v

(x) is the number density of the valen e

quarks, q

s

(x) is that of the sea quarks and e

f

is the orresponding ele tri harge.

The number density of a quark with avor f whi h has the momentum fra tion x

is alled Parton Density Fun tion (PDF).

2.3.1 The unpolarized PDF

The unpolarized PDF of quark q with avor f is des ribed as follows:

q

f

(x) = (q

v

(x) + q

s

(x))

f

; (2.33)

q

s

f

(x) = �q

f

(x): (2.34)

The q(x) onsists of the density of the valen e quarks q

v

(x) and the sea quarks

q

s

(x). q

s

(x) and the anti-quark density �q(x) are identi al. The PDF re e ts the

distribution of quark momentum in the nu leon.

Assuming that the heavy quarks an be negle ted and using the isospin symmetry

between the proton and neutron, we obtain the relations:

u

p

(x) = d

n

(x) � u(x); (2.35)

d

p

(x) = u

n

(x) � d(x); (2.36)

s

p

(x) = s

n

(x) � s(x): (2.37)

The valen e quarks are de�ned as:

u

v

(x) = u(x)� �u(x); (2.38)

d

v

(x) = d(x)�

�

d(x); (2.39)

and we an reprodu e the quantum number of the proton summing up all parton

densities:

Z

1

0

[u(x)� �u(x)℄dx = 2; (2.40)

Z

1

0

[d(x)�

�

d(x)℄dx = 1; (2.41)

Z

1

0

[s(x)� �s(x)℄dx = 0: (2.42)

2.3. THE QUARK PARTON MODEL 11

2.3.2 The polarized PDF

In the polarized QPM, the quark spin distribution �q

f

(x) is de�ned as follows:

�q

f

(x) = q

+

f

(x)� q

�

f

(x); (2.43)

where q

+

f

(x) is the distribution of quarks with spins in the same dire tions as the spin

of the nu leon, q

�

f

(x) is that with the opposite dire tion. When the spin dire tion

of the virtual photon and the quarks are longitudinally parallel or anti-parallel, we

an de�ne the polarized stru ture fun tions:

g

1

(x) =

1

2

X

f

e

2

f

(q

+

f

(x)� q

�

f

(x))

=

1

2

X

f

e

2

f

�q

f

(x); (2.44)

g

2

(x) = 0: (2.45)

In the simple QPM, g

2

(x) is approximated to zero. The value of g

1

(x) is most

sensitive to the quark spin distribution. This is explained in Fig.2.5. The in oming

polarized lepton emits a polarized virtual photon. This photon an be absorbed

only by a quark with the opposite spin dire tion be ause the �nal state of a quark

must have spin 1/2. In the top pi ture of Fig.2.5 we an measure the ross se tion

for beam and target polarization anti-parallel and in the bottom pi ture we an

measure that for polarization parallel. The top pro ess is sensitive to q

+

f

(x) and the

bottom is sensitive to q

�

f

(x). Sin e g

1

(x) is the di�eren e of q

+

f

(x) and q

�

f

(x), it

gives the information of the quark spin.

Spin of photon

Spin of nucleon

Spin of photon

Spin of nucleon

e+

e+

e+’

e+’

q (x)f-

q +(x)f

Figure 2.5: Diagram polarized DIS. The top �gure shows the pro ess whi h

is sensitive to quarks polarized parallel to the nu leon's spin, and

the bottom �gure is the pro ess for quarks with the anti-parallel

spin to the nu leon.


2.3.3 Measurement of g

1

(x)

To measure g

1

(x) in the DIS experiments, we de�ne the asymmetry of the polarized

ross se tions in the ases of the dire tion of the target polarization longitudinal

(� = 0

o

; 180

o

in Fig.2.3) and transversal (� = 90

o

; 270

o

in Fig.2.3) to that of the

in oming lepton:

A

k

=

�(180

o

)� �(0

o

)

�(180

o

) + �(0

o

)

; (2.46)

A

?

=

�(270

o

)� �(90

o

)

�(270

o

) + �(90

o

)

: (2.47)

This measured ross se tion asymmetry is related to the asymmetry in absorption

of the virtual photon whi h was emitted by the in ident lepton:

A

k

= D(A

1

+ �A

2

); (2.48)

A

?

= d(A

2

� �A

1

); (2.49)

where D; �; d; � are given by:

D =

y(2� y)

y

2

+ 2(1� y)(1 +R(x;Q

2

))

;

d = D

s

2�

1 + �

;

� =

1� y

1� y +

y

2

2

;

� =

2 (1� y)

2� y

;

� = �

1 + �

2�

;

=

Q

�

=

2Mx

Q

:

R(x;Q

2

) is the ratio of longitudinal (�

T

) to transverse (�

L

) absorption ross

se tion

1

of the virtual photon:

R(x;Q

2

) =

�

L

�

T

= (1 +

2

)

F

2

(x;Q

2

)

2xF

1

(x;Q

2

)

� 1: (2.50)

In the Bjorken limit, R(x;Q

2

) be omes zero. The polarized stru ture fun tions g

1

and g

2

an be dedu ed from A

1

and A

2

by:

A

1

=

g

1

�

2

g

2

F

1

; (2.51)

A

2

=

(g

1

+ g

2

)

F

1

: (2.52)

A

2

is limited by R(x;Q

2

) be ause it ontains the interferen e term �

I

between �

T

and �

L

whi h has to obey the triangular relation �

I

�

p

�

L

�

T

[11℄, leading to:

jA

2

j �

q

R(x;Q

2

): (2.53)

1

For photons \polarization" is usually used for the! E-�eld dire tion, not for the spin dire tion.

In this sense, transverse photon polarization means longitudinal spin dire tion.

2.4. SUM RULES 13

From the results of re ent experiment [12℄, it is lear that A

2

is lose to zero. If

g

2

and A

2

an be ignored, g

1

(x) is obtained by these simple relations:

A

1

�

A

k

D

; (2.54)

g

1

� A

1

� F

1

: (2.55)

In Bjorken limit, F

1

(x) and g

1

(x) are des ribed by:

F

1

(x) �

F

2

(x)

2x

; (2.56)

g

1

(x) �

A

k

D

F

2

(x)

2x

: (2.57)

Eq.2.56 is alled the Callan-Gross relation. The re ent measurement of g

1

(x) is

shown in Fig.2.6.

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

x

gp 1

HERMES ( 1997 )

SLAC E-143

Q2 = 2 GeV2

0.1 0.50.02x

HERMES ( 1997 )

SMC ( 1993+1996 )

Q2 = 10 GeV2

0.1 0.50.02

Figure 2.6: The re ent results of the spin dependent stru ture fun tion on the

proton g

p

1

(x). The solid ir le shows the �nal result for g

1

from

1997 data at HERMES, whi h are shown at Q

2

=2 GeV

2

(the

left �gure) and Q

2

=10 GeV

2

(the right �gure). The results from

SLAC-E143 (Q

2

=2 GeV

2

) and SMC (Q

2

=10 GeV

2

) are shown

for omparison.

2.4 Sum Rules

2.4.1 Gottfried Sum Rule

The Gottfried Sum Rule [13℄ is for the unpolarized stru ture fun tions of proton F

p

2

and neutron F

n

2

. From Eq.2.32, Eq.2.40 - Eq.2.42 we obtain the relation:

S

G

=

Z

1

0

1

x

(F

p

2

(x)� F

n

2

(x))dx;

=

1

3

+

2

3

Z

1

0

u

s

(x)� d

s

(x): (2.58)


If the unpolarized sea quark distributions is symmetri , S

G

=1/3 should be obtained.

The measured value by NMC is S

G

= 0.235 � 0.026 [14℄, whi h means this assump-

tion is not valid and an unpolarized sea quark avor asymmetry exists.

2.4.2 Bjorken Sum Rule

In polarized ase, the integral of g

1

(x;Q

2

) is derived from Eq.2.44:

�

1

(Q

2

) �

Z

1

0

g

1

(x;Q

2

)dx: (2.59)

With the assumption of iso-spin symmetry, the integrals of the proton and neutron

spin stru ture fun tions �

p

1

and �

n

1

are expressed as follows:

�

p

1

=

1

9

a

0

+

1

12

a

3

+

1

36

a

8

; (2.60)

�

n

1

=

1

9

a

0

�

1

12

a

3

+

1

36

a

8

; (2.61)

where a

0

, a

3

and a

8

are the SU(3) avor singlet, isotriplet and o tet isosinglet

ombinations. They are expressed by the quark distributions:

a

0

= (�u+��u) + (�d+�

�

d) + (�s+��s); (2.62)

a

3

= (�u+��u)� (�d +�

�

d); (2.63)

a

8

= (�u+��u) + (�d+�

�

d)� 2(�s+��s): (2.64)

(2.65)

Furthermore, assuming SU(3) avor symmetry, a

3

and a

8

an be obtained from the

weak oupling onstants F and D:

a

3

= F +D; (2.66)

a

8

= 3F �D: (2.67)

The values of F;D are measured in neutron and hyperon �-de ays. F and D are

obtained as follows [15℄:

F = 0:459� 0:008; (2.68)

D = 0:798� 0:008: (2.69)

From the di�eren e of the two polarized stru ture fun tions, the Bjorken sum

rule is derived as follows:

�

p

1

(Q

2

)� �

n

1

(Q

2

) =

1

6

�

�

�

�

�

�

g

A

g

V

�

�

�

�

�

� Corr (2.70)

=

1

6

� (F +D) � Corr; (2.71)

where g

A

; g

V

are the axial-ve tor and ve tor oupling onstants. Corr is the orre -

tion fa tor with higher order QCD.

The Bjorken Sum Rule [16℄ [17℄ uses the assumption of isospin invarian e only,

and it is therefore a good tool to test QCD. No existing experimental data show a

violation of this sum rule [18℄.

2.5. SEMI-INCLUSIVE MEASUREMENT 15

2.4.3 Ellis-Ja�e Sum Rule

If we assume that �s = 0 in the Bjorken Sum Rule, we an derive the moment of

g

p

1

and g

n

1

from Eq.2.60 and Eq.2.61:

�

p

1

=

1

12

�

�

�

�

�

�

g

A

g

V

�

�

�

�

�

�

�

1 +

5

3

�

3F �D

F +D

�

� Corr

=

1

18

(9F �D); (2.72)

�

n

1

=

1

12

�

�

�

�

�

�

g

A

g

V

�

�

�

�

�

�

�

�1 +

5

3

�

3F �D

F +D

�

� Corr

=

1

18

(6F � 4D): (2.73)

At Q

2

= 5GeV

2

, the following values are obtained for Ellis-Ja�e sum rule[19℄:

�

p

1

= 0:179� 0:009;

�

n

1

= �0:019� 0:009:

These al ulated values are not onsistent with the values from the experiment

[18℄. The measured values are learly smaller than the predi tion. This disagreement

shows a polarization of the strange quarks. The violation of Ellis-Ja�e Sum Rule is

the starting point of the study of the nu leon spin stru ture (the Spin Crisis). One

of the motivations for the HERMES experiment is to solve this puzzle.

2.5 Semi-In lusive Measurement

Now we return the onsideration of the DIS pro ess again. The DIS pro ess of

lepton l on nu leon N is shown in the previous se tion as follows:

l +N ! l

0

+ h+X; (2.74)

where l

0

is the s attered lepton, h is the dete ted hadron, X are other not de-

te ted parti les. In in lusive measurements, we dete t the s attered lepton only. In

semi-in lusive measurements, we dete t produ ed hadrons h in oin iden e with the

s attered lepton l

0

. In this measurement we an extra t the quark spin distributions

of ea h avor. The diagram of semi-in lusive DIS is shown in Fig.2.7. The virtual

photon emitted by the in ident lepton is absorbed by one quark in the target and

some hadrons are produ ed. We assume that the hadron with the largest momentum

is produ ed from the fragmentation of the stru k quark in the target. This hadron

is alled \leading hadron".

There are two types of produ ed hadrons, one is the \ urrent fragment" whi h

is produ ed from the fragmentation of the stru k quark and another is the \target

fragment" whi h is produ ed from the target remnant. These two an be kinemati-

ally separated by the Feynman s aling variable x

F

. We need the urrent fragments

and therefore sele t the hadron with x

F

> 0.

The fragmentation pro esses of the stru k quark annot be al ulated exa tly,

so it is parameterized using the experimental data [20℄ [21℄, whi h are so alled frag-

mentation fun tion D

h

f

(z). These fun tions de�ne the probability that the hadron

h is with momentum fra tion z produ ed from a stru k quark with avor f .


Figure 2.7: Diagram of semi-in lusive DIS. P

h

is the momentum of a pro-

du ed hadron in the DIS pro ess. X shows other produ ed par-

ti le whi h is not identi�ed.

The semi-in lusive DIS variables

P

jj

=

!

P

h

�

!

q

j

!

q

j

longitudinal momentum of a hadron

x

F

=

P

jj

j

!

q

j

'

2P

jj

W

Feynman s aling variable

z =

P � P

h

P � q

lab

=

E

h

�

fra tional energy transfer of the virtual photon arried

by a hadron h

Table 2.2: De�nition of semi-in lusive kinemati al values.

2.5.1 Fragmentation Fun tions

In the QPM, the hadron ross se tion �

h

in semi-in lusive DIS is expressed with

fragmentation fun tions and the unpolarized quark distributions:

1

�

tot

d�

h

dz

(x;Q

2

; z) =

P

f

e

2

f

q

f

(x;Q

2

)D

h

f

(x;Q

2

; z)

P

f

e

2

f

q

f

(x;Q

2

)

; (2.75)

where �

tot

is the ross se tion of in lusive DIS, e

f

is the quark harge and q

f

are the

unpolarized PDF. The fragmentation fun tions are normalized by the multipli ity

n

h

of all produ ed hadron:

X

f

Z

1

0

D

h

f

(Q

2

; z) dz = n

h

(Q

2

); (2.76)


and energy momentum onservation leads to:

X

h

Z

1

0

zD

h

f

(Q

2

; z) dz = 1: (2.77)

The fragmentation fun tions of quarks with avor f are determined for all

hadrons. Then the n

f

� n

h

types fragmentation fun tions exist, where n

f

is the

number of the quark types and n

h

is that of the hadron types. We assume iso-spin

symmetry and harge symmetry to redu e the number of D

h

f

(x). If we use these

two assumptions, for example for pion produ tion, three types of fragmentation

fun tions are obtained as follows:

D

+

(z) � D

�

+

u

(z) = D

�

�

d

(z) = D

�

+

�

d

(z) = D

�

�

�u

(z); (2.78)

D

�

(z) � D

�

+

d

(z) = D

�

�

u

(z) = D

�

+

�u

(z) = D

�

�

�

d

(z); (2.79)

D

s

(z) � D

�

+

s

(z) = D

�

�

s

(z) = D

�

+

�s

(z) = D

�

�

�s

(z); (2.80)

where the Q

2

dependen e of the fragmentation fun tions has been omitted. Here

we onsider only three quark types (u; d; s) and � as produ ed hadron. D

+

(z) is

alled favored fragmentation fun tions and D

�

(z) is alled unfavored. In the ase

of strange quarks, we de�ne D

s

(z) separately be ause of the mass of strange quark.

In theory, some fragmentation models are established. These models are shown in

the next se tion.

2.5.2 Fragmentation Models

Fragmentation fun tions are one of the most important variables when we study

semi-in lusive DIS pro esses, but they are not al ulated in perturbative QCD. So

some phenomenologi al fragmentation models have to be used for example in Monte

Carlo programs. Three fragmentation models are des ribed below, all of whi h are

based on the QPM.

Independent Fragmentation Model

The independent fragmentation model was onsidered to des ribe the fragmentation

pro ess in 1978, whi h is one of the �rst models of fragmentation by Field and

Feynmann [22℄. This model is based on the simple assumption that every quark

fragments independently. When a quark q

0

is stru k, it ombines with other anti-

quark �q

1

and a q

0

�q

1

pair is reated whi h is the �rst meson with energy fra tion z

0

.

The energy left z

1

= (1 � z

0

) is used for another pair q

1

�q

2

. These meson reation

pro ess is iterated down to a ertain energy uto� E

min

.

One of the parameters of this model is f(1 � z)dz whi h gives the probability

that a parti ular meson is reated �rst. A further parameter is the probability

f

for the reation of a q

f

�q

f

quark pair with same avor. The isospin invarian e gives

u

+

d

+

s

= 1 and �

u

=

d

. A ording to the measured K/�-ratio the

probability of s�s reation is

s

= 0:3 .

In this model the ratio between the favored and unfavored fragmentation fun -

tions is obtained as follows:

D

�

(z)

D

+

(z)

=

(1� z)

z + (1� z)

: (2.81)


If we negle t the strange quarks ( = 0:5) this ratio is expressed as D

�

=D

+

=

(1� z)=(1 + z).

The independent fragmentation an des ribe many properties of hadron frag-

mentation, but there are some inherent problems like the violation of olor and

avor quantum number onservation. Therefore other models have been developed

and the independent fragmentation model is usually not used any longer.

LUND String Model

The LUND String model [23℄ is the most su essful model today, whi h is based on

the QCD predi tion. It is similar to the independent fragmentation model, but now

the fragmentation is not independent, but the quarks are onne ted to the olor

�eld. With growing distan e between the quarks, the olor �eld is sket hed as a

tube and is alled a string. The energy is stored in this string of olor �eld. When

the energy ex eeds the mass of a q�q-pair, a new pair is produ ed and the string is

divided in two strings. The fragmentation pro eeds until the last pair is lose to the

mass of a olorless hadron.

In this model, there are main three parameters in the fun tion whi h is alled

LUND symmetri fragmentation fun tion:

f(z) =

(1� z)

a

z

� exp

�bm

2

?

z

!

; (2.82)

where m

?

is the transverse mass squared m

2

?

= m

2

+ p

2

?

, a and b are parameters

whi h are determined from experiment.

The LUND string model is used by the JETSET [24℄ library whi h is the basis

for Monte Carlo studies for HERMES. These parameters have to be tuned for the

onditions at HERMES [25℄. The numbers are shown below (Fig.2.8 and Tab.2.3).

In the present studies, these values are used for all MC produ tions.

a b < P

2

?

>

Default Value 0.30 0.58 0.36

Fitted Value 0.82 0.24 0.34

Table 2.3: Fitted parameters of LUND model for HERMES. The parameters

of the LUND are tuned for HERMES experiment. These �tted

valued are determined to reprodu e the spe tra from all hadrons.

Cluster Model

The Cluster model des ribes the fragmentation pro esses by the QCD perturbation

theory. The highly virtual partons go into olor-singlet lusters after the intera tion

between the photon and the parton whi h is alled \pre on�nement". The lusters

de ay into hadrons from simple phase spa e arguments. The Cluster model has the

advantage that it is most theoreti al and the adjustable parameters are the QCD

s ale parameters.


0

0.5

1

1.5

2

2.5

0 0.2 0.4 0.6 0.8 1

z

dN

h/d

z

JETSET:Data (Corr.)FittedDefault

Figure 2.8: Charged hadron spe tra orre ted for dete tor in uen es. The

solid line shows the MC data with the �tted parameters in

Tab.2.3. The dashed line is the MC data with the default pa-

rameters whi h is before the tuning of MC.

This model is used in the HERWIG Monte Carlo generator [26℄. The parameters

of this model are �tted for the HERMES experiment already. However the Cluster

model fragmentation are not onsistent with the HERMES data. Sin e the LUND

model has better agreement than the Cluster model, we use the LUND model with

�tted parameters for HERMES in our study.

2.5.3 Hadron Asymmetry

In the semi-in lusive measurement, A

h

jj

is de�ned as a semi-in lusive asymmetry for

hadron h a ording to Eq.2.46:

A

h

k

=

�

h

(180

o

)� �

h

(0

o

)

�

h

(180

o

) + �

h

(0

o

)

=

N

h

"#

�N

h

""

N

h

"#

+N

h

""

; (2.83)

where N

h

"#

is the number of dete ted hadrons in the ase of the beam and target

polarizations anti-parallel and N

h

""

is for the ase of parallel polarizations.

The semi-in lusive stru ture fun tions are de�ned as:

g

h

1

(x;Q

2

; z) =

1

2

X

f

e

2

f

D

h

f

(Q

2

; z)�q

f

(x;Q

2

); (2.84)

F

h

1

(x;Q

2

; z) =

1

2

X

f

e

2

f

D

h

f

(x;Q

2

; z)q

f

(x;Q

2

); (2.85)

where the assumption that the fragmentation pro ess is spin-independent is used.


From Eq.2.51 and Eq.2.50, we obtain the semi-in lusive spin asymmetry A

h

1

(x;Q

2

)

from g

h

1

(x;Q

2

; z) and F

h

1

(x;Q

2

; z) under the assumption of g

2

= 0:

A

h

1

(x;Q

2

) �

R

dz g

h

1

(x;Q

2

; z)

R

dz F

h

1

(x;Q

2

; z)

=

1 +R(x;Q

2

)

1 +

2

�

P

f

e

2

f

�q

f

(x;Q

2

)

R

dz D

h

f

(Q

2

; z)

P

f

e

2

f

q

f

(x;Q

2

)

R

dz D

h

f

(Q

2

; z)

; (2.86)

where we have to apply the relation of Eq.2.50 at HERMES kinemati range. This

hadron spin asymmetry A

h

1

(x;Q

2

) provides us the possibility to extra t quark po-

larizations.

21

Chapter 3

HERMES Experiment

3.1 Experimental Setup

The HERMES experiment is arried out at DESY in Hamburg, Germany. The

dete tor is lo ated in the East Hall of the HERA lepton-proton ollider ring. The

HERMES uses only the polarized lepton beam of HERA, together with a polarized

internal gas target. The intera tion between a polarized target and a polarized beam

provides the possibility to study the quark spin. Details of the HERA lepton storage

ring in luding the lo ations of the other three HERA experiments, ZEUS, H1 and

HERA-B, are shown in Fig 3.1.

3.2 The HERA Storage Ring

The Fig.3.1 shows the HERA lepton (ele tron or positron) ring. The ring an

be operated with lepton at an energy of 27.5 GeV. The beam be omes polarized

transverse to the beam plane due to the asymmetry in the emission of syn hrotron

radiation (Sokolov-Ternov me hanism [27℄). The beam polarization P (t) is given as

fun tion of time t:

P (t) = p

max

(1� e

�t=�

); (3.1)

where � is the polarization build up time and p

max

is the equilibrium polarization.

At HERA the beam rea hes 60% polarization in about 30 minutes after the beam

inje tion.

Be ause it is ne essary to obtain the longitudinal beam polarization at HERMES,

two spin rotators are lo ated up and downstream of the HERMES Spe trometer.

The �rst rotates the lepton spin to the longitudinal dire tion the se ond one rotates

ba k to transverse. The beam polarization is measured by a transverse polarimeter

in the West Hall of HERA and the polarization is onstant in the whole storage

ring. The longitudinal polarization of the HERA lepton beam is shown in Fig. 3:2 .

22 CHAPTER 3. HERMES EXPERIMENT

Polarized HERA Beam:

� Self-polarization by Sokolov-Ternov e�ect

� Spin Rotators { Longitudinal Spin

� P

B

� 40� 60%

O

W

NSZEUS H1

HERA-B

HERMES

beam directiontransverse polarimeter

spin rotator 1

spin rotator 2

longitudinal polarimeter

(Parallel talk by M. Spengos, Section 11, Saturday 4:45 pm)

Figure 3.1: The HERA storage ring. HERMES are lo ated with the other

three experiments, ZEUS, H1 and HERA-B. Be ause the longi-

tudinal beam polarization is needed for HERMES, the two spin

rotator is lo ated up and down stream of the HERMES experi-

ment. In 1995 - 1997, the positron beam is used and the ele tron

beam is operated during 1998.

0

20

40

60

80

6 6.5 7 7.5 8Time [hours]

Pol

ariz

atio

n [%

]

Longitudinal Polarimeter

Figure 3.2: The longitudinal polarization of HERA beam. The polarization

rea hes up to 60 % in about 30 minutes.

3.3. THE INTERNAL GAS TARGET 23

3.3 The Internal Gas Target

HERMES uses a polarized internal gas target, applying the novel te hnique of an

internal storage ell. The target gas is inje ted in an open-ended aluminum tube

inside the lepton storage ring. The thi kness of this aluminum is about 100 �m. The

polarized gas oming from an atomi beam sour e (ABS), where the polarization is

a hieved by Stern-Gerla h separation, is inje ted into the storage ell via a small

feed tube. The leaking gas from both open ends is pumped away by a di�erential

pump system. Fig 3.3 shows the design of the target hamber.

We used a helium3 (

3

He) target in 1995, a proton (p) target in 1996 and 1997

and a deuteron (D) target from 1998 to 2000. In this system the target density is

1.2 � 10

15

nu leons/ m

2

for

3

He and 7.5 � 10

13

nu leons/ m

2

for p and D.

Open−Ended Storage Cell

gas injection

fixed collimator

e−beam

exit window

pumps

suppressors

wake field

vacuum chamber

thin walledbeam pipe

29 mm

9.8 mm

Figure 3.3: The internal target of HERMES.


3.4 The HERMES Spe trometer

HERMES spe trometer has a large a eptan e and is a forward angle dete tor. This

spe trometer is shown in Fig.3.4 and Fig.3.5 [28℄. With the HERMES dete tor,

s attering angles of parti les are a epted within � 170 mrad in the horizontal

dire tion and between +/(-)40 mrad and +/(-)140 mrad in the verti al dire tion.

A spe trometer magnet and several dete tors for parti le tra king allow the de-

termination of parti le momenta. The tra king system onsists of mi rostrip gas

hambers, drift hambers and three proportional hambers in the magnet.

For semi-in lusive measurements in DIS, HERMES needs a very good parti le

identi� ation (PID), not only to separate lepton and hadrons, but also to identify

the hadrons. To a hieve this high quality parti le identi� ation, the HERMES

spe trometer is equipped with four spe ial PID dete tors : an ele tro magneti lead-

glass alorimeter, a preshower hodos ope, a transition radiation dete tor (TRD) and

Cherenkov Counter. The original threshold Cherenkov has been repla ed in 1998 by

a Ring Imaging Cherenkov Counter (RICH) to allow a lean hadron identi� ation

over the momentum rage of 2 � 15 GeV.

1

0

2

-1

-2

0 1 2 3 4 5 6 7 8 m

mDRIFT CHAMBERS

LUMINOSITY

CHAMBERSDRIFT

VC 1/2

FC 1/2

VERTEX CHAMBERS

TARGETCELL

DVC

MC 1-3

HODOSCOPE H0

MONITOR

BC 1/2

BC 3/4 TRD CALORIMETER

TRIGGER HODOSCOPE H1

MAGNET

PROP.CHAMBERS

FIELD CLAMPS

PRESHOWER (H2)

CERENKOV

STEEL PLATE

Figure 3.4: The HERMES Spe trometer. Two identi al dete tors are set

upper and lower of the beam line. The Cherenkov Counter is

operated as a threshold Cherenkov Counter during 1995 - 1997

data taking. Sin e 1998, a Ring Imaging Cherenkov Counter are

repla ed to identify produ ed hadrons.

3.4. THE HERMES SPECTROMETER 25

3He Target

ChambersVertex

ChambersFront Chambers

Magnet MagnetChambers

Back Cerenkov TRD Hodoscopes

Calorimeter

10 m

Figure 3.5: A three dimensional CAD diagram of the HERMES Spe trome-

ter.

3.4.1 The Tra king System

Spe trometer Magnet

The HERMES spe trometer has a H-type magnet with a de e ting power of 1.5 Tm

and it is usually operated at 1.3 Tm for the momentum measurement. The lepton

beam lines are prote ted by a horizontal septum plate with ompensator oils whi h

is arranged in the beam plane. The angular a eptan e is 40 mrad < j�

y

j < 140

mrad verti ally, and j�

x

j < 170 mrad horizontally, where the lower verti al limit is

given by the septum plate. The total a eptan e of s attering angles is 40 mrad

< j�j < 220 mrad.

Vertex and Front Chambers

In upper region of the spe trometer magnet, the mi ro-strip gas hambers whi h is

alled vertex hambers VC1/2 and the drift vertex hambers DVC and the drift front

hambers FC1/2 are installed to re onstru t the parti le vertex and to determine the

s attering angle. The VCs onsisted of 2 modules ea h with 6 planes, these planes

are made by glass bases of 300 �m thi kness whi h are overed with 7 �m aluminum

strips. These strips are separated by 193 �m and arranged in 3 mm distan e from a

athode plane. The gap is �lled with DiMethylEther(DME)/Ne mixture (50/50%)

gas. The eÆ ien ies of vertex hambers were around 95% for 1997 running, and the

resolution of about 65 �m (�) per plane was obtained.

The FC's is a pair of drift hambers, ea h hamber has one module with 6 planes.


The size of drift ell is 7 mm (3.5 mm maximum drift length), and the used gas is

Ar/CO

2

/CF

4

(Ar:CO

2

:CF

4

90:5:5) like in all other HERMES drift hambers.

The DVC's were onstru ted and installed downstream of the VC, be ause of

problems with the VC's, The DVC a eptan e is verti ally � 35 mrad � � 270

mrad and horizontally � 200 mrad. But the DVC drift ell is smaller than the FC's.

The gas omposition of DVC is Ar/CO

2

/CF

4

, the same as for the FC's.

Magnet Chambers

The proportional wire hambers MC1/2/3 are mounted in the gap of the spe -

trometer magnet. These were originally installed to analyze multiple tra ks in high

multipli ity events. It has turned out that they are also parti ularly useful for the

analysis of parti les with low energy whi h are de e ted by the magnet so that they

do not ome in the a eptan e, pion from the �-de ays, for example. Ea h hamber

onsists of 3 modules. The distan e between anode and athode planes is 4�0.03

mm. The gas is the same as the drift hamber's but the proportions of the mixture

is di�erent (Ar:CO

2

:CF

4

65:30:5).

Ba k Chambers

In the region downstream of the spe trometer magnet two sets of the ba k drift

hambers BC1/2 and BC3/4 are mounted. The di�eren e between BC1/2 and

BC3/4 is the dete tor size. The a tive areas of these modules depend on their

z-position and the a eptan e of the spe trometer. The BC1/2 a tive area is hor-

izontally 1880 mm and verti ally 520 mm, and the BC3/4's is 2890 mm and 710

mm. Ea h set of hambers has two modules with 6 planes. The eÆ ien y of the

drift hamber depends on the drift distan e and the typi al eÆ ien y is well above

99%. The details are shown in [29℄. These dete tors are shown in Fig.3.6.

Figure 3.6: S hemati view of the HERMES BC module.

3.4. THE HERMES SPECTROMETER 27

3.4.2 The Parti le Identi� ation System

As I mentioned before, a lean parti le identi� ation is very important at HERMES.

The PID is performed in two steps : �rst a lepton-hadron separation and se ond the

identi� ation of the hadron. Espe ially the se ond point is essential for the study of

the quark spin distribution. As for the lepton-hadron separation, the hadron reje -

tion fa tor (HRF) of the system is designed to be 10

4

at least, where HRF is de�ned

as the total number of hadrons divided by the number of hadrons misidenti�ed as

leptons.

This value depends on the eÆ ien y for the lepton identi� ation. The ontamina-

tion of the s attered lepton sample by hadrons is below 1% for the whole kinemati

range. This hadron reje tion is done by the alorimeter, the preshower hodos ope

and the TRD. The se ond step, the hadron identi� ation is done by the fourth PID

dete tor, the RICH. These four dete tors are des ribed in the following.

Calorimeter

The alorimeter (Fig.3.7) provides a �rst level trigger for s attered leptons. It on-

sists of two modules whi h have 420 blo ks ea h above and below the beam line.

All 840 blo ks are made by lead-glass blo ks with size of 9 � 9 � 50 m

3

and are

arranged in a 42 � 10 array for ea h module. Ea h blo k is readout by a photomul-

tiplier tube (PMT). Both the alorimeter walls an be moved 50 m verti ally from

the beam pipe to prevent radiation damage during beam inje tion or dump. The

energy resolution of this dete tor is parameterized by the following fun tion:

�E

E

= 1:5� 0:5 +

5:1� 1:1

q

E[GeV ℄

; (3.2)

where E is energy of leptons measured by the alorimeter. Combined with the

preshower ounter, the eÆ ien y for the dete tion of lepton is about 95%. The

alorimeter is des ribed in detail in [30℄.

90 (10 cells)

Electron beam

9 x 9 x 50 cm 3

378 (42 cells) cm

cm 20

48 cm

cm90 (10 cells)

cm

Figure 1: Isometric view of the HERMES calorimeter.

The light transmittance was measured for di�erent F101 LG block: as the

working wavelengths of light exceed 500 nm, the spectral distribution well matches

the spectral response of the green extended phototube Philips XP3461 chosen.

3 Measured characteristics of the calorimeter

Measurements with 1�30 GeV/c

10)

electron beams have been performed

at CERN and DESY with a 3�3 array of counters with the addition of a preshower

detector, consisting of 1 cm of lead + 1 cm of plastic scintillator. This con�guration

provides information on the longitudinal development of the electromagnetic shower

and then allows a better rejection of pions. At DESY each lead glass block was

also calibrated and equalized to �1% in a 3 GeV/c electron beam positioned at the

module center.

3.1 Linearity of the response

The absolute energy calibration measurements showed a good linearity over the

whole measured energy range, as represented in �g. 2: the linear �t reproduces the

data better than 1%.

Figure 3.7: Isometri view of the HERMES Calorimeter.


Hodos opes

There are two hodos opes H1 and H2, whi h are plasti s intillators. H2 is behind

the 11 mm Pb radiator (two radiation lengths) sandwi hed between two 1.3 mm

stainless steel sheets and used as a preshower ounter in front of the alorimeter.

The �rst level trigger is obtained by a ombination with the alorimeter and H1,

where H2 gives a dis rimination between leptons and hadrons. Both the hodos opes

onsist of 42 verti al s intillator modules ea h in the upper and lower dete tor. The

size of the modules is 9.3 � 91 � 1 m

3

and they are read out by PMTs of 5.2 m

diameter.

Transition Radiation Dete tor

The transition radiation dete tor is used for the dis rimination between leptons and

hadrons [31℄. Only leptons emit transition radiation in the HERMES energy region.

The HERMES TRD is a multiwire proportional hamber onstru ted by 6 mod-

ules above and below the beam. Ea h module has polyethylene/polypropylene �bers

as radiators and Xe/CH

4

gas (90:10) in the proportional hamber. The �bers used

for the radiators are 17-20 �m in diameter and have a material density of 0.059

g/ m

3

. The radiators are 6.35 m thi k, and Xe/CH

4

has been hosen be ause of

its eÆ ient X-ray absorption.

The hambers are 2.54 m thi k and the wire diameter was used to be unusually

75 �m large to allow operation at high voltage while limiting the gas gain to about

10

4

. The original purpose of the TRD is to obtain the pion reje tion fa tor (PRF)

of at least 100 for 90% lepton eÆ ien y at 5 GeV and above, a ordingly the energy

averaged PRF is about 300 [31℄ for 90% lepton eÆ ien y. When we analyze the data

as the fun tion of momentum, the PRF of 130 is obtained at 5 GeV. The design

goal is therefore a omplished.

Ring Imaging Cherenkov Counter

To study the quark spin stru ture, it is ne essary that produ ed hadrons (pion, kaon,

proton) are identi�ed separately. Until 1998, threshold Cherenkov Counter was used

to identify pions whi h are produ ed in DIS s attering. This threshold Cherenkov

Counter had good performan e for pion separation from hadrons. The eÆ ien y of

this ounter is above 99%. In 1998 a Ring Imaging Cherenkov Counter (RICH) has

been installed as a substitute for the threshold Cherenkov Counter. The HERMES

RICH is designed to identify pions, kaons and protons in a momentum region of 2

� 15 GeV. This RICH has two radiators, aerogel and C

4

F

10

gas. Measuring two

ring images produ ed by Cherenkov photons on the PMT plane, we an identify and

separate ea h parti le. The RICH is des ribed in detail in the next hapter.

29

Chapter 4

Ring Imaging Cherenkov Counter

In its original version the HERMES spe trometer had a threshold Cherenkov Counter

to dete t produ ed hadrons, espe ially pions. Although this threshold ounter had

good performan e for the pion dete tion, a new Ring Imaging Cherenkov Counter

(RICH) was proposed in 1997 [32℄ to identify also other hadrons produ ed in DIS.

The RICH was installed in 1998 and is operating sin e then. This RICH is designed

to dete t three types of hadrons : pions (�

+

; �

�

), kaons (K

+

; K

�

) and (anti-)protons

(p; �p).

The HERMES RICH has a very spe ial property that it is working with two

radiator, sili a aerogel and C

4

F

10

. Thanks to re ent developments, sili a aerogel is

available with high transmission and low refra tion index. If a hadron passes the

RICH, it emits Cherenkov light at di�erent angles for sili a aerogel and C

4

F

10

Gas.

Therefore the Cherenkov light forms two rings on the photomultiplier plane. We

an identify the hadron by determining the Cherenkov angle of the two rings. The

details are dis ussed in the following.

4.1 Dete tor Design

In this se tion, the fundamental prin iple of the RICH is des ribed. If a harged par-

ti le is moving with velo ity v in an opti al medium with refra tive index n and it's

velo ity v ex eeds the velo ity of light in the medium =n, it emits Cherenkov light

with the Cherenkov radiation angle �. The Cherenkov radiation angle is determined

as:

v = �; (4.1)

os � =

1

n�

: (4.2)

From this relation, we an get the parti le velo ity v if we an measure the

Cherenkov radiation angle � and know the refra tive index n of the material. Sin e

the parti le momentum is measured by the spe trometer magnet, it is possible to

identify the parti le whi h passes the RICH if we an measure the Cherenkov angle

�. The refra tive index of the radiator material determines the Cherenkov angle

produ ed by the harged parti les. At HERMES the RICH had to be installed in

a predetermined pla e where the threshold Cherenkov Counter was mounted. The

30 CHAPTER 4. RING IMAGING CHERENKOV COUNTER

θParticle

Cerenkov Angle

vt

ct/n

Cerenkov Photon

Figure 4.1: The diagram of Cherenkov radiation. A harged parti le with

the velo ity v whi h ex eeds the velo ity of light in the medium

an emit Cherenkov light with the Cherenkov angle �.

whole design of RICH is shown in Fig.4.2. As radiators, sili a aerogel and C

4

F

10

gas

were hosen be ause their re e tive indi es satisfy our demand.

The Cherenkov angles from aerogel and C

4

F

10

gas are shown in Fig.4.3 as a fun -

tion of parti le momentum. If the parti le momentum is above the threshold of the

Cherenkov radiation of this parti le, it emits the Cherenkov lights and makes ring

images on the photon dete tor plane whi h is then read out by arrayed photomul-

tipliers. One of the parameters whi h determines the performan e of RICH is p

max

,

whi h is the maximum separation momentum. This is the maximum momentum

for whi h the average Cherenkov angle of two parti les is separated by standard

deviations n

�

. The p

max

parameter of RICH is shown below [33℄:

p

max

=

v

u

u

t

m

2

2

�m

2

1

2k

f

n

�

; (4.3)

where m

1

; m

2

are masses of two parti les and k

f

= tan � � �

�

=

p

N is the RICH

dete tor onstant, N is the number of dete ted photons, � is the Cherenkov angle

and �

�

is the standard deviation of the Cherenkov angle distribution. The RICH at

HERMES is designed as n

�

= 4.652.

This RICH an identify parti les from two ring images made by Cherenkov pho-

tons. Besides this it an a t also as a threshold Cherenkov Counter in a parti ular

momentum region. This momentum region is indi ated in Fig.4.4. The light shaded

region in Fig.4.4 indi ates the region where the RICH a ts as a threshold Cherenkov

Counter, and the dark shaded region indi ates the region where we need to re on-

stru t the Cherenkov angle to identify parti les.

As the re onstru tion of the Cherenkov angles is most important for PID, we use

di�erent PID algorithms and ompare the re onstru ted angles. All the algorithms

need Monte Carlo simulations of RICH. Therefore a realisti MC is important. So

we have to study the properties of the RICH in detail and must in lude these into

the MC.

In the following se tion, we des ribe the properties and the MC of the RICH.

4.1. DETECTOR DESIGN 31

aluminum box

mirror array

soft steel platePMT matrix

aerogel tiles

Particle Track

Aer

ogel

Tile

Photo

n Det

ecto

r

Mirror[R=220(cm)]

6.2 (cm)

118 (cm)

86 (cm)

55 (cm)70 (cm)

Cherenkov Photon

C4F10Gas

Figure 4.2: S hemati view of Ring Imaging Cherenkov Counter at HER-

MES. Two identi al RICH is lo ated upper and lower of the beam

lime, these �gures show the upper RICH. The bottom �gure is

the side view of RICH. The aerogel tiles are assembled ba k of

the entran e window. The C

4

F

10

gas are �lled in the RICH. A

harged parti le emits Cherenkov photons by these two radia-

tors, and the photons are re e ted by the mirror to dete t the

ring images on the PMT array. The mirror are divided by eight

segments for one RICH.


0

0.05

0.1

0.15

0.2

0.25

0 2 4 6 8 10 12 14 16 18 20p (GeV)

θ (r

ad)

Aerogel (n=1.0304)

C4F10 Gas (n=1.0014)

e

e

π

π

K

K

p

p

Figure 4.3: The relation between Cherenkov angle and parti le momentum

for ele trons, pion, kaon and protons. The solid lines des ribe

the Cherenkov angle from aerogel and the dotted lines are for

that from C

4

F

10

gas. Due to using two radiators, the parti le

identi� ation in large momentum region be omes possible.

π/K aerogel

K/p aerogel

/K gasπ

K/p gas

effective threshold average angle

GeV10 155

Figure 4.4: Momentum ranges for hadron separation. Between the dashed

lines the hadrons an be separated.

4.2. STUDY OF RICH PROPERTY 33

4.2 Study of RICH Property

4.2.1 Aerogel Radiator

As mentioned before, the RICH has two radiators whi h are sili a aerogel made

from SiO

2

(Fig.4.5 and Fig.4.6) and C

4

F

10

gas. Espe ially, sili a aerogel is used as

a radiator for a RICH for the �rst time. Seeing Fig.4.2, there is a ontainer for

aerogel whi h is next to the entran e window. The entran e window is 187.7 m

wide and 46.4 m high, and the ontainer size is 200 m wide and 60 m high. As

exit window a 3.2 mm thi k UVT-lu ite is pla ed. The ontainer is made by 1mm

thi k aluminum and is �lled with dry N

2

gas. Aerogels in the shape of small tiles

are sta ked in it. The aerogel tiles are sta ked in 5 layers, 5 horizontal rows and

17 verti al olumns for the top and the bottom part of the RICH. These assemblies

of 425 aerogel tiles in total for one RICH a ts as radiator wall with a thi kness of

about 5.5 m as shown in Fig.4.9. To ut the di�used re e tion at the sides of the

aerogel tiles, bla k plasti spa ers are mounted between the tiles.

These aerogel tiles are produ ed at Matsushita Ele tri Works [34℄ [35℄. The

typi al size of aerogel tile is 11 � 11 � 1.1 m

3

, whi h is determined by the pro-

du tion pro ess. Besides, the form of one aerogel tile is not perfe t re tangular as

shown in Fig.4.8. The refra tive index of aerogel is around 1.0304, but it has some

deviations, whi h has e�e ts on the Cherenkov angle produ ed by parti les passing

through aerogel. The transmittan e of aerogel is measured already [36℄ [37℄. From

this study, the transmittan e is determined to be above 90 %. The transmittan e

T an be parameterized as a fun tion of wavelength � and aerogel thi kness t as

follow:

T = A � exp

�

�C � t

�

4

�

; (4.4)

where C and A are alled the Hunt parameters depending on the aerogel properties.

C des ribes the amount of Rayleigh s attering and A is in luded to a ount for

surfa e re e tions due to the di�eren e in the refra tive indi es between aerogel and

gas. The �tted parameters for HERMES are C � t = 0.0094 �m

4

and A = 0.964 [36℄.

The lu ite of the aerogel exit window absorbs photons whi h have wavelengths less

than 280 nm to remove Rayleigh s attered Cherenkov photons, whi h an be ome

the sour e of ba kground. As an be seen Eq.4.4, the number of Rayleigh s attered

photons in reases as the wavelength � be omes shorter. Sin e photons with short

wavelength have to be removed. This eÆ ien y is shown in Fig.4.10.

To modify the RICH Monte Carlo, we studied the details of the main aerogel

properties : the aerogel size and the refra tive indi es of the di�erent tiles, and

the des ription of the surfa e. We measured the relations between the temperature

and the index, and the dependen e of the index on the position within one aerogel

tile, be ause good knowledge on the refra tive index is most important to identify

parti les [38℄. The main results of aerogel property measurement are following.

The Distribution of the Refra tive Index

The refra tive index of aerogel is not exa tly onstant, but di�ers slightly from tile

to tile. To use aerogel as a radiator for the RICH, there are some onditions whi h


OSiSi

O

O

O

SiSi

O (SiO )2 n

~50nm1~2nm

Figure 4.5: Composition of sili a aerogel [34℄. The aerogel is made from SiO

2

hemi ally and has highly porous stru ture. The typi al parti le

size is 1 � 2 nm.

SiSi

Si

O

OO

O

OCH CHO 2 3H

(CH ) Si N Si(CH ) H

333 3

SiSi

Si

O

OO

O

OCHO 2CH3

CH3

CH3SiCH3

NH3

(SiO )2(SiO )2

Figure 4.6: Hydrophobi pro ess for sili a aerogels. Due to this new produ -

tions te hniques, lear and hydrophobi aerogels be ame avail-

able.


Figure 4.7: Photo of aerogel tile. Sili a aerogel produ ed with the new te h-

niques has a high transparen y and transmittan e. The typi al

size is shown in Fig.4.8.

Center

Edge

Aerogel Tile

11 (cm)

11 (cm)1 (cm)

Aerogel Tile

Center

Edge

1.5 (mm)

Figure 4.8: Typi al size of one aerogel tile. The typi al size of one aerogel

tile is 11 � 11 � 1 m. The hight of a aerogel edge is higher

than the entral part be ause of surfa e tension in the pro ess

produ ed sili a aerogels.


Aerogel Tile11cm+ 11cm+ 1cm

190cm (17 Tiles)

55cm(5 Tiles)

6.2cm(5 Tiles)

x

y

z

Figure 4.9: Aerogel wall as radiator in RICH. The assembly of aerogel tiles

as Cherenkov radiator is shown for one RICH. 850 aerogel tiles

are assembled.

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 100 200 300 400 500 600 700 800λ (nm)

The

ligh

t tra

nsm

issi

on th

roug

h lu

cite

Figure 4.10: EÆ ien ies for transmittan e though lu ite. The light with the

wavelength � less than 280 nm annot be go through lu ite,

whi h is in luded in MC as a � dependen e.


should be satis�ed. One ondition is that the average of refra tive index n

mean

is

between 1.028 and 1.032, and another is that the values of all aerogel refra tive

indi es are in the region of n

mean

� 0.002. In addition to these, there is a ondition

for the standard deviation � of refra tive indi es whi h omes from the resolution

of the Cherenkov angles. The e�e t on the resolution an be estimated as :

��

�

� 0:5%: (4.5)

From this relation, the resolution of the refra tive indi es are al ulated as :

�(n� 1)

n� 1

= 2

��

�

� 1%: (4.6)

The distribution of refra tive index has to be less than 1 %. The average of refra tive

index is about 1.03, therefore the standard deviation � of the refra tive index has

to be less than 3.0 � 10

�4

[38℄ [39℄ [40℄.

To sele t the aerogel tiles whi h an be used for the RICH, we measured the

refra tive indi es of 1332 aerogel tiles [38℄. The result of this measurement is that

the standard deviation � is 4.1 � 10

�4

, whi h didn't satisfy the above ondition.

But this problem was solved by separating the sample in two groups, be ause there

are two independent RICH dete tor in the upper and lower half of the HERMES

spe trometer. Fig.4.11 shows the distribution of the refra tive index for 1040 sele ted

aerogel tiles. The refra tive index for the two groups is shown in Fig.4.12, where

the original distribution has been divided at the entral value of refra tive index.

In this ase the standard deviations are 2.5 � 10

�4

and 2.6 � 10

�4

, whi h satisfy

the expe ted onditions. This pro edure allowed us to use the aerogel as radiator

material of the RICH.


n-1=3.02 10

RMS=4.1 10

xx

-2

-2

-

0

100

200C

ou

nts

0.028 0.03 0.032 0.034

n-1

Figure 4.11: Distribution of the refra tive index of 1040 sele ted aerogels.

The horizontal axis shows the value of n - 1, where n is the

refra tive index of aerogels. The verti al axis is for the number

of aerogel tiles.

n-1=2.99 10

RMS 2.5 10

xx

-2

-4

-

0

100

200

Cou

nts

0.028 0.03 0.032 0.034

n-1

n-1=3.06 10

RMS 2.5 10

xx

-2

-2

-

0

100

200

Cou

nts

0.028 0.03 0.032 0.034

n-1

Figure 4.12: The distribution of refra tive index n of aerogel whi h are di-

vided into two groups. The horizontal axis shows the value of n

- 1 as Fig.4.11. The left �gure is for the aerogel tiles with 1.0290

� n � 1.0302 and the right is for that with 1.0303� n � 1.0313.


The Geometry

The ondition on the size and the surfa e of the aerogel tiles is that the distributions

of the longitudinal and transverse sizes are within � 0.5 mm [38℄.

We measured the size of the aerogel tiles, and the distributions for all three

dimensions for the 1040 sele ted aerogel tiles are shown below in Fig.4.13.

The Des ription of Surfa e and Edge

The aerogel surfa e has to be at to obtain omplete ring images from Cherenkov

photons, whi h means that the deviation of the surfa e from at has to be less than

1.0 mm. Be ause the aerogel tile is made from sol, the edge of the tile is higher than

the enter of tile whi h was due to the surfa e tension in the produ tion pro esses.

We measured the shape of the surfa e of aerogel. We sele ted one aerogel tile to

study the surfa e. The result of this measurement is shown in Fig.4.14 together with

the �tted fun tion. From this �gure, we an see that the distan e between the edge

and the enter of the tile is about 1.5 mm. We need to tiles whi h the deviation

of the surfa e is less than 1.0 mm, although it seems that the tile is not satisfy

this ondition. But there is not so large in uen e to dete ted rings images, be ause

almost all part of aerogel surfa e is within 1.0 mm from at and only the part whi h

is distant in 5 mm from the edge has more deviation up to 1.5 mm. Beside, the

bla k plasti spa ers whi h are mounted between the tiles ut the photons whi h

pass through the edge of tile. Therefore the aerogel tiles an be satisfy our demand

and the fun tion in Fig.4.14 is used as the typi al model of the surfa e of aerogel

tile.


0

100

200

300

Cou

nts

110 112 114 116 (mm)

X dimension

Mean 114.0 mm

RMS 0.4 mm

0

100

200

300

Cou

nts

110 112 114 116 (mm)

Y dimension

Mean 114.0 mm

RMS 0.4 mm

0

100

200

Cou

nts

(mm)

Z dimension

Mean 11.3 mm

RMS 0.4 mm

8 10 12 14

Figure 4.13: Distributions of the size of the aerogel tiles. The distributions

for three dimensions are shown, the de�nition of the dimensions

are in Fig.4.9.


0 10 20 30 40 500

0.5

1

1.5

2

distance from the center (mm)

thic

kn

ess (

mm

)(rela

tive)

Figure 4.14: Des ription of the shape of aerogel surfa e. This is a measured

results for one tile and this �tted fun tions are used as a typi-

al shape of aerogel surfa e in MC. The horizontal axis shows

the distan e from the enter, this �gure show a half surfa e of

aerogel.

4.2.2 Mirrors

The mirror of of the both RICH dete tor halves is divided into eight segments whi h

are arrayed to make a part of spheri al surfa e. The size of one mirror segment is

252.4 m in the horizontal and 79.4 m in the verti al. It weights less than 13 kg.

The mirror re e tivity was already measured before the RICH was installed. That

study showed that the re e tivity is above 85 % for light in the 300 � 600 nm

range of wavelength (Fig.4.15). This re e tivity was measured for only one mirror

segment and we adapted this re e tivity to other segments. By this method, no

position dependen e of the mirror re e tivity is available. Besides, the mirror has

only a simple alignment be ause of the short time s ale of the RICH installation.

But the re e tivity and alignment of the mirror are important, be ause we need the

number of �red PMTs to identify parti les, and this number depends among others

on the re e tivity. Therefore we need to know the details of the mirror alignment

and have to he k the e�e ts on the PID.

I studied the details of the mirror re e tivity, in luding the e�e ts of the mirror

alignment and put the measured data into the MC. I dis uss my analysis of the

mirror re e tivity in Se .4.3.1.

4.2.3 Photon Dete tor Plane

The photomultiplier plane has a size of 120 m � 60 m and ontains an array of

PMTs, whi h are Philips XP1911 with a diameter of 18.6 mm (0.75 in h) . There

are 1934 PMTs in ea h RICH half in 73 olumns of alternately 26 and 27 PMTs

ea h. The minimum a tive photo- athode diameter is 15 mm.

This PMT has a broad quantum eÆ ien y urve that mat hes the Cherenkov


0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 100 200 300 400 500 600 700 800λ (nm)

Mirr

or R

efle

ctiv

ity

Figure 4.15: The wavelength dependen e of the mirror re e tivity for

Cherenkov photon. This dependen e is shown as fun tions

whi h is used in MC. Before MC modi� ation, this dependen e

is in luded for the mirror property.

light spe trum of aerogel well. It is also good for Cherenkov light from the gas

be ause the eÆ ien y extends into the UV region. The PMT elementary ell of the

array is a hexagon with a PMT is at the enter. The distan e between two adja ent

ell enters is 23.3 mm (Fig. 4.16).

The PMT photo- athodes over only 38 % of the area of the fo al plane. To

in rease the overage of PMT, a light-gathering one, made of a aluminized plasti

foil funnel was inserted onto ea h photo- athode. This PMT photo- athodes with

foil funnels then over 91 % of the surfa e. These funnels minimize the dead spa e

between the PMTs. They have a high re e tivity above � = 200 nm. The photon

olle tion eÆ ien y in reased to about 80% whi h was only 40 % before inserting

the funnels. Their e�e ts on the olle tion eÆ ien y are shown in Fig.4.17.

4.3 Monte Carlo Simulation for RICH

For parti le identi� ation, we need to re onstru t the Cherenkov angles, using the

lo ation �red PMTs, the information of the parti le tra ks and the photon hit posi-

tion on PMTs. As the PMT plane does not ompletely mat h the true mirror fo al

surfa e, the photon olle tion eÆ ien y is not 100 %. We need the omparison of

Monte Carlo with experimental results to identify the parti le type with these input

parameters. So it is very important that the MC an reprodu e the experimental

data exa tly. We an do a urate parti le identi� ation if we have a reliable MC

data set. Therefore I studied the MC simulation of the RICH and in luded ertain

additional e�e ts in the MC ode.

4.3. MONTE CARLO SIMULATION FOR RICH 43

PMT

PMT

PMT15 mm23.3 mm

photo cathode

funnel

Figure 4.16: S hemati pi ture of PMTs with funnels. The minimum a tive

photo- athode diameter is 15 mm. The funnels are atta hed to

minimize the dead spa e between the PMTs. The e�e t of these

funnels are shown in Fig.4.17.

Incident angle (rad)

Eff

icie

ncy

Direct

1-bounce

2-bounce

3-bounce

Total

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Figure 4.17: The photon olle tion eÆ ien y verses angle of in iden e.

The total eÆ ien y is marked by the solid points, the other

points show the omponents of the eÆ ien y as the number of

\boun es", whi h is re e tion from the foil surfa e. The in u-

en e of the funnels is found from the di�eren e between the total

eÆ ien y and the eÆ ien y for \dire t" dete tion.


4.3.1 Development of MC for RICH

Though the MC for the RICH had been available, it was not realisti enough be ause

the des ription of the properties of aerogel and the mirror has been mu h too simple.

For example, the refra tive index and the size of all the aerogel tiles were treated as

onstant, the surfa e of the aerogel was at and the mirror alignment was assumed to

be ideal. The Cherenkov angles of photons from aerogel and gas were not onsistent

with experimental data, the Cherenkov angle from MC being smaller than that of

the experimental data. Moreover, the number of �red PMTs from MC events was

less than that of the experimental data. The Cherenkov angle and the number of

the �red PMTs are primary values for the PID. Therefore I analyzed the di�erent

e�e ts of the RICH properties on the PID and in luded these e�e ts in the MC in

order to reprodu e the experimental data.

The properties whi h a�e t the PID are the following:

1. The aerogel properties

� The realisti size (x; y; z) and the surfa e of the aerogel tile

� The a tual refra tive index of all aerogel tiles

2. The Rayleigh s attering and the boundary orre tions

� The Rayleigh s attering whi h a�e ts the number of the �red PMTs

� The lo ation of the gap between the aerogel wall and the exit window of

the aluminum ontainer

3. The position dependen e of the mirror re e tivity

� The mirror re e tivity of ea h segment whi h ontains the orre tions

due to the mirror misalignment

In the next se tion, I des ribe the orre tion of the MC and results after the

above mentioned propertied are in luded.

The Aerogel Tile Properties

In Se .4.2.1, the properties of the aerogel tiles are des ribed. As mentioned before,

425 aerogel tiles are used for one RICH, but the size of all aerogel tiles is not exa tly

the same for all the tiles. Therefore, the measured sizes in all three dimensions

(x; y; z) of 850 aerogel tiles are in luded in the MC ode. In the same way, the

values of the refra tive indi es are in luded. As the result the Cherenkov angle

be ame dependent on the refra tive index of ea h aerogel tile. Be ause the aerogel

radiator wall is assembled by a large number of aerogel tiles, there are gaps between

the tiles. These gaps are �lled with dry N

2

gas, but there are no Cherenkov photon

produ ed when parti les go through this gaps. As they ause a de rease in the

number of �red PMTs, I in luded these boundary e�e ts in the MC ode (Fig.4.18).

After the measured sizes are in luded, a more realisti value for the surfa e of the

tiles had also to be in luded. So, the fun tion from Fig.4.14 has been implemented

in the MC ode.


Figure 4.18: The RICH in the MC. The aerogel wall as radiator is assembled

by 425 tiles, whi h are shown in the right �gure. The Cherenkov

photons produ ed by aerogel wall and gas are re e ted by the

mirror.

Before in luding these e�e ts, the standard deviation of the MC Cherenkov angle

from parti les whi h passed through the aerogel was smaller than that angle in the

experimental data.

Rayleigh S attering

The Cherenkov photon from a parti le whi h goes through aerogel is s attered by

Rayleigh s attering. The s attered photons are dete ted as ba kground events, so

the number of the �red PMTs be omes smaller. The number of s attered photons

is proportional to the thi kness of the radiator. The thi kness of one tile is de�ned

about 1.1 m in MC, however detailed measurement shows that the thi kness of

the main part of aerogel tile is 0.9 m, whi h is smaller than the edge thi kness,

as explained in Fig.4.8. Therefore the e�e tive distan e for Rayleigh s attering is

about 4.5 m. In the original MC version, the e�e tive distan e was 5.5 m whi h

redu ed the number of �red PMTs too mu h. Therefore I modi�ed this value of

the thi kness of the aerogel wall in the MC ode, and ompared the number of �red

PMTs with and without orre tion.

Position Dependen e of the Mirror Re e tivity

The Cherenkov photons are re e ted by the mirror of the RICH and they enter

the PMT arrays. If the mirror re e tivity is 100 %, all the produ ed Cherenkov

photons whi h hit the mirror an rea h the PMTs. The mirror is not perfe t in

reality, and the number of �red PMTs depends on the mirror re e tivity. Therefore

this re e tivity a�e ts on the performan e of the PID. As was mentioned before,

the wavelength dependen e of the mirror re e tivity is in luded in MC ode, whi h

was measured before the RICH installation, but this re e tivity is the result of

only one mirror segment and is applied to all the segments. This was obviously


not enough to reprodu e the experimental data. I studied the details of the mirror

re e tivity of all 16 segments by omparing MC data with the experimental data.

Then I in luded the position dependen e of the mirror re e tivity. Here, I de�ned

that the position dependen e of the mirror re e tivity in ludes all mirror e�e ts in

terms of the number of �red PMTs. Espe ially it should be noti ed that the position

dependen e of mirror re e tivity is not only all mirror e�e ts but also other e�e ts

from the unknown fa tors, whi h is eÆ ien y of funnels, light orre tion eÆ ien y

and PMT eÆ ien y. Beside the e�e ts among others are the mirror alignment of the

16 mirror segments, the slope of the tra k when a photon hits the mirror, and the

real mirror re e tivities. We de�ned that the eÆ ien y from all unknown fa tors are

the position dependen e of the mirror re e tivity to normalize for the experiment

data in this analysis. These ontribution of ea h e�e t should be analyzed more

a urately in the next step.

The above e�e ts on the PID, the aerogel properties and the Rayleigh s attering,

have an impa t on the Cherenkov angle of the photon whi h is produ ed by the

aerogel. On the other hand, this position dependen e of the mirror re e tivity is

relevant to the Cherenkov photons produ ed by both aerogel and C

4

F

10

gas. So both

numbers of �red PMTs have been he ked. In this analysis, the normalized number

of �red PMTs by photons from C

4

F

10

gas (N

gas

) are used sin e the Cherenkov angle

by photons from aerogel is larger than that from gas, therefore the diameter of the

Cherenkov ring by aerogel photons is larger than the size of one mirror segment.

While the diameter of the Cherenkov ring from gas are very small at the mirror,

whi h average diameter about is 5 m, hen e the number of �red PMTs by gas

photons are more suitable than that of by aerogel photons. The method of this

analysis is des ribed below [41℄.

1. divide the mirror surfa e into 5 m � 5 m bins

2. obtain the normalized number of �red PMTs by photons from C

4

F

10

gas (N

gas

)

at the point where a tra k hits the mirror in both experimental (N

gas

(EXP))

and MC (N

gas

(MC)) data

3. take the ratio of N

gas

(EXP) to N

gas

(MC) in ea h bin

4. al ulate the mean value of the ratio for ea h mirror segment

5. the mean value of the ratio is de�ned to be the re e tivity of this mirror

segment

In this analysis, I used the number of N

gas

to obtain the mirror re e tivity

be ause the Cherenkov angle of photons oming from C

4

F

10

gas is smaller than the

Cherenkov angle of photons from aerogel (see Fig.4.3). Then almost all Cherenkov

rings of photons from the gas are within one bin. I used N

gas

for ea h bin to

obtain the position dependen e of the mirror re e tivity. In addition to this, N

gas

is normalized as follows:

N

gas

=

N

pmt

gas

L

gas

=73

; (4.7)


where N

pmt

gas

is the number of �red PMTs from the gas before normalization, L

gas

is

the parti le tra k length through the gas and 73 is the mean value of L

gas

whi h

is obtained from the MC data. This is ne essary be ause the number of produ ed

Cherenkov photons is proportional to the parti le tra k length through the radia-

tor. The de�nition of L

gas

and the distribution of L

gas

are shown in Appendix.A

(Fig.A.1).

There are some onditions for data sele tion. First, I sele ted �

+

and �

�

parti les

with momentum of 4.5 � 8.5 GeV/ . The resulting Cherenkov angle from the gas is

then 0.03 � 0.07 rad. In addition, I used the parti le with N

gas

larger than 2. The

reason that �

+

and �

�

are sele ted is to avoid geometri al a eptan es e�e t due

to the parti le harge. Furthermore �

+

and �

�

an be most a urately identi�ed

among all the dete ted hadrons. The sele ted momentum range, 4.5 � 8.5 GeV/ ,

allows to obtain the purest �

+

and �

�

samples (see Fig.4.3). The photons from these

parti les are well distributed in a large region of the mirror. The sele tion uts of

the Cherenkov angle from the gas and N

gas

are made to assure that the dete ted

photons are really generated by a pion.

To obtain the mirror re e tivity in ea h bin, I needed the ratio between the

MC data, whi h have no photon absorption (all photon whi h hit the mirror are

re e ted, as the mirror re e tivity is 100 %) ex ept for the wavelength dependen e

of the re e tivity (Fig.4.15) and the experimental data. These ratios are all obtained

within the 5 � 5 m bins of the mirror. I determined the mean values of the ratios

in ea h mirror segment, whi h is the re e tivity of that segment in luding other

eÆ ien ies from unknown fa tors. Using this method, the results with the ratio of

N

gas

(EXP) to N

gas

(MC) at the mirror surfa e are shown in Appendix.A (Fig.A.2).

From these ratios, the mean value of the ratios for ea h mirror segment are evaluated,

as shown in Appendix.A (Fig.A.3).

I in luded these value for ea h mirror segment as the position dependen e of the

re e tivity into the MC ode to normalize the experimental data. The e�e ts of this

on the PID are shown in the next se tion (Se . 4.3.2).


4.3.2 PID EÆ ien y after Modi� ation of MC

In this se tion, the eÆ ien y for the PID and the omparison between this eÆ ien y

before and after in luding the RICH properties, whi h I dis ussed above. The \eÆ-

ien y" whi h is used in this se tion is de�ned below.

At �rst, the omparison of the Cherenkov angle between experimental data and

MC is shown. The aerogel tile properties whi h are in luded in MC a�e t the

Cherenkov angle. And the omparison of the number of �red PMTs, whi h is a�e ted

by the Rayleigh s attering and the Mirror re e tivities is dis ussed.

PID algorithm

The dete ted ring on the PMTs are not a omplete ir le be ause of the RICH

geometry. Therefore the re onstru ted parti le tra ks are needed to identify the

parti le type. At HERMES, a method whi h is alled \Inverse Ray Tra ing" (IRT)

is used for the re onstru tion of the Cherenkov angle from the tra k parameters and

the position of the PMTs. A detailed dis ussion of IRT is beyond the subje t of this

thesis. It is found in [42℄ [43℄ [44℄.

The MC data is used for the estimation of the eÆ ien y of the IRT method. To

do this estimation, it is ne essary to ompare the Cherenkov angle and the number

of �red PMTs between MC and experimental data. I'll show this omparison in the

next se tion, and it be omes obvious that in luding the RICH properties is ne essary

to reprodu e the experimental data.

Cherenkov Angle Resolution

The aerogel properties des ribed in Se .4.3.1 have been in luded step by step in

several MC produ tion to he k their e�e t on the Cherenkov angle. The MC

produ tions whi h I made are listed below.

1. Simplest simulation

This produ tion is the base for the others. In this MC, the aerogel radiator is

onsidered as one wall, whi h is not divided into tiles.

2. Rayleigh s attering

The Rayleigh s attering e�e t is added to the above.

3. Aerogel tile e�e ts

The aerogel tiles are in luded as 425 separate tiles, ea h di�erent size and

refra tive index. The surfa e of the aerogel tiles is also simulated.

4. Boundary orre tion of photon refra tion

The e�e t of the gap between aerogel tiles and the exit window is onsidered.

5. Mirror re e tivity

The position dependen e of the mirror re e tivity whi h ontain all mirror

alignments e�e ts is in luded.

The Cherenkov angles from aerogel and gas are shown Tab.4.1 for all above

MC produ tions and the experimental data. In Tab.4.1, �

�

aero

and �

�

gas

is the


type �

aero

� �

�

aero

[mrad℄ �

gas

� �

�

gas

[mrad℄

EXP 244 � 9.3 49.0 � 9.2

MC 1 235 � 6.3 54.0 � 7.6

MC 2 233 � 6.0 54.2 � 7.5

MC 3 234 � 6.5 54.0 � 7.6

MC 4 241 � 7.1 54.2 � 7.7

MC 5 241 � 7.1 54.0 � 7.5

Table 4.1: Comparison of the Cherenkov angle for ea h MC produ tion and

the experimental data. The deviation are found between the type

1 and type 3 of MC data.

standard deviation of the Cherenkov angle of photons from aerogel (gas). Be ause

the properties of the RICH are mainly dominated by the aerogel, the Cherenkov

angle from the gas was not a�e ted by these MC modi� ations. The Cherenkov

angle from aerogel be ame lose to the experimental value after in luding all e�e ts.

Fig.4.19 shows the omparison between the experimental data and the MC data.

The dark shaded histogram are for the MC data, and the light shaded histogram is

the experimental data. The MC data of the top �gure of Fig.4.19 is the produ tion

of type 1, the se ond from the top shows type 2, the third shows type 3, the forth

shows type 4 and the bottom shows type 5. We an see that the e�e t from the

boundary orre tion (type 4) has large in uen e to Cherenkov angles. The MC

produ tion an reprodu e the experimental data for the Cherenkov angle of the

photon from aerogel and gas rather well though some dis repan y remains. We an

see better agreement of the Cherenkov angle in the re ent results, whi h is shown

in Appendix.A (Fig.A.4).


0

100

200

300

400

500

600

0.2 0.225 0.25 0.275 0.3

θaero (rad)

coun

ts

0

100

200

300

400

500

0.02 0.04 0.06 0.08

θgas (rad)

coun

ts

0

100

200

300

400

500

600

0.2 0.225 0.25 0.275 0.3

θaero (rad)

coun

ts

0

100

200

300

400

500

0.02 0.04 0.06 0.08

θgas (rad)

coun

ts

0

100

200

300

400

500

600

0.2 0.225 0.25 0.275 0.3

θaero (rad)

coun

ts

0

100

200

300

400

500

0.02 0.04 0.06 0.08

θgas (rad)

coun

ts

0

100

200

300

400

500

600

0.2 0.225 0.25 0.275 0.3

θaero (rad)

coun

ts

0

100

200

300

400

500

0.02 0.04 0.06 0.08

θgas (rad)

coun

ts

0

100

200

300

400

500

600

0.2 0.225 0.25 0.275 0.3

θaero (rad)

coun

ts

0

100

200

300

400

500

0.02 0.04 0.06 0.08

θgas (rad)

coun

ts

Figure 4.19: Comparison of the Cherenkov angles between the experimental

data (light shaded histograms) and the MC data (dark shaded

histograms). The left �gure shows the Cherenkov angle from

aerogel and the right �gure shows that from the gas.


Number of Fired PMTs

As same as the Cherenkov angle, the number of �red PMTs is an essential value for

the PID. I he ked the number of �red PMTs by photons from aerogel (N

pmt

aero

) and

from gas (N

pmt

gas

) with MC and experimental data.

The used MC data are the same as in the previous se tion. The ontribution of

ea h e�e t to the number of �red PMTs is shown in Tab.4.2.

type N

pmt

aero

N

pmt

gas

EXP 11.3 11.4

MC 1 11.2 15.6

MC 2 7.0 13.2

MC 3 13.8 15.1

MC 4 16.0 15.2

MC 5 11.2 13.1

Table 4.2: Comparison of the number of �red PMTs for ea h MC produ tion

and experimental data.

Comparing the experimental data and the MC data of type 5, the MC data after

in luding all e�e ts are lose to the experimental data, espe ially the e�e t of the

mirror re e tivity (see the di�eren e between type 4 and type 5) has largest in uen e

on the number of PMTs. Fig.4.20 shows the omparison of the experimental data

and the MC data for the number of PMTs. Again, the dark shaded histograms are

the MC data. The order of �gures are same as Fig.4.19.


0

100

200

300

400

500

600

0 5 10 15 20

Naero

coun

ts

0

100

200

300

400

500

600

700

800

0 5 10 15 20

Ngas

coun

ts

0

100

200

300

400

500

600

0 5 10 15 20

Naero

coun

ts

0

100

200

300

400

500

600

700

800

0 5 10 15 20

Ngas

coun

ts

0

100

200

300

400

500

600

0 5 10 15 20

Naero

coun

ts

0

100

200

300

400

500

600

700

800

0 5 10 15 20

Ngas

coun

ts

0

100

200

300

400

500

600

0 5 10 15 20

Naero

coun

ts

0

100

200

300

400

500

600

700

800

0 5 10 15 20

Ngas

coun

ts

0

100

200

300

400

500

600

0 5 10 15 20

Naero

coun

ts

0

100

200

300

400

500

600

700

800

0 5 10 15 20

Ngas

coun

ts

Figure 4.20: Comparison the number of �red PMTs between the experi-

mental data (light shaded histograms) and the MC data (dark

shaded histograms). The left �gure shows N

pmt

aero

and the right

�gure shows N

pmt

gas


The mirror re e tivity a�e ts not only N

pmt

aero

but also N

pmt

gas

. In this respe t the

mirror re e tivity is di�erent from other e�e ts. I studied the ontribution of the

mirror re e tivity to N

pmt

aero

and N

pmt

gas

in more detail. I de�ne the \EÆ ien y", as the

ratio between the number of �red PMTs in the experiment and in the MC. Therefore,

the eÆ ien y of aerogel is de�ned as N

pmt

aero

(EXP)/N

pmt

aero

(MC) and the eÆ ien y of

gas is de�ned as N

pmt

gas

(EXP)/N

pmt

gas

(MC). These eÆ ien ies must be almost 1 if the

MC reprodu es the experimental data a urately. This value is one riterion for

the quality of the MC. Additionally, I he ked the �

x

and �

y

dependen e of the

eÆ ien ies, where �

x

and �

y

are the slopes of the tra ks at the target.

Fig.4.21 shows the eÆ ien y versus the slopes for aerogel and gas before the

MC modi� ation, where the simple MC produ tion is used for the omposition with

experimental data. In Fig.4.21, the regions for �

x

and �

y

are -0.14 < �

x

< 0.14 and

0.04 < �

y

< 0.16. It an be seen from Fig.4.21 that the ratio of the number of

�red PMTs is far from 1, about 0.4 � 0.7, and there are large u tuations in the

eÆ ien ies depending on �

x

and �

y

. This shows that some e�e ts whi h are on both

N

pmt

aero

and N

pmt

gas

are missing in MC. Therefore I ompared the experimental data and

the MC produ tion of type 5. Fig.4.22 shows this results.

Fig.4.22 shows that the mirror re e tivity has a strong e�e t to the number of

�red PMTs. The u tuations observed in the eÆ ien ies before the MC modi� ation

be ame small, and the both eÆ ien ies in reased by the MC modi� ation although

their value is only 0.7 � 0.9. Even after in luding the mirror re e tivity, there are

some di�eren e between N

pmt

aero(gas)

(EXP) and N

pmt

aero(gas)

(MC).

-0.15-0.1-0.05 0 0.05 0.1 0.15

0.040.06

0.080.1

0.120.14

0.160

0.2

0.4

0.6

0.8

1

θx (rad)

θy (rad)

Eff

icie

ncy

(ae

rog

el)

-0.15-0.1-0.05 0 0.05 0.1 0.15

0.040.06

0.080.1

0.120.14

0.160

0.10.20.30.40.50.60.70.8

θx (rad)

θy (rad)

Eff

icie

ncy

(g

as)

Figure 4.21: The �

x

and �

y

dependen e of the eÆ ien y for aerogel and gas

before MC modi� ation. The left �gure shows eÆ ien ies for

aerogel and the right �gure shows that for gas.


-0.15-0.1-0.05 0 0.05 0.1 0.15

0.040.06

0.080.1

0.120.14

0.160

0.10.20.30.40.50.60.70.80.9

θx (rad)

θy (rad)

Eff

icie

ncy

(ae

rog

el)

-0.15-0.1-0.05 0 0.05 0.1 0.15

0.040.06

0.080.1

0.120.14

0.160

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

θx (rad)

θy (rad)

Eff

icie

ncy

(g

as)

Figure 4.22: The �

x

and �

y

dependen e of the eÆ ien y for aerogel and gas

after MC modi� ation. The left �gure shows eÆ ien ies for

aerogel and the right �gure shows that for gas.

I onsidered that the ause of this is the limited a eptan e of the PM tubes.

I tried to use the Cherenkov photons from the gas to study the mirror re e tivity,

but it is impossible to get the real number of dete ted photons from N

pmt

gas

be ause

the Cherenkov ring made by the gas photons is very small (the average diameter is

about 5 m at the mirror). As it is possible that some Cherenkov gas photons hit

the same PMT, the number of �red PMTs is not dire tly proportional to the mirror

re e tivities (see Fig.4.23).

To prove this spe ulation, I studied the relation between the number of �red

PMTs and the mirror re e tivities with MC. I produ ed the MC data hanging

the mirror re e tivities of all mirror segments from 0 to 1, and made plots of the

mean value of the number of �red PMTs versus the mirror re e tivity. In these

produ tions, I used �

+

and �

�

parti les. The results of this study are shown below

in Fig.4.24. Fig.4.24 shows that N

pmt

aero

is proportional to the mirror re e tivity, but

N

pmt

gas

isn't. From these results, 2 or more photons have to hit the same PMT if the

number of photons from gas is more than 12.

From the above detailed study on the number of �red PMTs, it is ne essary that

this non-linearity has to be onsidered to obtain the a urate mirror re e tivities.

However, the a urate number of dete ted photons, whi h is not the number of �red

PMTs, annot be obtain from experimental data. Therefore this non-linearity has

to be onsidered in future improvements of the MC. In spite of the last mentioned

restri tion, the MC have be ame lose to the experimental one by in luding the

RICH properties. This is bene� ial for the analysis of the RICH data, espe ially for

the extra tion of the spin asymmetries whi h is dis ussed in the following hapter.


Figure 4.23: The Cherenkov rings of aerogel and gas on the PMT plane.

The smaller ring is made by photon from gas. Some Cherenkov

photons from gas an hit the same PMT. Therefore the mirror

re e tivity is not proportional to the number of �red PMTs.

0

1

2

3

4

5

6

7

8

9

10

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

1.94+8.07*R

(R=Reflectivity)

Mirror Reflectivity

Nae

ro

0

2

4

6

8

10

12

14

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

2.86+13.5*R

7.18+6.41*R

(R=Reflectivity)

Mirror Reflectivity

Nga

s

Figure 4.24: The mirror re e tivity versus the number of �red PMTs by pho-

ton from aerogel and gas with MC. The left �gure shows that

the re e tivity vs mean value of N

pmt

aero

and the right �gure shows

that the re e tivity vs mean value of N

pmt

gas

.

57

Chapter 5

Extra tion of Polarized Quark

Distributions

This hapter des ribes the formalism for the extra tion of polarized quark distribu-

tions, espe ially the avor asymmetry of the light sea quark : ��u � �

�

d. In this

analysis, spin asymmetries whi h are obtained from experiment and the parameter-

ization of unpolarized quark distributions (Parton Distribution Fun tion) whi h are

used for the purity generation are needed. All available in lusive and semi-in lusive

asymmetries measured by HERMES are used in this study and various types of

quark distributions are extra ted.

5.1 The Purity Method

5.1.1 De�nition of Purity

For the extra tion of quark polarizations, we need semi-in lusive spin asymmetries

A

h

1

(x) obtained from the experiment. In the limit of largeQ

2

, A

h

1

(x) an be expressed

from Eq.2.86 as follows:

A

h

1

(x;Q

2

) =

1 +R(x;Q

2

)

1 +

2

�

P

f

e

2

f

�q

f

(x;Q

2

)

R

dzD

h

f

(Q

2

; z)

P

f

e

2

f

q

f

(x;Q

2

)

R

dzD

h

f

(Q

2

; z)

; (5.1)

where the spin asymmetry is related to the polarized and the unpolarized PDF.

As several in lusive and semi-in lusive spin asymmetries are now available from

HERMES, we an extra t polarized quark distributions with help of the unpolarized

PDF and fragmentation fun tions.

Spin asymmetries from experiment are obtained for ea h x

i

bin and are on-

sidered to be independent of Q

2

. F

1

and g

1

are dependent on Q

2

. However, for

the measured spin asymmetries whi h is approximate to the ratio of g

1

and F

1

, no

signi� ant Q

2

dependen e has been observed [45℄ [46℄. Therefore the relation whi h

is used in pra ti al analysis reads:

A

h

1

(x

i

) = C

i

�

X

f

P

h

f

(x

i

)

�q

f

(x

i

)

q

f

(x

i

)

; (5.2)

58 CHAPTER 5. EXTRACTION OF POLARIZED QUARK DISTRIBUTIONS

where C

i

is de�ned as C

i

= (1+R(x

i

))=(1+

2

i

) and the kinemati variablesR(x

i

; Q

2

)

and

i

denote the mean value in the x

i

bin. P

h

f

(x

i

) is de�ned as:

P

h

f

(x

i

) =

e

2

f

q

f

(x

i

)

~

D

h

f

(x

i

; z)

P

f

e

2

f

q

f

(x

i

)

~

D

h

f

(x

i

; z)

; (5.3)

where

~

D

h

f

(x

i

; z) is the e�e tive fragmentation fun tion whi h in ludes the dete tor

a eptan e. They are the probability that one hadron h with energy fra tion z is

dete ted when the virtual photon is s attered by a quark with avor f . The values

of z

i

are determined for ea h x

i

bin. Therefore the purities depend on a single

parameter x

i

in this de�nition. This is alled the \quark avor purity" introdu ed

in [47℄. Introdu ing this purities P

h

f

(x

i

), we an express the spin asymmetry A

h

1

(x

i

)

as a linear ombination of quark polarizations. They represent the probabilities that

a hadron h is produ ed from a quark with avor f . A ording to the de�nition of

purity, the relation holds for every x

i

bin:

X

f

P

h

f

(x

i

) = 1: (5.4)

A tually, we use the equation Eq.5.2 to extra t the quark polarizations. If we

measure spin asymmetries for several types of hadrons, the quark polarizations an

be al ulated by solving the matrix equation:

A = C

i

� P Q; (5.5)

where A is the ve tor of measured spin asymmetries, Q is the ve tor of ea h quark

polarization and P is the purity matrix whi h ontains the purities of ea h quark

avor as matrix elements. They are de�ned as:

A =

0

B

�

A

h

1

1

(x

i

)

.

.

.

A

h

m

1

(x

i

)

1

C

A

; Q =

0

B

�

�q

f

1

=q

f

1

(x

i

)

.

.

.

�q

f

n

=q

f

n

(x

i

)

1

C

A

; (5.6)

P =

0

B

B

�

P

h

1

f

1

(x

i

) : : : P

h

1

f

m

(x

i

)

.

.

.

.

.

.

.

.

.

P

h

m

f

1

(x

i

) : : : P

h

m

f

n

(x

i

)

1

C

C

A

: (5.7)

Here f

i

is the quark avor : f

i

= fu; �u; d;

�

d; s; �sg , and h

i

is the hadron type:

h

n

= f�

+

; �

�

K

+

; K

�

; p; �p:::g. In present measurements of HERMES pions, Kaon

and protons an be dete ted, and we onsider these three types of hadrons in the

following. In the ideal ase, Eq.5.5 an be solved when the number of hadron types

(the number of asymmetries) m is equal to the number n of avor types. But, as we

will see later, we have to use more asymmetries than avor types be ause there are

orrelations between some asymmetries. These orrelations mean that the number

of independent asymmetries are smaller than the number of available asymmetries

m. In this analysis all available asymmetries for the extra tion of quark distributions

are used. A detailed dis ussion of the asymmetries will follow in Se .5.5.

5.1. THE PURITY METHOD 59

The formalism of quark polarization extra tion with purities an be extended to

the in lusive measurement, where we dete t the s attered lepton only. In this ase

the purity for the s attered lepton is des ribed as:

P

e

f

(x

i

) =

e

2

f

q

f

(x

i

)

P

f

e

2

f

q

f

(x

i

)

; (5.8)

be ause the fragmentation fun tion of the in lusive ele tron D

e

f

(x

i

) is equal to 1 in

every x

i

bin :

R

dzD

e

f

(x

i

; z) = 1. Therefore we an in lude the in lusive asymmetry

in this analysis.

To extra t the quark distributions we have to solve above matrix equation Eq.5.5.

With a ve tor of measured asymmetries A, �

2

an be de�ned as follows:

�

2

= (A� C

i

� P Q)

T

V

�1

A

(A� C

i

� P Q): (5.9)

�

2

is minimized if Q be omes a solution of the matrix equation Eq.5.5. In Eq.5.9,

V

A

is a ovarian e matrix whi h is introdu ed to onsider the orrelations between

the measured asymmetries in A, and is des ribed in detail in Appendix B.1 . V

A

an be expressed by orrelation oeÆ ients between asymmetries as follows:

(V

A

)

ij

= �(A

h

i

1

; A

h

j

1

) ÆA

h

i

1

ÆA

h

j

1

; (5.10)

where �(A

h

i

1

; A

h

j

1

) is a orrelation oeÆ ient between asymmetry of hadron h

i

(A

h

i

1

)

and hadron h

j

(A

h

j

1

). ÆA

h

i

1

and ÆA

h

j

1

denote the un ertainties of the asymmetries of

types i and j.

Under some assumptions, �(A

h

i

1

; A

h

j

1

) an be repla ed by the respe tive orrela-

tion oeÆ ient between the number of hadrons, whi h is des ribed in Appendix B.1.

A ording to this relation, the ovarian e matrix reads:

(V

A

)

ij

= �(N

h

i

; N

h

j

) ÆA

h

i

1

ÆA

h

j

1

; (5.11)

�(N

h

i

; N

h

j

) =

hn

h

i

n

h

j

i

q

h(n

h

i

)

2

ih(n

h

j

)

2

i

; (5.12)

where hn

h

i

i is an averaged parti le multipli ity for hadron h

i

, whi h is the number

of hadrons h

i

in one DIS event.

The statisti al errors on the extra ted quark distributions Q is expressed by

ovarian e matrix of Q. This is shown below:

V

Q

= C

�1

i

(P

T

V

�1

A

P)

�1

: (5.13)

With this ovarian e matrix, the statisti al errors on quark distributions are

found to be :

Æ

�q

f

i

q

f

i

!

=

q

(V

Q

)

ii

: (5.14)

The quark distributions are extra ted with this method whi h I have mentioned

above. To do this analysis, the Singular Value De omposition (SVD) method [48℄


is used, whi h is the method for solving most linear least-squares problems and

obtaining singular values. In this analysis the number of asymmetries is not stable

be ause I tried to use several sets of input asymmetries for extra tion of quark

distributions. In the ase where the number of input asymmetries is larger than

that of the extra ted quark distributions, SVD an solve the matrix equation Eq.5.5.

Therefore this is a powerful tool for solving this matrix. The detail of the SVD

method is des ribed in Appendix.B.2.

An a urate determination of the systemati errors has been omitted in this

study, but it will be mentioned only summarily later in Se .6.4.

5.2 Evaluation of Purities

To solve the matrix equation Eq.5.5, I need measured spin asymmetries and purity

matrix. In this analysis, asymmetries whi h have seen obtained by other studies

[49℄ [50℄ are used as input, and the details of all used asymmetries are des ribed in

Se .5.5. The purities onsist of unpolarized quark distributions and fragmentation

fun tions whi h in lude of the a eptan e the HERMES dete tors. They are pro-

du ed by MC simulation of the unpolarized DIS pro ess, whi h is made by three

steps. The �rst two steps generate the a tual DIS events and the unpolarized PDF

q(x). The fragmentation fun tions D

h

q

(x; z) whi h are independent of the a eptan e

of HERMES are ne essary as input for MC. The third step gives e�e ts from the

dete tor a eptan e. In the following, the detail of ea h step is des ribed.

In a �rst step, the PEPSI [51℄ generator, whi h is based on LEPTO [52℄, produ ed

DIS events with unpolarized PDF, based on the CTEQ4LQ [53℄ and the GRV95 [54℄

parameterizations. In the main analysis, CTEQ4LQ parameterizations are used.

The data whi h have been produ ed with GRV95 are used to estimate the systemati

errors from the purities. LEPTO sele ts the DIS kinemati s plane a ording to the

lepton-nu leon s attering ross se tion and de ides whi h quark is s attered by using

the unpolarized PDF.

In the se ond step, the JETSET generator [24℄, whi h is part of LEPTO [52℄,

pro esses hadronization and gives the �nal state of hadrons in DIS. JETSET ontains

di�erent kind of hadronization programs whi h are based on the Independent model

(Se .2.5.2), the LUND String model (Se .2.5.2) and the Cluster model (Se .2.5.2)

and there are default model parameters for ea h model. As I mentioned before in

Se .2.5.2, the LUND String model is the best to simulate hadronization nowadays,

so it has been used to produ e the �nal state hadrons in MC and the parameters for

LUND model have been tuned for HERMES (Fig.2.8 and Tab.2.3) [25℄ to reprodu e

the experimental hadron spe tra. The produ tion with other hadronization models

an be used to estimate systemati errors from fragmentation fun tions, but this

hasn't been done in this study.

In the last step, the simulation of e�e ts from the HERMES dete tors is per-

formed. A detailed model of the HERMES dete tors is available in the dete tor

simulation program, whi h is named HMC [55℄ and is based on GEANT [56℄. I

modi�ed some parts of this HMC for the RICH as mentioned in the previous se -

tions, but to simulate the response of all dete tors is time onsuming, so it takes

long to produ e DIS events. Therefore I used a simple box a eptan e model for

5.2. EVALUATION OF PURITIES 61

simulation of the dete tors, whi h uses a momentum look-up table. It in ludes the

momentum ut-o�s, an generate a urate geometry of parti le tra ks behind the

magnet, and the same uts of event sele tion whi h is used in analysis of experimen-

tal data are applied.

From this MC data, the purities are obtained from the equation below:

P

h

f

(x

i

) =

N

h

f

(x

i

)

P

f

N

h

f

(x

i

)

; (5.15)

where N

h

f

(x) is the number of hadron h whi h is produ ed when the quark in the

nu leon with avor f is s attered. The z dependen e of N

h

f

(x) is ignored by sum-

ming up yields for this variable. This relation in ludes the fragmentation fun tions

whi h ontain also the dete tor a eptan e of HERMES. This e�e tive fragmenta-

tion fun tions are:

~

D

h

f

(x

i

; z) =

N

h

f

(x

i

; z)

re

P

h

N

h

f

(x

i

; z)

gen

; (5.16)

where

P

h

N

h

f

(x; z)

gen

is the number of events whi h a quark with avor f is generated

and N

h

f

(x; z)

re

is the number of re onstru ted events with quark avor f .

A ording to this relation, I generated purities P

h

f

for a proton and a neutron

target, where the type of hadrons h is h = fh

+

; h

�

; �

+

; �

�

K

+

; K

�

g and the type

of quark avors f = fu; d; �u;

�

d; s; �sg. The hadron types of h

+

and h

�

in lude all

produ ed hadrons as f�

+

; �

�

; K

+

; K

�

; p; �p; ::::g. Similarly the in lusive purities P

e

f

are generated. The reason that I need purities for two di�erent targets proton and

neutron, is to use all available asymmetries on proton,

3

He and deuterium target.

The details of this topi are des ribed in Se .5.4. As I have mentioned before,

the unpolarized parameterization whi h is used for purity generation is CTEQ4LQ.

Fig.5.1-Fig.5.7 (Fig.5.8-Fig.5.14) show the purities for proton (neutron) target gen-

erated with CTEQ4LQ, ompared to the purities generated with GRV95.

These �gures show the probability that one hadron h is produ ed tagging a

quark with avor f in ea h x bin. For all hadrons, the high probabilities of hadron

produ tion from the valen e quarks (u; d) are shown, in parti ular a high probability

for u quarks. Beside we an see whi h hadron is sensitive to the sea quarks : �u;

�

d; s; �s.

For the ase of the proton target, �

+

is sensitive to

�

d be ause �

+

onsists of u and

�

d (Fig.5.4). Similarly, �

�

is sensitive to �u (Fig.5.5). Fig.5.6 and Fig.5.7 show that

kaons are important to study of strange quarks. K

+

, whi h ontains u and �s quarks,

has a high probability that they are produ ed from �s quarks (Fig.5.6). Espe ially,

K

�

is so sensitive to �u and s quarks than other hadrons (Fig.5.7) be ause it is made

of two types sea quarks �u and s.

For the ase of the neutron target, the purities have a high value for d quarks

than that of the proton target. Fig.5.11 - Fig.5.14 show a high probability of

�

d

quarks. As the proton target, we an see that K

�

is so sensitive to s quarks from

Fig.5.14.

Comparing the purities generated with CTEQ4LQ and GRV, there are almost

no di�eren e for the u; d; �u and

�

d quarks in all the �gures. On the other hand, the

signi� ant deviations are shown for the purities of s and �s quarks. These deviations

ome from the di�eren e of s and �s quark distributions between CTEQ4LQ and


GRV. The CTEQ4LQ parameterization is a set with a low starting value of Q

2

and

it has more information of the sea quarks than GRV. The deviations of the purities

of sea quarks show these informations. The extra tion of sea quark distributions is

one of the main purposes, hen e the purities generated with CTEQ4LQ are used in

this study.

10-3

10-2

10-1

1

GRVPu

e+ CTEQ Pde+ Ps

e+

10-2

10-1

10-3

10-2

10-1

1

Pu–e+

10-2

10-1

Pd–e+

10-2

10-1 x

Ps–e+

Figure 5.1: In lusive (e

+

) purities for proton target


10-3

10-2

10-1

1

GRVPu

h+ CTEQ Pdh+ Ps

h+

10-2

10-1

10-3

10-2

10-1

1

Pu–h+

10-2

10-1

Pd–h+

10-2

10-1 x

Ps–h+

Figure 5.2: h

+

purities for proton target

10-3

10-2

10-1

1

GRVPu

h- CTEQ Pdh- Ps

h-

10-2

10-1

10-3

10-2

10-1

1

Pu–h-

10-2

10-1

Pd–h-

10-2

10-1 x

Ps–h-

Figure 5.3: h

�



10-3

10-2

10-1

1

GRVPu

π+ CTEQ Pdπ+ Ps

π+

10-2

10-1

10-3

10-2

10-1

1

Pu–π+

10-2

10-1

Pd–π+

10-2

10-1 x

Ps–π+

Figure 5.4: �

+


10-3

10-2

10-1

1

GRVPu

π- CTEQ Pdπ- Ps

π-

10-2

10-1

10-3

10-2

10-1

1

Pu–π-

10-2

10-1

Pd–π-

10-2

10-1 x

Ps–π-

Figure 5.5: �

�



10-3

10-2

10-1

1

GRVPu

K+ CTEQ PdK+ Ps

K+

10-2

10-1

10-3

10-2

10-1

1

Pu–K+

10-2

10-1

Pd–K+

10-2

10-1 x

Ps–K+

Figure 5.6: K

+


10-3

10-2

10-1

1

GRVPu

K- CTEQ PdK- Ps

K-

10-2

10-1

10-3

10-2

10-1

1

Pu–K-

10-2

10-1

Pd–K-

10-2

10-1 x

Ps–K-

Figure 5.7: K

�



10-3

10-2

10-1

1

GRVPu

e+ CTEQ Pde+ Ps

e+

10-2

10-1

10-3

10-2

10-1

1

Pu–e+

10-2

10-1

Pd–e+

10-2

10-1 x

Ps–e+

Figure 5.8: In lusive (e

+

) purities for neutron target


10-3

10-2

10-1

1

GRVPu

h+ CTEQ Pdh+ Ps

h+

10-2

10-1

10-3

10-2

10-1

1

Pu–h+

10-2

10-1

Pd–h+

10-2

10-1 x

Ps–h+

Figure 5.9: h

+

purities for neutron target

10-3

10-2

10-1

1

GRVPu

h- CTEQ Pdh- Ps

h-

10-2

10-1

10-3

10-2

10-1

1

Pu–h-

10-2

10-1

Pd–h-

10-2

10-1 x

Ps–h-

Figure 5.10: h

�



10-3

10-2

10-1

1

GRVPu

π+ CTEQ Pdπ+ Ps

π+

10-2

10-1

10-3

10-2

10-1

1

Pu–π+

10-2

10-1

Pd–π+

10-2

10-1 x

Ps–π+

Figure 5.11: �

+


10-3

10-2

10-1

1

GRVPu

π- CTEQ Pdπ- Ps

π-

10-2

10-1

10-3

10-2

10-1

1

Pu–π-

10-2

10-1

Pd–π-

10-2

10-1 x

Ps–π-

Figure 5.12: �

�



10-3

10-2

10-1

1

GRVPu

K+ CTEQ PdK+ Ps

K+

10-2

10-1

10-3

10-2

10-1

1

Pu–K+

10-2

10-1

Pd–K+

10-2

10-1 x

Ps–K+

Figure 5.13: K

+


10-3

10-2

10-1

1

GRVPu

K- CTEQ PdK- Ps

K-

10-2

10-1

10-3

10-2

10-1

1

Pu–K-

10-2

10-1

Pd–K-

10-2

10-1 x

Ps–K-

Figure 5.14: K

�



5.3 Assumptions on Sea Quark Polarizations

As I mentioned before, I need at least six asymmetries whi h must be independent of

ea h other in order to extra t quark distributions of all avors : f = fu; d; �u;

�

d; s; �sg.

At HERMES the asymmetries from 1995 � 1998 data taking are available, but

the number of independent asymmetries is not suÆ ient to extra t all the types of

quark polarizations a urately. Therefore two kinds of assumptions on the sea quark

polarizations or distributions are used to de rease the number of extra ted quark

types and avoid this problem. The �rst type assumes to be the same polarizations

for all sea quark avors, whi h has already been used in other thesis [57℄ [49℄ [58℄

[59℄. The se ond type, whi h is introdu ed �rst time in this analysis, is based on

the Chiral Quark Soliton Model [6℄. The details are shown in Se 5.3.2.

In the following, I will des ribe these two kinds of assumptions and types of

quark polarizations or distributions whi h are extra ted in ea h ase.

5.3.1 Assumption of Polarization Symmetri Sea

At �rst, the term of \quark polarization" and \quark distribution" have to be de�ned

learly. The quark polarization is de�ned that the ratio between the unpolarized

quark distributions and the polarized quark distributions as �q

f

=q

f

(x). The quark

distribution expresses the x distribution of the polarized quarks as �q

f

(x).

The �rst assumption is that all sea quark polarizations are the same as follows:

�q

s

(x)

q

s

(x)

=

��u(x)

�u(x)

=

�

�

d(x)

�

d(x)

=

�s(x)

s(x)

=

��s(x)

�s(x)

: (5.17)

With this assumption, we an extra t two types of quark polarizations. One is the

extra tion of the valen e quark polarizations, named \valen e de omposition" and

another is the extra tion of ea h quark avor polarizations, named \ avor de om-

position". They are shown next.

Valen e De omposition

The spin asymmetry of hadron h an be de omposed into ontributions from valen e

quarks with Eq.5.17:

A

h

1

(x) = P

h

1

(x) �

�u

v

(x)

u

v

(x)

+ P

h

2

(x) �

�d

v

(x)

d

v

(x)

+ P

h

3

(x) �

�q

s

(x)

q

s

(x)

; (5.18)

where the oeÆ ient C

i

= (1+R)=(1+

2

i

) whi h is multiplied to the purity matrix

(Eq.5.5) is omitted for brevity. The purities in Eq.5.18 are de�ned as:

P

h

1

(x) = P

h

u

(x) �

u(x)� �u(x)

u(x)

; (5.19)

P

h

2

(x) = P

h

d

(x) �

d(x)�

�

d(x)

d(x)

; (5.20)

P

h

3

(x) = P

h

u

(x) �

�u(x)

u(x)

+ P

h

�u

(x) + P

h

d

(x) �

�

d(x)

d(x)

+ P

h

�

d

(x)

+ P

h

s

(x) + P

h

�s

(x): (5.21)

5.3. ASSUMPTIONS ON SEA QUARK POLARIZATIONS 71

where P

h

q

(x) (q = u; �u; d;

�

d; s; �s) are the original purities de�ned in Eq.5.3.

The extra ted quark polarization ve tor Q is as follows:

Q =

�u

v

(x)

u

v

(x)

;

�d

v

(x)

d

v

(x)

;

�q

s

(x)

q

s

(x)

!

: (5.22)

Therefore with these de�nition of purities of Eq.5.19-Eq.5.21, I obtained the

valen e quark polarizations : �u

v

=u

v

and �d

v

=d

v

.

Flavor De omposition

Similarly, the spin asymmetry an be de omposed into ontributions from di�erent

quark avors with Eq.5.17:

A

h

1

(x) = P

h

1

(x) �

�u(x) + ��u(x)

u(x) + �u(x)

+ P

h

2

(x) �

�d(x) + �

�

d(x)

d(x) +

�

d(x)

(5.23)

+ P

h

3

(x) �

�q

s

(x)

q

s

(x)

; (5.24)

where the purities are de�ned as:

P

h

1

(x) = P

h

u

(x) �

u(x) + �u(x)

u(x)

; (5.25)

P

h

2

(x) = P

h

d

(x) �

d(x) +

�

d(x)

d(x)

; (5.26)

P

h

3

(x) = �P

h

u

(x) �

�u(x)

u(x)

+ P

h

�u

(x)� P

h

d

(x) �

�

d(x)

d(x)

+ P

h

�

d

(x)

+ P

h

s

(x) + P

h

�s

(x): (5.27)

In this ase, the extra ted quark polarization ve tor Q is as follows:

Q =

�u(x) + ��u(x)

u(x) + �u(x)

;

�d(x) + �

�

d(x)

d(x) +

�

d(x)

;

��q(x)

q(x)

!

: (5.28)

With these relations, the ontributions of the quark avors u and d to the spin of

the nu leon an be extra ted.

Together with the assumption on the sea quark polarization of Eq.5.17, three

independent asymmetries are needed at least.

5.3.2 Assumption of Sea Quark from CQSM

Another new assumption is about the sea quark distributions, whi h is based on

the Chiral Quark Soliton Model (CQSM). A simple relation between the polarized

anti-quark distributions is:

��u(x) + �

�

d(x)� 2��s(x) =

3F �D

F +D

h

��u(x)��

�

d(x)

i

; (5.29)

where F and D are the weak oupling onstants in neutron and hyperon �-de ays

(Eq.2.69), and whi h is based on CQSM in the large-N

limit (N

is the number of

olors) [6℄.


One of the main purposes of this study is the extra tion of avor the asymmetry

of the light sea quark : ��u��

�

d. Therefore it is not desirable to use the assumptions

in Eq.5.17 to extra t ��u � �

�

d. I introdu ed the �rst part of Eq.5.30 as a new

assumption to substitute Eq.5.17.

So the new assumptions are as follows:

��s(x) =

1

2

�

��u(x) + �

�

d(x)�

3F �D

F +D

h

��u(x)��

�

d(x)

i

�

;

�s(x)

s(x)

=

��s(x)

�s(x)

: (5.30)

With these new assumptions, I an extra t quark distributions dire tly, espe ially

the avor asymmetry of the light sea quarks.

Flavor Asymmetry of Light Sea Quarks

With Eq.5.30, the spin asymmetry an be expressed as follows:

A

h

1

(x) = P

h

1

(x) [�u(x) + ��u(x)℄ + P

h

2

(x)

h

�d(x) + �

�

d(x)

i

+P

h

3

(x)

h

��u(x) + �

�

d(x)

i

+ P

h

4

(x)

h

��u(x)��

�

d(x)

i

;

where the purities are de�ned as:

P

h

1

(x) = X

h

u

(x); (5.31)

P

h

2

(x) = X

h

d

(x); (5.32)

P

h

3

(x) = X

h

s

(x) +

3

2

X

h

0

(x); (5.33)

P

h

4

(x) = X

h

3

(x)�

3F �D

F +D

�

X

h

s

(x) +

1

2

X

h

0

(x)�X

h

8

(x)

�

; (5.34)

(5.35)

where oeÆ ients of X

h

(x) are shown below:

X

h

u

(x) =

P

h

u

(x)

u(x)

; (5.36)

X

h

d

(x) =

P

h

d

(x)

d(x)

; (5.37)

X

h

0

(x) =

1

3

(

�

P

h

u

(x)

u(x)

+

P

h

�u

(x)

�u(x)

�

P

h

d

(x)

d(x)

+

P

h

�

d

(x)

�

d(x)

�

P

h

s

(x)

s(x)

+

P

h

�s

(x)

�s(x)

)

;(5.38)

X

h

3

(x) =

1

2

(

�

P

h

u

(x)

u(x)

+

P

h

�u

(x)

�u(x)

+

P

h

d

(x)

d(x)

�

P

h

�

d

(x)

�

d(x)

)

; (5.39)

X

h

8

(x) =

1

6

�

�

P

h

u

(x)

u(x)

+

P

h

�u

(x)

�u(x)

�

P

h

d

(x)

d(x)

+

P

h

�

d

(x)

�

d(x)

+ 2

P

h

s

(x)

s(x)

�

P

h

�s

(x)

�s(x)

!

�

: (5.40)

The ratio of F and D are obtained from [60℄ : F=D = 0:492� 0:083. The P

h

(x) in

the de�nitions of X

h

(x) above are the purities that are de�ned originally in Eq.5.3.


In this analysis, the elements of the purity matrix P is a ombination of the original

purities (Eq.5.3) and the unpolarized PDF.

Here the extra ted quark ve tor Q is as follows:

Q =

�

�u(x) + ��u(x); �d(x) + �

�

d(x); ��u(x) + �

�

d(x); ��u(x)��

�

d(x)

�

:(5.41)

One of the hara teristi of this analysis with the new assumptions is that the

quark distributions an be extra ted dire tly, not only the polarization whi h is

the ratio between the polarized and the unpolarized distributions. Additionally,

using the results of the avor asymmetry extra tion Eq.5.41, it is possible to extra t

almost all avor quark distributions : �u;�d;��u;�

�

d;�s = ��s . This analysis is

des ribed in the next.

It has to be mentioned that due to the ombination with the unpolarized PDF,

whi h are Q

2

dependent, also the matrix P and the quark ve tor Q be ome Q

2

dependent in this avor asymmetry extra tion. As the measured asymmetries are

onsidered to be in the �rst order Q

2

independent, the PDF in the ombination of

the original purities (Eq.5.36 - Eq.5.40) at a �xed Q

2

of Q

2

=2.5 GeV

2

has been

used to get the Q ve tor also at this �xed Q

2

. The Q

2

dependen e of the purities

whi h are used to extra t the avor asymmetry (Eq.5.31 - Eq.5.34) are shown in

Fig.5.15 and Fig.5.16. The purities are al ulated at �xed value Q

2

of Q

2

=1.2, 2.5

and 10 GeV

2

. The solid ir le shows the purities extra ted using the mean value

of Q

2

in ea h x bin. At small x, there are no Q

2

dependen e for all ombination

purities P

h

1

� P

h

4

. The small deviation an be observed in the high x bins, although

these deviations don't have in uen e on the extra ted quark distributions in this

HERMES pre ision.

Single Quark Distribution

With results of the above analysis, as has been suggested, I an obtain the quark

distributions of almost all avors, where the distribution of �s(x) and ��s(x) are

de�ned to be the same.

First, I extra t �u(x);��u(x);�d(x);�

�

d(x) from the results of Eq.5.41. With

��u+�

�

d and ��u��

�

d of Eq.5.41 and the �rst relation of Eq.5.30 the strange quark

distributions an be extra ted : ��s(x) = �s(x). With this method, the quark

distributions of ea h other avor an be obtained.


-1

-0.75

-0.5

-0.25

0

0.25

0.5

0.75

1Pe+

P1 ∆(u+u–)

Q2 dep

Q2=1.2Q2=2.5Q2=10

P2 ∆(d+d–) P3 ∆(u

–+d

–) P4 ∆(u

–-d

–)

-1

-0.75

-0.5

-0.25

0

0.25

0.5

0.75

1Ph

+

10-2

10-1

-1

-0.75

-0.5

-0.25

0

0.25

0.5

0.75

1Ph

-

x10

-210

-1

x10

-210

-1

x10

-210

-1

x

Figure 5.15: The Q

2

dependen e of the purities for avor asymmetry extra -

tion for the proton target. These purities are ombination of the

original purities (Eq.5.3) and unpolarized PDF, whi h is used

CTEQ4LQ.


-1

-0.75

-0.5

-0.25

0

0.25

0.5

0.75

1Pe+

P1 ∆(u+u–)

Q2 dep

Q2=1.2Q2=2.5Q2=10

P2 ∆(d+d–) P3 ∆(u

–+d

–) P4 ∆(u

–-d

–)

-1

-0.75

-0.5

-0.25

0

0.25

0.5

0.75

1Ph

+

10-2

10-1

-1

-0.75

-0.5

-0.25

0

0.25

0.5

0.75

1Ph

-

x10

-210

-1

x10

-210

-1

x10

-210

-1

x

Figure 5.16: The Q

2

dependen e of the purities for avor asymmetry extra -

tion for the neutron target. These purities are ombination of

the original purities (Eq.5.3) and unpolarized PDF, whi h is

used CTEQ4LQ.


5.4 Interpretation of Asymmetries on the

3

He and

D Targets

At HERMES, proton (p), helium3 (

3

He) and deuteron (D) have been used as targets.

The quark distributions are de�ned on the proton and on the neutron are related

by isospin symmetry : u

p

= d

n

and �u

p

=

�

d

n

. As a polarized neutron target is not

available,

3

He and D targets are used as neutron targets. To analyze the asymmetries

on the

3

He and the D targets, we have to translate these experimental asymmetries

into the asymmetries on the neutron target. In this study, two methods are used to

translate the measured asymmetries on the

3

He and the D target, whi h are based

on the same relation and are identi al essentially. The number of input asymmetries

whi h an be used in ea h method is di�erent. The details are des ribed in the

following.

5.4.1 Asymmetry on the

3

He Target

The

3

He onsists of two protons and one neutron, If the protons have anti-parallel

spins so that the total proton spin is zero (S-state), whi h is the most probable

on�guration, the spin of

3

He is entirely arried by the neutron. But the e�e tive

spin of the neutron in

3

He seen in the measurement is diluted by the two protons

be ause of the angular momentum in the

3

He. So for a ompletely polarized

3

He,

the e�e tive polarizations of the protons and the neutron have to be onsidered.

The e�e tive polarization of the proton is p

p

3

He

= - 0.028 and that of the neutron is

p

n

3

He

= 0.86 [61℄.

With the value of these e�e tive polarizations, the measured asymmetry on the

3

He target are expressed by the asymmetry on the proton and the neutron in every

x bin as:

A

h

1;

3

He

= f

h

p;

3

He

p

p

3

He

A

h

1;p

+ f

h

n;

3

He

p

n

3

He

A

h

1;n

; (5.42)

where f

h

p;

3

He

and f

h

n;

3

He

are the dilution fa tors whi h give the probability of s attering

on one of the two protons or the neutron. From this de�nition, f

h

p;

3

He

+ f

h

n;

3

He

= 1 is

obvious. These fa tors an be expressed as follows:

f

h

p;

3

He

=

2�

h

p

�

h

3

He

= 2

n

h

p

n

h

3

He

�

�

p

�

3

He

= 2

n

h

p

n

h

3

He

�

F

p

2

2F

D

2

+ F

p

2

; (5.43)

where �

h

p

and �

h

3

He

are the integrated unpolarized ross se tions of the hadron h on

the proton target and on the

3

He target, respe tively. �

p

and �

3

He

are the in lusive

ross se tions on ea h target. n

h

p

and n

h

3

He

are the multipli ities of hadron h on ea h

target. In Eq.5.43, these relations are used:

�

h

3

He

= 2�

h

p

+ �

h

n

; (5.44)

F

n

2

= 2F

D

2

� F

p

2

; (5.45)

5.4. INTERPRETATION OF ASYMMETRIES ON THE

3

HE AND D

TARGETS 77

where �

h

n

is the integrated unpolarized ross se tion of the hadron h on the neutron

target.

With the relation of Eq.5.42 and the oeÆ ients from Eq.5.43, we an trans-

late the asymmetry on the

3

He target to a ombination of that on the proton and

the neutron. In the followings, I will des ribe the pra ti al way of using the

3

He

asymmetries as input asymmetries with Eq.5.42.

5.4.2 Asymmetry on the D Target

In ase of the D target, we an onsider the translation of the asymmetry on the D

target similarly to the ase of the

3

He target. When the deuteron is in the S-state, the

spins of the proton and the neutron are aligned parallel with no angular momentum.

But the proton and the neutron polarization are diluted by the existen e of D-state

in whi h one of the nu leons is anti-aligned. This e�e tive polarizations of the proton

and the neutron are p

p

D

= p

n

D

= 0.925

1

[62℄ [63℄. This e�e t has to be onsidered in

the treatment of the asymmetry on the D target. Here f

h

p;D

+ f

h

n;D

= 1 is also valid.

In this ase, the asymmetry is expressed by the asymmetry on the proton and the

neutron as:

A

h

1;D

= f

h

p;D

p

p

D

A

h

1;p

+ f

h

n;D

p

n

D

A

h

1;n

; (5.46)

and the dilution fa tor f

h

p;D

an be de�ned as:

f

h

p;D

=

�

h

p

�

h

D

=

n

h

p

n

h

D

�

�

p

�

D

=

n

h

p

n

h

D

�

F

p

2

2F

D

2

; (5.47)

where �

h

D

is the integrated unpolarized ross se tion of the hadron h on the D target

and �

D

is the in lusive ross se tion. n

h

D

is the multipli ity of hadron h on the D

target. Similarly, Eq.5.45 and this relation are used:

�

h

D

= �

h

p

+ �

h

n

; (5.48)

whi h is the substitute for Eq.5.44.

These relations are used in the next se tion.

5.4.3 Method to use Asymmetry on

3

He and D

In the previous se tions, the expressions of the asymmetry on the

3

He and the D

target are shown. Here I des ribe how to apply these relations to extra t the quark

distributions.

When I use two types of asymmetries on the proton and the neutron target as

input of Eq.5.5, the matrix equation is expanded as follows:

�

A

p

A

n

�

= C

i

�

�

P

p

P

n

�

�Q; (5.49)

1

They are almost equal the D-State probability, !

D

=0.058 � 0.01.


whereA

p

andA

n

are asymmetries on the proton and the neutron target respe tively,

and P

p

and P

n

are purity matri es on the respe tive target. Sin e the asymmetries

on a free neutron target are not available experimentally, as mentioned before, asym-

metries on the

3

He and the D target should be used, whi h are substitutes for that

on the neutron with the relations of Eq.5.42 and Eq.5.46. Be ause the MC gen-

erator on the

3

He and the D target annot reprodu e the experimental data, the

asymmetries have to be al ulated for that on the neutron.

There are two methods to apply the above relations to Eq.5.49. The one is

the method to use the mixing matrix whi h is shown in the following [57℄ [49℄ [58℄

[59℄. Another is the new method, whi h is introdu ed in this study, making new

de�nitions of the purity for the

3

He and the D target. Both methods are based

on the same relation, however, the kind of asymmetries whi h an be used in ea h

method is di�erent. With the mixing matrix method, the in lusive, h

+

and h

�

asymmetries on all target (p,

3

He and D) an be used. On the other hand, with

the new de�nition of purity, �

+

, �

�

, K

+

and K

�

asymmetries on the D target

also an be in luded to extra t quark distributions. Therefore, we an use up to

13 types of asymmetries with the new de�nition of purity. Although there is a

ondition to use the new de�nition of purity, whi h is des ribed later. To study

of quark distributions, in parti ular sea quarks, the pion and kaon asymmetries

are onsidered to be important, be ause they ontain sea quarks. Espe ially K

�

onsists of two sea quarks, �u and s, so it is seems to be sensitive to sea quark

distributions. Therefore, we sele t the method with the new de�nition of purity to

al ulate moment �nally, but both results are shown as follows. The details of both

methods are des ribed in the following se tions.

The Mixing Matrix

From Eq.5.42 and Eq.5.46, the asymmetries on the neutron target are obtained from

the measured asymmetries as follows:

� the ase of the

3

He target

A

h

1;n

=

1

f

h

n;

3

He

p

n

3

He

�

�

A

h

1;

3

He

� f

h

p;

3

He

p

p

3

He

A

h

1;p

�

=

1

p

n

3

He

(1� f

h

p;

3

He

)

�

�

A

h

1;

3

He

� f

h

p;

3

He

p

p

3

He

A

h

1;p

�

: (5.50)

� the ase of the D target

A

h

1;n

=

1

f

h

n;D

p

n

D

�

�

A

h

1;D

� f

h

p;D

p

p

D

A

h

1;p

�

=

1

p

n

D

(1� f

h

p;D

)

�

�

A

h

1;D

� f

h

p;D

p

p

D

A

h

1;p

�

: (5.51)

If I use Eq.5.50 and Eq.5.51 in the matrix equation of Eq.5.49, the following

expansion is a hieved:

0

B

�

A

p

A

3

He

A

D

1

C

A

= C

i

� N �

0

B

�

P

p

P

n

P

n

1

C

A

�Q; (5.52)


3

HE AND D

TARGETS 79

where A

3

He

and A

D

are the asymmetries on the

3

He and the D target, and N is

alled \the mixing matrix" whi h is introdu ed to al ulate A

3

He

and A

D

to A

n

with Eq.5.50 and 5.51.

The elements of the mixing matrix N are shown below:

N =

0

B

B

B

B

B

B

B

B

B

B

B

B

B

B

B

B

B

�

1

.

.

.

1 0

a

h

1

3

He

b

h

1

3

He

.

.

.

.

.

.

a

h

m

3

He

b

h

m

3

He

a

h

1

D

b

h

1

D

.

.

.

0

.

.

.

a

h

m

D

b

h

m

D

1

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

A

(5.53)

where

a

h

i

3

He

� f

h

i

p;

3

He

p

p

3

He

; b

h

i

3

He

� (1� f

h

i

p;

3

He

) p

n

3

He

; (5.54)

a

h

i

D

� f

h

i

p;D

p

p

D

; b

h

i

D

� (1� f

h

i

p;D

) p

n

D

; (5.55)

where h

i

is the hadron type of asymmetries. With the mixing matrixN , asymmetries

on the

3

He and the D target an be in luded in this analysis, although there is a

ondition to use this method. This method is allowed only if the orresponding

asymmetries on the proton target exists for all used asymmetries on the helium3

and deuteron target. Now the available data on the proton target are the in lusive

and the harged hadron(h

+

, h

�

) asymmetries, and there is no data for identi�ed

pions, kaons and protons, sin e the RICH had not yet been installed during the data

taking time in the proton target. However, sin e the �, K and p asymmetries are

onsidered to be sensitive to the sea quark polarization, the in lusion of deuteron

data, where �, K and p are identi�ed with RICH, are quite important. Be ause

these pion and kaon asymmetries annot be used in this mixing matrix method, I

onsidered another method to use these additional asymmetries. This new method

is des ribed in the next part.

The New De�nition of Purity for the

3

He and the D Target

Here a new de�nition of the purities on the

3

He and the D target is introdu ed to use

the purities for the proton and the neutron target. As the mixing matrix method,

the Eq.5.42 and Eq.5.46 are based on this new method and expressed as follows:

� the ase of the

3

He target

A

h

1;

3

He

= f

h

p;

3

He

p

p

3

He

A

h

1;p

+ f

h

n;

3

He

p

n

3

He

A

h

1;n

= a

h

3

He

A

h

1;p

+ b

h

3

He

A

h

1;n

= a

h

3

He

X

f

P

h

f;p

�q

f

q

f

+ b

h

3

He

X

f

P

h

f;n

�q

f

q

f

=

X

f

�

a

h

3

He

� P

h

f;p

+ b

h

3

He

� P

h

f;n

�

�q

f

q

f


=

X

f

P

h

f;

3

He

�q

f

q

f

; (5.56)

where P

h

f;p

, P

h

f;n

and P

h

f;

3

He

are purities of parti le h for the proton, the neutron

and the helium3 target respe tively. To obtain Eq.5.56, the purities for the

helium3 target are de�ned as:

P

h

f;

3

He

�

�

a

h

3

He

� P

h

f;p

+ b

h

3

He

� P

h

f;n

�

: (5.57)

� the ase of the D target is similarly to the ase of the

3

He target

A

h

1;D

= f

h

p;D

p

p

D

A

h

1;p

+ f

h

n;D

p

n

D

A

h

1;n

= a

h

D

A

h

1;p

+ b

h

D

A

h

1;n

= a

h

D

X

f

P

h

f;p

�q

f

q

f

+ b

h

D

X

f

P

h

f;n

�q

f

q

f

=

X

f

�

a

h

D

� P

h

f;p

+ b

h

D

� P

h

f;n

�

�q

f

q

f

=

X

f

P

h

f;D

�q

f

q

f

; (5.58)

where P

h

f;D

are the purities for the deuteron target, and they are de�ned as:

P

h

f;D

�

�

a

h

D

� P

h

f;p

+ b

h

D

� P

h

f;n

�

: (5.59)

With the Eq.5.56 and Eq.5.58, all available asymmetries in luding pion and

kaon asymmetries an be used. We need to know the multipli ities of parti le h

on all targets to estimate the new rede�ned purities on the

3

He and the D target.

However, sin e there are no pion and kaon data for the proton target, we assume on

the multipli ities on the proton as follows:

n

h

p

n

h

3

He

=

n

h

p

n

h

D

= 1: (5.60)

Using Eq.5.60, the oeÆ ients are hanged like below with Eq.5.43 and Eq.5.47:

a

h

3

He

=

2F

p

2

2F

D

2

+ F

p

2

� p

p

3

He

; (5.61)

b

h

3

He

=

1�

2F

p

2

2F

D

2

+ F

p

2

!

� p

n

3

He

; (5.62)

a

h

D

=

F

p

2

2F

D

2

� p

p

D

; (5.63)

b

h

D

=

1�

F

p

2

2F

D

2

!

� p

n

D

: (5.64)

These above oeÆ ients don't depend on the experimental variables, but they

depend on the parameterizations of the stru ture fun tions : F

p

2

and F

d

2

. There is


3

HE AND D

TARGETS 81

no fundamental di�eren e from the original method with the mixing matrix, but

this method has more exibility. In this study, the stru ture fun tions are obtained

from [64℄.

Fig.5.17 shows the multipli ity ratios for the proton and the deuteron targets.

The ratios are lose the one, whi h justi�es the assumptions in Eq.5.60. As a

omparison of the result with this method and that with the mixing matrix method

shows, the deviations of the ratios from one an be negle ted. In addition, with the

preliminary data of the pion and kaon multipli ities on the proton target, whi h are

obtained from the unpolarized data [65℄, the results with the assumption Eq.5.60

are ompared with that using real multipli ities. The deviation between two results

are not observed and they are almost identi al in the present statisti s.

With the assumption Eq.5.60, the new de�nition of the purity for the

3

He and

the D target is available for all hadron asymmetries. Now the asymmetries of pion

and kaon on the deuteron target whi h have been identi�ed by the RICH an be

used to extra t quark polarizations. The results of the analysis performed with

all available asymmetries on all the targets for both assumptions on the sea quark

distributions (Eq.5.17 and Eq.5.30) are shown in Se .6.1.

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

nph+ / nD

h+

Rat

io o

f h

+ Mul

tiplic

ity

10-2

10-1

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

nph- / nD

h-

x

Rat

io o

f h

- Mu

ltip

licit

y

Figure 5.17: The ratio of the multipli ities on the proton and the deuteron

target. The top �gure shows the ratio of h

+

multipli ities and

the bottom that of h

�

. The data on the proton target are from

1997 [66℄ and on the D target from 1998 [67℄.


5.5 The Set of Analyzed Asymmetries

In this se tion, all asymmetries measured by HERMES for di�erent parti les and

di�erent targets whi h have been used as an input in this analysis are presented.

5.5.1 1995 Asymmetries on the

3

He Target

In 1995, only a threshold Cherenkov Counter with a limited momentum range for the

hadron identi� ation was operated. Therefore, in lusive, h

+

and h

�

asymmetries

on the

3

He target are available. They are shown in Fig.5.18. At HERMES, the

binning in x onsists of nine bins and the de�nition of the binning in x is shown in

Appendix.C.1. Here, the values obtained from [49℄ have been used, whi h are given

in Tab.C.2.

10-2

10-1

-0.3

-0.2

-0.1

0

0.1

0.2

0.3A1

A1e+ (3He 95) A1

h+ (3He 95)

10-2

10-1

x

A1h- (3He 95)

Figure 5.18: The in lusive and semi in lusive asymmetries on the

3

He target

from 1995. The harged hadron asymmetries are shown as the

semi in lusive data. For ea h point the error bars represent the

statisti al un ertainty. The systemati un ertainty is not shown

in this �gure.

In 1995, the positron beam are used at HERA. The harged hadron asymmetries

on the

3

He target are measured for the �rst time. The in lusive asymmetry in

Fig.5.18 shows slightly negative, and the hadron asymmetries are almost zero. The

S-state, the two proton spin is zero, is dominated the wave fun tion of the

3

He

nu leus, therefore the measured asymmetries are almost aused by the neutron.

5.5. THE SET OF ANALYZED ASYMMETRIES 83

5.5.2 1996 and 1997 Asymmetries on the Proton Target

In 1996 and 1997, the threshold Cherenkov Counter was also used, So only in lusive,

h

+

and h

�

asymmetries are available on the proton target. Fig.5.5.2 and Fig.5.5.2

show the 1996 and 1997 asymmetries. These asymmetries have been also obtained

from [49℄. The numeri al values of these asymmetries are shown in Tab.C.3 and

Tab.C.4. Both asymmetries are on the proton target, although, they are treated

separately be ause their systemati un ertainty are di�erent.

10-2

10-1

-0.2

0

0.2

0.4

0.6

0.8

1A1

A1e+ (p 96) A1

h+ (p 96)

10-2

10-1

x

A1h- (p 96) 10

-210

-1

-0.2

0

0.2

0.4

0.6

0.8

1A1

A1e+ (p 97) A1

h+ (p 97)

10-2

10-1

x

A1h- (p 97)

Figure 5.19: The in lusive and semi in lusive asymmetries on the proton

target from 1996 and 1997. The harged hadron asymmetries

are shown as the semi in lusive data. For ea h point the er-

ror bars represent the statisti al un ertainty. The systemati

un ertainty is not shown in this �gure.

The in lusive asymmetries on the proton target show positive value and in rease

as x in rease. In the se ond bin from the high x, the h

+

asymmetry in 1996 shows

a strange behavior, however, it is not physi al phenomenon. This behavior doesn't

be shown in 1997, the statisti in 1997 is more than that in 1996. The same is true

for h

�

asymmetries in high x bins. The negative hadron asymmetries show smaller

positive value than that of orresponding positive hadron asymmetries. It gives a

hint that u quarks have positive and d quarks have negative polarizations.


5.5.3 1998 Asymmetries on the D Target

In 1998, the RICH was started to dete t the s attered hadrons, So in lusive, h

+

,

h

�

, �

+

, �

�

, K

+

and K

�

asymmetries are available on the D target. Fig.5.5.3 shows

these asymmetries. These asymmetries have been extra ted in [50℄. The numeri al

values of these asymmetries are shown in Tab.C.5 and Tab.C.6.

10-2

10-1

-0.2

0

0.2

0.4

0.6

0.8

1A1

A1e- (D 98) A1

h+ (D 98)

10-2

10-1

x

A1h- (D 98)

-0.2

0

0.2

0.4

0.6

0.8

1A1

A1π+ (D 98) A1

K+ (D 98)

10-2

10-1

-0.2

0

0.2

0.4

0.6

0.8

1A1

π- (D 98)

10-2

10-1

x

A1K- (D 98)

Figure 5.20: The in lusive and semi in lusive asymmetries on the D target

from 1998. The harged hadron, pion and kaon asymmetries

are shown as the semi in lusive data. For ea h point the error

bars represent the statisti al un ertainty. The systemati un er-

tainty is not shown in this �gure. The harged kaon asymmetries

are measured �rst time at HERMES.

The in lusive asymmetry on the D target shows positive and slightly in reasing

as x in rease. The positive value of the in lusive asymmetry is smaller than that on

the proton target. As the data in 1996, the positive hadron asymmetry shows a small

value in the se ond high x bin, although, it is aused by the statisti s. The positive

hadron, pion and kaon asymmetries show a slightly positive, and they with negative

harge are near to zero. The pion asymmetries are similar to that of hadrons, it

shows that the ontribution of pions to hadrons are most dominate. The proton and

anti-proton asymmetries on the D target an be measured with RICH, and they

are already obtained. Though they are not used to extra t the quark distributions,

sin e the knowledge of the fragmentation fun tion of proton and anti-proton are

poor and the protons whi h are identi�ed with RICH have a ontamination by the

proton de ayed from the lambda parti le. Additionally the proton and anti-proton

is not so large statisti s, hen e we don't use the proton and anti-proton asymmetries

in this study.

5.5. THE SET OF ANALYZED ASYMMETRIES 85

5.5.4 The Sets of Input Asymmetries

The following shows the set of asymmetries whi h has been used for ea h method

on the proton, the

3

He and the D target. In 1995-1997, HERA has been operated

with positron and in 1998 with ele tron.

The Method with the Mixing Matrix

In the mixing matrix method, the asymmetries of parti le h on the proton target

is ne essary for that on the

3

He and the D target, respe tively. As I mentioned

before, be ause only the in lusive, h

+

and h

�

asymmetries on the proton target are

available, I an use only the asymmetries of these parti le for all targets. With the

expanded matrix of Eq.5.52, the sets of input asymmetries are shown below:

A

p

=

0

B

B

B

B

B

B

B

B

B

�

A

e

+

1;p

(96)

A

h

+

1;p

(96)

A

h

�

1;p

(96)

A

e

+

1;p

(97)

A

h

+

1;p

(97)

A

h

�

1;p

(97)

1

C

C

C

C

C

C

C

C

C

A

; A

3

He

=

0

B

�

A

e

+

1;

3

He

(95)

A

h

+

1;

3

He

(95)

A

h

�

1;

3

He

(95)

1

C

A

; A

D

=

0

B

�

A

e

�

1;D

(98)

A

h

+

1;D

(98)

A

h

�

1;D

(98)

1

C

A

; (5.65)

where A

j

1;i

(k) is the asymmetry A

1

of parti le j on the target i of the year k. The

asymmetries of the year 1996 and 1997 are treated separately be ause the systemati

errors oming from the target polarization are di�erent.

The Method with the New De�nition of Purity for the

3

He and D Target

With the new de�nition of the purities for the

3

He and the D targets, I an use more

asymmetries as input be ause it is not ne essary to have the asymmetries on the

proton target of all the parti le types. This gives the following set of asymmetries:

A

p

=

0

B

B

B

B

B

B

B

B

B

�

A

e

+

1;p

(96)

A

h

+

1;p

(96)

A

h

�

1;p

(96)

A

e

+

1;p

(97)

A

h

+

1;p

(97)

A

h

�

1;p

(97)

1

C

C

C

C

C

C

C

C

C

A

; A

3

He

=

0

B

�

A

e

+

1;

3

He

(95)

A

h

+

1;

3

He

(95)

A

h

�

1;

3

He

(95)

1

C

A

; A

D

=

0

B

B

B

B

B

B

B

�

A

e

�

1;D

(98)

A

�

+

1;D

(98)

A

�

�

1;D

(98)

A

K

+

1;D

(98)

A

K

�

1;D

(98)

1

C

C

C

C

C

C

C

A

: (5.66)

The sets of asymmetries on the proton and the

3

He targets are the same as the

previous ones, but additionally I an use the asymmetries of �

+

, �

�

, K

+

and K

�

on

the D target. Here I didn't use the h

+

and h

�

asymmetries of the D target be ause

there are large orrelations between the asymmetries of the hadrons and that of

pions and kaons.

87

Chapter 6

Analysis and Results

6.1 Results for the Polarized Quark Distributions

In this se tion, the results for quark polarizations �q=q and distributions �q are

shown whi h have been obtained with the two methods des ribed in Se .5.4.3. There

are four types of extra ted quark polarizations and distributions (see Se .5.3.1 and

Se .5.3.1).

Before we extra t quark polarization from experimental data, we he ked the

extra tion method with MC data. This means that the quark polarizations extra ted

with this method should reprodu e the polarized PDF whi h is put into MC as input

data. First, the input data su h as in lusive and harged hadron asymmetries is

produ ed by using polarized PDF with MC. We use GRSV parameterization [68℄

as polarized PDF. Next, the purity matrix is produ ed by using CTEQ4LQ [53℄

or GRV [54℄ as unpolarized PDF. The quark polarization is determined from the

simulated asymmetries and purities with this method. If the method is orre t, the

obtained results from MC data should reprodu e the GRSV polarized PDF whi h

is used as input of MC.

Additionally, the omparison between the mixing matrix method and the new

de�nition of purity are shown, where in lusive and harged hadrons asymmetries on

all targets are used as input (see Eq.5.65) for both method. Comparing the both

results, the in uen e from the assumption of Eq.5.60 an be he ked, and we an see

that the assumption on the ratio of the multipli ities (Eq.5.60) does not have large

in uen e on the extra ted quark distributions. The results of this study is shown in

the following.

Here Fig.6.1 shows the results of the avor de omposition from MC. There are

two types of quark polarizations, whi h are the result with the mixing matrix and

that with the new de�nition of purity. In Fig.6.1, the solid line shows the GRSV

parameterization [68℄ whi h is used for produ tion of the MC data. We an see that

the extra ted quark distributions in Fig.6.1 reprodu e the input GRSV parameter-

izations for the both results. From the omparison between the results with the

mixing matrix method and the new de�nition of purity, it be omes lear that the

deviations between these two methods are very small. Therefore the result with the

assumption on the ratio of the multipli ities is onsistent with that without this

assumption, and almost an re e t the realisti quark distributions.

The �gure of the extra ted avor asymmetry from the MC is shown in Fig.6.2.

88 CHAPTER 6. ANALYSIS AND RESULTS

-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

(∆u

+∆u–

)/(u

+u–)

mixing matrixnew definition of purity

GRSV

-1

-0.8-0.6

-0.4

-0.2

00.2

0.4

0.6

0.81

(∆d

+∆d–

)/(d

+d–)

GRSV

10-2

10-1

-1-0.8

-0.6

-0.4

-0.20

0.2

0.4

0.60.8

1

x

∆qs/

qs

GRSV

Figure 6.1: The avor de omposition from MC. The two results with the

mixing matrix method (Se .5.4.3) and the new de�nition of pu-

rity (Se .5.4.3) are shown for omparison. q

s

is de�ned as sea

quarks : q

s

= f�u;

�

d; s; �sg.

From this results, it is shown that the reprodu tion of the input GRSV PDF is

a hieved for the quark distribution extra tion. For both the avor de omposition

and the avor asymmetry extra tion, there is no large di�eren e between the results

with the mixing matrix and the new de�nition of purity.

From Fig.6.1 and Fig.6.2, the methods whi h are des ribed in Se .5.3.1 and

Se .5.3.2 are reliable to obtain the quark distributions and polarizations. In addition,

it is lear that the systemati un ertainties whi h ome from the analysis method is

small. In the next se tion, the extra ted quark distributions from the experimental

results are shown.

6.1. RESULTS FOR THE POLARIZED QUARK DISTRIBUTIONS 89

-1-0.5

00.5

11.5

22.5

33.5

∆u+∆

u–


GRSV

-3-2.5

-2-1.5

-1-0.5

00.5

1

∆d+∆

d–

GRSV

-3

-2

-1

0

1

2

3

∆u–+∆

d–

GRSV

10-2

10-1

-3

-2

-1

0

1

2

3

x

∆u–-∆

d–

GRSV

Figure 6.2: The avor asymmetry extra tion of light sea quarks from MC.

The two results with the mixing matrix method (Se .5.4.3) and

the new de�nition of purity (Se .5.4.3) are shown for omparison.


6.1.1 The Valen e De omposition

Fig.6.3 and Fig.6.4 are the results of the valen e de omposition whi h are dis ussed

in Se .5.3.1. Fig.6.3 is obtained by solving the matrix equation Eq.5.5. Fig.6.4

shows the values times x and unpolarized PDF. The purities whi h were produ ed

from CTEQ4LQ PDF are used as input for Eq.5.5, like for all the following results.

The numeri al values of these results are listed in Tab.C.14 and Tab.C.16.

-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

∆uv/

uv


-1

-0.8-0.6

-0.4

-0.2

00.2

0.4

0.6

0.81

∆dv/

dv

10-2

10-1

-1-0.8

-0.6

-0.4

-0.20

0.2

0.4

0.60.8

1

x

∆qs/

qs

Figure 6.3: The extra ted quark polarizations in valen e de omposition from

experimental data. The solid ir les show the results with the

mixing matrix using in lusive and harged hadron asymmetries.

The open squares is that with the new de�nition of purities us-

ing in lusive, harged hadron, harged pion and harged kaon

asymmetries. In this de omposition, assumption of polarization

symmetri sea is used.


-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6x

∆uv


-0.2

-0.1

0

0.1

0.2

x ∆d

v

10-2

10-1

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

x

x ∆q

s

Figure 6.4: The quark distributions obtained from the valen e de omposition

result at Q

2

= 2.5 GeV

2

. The solid lines express the positivity

limits whi h are unpolarized PDF, where CTEQ4LQ parameter-

izations are shown. The two results with the mixing matrix and

the new de�nition of purity are obtained.

The solid ir le shows the result with the mixing matrix to treat asymmetries

on the

3

He and D target (Se .5.4.3) , and the open square shows the result with the

new de�nition of purity (Se .5.4.3). In Fig.6.4, the solid urves show the positivity

limit j�q=qj � C

�1

i

obtained from Eq.5.2, the quark polarizations don't ex eed this

limit. q

s

expresses the sea quarks whi h in lude f�u;

�

d; s; �sg.

Fig.6.3 shows that the polarization of u

v

quarks is positive over the full x region,

and there is a slight in rease with in reasing x. The polarization of d

v

quarks is


negative in the small x region, but it is not obvious in the highest x bins. In ase

of the sea quarks q

s

, the polarization is at and onsistent with zero over the whole

in x range. Be ause of the de reasing ontribution of the sea quarks in the high x

range, our data are not sensitive in this region and therefore the highest two x bins

have been omitted.

Comparing the two methods of treating the asymmetries on the

3

He and the

D target (solid ir le and open square), there are only small di�eren es, whi h are

in the same order of magnitude as the di�eren e seen in the MC simulation. It it

no large e�e t on the valen e quarks to use the asymmetries of pions and kaons

instead of general hadron asymmetries. For the polarization of the valen e quarks,

the extra ted values are stable and not sensitive to the extra tion method.


6.1.2 The Flavor De omposition

The results for the avor de omposition are presented in Fig.6.5 and Fig.6.6, where

the purities obtained from CTEQ4LQ [53℄ have been used. The numeri al values of

these results are listed in Tab.C.18 and Tab.C.20. We an obtain the ontributions

of ea h quark avor in this de omposition espe ially that of u(= u+�u) and d(= d+

�

d)

quarks. As same as the valen e de omposition results, they show a omparison of

the two extra tion methods.

-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

(∆u

+∆u–

)/(u

+u–)


-1

-0.8-0.6

-0.4

-0.2

00.2

0.4

0.6

0.81

(∆d

+∆d–

)/(d

+d–)

10-2

10-1

-1-0.8

-0.6

-0.4

-0.20

0.2

0.4

0.60.8

1

x

∆qs/

qs

Figure 6.5: The extra ted quark polarizations in avor de omposition from

experimental data. The solid ir les show the results with the

mixing matrix using in lusive and harged hadron asymmetries.

The open squares is that with the new de�nition of purities us-

ing in lusive, harged hadron, harged pion and harged kaon

asymmetries. In this de omposition, assumption of polarization

symmetri sea is used.


Looking at the results of u quark in Fig.6.5, the positive value of the u quark

polarization is shown and there is almost no di�eren e between the two methods.

The statisti errors are also small for both results, therefore it is lear that the

polarization of the u quark is positive and in reasing as x in reases.

For the d quark polarization, the two results are almost the same and there is

not obvious di�eren e. They show negative values in almost all x bins, but it is not

lear for the highest bin of x. From Fig.6.6, the d quark distribution almost ex eeds

the positivity limit in the highest bin of x.

In this ase, q

s

are de�ned as q

s

= �u +

�

d + s + �s. About the sea quarks of

q

s

, it shows zero or a slightly negative value in the small x region. The valen e

quark polarization an be extra ted without depending the extra tion method and

the type of the used asymmetries. I also al ulated the moments of �u + ��u and

�d+�

�

d. They are shown in Se .6.2.

In the valen e and the avor de omposition, the ontributions of the valen e

quarks to the nu lear spin an be extra ted separately, but that of the sea quarks

annot be obtained learly. Hen e I analyzed the sea quark distributions with the

new assumptions of Eq.5.30. The results are shown in the following se tions.


-0.2

0

0.2

0.4

0.6

x (∆

u+∆

u–)


-0.3

-0.2

-0.1

0

0.1

0.2

0.3

x (∆

d+∆

d–)

10-2

10-1

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

x

x ∆q

s

Figure 6.6: The quark distributions obtained from the avor de omposition

result at Q

2

= 2.5 GeV

2

. The solid lines express the positivity

limits whi h are unpolarized PDF, where CTEQ4LQ parameter-

izations are shown. The two results with the mixing matrix and

the new de�nition of purity are obtained.


6.1.3 Flavor Asymmetry of Light Sea Quarks

Here I show the results for the avor asymmetry extra tion of light sea quarks, whi h

are the main subje t of this study based on the new extra tion method assuming

CQSM, we obtained the polarized sea quark distribution in luding ��u � �

�

d as

shown in Fig.6.7 and Fig.6.8. It should be noti ed that the results with the new

de�nition of purity in lude pion and kaon asymmetries measured with RICH of the

�rst time at HERMES, although that of the mixing matrix only use the in lusive

and harged hadron asymmetries. These results are obtained by solving Eq.5.5,

although in this ase the purities are di�erent between the previous analysis of the

valen e and the avor de omposition. The purities whi h are used here in lude the

unpolarized PDF from CTEQ4LQ (see Eq.5.31-Eq.5.40), therefore it is possible to

extra t the polarized quark distributions dire tly, not the quark polarizations. As I

mentioned earlier, the quark distributions are extra ted at Q

2

=2.5 GeV

2

.

When we look at the distributions of the u and d quarks, �u+��u and �d+�

�

d,

of Fig.6.7, they are onsistent in two extra tion methods and there is almost no

di�eren e between the solid ir le and the open square points. The distribution of

�u + ��u is positive values in all measured region, and whi h de reases with the x

in reasing. While the result of the �d + �

�

d is negative, whi h is opposite to the

ase of u quark, and it in reases with the x in reasing. From Fig.6.8, all valen e

and sea quark distributions, multiplied x, lie within the positivity limit.

However, there are some di�eren es in the distributions of the sea quarks in

Fig.6.7 between the two extra tion methods (see the s ale of Fig.6.7 arefully, whi h

is large omparing the �gures for the valen e quarks). For the distribution of the

��u + �

�

d, there might be a deviation between the two methods in small x region.

The result with the mixing matrix, whi h is shown with the solid ir le, tells us that

this distribution is slightly negative in the middle x region but it in reases to zero

in the low x bin. The result with the new de�nition of purities whi h is indi ated

with the open square. However, it is almost negative in the middle x and does not

in rease in lowest x bin.

The avor asymmetry of light sea quark ��u � �

�

d an be extra ted dire tly in

this analysis. The distribution of ��u ��

�

d has a slight positive value. Comparing

the solid ir le and the open square, the small di�eren e is observed. In the lowest x

bin, the both result shows almost zero for the avor asymmetry, although using the

new de�nition of purity, the avor asymmetry in reases sightly in the lowest x bin.

Using the pion and kaon asymmetries whi h ontain additional information about

the sea quarks seems to have the biggest in uen e in the low x region as expe ted.

From the result with the new de�nition of purity (the open square), we an see that

��u � �

�

d is almost zero, but slightly positive. However, statisti pre ision is not

enough yet to really �nd a signi� ant di�eren e between the two approa hes.

For both the distribution of ��u+�

�

d and ��u��

�

d, it might be possible that the

asymmetries of pion and kaon ause these di�eren e of the sea quark distribution

between the two extra tion methods in the low x region. However, the more number

of asymmetries and high statisti s are ne essary to obtain a urate sea distributions,

in parti ular for pions and kaons.


-1-0.5

00.5

11.5

22.5

33.5

∆u+∆

u–mixing matrixnew definition of purity

-3-2.5

-2-1.5

-1-0.5

00.5

1

∆d+∆

d–

-3

-2

-1

0

1

2

3

∆u–+∆

d–

10-2

10-1

-3

-2

-1

0

1

2

3

x

∆u–-∆

d–

Figure 6.7: The extra ted quark distributions in luding avor asymmetry

of light sea quarks with the new assumptions based on CQSM

(Se .5.3.2) from experimental data, whi h is obtained at Q

2

=

2.5 GeV

2

. The solid ir les show the results with the mixing ma-

trix using in lusive and harged hadron asymmetries. The open

squares is that with the new de�nition of purities using in lusive,

harged hadron, harged pion and harged kaon asymmetries.


-0.2

0

0.2

0.4

0.6x

(∆u

+∆u–

)mixing matrixnew definition of purity

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

x (∆

d+∆

d–)

-0.4

-0.2

0

0.2

0.4

x (∆

u–+∆

d–)

10-2

10-1

-0.4

-0.2

0

0.2

0.4

x

x (∆

u–-∆

d–)

Figure 6.8: The quark distributions times x from the avor asymmetry ex-

tra tion, whi h is obtained at Q

2

= 2.5 GeV

2

. The solid lines

express the positivity limits whi h are unpolarized PDF, where

CTEQ4LQ parameterizations are shown. The two results with

the mixing matrix and the new de�nition of purity are obtained.


6.1.4 Single Quark Distribution

From the results of avor asymmetry extra tion, whi h is shown in the previous

se tion (Se .6.1.3), the quark distribution for ea h single quark avor an be ex-

tra ted (see Se .5.3.2). The results are shown in Fig.6.9. For the valen e quarks,

�u and �d, there is almost no di�eren e between the two extra tion methods. The

moments of the both results are onsistent, whi h is shown in Se .6.2.

It makes lear that the absolute value of �u is positive in the measured x region

and that of �d is negative. in the same x region. From Fig.6.10, the d quark distri-

bution times x ex eeds the positivity limit in the se ond highest x bin. Therefore

we an say that the se ond point of �d in Fig.6.9 is a little o� the statisti al u -

tuation. The statisti al deviation of the data in this x bin an already be seen in

the asymmetries on all targets. The systemati un ertainty needs to be evaluated

in the next step. Even with this un ertainty, the distribution of u and d quarks an

be extra ted stable.

The distribution of ��u is almost zero over the measured x region. The om-

parison of the two methods shows that there is almost no di�eren e in any x bin.

The distributions of �

�

d and ��s seem to be a little di�erent for the two extra tion

methods, espe ially there might be di�eren e in low x region, whi h shows that the

di�eren e in ��u ��

�

d and ��u + �

�

d between the methods is totally driven by the

di�eren e in �

�

d. It is also lear that the ontribution of the

�

d quark dominates the

avor asymmetry, ��u � �

�

d, sin e ��u itself is almost zero in all the measured x

region.

There is another point to mention on erning the ��s quark distribution. In this

study, it is assumed that the distributions of �s and ��s are equal. Hen e the result

for the ��s distribution also shows that of �s quarks. From the results with the new

de�nition of purity (the open square), we an see that it is slightly negative in all

x bins (negle ting the three bins of high x). The moment of �s quark distribution

shows this e�e t learly, whi h is shown in Se .6.2. Fig.6.10 shows that no data

points, within their errors, lie outside the positivity limit.

There are some previous determinations of the distribution of ��s [69℄ before our

analysis. Some results show slight negative values of ��s distribution and moments.

In the present study, it also shows the slight negative value of the x dependen e and

moments whi h are des ribed in Se .6.2. With the two methods, the mixing matrix

method and the new de�nition of purity, I obtained the result onsistent with the

previous studies. The omparison between this results and the theoreti al predi tion

are shown in the following se tion (Se .6.2.3).


-1-0.5

00.5

11.5

22.5

33.5

4∆u mixing matrixnew definition of purity

-3-2.5

-2-1.5

-1-0.5

00.5

1∆d

-2-1.5

-1-0.5

00.5

11.5

2∆u–

-2-1.5

-1-0.5

00.5

11.5

2∆d–

10-2

10-1

-2-1.5

-1-0.5

00.5

11.5

2

x

∆s–

Figure 6.9: The extra ted single distributions from the results of the a-

vor asymmetry, whi h is obtained at Q

2

= 2.5 GeV

2

. The solid

ir les show the results with the mixing matrix using in lusive

and harged hadron asymmetries. The open squares is that with

the new de�nition of purities using in lusive, harged hadron,

harged pion and harged kaon asymmetries.


-0.1

00.1

0.2

0.30.4

0.5

0.6

x ∆u mixing matrix

new definition of purity

-0.2

-0.1

0

0.1

0.2

x ∆d

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

x ∆u

–

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

x ∆d

–

10-2

10-1

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

x

x ∆s

–

Figure 6.10: The single quark distributions times x, whi h is obtained at Q

2

= 2.5 GeV

2

. The solid lines express the positivity limits whi h

are unpolarized PDF, where CTEQ4LQ parameterizations are

shown. The two results with the mixing matrix and the new

de�nition of purities are obtained.


6.1.5 Results for the di�erent Purities

In the analysis des ribed above, I used the purities whi h are produ ed from CTEQ4LQ

[53℄ unpolarized PDF. However, it is important to he k the stability of the result

for extra ted quark polarization. Therefore, I tested with the value of the purities

whi h is produ ed by several di�erent unpolarized PDF, su h as CTEQ4LQ and

GRV. Then quark polarization is determined by using two purities based on CTEQ

and GRV. Fig.6.11 shows the result of the avor de omposition with the mixing

matrix by the CTEQ4LQ and the GRV purities. There is almost no deviation

whi h omes from the di�eren e of PDF be ause the used purities by CTEQ4LQ

and GRV are onsistent, espe ially for the valen e quarks. This omparison tells us

that the produ ed purities don't give an signi� ant ontribution to the systemati

un ertainty. This is also lear for Fig.6.12, whi h shows that the avor asymmetry

of light sea quarks does not depend on the purities used.

-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

(∆u

+∆u–

)/(u

+u–)

mixing matrix (CTEQ)mixing matrix (GRV)

-1

-0.8-0.6

-0.4

-0.2

00.2

0.4

0.6

0.81

(∆d

+∆d–

)/(d

+d–)

10-2

10-1

-1-0.8

-0.6

-0.4

-0.20

0.2

0.4

0.60.8

1

x

∆qs/

qs

Figure 6.11: Comparison of the avor de omposition using purities from

CTEQ4LQ and GRV. The solid ir les are results using pu-

rities produ ed from CTEQ4LQ. The open ir les are that from

GRV.


-1-0.5

00.5

11.5

22.5

33.5

∆u+∆

u–

mixing matrix (CTEQ)mixing matrix (GRV)

-3-2.5

-2-1.5

-1-0.5

00.5

1

∆d+∆

d–

-3

-2

-1

0

1

2

3

∆u–+∆

d–

10-2

10-1

-3

-2

-1

0

1

2

3

x

∆u–-∆

d–

Figure 6.12: Comparison of the avor asymmetry extra tion using purities

from CTEQ4LQ and GRV. The solid ir les are results using

purities produ ed from CTEQ4LQ. The open ir les are that

from GRV.


6.2 Moments of Polarized Quark Distributions

In the previous se tions, the results of the extra tion of the quark polarizations and

distributions are shown for ea h extra tion type. To determine the moments of the

extra ted quark distributions, they are �tted with parameterized fun tions. The

used fun tional forms are as follows:

�q(x) = N

q

x

�

q

� q(x); (6.1)

�q

s

(x) = N

q

s

x

�

q

s

(1� x)

�

q

s

� q

s

(x); (6.2)

where q is the valen e quark : u; u

v

; d; d

v

, and q

s

is the sea quark : �u;

�

d; s; �s. These

forms are for the GRSV parameterizations [68℄. N

q

, �

q

and �

q

are �t parameters.

The CTEQ4 PDF at Q

2

0

= 2:5 GeV

2

, whi h is the Q

2

mean value at HERMES, are

used for the unpolarized PDF in Eq.6.1 and Eq.6.2.

The n-th moment is de�ned as:

�

(n)

q

f

(Q

2

0

) �

Z

1

0

x

n�1

�q

f

(x;Q

2

0

)dx: (6.3)

Eq.6.3 is integral over the full x range, 0 < x < 1, though the experimental data are

available only in a limited x region whi h is 0.023 < x < 0.6 at HERMES.

The pra ti al al ulation of the moments is done as follows. There is a little

di�eren e between the ase of the valen e and avor de omposition (�rst ase) and

that of avor asymmetry and ea h quark distribution extra tion (se ond ase), be-

ause in the �rst ase, the results are obtained as the quark polarizations �q

f

=q

f

but in the se ond ase, the results are the quark distributions �q

f

.

The �rst moment in the measured x region is obtained as:

�q

f

(Q

2

0

) =

X

i

�q

f

q

f

�

�

�

�

�

i

Z

x

i+1

x

i

q

f

(x;Q

2

0

)dx

!

; (6.4)

where, in the �rst ase, (�q

f

=q

f

)j

i

is the results from the experiment, and in the

se ond ase, it is the value of the results divided by the unpolarized CTEQ4LQ PDF

[53℄. (�q

f

=q

f

)j

i

is onstant within ea h bin (x

i

; x

i+1

) and q

f

(x) is obtained by the

CTEQ4LQ parameterization. Outside the measured region, the �tted parameters

of N

q

, �

q

and �

q

are used to determine the moments. In the high x region of 0.6

< x < 1, extrapolations of the �tted fun tions (Eq.6.1 or Eq.6.2) are required, and

the moments are al ulated as this integrals. In the low x region of 0 < x < 0.023,

there is no lear predi tion to determine polarized quark distributions. Therefore

we applied a simple Regge parameterization [70℄ [71℄ of �q

f

(x) / x

��

with �=0

�tted to the data for x < 0.075. The moments in the low x region are al ulated as

the integral of extrapolations with this Regge parameterization, whi h is onstant

in this study.

To obtain the moments for the sea quarks, there is a point that we should noti e.

As I have mentioned before, almost all sea quark polarizations and distributions

ex eed the positivity limit in the high x region, hen e I negle ted three points in the

high x region to al ulate the moment. Therefore the method of determination of the

moment in high x region, whi h is des ribed above, is only applied to the moment

al ulation for valen e quarks. In this study, the results with the new de�nition of

6.2. MOMENTS OF POLARIZED QUARK DISTRIBUTIONS 105

purities are used to al ulate the moments for extra ted quarks be ause these results

in lude informations from pion and kaon asymmetries.

The parameterized quark distributions and moments are shown in the next se -

tions for the results with the new de�nition of purity, and the omparison with the

theoreti al predi tion will be dis ussed. Only the �tted fun tions by Eq.6.1 are used

to obtain the moments in the high x region, 0.6 < x. Therefore the best �t for

valen e quarks by Eq.6.1 are shown in the following.

6.2.1 The Valen e and Flavor De omposition

The �t results for valen e quarks extra ted from the valen e and avor de omposi-

tion are shown in Fig.6.13, whi h are the results with the new de�nition of purity.

�u + ��u and �d + �

�

d extra ted from the avor de omposition in lude the sea

quarks, although they are �tted by Eq.6.1 be ause u and d quarks are dominant.

For the results with the mixing matrix, the moments an be seen in Appendix.C.5.

And the numeri al values of the �rst moment, extra ted from the results of Fig.6.13

by using Eq.6.4, are shown in Tab.6.1 and Tab.6.2, respe tively. The �tted pa-

rameters, N

q

; �

q

and �

q

whi h are used to al ulate the moments are shown in

Appendix.C.5 (Tab.C.29 and Tab.C.33).

-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

x ∆u

v

Best fit

GRSV / (1+R)

10-2

10-1

-0.2

-0.1

0

0.1

0.2

x

x ∆d

v

-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

x (∆

u+∆

u–)

Best fit

GRSV / (1+R)

10-2

10-1

-0.2

-0.15

-0.1

-0.05

0

0.05

0.1

x

x (∆

d+∆

d–)

Figure 6.13: The polarized valen e quark distributions whi h are �tted by the

fun tion of Eq.6.1. The left �gures are shown the valen e quarks

extra ted by the valen e de omposition. The right �gures are

extra ted by the avor de ompositions. The thi k lines are the

result of a best �t and the thin lines are obtained from GRSV

parameterization orre ted (1 +R).

In Fig.6.13, the thi k line shows the result of a �t with the fun tions of Eq.6.1 and

the thin line represents the GRSV PDF [68℄ whi h is orre ted by the term (1+R) for

the omparison with this results. The �t of the extra ted u quark distributions, x�u

and x(�u+��u) show good agreement with the orre ted GRSV PDF, espe ially the

�u+��u is onsistent in the full x region. For the d quark distributions, x(�d+�

�

d),

whi h is extra ted by the avor de omposition, is onsistent with GRSV, but x�d

v


does not agree in the ase of the valen e de omposition. They are still ompatible

with GRSV within the statisti errors.

measured region low-x high-x total integral

�u

v

0.56 � 0.09 0.04 0.02 0.62 � 0.09

�d

v

-0.14 � 0.15 -0.02 0.00 -0.17 � 0.15

�q

s

-0.03 � 0.05 -0.00 0.00 -0.04 � 0.05

Table 6.1: The �rst moment for the valen e de omposition


�u+��u 0.52 � 0.04 0.04 0.02 0.57 � 0.04

�d+�

�

d -0.21 � 0.07 -0.03 0.00 -0.24 � 0.07

�q

s

-0.03 � 0.05 -0.00 0.00 -0.04 � 0.05

Table 6.2: The �rst moment for the avor de omposition

The moments of the valen e and the avor de omposition results are al ulated

for three x regions, one is the measured x region 0.023 < x < 0.6, others are the

low x and high x region. The moments for ea h region are given by the method

whi h is explained before. The total moment in the region of 0 < x < 1, are

obtained summing up the moments of the three regions. These total moments an

be ompared with the theoreti al predi tions and the result by SMC [72℄. This is

dis ussed in Se .6.2.3.

6.2.2 Flavor Asymmetry and Single Quark Distribution

Here, the �t results for valen e quarks from the avor asymmetry extra tion and the

single quark distribution are shown in Fig.6.14, whi h have been a hieved with the

new de�nition of the purity. For the results with the mixing matrix, the numeri al

values of the �rst moment and the �tted parameters are shown Appendix.C.5. The

�tted parameters for this results also are shown in Appendix.C.5 (Tab.C.37 and

Tab.C.41).

The de�nitions of the urves in Fig.6.14 are the same as for the �gure for the

previous valen e and the avor de omposition. In this ase, the �ts to the u quark

distributions �u and �u + ��u show onsisten y with the GRSV PDF in the full

x region. �d and �d + �

�

d are in agreement with GRSV PDF within the statisti

errors, similarly to the �t for the valen e and avor de omposition.

The al ulated moments for the avor asymmetry and all quark distributions

are shown for three x regions in Tab.6.3 and Tab.6.4. The total moments of �u +

��u and �d + �

�

d show a good onsisten y with the moments extra ted with the

avor de omposition in Tab.6.2. These moments are ompared with the theoreti al

predi tions in the following se tion.


-0.1

0

0.1

0.2

0.3

0.4

0.5

x (∆

u+∆

u–)

Best fit

GRSV / (1+R)

10-2

10-1

-0.2

-0.15

-0.1

-0.05

0

0.05

0.1

x

x (∆

d+∆

d–)

-0.1

0

0.1

0.2

0.3

0.4

0.5

x ∆u Best fit

GRSV / (1+R)

10-2

10-1

-0.2

-0.15

-0.1

-0.05

0

0.05

0.1

xx

∆d

Figure 6.14: The polarized valen e quark distributions whi h are �tted by

the fun tion of Eq.6.1. The left �gures show the valen e quarks

extra ted by the avor asymmetry extra tion. The right �gures

are results for the single quark distribution. The thi k lines

are the result of a best �t and the thin lines are obtained from

GRSV parameterization orre ted by (1 +R).


�u+��u 0.53 � 0.05 0.04 0.02 0.59 � 0.05

�d+�

�

d -0.19 � 0.08 -0.03 0.00 -0.22 � 0.08

��u+�

�

d -0.09 � 0.08 -0.01 0.00 -0.10 � 0.08

��u��

�

d 0.09 � 0.09 0.02 0.00 0.11 � 0.09

Table 6.3: The �rst moment for avor asymmetry extra tion


�u 0.54 � 0.12 0.04 0.02 0.60 � 0.12

�d -0.18 � 0.13 -0.03 0.00 -0.21 � 0.13

��u 0.00 � 0.06 0.00 0.00 0.00 � 0.06

�

�

d -0.09 � 0.06 -0.02 0.00 -0.10 � 0.06

��s -0.06 � 0.04 -0.01 0.00 -0.07 � 0.04

Table 6.4: The �rst moment for single quark distribution


6.2.3 Comparison of Results with Predi tions

The obtained moments of quark distributions an be ompared to the theoreti al

predi tions or other experimental results. Here the moment of this study whi h has

been extra ted with the new de�nition of purity is used.

There is one experimental measurement of polarized quark distributions and mo-

ments by SMC [72℄. From this experiment, the results of the valen e de omposition

are available, with the assumption of SU(3) symmetri sea : ��q(x) = ��u(x) =

�

�

d(x) = �s(x) = ��s(x) . It is possible to ompare our results and the SMC results

whi h are extrapolated to Q

2

= 2.5 GeV

2

and integrated over the HERMES x range,

0.023 < x < 0.6. This omparison is shown in Tab.6.5, where the systemati errors

are omitted. Only the statisti al errors are shown. For this study in Tab.6.5, �u

v

and �d

v

are obtained by the results of the valen e de omposition in Tab.6.1, in

addition to ��u and �

�

d are the results from Tab.6.4. There is good agreement with

the SMC results, espe ially for �u

v

. There are some deviations in the sea quark

distribution, ��u and �

�

d, be ause the assumptions on the sea quarks whi h are used

in ea h analysis are signi� antly di�erent.

In the next, the omparison with the theoreti al predi tions is dis ussed. There

are two predi tions whi h an be ompared with our results, whi h are a hieved

by the latti e QCD and the SU(3) analysis within errors. Tab.6.6 shows these

omparisons. Here the values of the latti e QCD are al ulated for Q

2

=5 GeV

2

[73℄.

The present results show a lower value than the predi tion by the latti e QCD. The

al ulation of latti e QCD is performed in the quen hed approximation.

This study SMC

�u

v

0.56 � 0.09 0.59 � 0.08

�d

v

-0.14 � 0.15 -0.33 � 0.11

��u 0.00 � 0.06 0.02 � 0.03

�

�

d -0.09 � 0.06 0.02 � 0.03

Table 6.5: Comparison of the �rst moment between this study and the results

by SMC. These value are for the HERMES measured region, 0.023

< x < 0.6, and obtained at Q

2

= 2.5 GeV

2

This study Latti e QCD

�u

v

0.62 � 0.09 0.84 � 0.05

�d

v

-0.17 � 0.15 -0.25 � 0.02

Table 6.6: Comparison of the �rst moment between this study and the pre-

di tion by the latti e QCD. The theoreti al value by the latti e

QCD is obtained at Q

2

= 5 GeV

2

.


This study SU(3)

1. �u+��u 0.57 � 0.04 0.66 � 0.03

�d+�

�

d -0.24 � 0.07 -0.35 � 0.03

2. �u+��u 0.59 � 0.05 0.66 � 0.03

�d+�

�

d -0.22 � 0.08 -0.35 � 0.03

3. �u+��u 0.60 � 0.13 0.66 � 0.03

�d+�

�

d -0.31 � 0.14 -0.35 � 0.03

�s+��s -0.14 � 0.06 -0.08 � 0.02

Table 6.7: Comparison of the �rst moment between this study and the pre-

di tion by the SU(3) analysis. These theoreti al values are ob-

tained at Q

2

= 2.5 GeV

2

.

For the moments of the avor de omposition results, the predi tion of the SU(3)

analysis an be ompared. In Tab.6.7, the extra tion method is shown in the �rst

olumn. The moments in row No.1 are obtained with the avor de omposition

results (Tab.6.2), and in row No.2 the results from the avor asymmetry extra tion

are shown (Tab.6.3). In row No.3 the moments al ulated from the extra ted results

of the single quark distribution (Tab.6.4) are shown. For row No.3, the moment of

the strange quarks is listed, whi h is obtained from the ��s moment in Tab.6.4.

In the extra tion of single quark distributions, the assumption is used that the

distributions of �s and ��s are equal, hen e �s + ��s = 2 ��s, whi h is used to

al ulate the value in Tab.6.7.

For row No.1 of the avor de omposition results, both moments are onsistent

with the predi tion of SU(3) analysis within errors. The same is true for the moments

extra ted by the avor asymmetry extra tion method. Additionally, the results

obtained by single quark distribution in row No.3 are in good agreement with the

SU(3) predi tions. Espe ially the valen e quarks show the best agreement, and

the moment of the strange quarks shows a signi� antly negative value and this is

onsistent with the value of the predi tion. From these, the sum of the quark spin

ontributions �� = �u +��u +�d +�

�

d + �s +��s = 0.23 � 0.11 in the ase of

No.3.


6.3 Flavor Asymmetry Extra tion with Another

Method

In the previous se tion, the avor asymmetry of light sea quarks with the purity

matrix method is extra ted. This avor asymmetry was predi ted by the CQSM.

Besides it was extra ted by another method from the asymmetries of HERMES [9℄

[10℄. In the other method, we don't need to use assumptions on the polarized quark

distributions, and use the relations whi h is based on QPM only (Eq.2.78-Eq.2.80

and Eq.2.86). We estimated the avor asymmetry of light sea quarks with this

method using the asymmetries of HERMES, and ompared it with the results ob-

tained in the previous se tion. Additionally, the dependen e of the avor asymmetry

on the fragmentation fun tions is studied. Only the parameterized fragmentation

fun tions de�ned as the favored one and the unfavored one are available. Therefore

the fragmentation fun tions of strange quarks D

s

are onsidered to be the same as

the unfavored ones, but I estimated the ontribution of D

s

on the avor asymmetry

with the theoreti al model of the fragmentation fun tions in [74℄.

6.3.1 Another Method for Flavor Asymmetry Extra tion

A ording to [9℄ [10℄, the asymmetry of hadron h on the proton target is written by:

A

h

1;p

(x) =

1 +R(x)

1 +

2

�

P

f

e

2

f

n

�q

f

(x)D

h

f

+�q

�

f

(x)D

h

�

f

o

P

f

e

2

f

n

q

f

(x)D

h

f

+ q

�

f

(x)D

h

�

f

o

; (6.5)

where f shows the quark avor : f = fu; d; sg. In Eq.6.5, the avor of quarks and

anti-quarks is shown separately. D

h

f

(x) is the integrated fragmentation fun tion in

the measured region, whi h is D

h

f

=

R

1

z

min

dzD

h

f

(z).

The fragmentation fun tions are lassi�ed in the following:

(D

�

)

+

� D

�

+

u

= D

�

+

�

d

= D

�

�

d

= D

�

�

�u

; (6.6)

(D

�

)

�

� D

�

+

d

= D

�

+

�u

= D

�

�

u

= D

�

�

�

d

= D

�

+

s

= D

�

+

�s

= D

�

�

s

= D

�

�

�s

; (6.7)

(D

K

)

+

� D

K

+

u

= D

K

+

�s

= D

K

�

�u

= D

K

�

s

; (6.8)

(D

K

)

�

� D

K

+

d

= D

K

+

s

= D

K

+

�u

= D

K

+

�

d

= D

K

�

u

= D

K

�

d

= D

K

�

�

d

= D

K

�

�s

; (6.9)

(D

p

)

+

� D

p

u

= D

p

d

= D

�p

�u

= D

�p

�

d

; (6.10)

(D

p

)

�

� D

p

s

= D

p

�u

= D

p

�

d

= D

p

�s

= D

�p

u

= D

�p

d

= D

�p

s

= D

�p

�s

; (6.11)

where (D

h

)

+

and (D

h

)

�

are the favored and the unfavored fragmentation fun tions

of hadron h, respe tively. Returning to Eq.6.5, this relations an be rewritten as:

X

f

e

2

f

n

�q

f

(x)D

h

f

+�q

�

f

(x)D

h

�

f

o

=

1 +

2

1 +R(x)

� A

h

1;p

(x)

2

4

X

f

e

2

f

n

q

f

(x)D

h

f

+ q

�

f

(x)D

h

�

f

o

3

5

� �N

h

p

(x); (6.12)

6.3. FLAVOR ASYMMETRY EXTRACTION WITH ANOTHER METHOD 111

where �N

h

p

(x) refers to the spin-dependent produ tion pro esses of harged hadrons

with the proton target. Similarly, we an get �N

h

n

(x) for the neutron target. Form-

ing a ombination of �N

h

p

(x) and �N

h

n

(x), we obtain the following relation:

��u(x)��

�

d(x) =

�N

h

+

p

(x)��N

h

+

n

(x) + �N

h

�

p

(x)��N

h

�

n

(x)

2I

1

�

�N

h

+

p

(x)��N

h

+

n

(x)��N

h

�

p

(x) + �N

h

�

n

(x)

2I

2

;(6.13)

where I

1

and I

2

are de�ned as the ombination of the fragmentation fun tions:

I

1

= 3(D

�

)

+

+ 4(D

K

)

+

+ 3(D

p

)

+

+ 3(D

�

)

�

+ 2(D

K

)

�

+ 3(D

p

)

�

; (6.14)

I

2

= 5(D

�

)

+

+ 4(D

K

)

+

+ 3(D

p

)

+

� 5(D

�

)

�

� 4(D

K

)

�

� 3(D

p

)

�

: (6.15)

From Eq.6.13, if we have N

h

p

(x) and N

h

n

(x) whi h an be obtained by experi-

mental asymmetries on the respe tive target, the avor asymmetry of light sea the

quarks an be extra ted without the help of MC and the additional assumptions.

Experimentally, I have to use the asymmetry on the D target to obtain �N

h

n

, so

it is ne essary to hange Eq.6.12 for the pro ess on the D target. In ase of the D

target, Eq.6.12 reads as follows:

X

f

e

2

f

n

�q

f

(x)D

h

f

+�q

�

f

(x)D

h

�

f

o

=

1 +

2

(1 + R(x))(1� 1:5 !

D

)

� A

h

1;D

(x)

2

4

X

f

e

2

f

n

q

f

(x)D

h

f

+ q

�

f

(x)D

h

�

f

o

3

5

� �N

h

D

(x); (6.16)

where !

D

is the D-state probability, !

D

= 0.058 [62℄ [63℄. Using Eq.6.16, Eq.6.13 is

rewritten as:

��u(x)��

�

d(x) =

2 �N

h

+

p

(x)��N

h

+

D

(x) + 2 �N

h

�

p

(x)��N

h

�

D

(x)

2I

1

�

2 �N

h

+

p

(x)��N

h

+

D

(x)� 2 �N

h

�

p

(x) + �N

h

�

D

(x)

2I

2

:(6.17)

With the relation Eq.6.17 and the asymmetries of hadrons at HERMES on the pro-

ton and the D target, I an extra t ��u��

�

d dire tly. The detail of the fragmentation

fun tions whi h are used in this analysis is des ribed in the following se tion.

In this method, the Q

2

dependen e of the avor asymmetry have been negle ted

be ause no signi� ant Q

2

dependen e has been observed in the measured spin asym-

metries. Therefore the unpolarized PDF at the �xed Q

2

=4 GeV

2

is used, whi h is

the same value as the previous analysis in [9℄ [10℄.

6.3.2 Theoreti al Model of the Fragmentation Fun tions

For the extra tion of the avor asymmetry in [9℄ [10℄, the fragmentation fun tions

parameterized from the EMC experiment [21℄ are used. These are the favored and


the unfavored fragmentation fun tions of pion, kaon and proton : (D

�

)

+

; (D

�

)

�

;

(D

K

)

+

; (D

K

)

�

; (D

p

)

+

; (D

p

)

�

. The fragmentation fun tions of strange quarks are

onsidered to be the same as the unfavored fragmentation fun tions. They are

not obtained from experiment yet be ause the poor statisti s of data. But the

ontributions of strange quark to the fragmentation fun tions should be treated

separately from the unfavored fragmentation fun tions, like in Eq.2.80. Therefore

the spe tator model for the strange quark fragmentation fun tion is used whi h will

be des ribed in the next se tion. In spite of using the strange quark fragmentation

fun tion, Eq.6.14 and Eq.6.15 are not hanged.

The Fragmentation Fun tions in the Spe tator Model

Be ause the fragmentation fun tions of strange quarks are not obtained from the

experiment, I used that from the spe tator model des ribed in [74℄. In this paper,

the model parameters of pion fragmentation are obtained by �tting the pion data of

EMC [21℄. This way, the fragmentation fun tions of the strange quarks are obtained

whi h are di�erent from the unfavored fragmentation fun tions. In this model,

the fragmentation fun tions of pion and kaon are estimated. These fragmentation

fun tions are shown in Fig.6.3.2.

With these parameterized fragmentation fun tions, I studied the avor asym-

metry of light sea quarks from Eq.6.17. To study the ontribution of the strange

quarks to the avor asymmetry, I used several sets of fragmentation fun tions as

input. These sets are shown below:

1. The EMC parameterizations

(D

�

)

+

; (D

�

)

�

; (D

K

)

+

; (D

K

)

�

; (D

p

)

+

; (D

p

)

�

: [ EMC parameterization ℄

2. The Spe tator model parameterizations

(D

p

)

+

; (D

p

)

�


(D

�

)

+

; (D

�

)

�

; (D

�

)

�

s

; (D

K

)

+

; (D

K

)

�

; (D

K

)

+

s

; (D

K

)

�

s

: [ Spe tator model ℄

3. The ontribution of the strange quarks

(D

�

)

+

; (D

�

)

�

; (D

K

)

+

; (D

K

)

�

; (D

p

)

+

; (D

p

)

�


(D

�

)

�

s

; (D

K

)

+

s

; (D

K

)

�

s


4. The pion ontributions

(D

K

)

+

; (D

K

)

�

; (D

p

)

+

; (D

p

)

�


(D

�

)

+

; (D

�

)

�


5. The kaon ontributions

(D

�

)

+

; (D

�

)

�

; (D

p

)

+

; (D

p

)

�


(D

K

)

+

; (D

K

)

�


where (D

�

)

s

and (D

K

)

s

are the pion and kaon fragmentation fun tions of the strange

quarks respe tively. Comparing the set 1 and the set 2, we an estimate the frag-

mentation model dependen e of the avor asymmetry, and omparing the set 1 and

the set 3, the dependen e on the fragmentation fun tions of the strange quarks is

obtained. Additionally, omparing the set 1 and set 4(5), we obtain the ontribu-

tions of the pion(kaon) fragmentation fun tions. This analysis is useful to estimate

the systemati error from the fragmentation model.


Using the asymmetries on the proton ('97) and the deuteron ('98) and the above

sets of the fragmentation fun tions, the avor asymmetry with Eq.6.17 is studied.

The result are shown in the following and omparison with that from the purity

method with the mixing matrix. It should be noted that the geometri al a eptan e

of HERMES dete tor, whi h ould be z dependent, is not taken into a ount in the

present analysis.

Figure 6.15: Fragmentation fun tions in the Spe tator model. The pion and

kaon fragmentation fun tions of strange quarks are extra ted by

�tting model parameters to EMC data.


6.3.3 Results and Comparison with the Purity Matrix Method

When I extra ted the avor asymmetry by solving Eq.6.17, I applied 5 sets of frag-

mentation fun tions, whi h have been already des ribed. The extra ted ��u � �

�

d

distributions are shown in Fig.6.16.

-4

-2

0

2

4

∆u–-∆

d–

EMC parametrizations

-4

-2

0

2

4

∆u–-∆

d–

Spectator model parametrizations

10-2

10-1

-4

-2

0

2

4

x

∆u–-∆

d–

contribution of the strange quarks

-4

-2

0

2

4

∆u–-∆

d–

pion contributions

10-2

10-1

-4

-2

0

2

4

x

∆u–-∆

d–

kaon contributions

Figure 6.16: Flavor asymmetry of light sea quarks extra ted from Eq.6.17.

These �gures are obtained by di�erent sets of fragmentation

fun tions.

In this analysis, only the asymmetries of the two years are used as input. There-

fore the statisti errors be ame larger than the results of the purity methods. From

Fig.6.16, in spite of using di�eren e fragmentation fun tions, these �gures don't

show any di�eren e, but they are very onsistent. From these omparisons, it an

be on luded that the fragmentation fun tions don't have large in uen e on the

avor asymmetry of light sea quarks. It also shows that the systemati errors whi h

ome from the un ertainty on the fragmentation fun tions is not dominant in the

total systemati errors.

In Fig.6.17, the omparison of the result of the purity method with the mixing

matrix based on the in lusive and hadron asymmetries and this dire t extra tion

is shown. From Fig.6.17, the un ertainty from the MC and from the assumptions

on the sea quarks is estimated in the purity method be ause the dire t extra tion

method does not depend on the MC or any assumptions, only on fragmentation

fun tions.

Fig.6.17 indi ates that the avor asymmetry extra ted by the dire t extra tion

shows good agreement with the result of the purity method. For the dire t extra tion


10-2

10-1

-3

-2

-1

0

1

2

3

x

∆u–-∆

d–

purity method (mixing matrix)direct extraction (EMC)

10-2

10-1

-0.4

-0.2

0

0.2

0.4

x

x (∆

u–-∆

d–)

purity method (mixing matrix)direct extraction (EMC)

Figure 6.17: Comparison between the dire t extra tion method and the pu-

rity matrix method. The result of the dire t extra tion method

with the EMC parameterization are shown. The upper �gure

is shown as avor asymmetry of light sea quarks extra ted two

di�erent method. The lower �gure is the results times x.

results, the statisti errors are larger than the purity method. Be ause the dire t

extra tion is less statisti s than the purity matrix method. The bottom �gure of

Fig.6.17 shows x times the avor asymmetry. In two bins of the high x, they ex eed

the positivity limit for the result of the dire t extra tion method. Hen e the negle t

of the sea quark distribution in high x bins an be justi�ed by the other analysis.

Comparing the result of the purity method and that of the dire t extra tion

method, the existen e of a avor asymmetry of light sea quarks ��u��

�

d is on�rmed.

It is indi ated that it is positive and there are no large ontributions to the systemati

un ertainties by MC, the fragmentation fun tions and the assumptions on the sea

quark distributions.


6.4 Simple Estimation of Systemati Errors

In this study, the a urate extra tion of the systemati un ertainty has been skipped,

although the simple estimations are studied. In this se tion, these estimations are

summarized.

The systemati errors of the extra ted quark polarizations and distributions an

be separated into two parts, one omes from the un ertainty of the experiment and

the other omes from that of the analysis method. The experimental un ertainty

arises by di�erent auses as follows:

� Beam polarization

� Target polarization

� Yield u tuations

� Cross se tion ratio R

� Spin stru ture fun tion g

2

� Smearing orre tions

� Radiative orre tions

� Error on the measured asymmetries

The analysis method un ertainty are given as follows:

� Unpolarized PDF

� Method of using asymmetries on the

3

He and the D target

� Fragmentation Fun tions

In this study, we only dis ussed the analysis method un ertainty. For the un er-

tainty from the unpolarized PDF, we an estimate by omparing the purities gen-

erated with CTEQ4LQ and GRV. Additionally, the results for the di�erent purities

are shown in Fig.6.11 and Fig.6.12. From these omparisons, the un ertainty from

PDF are not observed signi� antly. Some deviations an be seen in the omparison

of the purities. However, the extra ted quark distributions are almost identi al.

The un ertainty from the interpretation of asymmetries on the

3

He and the D

target is already dis ussed. There is almost no in uen e on the systemati un er-

tainty from the di�erent method, the mixing matrix method and the new de�nition

of purity. We an see it omparing the both results with MC data in Fig.6.1 and

Fig.6.2. The deviation annot be observed learly.

In Se .6.3, the un ertainty from the fragmentation models are shown. The re-

sults with the di�erent fragmentation fun tions are almost the same. We an see

that the di�eren e of fragmentation fun tions doesn't have large ontribution to the

systemati un ertainty. This is shown in Fig.6.16. From these observations, it an

be on luded that the systemati un ertainty from the analysis method are small.

The main ontribution to the systemati un ertainty omes from the experimen-

tal un ertainty. Therefore the range of the systemati errors are shown from the

previous analysis [69℄ for referen e. In [69℄, the systemati errors are obtained as:

� for valen e de omposition

0:026 <

�u

v

u

v

< 0:071; (6.18)

0:098 <

�d

v

d

v

< 0:437: (6.19)

6.4. SIMPLE ESTIMATION OF SYSTEMATIC ERRORS 117

� for avor de omposition

0:007 <

�u+��u

u+ �u

< 0:039; (6.20)

0:016 <

�d+�

�

d

d+

�

d

< 0:129: (6.21)

For the present analysis, the systemati un ertainty an be expe ted to be not

so di�erent from these values.

119

Chapter 7

Con lusions and Summary

The main subje t of this thesis is the extra tion of the avor asymmetry of light sea

quarks, whi h is the asymmetry between the polarized �u and

�

d quark distribution :

��u��

�

d. Another subje t is to extra t of the sea quark distribution.

The purity matrix method has been used to extra t the quark distributions,

whi h needs several spin asymmetries from the experiment as input. The quark

distributions have been studied by this method, espe ially for the valen e quarks.

To study the sea quark distributions, more input asymmetries are ne essary. Fortu-

nately at HERMES, there are several in lusive asymmetry, h

+

and h

�

asymmetries

on the proton and helium3 target available. Sin e 1998, a Ring Imaging Cherenkov

Counter repla ed the threshold Cherenkov Counter and more asymmetries be ame

available. In the present study, these pion and kaon asymmetries on the deuteron

target in 1998 have been used.

In this work, the eÆ ien ies of the HERMES RICH have been studied �rst.

This RICH has two Cherenkov radiators, sili a aerogel and C

4

F

10

gas. I estimated

the e�e ts oming from the radiators and dete tor geometries. The results for the

eÆ ien ies have been in luded in the Monte Carlo simulation, whi h is essential

to study the experimental data a urately. It be ame possible to reprodu e the

experimental data by the updated MC mu h better than before. To a hieve a more

a urate analysis of the experimental data, it is ne essary to understand the RICH

performan e in detail. This update is an important ontribution to the extra tion

of the pion and kaon asymmetries in 1998.

Se ondly, using the asymmetries available at HERMES, I extra ted the quark

distributions by the purity matrix method. To use this method, I introdu ed new

assumptions to study the sea quark distributions, whi h are based on the Chiral

Quark Soliton Model. Additionally, I proposed a new method to treat the asym-

metries on the helium3 and the deuteron target. Therefore it be ame possible to

in lude the pion and kaon asymmetries in this analysis. Finally, the valen e and

the sea quark distributions, the avor asymmetry of the light sea quarks and their

moments were obtained. It is on�rmed that �u and �u

v

are positive and �d

shows negative values. For the avor asymmetry, ��u��

�

d is slightly positive and

its moment has also a slight positive value. ��u is almost zero, and both �

�

d and

��s(= �s) show slight negative values. The moments of this results are almost on-

sistent with the theoreti al predi tion and the other experimental results. However,

more statisti s is needed to really larify the sea quark distributions. The present

120 CHAPTER 7. CONCLUSIONS AND SUMMARY

method to extra t the avor asymmetry is a good tool for the study of the nu leon

spin ontent and gave a hint to several quark distributions. Both results from the

mixing matrix method and from the new de�nition of purities are onsistent. The

extra ted results don't depend on the extra tion methods. It has been on�rmed

that the avor asymmetry is not sensitive to the un ertainties whi h ome from

the parameterizations of the unpolarized quark distributions and the fragmentation

fun tions. The main on lusions of this thesis are summarized below:

1. An Analysis method to extra t avor asymmetry of light sea quarks ��u��

�

d

is proposed and examined. The single quark distributions �u;�d; ��u;�

�

d;

�s = ��s are extra ted at the same time.

2. Flavor asymmetry of light sea quarks is extra ted, whi h shows almost zero or

slightly positive. Their �rst moments are also slight positive values.

3. Single quark distributions are obtained. Espe ially the x dependen e of ��u;�

�

d;

and �s = ��s are extra ted. ��u is almost zero while �

�

d and �s = ��s and

their moments are slightly negative.

4. It is found that data of the pion and kaon asymmetries are important to study

of the polarized sea quark distribution.

5. There are small systemati un ertainties whi h ome from the unpolarized

PDF, the extra tion methods and the fragmentation fun tions.

As an outlook to future the following omments an be made. Be ause the data

has been taken with RICH sin e 1998, asymmetries with more statisti s will be ome

available in the near future, espe ially for the pion and kaon asymmetries. These

pion and kaon asymmetries are sensitive to the sea quarks. Therefore it is expe ted

that an a urate study of the sea quark distributions be omes possible. Also the

avor asymmetry of light sea quarks will be determined more pre isely. If enough

di�erent types of asymmetries are obtained, the single quark distributions an be

extra ted without any assumptions on the quark distributions. These studies based

on the experimental data olle ted with the RICH will give more a urate results

to the polarized quark distributions and have a large ontribution to solve the spin

problem of the nu leon.

121

Appendix A

Detail of MC Development

The details of MC development is des ribed in following. To normalize the exper-

imental data, the photon olle tion eÆ ien ies are in luded as the position depen-

den e of the mirror re e tivity. The analysis method to obtain the mirror re e tivity

for ea h segment is des ribed in Se .4.3.1. The L

gas

is used for normalization of the

number of �red PMTs. The de�nition and distribution of L

gas

are shown in Fig.A.1.

0

200

400

600

800

1000

30 40 50 60 70 80 90 100

Figure A.1: De�nition and Distribution of L

gas

for MC data. The left �gure

shows the de�nition of L

gas

, whi h is determined as the distan e

between the position that the parti le tra k goes into gas and

the hit point of the tra k on the mirror. The right �gure is the

distribution of L

gas

. The mean value of L

gas

is 73, whi h is used

for the normalization of the parti le tra k length.

122 APPENDIX A. DETAIL OF MC DEVELOPMENT

In Fig.A.2 the ratio between N

gas

(EXP) and N

gas

(MC) is shown, whi h is used to

al ulate the mirror re e tivity for ea h segment as the photon olle tion eÆ ien y.

20

30

40

50

60

70

80

-100 -80 -60 -40 -20 0 20 40 60 80 1000

0.2

0.4

0.6

0.8

1

1.2

x

y

x

y

x

y

-80

-70

-60

-50

-40

-30

-20

-100 -80 -60 -40 -20 0 20 40 60 80 10000.10.20.30.40.50.60.70.80.91

x

y

x

y

x

y

Figure A.2: The ratio between N

gas

(EXP) and N

gas

(MC). The parti les are

�

+

and �

�

with momentum of 4.5 < p < 8.5 (GeV/ ). This ratio

gives the informations of eÆ ien ies from the unknown fa tors,

for example, the position dependen e of the mirror re e tivity,

the mirror alignment, the slope of the in ident photon, the e�e t

from funnels, the PMT eÆ ien ies.

123

The mean value of the ratio whi h is shown in Fig.A.2 for ea h mirror segment

al ulated as Fig.A.3. These values are obtained in order to reprodu e the exper-

imental data. They an be onsidered as the photon olle tion eÆ ien ies. These

values are in luded in MC as the mirror re e tivities for ea h segment. The mirror

re e tivities and alignments, the funnels and other unknown fa tors an ontribute

to these values.

20

30

40

50

60

70

80

90

-100 -75 -50 -25 0 25 50 75 100

0.42± 0.06

0.78± 0.03

0.78± 0.03

0.66± 0.04

0.57± 0.05

0.67± 0.04

0.65± 0.04

0.58± 0.04

-90

-80

-70

-60

-50

-40

-30

-20

-100 -75 -50 -25 0 25 50 75 100

0.59± 0.04

0.76± 0.03

0.60± 0.04

0.55± 0.05

0.51± 0.05

0.63± 0.04

0.71± 0.04

0.45± 0.06

Figure A.3: The position dependen e of upper and lower mirror re e tivity.

These values in lude not only the mirror re e tivities but also

all other eÆ ien ies from unknown fa tors.

124 APPENDIX A. DETAIL OF MC DEVELOPMENT

The omparison between the experimental data and MC data after modi� ation

is shown in Se .4.3.2. We an see that the new MC gives a good reprodu tion om-

paring before modi� ation. The latest omparison of the Cherenkov angle between

the experimental data and the MC give a better result as shown in Fig.A.4.

0

0.01

0.02

0.03

0.04

0.05

0.2 0.21 0.22 0.23 0.24 0.25 0.26 0.27 0.28 0.29 0.3Θ [rad ]

0

0.01

0.02

0.03

0.04

0.05

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1Θ [rad ]

Figure A.4: Re ent Comparison of the Cherenkov angles between the experi-

mental data (points) and the MC data (histogram). They shows

the normalized distributions of re onstru ted angles for single,

low ba kground ele trons (p > 5). The left �gure shows a om-

parison of the Cherenkov angle from aerogel, and the right �gure

shows that from gas.

Fig.A.4 shows a good agreement in the entral part of the distribution. The tails

of the �gure for aerogel angles are due to several e�e ts from the ba kground. The

auses of ba kgrounds are dis ussed in [44℄ and are in luding in MC.

125

Appendix B

Detail of Analysis

B.1 Covarian e Matrix V

A

As the number of dete ted parti les is Poisson distributed, the varian e of the in-

lusive leptons �

2

N

e

+

is al ulated as:

�

2

N

e

+

= hN

2

e

+

i � hN

e

+

i

2

= N

2

e

+

; (B.1)

where N

e

+

is the number of in lusive leptons. For the semi-in lusive measurement,

the varian e of the semi-in lusive hadrons �

2

N

h

is obtained as:

�

2

N

h

= hN

2

h

i � hN

h

i

2

= N

2

e

+

h(n

h

i

)

2

i = N

h

h(n

h

i

)

2

i

hn

h

i

i

; (B.2)

where n

h

i

is the number of hadrons h per event and N

h

is the number of dete ted

hadrons.

Using the varian e, the statisti al ovarian es Cov(N

h

1

; N

h

2

) for the parti le

ount rates are al ulated as:

Cov(N

h

1

; N

h

2

) � (V

N

)

12

= hN

h

1

N

h

2

i � hN

h

1

ihN

h

2

i (B.3)

= N

e

+

hn

h

1

i

n

h

2

i

i; (B.4)

Cov(N

e

+

; N

h

) � (V

N

)

eh

= N

+

e

n

h

; (B.5)

where n

e

+

i

=1 is used. The orrelation oeÆ ients of the parti le ount rate are :

Cor(N

h

1

; N

h

2

) � �(N

h

1

; N

h

2

) =

(V

N

)

12

�

N

h

1

�

N

h

2

(B.6)

=

hn

h

1

i

n

h

2

i

i

q

h(n

h

1

i

)

2

(n

h

2

i

)

2

i

; (B.7)

Cor(N

e

+

; N

h

) � �(N

e

+

; N

h

) =

n

h

i

q

h(n

h

i

)

2

i

: (B.8)

The ovarian es (V

A

)

ij

of the asymmetries A

h

i

1

and A

h

j

1

are obtained from the

ovarian es (V

N

)

ij

of the parti le ount rates as follows:

126 APPENDIX B. DETAIL OF ANALYSIS

(V

A

)

ij

=

X

�;�

�A

h

i

1

�N

h

�

�A

h

j

1

�N

h

�

(V

N

)

��

=

�A

h

i

1

�N

h

i

+

�A

h

j

1

�N

h

j

+

(V

N

)

ij

+

+

�A

h

i

1

�N

h

i

�

�A

h

j

1

�N

h

j

�

(V

N

)

ij

�

'

1

2

2

4

�A

h

i

1

�N

h

i

+

�A

h

j

1

�N

h

j

+

+

�A

h

i

1

�N

h

i

�

�A

h

j

1

�N

h

j

�

3

5

(V

N

)

ij

'

�A

h

i

1

�N

h

i

�

�A

h

j

1

�N

h

j

�

(V

N

)

ij

; (B.9)

where Greek indi es run over both spin states. In Eq.B.9, two fa ts are used, one is

that the asymmetry for one parti le type only depends on the ount rates for this

parti le in both spin states whi h are denoted as N

h

i

+

and N

h

i

�

, another is that the

two spin states are statisti ally un orrelated su h as (V

N

)

i

+

i

�

. Furthermore two

approximations are used whi h are N

h

i

�

�

1

2

N

h

i

and

�

�

�

�

�A

h

i

1

�N

h

i

+

�

�

�

�

�

�

�

�

�

�A

h

i

1

�N

h

i

�

�

�

�

�

. With these

assumptions, the errors on the asymmetries �

A

h

i

1

be omes:

�

A

h

i

1

2

=

�A

h

i

1

�N

h

i

�

!

2

�

N

h

i

2

; (B.10)

where �

N

h

i

is the varian e of the number of dete ted hadrons N

h

i

.

From Eq.B.10, the ovarian es of the asymmetries (V

A

)

ij

is written as:

(V

A

)

ij

'

�

A

h

i

1

�

A

h

j

1

�

N

h

i

�

N

h

j

(V

N

)

ij

: (B.11)

From the de�nition of orrelations of the di�erent types of asymmetries,

�(A

h

i

1

; A

h

j

1

) � (V

A

)

ij

=(�

A

h

i

1

�

A

h

j

1

) and Eq.B.11, the relation that the orrelations of

asymmetries and parti le ount rates are equal:

�(A

h

i

1

; A

h

j

1

) ' �(N

h

i

; N

h

j

); (B.12)

has been dedu ed.

Therefore in the pra ti al analysis, the orrelations of parti le ount rates are

substitutes for that of the asymmetries. The orrelations of parti le ount rates

whi h are used in this study are shown in the next Appendix.C.3, where the ount

rates are given for in lusive and semi-in lusive DIS events.

B.2. SINGULAR VALUE DECOMPOSITION 127

B.2 Singular Value De omposition

\Singular Value De omposition"(SVD) is the method for solving the matrix equa-

tion (Eq.5.5). This method an obtain the solution whi h minimizes �

2

in Eq.5.9.

However the Cholesky de omposition is ne essary to apply the SVD method. The

Cholesky de omposition is expressed as:

V

�1

A

= L

T

L; (B.13)

where L is a triangular matrix. This de omposition an be applied be ause the

ovarian e matrix V

A

is symmetri and positive. With L, A

0

and P

0

de�ned from

A and P as follows:

A

0

= LA;

P

0

= LP: (B.14)

With the relations Eq.B.14, �

2

is expressed as:

�

2

= (A

0

� C

i

� P

0

Q)

T

(A

0

� C

i

� P

0

Q): (B.15)

The SVD method is applied to Eq.B.15. From a mathemati al theorem, any

m�n matrix P an be written as a produ t of an m�n olumn-orthogonal matrix

U , an m � n diagonal matrix W whi h has positive or zero elements. Therefore P

is expressed with a n� n orthogonal matrix Z as:

P

ij

=

n

X

k=1

U

ik

W

k

Z

jk

; (B.16)

where the diagonal of W is denoted by the ve tor W . From Eq.B.16, the solution

an be obtained as:

Q = Z � [diag(1=w

i

)℄ � (U

T

�A); (B.17)

whi h minimizes �

2

. This method an be applied to the ase in whi h there are

more equations than unknowns without modi� ation.

The the ovarian es between the extra ted quark distributions and the asymme-

tries are obtained as:

V

Q

= C

�1

i

� (P

�1

V

A

(P

T

)

�1

)

= C

�1

i

� (P

T

V

�1

A

P)

�1

(B.18)

= C

�1

i

� (P

0�1

(P

0�1

)

T

);

whi h is obtained by Eq.B.13 and the following relation:

V

Mx

=M V

x

M

T

; (B.19)

where V

Mx

(V

x

) is a ovarian e matrix of Mx(x). And from Eq.B.17,

(V

Q

)

ij

=

n

X

k=1

Z

0

ik

Z

0

jk

W

0

k

: (B.20)

129

Appendix C

Table of Results

C.1 De�nition of the Binning in x

Bin number Range in x hxi

1 0.023 � 0.040 0.033

2 0.040 � 0.055 0.047

3 0.055 � 0.075 0.065

4 0.075 � 0.100 0.087

5 0.100 � 0.140 0.119

6 0.140 � 0.200 0.168

7 0.200 � 0.300 0.244

8 0.300 � 0.400 0.342

9 0.400 � 0.600 0.464

Table C.1: x binning de�nition. The �rst olumn shows the number of ea h

bin, their limits are de�ned in the se ond olumn. The mean

value hxi of x for the in lusive asymmetry on the D target is

shown in the third olumn.

130 APPENDIX C. TABLE OF RESULTS

C.2 Input Asymmetries

C.2.1 1995 Asymmetries on the

3

He Target

Bin x Q

2

A

e

+

1

� stat. � sys.

1 0.033 1:220 -0.036 � 0.013 � 0.004

2 0.048 1:470 -0.009 � 0.014 � 0.004

3 0.065 1:730 -0.028 � 0.015 � 0.004

4 0.087 2:000 -0.025 � 0.018 � 0.004

5 0.118 2:310 -0.034 � 0.019 � 0.004

6 0.166 2:660 -0.038 � 0.023 � 0.005

7 0.239 3:070 -0.006 � 0.030 � 0.008

8 0.338 3:790 0.078 � 0.051 � 0.012

9 0.450 5:250 -0.024 � 0.074 � 0.019

Bin x Q

2

A

h

+

1

� stat. � sys.

1 0.033 1:210 -0.051 � 0.032 � 0.006

2 0.048 1:450 -0.011 � 0.033 � 0.006

3 0.065 1:750 -0.030 � 0.034 � 0.006

4 0.087 2:140 -0.035 � 0.039 � 0.006

5 0.118 2:710 -0.024 � 0.041 � 0.007

6 0.165 3:680 0.006 � 0.049 � 0.008

7 0.238 5:180 -0.155 � 0.067 � 0.009

8 0.337 7:210 -0.092 � 0.138 � 0.013

9 0.447 9:810 -0.110 � 0.289 � 0.020

Bin x Q

2

A

h

�

1

� stat. � sys.

1 0.033 1:210 -0.079 � 0.037 � 0.006

2 0.048 1:450 0.021 � 0.039 � 0.006

3 0.065 1:750 -0.021 � 0.042 � 0.006

4 0.087 2:140 0.027 � 0.049 � 0.006

5 0.118 2:710 0.009 � 0.053 � 0.006

6 0.165 3:680 -0.022 � 0.066 � 0.007

7 0.238 5:180 0.064 � 0.092 � 0.009

8 0.337 7:210 -0.229 � 0.199 � 0.012

9 0.447 9:810 0.308 � 0.648 � 0.019

Table C.2: The 1995 asymmetries on the

3

He target. These data are from

[49℄.

C.2. INPUT ASYMMETRIES 131

C.2.2 1996 Asymmetries on the Proton Target

Bin x Q

2

A

e

+

1

� stat. � sys.

1 0.033 1:210 0.086 � 0.012 � 0.006

2 0.047 1:470 0.097 � 0.015 � 0.008

3 0.065 1:720 0.121 � 0.015 � 0.011

4 0.087 2:000 0.169 � 0.017 � 0.013

5 0.119 2:310 0.214 � 0.019 � 0.018

6 0.168 2:660 0.234 � 0.022 � 0.025

7 0.244 3:080 0.326 � 0.028 � 0.032

8 0.342 3:770 0.375 � 0.048 � 0.039

9 0.465 5:250 0.574 � 0.068 � 0.050

Bin x Q

2

A

h

+

1

� stat. � sys.

1 0.033 1:200 0.083 � 0.031 � 0.008

2 0.047 1:450 0.100 � 0.032 � 0.011

3 0.065 1:750 0.149 � 0.034 � 0.013

4 0.087 2:140 0.172 � 0.036 � 0.016

5 0.118 2:720 0.251 � 0.038 � 0.020

6 0.166 3:680 0.227 � 0.047 � 0.027

7 0.239 5:180 0.480 � 0.064 � 0.032

8 0.338 7:260 0.234 � 0.130 � 0.041

9 0.449 9:810 0.891 � 0.261 � 0.052

Bin x Q

2

A

h

�

1

� stat. � sys.

1 0.033 1:210 0.090 � 0.039 � 0.004

2 0.047 1:460 0.106 � 0.042 � 0.004

3 0.065 1:750 0.064 � 0.043 � 0.006

4 0.087 2:140 0.165 � 0.049 � 0.008

5 0.118 2:700 0.256 � 0.052 � 0.011

6 0.166 3:670 0.284 � 0.068 � 0.017

7 0.239 5:160 0.276 � 0.093 � 0.024

8 0.338 7:230 0.438 � 0.197 � 0.034

9 0.449 9:750 0.108 � 0.413 � 0.050

Table C.3: The 1996 asymmetries on the proton target. These data are from

[49℄.


C.2.3 1997 Asymmetries on the Proton Target

Bin x Q

2

A

e

+

1

� stat. � sys.

1 0.033 1:210 0.075 � 0.008 � 0.005

2 0.047 1:470 0.107 � 0.009 � 0.007

3 0.065 1:720 0.116 � 0.010 � 0.009

4 0.087 1:990 0.162 � 0.011 � 0.012

5 0.119 2:300 0.191 � 0.011 � 0.015

6 0.168 2:660 0.252 � 0.014 � 0.021

7 0.245 3:060 0.325 � 0.018 � 0.027

8 0.342 3:740 0.486 � 0.030 � 0.034

9 0.465 5:160 0.640 � 0.042 � 0.050

Bin x Q

2

A

h

+

1

� stat. � sys.

1 0.033 1:210 0.089 � 0.018 � 0.006

2 0.047 1:460 0.116 � 0.020 � 0.009

3 0.065 1:750 0.133 � 0.020 � 0.011

4 0.087 2:140 0.185 � 0.022 � 0.014

5 0.118 2:700 0.250 � 0.024 � 0.017

6 0.165 3:670 0.258 � 0.029 � 0.024

7 0.238 5:160 0.400 � 0.040 � 0.028

8 0.339 7:230 0.490 � 0.081 � 0.035

9 0.447 9:750 0.556 � 0.164 � 0.047

Bin x Q

2

A

h

�

1

� stat. � sys.

1 0.033 1:210 0.034 � 0.024 � 0.003

2 0.047 1:460 0.090 � 0.025 � 0.004

3 0.065 1:750 0.067 � 0.026 � 0.005

4 0.087 2:140 0.021 � 0.030 � 0.007

5 0.118 2:700 0.155 � 0.033 � 0.010

6 0.165 3:670 0.166 � 0.041 � 0.015

7 0.238 5:160 0.197 � 0.059 � 0.021

8 0.339 7:230 0.656 � 0.122 � 0.030

9 0.447 9:750 0.120 � 0.249 � 0.045

Table C.4: The 1997 asymmetries on the proton target. These data are from

[49℄.

C.2. INPUT ASYMMETRIES 133

C.2.4 1998 Asymmetries on the D Target

Bin x Q

2

A

e

+

1

� stat. � sys.

1 0.033 1:216 0.010 � 0.010 � 0.001

2 0.047 1:474 0.014 � 0.011 � 0.002

3 0.065 1:727 0.022 � 0.012 � 0.004

4 0.087 2:002 0.037 � 0.014 � 0.007

5 0.119 2:319 0.084 � 0.015 � 0.012

6 0.168 2:682 0.136 � 0.018 � 0.021

7 0.244 3:085 0.186 � 0.023 � 0.037

8 0.342 3:770 0.309 � 0.038 � 0.062

9 0.464 5:215 0.269 � 0.052 � 0.092

Bin x Q

2

A

h

+

1

� stat. � sys.

1 0.033 1:209 -0.009 � 0.026 � 0.001

2 0.048 1:459 0.053 � 0.027 � 0.005

3 0.065 1:772 0.011 � 0.028 � 0.003

4 0.087 2:169 0.036 � 0.032 � 0.006

5 0.118 2:744 0.124 � 0.035 � 0.013

6 0.165 3:728 0.091 � 0.042 � 0.013

7 0.238 5:216 0.269 � 0.058 � 0.028

8 0.337 7:298 0.041 � 0.120 � 0.022

9 0.449 9:974 0.562 � 0.237 � 0.063

Bin x Q

2

A

h

�

1

� stat. � sys.

1 0.033 1:207 0.008 � 0.030 � 0.001

2 0.047 1:467 -0.008 � 0.032 � 0.002

3 0.064 1:779 -0.046 � 0.034 � 0.005

4 0.087 2:186 -0.042 � 0.040 � 0.006

5 0.118 2:787 -0.007 � 0.043 � 0.007

6 0.165 3:742 -0.011 � 0.054 � 0.009

7 0.238 5:226 0.129 � 0.076 � 0.018

8 0.337 7:241 0.029 � 0.160 � 0.021

9 0.449 10:002 0.302 � 0.314 � 0.044

Table C.5: The 1998 asymmetries on the D target. These data are from [50℄.


Bin x Q

2

A

�

+

1

� stat. � sys.

1 0.033 1:209 -0.024 � 0.030 � 0.002

2 0.047 1:460 0.048 � 0.032 � 0.005

3 0.065 1:775 0.005 � 0.033 � 0.003

4 0.087 2:176 -0.016 � 0.038 � 0.005

5 0.118 2:753 0.156 � 0.041 � 0.016

6 0.165 3:746 0.086 � 0.050 � 0.012

7 0.238 5:215 0.206 � 0.070 � 0.024

8 0.337 7:301 0.032 � 0.145 � 0.021

9 0.449 10:062 0.535 � 0.291 � 0.060

Bin x Q

2

A

�

�

1

� stat. � sys.

1 0.033 1:206 0.020 � 0.032 � 0.002

2 0.047 1:466 -0.001 � 0.035 � 0.002

3 0.064 1:774 -0.038 � 0.037 � 0.005

4 0.087 2:173 -0.045 � 0.043 � 0.006

5 0.118 2:773 -0.009 � 0.047 � 0.007

6 0.165 3:712 -0.016 � 0.059 � 0.010

7 0.237 5:189 0.093 � 0.082 � 0.016

8 0.337 7:210 -0.027 � 0.172 � 0.021

9 0.451 10:032 0.095 � 0.337 � 0.037

Bin x Q

2

A

K

+

1

� stat. � sys.

1 0.034 1:209 0.050 � 0.071 � 0.005

2 0.048 1:457 0.113 � 0.072 � 0.007

3 0.065 1:760 0.066 � 0.075 � 0.006

4 0.087 2:185 0.179 � 0.085 � 0.015

5 0.118 2:856 0.012 � 0.088 � 0.006

6 0.165 3:904 0.111 � 0.105 � 0.014

7 0.239 5:455 0.390 � 0.146 � 0.034

8 0.336 7:480 0.029 � 0.305 � 0.024

9 0.450 9:704 0.530 � 0.617 � 0.070

Bin x Q

2

A

K

�

1

� stat. � sys.

1 0.034 1:212 -0.173 � 0.102 � 0.011

2 0.047 1:463 -0.012 � 0.104 � 0.002

3 0.064 1:776 -0.103 � 0.110 � 0.008

4 0.086 2:242 -0.056 � 0.126 � 0.008

5 0.117 2:881 -0.025 � 0.137 � 0.006

6 0.165 3:942 -0.065 � 0.174 � 0.008

7 0.237 5:450 0.183 � 0.254 � 0.019

8 0.341 7:590 -0.081 � 0.555 � 0.033

9 0.432 9:461 2.757 � 0.960 � 0.217

Table C.6: The 1998 asymmetries on the D target. These data are from [50℄.

C.3. CORRELATION COEFFICIENTS FOR PARTICLE COUNT RATES 135

C.3 Correlation CoeÆ ients for Parti le Count

Rates

Bin �(N

e

+

; N

h

+

) �(N

e

+

; N

h

�

) �(N

h

+

; N

h

�

)

1 0.446 0.395 0.128

2 0.492 0.417 0.137

3 0.507 0.411 0.130

4 0.497 0.386 0.118

5 0.451 0.336 0.105

6 0.364 0.260 0.096

7 0.260 0.177 0.083

8 0.183 0.120 0.067

9 0.127 0.081 0.054

Table C.7: The orrelation oeÆ ients of ount rates for the

3

He target in

1995. These data are from [49℄.

Bin �(N

e

+

; N

h

+

) �(N

e

+

; N

h

�

) �(N

h

+

; N

h

�

)

1 0.452 0.394 0.130

2 0.500 0.414 0.140

3 0.517 0.406 0.134

4 0.509 0.379 0.120

5 0.464 0.328 0.107

6 0.375 0.253 0.098

7 0.267 0.171 0.083

8 0.188 0.115 0.067

9 0.130 0.076 0.051

Table C.8: The orrelation oeÆ ients of ount rates for the proton target in

1997. These data are from [49℄.


Bin �(N

e

+

; N

h

+

) �(N

e

+

; N

h

�

) �(N

e

+

; N

�

+

)

1 0.374 0.325 0.324

2 0.402 0.344 0.345

3 0.401 0.337 0.343

4 0.385 0.313 0.326

5 0.353 0.281 0.298

6 0.292 0.226 0.244

7 0.215 0.166 0.179

8 0.155 0.119 0.129

9 0.106 0.081 0.085

Bin �(N

e

+

; N

�

�

) �(N

e

+

; N

K

+

) �(N

e

+

; N

K

�

)

1 0.299 0.140 0.097

2 0.316 0.155 0.107

3 0.310 0.157 0.106

4 0.290 0.149 0.097

5 0.261 0.135 0.085

6 0.210 0.113 0.067

7 0.155 0.082 0.047

8 0.110 0.060 0.033

9 0.076 0.043 0.024

Table C.9: The orrelation oeÆ ients of ount rates of in lusive events for

the D target in 1998. These data are from [67℄.

Bin �(N

h

+

; N

h

�

) �(N

h

+

; N

�

+

) �(N

h

+

; N

�

�

) �(N

h

+

; N

K

+

) �(N

h

+

; N

K

�

)

1 0.129 0.868 0.116 0.140 0.097

2 0.129 0.861 0.113 0.155 0.107

3 0.127 0.858 0.111 0.157 0.106

4 0.115 0.852 0.101 0.149 0.097

5 0.102 0.848 0.088 0.135 0.085

6 0.088 0.839 0.078 0.113 0.067

7 0.070 0.837 0.062 0.082 0.047

8 0.051 0.832 0.042 0.060 0.033

9 0.050 0.814 0.039 0.043 0.024

Table C.10: The orrelation oeÆ ients of ount rates of semi-in lusive h

+

for the D target in 1998. These data are from [67℄.

C.3. CORRELATION COEFFICIENTS FOR PARTICLE COUNT RATES 137

Bin �(N

h

�

; N

�

+

) �(N

h

�

; N

�

�

) �(N

h

�

; N

K

+

) �(N

h

�

; N

K

�

)

1 0.115 0.924 0.048 0.297

2 0.112 0.919 0.053 0.313

3 0.111 0.922 0.051 0.313

4 0.102 0.926 0.045 0.309

5 0.090 0.927 0.042 0.304

6 0.077 0.928 0.037 0.296

7 0.062 0.931 0.029 0.287

8 0.048 0.928 0.024 0.279

9 0.041 0.936 0.017 0.282

Table C.11: The orrelation oeÆ ients of ount rates for semi-in lusive h

�

for the D target in 1998. These data are from [67℄.

Bin �(N

�

+

; N

�

�

) �(N

�

+

; N

K

+

) �(N

�

+

; N

K

�

)

1 0.116 0.016 0.020

2 0.111 0.016 0.023

3 0.110 0.017 0.024

4 0.101 0.016 0.023

5 0.088 0.017 0.019

6 0.076 0.018 0.016

7 0.061 0.012 0.013

8 0.045 0.011 0.013

9 0.040 0.016 0.015

Bin �(N

�

�

; N

K

+

) �(N

�

�

; N

K

�

) �(N

K

+

; N

K

�

)

1 0.027 0.010 0.063

2 0.030 0.011 0.067

3 0.028 0.008 0.072

4 0.025 0.010 0.059

5 0.023 0.011 0.057

6 0.023 0.010 0.043

7 0.016 0.009 0.044

8 0.009 0.000 0.051

9 0.009 0.000 0.030

Table C.12: The orrelation oeÆ ients of ount rates for semi-in lusive �

and K for the D target in 1998. These data are from [67℄.


C.4 Results of Quark Polarizations and Distribu-

tions

C.4.1 Valen e De omposition

Results with the Mixing Matrix

Bin

�u

v

u

v

�d

v

d

v

�q

s

q

s

1 0.184 � 0.104 -0.437 � 0.220 0.029 � 0.082

2 0.201 � 0.075 -0.341 � 0.177 0.024 � 0.081

3 0.292 � 0.060 -0.158 � 0.157 -0.098 � 0.087

4 0.376 � 0.056 -0.106 � 0.166 -0.208 � 0.114

5 0.309 � 0.050 -0.119 � 0.174 -0.116 � 0.153

6 0.345 � 0.048 0.037 � 0.206 -0.311 � 0.259

7 0.367 � 0.038 -0.211 � 0.209 -0.107 � 0.444

8 0.472 � 0.050 0.289 � 0.369 |

9 0.694 � 0.066 -0.996 � 0.665 |

Table C.13: Results of valen e de omposition with the mixing matrix. Only

statisti errors are shown.

Bin

�

�u

v

u

v

;

�d

v

d

v

� �

�u

v

u

v

;

�q

s

q

s

� �

�d

v

d

v

;

�q

s

q

s

�

�

2

=ndf

1 0.863 -0.981 -0.935 0.701

2 0.797 -0.966 -0.911 0.675

3 0.748 -0.952 -0.896 0.500

4 0.745 -0.948 -0.899 1.525

5 0.743 -0.941 -0.906 1.564

6 0.723 -0.926 -0.910 1.583

7 0.510 -0.839 -0.859 1.805

8 0.370 | | 2.421

9 0.353 | | 0.748

Table C.14: Correlations among Results of the valen e de omposition with

the mixing matrix. The last olumn shows the minimum value

of �

2

in Eq.5.9.

C.4. RESULTS OF QUARK POLARIZATIONS AND DISTRIBUTIONS 139

Results with the New De�nition of Purity

Bin

�u

v

u

v

�d

v

d

v

�q

s

q

s

1 0.230 � 0.111 -0.332 � 0.233 -0.010 � 0.088

2 0.193 � 0.078 -0.358 � 0.182 0.033 � 0.085

3 0.304 � 0.060 -0.133 � 0.156 -0.114 � 0.088

4 0.371 � 0.055 -0.134 � 0.159 -0.193 � 0.110

5 0.318 � 0.048 -0.070 � 0.160 -0.153 � 0.143

6 0.344 � 0.045 0.029 � 0.184 -0.301 � 0.233

7 0.373 � 0.037 -0.175 � 0.196 -0.194 � 0.422

8 0.470 � 0.049 0.168 � 0.347 |

9 0.630 � 0.061 -1.517 � 0.579 |

Table C.15: Results of valen e de omposition with the new de�nition of pu-

rity. Only statisti errors are shown.

Bin

�

�u

v

u

v

;

�d

v

d

v

� �

�u

v

u

v

;

�q

s

q

s

� �

�d

v

d

v

;

�q

s

q

s

�

�

2

=ndf

1 0.878 -0.983 -0.943 1.058

2 0.810 -0.969 -0.916 0.656

3 0.749 -0.953 -0.895 0.374

4 0.729 -0.946 -0.890 1.686

5 0.708 -0.935 -0.889 1.651

6 0.673 -0.914 -0.890 1.221

7 0.482 -0.832 -0.847 1.666

8 0.342 | | 1.943

9 0.267 | | 1.275

Table C.16: Correlations among Results of the valen e de omposition with

the new de�nition of purity. The last olumn shows the mini-

mum value of �

2

in Eq.5.9.


C.4.2 Flavor De omposition


Bin

�u+��u

u+�u

�d+�

�

d

d+

�

d

�q

s

q

s

1 0.098 � 0.009 -0.114 � 0.025 0.029 � 0.082

2 0.118 � 0.010 -0.109 � 0.028 0.024 � 0.081

3 0.140 � 0.011 -0.123 � 0.030 -0.098 � 0.087

4 0.195 � 0.012 -0.160 � 0.036 -0.208 � 0.114

5 0.212 � 0.013 -0.118 � 0.041 -0.116 � 0.153

6 0.252 � 0.016 -0.100 � 0.053 -0.311 � 0.259

7 0.334 � 0.019 -0.180 � 0.075 -0.107 � 0.444

8 0.416 � 0.033 -0.080 � 0.168 |

9 0.661 � 0.047 -1.173 � 0.341 |

Table C.17: Results of avor de omposition with the mixing matrix. Only

statisti errors are shown.

Bin

�

�u+��u

u+�u

;

�d+�

�

d

d+

�

d

� �

�u+��u

u+�u

;

�q

s

q

s

� �

�d+�

�

d

d+

�

d

;

�q

s

q

s

�

�

2

=ndf

1 -0.778 0.027 -0.238 0.701

2 -0.745 -0.037 -0.254 0.675

3 -0.724 -0.067 -0.267 0.500

4 -0.703 -0.107 -0.286 1.525

5 -0.686 -0.132 -0.303 1.564

6 -0.715 -0.100 -0.234 1.583

7 -0.741 0.094 0.111 1.805

8 -0.636 | | 2.421

9 -0.546 | | 0.748

Table C.18: Correlations among Results of the avor de omposition with the

mixing matrix. The last olumn shows the minimum value of

�

2

in Eq.5.9.



Bin

�u+��u

u+�u

�d+�

�

d

d+

�

d

�q

s

q

s

1 0.097 � 0.009 -0.109 � 0.025 -0.010 � 0.088

2 0.118 � 0.010 -0.109 � 0.028 0.033 � 0.085

3 0.141 � 0.011 -0.122 � 0.030 -0.114 � 0.088

4 0.196 � 0.012 -0.165 � 0.035 -0.193 � 0.110

5 0.211 � 0.013 -0.108 � 0.040 -0.153 � 0.143

6 0.252 � 0.016 -0.101 � 0.052 -0.301 � 0.233

7 0.333 � 0.019 -0.181 � 0.074 -0.194 � 0.422

8 0.424 � 0.032 -0.102 � 0.163 |

9 0.662 � 0.045 -0.866 � 0.318 |

Table C.19: Results of avor de omposition with the new de�nition of purity.

Only statisti errors are shown.

Bin

�

�u+��u

u+�u

;

�d+�

�

d

d+

�

d

� �

�u+��u

u+�u

;

�q

s

q

s

� �

�d+�

�

d

d+

�

d

;

�q

s

q

s

�

�

2

=ndf

1 -0.775 0.034 -0.262 1.058

2 -0.740 -0.050 -0.254 0.656

3 -0.722 -0.093 -0.243 0.374

4 -0.707 -0.136 -0.241 1.686

5 -0.696 -0.170 -0.230 1.651

6 -0.722 -0.131 -0.160 1.221

7 -0.734 0.067 0.147 1.666

8 -0.630 | | 1.943

9 -0.566 | | 1.275

Table C.20: Correlations among Results of the avor de omposition with the

new de�nition of purity. The last olumn shows the minimum

value of �

2

in Eq.5.9.


C.4.3 Flavor Asymmetry of Light Sea Quarks


Bin �u+��u �d+�

�

d ��u+�

�

d ��u��

�

d

1 1.982 � 0.327 -2.102 � 0.476 0.149 � 1.264 0.245 � 1.409

2 1.941 � 0.235 -1.337 � 0.369 -0.565 � 0.756 1.307 � 0.909

3 1.675 � 0.185 -1.068 � 0.292 -0.748 � 0.538 0.582 � 0.659

4 1.677 � 0.158 -1.025 � 0.255 -0.553 � 0.455 -0.005 � 0.539

5 1.503 � 0.124 -0.493 � 0.204 -0.557 � 0.349 0.680 � 0.401

6 1.289 � 0.096 -0.256 � 0.160 -0.605 � 0.270 0.706 � 0.335

7 0.931 � 0.073 -0.196 � 0.124 0.125 � 0.149 -0.377 � 0.242

8 0.957 � 0.085 -0.343 � 0.141 | |

9 0.497 � 0.055 -0.138 � 0.105 | |

Table C.21: Results of avor asymmetry extra tion with the mixing matrix.


Bin

�

�u+��u;�d+�

�

d

� �

�u+��u;��u+�

�

d

� �

�u+��u;��u��

�

d

�

1 -0.219 -0.693 0.820

2 -0.284 -0.657 0.757

3 -0.296 -0.659 0.733

4 -0.260 -0.681 0.733

5 -0.256 -0.670 0.711

6 -0.434 -0.589 0.652

7 -0.791 -0.376 0.619

8 -0.842 | |

9 -0.815 | |

Bin

�

�d +�

�

d;��u+�

�

d

� �

�d+�

�

d;��u��

�

d

� �

��u+�

�

d;��u��

�

d

�

�

2

=ndf

1 -0.367 0.156 -0.714 0.786

2 -0.355 0.143 -0.709 0.493

3 -0.353 0.151 -0.743 0.472

4 -0.376 0.176 -0.764 1.685

5 -0.396 0.195 -0.767 1.446

6 -0.272 0.090 -0.763 1.287

7 0.230 -0.389 -0.690 1.723

8 | | | 1.282

9 | | | 0.391

Table C.22: Correlations among Results of the avor asymmetry extra tion

with the mixing matrix. The last olumn shows the minimum

value of �

2

in Eq.5.9.



Bin �u+��u �d+�

�

d ��u+�

�

d ��u��

�

d

1 2.178 � 0.349 -1.897 � 0.484 -0.979 � 1.405 1.202 � 1.520

2 1.911 � 0.237 -1.360 � 0.366 -0.435 � 0.760 1.173 � 0.907

3 1.655 � 0.181 -1.088 � 0.286 -0.676 � 0.496 0.451 � 0.618

4 1.604 � 0.148 -1.096 � 0.245 -0.257 � 0.381 -0.342 � 0.475

5 1.487 � 0.114 -0.495 � 0.193 -0.504 � 0.272 0.635 � 0.340

6 1.234 � 0.089 -0.300 � 0.154 -0.350 � 0.203 0.412 � 0.278

7 0.929 � 0.072 -0.189 � 0.123 0.091 � 0.133 -0.357 � 0.230

8 0.914 � 0.081 -0.273 � 0.132 | |

9 0.465 � 0.052 -0.045 � 0.091 | |

Table C.23: Results of avor asymmetry extra tion with the new de�nition

of purity. Only statisti errors are shown.

Bin

�

�u+��u;�d+�

�

d

� �

�u+��u;��u+�

�

d

� �

�u+��u;��u��

�

d

�

1 -0.137 -0.736 0.843

2 -0.291 -0.664 0.760

3 -0.369 -0.636 0.713

4 -0.401 -0.623 0.686

5 -0.444 -0.593 0.644

6 -0.578 -0.493 0.576

7 -0.795 -0.359 0.616

8 -0.827 | |

9 -0.793 | |

Bin

�

�d +�

�

d;��u+�

�

d

� �

�d+�

�

d;��u��

�

d

� �

��u+�

�

d;��u��

�

d

�

�

2

=ndf

1 -0.405 0.211 -0.760 1.101

2 -0.340 0.129 -0.708 0.546

3 -0.294 0.081 -0.704 0.384

4 -0.276 0.052 -0.685 1.851

5 -0.255 0.024 -0.660 1.514

6 -0.158 -0.055 -0.635 1.190

7 0.251 -0.414 -0.649 1.595

8 | | | 1.397

9 | | | 0.520

Table C.24: Correlations among Results of the avor asymmetry extra tion

with the new de�nition of purity. The last olumn shows the

minimum value of �

2

in Eq.5.9.


C.4.4 Single Quark Distributions


Bin �u �d

1 1.785 � 1.002 -2.299 � 1.060

2 1.570 � 0.636 -1.709 � 0.697

3 1.758 � 0.464 -0.986 � 0.516

4 1.956 � 0.387 -0.746 � 0.435

5 1.442 � 0.293 -0.555 � 0.335

6 1.239 � 0.236 -0.306 � 0.268

7 1.057 � 0.160 -0.070 � 0.189

8 0.639 � 0.184 -0.661 � 0.215

9 0.623 � 0.123 -0.012 � 0.151

Bin ��u �

�

d ��s

1 0.197 � 0.947 -0.048 � 0.947 0.035 � 0.671

2 0.371 � 0.591 -0.936 � 0.591 -0.491 � 0.405

3 -0.083 � 0.425 -0.665 � 0.425 -0.467 � 0.289

4 -0.279 � 0.353 -0.274 � 0.353 -0.276 � 0.243

5 0.061 � 0.266 -0.618 � 0.266 -0.387 � 0.186

6 0.051 � 0.215 -0.655 � 0.215 -0.415 � 0.145

7 -0.126 � 0.142 0.251 � 0.142 0.123 � 0.084

8 | | |

9 | | |

Table C.25: Results of single quark distribution with the mixing matrix.




Bin �u �d

1 2.067 � 1.092 -2.008 � 1.142

2 1.541 � 0.637 -1.729 � 0.696

3 1.768 � 0.435 -0.975 � 0.489

4 1.903 � 0.339 -0.797 � 0.391

5 1.421 � 0.246 -0.561 � 0.291

6 1.203 � 0.194 -0.331 � 0.231

7 1.062 � 0.151 -0.056 � 0.181

8 0.668 � 0.164 -0.519 � 0.195

9 0.620 � 0.103 0.109 � 0.127

Bin ��u �

�

d ��s

1 0.112 � 1.035 -1.091 � 1.035 -0.681 � 0.743

2 0.369 � 0.592 -0.804 � 0.592 -0.405 � 0.406

3 -0.112 � 0.396 -0.564 � 0.396 -0.410 � 0.267

4 -0.300 � 0.305 0.042 � 0.305 -0.074 � 0.205

5 0.066 � 0.218 -0.570 � 0.218 -0.353 � 0.147

6 0.031 � 0.172 -0.381 � 0.172 -0.241 � 0.111

7 -0.133 � 0.133 0.224 � 0.133 0.103 � 0.076

8 | | |

9 | | |

Table C.26: Results of single quark distribution with the new de�nition of

purity. Only statisti errors are shown.


C.5 Moments of Polarized Quark Distributions

C.5.1 Valen e De omposition


N

q

�

�q

�

2

/ndf

�u

v

0.77 � 0.10 0.42 � 0.08 1.53/9

�d

v

-0.01 � 0.01 -1.24 � 0.28 0.57/9

Table C.27: Fit parameters of the valen e quarks in the valen e de omposi-

tion with the mixing matrix.


�u

v

0.56 � 0.09 0.04 0.02 0.62 � 0.09

�d

v

-0.14 � 0.16 -0.03 0.00 -0.16 � 0.16

�q

s

-0.03 � 0.05 -0.00 0.00 -0.03 � 0.05

Table C.28: First moment of polarized quark distributions in the valen e

de omposition with the mixing matrix.


N

q

�

�q

�

2

/ndf

�u

v

0.71 � 0.09 0.37 � 0.08 1.08/9

�d

v

-0.01 � 0.01 -1.01 � 0.28 1.17/9

Table C.29: Fit parameters of the valen e quarks in the valen e de omposi-

tion with the new de�nition of purity.


�u

v

0.56 � 0.09 0.04 0.02 0.62 � 0.09

�d

v

-0.14 � 0.15 -0.02 0.00 -0.17 � 0.15

�q

s

-0.03 � 0.05 -0.00 0.00 -0.04 � 0.05

Table C.30: First moment of polarized quark distributions in the valen e

de omposition with the new de�nition of purity.

C.5. MOMENTS OF POLARIZED QUARK DISTRIBUTIONS 147

C.5.2 Flavor De omposition


N

q

�

�q

�

2

/ndf

�u+��u 0.90 � 0.06 0.67 � 0.03 1.89/9

�d+�

�

d -0.21 � 0.07 0.19 � 0.12 1.53/9

Table C.31: Fit parameters of the valen e quarks in the avor de omposition

with the mixing matrix.


�u+��u 0.52 � 0.04 0.04 0.02 0.57 � 0.04

�d+�

�

d -0.22 � 0.08 -0.03 0.00 -0.26 � 0.08

�q

s

-0.03 � 0.05 -0.00 0.00 -0.03 � 0.05

Table C.32: First moment of polarized quark distributions in the avor de-

omposition with the mixing matrix.


N

q

�

�q

�

2

/ndf

�u+��u 0.92 � 0.06 0.68 � 0.03 1.94/9

�d+�

�

d -0.20 � 0.07 0.19 � 0.12 1.05/9

Table C.33: Fit parameters of the valen e quarks in the avor de omposition

with the new de�nition of purity.


�u+��u 0.52 � 0.04 0.04 0.02 0.57 � 0.04

�d+�

�

d -0.21 � 0.07 -0.03 0.00 -0.24 � 0.07

�q

s

-0.03 � 0.05 -0.00 0.00 -0.04 � 0.05

Table C.34: First moment of polarized quark distributions in the avor de-

omposition with the new de�nition of purity.


C.5.3 Flavor Asymmetry of Light Sea Quarks


N

q

�

�q

�

2

/ndf

�u+��u 0.89 � 0.08 0.63 � 0.05 1.31/9

�d+�

�

d -0.17 � 0.06 0.12 � 0.14 0.81/9

Table C.35: Fit parameters of the valen e quarks in the avor asymmetry

extra tion with the mixing matrix.


�u+��u 0.54 � 0.05 0.04 0.02 0.60 � 0.05

�d+�

�

d -0.21 � 0.09 -0.03 0.00 -0.25 � 0.09

��u+�

�

d -0.09 � 0.09 -0.01 0.00 -0.11 � 0.09

��u��

�

d 0.10 � 0.10 0.02 0.00 0.12 � 0.10

Table C.36: First moment of polarized quark distributions in the avor asym-

metry extra tion with the mixing matrix.


N

q

�

�q

�

2

/ndf

�u+��u 0.83 � 0.08 0.61 � 0.05 1.03/9

�d+�

�

d -0.18 � 0.07 0.16 � 0.14 0.52/9

Table C.37: Fit parameters of the valen e quarks in the avor asymmetry

extra tion with the new de�nition of purity.


�u+��u 0.53 � 0.05 0.04 0.02 0.59 � 0.05

�d+�

�

d -0.19 � 0.08 -0.03 0.00 -0.22 � 0.08

��u+�

�

d -0.09 � 0.08 -0.01 0.00 -0.10 � 0.08

��u��

�

d 0.09 � 0.09 0.02 0.00 0.11 � 0.09

Table C.38: First moment of polarized quark distributions in the avor asym-

metry extra tion with the new de�nition of purity.

C.5. MOMENTS OF POLARIZED QUARK DISTRIBUTIONS 149

C.5.4 Single Quark Distributions


N

q

�

�q

�

2

/ndf

�u 0.81 � 0.18 0.53 � 0.12 0.56/9

�d -0.13 � 0.07 -0.09 � 0.21 1.11/9

Table C.39: Fit parameters of the valen e quarks in the single quark distri-

bution with the mixing matrix.


�u 0.54 � 0.13 0.04 0.02 0.59 � 0.13

�d -0.22 � 0.14 -0.03 0.00 -0.25 � 0.14

��u 0.01 � 0.07 0.00 0.00 0.01 � 0.07

�

�

d -0.10 � 0.07 -0.02 0.00 -0.11 � 0.07

��s -0.06 � 0.05 -0.01 0.00 -0.07 � 0.05

Table C.40: First moment of polarized quark distributions in the single quark

distribution with the mixing matrix.


N

q

�

�q

�

2

/ndf

�u 0.83 � 0.17 0.54 � 0.11 0.68/9

�d -0.12 � 0.07 -0.12 � 0.21 0.98/9

Table C.41: Fit parameters of the valen e quarks in the single quark distri-

bution with the new de�nition of purity.


�u 0.54 � 0.12 0.04 0.02 0.60 � 0.12

�d -0.18 � 0.13 -0.03 0.00 -0.21 � 0.13

��u 0.00 � 0.06 0.00 0.00 0.00 � 0.06

�

�

d -0.09 � 0.06 -0.02 0.00 -0.10 � 0.06

��s -0.06 � 0.04 -0.01 0.00 -0.07 � 0.04

Table C.42: First moment of polarized quark distributions in the single quark

distribution with the new de�nition of purity.

LIST OF FIGURES 151

List of Figures

2.1 The diagram of DIS . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2.2 De�nition of DIS kinemati variables . . . . . . . . . . . . . . . . . . 5

2.3 De�nition of s attering angles in DIS . . . . . . . . . . . . . . . . . . 5

2.4 DIS Pro ess in Breit-Frame . . . . . . . . . . . . . . . . . . . . . . . 9

2.5 Diagram of polarized DIS . . . . . . . . . . . . . . . . . . . . . . . . 11

2.6 The re ent results of g

1

(x) on the proton . . . . . . . . . . . . . . . . 13

2.7 Diagram of semi-in lusive DIS . . . . . . . . . . . . . . . . . . . . . . 16

2.8 Charged hadron spe tra orre ted for dete tor in uen es . . . . . . . 19

3.1 The HERA storage ring . . . . . . . . . . . . . . . . . . . . . . . . . 22

3.2 The longitudinal polarization of HERA beam . . . . . . . . . . . . . 22

3.3 The internal target of HERMES . . . . . . . . . . . . . . . . . . . . . 23

3.4 The HERMES Spe trometer . . . . . . . . . . . . . . . . . . . . . . . 24

3.5 A three dimensional CAD diagram of the HERMES Spe trometer . . 25

3.6 S hemati view of the HERMES BC module . . . . . . . . . . . . . . 26

3.7 Isometri view of the HERMES Calorimeter . . . . . . . . . . . . . . 27

4.1 The diagram of Cherenkov radiation . . . . . . . . . . . . . . . . . . 30

4.2 S hemati view of Ring Imaging Cherenkov Counter at HERMES . . 31

4.3 The relation between Cherenkov angle and parti le momentum . . . . 32

4.4 Momentum ranges for hadron separation . . . . . . . . . . . . . . . . 32

4.5 Composition of sili a aerogel . . . . . . . . . . . . . . . . . . . . . . . 34

4.6 Composition of sili a aerogel . . . . . . . . . . . . . . . . . . . . . . . 34

4.7 Photo of aerogel tile . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

4.8 Typi al size of one aerogel tile . . . . . . . . . . . . . . . . . . . . . . 35

4.9 Aerogel wall as radiator in RICH . . . . . . . . . . . . . . . . . . . . 36

4.10 EÆ ien ies for transmittan e though lu ite . . . . . . . . . . . . . . . 36

4.11 Distribution of the refra tive index of 1040 sele ted aerogels . . . . . 38

4.12 The distribution of refra tive index n of aerogel in two groups . . . . 38

4.13 Distributions of the size of the aerogel tiles . . . . . . . . . . . . . . . 40

4.14 Des ription of the shape of aerogel surfa e . . . . . . . . . . . . . . . 41

4.15 The wavelength dependen e of the mirror re e tivity . . . . . . . . . 42

4.16 S hemati pi ture of PMTs with funnels . . . . . . . . . . . . . . . . 43

4.17 The photon olle tion eÆ ien y verses angle of in iden e . . . . . . . 43

4.18 The RICH in the MC . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

4.19 Comparison of the Cherenkov angles between the experimental and

the MC data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

152 LIST OF FIGURES

4.20 Comparison of the number of �red PMTs between the experimental

data and the MC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

4.21 The �

x

and �

y

dependen e of the eÆ ien y before MC modi� ation . 53

4.22 The �

x

and �

y

dependen e of the eÆ ien y after MC modi� ation . . 54

4.23 The Cherenkov rings of aerogel and gas on the PMT plane . . . . . . 55

4.24 The mirror re e tivity versus the number of �red PMTs with MC . . 55

5.1 In lusive (e

+

) purities for proton target . . . . . . . . . . . . . . . . . 62

5.2 h

+

purities for proton target . . . . . . . . . . . . . . . . . . . . . . . 63

5.3 h

�


5.4 �

+


5.5 �

�


5.6 K

+

purities for proton target . . . . . . . . . . . . . . . . . . . . . . 65

5.7 K

�

purities for proton target . . . . . . . . . . . . . . . . . . . . . . 65

5.8 In lusive (e

+

) purities for neutron target . . . . . . . . . . . . . . . . 66

5.9 h

+

purities for neutron target . . . . . . . . . . . . . . . . . . . . . . 67

5.10 h

�


5.11 �

+


5.12 �

�


5.13 K

+


5.14 K

�


5.15 The Q

2

dependen e of the purities for the proton target . . . . . . . . 74

5.16 The Q

2

dependen e of the purities for the neutron target . . . . . . . 75

5.17 The ratio of the multipli ities on the proton and the deuteron target . 81

5.18 The in lusive and semi in lusive asymmetries on the

3

He target . . . 82

5.19 The in lusive and semi in lusive asymmetries on the proton target . . 83

5.20 The in lusive and semi in lusive asymmetries on the D target . . . . 84

6.1 The avor de omposition from MC . . . . . . . . . . . . . . . . . . . 88

6.2 The avor asymmetry extra tion from MC . . . . . . . . . . . . . . . 89

6.3 The extra ted quark polarizations in valen e de omposition from ex-

perimental data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

6.4 The quark distributions obtained from the valen e de omposition result 91

6.5 The extra ted quark polarizations in avor de omposition from ex-

perimental data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

6.6 The quark distributions obtained from the avor de omposition result 95

6.7 The extra ted quark distributions in luding avor asymmetry of light

sea quarks. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

6.8 The quark distributions times x from the avor asymmetry extra tion. 98

6.9 The extra ted single quark distributions. . . . . . . . . . . . . . . . . 100

6.10 The single quark distributions times x from the avor asymmetry

extra tion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

6.11 Comparison of the avor de omposition using purities from CTEQ4LQ

and GRV . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

6.12 Comparison of the avor asymmetry extra tion using purities from

CTEQ4LQ and GRV . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

6.13 The �tted polarized valen e quark distributions . . . . . . . . . . . . 105

LIST OF FIGURES 153

6.14 The �tted polarized valen e quark distributions . . . . . . . . . . . . 107

6.15 Fragmentation fun tions in the Spe tator model . . . . . . . . . . . . 113

6.16 Flavor asymmetry of light sea quarks extra ted from Eq.6.17 . . . . . 114

6.17 Comparison between the dire t extra tion method and the purity

matrix method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

A.1 De�nition and Distribution of L

gas

for MC data . . . . . . . . . . . . 121

A.2 The ratio between N

gas

(EXP) and N

gas

(MC) . . . . . . . . . . . . . . 122

A.3 The position dependen e of mirror re e tivity . . . . . . . . . . . . . 123

A.4 Re ent Comparison of the Cherenkov angles between the experimen-

tal and the MC data . . . . . . . . . . . . . . . . . . . . . . . . . . . 124

LIST OF TABLES 155

List of Tables

2.1 De�nition of DIS kinemati al values . . . . . . . . . . . . . . . . . . . 4

2.2 De�nition of semi-in lusive kinemati al values . . . . . . . . . . . . . 16

2.3 Fitted parameters of LUND model for HERMES . . . . . . . . . . . . 18

4.1 Comparison of the Cherenkov angle for ea h MC produ tion and the

experimental data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

4.2 Comparison of the number of �red PMTs for ea h MC produ tion

and experimental data . . . . . . . . . . . . . . . . . . . . . . . . . . 51

6.1 The �rst moment for the valen e de omposition . . . . . . . . . . . . 106

6.2 The �rst moment for the avor de omposition . . . . . . . . . . . . . 106

6.3 The �rst moment for avor asymmetry extra tion . . . . . . . . . . . 107

6.4 The �rst moment for single quark distribution . . . . . . . . . . . . . 107

6.5 Comparison of the �rst moment between this study and the results

by SMC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108

6.6 Comparison of the �rst moment between this study and the predi tion

by the latti e QCD. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108

6.7 Comparison of the �rst moment between this study and the predi tion

by the SU(3) analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

C.1 x binning de�nition . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129

C.2 The 1995 asymmetries on the

3

He target . . . . . . . . . . . . . . . . 130

C.3 The 1996 asymmetries on the proton target . . . . . . . . . . . . . . 131

C.4 The 1997 asymmetries on the proton target . . . . . . . . . . . . . . 132

C.5 The 1998 asymmetries on the D target . . . . . . . . . . . . . . . . . 133

C.6 The 1998 asymmetries on the D target . . . . . . . . . . . . . . . . . 134

C.7 The orrelation oeÆ ients of ount rates for the

3

He target in 1995 . 135

C.8 The orrelation oeÆ ients of ount rates for the proton target in 1997135

C.9 The orrelation oeÆ ients of ount rates of in lusive events for the

D target in 1998 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136

C.10 The orrelation oeÆ ients of ount rates of semi-in lusive h

+

for the

D target in 1998 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136

C.11 The orrelation oeÆ ients of ount rates for semi-in lusive h

�

for

the D target in 1998 . . . . . . . . . . . . . . . . . . . . . . . . . . . 137

C.12 The orrelation oeÆ ients of ount rates for semi-in lusive � and K

for the D target in 1998 . . . . . . . . . . . . . . . . . . . . . . . . . 137

C.13 Results of valen e de omposition with the mixing matrix . . . . . . . 138

156 LIST OF TABLES

C.14 Correlations among Results of the valen e de omposition with the

mixing matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138

C.15 Results of valen e de omposition with the new de�nition of purity . . 139

C.16 Correlations among Results of the valen e de omposition with the

new de�nition of purity . . . . . . . . . . . . . . . . . . . . . . . . . . 139

C.17 Results of avor de omposition with the mixing matrix . . . . . . . . 140

C.18 Correlations among Results of the avor de omposition with the mix-

ing matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140

C.19 Results of avor de omposition with the new de�nition of purity . . . 141

C.20 Correlations among Results of the avor de omposition with the new

de�nition of purity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141

C.21 Results of avor asymmetry extra tion with the mixing matrix . . . . 142

C.22 Correlations among Results of the avor asymmetry extra tion with

the mixing matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142

C.23 Results of avor asymmetry extra tion with the new de�nition of purity143

C.24 Correlations among Results of the avor asymmetry extra tion with

the new de�nition of purity . . . . . . . . . . . . . . . . . . . . . . . 143

C.25 Results of single quark distribution with the mixing matrix . . . . . . 144

C.26 Results of single quark distribution with the new de�nition of purity . 145

C.27 Fit parameters of the valen e quarks in the valen e de omposition

with the mixing matrix . . . . . . . . . . . . . . . . . . . . . . . . . . 146

C.28 First moment of polarized quark distributions in the valen e de om-

position with the mixing matrix . . . . . . . . . . . . . . . . . . . . . 146

C.29 Fit parameters of the valen e quarks in the valen e de omposition

with the new de�nition of purity . . . . . . . . . . . . . . . . . . . . . 146

C.30 First moment of polarized quark distributions in the valen e de om-

position with the new de�nition of purity . . . . . . . . . . . . . . . . 146

C.31 Fit parameters of the valen e quarks in the avor de omposition with

the mixing matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147

C.32 First moment of polarized quark distributions in the avor de ompo-

sition with the mixing matrix . . . . . . . . . . . . . . . . . . . . . . 147

C.33 Fit parameters of the valen e quarks in the avor de omposition with

the new de�nition of purity . . . . . . . . . . . . . . . . . . . . . . . 147

C.34 First moment of polarized quark distributions in the avor de ompo-

sition with the new de�nition of purity . . . . . . . . . . . . . . . . . 147

C.35 Fit parameters of the valen e quarks in the avor asymmetry extra -

tion with the mixing matrix . . . . . . . . . . . . . . . . . . . . . . . 148

C.36 First moment of polarized quark distributions in the avor asymmetry

extra tion with the mixing matrix . . . . . . . . . . . . . . . . . . . . 148

C.37 Fit parameters of the valen e quarks in the avor asymmetry extra -

tion with the new de�nition of purity . . . . . . . . . . . . . . . . . . 148

C.38 First moment of polarized quark distributions in the avor asymmetry

extra tion with the new de�nition of purity . . . . . . . . . . . . . . . 148

C.39 Fit parameters of the valen e quarks in the single quark distribution

with the mixing matrix . . . . . . . . . . . . . . . . . . . . . . . . . . 149

C.40 First moment of polarized quark distributions in the single quark

distribution with the mixing matrix . . . . . . . . . . . . . . . . . . . 149

LIST OF TABLES 157

C.41 Fit parameters of the valen e quarks in the single quark distribution

with the new de�nition of purity . . . . . . . . . . . . . . . . . . . . . 149

C.42 First moment of polarized quark distributions in the single quark

distribution with the new de�nition of purity . . . . . . . . . . . . . . 149

BIBLIOGRAPHY 159

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A knowledgements

During my study at the Tokyo Institute of Te hnology, I was helped by a lot of

people. I'd like to express my thanks to all people who supported me.

First of all, I would like to thank Prof. Toshi-Aki Shibata who gave me a han e

to study the spin physi s at HERMES. I am indebted to Dr. Yasuhiro Sakemi for

his valuable advi e and suggestions during in my entire analysis. I wish to express

my thanks to Dr. Stefan Bernreuther for reading the entire text in its original form

and giving helpful omments. My thanks are also due to the members of Tokyo

group with whom I worked. Espe ially, Dr. Hideyuki Kobayashi gave me many

advi e, whi h was helpful for me. Thanks are due to Mr. Shigenobu Yoneyama who

provide me with his asymmetries data on the D target in 1998. I'd like to thank Mr.

Kentaro Suetsugu who analyzed the RICH performan e. His analysis and advi e

have been helpful for my study. I am indebted to all members in my laboratory.

Additionally, I thank the members of the HERMES Collaboration. Parti ularly,

the members of the RICH group and the �q analysis group gave me valuable sug-

gestions and riti isms. Thanks to all of them.

CONTENTS · Lab. t Departmen of ysics Ph o oky T Institute of hnology ec T ebruary F 2001. Abstract...

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