博士論文
Search for Muon Neutrino Oscillationsin Kamiokande and Super-Kamiokande(カミオカンデとスーパーカミオカンデにおけるミューオン・ニュートリノ振動の探索)

畠山 衆一郎
平成 9 年

Acknowledgments
At rst, I would like to express great appreciation to Prof. A.Suzuki. He introduced me to theneutrino physics and the theme of this thesis. Prof. M.Mori gave me helpful suggestions about theanalysis of the upward going muon. Dr. T.Hasegawa indicated me useful information about thecross section of charged current interaction and had some important discussions to me about them.
I would like to thank Prof. Y.Totsuka, spokesman of Kamiokande and Super-Kamiokande experiments. His deep insight into physics and experiments was indispensable to Kamiokande andSuper-Kamiokande experiments. I also thank to ICRR sta s, Prof. Y.Suzuki, Prof. T.Kajita,
Prof. M.Nakahata, Dr. K.Inoue, Dr. K.Martens, Dr. Y.Fukuda, Dr. Y.Itow, Dr. Y.Takeuchi, Dr.
M.Miura, Dr. M.Shiozawa, Dr. Y.Koshio.
Dr. M.Takita and the other colleagues in the upward going muon analysis group in SuperKamiokande had many valuable discussions to me. Dr. Y.Oyama and Dr. T.Hara gave me manysuggestions about the analysis of upward going muon in Kamiokande-II,III.
I would like to thank M.Koga, K.Fujita and the other graduate students and sta s in Tohoku
University, and the other graduate students and sta s in Kamiokande and Super-Kamiokande collaboration group. Special thanks to C.Mauger. He kindly checked my english.
At last, I would like to thank my parents and sisters. I have been able to concentrate on mystudy with their continuous help and supports.

1

Contents

1

Introduction

1.1 Theoretical Background of Neutrino Oscillations . . . . . . . . . . . . . . . . . . .
1.1.1 Neutrino Oscillations in Vacuum . . . . . . . . . . . . . . . . . . . . . . .
1.1.2 Neutrino Oscillations in Matter . . . . . . . . . . . . . . . . . . . . . . . .
1.2 Searches for Neutrino Oscillations . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2.1 Reactor and Accelerator Neutrino Experiments . . . . . . . . . . . . . . . .
1.2.2 Solar Neutrino Experiments . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2.3 Atmospheric Neutrino Experiments . . . . . . . . . . . . . . . . . . . . . .
1.2.4 Upward Through Going Muons . . . . . . . . . . . . . . . . . . . . . . . .
1.3 Oscillation Search by Upward-Through-Going Muons . . . . . . . . . . . . . . . . .

4
5
7
12
12
17
21
24
27

2

Kamiokande and Super-Kamiokande Detectors

28

3

Event Selection

40

2.1 Kamioka Underground Laboratory . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2 Brief History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3 The Detector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3.1 Ring-Imaging Water Cherenkov Detector . . . . . . . . . . . . . . . . . . .
2.3.2 Detector Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3.3 20 Inch-Diameter PMT . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3.4 Water Puri cation System and Radon Free Air System . . . . . . . . . . . .
2.3.5 Electronics and DAQ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3.6 Real Time Data Reductions . . . . . . . . . . . . . . . . . . . . . . . . . .
3.1 Event Selection Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2 Typical Muon Events . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.3 Automatic Muon Directional Fitting . . . . . . . . . . . . . . . . . . . . . . . . . .
3.3.1 Muon Fitting for Kamiokande-III . . . . . . . . . . . . . . . . . . . . . . .
3.3.2 1st Muon Fitting for Super-Kamiokande . . . . . . . . . . . . . . . . . . .
3.3.3 2nd Muon Fitting for Super-Kamiokande . . . . . . . . . . . . . . . . . . .
3.4 Total Q and Event Classi cation . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.5 Other Q Cuts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.6 Muon Zenith Angle Selection . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.6.1 Zenith Angle Selection for Kamiokande-III . . . . . . . . . . . . . . . . . .
3.6.2 Zenith Angle Selection for Super-Kamiokande . . . . . . . . . . . . . . . .
3.7 Eye Scan & Manual Muon Fitting . . . . . . . . . . . . . . . . . . . . . . . . . .
3.8 Summary of the Event Selection . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2

4

28
28
30
30
31
34
36
37
37

40
44
56
56
61
62
69
73
73
73
74
74
77

CONTENTS

3

4

Observed Flux of Upward Through Going Muons

78

5

Expected Flux of Upward Through Going Muons

92

6

Results and Discussions

110

7Conclusion

122

A

The Analysis by Another Energy Threshold

123

4.1
4.2
4.3
4.4
4.5
4.6

Live Time Calculation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
E ective Area . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Selection Eciency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Contamination by Cosmic-Ray Muons . . . . . . . . . . . . . . . . . . . . . . . .
Systematic Errors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Observed Flux . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.6.1 Upward Through Going Muon Flux of Kamiokande-II+III . . . . . . . . . . .
4.6.2 Upward Through Going Muon Flux of Super-Kamiokande . . . . . . . . . .

79
81
82
83
87
90
90
91

5.1 Calculation of Upward Through Going Muon Flux . . . . . . . . . . . . . . . . . . 92
5.2 Atmospheric Muon Neutrino Flux . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
5.2.1 Primary Cosmic-Ray Fluxes . . . . . . . . . . . . . . . . . . . . . . . . . . 94
5.2.2 Production and Decay of Hadrons . . . . . . . . . . . . . . . . . . . . . . . 96
5.2.3 Comparison of Several Calculations . . . . . . . . . . . . . . . . . . . . . . 96
5.2.4 Uncertainties of Atmospheric Neutrino Fluxes . . . . . . . . . . . . . . . . . 96
5.3 Neutrino Nucleon Cross Section . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
5.3.1 Cross Section Formula for CC Interaction . . . . . . . . . . . . . . . . . . . 100
5.3.2 QCD Parton Distribution Functions . . . . . . . . . . . . . . . . . . . . . . 100
5.3.3 Comparison with Experimental Data . . . . . . . . . . . . . . . . . . . . . 103
5.4 Muon Energy Loss in Rock . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
5.5 Application to Kamiokande/Super-Kamiokande Detector . . . . . . . . . . . . . . 106
5.6 Calculated Upward Through Going Muon Fluxes . . . . . . . . . . . . . . . . . . . 108
6.1
6.2
6.3
6.4
6.5

Comparison between Expected and Observed Flux without Oscillations . . . . . . . 110
Method to Examine Neutrino Oscillations . . . . . . . . . . . . . . . . . . . . . . . 113
 $  Oscillations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
e $  Oscillations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
Comparison with Other Experiments . . . . . . . . . . . . . . . . . . . . . . . . . 119

Chapter 1
Introduction

1.1 Theoretical Background of Neutrino Oscillations
Elementary particle physics has been developed by interactive activities between theories and experiments. At present, the \Standard Model" formulated by Weinberg-Salam is accepted by many
physicists not only because it has a mathematically beautiful formulation but also has been tested
by many experiments to incredible accuracy. Is our goal then to complete the Standard Model ? Let
us look at neutrinos, here is essentially unsolved problems. Neutrino masses, if any, are abnormallysmall in comparison with other fermions. In case of the photon, its zero mass is assured by charge
conservation law, but it cannot be applied to the neutrino.
On the other hand, many models which are trying to unify the weak, electromagnetic and strong
interactions predict nite neutrino masses. In this viewpoint, neutrinos are the members of thesame multiplet including quarks and charged leptons. So it may be natural that neutrinos havenite masses. Table 1.1 shows the list of fermion masses (or the upper limits) from the recent
experimental results.

< 15eVme me =0.5 MeV
mu =28 MeV
md =515 MeVm < 170keVm =106 MeV
mc =1.06 GeV
ms =100300 MeV< 24MeVm m =1.777 GeV
mt =180612 GeV
mb =4.14.5 GeV

Table 1.1: List of fermion masses [1].
Many experiments have been carrying out to measure neutrino masses. In a direct method, theupper limit of the electron neutrino mass was derived from precise measurement of the end-point
energy of the tritium decay spectrum . The upper limit of the muon neutrino mass comes frommeasuring the momentum of muons produced in the decay of pions at rest . The upper limit of thetau neutrino mass was obtained by the end-point shape of the invariant mass spectrum of 5 pions
decay made of  0 ,  0 ! 30 + 2 + +  .
On the other hand, another method to test the nite neutrino mass was considered in relatively
early time by assuming the mixing of neutrino avors in the weak interaction. If neutrinos havethe nite mass, it is possible the mass eigenstates are di erent from the weak interaction avor
eigenstates. In such a situation, a neutrino changes its avor to another one while it is traveling.
This phenomenon is called \neutrino oscillations".

4CHAPTER 1.

5

INTRODUCTION

Intuitively, the neutrino oscillations can be explained as follows:each neutrino mass eigenstate propagates by a di erent phase velocity, so the phase of each avor
eigenstate is gradually shifting. Therefore a neutrino appears with di erent avor eigenstates fromits initial one along traveling.
The probability that one neutrino changes to another avor neutrino depends on the di erenceof masses of two neutrinos and the mixing angle parameter between them. So if the neutrino
oscillations is measured, one can get the evidence of the nite neutrino masses.

1.1.1 Neutrino Oscillations in Vacuum
In general, the avor eigenstates, e , and  are not the same as the mass eigenstates, 1 , 2 , 3
for the massive neutrinos. The transformation between those eigenstates is written by,j i = U j jj i ;(1.1)

where U is the transformation matrix between mass eigenstates and avor eigenstates.
is theindex for avor eigenstates, = e; ;  and j is the index for mass eigenstates, j = 1; 2; 3.
Neutrino oscillations in vacuum is one of remarkable features of Quantum Mechanics. Timeevolution of states obey the Schrodinger equation:idj i = Ej jj i ;
dt j

(1.2)

where Ej is the energy of j . The solution for the neutrino wave function can be written asj (t)i = e0iEj tj (0)i :
j

j

(1.3)

Neutrinos are generated as a avor eigenstate at the weak decay of particle and are observed
as a avor eigenstate via weak interactions, so both of initial and nal states are avor eigenstates.
Hence the Schrodinger equation, Eq.(1.2), is rewritten using Eq.(1.1) asi

dj i = U j Ej Ujy j i ;
dt

(1.4)

and its solution, Eq.(1.3), is presented byasj (t)i = U j e0iEj t Ujy j (0)i :(1.5)

P ( !  ) = jh (t)j (0)ij2= jh (0)jU j e0iEj t Ujy j (0)ij2 :(1.6)

If an  is produced at t = 0, the probability of detecting this neutrino at t = t can be written

To simplify the problem, we consider only two avors. For example, if one assumes the neutrino
mixing between  and  , a transformation matrix is expressed as

v0 sin v0
U = 0cossin v0 cos v0
"

#

;

(1.7)

CHAPTER 1.

6

INTRODUCTION

where v0 is the mixing angle in vacuum between  and  like the Cabibbo angle in the case of thequark avor mixing. Then Eq.(1.4) is expressed as

d
i
dt

by

"

ji
j i

#

= U E02 E03 U y jj ii"#"
E2 + E3 1 0
E3 0 E2 0 cos 2v0=
0 1 + 2sin 2v0
2
"

#

"

#

sin 2v0
cos 2v0#! "ji : (1.8)
j i#The term (E2 + E3 )=2 is a common phase, so it can be omitted. Hence Eq.(1.8) can be written

d
i
dt

"

ji = E3 0 E2 0 cos 2v0 sin 2v0j i
sin 2v0 cos 2v0
2
#

"

#"

ji :
j i
#

(1.9)

The probability of Eq.(1.6), is expressed as

v0 sin v0e0iE t 0
P ( !  ) = [1 0] 0cos
00sin v cos v
0 e0iE t= e0iE t cos2 v0 + e0iE t sin2 v0 2 :
"

#"

2

#"

3

2

cos v0 0 sin v0
sin v0 cos v0

3

#"

1
0

#

2

(1.10)

If neutrino masses are suciently smaller than their momenta, the following approximation canbe applied,m2
1(1.11)
E = (p2 + m2 ) 2  p + j ;
j

j

2p

where mj is the mass of j and p is the momentum of  .
Applying the above approximation Eq.(1.9) is reduced to

d
i
dt

"

ji =  0 cos 2v0 sin 2v0j i
Lv sin 2v0 cos 2v0
#

"

#"

ji :j i#(1.12)

Here, Eq.(1.10) is also reduced to

Pv ( !  )  1 0 sin2 2v0 sin2

1m0 2t4p= 1 0 sin2 2v0 sin2 L;
Lv(1.13)

where L is the propagation length in the time interval of t (L = t for neutrinos in natural units),
Lv is the oscillation length in vacuum de ned as

E (GeV) 14E
=
2
:48
0
2
m
1m0 2 (eV2) (km) ;1m0 2  m23 0 m22 :Lv(1.14)(1.15)

Eq.(1.13) indicates that if  and  have nite di erent masses and there is a non-zero
mixing angle between them, the number of observed  events are less than that of generated 
events.

CHAPTER 1.

7

INTRODUCTION

1.1.2 Neutrino Oscillations in Matter
Neutrino oscillations in matter was rst proposed by S.P.Mikheyev and A.Yu.Smirnov based on thetheory advocated by L.Wolfenstein, hence, that is often called the MSW e ect[2],[3].
Electron neutrinos propagating through matter receive an additional potential energy of Ve by
the charged current interaction with electrons in matter(Fig.1.1):

Ve =

p

2GF Ne ;(1.16)

where GF is the Fermi coupling constant, Ne is the electron density in matter.
(b)

(a)
νe

w

νe, µ,τνe, µ,τ

e-

z0

+

νe

e-

e-

e-

Figure 1.1: Feynman diagrams of the neutrino scattering with electron, (a) a charged current
interaction of e, and (b) a neutral current interaction of e; ;  .
Adding Ve to Eq.(1.4), the neutrino propagation in matter is given asdj i = ( U j Ej Ujy + Ve ) j i :(1.17)
dt
When a mixing angle for e and  is expressed by v , a transformation matrix is de ned byi

v sin v
U = 0cossin v cos v"#(1.18);and then Eq.(1.17) is represented as

d
i
dt

"

je i
E1 0
Ve 0
y
j  i = U 0 E 2 U + 0 0
#

"

#

"

je i
ji :(1.19)je i ;
ji

(1.20)#! "#Omitting common phase, Eq.(1.19) reduces to

d
i
dt

"

je i = ji Lv
#

"

Lv
Le

0 cos 2v sin 2v
sin 2v cos 2v# "#where Le is the oscillation length in matter de ned as

Le 

p

2 = 1:624 2 104 (km) ;GF Ne(1.21)

CHAPTER 1.

8

INTRODUCTION

3

Electron Density ( NA × mol / cm )Electron Density in the Earth
6

5

4

3

2

1

0

0.2

0.4

0.6

0.8

1

R/Rearth

Figure 1.2: Electron density distribution in the earth[6]. Horizontal axis a is radial coordinate
in units of the earth radius. Vertical axis is the electron density in units of NA mol, where NA
is the Avogadoro number.
where Ne is the electron density in matter in units of the Avogadro's number per cm3 . Fig.1.2 showsthe electron density distribution in the earth as a function of the earth radius.
The mixing matrix in matter, Um , can be parameterized analogous to the vacuum mixing parameterization as"#cos m sin m
0 sin m cos m ;Um =(1.22)

where m is the mixing angle in matter.
Similarly to Eq.(1.12), the Schrodinger equation of neutrino in matter is expressed as

d
i
dt

"

 0 cos 2m sin 2mjeiji = Lm sin 2m cos 2m
#

"

# "

je i
ji ;#(1.23)

Comparing Eq.(1.20) and Eq.(1.23), the relation between the oscillation lengthes in vacuum andin matter is given by

Lv;
Lm = q 2
sin 2v + ( LLve 0 cos 2v )2and the mixing angles in vacuum and in matter have the following relation,2
sin2 2m = sin2 2 +sin( Lv20v cos 2 )2 ;v
v
Le

(1.24)(1.25)

CHAPTER 1.

9

INTRODUCTION

when m2 > m1 .
Assuming 1m2 is positive, Eq.(1.25) shows that the neutrino mixing is resonantly enhanced
for neutrinos at Lv =Le = cos 2v , even if the intrinsic mixing, sin2 2v , is small. This is the key
point of the MSW e ect.
In terms of the electron density, the resonance condition is represented as
1m2

Ne = Ne;crit p
2 2GF E cos 2v ;(1.26)

where Ne;crit is the critical electron density of the resonance condition, 1m2 = m2 0m1 . Numericallythe resonance condition is represented as) 1 Ne(cm3) = cos 2v :(1.27)
1:53 2 1004 2 E (GeV
1m2(eV2 )Fig.1.3 shows how the e ective neutrino mass squared and the e ective mixing angle change
according to the electron density in matter. As can be seen in Fig.1.3, if Ne  Ne;crit , m ' v ,neutrinos oscillate with a oscillation length of Lv , as in vacuum. For Ne  Ne;crit , m ' =2, theoscillation length in matter Lm which is de ned in Eq.(1.24) is much shorter than Lv for a large Ne .
(b)

(a)
2

m

νe

m2 ν
µ

νµ

θm
π/2

π/4

νe
m1
0

Ne,critNe

0

Ne,critNe

Figure 1.3: (a) E ective neutrino mass squared in the medium with electron density Ne. Ne;critis the crossing point de ned by Eq.(1.26). (b) Mixing angle m in the medium as a function of
Ne , corresponding to the case shown in (a).
If e is produced in the region Ne > Ne;crit and propagates into the region Ne < Ne;crit , thestate follows the upper branch given in Fig.1.3-(a), then e fully converts into  , provided thatthe density gradient, d(ln Ne )=dr, is suciently small that neutrino conversion occurs adiabatically.
This adiabatic condition may be derived from(energy gap) 2 (transition time)  h ;where the energy gap E isE =

1 (1m2 ) sin 2 ;2E

(1.28)(1.29)

CHAPTER 1.

10

INTRODUCTION

and the transition time in the level crossing region is

t =

=
=

r
c

1 dNe 01 Ne
!

Ne dr
Ne
! 01
1 dNe A ;Ne dr
A

(1.30)

pwhere A  2 2EGF Ne . Since \resonance" occurs at A = 1m2 cos 2 and its width is A 
1m2 sin 2, we nd that Eq.(1.28) is written

1 dNe  1m2 sin2 2v :Ne dr
2E cos 2v(1.31)The probability for e ! e is easily calculated by averaging out the time-varying part,P (e ! e ) = sin2 v sin2 ~v + cos2 v cos2 ~v= 12 (1 + cos 2v cos 2~v ) ;(1.32)

(1.33)

where ~v is the mixing angle at the initial point.
If the adiabatic condition Eq.(1.31) is not satis ed, the state of the upper branch undergoes atransition to the lower branch while passing through the crossing point with the probability given by

Pf = exp(0

where

2 );
2 sin2 2v= 2E cos12m(1=Ne )(dNe =dr )(1.34)

is the ratio of the right-hand side to the left-hand side of inequality Eq.(1.31). This is a straightforward application of the well-known Landou-Zener formula for level crossings [4].
With this Pf , we write the probability for e ! e for general caseP (e ! e ) = (sin2 v sin2 ~v + cos2 v cos2 ~v )(1 0 Pf ) + (sin2 v cos2 ~v + cos2 v sin2 ~v )Pf= 21 + ( 21 0 Pf ) cos 2v cos 2~v :For Ne ! 1; cos 2~v ! 01, and this reduces to(1.35)P (e ! e) = sin2 v + cos 2v Pf :(1.36)

In summary, a virtually complete conversion of e to  takes place when the three conditions
are satis ed:(i) Ne > Ne;crit : see Eq.(1.26), which gives 1m2 < const (resonance condition)> const : see Eq.(1.31), (for a small v ; adiabatic condition)(ii) 1m2 sin2 2v CHAPTER 1.

11

INTRODUCTION

νe

2 ~

∆m

log( ∆ m2 )∆m2 ~ G F N e Eν

νµ

d
( ln

dr

N
)E

e

1

ν

z

log z

Figure1.4: Schematic gure of the MSW triangle. The abscissa is the logarithm of z =2
sin 2v = cos 2v . Complete conversion of e !  takes place inside the triangle.
(iii) sin2 v < 1 : see Eq.(1.35).
These three conditions complete a rectangular triangle (which we refer to as the MSW triangle) inthe (1m2 ; sin2 2v ) plane (see Fig.1.4).
For antineutrinos, the potential energy by the charged current interaction with positrons inmatter has an opposite sign,p

Ve = 0 2GF Ne :(1.37)

So sin2 2m and Lm became as follows,

Lm =

Lv

sin2 2v + ( LLve + cos 2v )22
sin2 2m = sin2 2 +sin( Lv2+v cos 2 )2 :
v
v
Le
q

;

(1.38)(1.39)

CHAPTER 1.

12

INTRODUCTION

1.2 Searches for Neutrino Oscillations
Experiments for neutrino oscillations can be done with various neutrino sources:
Arti cial sources: Neutrinos from nuclear reactors (e) Low energy neutrinos from meson factories (e ; ; ) High energy neutrinos from accelerators (; )Natural sources: Neutrinos generated inside the sun by nuclear fusions { Solar neutrinos (e ) Neutrinos generated from interactions between cosmic-rays and the atmosphere { Atmospheric
neutrinos (e ; e;  ;  )These neutrino sources are summarized in Table 1.2, together with their energy regions, and thepossible typical observation distances L, and the typical mass squared di erence squared 1m2 towhich they are sensitive (Assuming the best possible experimental conditions with small statisticalerrors or large mixing angles).

 source

ReactorMeson Factory
Accelerator
Solar
AtmosphericE (MeV) L (m)140103
0.1  10103  106

102
102
104
1011
107

1m2 (eV2)

1002
1001
1001
10010  1004
1004  1

Table 1.2: Sensitivity of various neutrino sources to neutrino oscillations.
1.2.1 Reactor and Accelerator Neutrino Experiments
In this subsection, we review the neutrino oscillations experiments which use arti cial sources.

1. Reactor Experiments: Reactor neutrino experiments use electron antineutrinos producedin the core of nuclear reactors. The signal of e is detected by the reaction of e + p ! e+ + n.
To eliminate accidental e events, the reactor-o data are subtracted from the reactor-on data.
The existing experiments are Gosgen[24], Bugey [25] and Chooz[26] which already presented theirresults. Palo Verde will take data soon, Kam-LAND 1 [27] will start in the year 2001.
The positron spectrum from the reaction e + p ! e+ + n has been measured at L = 37:9m
from the core of the Gosgen power reactor to search for neutrino oscillations of the type e $ x in
the low 1m2 parameter range. Fig.1.5 is energy spectra of detected positrons which are generated
by electron antineutrinos produced by Gosgen reactor. The results up to now are consistent with
1

Kamioka Liquid scintillator AntiNeutrino Detector. The experiment is projected by Tohoku University.

CHAPTER 1.

INTRODUCTION

13

the no oscillation hypothesis. Upper limits of 1m2  0:016 eV2 (90% CL) for full mixing, andsin2 2  0:17 (90% CL) in the limit of large 1m2 were obtained. The ratio of the integrated
experimental yield to that predicted for the case of no oscillations was 1.05 6 0.02(stat) 6 0.05(syst;
68%CL) in the Gosgen experiment.
Bugey, using detection modules lled with 6 Li-loaded liquid scintillator reported on high statistics
measurements of neutrino energy spectra carried out at 15, 40 and 95 meters away from a 2800
Megawatt reactor. No oscillations have been observed. Exclusion zones for oscillation parameters
are deduced from the observed consistency of the spectra at the three distances. The minimum
excluded values of 1m2 and sin2 2 parameters are 1 2 1002 eV2 and 2 2 1002 (at 90% CL),
respectively.
Initial results are presented from Chooz, a long-baseline reactor-neutrino vacuum-oscillations experiment. Electron antineutrinos from the reactors were detected by a liquid scintillation calorimeter
located at a distance of about 1 km. The detector was constructed in a tunnel protected fromcosmic-rays by a 300 MWE rock overburden. From the statistical agreement between detected
and expected neutrino event rates, the Chooz experiment nds (at 90%CL) no evidence for neutrino oscillations in the e disappearance mode for the parameter region given approximately by
1m2 > 0:9 2 1003 eV2 for maximum mixing and sin2 2 > 0:18 for large 1m2.

Figure 1.5: Results of the Gosgen reactor experiment. (a) Positron energy spectra for reactor-on
and reactor-o . Bin width = 0.305 MeV. The errors shown are statistical. The contribution of
the accidentals is indicated by the dashed curve. (b) Experimental positron spectrum obtainedby subtracting reactor-o from reactor-on spectra. The solid curve represents the predicted
positron spectrum assuming no neutrino oscillations. ( gure is taken from [24].)
2. Meson Factory Experiments: The meson factory experiments use low energy neutrinos(several tens of MeV) which are generated by the decay of stopping + mesons. The on-going

CHAPTER 1.

14

INTRODUCTION

experiments are KARMEN2 [19], LSND3 [20].
KARMEN is situated at the beam stop neutrino source ISIS. It provides  's, e 's and  's inequal intensities from the + ! + -decay at rest. The oscillation channels  ! e and  ! eare investigated with a 56 t liquid scintillation calorimeter at a mean distance of 17.6 m from the -source. No evidence for oscillations was found with KARMEN, resulting in 90% CL exclusion
limits of sin2 2 < 8:5 2 1003 ( ! e) and sin2 2 < 4:0 2 1002 ( ! e) for 1m2 > 1 eV2 .
LSND carried out at the Los Alamos Meson Factory searches for  ! e oscillations by using from + decay at rest. The e are detected via the reaction ep ! e+ n, correlated with a fromnp ! d (E = 2.2 MeV). The use of tight cuts to identify e+ events with correlated rays yields 22
events with e+ energy between 36 and 60 MeV and only 4.6 60:6 background events. If attributedto  ! e oscillations, this corresponds to an oscillation probability of (0:3 6 0:12 6 0:05)%.

3. High Energy Accelerator Experiments: High energy accelerator experiments use highenergy neutrinos(several GeV) which are generated by high energy  6 ; K 6 mesons in ight decays.
These high energy mesons are produced by high energy proton beams focused on a xed target.
The produced mesons are momentum selected to make a narrow neutrino energy spectrum { called\narrow-band neutrino beam", or are not selected by momentum to get a high intensity of neutrinoux { called \wide-band neutrino beam". The typical neutrino beam lines are listed in Table 1.3.
accelerator
nameprotonenergydecay neutrino experiment/
pipe
ight
detector
lengthlengthname
450(GeV)414(m) 822(m)CHORUS
NOMAD
800(GeV) 140(GeV) 542(m) 1480(m)
CCFR

CERNSPSFNALTevatoron
FNAL-Main 120(GeV)-Injector
KEK12(GeV)PS

neutrinomeanenergy27(GeV)

12(GeV)
1(GeV)

comments

wide-band(running)
narrow-band(running)
800(m) 950(m)COSMOS
wide-band
700(km)MINOS(will run 2000)
200(m) 200(km)Superwide-band
Kamiokande (will run 1998)

Table 1.3: On-going/under-construction neutrino beam lines. FNAL-MI to MINOS and KEKPS to Super-Kamiokande will have such a long neutrino ight path, they are called \LongBase-Line" experiments.
The existing experiments are CHORUS-NOMAD, CDHSW4 [23], BNL-E776[21], CCFR 5 [22].
CDHSW has searched for  oscillations by comparing the rates of  charged-current interactions
in two detectors located 130 and 885 m from the target, which was struck by a 19.2 GeV/c proton
beam from the CERN Proton Synchrotron. No evidence for  oscillations was found. At the 90%CL,
1m2 values between 0.26 and 90 eV2 are excluded
for maximal mixing. The most restrictive limit
2
on the neutrino mixing-angle parameter sin 2 is 0.053 at 1m2 = 2:5 eV2 .

Karlsruhe Rutherford Medium Energy Neutrino experimentLiquid Scintillator Neutrino Detector
4 CERN, Dortmund, Heidelberg, Saclay, Warsaw
5 Cincinnati, Columbia, Fermi, Rochester
23

CHAPTER 1.

INTRODUCTION

15

BNL-E776 at the Brookhaven National Laboratory has searched for the appearance of e(e), 1
km from the source of a wide-band  ( ) beam. The experiment used a total of 3 2 1019 protons
on target from the Alternating Gradient Synchrotron. No excess of e or e over the expectedbackground was detected. The 90% CL limits obtained are 1m2  7:5 2 1002 eV2 for maximal
mixing, and sin2 2  0:003 for large 1m2 .
CCFR at Fermilab presented limits on  ( ) ! e (e ) oscillations. Neutrino energies range
from 30 to 600 GeV with a mean of 140 GeV, and  ight lengths vary from 0.9 to 1.4 km. The
result excludes oscillations in the region with sin2 2 > 1:8 2 1003 for large 1m2 (> 1000 eV2 ) and
1m2 > 1.6 eV2 for sin2 2 = 1. This result is the most stringent limit to date for 1m2 > 25 eV2
and it excludes the high 1m2 oscillation region favored by the LSND experiment. The  to ecross section ratio was measured as a test of  ( ) ! e (e ) universality to be 1.026 6 0.025(stat)
6 0.049(syst).

Figure 1.6: Left gure is calculated spectra for ; e; ; e per proton on target(POT) per m2
at the BNL-E776 experiment, (a) the horn at positive polarity and (b) the horn at negative
polarity. Right gure is the calculated neutrino energy spectra for the CCFR experiment.
( gures are taken from [21], [22].)
90% CL exclusion curves for  ( ) ! e (e ) and  !  oscillations from various accelerator
and reactor experimants are presented in Fig.1.7, Fig.1.8, respectively.

CHAPTER 1.

16

INTRODUCTION

Figure 1.7: 90% CL exclusion curves and limits of e $  oscillations from various neutrino
experiments. Here, Palo Verde and Kam-LAND contour plots show the expected ones. (fromG.Gratta, stanford-HEP-97-03.)νµ ↔ ντ3∆m,2 (eV2)

10

CHORUS
NOMAD

10

2

CDHSW

10

1

-1

10
10

-3

10

-2

10

-1

1
sin2 2θ,vFigure 1.8: 90% CL exclusion curves and limits of  $  oscillations from CHORUS, NOMAD, CDHSW experiments.

CHAPTER 1.

17

INTRODUCTION

1.2.2 Solar Neutrino Experiments
The sun is an intense source of low energy (E < 15MeV) electron neutrinos(e ) that are productsof the nuclear fusion processes in its higher temperature central region. The underlying nuclear
fusion processes are in the following, p-p Chain! 2H + e+ + e (E < 0:42MeV)(1.40)

p + e0 + p
2H + p! 2H + e (E = 1:44MeV)! 3He +
3 He +3 He ! 4 He + 2p

(1.41)(1.42)(1.43)

3 He +4 He ! 7 Be +7 Be + e0 ! 7 Li + e (E =0:86MeV(90%))
0:38MeV(10%)
7 Li + p ! 24 He(1.44)(1.45)(1.46)

7 Be + p
8B
8 Be3! 8B +! 8Be3 + e+ + e (E < 15MeV)! 24 He(1.47)(1.48)(1.49)

3 He + p! 4He + e+ + e; (E < 18:8MeV)(1.50)

12 C + p
13 N
13 C + p
14N + p
15 O
15N + p

! 13N +
! 13C + e+ + e (E < 1:2MeV)! 14N +
! 15O +! 15C + e+ + e (E < 1:7MeV)! 12C +4 He :(1.51)(1.52)(1.53)(1.54)(1.55)(1.56)

p+p

or

or

or

or

 CNO Cycle

Processes (1.40) to (1.56) produce electron neutrinos which are called, respectively, pp; pep and
7 Be,8 B,13 N and 15 O neutrinos. pep and 7 Be neutrinos are monoenergetic. Other neutrinos havecontinuous energy spectra. Fig.1.9 shows the calculated uxes of solar neutrinos on the earth.
In the late 1960s, a unique experiment which aimed to detect solar neutrinos were started, andit rst succeeded in the detection of the astronomical neutrinos. The experiment was started by
R.Davis et al. in the Homestake Gold Mine at Lead, South Dakota, and its goal was to observe
the ux of the electron neutrinos. In order to detect the solar neutrinos, they used a radiochemical
method based upon the inverse beta process of e +37 Cl ! e0 +37 Ar, of which the energy threshold
was 0.814 MeV. The total solar neutrino ux for the chlorine experiment was predicted to be 7.5

CHAPTER 1.

INTRODUCTION

18

Figure 1.9: Calculated uxes of various solar neutrinos on the earth as a function of energy.
7 Be and pep neutrinos are mono-energetic and their uxes are in unit of cm02sec01 .
SNU (Solar Neutrino Unit: 1 SNU  10036 captures/target atom/sec). The solar neutrinos weredetected at rst in 1968, and the observed rate was below 3 SNU. The Homestake experimentproved that the origin of solar energy was nuclear fusion. However, it also reported the unexpected
result in the framework of the solar model. They observed less than half of the solar neutrino ux
predicted from the solar model. This experiment continued to observe solar neutrinos and reported
the de cit for more than twenty years. The de cit of the solar neutrino ux has been called the\solar neutrino problem", and well known in the eld of both particle physics and astrophysics.
The second experiment to observe the solar neutrino ux was Kamiokande-II, which started its
observation in January 1987. Kamiokande-II was an imaging water Cherenkov detector with 4500
tons of water, and detected solar neutrinos above 7.5 MeV via the reaction e + e0 ! e0 + e .
Kamiokande-II was the rst experiment which had the ability to measure the direction and theenergy of solar neutrinos in real time. Thus, it was rst revealed from the data of Kamiokande-IIthat the neutrinos undoubtedly come from the direction of the sun. Fig. 1.10 shows the directional
distribution of solar neutrino events observed by Kamiokande-II and the next period of experiment
Kamiokande-III. The 8 B solar neutrino ux observed by Kamiokande-II,III in 1966 live days was
0:48 6 0:03(stat.) 6 0:07(syst.) times the prediction by Bahcall and Ulrich in 1988. Thus, the de citof the solar neutrinos was con rmed by Kamiokande-II,III.
Thereafter, two radiochemical experiments to measure the pp neutrinos, namely SAGE and
GALLEX were started and their rst results were reported 1991 and 1992, respectively. An inverse-decay reaction of e +71 Ga ! e0 +71 Ge is used in both experiments, and its energy threshold is
0.2332 MeV which makes those experiments sensitive to the pp neutrinos. The observed pp neutrinouxes were 20+15
020 (stat.) 6 9(syst.) (SNU) by SAGE(1991) and 83 6 19(stat.) 6 8(syst.) (SNU) by
GALLEX(1992), though the total rate for the 71 Ga experiment is predicted as 132 SNU. The pp

CHAPTER 1.

19

INTRODUCTION

Figure 1.10: Distribution of cos sun for the entire data of the nal sample of Kamiokande-II
and Kamiokande-III(1966 days). The solid and dashed line correspond to the expected signalof the SSM and the best t(SSM 2 0.48, respectively. ( gure is taken from [5].)and pep neutrino uxes are thought to be theoretically robust and all the predicted uxes of pp and
pep neutrinos are essentially the same among recent Standard Solar models(SSMs), and the sumof those neutrino uxes is estimated to be 74 6 1 SNU. If the standard electroweak theory and theprediction of pp and pep neutrino uxes are correct, the results from 71Ga experiments imply thede cit of other solar neutrinos such as the 7 Be or 8 B neutrinos.
Table 1.4 and Table 1.5 show the summary of experimental results and the two di erent SSM
predictions on the solar neutrino ux, where the unit is SNU for 37 Cl and 72 Ga experiments and /cm2/sec for e0(H2O) target experiment, respectively.
Fig.1.11 shows the allowed e $  oscillation parameter regions obtained by the Homestake,Kamiokande, and Ga experiments, and the combined data, where SSM are (a) by Bahcall & Pinsonneault and (b) by Turck-Chieze & Lopes.
Target(Exp.)
37 Cl(Homestake)
71 Ga(SAGE)
71 Ga(Gallex)e0 (H2 O, Kamiokande)

Data

SSM prediction

Data/SSM

2:28 6 0:23
8:0 6 3:0
0:29 6 0:03
+18+5
+21
:04
7301607
131:5017
0:560+00::1214+0
00:05
79 6 10 6 6
131:5+21
0:60 6 0:08 6 0:05
017
6
6
(2:7 6 0:2 6 0:3) 2 10 (5:7 6 2:5) 2 10 0:48 6 0:03 6 0:06
Table 1.4: SSM = Bahcall and Pinsonneault Model

CHAPTER 1.

20

INTRODUCTION

Target(Exp.)
37 Cl(Homestake)
71 Ga(SAGE)
71 Ga(Gallex)e0 (H2 O, Kamiokande)

Data

SSM prediction

Data/SSM

2:28 6 0:23
6:4 6 1:4
0:36 6 0:04
15+0:04
73+18+5123
6
7
0:590+00::13
01607
00:06
79 6 10 6 6
123 6 70:64 6 0:08 6 0:05
(2:7 6 0:2 6 0:3) 2 106 (4:4 6 1:1) 2 106 0:62 6 0:04 6 0:08
Table 1.5: SSM = Turck-Chieze and Lopes Model

Figure 1.11: The allowed parameter regions of the Homestake, Kamiokande, and Ga experiments, and the combined data. Here SSM's are (a) by Bahcall & Pinsonneault and (b) by
Turck-Chieze & Lopes. The SSM uncertainties are included in the analysis. For the Bahcall &PinsonneaultSSM, the 1 value of the SSM uncertainties is assumed to be 1/3 of their 'e ective
3' error. The earth e ect on the regeneration of e is included. The excluded region by the
Kamiokande day/night data are also shown. ( gures are taken from Ref.[7].)CHAPTER 1.

21

INTRODUCTION

1.2.3 Atmospheric Neutrino Experiments
Atmospheric neutrinos are produced through decays of charged pions and kaons which are secondary
particles in the primary cosmic-ray interactions in the atmosphere. Pions, which are the dominantsources of atmospheric neutrinos, decay before interacting with air nuclei due to the small density athigh altitude. Consequently the following decay chain is the main process of atmospheric neutrino
production:p(He) + air nucleus ! 6 + X ;(1.57)
 6 ! 6 +  ( ) ;(1.58)
66
 ! e + e(e) +  ( ) :(1.59)
Thus, it is expected that the ux ratio of ( +  )=(e + e ) is  2. This ux ratio's calculations from di erent groups agree to within 5%, although the absolute ux values are with a 20%
uncertainty.
There are two experimental methods of studying atmospheric neutrinos in massive undergroundand underwater detectors:

1. Detecting neutrino interactions occurring in the detector, which is called fully or partiallycontained events, depending on whether all produced particles stop inside the detector or not.
See Fig.1.12.
2. Detecting upward-going muons produced by neutrino interactions in nearby rock surrounding
the detector.

b) partially contained
eventa) fully containedeventνµ (νe)νµ (νe)µ(e)µ(e)

detectordetector: vertex point: stopping point

Figure 1.12: Schematic gure of fully and partially contained events. The vertex points ofneutrino interactions are within the detector. \Fully" and \partially" are categorized by theirstopping points.

CHAPTER 1.

22

INTRODUCTION

In Kamiokande event types are grouped by the mean energy of neutrinos: \sub-GeV"(< E >0:7GeV), where fully-contained events are the main component; \multi-GeV"(< E > 6GeV),
where fully-and partially-contained events are main components; and < E > 100GeV upward-

through-going muons. In this subsection, \sub-GeV" and \multi-GeV" data are shown. Upwardthrough-going muons which are taken in this thesis for the neutrino oscillation analysis are describedin the Section 1.2.4.
Up to now there are ve results of atmospheric neutrinos taken in di erent underground detectors.
Two are the water Cherenkov type of experiments, IMB and Kamiokande. The other three aretracking detectors, NUSEX, Frejus and Soudan-II. All experiments furnish the ability to identify theparticle type of electron-like or muon-like. Thus, it is possible to compare the R(=e) ratio observedin all these experiments, where R(=e) is de ned as follows,

R(=e) =( + )=(e + e)Data :( + )=(e + e)MC(1.60)

Table 1.6 lists the observed and expected number of events, and R(=e).
Experiment

Exposure
Data (events) MC (events) ratio of Data and MC(kton1year)ee
R(=e)+0
Kamiokande
7.7
248
234
227.6 356.8
0:60 0+00:::060508 6 0:05 a
8.2(full),6.0(part) 98
135
66.5
162.2
0:57 00:07 6 0:07 b
IMB-3
7.7
325
182
257.3 268.0
0:54 6 0:05 6 0:12
:35
Nusex
0.74
18
32
20.5
36.8
0:99 +0
00:25 6 smallFrejus
2.0
75
125
81.4
136.2
1:00 6 0:15 6 0:08
:05
Soudan-II
1.52
49.4
37.0
45.3
47.1
0:72 6 0:19 +0
00:07

Table 1.6: Results of the contained event analysis of each experiment. a: sub-GeV, b: multiGeV.
Fig.1.13 shows the zenith angle (cos 2) distribution of e-like and -like events from the Kamiokande
experiment. Data points look somewhat deviated from the MC predictions. The dependence of theratio R(=e) on cos 2 illustrates this more clearly as shown in Fig.1.13(c). One sees a gradualdecrease with respect to cos 2 from 1 to -1. This dependence is well- tted by the expectation fromneutrino oscillations as shown by histograms in Fig.1.13(c), where the dashed histogram corresponds
to the best- t value for  $  oscillations, and the dotted one for  $ e oscillations.
In terms of IMB-3 [11], [12] experiments, the result supports a small R(=e) ratio. On theother hand, two tracking experiments of NUSEX [15] and Frejus [14] show R(=e)  1, whichmeans that there is no atmospheric neutrino anomaly. Since the statistical signi cance is not highin NUSEX and Frejus, these results are still consistent with those of Kamiokande and IMB-3 at the
3  level. The Soudan-II result [13] is also smaller than 1, although its statistical signi cance is
not enough. It is noted that the detection threshold momentum of Soudan-II is  100 MeV/c anddE/dx information is available, unlike NUSEX and Frejus.

CHAPTER 1.

INTRODUCTION

23

Figure 1.13: Zenith angle distribution of e-like(a) and -like(b) multi-GeV events from the
Kamiokande experiment. The histograms are MC prediction. The zenith angle distribution ofR(=e) is shown in (c), where the histograms are the expectations from the neutrino oscillations
with the best t values of (1m2; sin2 2). ( gure is taken from [16].)

Figure 1.14: 90% CL allowed parameter regions from the Kamiokande experiment for the casesof  $ e (left) and  $  (right). The best- t values are indicated by dash-crosses (subGeV data), full-crosses (multi-GeV data) and stars (sub- and multi-GeV data combined). The
other curves are the 90% CL excluded regions from the previous experiments. ( gure is takenfrom [16].)

CHAPTER 1.

24

INTRODUCTION

1.2.4 Upward Through Going Muons
In Section 1.2.3, we mentioned upward-going muons originate from muon neutrinos which are generated predominantly in the atmosphere. In this section, we describe the features of upward-going
muon events. The previous and on-going experiments of upward-going muons are described in
Chapter 6.
Upward-going muons are grouped by two types in terms of their detection. One is \throughgoing" muons which are punching through the detector and the other is \stopping" muons which
are stopped in the detector. Fig.1.15 shows schematic diagram of an upward-through-going muon
event and an upward-stopping muon event.
a) upward through going
muon eventb) upward stopping
muon event

µ
νµ

µ
νµ

detectordetector: vertex point: stopping point

Figure 1.15: Schematic gure of upward stopping and through-going muon events. The vertex
points of neutrino interactions are in the rock outside the detector. \Stopping" and \throughgoing" are categorized by their stopped points.
The reason why we select only \upward" muons is that almost all \downward" muons are cosmicray muons 6 which are passing through the rock covering the detector and we cannot distinguish
the neutrino induced muons from them. Moreover why we use muons is that electrons generated
from electron neutrinos can not reach to the detector due to their short radiation length.
The generation of an atmospheric neutrino and their detectinon as an upward-through-going
muon is represented in the schematic diagram in Fig.1.16, where a muon neutrino (or muon antineutrino) interacts with a nucleon in the rock at nearby the detector through the charged current
weak interaction, ( ) + N ! 0(+) + N 0 + X :(1.61)The detailed description of the generation of upward-through-going muons is in Chapter 5, where
the expected ux of upward-through-going muons detected in the underground detector is calculated.
6 The dominant component of cosmic-ray muons are atmospheric muons which are generated directry in theatmosphere by the reaction of Eq. (1.58).

CHAPTER 1.

25

INTRODUCTIONµ−( Upward ThroughGoing Muon )νµµ−detector

W
N

N’

νµ
the earthπ+µ+nucleus in the atmosphare

prompt cosmic ray(H,He,etc...)Figure 1.16: Schematic gure for generation and detection of an upward through going muon.
In this example, a prompt cosmic-ray { H, He, etc. { interacts with a nucleus in the atmosphere{ N, O { and generates a muon neutrino by the following reactions, ACR + Aair ! + +anything,+ ! + +  . A produced muon neutrino goes through the earth and interacts with a nucleus
in the rock near the detector by CC weak interaction,  + N ! 0 + N 0 + anything. A
produced muon travels in the rock and penetrates through the detector.
The mean energy of detected upward-through-going muons is relatively higher than that ofcontained event's muons because they travel 0  several 100 m in the rock. The energy spectra ofparent muon neutrinos which generate upward-through-going/stopping muons are shown in Fig.1.17
together with those of contained event's muons.
Here we summarize the features of upward-through-going muon events detected in deep underground water Cherenkov detector in contrast to contained events.
1. The only particle we must identify is single muon which is generated by CC interaction,
Eq.(1.61) because neutrino-induced electron and pion and leptons decayed from it are stopped
in the rock before they reach the detector.
2. There are no background events except for cosmic-ray muons contaminating from an almost
horizontal downward direction.
3. The neutrino direction is determined accurately.

CHAPTER 1.

INTRODUCTION

26

Figure 1.17: Approximate energies of atmospheric muon neutrinos contributing to the di erent
event types of contained, upward stopping muon, and upward through going muon events bysolid line, and multi-GeV event by dash line. ( gure is taken from [8].)4. The information of muon energy deposited is obtained only within the detector, so the energyof the parent neutrino for each event is unknown.
From the feature 1 and 2, event identi cation becomes very simple and reliable, so the selectioneciency of upward through going muons is very good 7 . Although feature 1 is a weak point of
upward through going muon events because when we perform oscillation analysis, we can't use thee = ux ratio to compare to the expected one, but the gradient of zenith angle distributions has
a small uncertainty so in the oscillation analysis, we normalize the absolute value of the expected
ux when comparing with the observed one.(see Chapter 6.)
Feature 3 is caused by two reasons. One is that the scattering angles between parent neutrinos
and induced muons are very small because we use high energy( 100GeV) neutrinos. Another isthat the angular resolution is very accurate 8 because the determination of muon direction is reliable
especially for upward-through-going muons. (see Chapter 3.)
Feature 4 is because we don't know the total energy deposited from neutrino induced muonsbecause we can't know the points where they are created and stop. Therefore, in the oscillation
analysis, we have not compared the energy spectrum of uxes but the angular distribution of thembetween the observed and expected ux.
78

more than 99 % for Super-Kamiokande 1:7 for Super-KamiokandeCHAPTER 1.

27

INTRODUCTION

1.3 Oscillation Search by Upward-Through-Going Muons
We reviwed various experiments of neutrino oscillations in the previous sections. In the case of
e $  oscillations, 1m2 >for by reactor, meson factory and high 1003 eV2 regions2 are searched
03energy accelerator neutrino experiments. sin 2  10 parameter regions remain. On the other04 eV2 regions are searched for by the solar neutrinos. 1m2  1005 eV2hand, 10010 <
1
m2 <
10


at sin2 2  0:5 or sin2 2  1002 parameter regions are remained.
In the case of  $  oscillations, it can not be searched for by either reactor experiments
nor solar neutrino experiments because the source neutrinos consist of only e or e , respectively.
1m2 0 2 > 1001 eV2 is searched for by meson factory and high energy accelerator neutrino experiments.
sin 20  1003 parameter regions are remained.
Both for e $  and  $  oscillations, 1004 <regions 21m2(12m0 20 ) < 1 eV2 parameter202>
are searched for by atmospheric neutrino experiments. sin 2(sin 2 )  0:4, 1m (1m )  1002
parameter regions remain.
In this thesis, upward-through-going muons originating from atmospheric muon neutrinos aretaken for analysis of muon neutrino oscillations, and the neutrino oscillation hypothesis in the1003 < 1m2 (1m0 2) < 1 eV2 parameter regions is tested. Fig.1.18 shows oscillation parameterregions searched by various neutrino experiments. The hatched regions can be searched for using
upward-through-going muons in this thesis.
νe

νµ
10

2

10

0

10

-2

-4

,
∆m 2

2

∆ m2

10

νµ

reactor, accelerator ν

accelerator ν

10

0

10

-2

10

-4

10

10

-6

10

-6

10

-8

10

-8

10

-10

10

-10

atmospheric ν

solar ν00.5

sin 2 2θ1

ντ

atmospheric ν

0

0.5

,1
sin 2 2θFigure 1.18: The oscillation parameter regions searched for various neutrino experiments. The
hatched regions are searched by using upward-through-going muons in this thesis.

Chapter 2
Kamiokande and Super-Kamiokande
Detectors

2.1 Kamioka Underground Laboratory
The Kamioka Underground Laboratory is located in the Mozumi Mine in Kamioka Town, which
is operated by Kamioka Mining and Smelting Company 1 . Kamioka Town, about 250 km west ofTokyo, lies in the northern region of Gifu Prefecture, bordering on Toyama Prefecture. The exact
location of the laboratory is latitude 36.42 N and longitude 137.31 E; the geomagnetic latitude is25.8 N. The laboratory has an elevation of 350 m, but it lies nearly under the top of Mt. Ikenoyama,
1,350 m high, as shown in Fig.2.1. Therefore, the laboratory has a rock overburden of about 1,000
m in the vertical direction.

2.2 Brief History
The original motivation which stimulated the construction of the Kamiokande detector was thesearch for nucleon decay 2 . As yet, it has not been discovered, with only the lower limits of thenucleon partial lifetimes having been determined. However, it was realized at the very beginningof the detector operation that the energy spectrum of decay electrons from the stopping muons
could be measured down to a kinematic energy of  12 MeV. The trigger rate below this energy
rapidly increased because of radioactive impurities dissolved in the water. Therefore, if they could
be reduced, the Kamiokande detector could be converted to a real-time solar neutrino detector 3 .
The Kamiokande detector has been continually improved. The rst phase of the experiment(July 1983 - November 1985) is called Kamiokande-I. It was initially operated with no anticounter
layer, which was constructed later in October - December 1984 (top and bottom sections, both
inside the main detector tank) and in October 1985 ( barrel section, between the rock cavity andthe main detector tank).
The second phase, Kamiokande-II, was aimed at observing solar neutrinos, and has been operated
since November 1985 with a new electronics system capable of recording the arrival time of eachphotomultiplier tube (PMT) signal in addition to its charge. The continued e ort to purify water1
2
3

The mine is closed now.
Kamioka Nucleon Decay Experiment
Kamioka Neutrino Detection Experiment28

CHAPTER 2.

KAMIOKANDE AND SUPER-KAMIOKANDE DETECTORS

29

Figure 2.1: Topographical map of the sites of the Kamiokande and Super-Kamiokande detectors
and, in particular, that of removing radioactive contamination from water should be specially noted,
since it has been essential for the successful observation of low energy neutrinos.
In June 1988 the detector performance of Kamiokande-II was improved by doubling the PMT
gain. The e ects of the PMT gain increase are: (i) improved energy resolution for low energy electrons and (ii) improvement in the event vertex reconstruction. These improvements were achieved
owing to the increase ( 20%) of hit PMTs corresponding to the unit deposited energy by electron.
This led to a signi cant reduction of background events.
The second phase was completed in April 1990. Up to then, since more than 10% of the PMTs
had been out of order due to various reasons, the detector was emptied and dead PMTs were
replaced. At this occasion, the entire electronics were replaced with a more compact, low-powersystem. A re ector was also introduced to each PMT in the inner detector to collect more light.
Observation was resumed in December 1990; this phase is called Kamiokande-III.
In December 1991, construction of Super-Kamiokande started and ocial runs of Super-Kamiokande
started in April 1996. Kamiokande-III was running until July 1996, but actual analysis was terminated, in terms of upward through going muons, in May 1995.
In this thesis, data of Kamiokande-III and Super-Kamiokande are analysised. Data of KamiokandeII was already analysised [50], but it was used to combine with the Kamiokande-III result in thisthesis.

CHAPTER 2.

KAMIOKANDE AND SUPER-KAMIOKANDE DETECTORS

30

2.3 The Detector
2.3.1 Ring-Imaging Water Cherenkov Detector
The Kamiokande and Super-Kamiokande detectors are cylindrical large Ring-imaging Water Cherenkov
Detector. In this section, the principle of Ring-imaging Water Cherenkov Detector will described.
A charged particle propagating in water emits cherenkov photons when its velocity (v) exceeds
the velocity of light in water(c=n);

 vc

(2.1)

cos  = n1 :(2.2)

n > 1;

where n(= 1.344) is the refraction index of water. The light emission process is analogous to shock
wave production. Cherenkov photons are emitted in a circular cone with a half opening angle of to the particle trajectory,In the case of the relativistic limit ( = 1),  has an asymptotic value of 41.9 . The spectrumof cherenkov light is quantum-mechanically calculated and the number of photons emitted per unit
path length per unit wave length is,dN= 2  2
dxd

10 1

!

n2 2

(2.3)

where is the ne structure constant, x the path length of the charged particle and  the wave
length of cherenkov light. The threshold momentum of cherenkov light emission depends on its
mass due to the condition of Eq.(2.1).
A ring-imaging water cherenkov detector detects chrenkov lights emitted in a massive detector
volume of water by charged particles, with PMTs arrayed two-dimensionally on the detector walls
surrounding the water. A cherenkov ring image is projected onto the detector walls. The particle
trajectry and energy are determined, using the charge and timing information of each hit PMT.
In the case of atmospheric neutrinos, a typical reaction ( quasi-elastic scattering) is as follows, + N ! ` + N0
where  is incident neutrino, N and N 0 are nucleons, ` is the accompanied lepton( e or  ).

CHAPTER 2.

KAMIOKANDE AND SUPER-KAMIOKANDE DETECTORS

31

2.3.2 Detector Overview
The components of detector are almost the same between Kamiokande and Super-Kamiokande.
The detectors consist of a cylindrical steel tank lled with pure water, water-air-puri cation system,
20-inch/8-inch diameter PMTs attached on wall of detector, electronics system, and online dataacquisition (DAQ) system.
Detector itself is divided in two parts. One is the inner detector (main detector) in which weobserve the desired events such as solar/atmospheric neutrinos or proton decay, and another is theouter detector (anti detector) to veto incoming events from outside of the detector for contained
events analysis. For the detector through going muon events analysis, signal of outer detector isused as trigger instead of veto.
The speci cation of Kamiokande-(II,III) and Super-Kamiokande detectors are summarized inTable2.1. The schematic views of detector are shown in Fig.2.2 for Kamiokande-(II-III), Fig.2.3 forSuper-Kamiokande. The inside photograph of Super-Kamiokande is shown in Fig. 2.4.
In the following text, the detail of detector speci cation is described about Super-Kamiokande.
That of Kamiokande already has many descriptions, for example Ref.[31],[5],[32], so it is not referred
to in this thesis.

Parameters
Super-Kamiokande Kamiokande-(II,III)
Remarks
Total size
42mh239m
16.1mh215.6m
Total mass
50,000 t
4,500 tNumber of PMTs 11,146 (20 inch)
948 (20 inch)
inner detector
1,885 (8 inch)
123 (20 inch)
anti detector
Photosensitive
40%
20%
coverage
PMT timing
3 nsec4 nsecat 1
resolution
photoelectronppEnergy resolution
2.6%/ E
3.6%/ Ee of E (GeV)
2.5%p
4%p
(<1 GeV)
16%/ E
20%/ Ee (<20 MeV)
Position
50 cm
110 cm10 MeV
resolution
 10 cm
15 cm
p ! e+0Angular28
28
10 Mev
resolution
 1
 2
Through going 
Eth (trigger)
45 MeV
5.2 MeVEth (analysis)
5 MeV
7.5 MeVsolar e/ separation
99%
9861 %
0.03< pe <1.33 GeV/c
0.2< p <1.5 GeV/c

Table 2.1: Performance of Super-Kamiokande detector and Kamiokande-(II,III) detector.

CHAPTER 2.

KAMIOKANDE AND SUPER-KAMIOKANDE DETECTORS

Figure 2.2: Schematic view of Kamiokande-(II,III) Detector

Figure 2.3: Schematic view of Super-Kamiokande Detector

32

CHAPTER 2.

KAMIOKANDE AND SUPER-KAMIOKANDE DETECTORS

33

Figure 2.4: The inside view of Super-Kamiokande. This photograph was taken when thedetector was under construction (Feb. 1996). Pure water was lled to the 3/5 level of the tankat this time. The man on the boat was wiping o dust on the surface of PMTs.

CHAPTER 2.

KAMIOKANDE AND SUPER-KAMIOKANDE DETECTORS

34

2.3.3 20 Inch-Diameter PMT
20 inch-diameter PMTs attached on the surface of inner detector were developed for Kamiokande
and improved for Super-Kamiokande by HAMAMATSU Photonics Co. Ltd. in cooperation withmembers of Kamiokande[33]. Fig.2.5 presents a cross-section picture of 20 inch-diameter PMT for
Super-Kamiokande. The gain of this PMT is about 107 with the applied voltage of about 2000V.
The strong feature of this PMT is it has acceptance of large area and wide angle due to it's big
diameter and sphere shaped photosensitive area. The quantum eciency of this PMT is goodenough in the range of 300 nm  600 nm which corresopnding to wave length of cherenkov light in
the water (Fig.2.6). To get good energy resolution for low energy events ( 5 MeV), a 1 p.e. levelsensitivity to the output signal is required. The 1 p.e. peak in the pulse height distribution can be
distinguished from dark current(Fig.2.7). Since photocasode and rst dinode has 30 cm distance,
trajectory of photoelectrons are a ected by external magnetic eld such as geomagnetic eld, andit may cause bad time resolution. In order to compensate the geomagnetic eld, electric wires arelaid on the surface of the cavity around detector to compensate for the magnetic eld.

Figure 2.5: Cross section view of 20 inch-in-diameter PMT

CHAPTER 2.

KAMIOKANDE AND SUPER-KAMIOKANDE DETECTORS

Figure 2.6: Quantum Eciencies of several 20 inch PMTs

Figure 2.7: Pulse height distribution corresponding to 1 p.e. of 20 inch PMT.

35

CHAPTER 2.

KAMIOKANDE AND SUPER-KAMIOKANDE DETECTORS

36

2.3.4 Water Puri cation System and Radon Free Air System

HEAT EXCHNGERIn order to collect cherenkov photons eciently, the water, as a mediator of cherenkov photons,
must be kept as transparent as possible. Futhermore, radioactive elements dissolved in the water,which is serious background for solar neutrino events, should be removed. For such purposes, awater puri cation system and radon free air system is necessary.
Fig.2.8 shows the water puri cation and radon free air system for Super-Kamiokande.
The 50,000 tons of water used in Super-Kamiokande is made from the water in the mine and it
goes through the water puri cation system to get rid of impurities. When pure water was lled in
the tank, it is circulated among the detector and water puri cation system to get higher purity ofwater.

VACUUM
DEGASIFIERIONEXCHANGERPUMPCARTRIDGE
POLISHERULTRAFILTER

Radon-free
air system

SK
TANK

PUMP
FILTER( 1µ mNom. )UV STERILIZERREVERSE OSMOSISBUFFER TANK

Figure 2.8: Block diagram for water puri cation system and radon free air system

CHAPTER 2.

KAMIOKANDE AND SUPER-KAMIOKANDE DETECTORS

37

2.3.5 Electronics and DAQSuper-Kamiokande has many channels of PMT signals, therefore the electronics and DAQ systemhave a complicated hierarchy. The block diagram of Electronics and DAQ (Data AcQuisition system)
are shown in Fig.2.9. It is explained how PMT's signal go to data taking workstation as an event
information in the following text.
Getting PMT's signal and converting them to charge and timing information are performed by
ATM (Analog Timing Module). ATM has functions of ADC (Analog to Digital Converter), TDC(Timing to Digital Converter), Ampli er and Discriminater. It is based on TKO (TRISTAN/KEK
Online) speci cation and designed for Super-Kamiokande.
Each ATM has 12 inputs from PMTs, and signal of each input is ampli ed and divided into
several signals. One is discriminated and the TTL logic output of all channels are summed to makea HITSUM signal. The other two signals are converted by TAC/QAC (timing-to-analog converter/ charge-to-analog converter) to a signal with a pulse height corresponding to it's timing/charge
quantity. The HITSUM signal for each ATM goes through a trigger making module and given to aworkstation to make a global trigger. The signal of the global trigger is divided and goes through
GONG(Go or Not Go) module and given to all ATMs. When an ATM gets a global trigger signal, it
converts the timing/charge signal to a digital value by ADC and write them to FIFO (First In First
Out) memories. The timing/charge information from the FIFO are gotten by the VME interface
board called SCH (super control header) and saved in the memory bu ers called SMP (super memory
partner) and given to data taking workstations.
There are 8 data taking workstations and data of each workstatin is transported through FDDI
and collected to one workstation and reconstructed as an event information.

2.3.6 Real Time Data Reductions
In Super-Kamiokande, the data transfer size is huge (6 Gbyte/hour) because of the large numberof PMT channels (11146), and high trigger counting ratio ( 10Hz). Therefore, data of SuperKamiokande is reduced by oine computers in real time. Fig.2.10 is a block diagram of the realtime data reductions. In the following text, each process of the real time reduction is explained.
The reconstructed data are transferred from online computer in the mine to the oine computerin the oce out of the mine by FDDI. Then data are reformated to ZEBRA 4 format and saved ona ram disk temporally.
The reformated les are loaded on the memory of a super-computer and converts the timing/charge information from ADC/TDC counts to real physical unit of nano sec/photoelectron. We
called this procedure TQREAL process.
After TQREAL process, the data are distributed to several workstations and subjected to thevarious reductions for each analysis, USA, LOW, MU, MID, SN, etc.
USA is the copy process to tape which belongs to the o site analysis group 5 , LOW is the
reduction process for the low energy analysis { i.e. solar neutrino analysis, MU is the reduction
process of muon events for upward through going muons and the radioactive spallation cut of low
energy analysis, MID is the reduction process for the middle energy analysis { i.e. atmospheric
neutrino analysis as contained events, and the nucleon decay analysis. SN is a watch process forsuper nova.
After those reduction processes, the data are saved in the MTL (Magnetic Tape Library).
45Fortran based data bank system software belong to the CERN program library.
This analysis group is composed of members from universities and laboratories in United States.

CHAPTER 2.

38

KAMIOKANDE AND SUPER-KAMIOKANDE DETECTORS

20-inch PMT
SCH
ATM
x 20
x 240

ATM
GONG

20-inch PMTSCHATMinterface
SUN classic

x 20
x 240

SMP

ATM

SMP

GONG

online CPU(slave)

SMP
SMP
SMP
SMP

x6

Super Memory Partner

VME
20-inch PMTSCH

SUN classic

ATM

online CPU(slave)

x 20
x 240

SUN classic

ATM
GONG

online CPU(host)

Analog Timing Module

online CPU(slave)
TKO

FDDI

SUN classic

SUN sparc10

online CPU(slave)
SUN classic

online CPU(slave)
SUN classic

FDDI

online CPU(slave)
SUN classic

online CPU(slave)

online CPU(slave)interface
SUN classic
SMP
SMP

SUN classic

online CPU(slave)

SMP
SMP
SMP
SMP

Super Memory Partnerinterface

VME
VME

interrupt reg.
TRG

20-inch PMT
SCH
TRIGGERATM

TRIGGER

x 20
x 240

PROCESSORHIT INFORMATION
ATM
GONG

Analog Timing ModuleTKO

PMT x 11200

ATM x ~1000

SMP x 48

online CPU(slave) x 9

Figure 2.9: DAQ Block Diagram of Super KamiokandeCHAPTER 2.

KAMIOKANDE AND SUPER-KAMIOKANDE DETECTORS

Online Format Data

(in the mine)

Reformat

Offline Computer
Super SUPARK 20
Super Computer
VPX 210(out the mine)
Ram Disk
75Mbyte/10min

TQREALMTL
200Mbyte/10min
Flow
Disk

USA

LOW

MU

MID

SN

etc

Work Stations
Super SPARK 20

USA

LOW

MU

MID

SN

etc

US storage

Figure 2.10: Realtime Oine Reduction Process

39Chapter 3
Event Selection
In this thesis, upward through going muon events which originate from atmospheric neutrinos areused to search for muon neutrino oscillations. The method to select upward through going muon
events is described in this chapter.

3.1 Event Selection Overview
Kamiokande and Super-Kamiokande detectors consist of pure water for a target material and a lotof PMTs for detecting Cherenkov light. The information used for event reconstruction is as follows:
1. Charge of PMT signal
2. Timing of PMT signal
3. Geometrical pattern of hit PMTs
Generally speaking, charge information is used to classify the event energy. On the other handtiming information and geometrical pattern are used to reconstruct the event vertex point and its
direction.
The primitive quantities for event selection are:

 Total Q

Sum of charge of all PMTs in the inner detector. The unit of charge is photoelectrons, It isalso called \Total PE".

 Max QThe charge of the PMT which has the maximum charge among all the PMTs.

 NhitNumber of hit PMTs in one event, where \hit PMTs" means PMTs whose charges are greaterthan the threshold charge. A threshold charge is de ned for each analysis.

 GDNS

Standard quantity to obtain goodness in the directional tting in which the relative timing
information of the PMTs is used. The de nition of GDNS is in section 3.6.2.

In the case of selection of the upward through going muon events, other quantities(functions)for muon directional tting are de ned. They are de ned in section 3.3.1.

40

CHAPTER 3.

41

EVENT SELECTION

The event selection process is roughly divided into three steps;
1. Total Q cut.
2. Automatic directional tting and then zenith angle selection, cos  < 0:1,
where  is de ned in Fig.3.1.
3. Eye scan cut to get rid of junk events and then manual directional tting to obtain the correct
muon direction.

verticalupwardZθ : zenith
ϕ : azimuth

θydetectorϕhorizontal
plane
µ

x

Figure 3.1: De nition of zenith and azimuth angle.
The ow chart to select the upward through going muon events is shown in Fig.3.2 for KamiokandeIII and Fig.3.3 for Super-Kamiokande. Each step is explained in the latter sections.

CHAPTER 3.

42

EVENT SELECTION

Raw Data
5m < track length < 20 m
6000 p.e.< Total Q <30000 p.e.

yeseliminatecontained events

Anti Total Q > 0 p.e.
yesMuon Fitting

bad

good

θ > 80 oyesoθ > 90for good Fit events

Upmu
File

yes
Final Upmu
SampleManual Fit

Eye Scan
Cut

Figure 3.2: The ow chart of event selection for the upward through going muon events in
Kamiokande-III.

CHAPTER 3.

43

EVENT SELECTION

1st reduction(real time)

2nd reduction(not real time)~ 10 Hz
Raw Data~ 5m track length
Eµ > 1.3 GeV~ 2m track length
E µ > 400 MeV

Total Q >12000 p.e.

Total Q >5000 p.e.

eliminate multi bundle
cosmic ray muonsyes
Detector Punch
Through Condition

yes

2nd Muon Fitting

Max Q > 235 p.e.
yes

~ 1.8 Hzcos θ < 0.1

bad
1st Muon Fitting

yes
good

muon vectorFilecos θ < 0.0
cos θ < 0.1

no
yes

yes
~ 1.2 k/day
2nd Upmu
File

Horizontal
Mu File1st Upmu
File

yes
Final Upmu
SampleManual Fit

Eye Scan
Cut

Figure 3.3: The ow chart of event selection for the upward through going muon events inSuperk-Kamiokande.

CHAPTER 3.

EVENT SELECTION

44

3.2 Typical Muon Events
In order to see the geometrical pattern of Cherenkov light, we use the detector-spread-shaped event
display. For example, a detector-punch-through muon event like Fig.3.4 looks like Fig.3.5 by the
Kamiokande-III event display, and like Fig.3.6 by the Super-Kamiokande event display.
The spread-shape of the inner detector is represented by two circles and one rectangle whichcorrespond to the top, bottom and barrel surface of the cylinder-shaped detector. Thr spread-shapeof the anti detector is also shown at the upper left position of the event display which has a similar
shape to the inner one. \ " marks on the detector surface represent PMTs attached on the detector
surface. The size of the \ " is proportional to the pulse height of corresponding PMT. \2" markson the detector surface represent dead PMT positions.
In Fig.3.5, we can a see a ring-shaped Cherenkov light pattern from a cosmic-ray muon. In thisevent, the entrance and exit positions of the muon are tted and marked by \3". The edge lineof Cherenkov ring is also drawn by assuming 42 opening angle from the tted muon track. In the
spread-shape of the anti detector, we can see the hit PMTs corresponding to the entrance and exit
positions at the inner detector.
Fig.3.6 shows the event display of Super-Kamiokande. This event is a typical single-track cosmicray muon. The display form is similar to Kamiokande-III except that dead PMT's are marked by asmall \3".
Muon events are categorized by their geometrical pattern of Cherenkov light as follows:
1. single-track muon event with no shower interactions { so called clean event.
(Fig.3.5 for Kam-3, Fig.3.6 for S-Kam)
2. multiple-bundle muon event { so called multi event.
(Fig.3.7, Fig.3.8 for S-Kam)
3. muon which stopped in the inner detector { so called stopping event.
(Fig.3.9 for S-Kam)
4. muon event with hard electromagnetic and/or hadronic showers { so called hard event.
(Fig.3.10 for S-Kam)
5. muon event crossing the detector edge { so called edge clipping event.
(Fig.3.11, Fig.3.12 for S-Kam)

CHAPTER 3.

45

EVENT SELECTION

Kam-3/S-Kam detector

42 oCherenkov Light
hit regionµFigure 3.4: Schematic view of detector-punch-thorough muon event. Cherenkov light of a muon
makes a ring pattern in the detector.

CHAPTER 3.

EVENT SELECTION

46

Figure 3.5: The detector-spread event display of Kamiokande-III. This is a typical single track
cosmic-ray muon. In this event, the entrance and exit position of the muon are tted andmarked by \3". The edge line of Cherenkov ring is also drawn by assuming 42opening angle
from tted muon track. \ " marks on detector's surface represent PMTs attached on detector
wall. The size of \ " is proportional to pulse height of corresponding PMT. \2" marks ondetector's surface represent dead PMTs.

CHAPTER 3.

47

EVENT SELECTION

NUM
33
RUN
1743
EVENT
188
DATE 96-May-31
TIME
21:43:51
TOT PE: 74546.4
MAX PE:
251.8
NMHIT : 8374
ANT-PE: 1719.3
ANT-MX:
43.0
NMHITA:
239

RunMODE:NORMALTRG ID :00001011
T diff.:0.467E+05us: 46.7ms
FSCC:
0
TDC0: -1129.0
Q thr. :0.0
BAD ch.: masked
SUB EV : 0/ 0

Figure 3.6: The detector-spread event display of Super-Kamiokande. This is a typical event ofa single track cosmic-ray muon. The display form is similar to Kamiokande-III's one except
dead PMT's are marked by small \3".

CHAPTER 3.

48

EVENT SELECTION

NUM
1
RUN
1743
EVENT
5636
DATE 96-May-31
TIME
21:52:27
TOT PE: 47137.5
MAX PE:
282.0
NMHIT : 4651
ANT-PE: 2394.5
ANT-MX:
56.2
NMHITA:
313

RunMODE:NORMALTRG ID :00001011
T diff.:0.742E+05us: 74.2ms
FSCC:
0
TDC0: -1127.5
Q thr. :0.0
BAD ch.: masked
SUB EV : 0/ 0

Figure 3.7: Two track cosmic-ray muon event (Super-Kamiokande).

CHAPTER 3.

49

EVENT SELECTION

NUM
2
RUN
1743
EVENT
2699
DATE 96-May-31
TIME
21:48: 0
TOT PE:203043.1
MAX PE:
272.9
NMHIT : 10618
ANT-PE: 4467.0
ANT-MX:
78.0
NMHITA:
514

RunMODE:NORMALTRG ID :00001011
T diff.:0.789E+05us: 78.9ms
FSCC:
0
TDC0: -1132.2
Q thr. :0.0
BAD ch.: masked
SUB EV : 0/ 0

Figure 3.8: Four track cosmic-ray muon event. Edge lines of Cherenkov rings are not drawn.
(Super-Kamiokande)CHAPTER 3.

50

EVENT SELECTION

NUM
448
RUN
1743
EVENT
211
DATE 96-May-31
TIME
21:43:58
TOT PE: 19032.1
MAX PE:
70.7
NMHIT : 5584
ANT-PE:
552.2
ANT-MX:
39.4
NMHITA:
90

RunMODE:NORMALTRG ID :00001011
T diff.:0.356E+05us: 35.6ms
FSCC:
0
TDC0: -1128.0
Q thr. :0.0
BAD ch.: masked
SUB EV : 0/ 0

Figure 3.9: Typical event of a stopping cosmic-ray muon (Super-Kamiokande).

CHAPTER 3.

51

EVENT SELECTION

NUM
1
RUN
1743
EVENT
2134
DATE 96-May-31
TIME
21:47: 4
TOT PE:1198176.1
MAX PE:
308.7
NMHIT : 11076
ANT-PE: 1052.0
ANT-MX:
53.6
NMHITA:
156

RunMODE:NORMALTRG ID :00001011
T diff.:0.356E+05us: 35.6ms
FSCC:
0
TDC0: -1132.5
Q thr. :0.0
BAD ch.: masked
SUB EV : 0/ 0

Figure 3.10: Typical event of a hard showered cosmic-ray muon(Super-Kamiokande).

CHAPTER 3.

52

EVENT SELECTION

NUM
112
RUN
1743
EVENT
573
DATE 96-May-31
TIME
21:44:29
TOT PE: 6099.8
MAX PE:
254.1
NMHIT : 1421
ANT-PE:
999.7
ANT-MX:
48.5
NMHITA:
191

RunMODE:NORMALTRG ID :00001011
T diff.:0.206E+06us: 206.
ms
FSCC:
0
TDC0: -1128.2
Q thr. :0.0
BAD ch.: masked
SUB EV : 0/ 0

Figure 3.11: Typical event of an edge-clipping cosmic-ray muon (Super-Kamiokande).

CHAPTER 3.

53

EVENT SELECTION

NUM
285
RUN
1743
EVENT
1640
DATE 96-May-31
TIME
21:46:14
TOT PE: 5400.5
MAX PE:
265.3
NMHIT :
624
ANT-PE: 1566.1
ANT-MX:
46.2
NMHITA:
186

RunMODE:NORMALTRG ID :00001011
T diff.:0.175E+06us: 175.
ms
FSCC:
0
TDC0: -1127.5
Q thr. :0.0
BAD ch.: masked
SUB EV : 0/ 0

Figure 3.12: Another event of an edge-clipping cosmic-ray muon (Super-Kamiokande).

CHAPTER 3.

54

EVENT SELECTION

In order to categorize the event types, relative timing information T and charge information Q of
PMTs are useful. Fig.3.13 shows T and Q distributions of a clean cosmic-ray muon event. Relative
timing distribution has two components. One is a peak distribution in the 950  1150 nsec which> 1150 nsec whichcorresponds to the direct Cherenkov light. The other is a broad tail shown in T corresponds to the light re ected on the PMTs or the detector surface.
In terms of the charge distribution, a peak near Q  0 p.e. is from hit PMTs from re ection
light or dark pulse. For detector punch-through-muon events, the charge of the PMTs which is close
to the muon exit position saturates at  250 p.e.
Fig.3.14 shows the T and Q distributionas of a stopping event. Since there is no exit point in
the stopping muon event, we can see no charges saturate.
Fig.3.15 shows the T and Q distributions of a hard event. The relative timing distribution doesn't
have a broad tail of re ected light because almost all the PMT's hits from direct light are due tothe cascade shower. Also the charge distribution has no re ection component. A lot of Q's are
saturated.

num. of PMTs

run_1743_event_188
800
700
600
500
400
300
200
100
0

800

900

1000

1100

1200

1300

num. of PMTs

T

10

3

10

2

1400

1500

relative time (nsec)

10

1
0

50

100

150

Q

200

250

charge (p.e.)

Figure 3.13: Upper histogram is relative timing distribution of inner PMTs for a typical single
track cosmic-ray muon event which corresponds to the event of Fig.3.6 (Super-Kamiokande).
Unit of T is nano seconds. Lower histogram is the charge distribution of inner PMTs for thesame event. Unit of Q is photoelectron.

CHAPTER 3.

55

EVENT SELECTION

num. of PMTs

run_1743_event_211
800
700
600
500
400
300
200
100
0

800

900

1000

1100

1200

1300

num. of PMTs

T
10

3

10

2

1400

1500

relative time (nsec)

10

1
0

10

20

30

40

50

60

70

charge (p.e.)

Q

Figure 3.14: T and Q distribution of typical stopping cosmic-ray muon event. This event
corresponds to the event of Fig.3.9 (Super-Kamiokande).

num. of PMTs

run_1743_event_2134

1600
1400
1200
1000
800
600
400
200
0

800

900

1000

1100

1200

num. of PMTs

10

1300

1400

1500

relative time (nsec)

T

2

10

1
0

50

100

150

200

Q

250

300

charge (p.e.)

Figure 3.15: T and Q distribution of typical hard showered cosmic-ray muon event. This event
corresponds to the event of Fig.3.10 (Super-Kamiokande).

CHAPTER 3.

EVENT SELECTION

56

3.3 Automatic Muon Directional Fitting
In order to select upward through going muon events it is necessary to t the muon track. The
muon track is determined by the muon entrance and exit positions on the inner detector wall. There
are two ways to t the entrance and exit positions. One is by human's eyes where two positions
are determined manually by viewing the event with the detector-spread event display. The other is
automatic track nding using T,Q information. Since almost all of the muon events are cosmic-ray
muons coming from above the horizontal direction( cos  > 0), we have to eliminate these eventsas eciently as possible. In the pre-reduction process for selecting upward through going muon
events, automatic muon directional tting is applied. The automatical method to determine themuon direction has basically 3 steps for both Kamiokande-III and Super-Kamiokande.
1. muon entrance nding
2. muon exit nding
3. muon track determination
For Super-Kamiokande, the muon event rate becomes much higher than that of Kamiokande,because of large detector acceptance 1 . So two muon tting processes are applied(see Fig.3.3). The
former is called the 1st muon tting which runs fast (1  2 sec/event) so that it is used in
realtime along with the online data taking. The latter is called the 2nd muon tting which nds
the muon track more precisely in the oine analysis but more slowly (1  10 sec/event).
In the following sections, automatic muon tting for Kamiokande-III and Super-Kamiokande are
explained.

3.3.1 Muon Fitting for Kamiokande-III
Processes of the muon tting for Kamiokande-III are in the following.
1. muon entrance nding:Candidates of muon entrance the position are determined by positions of PMTs which are hit
relatively early.
2. muon exit nding:Candidates of muon exit position are determined by the positions of PMTs which have maximum charges.
3. muon track determination:
Apply tting evaluation functions to each muon track which is reconstructed by entranceexit candidates, then a track which reduces the best evaluation value is selected.
There are two tting evaluation functions performed at the muon track determination. One isbased on charge information and is called RCONE which means Ratio of hit PMT numbers insideand outside of Cherenkov cone. RCONE is de ned by the following formula,

N
N
RCONE = 1 0 hit42 + hit70 :(3.1)
NPMT42 NPMT70
1 In Kamiokande-III counting rate of cosmic-ray muon event is about 0.37 Hz, while in Super-Kamiokande it
is about 2.0 Hz.

CHAPTER 3.

57

EVENT SELECTION

where Nhit42 is the number of hit PMTs in 42 Cherenkov cone, NPMT42 is the number of PMTs in
42 Cherenkov cone, Nhit70 is the number of hit PMTs between 70 cone and 42 cone, NPMT70 is
number ofpPMTs between 70 cone and 42 cone. The threshold charge of Nhit is set to an empiricalformula, total Q=100. The term Nhit42 =NPMT42 represents the \ lledness" of the Cherenkov ringand the term of Nhit70 =NPMT70 represents the \sharpness" of the edge of the Cherenkov ring. A
smaller RCONE means a better muon t.
Another tting evaluation function is based on the relative hit timing information and called
TOF- which means deviation( ) of the timing distribution subtracted Time of Flight of the muonand Cherenkov light.
TOF- is de ned by the following formula,
TOF- = RMS(Ti 0 TOFi )

(3.2)

l

(3.3)

l
c=n

TOFi = (i) + p(i)cwhere index i is the PMT number, Ti is the timing information of the i-th PMT, l(i) is the distancefrom the entrance point of the muon to the emission point of the Cherenkov photon which willreach the i-th PMT, lp(i) is the distance from the emission point of the Cherenkov photon to thei-th PMT, c is the light velocity in vacuum, n is the refractive index of water. Fig.3.16 shows how
to de ne l(i) and lp(i) . RMS is the root mean square function de ned byq

RMS(xi ) =

N x2
i i

P

N

:

(3.4)

A smaller TOF- means a better muon t.
Kam-3/S-Kam detectorl µ (i)

oto
ph

µ

n

42 o

l p(i)

PMT i

Figure 3.16: Positional correlation between muon and emitted Cherenkov photon.
These functions have good convergence to the true position of the muon entrance/exit point.
Fig.3.17 and Fig.3.18 shows convergence of RCONE, TOF- for Kamiokande-III, respectively. The
position which makes RCONE and TOF- a minimum value is taken as best t position for themuon entrance/exit.

CHAPTER 3.

58

EVENT SELECTION

Rcone

Convergency of Rcone
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0

2

4

6

8

10

12

14

Rcone

distance to entrance (m)
1.4
1.2
1
0.8
0.6
0.4
0.2
0

2

4

6

8

10

12

14

distance to exit (m)Figure 3.17: The upper gure shows the convergence of RCONE to the muon entrance. The
lower gure shows the convergence of RCONE to the muon exit. (Kamiokande-III)

TOFSGM (nsec)

Convergency of TOF-sigma
22.5
20
17.5
15
12.5
10
7.5
5
2.5
0

2

4

6

8

10

12

14

TOFSGM (nsec)

distance to entrance (m)
7
6
5
4
3
2
0

2

4

6

8

10

12

14

disrance to exit (m)Figure 3.18: The upper gure shows convergence of TOF- to the muon entrance. The lower
gure shows convergence of TOF- to the muon exit. (Kamiokande-III)

CHAPTER 3.

59

EVENT SELECTION

When one selects upward through going muon events, cosmic-ray muons are eliminated by theirzenith angles, cos   0, so good angular resolution for zenith angle directions is necessary. The
zenith angle deviation is de ned as

1 = jMF 0 MCj

(3.5)

where MF is the zenith angle of the muon reconstructed by automatic muon tting, MC is thezenith angle of the muon for the MC initial parameter.
1 distribution is tted by a gaussan function,

1 1 2 ;(3.6)
Gzenith(1) = c(1) 1 exp 0
2 (1)c(1) = c0 1 e0a 1 ;(3.7)
2
(1) = 0 + a2 1 1 ;(3.8)
where scale factor c(1) and deviation sigma  (1) are dependent on 1, therefore, there are 4
1

0

! 1

@

A

2

free parameters, c0 ; 0 ; a1 ; a2 .
Fig.3.19 shows the zenith angle deviation of muon tting. From this distribution we have,c0 =473 6 12 ,
0 = 2.66 6 0.06 
a1 = 0.0072 6 0.0009 , a2 = 0.0165 6 0.0015
The zenith angle resolution of muon tting is now reduced by the following formula,c(x) 1 (x) dx
R:c(x) dx(3.9)

 = 3:68 6 0:07 :(3.10)

 =We gotR

Because the Kamiokande-III/Super Kamiokande detectors have cylindrical shape, the geometricalacceptance is not symmetric for zenith angle direction. We have to check zenith angle dependenceof angular resolution. Fig.3.20 shows that dependence. We can see no signi cant zenith angle
dependence.
Moreover muon track reconstruction is suspected to be dependent on the muon track length forthe same reason. We also check the track length dependence of the angular deviation. Fig.3.21
shows that dependence.
We can see the inverse proportional correlation between the track length and the zenith angle
deviation. The origin of this dependence is that if the muon track length is short, the number of
PMTs used in the muon track reconstruction become relatively few (see Fig.3.12), and that makes
the large deviation to tting evaluation functions de ned by Eq.(3.1), Eq.(3.2).
The correlation becomes constant when the track length is more than 7 m. Therefore we adopted
a 7 m track length cut for the nal upmu samples.

CHAPTER 3.

5

EVENT SELECTION

500

400

300

200

100

0
0

0.2

0.3

10

0.4

15

ID
Entries
Mean
RMS
P1
P2
P3
P4

20

30

100
3634
3.187
3.575
46.95 / 51
473.2
2.656
0.7158E-02
0.1651E-01

25

zenith angle deviation ∆θ (deg.)

0.5

0.6

0.8

0.9

zenith angle cos θ

0.7

Zenith angle dependence of zenith angle deviation

0.1

1

60

Figure 3.19: Zenith angle deviation of muon direction reconstructed by automatic muon tting
compared with MC muon direction as reference. (Kamiokande-III)

6

5

4

3

2

1

0

0

Figure 3.20: Zenith angle dependence of zenith angle resolution  (Kamiokande-III)zenith angle resolution σθ (deg.)CHAPTER 3.

EVENT SELECTION

2

2.5

5

7.5

10

15

17.5

20

muon track length (m)12.5

Track length dependence of zenith angle deviation
10

10

1
0

61

The angular deviation of 1st muon tting is estimated by using muon events which are createdby Monte Carlo detector simulation.
The zenith angle deviation of 1st muon tting is shown in Fig.3.25. We can see angular deviation
becomes half in comparison with Kamiokande-III's one because PMT coverage density of SuperKamiokande is two times denser than that of Kamiokande-III.

The algorithm of muon track determination for 1st muon tting is simple and fast. It nds thecenter of mass positions for each entrance/exit candidates and just joins them. Approximationsof the center of mass position to the true muon entrance /exit points are very good (seeFig.3.23, Fig.3.24).

3. muon track determination:

The candidates of muon exit position are determined by the positions of PMTs which havemaximum charges.

2. muon exit nding:

The candidates of muon entrance position are determined by the PMT positions which are hit
relatively early.

1. muon entrance nding:

The process of 1st muon tting for Super-Kamiokande is as follows,

3.3.2 1st Muon Fitting for Super-Kamiokande

Figure 3.21: Track length dependence of the zenith angle deviation 1. We applied a 7m track
length cut to get good zenith angle resolution. (Kamiokande-III)zenith angle deviation ∆θ (deg.)CHAPTER 3.

EVENT SELECTION62

The zenith angle deviation of 1st muon tting is shown in Fig.3.25. The zenith angle deviation
is tted by function Gzenith (1) which is de ned by Eq.(3.6).
From this distribution we have,c0 =255 6 12,
0 = 1.24 6 0.04 
a1 = 0.0131 6 0.0023, a2 = 0.0264 6 0.0022
We have the zenith angle resolution for 1st muon tting by Eq.(3.9)

 = 2:23 6 0:05 :(3.11)

3.3.3 2nd Muon Fitting for Super-Kamiokande
The process of 2nd muon tting for Super-Kamiokande is as follows,
1. muon entrance nding:
The candidates of muon entrance position are determined by the PMT positions which are hit
relatively early.
2. muon exit nding:
The candidates of muon exit position are determined by the positions of PMTs which havemaximum charges.
3. muon track determination:
Muon entrance/exit candidates positions are clustered and for each cluster, tting evaluation functions de ned by Eq.(3.1), (3.2) are applied, then the track which reduces the best
evaluation value is selected.
The tting evaluation functions, RCONE,TOF- , are same as those of Kamiokande-III. The
convergence of the tting evaluation function for 2nd muon tting of Super-Kamiokande is shownin Fig.3.26 and Fig.3.27. We can see good convergence both of RCONE and TOF- .
The zenith angle deviation for 2nd muon tting of Super-Kamiokande is shown in Fig.3.28.
From this distribution we got,c0 =273 6 13,
0 = 1.14 6 0.04
a1 = 0.0189 6 0.0039, a2 = 0.0313 6 0.0032
We got zenith angle resolution,

 = 1:97 6 0:06 :(3.12)

Zenith angle dependence of angular resolution for 2nd muon tting is checked. Fig.3.29 showthat dependence. We can see no signi cant zenith angle dependence on it.
Muon track length dependence of the zenith angle deviation is shown in Fig.3.30. We can see
an inverse proportional correlation between track length and zenith angle deviation. Although thecorrelation becomes constant when the track length is more than 7 m. Therefore we adopted a 7
m track length cut to 2nd muon tting.

CHAPTER 3.

63

EVENT SELECTION

NUM
19
RUN
0
EVENT
19
DATE 96-Sep-14
TIME
0:31: 9TOT PE: 83052.3
MAX PE:
251.8
NMHIT : 9347
ANT-PE: 8647.3
ANT-MX:
414.0
NMHITA:
512

RunMODE:MonteCarloTRG ID :00000000
T diff.:
0.
us
:
0.
ms
FSCC:
0
TDC0: 14910.0
Q thr. :0.0
BAD ch.: masked
SUB EV : 0/ 0

Figure 3.22: Single muon event generated by detector MC simulation. Muon entrance and exit
positions marked by \3" are extracted from initial position and momentum of the muon in thisMC event. The edge line of the Cherenkov ring is drawn in this event display.

CHAPTER 3.

64

EVENT SELECTION

NUM
19
RUN
0
EVENT
19
DATE 96-Sep-14
TIME
0:31: 9TOT PE: 83052.3
MAX PE:
251.8
NMHIT : 9347
ANT-PE: 8647.3
ANT-MX:
414.0
NMHITA:
512

RunMODE:MonteCarloTRG ID :00000000
T diff.:
0.
us
:
0.
ms
FSCC:
0
TDC0: 14910.0
Q thr. :0.0
BAD ch.: masked
SUB EV : 0/ 0

Figure 3.23: Only PMTs which are hit relatively early are drawn in this event display. The
data are generated by MC (same to Fig.3.22). We can see clearly that the position of fast hit
PMTs well agrees with the position of the muon entrance.

CHAPTER 3.

65

EVENT SELECTION

NUM
19
RUN
0
EVENT
19
DATE 96-Sep-14
TIME
0:31: 9TOT PE: 83052.3
MAX PE:
251.8
NMHIT : 9347
ANT-PE: 8647.3
ANT-MX:
414.0
NMHITA:
512

RunMODE:MonteCarloTRG ID :00000000
T diff.:
0.
us
:
0.
ms
FSCC:
0
TDC0: 14910.0
Q thr. : 237.0
BAD ch.: masked
SUB EV : 0/ 0

Figure 3.24: Only PMTs which are charge saturated (Q> 235) are drawn in this event display.
The data are generated by MC (same to Fig.3.22). We can see clearly that the position of thesaturated Q PMTs well agrees with the position of muon exit.

CHAPTER 3.

66

EVENT SELECTION

ID
Entries
Mean
RMS

100
994
1.867
2.700
29.96 / 32
254.6
1.239
0.1308E-01
0.2644E-01

250
P1
P2
P3
P4

200

150

100

50

0

0

2

4

6

8

10

12

14

16

18

20

zenith angle deviation ∆θ (deg.)

Figure 3.25: Zenith angle deviation of muon direction reconstructed by automatic 1st muontting compared with MC muon direction as reference. (Super-Kamiokande)

Rcone

Convergency of Rcone
0.8
0.7
0.6
0.5
0.4
0.3
0.2
5

10

15

20

25

30

35

40

Rcone

distance to entrance (m)
1.6
1.4
1.2
1
0.8
0.6
0.4
0.2
0

5

10

15

20

25

30

35

40

distance to exit (m)Figure 3.26: The upper gure shows convergence of RCONE to the muon entrance. The lower
gure shows convergence of RCONE to the muon exit. (Super-Kamiokande)CHAPTER 3.

67

EVENT SELECTIONTOFSGM (nsec)

Convergency of TOF-sigma
30
25
20
15
10
5
5

10

15

20

25

30

35

TOFSGM (nsec)

distance to entrance (m)
20
18
16
14
12
10
8
6
4
2
0

5

10

15

20

25

30

35

40

disrance to exit (m)Figure 3.27: The upper gure shows the convergence of TOF- to the muon entrance. The
lower gure shows the convergence of TOF- to the muon exit. (Super-Kamiokande)ID
Entries
Mean
RMS

100
994
1.775
2.625
33.97 / 29
272.9
1.137
0.1892E-01
0.3126E-01

250
P1
P2
P3
P4

200

150

100

50

0

0

2

4

6

8

10

12

14

16

18

20

zenith angle deviation ∆θ (deg.)

Figure 3.28: The zenith angle deviation of muon direction reconstructed by the automatic 2nd
muon tting compared with MC muon direction as reference. (Super-Kamiokande)CHAPTER 3.

0

0.1

0.2

0.3

0.4

0.5

0.6

0.8

0.9

1

zenith angle cos θ

0.7

Zenith angle dependence of zenith angle deviation

EVENT SELECTION

4

3.5

3

2.5

2

1.5

1

0.5

0

10

1

10

10

15

20

25

35

40

45

50

Muon Track Length (m)30Zenith Angle Deviation of Automatic Muon Fit

5

68

Figure 3.30: The track length dependence of the zenith angle deviation of 2nd muon tting.
We applied a 7 m track length cut to get good zenith angle resolution. (Super-Kamiokande)

0-1Figure 3.29: The zenith angle dependence of the zenith angle resolution  for 2nd muon tting.
(Super-Kamiokande)

zenith angle resolution σθ1 (deg.)
Zenith Angle Deviation (deg.)CHAPTER 3.

EVENT SELECTION

69

3.4 Total Q and Event Classi cation
As has been discussed previously, events taken by Kamiokande-III and Super-Kamiokande are classi ed by Total Q. Fig.3.31 and Fig.3.32 shows Total Q distribution of Kamiokande-III and SuperKamiokande respectively.
There are two peaks in the total Q distribution. The lower one dominantly consists of ; -rays
from radioactive sources in the water or from excited nuclei which are spalled by cosmic-ray muons.
The Cherenkov light intensity of this peak corresponds to energy deposits of electrons which have1  50 MeV. Solar neutrino events are hidden in this energy region.
The valley region between two peaks dominantly consists of Cherenkov light from cosmic-ray
muons with track lengths which are relatively short such as detector edge clipping muons andstopping muons. Cherenkov light from electrons, muons and pions which are generated in thedetector by atmospheric neutrinos are also in this region. The energy deposit of electrons andmuons in this region is 100 MeV  10 GeV. The fully contained events of atmospheric neutrinos
and, if they exist, proton decay events are hidden in this energy region.
The higher peak consists of the Cherenkov light from cosmic-ray muons which punch throughdetector. The energy deposit of muons in this region is 1 GeV  100 GeV. The upward throughgoing muon events are hidden in this energy region.
Total Q distribution is proportional to the muon track length because the muon energy deposit
is proportional to it's track length. Fig.3.33 and Fig.3.34 shows the correlation between total Q andmuon track length in Kamiokande-III and Super-Kamiokande. Distributions of total Q divided by
the track length are also shown in the gures.
The peak value is tted by a gaussian function, and it reduces to this formula between total Q
and muon track length,
Kamiokande-III :total Q(p.e.) = (13:7 6 1:4) 2 track length(cm) ;(3.13)

total Q(p.e.) = (22:5 6 2:2) 2 track length(cm) :(3.14)

Super-Kamiokande :The reason for the di erence in the coecient between Kam-3 and S-Kam is that PMT gain
and PMT coverage density is increased from Kam-3 to S-Kam.

CHAPTER 3.

70

EVENT SELECTION

Total Q distribution of Kamiokande-III
3000

6000 p.e.(5 m track length)
2500

30000 p.e.
(22 m track length)

2000

1500

1000

500

0

0

1

2

3

4

5

6

7

log10(Total Q) (p.e.)

Figure 3.31: Total Q distribution of Kamiokande-III.

Total Q distribution of Super-Kamiokande
30000

25000

20000

muon events
15000

10000

5000

0

1

2

3

4

5

6

7

log10(Total Q) (p.e.)

Figure 3.32: Total Q distribution of Super-Kamiokande.

CHAPTER 3.

71

EVENT SELECTION

Total Q .vs. Muon track length (Kam-3)
ID
Entries
Mean
RMS

90
80
70
60
50
40
30
20
10
0

100
830
15.37
4.246
/ 10
85.15
13.68
1.376

23.70
Constant
Mean
Sigma

5

10

15

20

25

30

35

40

3

Total Q (10 p.e.)

Total Q/track length (p.e./cm)
50
45
40
35
30
25
20
15
10
5
0

0

2

4

6

8

10

12

14

16

18

20

track length (m)Figure 3.33: The upper gure shows the ratio of Total Q and muon track length, and the lower
gure shows the correlation plot between Total Q and the muon track length in Kamiokande-III.

Total Q .vs. Muon track length (S-Kam)
400

ID
Entries
Mean
RMS

350
300

11.98

250
Constant
Mean
Sigma

200
150

100
2657
24.81
5.493
/ 11
360.9
22.53
2.202

100
50
0

10

20

30

40

50

60

70

3

Total Q (10 p.e.)

Total Q/track length (p.e./cm)
225
200
175
150
125
100
75
50
25
10

20

30

40

50

track length (m)Figure 3.34: Same as Fig.3.33 in Super-Kamiokande.

CHAPTER 3.

72

EVENT SELECTION

Actually, a muon deposits it's energy not only by ionization but also by electromagnetic cascadeshower { bremsstrahlung, direct pair production and photo-nuclear interactions. This additionalenergy deposition appears as a Landu uctuation tail in the Total Q/muon track length distribution.
Selecting upward through going muon events, we perform a total Q cut to select detector throughgoing muon events.
For Kamiokande-III, we set total Q cut value as

6000(p.e.) < total Q < 30000(p.e.) :(3.15)The lower limit, 6000 p.e., corresponds to 4  5 m muon track length is deduced from Eq.(3.13).

From study of muon directional tting we found the angular resolution of muon directions is badwhen the muon track length is short, so we adapted 7m track length cut (see Fig.3.21). At the
pre-reduction phase, we took 4  5 m cut for safty.
The upper limit, 30000 p.e., corresponds to 20  25 m muon track length which is as long as
maximum passing length for Kamiokande-III detector. Although actually, the total Q of hard EM
showered muon events is larger than 30000 p.e., so hard events may be lost in this cut. We estimate
how many hard events are lost by MC events. Fig.3.35 shows the eciency of total Q cut for various
muon energies.

efficiency

Energy dependence of total Q cut efficiency
1

0.8

0.6

0.4

0.2

0
1

10

10

2

10

3

10

4

muon energy (GeV)

Figure 3.35: Eciency of total Q cut of Kamiokande-III for various muon energies.
For Super-Kamiokande, we set total Q cut value for the 1st reduction as

5000(p.e.) < total Q ;(3.16)

CHAPTER 3.

73

EVENT SELECTION

and for the 2nd reduction as

12000(p.e.) < total Q :(3.17)The 5000 p.e. cut for the 1st reduction corresponds to 2  2:5 m muon track length is deduced

from Eq.(3.14). The saved data passing this reduction is used not only for pward thorough going
muon analysis but also other muon analyses.
The 12000 p.e. cut for the 2nd reduction corresponds to 4:9  5:9 m muon track length. The
angular resoluation of muon direction is bad when muon track length is short. The 7m track length
cut is adapted(see Fig.3.30). At the pre-reduction phase, we took 4:9  5:9 cut for safty. In SuperKamiokande, the upper limit of total Q cut is not set in order to remain hard showered events. So
the eciency of the total Q cut in Super-Kamiokande is 100 %.

3.5 Other Q Cuts
For upward through going muons coming from out side the detector, PMTs in anti detector musthit. On the other hand, muons from fully contained events are generated in the inner detector, so
the PMTs in the anti detector must not hit.
In Kamiokande-III, to eliminate contained events, we set following condition after the total Q
cut,
0(p.e.) < anti total Q :(3.18)In Super-Kamiokande, anti total Q cut is not applied because in our analysis, especially in theearly time of the experiment, the DAQ system of the anti detector was not stable. Nevertheless, amaximum Q cut,
235(p.e.) < Max Q :(3.19)
is applied in the 1st reduction of Super-Kamiokande. This condition is set by assuming the charge of
PMTs at the muon exit must saturate if it is a penetrating muon event. For this condition, almost
all single-ring stopping muon events are rejected.

3.6 Muon Zenith Angle Selection
3.6.1 Zenith Angle Selection for Kamiokande-III
After total Q cut and anti total Q cut, automatic muon tting described in section 3.3.1 is applied.
Goodness of muon tting is estimated by the following conditions,

RCONE < 2000=total Q ;TOF- < 7 nsec ;(3.20)(3.21)

these conditions are deduced empirically.
After estimating goodness of muon tting, zenith angle selection of the remaining events is
performed,
 > 80 :(3.22)All remained bad condition events are saved. Finally, good condition events are selected as follows,

 > 90 :(3.23)

CHAPTER 3.

74

EVENT SELECTION

3.6.2 Zenith Angle Selection for Super-Kamiokande
After total Q cut and max Q cut, 1st muon tting described in section 3.3.2 is performed. Goodnessof muon tting is estimated by the follwing conditions,

Lenter > 3(m) & Lexit < 3(m)& GDNS > 0:88

(3.24)(3.25)

where Lenter is the distance from the max Q PMT position to the muon entrance position, Lexit
is the distance from the max Q PMT position to the muon exit position. These conditions areapplied because the max Q PMT position is identical to muon exit position. GDNS is de ned in thefollowing formula,!
X 1(
ti 0 T )2 X 1
exp 0 2 1 (~ 1 a) = 2
(3.26)GDNS =
2
i

i

i

i

where ti = Ti 0 TOFi is de ned in Eq.(3.2), (3.3), T is time when muon enter into inner detector,
i is timing resolution of i-th PMT, ~ is mean of i , a is scaling factor. GDNS represents 2estimation quantity which are normalized by exponential function, so it has range of 0  1. GDNS
! 1 means better tting.
After estimating goodness of muon tting, remained good condition events are performed zenith
angle selection as follows,cos  < 0:1 :(3.27)All remained bad condition events are saved.
After 1st muon tting, total Q cut of 2nd reduction is applied, then 2nd muon tting is applied.
All tted events are selected by zenith angle same as 1st reduction,cos  < 0:1 :(3.28)

3.7 Eye Scan & Manual Muon Fitting
Even though zenith angle selection is performed, some events which are not upward through going
muons remain.
Almost all such events are cosmic-ray muons with bad ts. Others are pedestal events 2 , PMT
ashing events 3 and contained events.
Table 3.1, 3.2 shows the fraction of event type remained before the eye scan phase for KamiokandeIII and Super-Kamiokande.
Upward through going muon events selected by eye scan are nally tted manually. The manual muon directional tting is performed by selecting muon entrance/exit points by looking at thedetector-spread event display. We estimate angular deviation of manual directional tting by comparing with MC muon events.
Zenith angle deviation of manual tting is de ned as follows

1MA = jMA 0 MCj ;(3.29)

Event that a part of channels do not take real signal but null signal to take the pedestal of their channels.
Event that one PMT is ashing because it's dinode is damaged. If it is often ashing, that channel will bekilled.
2

3

CHAPTER 3.

75

EVENT SELECTION

event typeclean event
multi event
cosmic-ray muon edge clipping event
hard event
stopping event
othersupward through going muon

ratio(%)
28
27
25
14
3
2
1

Table 3.1: Left event type after automatic zenith angle selection in Kamiokande-III.
event typemulti event
edge clipping event
cosmic-ray muon
hard event
stopping event
clean event
othersupward through going muon

ratio(%)
55
12
13
7
1
7
5

Table 3.2: Left event type after automatic zenith angle selection in Super-Kamiokande.
where MA is zenith angle of muon reconstructed by manual muon tting, MC is zenith angle ofmuon for MC initial parameter.
Fig. 3.36 and Fig. 3.37 are zenith angle deviations of Kamiokande-III and Super-kamiokande,
respectively. We tted these deviation distribution by function Gzenith (1) de ned by Eq.(3.6)
with the condition a1 = a2 = 0 because they have no broad tail components so that man does't
act so terrible mistake in the muon directional tting.
For Kamiokande-III, the tting results are c0 = 30.5 6 3.9, 0 = 1.89 6 0.08  .
Zenith angle resolution,  MA , is MA = 1:89 6 0:08 :(3.30)For Super-Kamiokande, the tting results are c0 = 67.1 6 6.5, 0 = 0.90 6 0.06  .
Zenith angle resolution,  MA , is MA = 0:90 6 0:06 :(3.31)

CHAPTER 3.

76

EVENT SELECTION

ID
Entries
Mean
RMS

35

20.85
P1
P2

30

100
200
1.698
1.435
/ 14
30.52
1.887

25

20

15

10

5

0

0

1

2

3

4

5

6

7

8

9

10

full angular deviation ∆θMC-MA (deg.)

Figure 3.36: Zenith angle deviation of manual directional reconstruction for Kamiokande-III.

ID
Entries
Mean
RMS

70

60

100
199
0.79100.65389.914 / 867.05
0.8997

P1
P2

50

40

30

20

10

0

0

1

2

3

4

5

6

7

8

9

10

zenith angle deviation ∆θMC-MA (deg.)

Figure 3.37: Zenith angle deviation of manual directional reconstruction for Super-Kamiokande.

CHAPTER 3.

77

EVENT SELECTION

3.8 Summary of the Event Selection
In Kamiokande-III, total 188 events of upward through going muon events are found in live time1332.5 days. Table 3.3 shows the summary of event selection for upward through going muon in
Kamiokande-III 4 5 .

total events
after total Q cut
after zenith90 < 
angle cut
80 <   90
after eye scan (cos  < 00:04)

number of event
1:4 2 108
2:9 2 107
23248
409536
188

Table 3.3: Summary of event selection of upward through going muon events in Kamiokande-III
In Super-Kamiokande, total 413 events of upward through going muon events are found in live
time 363 days. Table 3.4 shows the summary of event selection for upward through going muon inSuper-Kamiokande.

total events
after 1st reduction
after 2ndcos   0
reduction 0 < cos  < 0:1
after eye scannumber of event
3:5 2 108
2:1 2 105
6629
22511
413

Table 3.4: Summary of event selection of upward through going muon events in SuperKamiokande

4

In case of Kamiokande-III, the selection eciency of horizontal direction is not so good, we adapted cos  <00:04 (see Section 4.3).

Actually, we have two independent analysis in Kamiokande-III (Ref.[32]), the nal data are combined bothof them.
5Chapter 4
Observed Flux of Upward ThroughGoing Muons
he calculation of observed upward through going muon ux requires the following terms,

 Detector Live Time : The observation time which is used for analysis of upward throughgoing muons.

 E ective Area : Projected area of the detector on a plane normal to the direction of themuon track, where the muon track length is longer than 7 m.

 Selection Eciency : Eciency of selecting upward through going muons for one direction.
The di erential ux of upward through going muons is de ned as follows,d8!

= T 1 N1 iS 1 " ;d cos  ii i i
2 = 2 2 0:1
i=
0:06 (for i=10 in Kam-II,III) ;
Nbin

(4.1)
(4.2)

i = Int [(cos j + 1) 2 10] + 1 ;(4.3)
where the index , i, represents the i-th bin of zenith angle, Int(x) is the conversion function 1 fromreal numbers to integers, cos j is the cosine of the zenith angle of the j -th upward through going
muon, T is the detector live time, i is the solid angle of the i-th bin, Nbin (= 10) is the numberof bins, Si is the e ective area of the i-th bin, "i is the selection eciency of the i-th bin, Ni is thenumber of upward through going muons in the i-th bin.
The total ux of upward through going muons is de ned by averaging di erential uxes,

8tot =
1

NX
bin

1

i=1 NbinThis function round o under oating point part of x.

78

d8

!

d cos  i

:

(4.4)

CHAPTER 4.

79

OBSERVED FLUX OF UPWARD THROUGH GOING MUONS

4.1 Live Time Calculation
The total live time for upward through going muon analysis is calculated by

T=

X

i

(Tirun 0 Tidead 0 Tiped);(4.5)

where the index i represents the i-th observation run, Tirun is the run time, Tidead is the electronics
dead time, Tiped is the pedestal event time.
In Kamiokande-III, the total live time is

T = 1332:5 days.
The eciency of live time { T=T real is 86:8%, where T real is the elapsed time from the beginning

analysis run. Fig.4.1 shows the long term variation of the live time eciency of Kamiokande-III.

T/T

real

Live Time Efficiency of Kamiokande-III
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1

,

,

91

0

0

,

92

200

400

93

600

800

,

,

94

95 (year)1000 1200 1400 1600
realT

(total days)

Figure 4.1: Long term variation of live time eciency of Kamiokande-III.

CHAPTER 4.

OBSERVED FLUX OF UPWARD THROUGH GOING MUONSIn Super-Kamiokande, the total live time 2 isT = 363:3 days.

The eciency of the live time { T=T real is 81:3%.
Fig.4.2 shows the long term variation of the live time eciency of Super-Kamiokande.

T/T

real

Live Time Efficiency of Super-Kamiokande
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1

,

,

Apr 960050,

Apr 97 Jun 97
100

150

200

250

300

350

400

T

(total days)

real

450

Figure 4.2: Long term variation of live time eciency of Super-Kamiokande.

2This is current status. Observation in Super-Kamiokande is still continuing.

80CHAPTER 4.

OBSERVED FLUX OF UPWARD THROUGH GOING MUONS

81

4.2 E ective Area
The e ective area for an upward through going muon from a given direction is de ned to be theprojected area of the detector on a plane normal to the direction of the muon track. In this analysis,
upward through going muons with a track length longer than 7m are selected. so the projected areawith the track length longer than 7m is de ned as the e ective area. Fig.4.3 shows a schematic
illustration of the e ective area.
The calculation method of the e ective area is as follows. A large enough plane is taken near thedetector. Two-dimensional grid points are de ned on this plane every 10cm step, and a line is drawn
from each point vertically to the detector. If the length of the line cutting the two intersecting points
with the inner detector walls is longer than 7m, the corresponding grid point is counted as part ofthe detection area with the track length longer than 7m. Therefore, the e ective area becomes(number of grid points) 2 (100cm2 ).
The zenith angle in the direction of the area is changed degree by degree and this calculation
procedure is repeated. The calculation result of the e ective area as a function of cosine zenith
angle is shown in Fig.4.4.

Effective Area
Projection Plane

7 m

Detector

Muon Trackin Detector

Figure 4.3: Schematics of the e ective area for upward through going muons which track length
is longer than 7m.

CHAPTER 4.

OBSERVED FLUX OF UPWARD THROUGH GOING MUONS

82

Effective Area (7m track cut)2

Effective Area (m )

1400

1200

Super-Kamiokande

1000

800

600

400

Kamiokande-II,III
200

0

-1

-0.9

-0.8

-0.7

-0.6

-0.5

-0.4

-0.3

-0.2

-0.1

0

cosθ

Figure 4.4: Zenith angle dependence of e ective area of 7m track length cut for upward throughgoing muon.

4.3 Selection Eciency
The selection eciency of the zenith angle cut by muon tting of the i-th cosine bin is de ned asfollows,"i =Nileft;Ni(4.6)

where the index i means the i-th bin of the cosine zenith angle, Ni is the number of events in thei-th bin, Nileft is the number of remaining events. Nileft is estimated by using the MC data.

bin i=1 i=2 i=3 i=4 i=5 i=6 i=7 i=8 i=9cos  -0.95 -0.85 -0.75 -0.65 -0.55 -0.45 -0.35 -0.25 -0.15
Ni471 507 529 522 598 676 746 870 972
Nileft 469
507 525 517 591 675 744 859 966"i0.996 1.000 0.992 0.990 0.988 0.999 0.997 0.987 0.994

i=10

-0.05
1244
1135
0.912

Table 4.1: Selection eciency of zenith angle cut by muon tting of Kamiokande-III

CHAPTER 4.

83

OBSERVED FLUX OF UPWARD THROUGH GOING MUONS

Table 4.1 shows the selection eciency for each cosine bin in Kamiokande-III 3 . The selectioneciency at the most horizontal cosine bin ( i = 10 ) is not so good compared to other bins, so
we change the binning from 00:1 < cos  < 0 to 00:1 < cos  < 00:04. Therefore, in the caseof Kamiokande-III, the solid angle of the most horizontal bin is changed to i=10 = 2 2 0:06 (seeEq.(4.2)). The selection eciency of i = 10 bin becomes the followingcos  -0.07
Ni=10 717
Nileft=10 689"i=10 0.961
In the case of Super-Kamiokande, the event selection process has two parts and we estimate eachselection eciency independently and multiply them, "i = "i;1 2 "i;2 , where "i;1 ; "i;2 are selectioneciencies of 1st and 2nd zenith angle cut respectively. Table 4.2 show the selection eciency ofleft
1st and 2nd zenith angle cut, where Ni;1 ; Ni;2 and Ni;left
1 ; Ni;2 are the total number of events andnumber of remaining events of 1st, 2nd zenith angle cut, respectively. The selection eciency ofthe most horizontal bin is not so bad even if we take 00:1 < cos  < 0 for the most horizontal bin(i=10).

bin i=1
cos  -0.95
Ni;1 10000
Ni;2 10000
Ni;left
1 9986
left
Ni;2
9984
"i;1
0.999
"i;2
0.998
"i
0.997

i=2
-0.85
10000
10000
9979
9981
0.998
0.998
0.996

i=3
-0.75
10000
10000
9980
9984
0.998
0.998
0.996

i=4
-0.65
10000
10000
9975
9980
0.998
0.998
0.996

i=5
-0.55
10000
10000
9971
9972
0.997
0.997
0.994

i=6
-0.45
10000
10000
9974
9969
0.997
0.997
0.994

i=7
-0.35
10000
10000
9964
9967
0.996
0.997
0.993

i=8
-0.25
10000
10000
9944
9947
0.994
0.995
0.989

i=9
-0.15
10000
10000
9892
9916
0.989
0.992
0.981

i=10

-0.05
10000
10000
9767
9788
0.977
0.979
0.956

Table 4.2: Selection eciency of 1st, 2nd and combined zenith angle cut of Super-Kamiokande

4.4 Contamination by Cosmic-Ray Muons
One of the features which is worthy of special mention for upward through going muon events isthat there are no background events except cosmic-ray muons which come from an almost horizontal downward direction contaminating the neutrino induced upward going muon events. So it isimportant to estimate the contamination of cosmic-ray muon events.
The method of how to estimate the contamination of cosmic-ray muons is explained for theSuper-Kamiokande detector in the following text.

Actually, we have two independent analysis in Kamiokande-III (Ref.[32]), the selection eciency are combined both of them.
3

CHAPTER 4.

OBSERVED FLUX OF UPWARD THROUGH GOING MUONS

84

At rst we checked the angular distribution of cosmic-ray muon by both of real observed data andMC simulated data. Fig.4.5 shows azimuth(ter ) 4 and zenith(cos ) angle distribution of cosmic-ray
muon observed in the period of  35 minutes.
We can see that the distribution of the observed cosmic-ray muons agrees well with MC simulated
one which take into account the slant depth of rock around the detector. Especially, in terms of
azimuth angle direction, the geometrical thickness of the mountain is re ected in the rate of cosmicray muons arriving at the detector. On the other hand, the rate of neutrino induced muon events is

2500
2000
1500
1000
500
0

0

50

100

150

200

250

300

350

azimuth(deg.)

1600
1400
1200
1000
800
600
400
200
0

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

cos θ

Figure 4.5: Azimuth angle distribution of cosmic-ray muon direction integrated over all zenith
angle (upper gure) and zenith angle distribution of cosmic-ray muon direction integrated over
all azimuth angle (lower gure) observed in Super-Kamiokande. Cross point represents real
data and histogram represents MC simulated data which are taken account of slant depth ofrock around detector.
not a ected by the mountain shape. Fig.4.6 is -cos  distribution of upward through going muonand horizontal cosmic-ray muon events observed in this analysis. The two thick clusters in the regionof cos  > 0 and 60 <  < 240 are cosmic-ray muons from relatively thin rock depth directions.
To distinguish neutrino induced events and cosmic-ray muon events, we divide them into two
regions as follows,
region (1) :   60 ;  > 240 ,region (2) : 60 <   240 .

ter is terrestrial coordinate azimuth angle, on the other hand  is detector coodinate azimuth angle de ned
in Fig.3.1. There is correlation between  and ter , ter = 221 0 .
4

CHAPTER 4.

OBSERVED FLUX OF UPWARD THROUGH GOING MUONS

85

Fig.4.7 shows  projection of Fig.4.6. Vertical axis is logarithm scale. The border line of thedivided regions is drawn in it.
For each  region, the cos  distribution around the horizontal direction is plotted in Fig.4.8. It
is suspected that almost all of events in region (1) are neutrino induced muons because the rate is
the same order both of cos   0 and cos  > 0 muons. On the other hand, cos  > 0 muons inregion (2) show an exponentially increasing rate proportional to cos , so they are suspected to becosmic-ray muons. The slope of the zenith angle distribution of cosmic-ray muons in region (2) is
tted by an exponential function, ea+b(cos ) , and we get a = 1:06 6 0:12, b = 81:7 6 2:0 with tting
condition 2 =n = 0:55. The contaminating cosmic-ray muon events are estimated by integrating
the expoential slope in the upward going range,
Z 0
Ncont = ea+b(cos ) d(cos ) :(4.7)
01
We got the number of contaminating cosmic-ray muon events,

Ncont ' 3:5 6 0:3 :

(4.8)

cosθ

Azimuth-Zenith distribution of Through Going Muon (cosθ≤0.08)
Super-Kamiokande, Apr1996-Jun1997, 363 live days, mu track>7m

0

-0.2

-0.4

-0.6

-0.8

-1

0

50

100

150

200

250

300

350

φ

Figure 4.6: -cos  distribution of upward through going muon events and cosmic-ray muon
events (cos  < 0:08),

CHAPTER 4.

OBSERVED FLUX OF UPWARD THROUGH GOING MUONS

86

Azimuth distribution of Through Going Muon (cosθ≤0.08)
Super-Kamiokande, Apr1996-Jun1997, 363 live days, mu track>7m
10

3

10

2

(1)

(2)

(1)

10

0

50

100

150

200

250

300

350

φ

Figure 4.7:  projection of Fig.4.6. It is divided into two regions, region (1) :   60;  > 240, region (2) : 60 <   240 .

Zenith distribution of Through Going Muon (-0.1≤cosθ≤0.08)
Super-Kamiokande, Apr1996-Jun1997, 363 live days, mu track>7m
10

4

10

3

region (1)region (2)

10

2

10

1

10

-1

-0.1

-0.08

-0.06

-0.04

-0.02

0

0.02

0.04

0.06

0.08

cosθ

Figure 4.8: cos  distribution around horizontal direction in (1) { plotted by lled triangle, andregion (2) { plotted by circle.

CHAPTER 4.

OBSERVED FLUX OF UPWARD THROUGH GOING MUONS

87

4.5 Systematic Errors
One of sources of the experimental systematic error originates from the uncertainty of the parent
neutrino direction and deviation of the muon track direction. Fig.4.9 shows a schematic view ofthe source of angular uncertainties. 1 stands for the scattering angle between muon neutrino
and muon produced by charged current neutrino-nucleon interaction. 1mul corresponds to thede ected angle caused by coulomb multiple scattering of a muon traveling through the rock.

Detectorµ∆θ mulµ∆θ νµRock layerµνµFigure 4.9: De nition of angular uncertainties,which a ect the observed upward through going
muon ux as systematic errors.
1 depends on the parent neutrino energy and 1mul depends on the induced muon energy.

Unfortunately, the muon energy at the production point and the parent neutrino energy are unknown
because we can only measure the muon energy deposited in the inner detector. Therefore, the MCtechnique was introduced in order to estimate these uncertainties.
1 is approximately given by Berezenskii et al.[35]:1 2:6

s

100(GeV) ;

(4.9)

E

where E is the parent neutrino energy.
1 is also estimated by PYTHIA 5.71 [34] MC event generator. Fig.4.10 shows 1 as afunction of neutrino energy. Eq.(4.9) agrees well with PYTHIA's estimation.
Then, 1mul in the matter for muons is estimated approximately as follows:
v
u
u
t

MeV
x(cm)
x(cm)
1mul = cp13:(6MeV/c
1+
0:038 ln;) X0(cm)
X0 cm

"

#

(4.10)

CHAPTER 4.

OBSERVED FLUX OF UPWARD THROUGH GOING MUONS

88∆θνµ (deg.)Eν dependence of ∆θνµ
4

3.5

3

PYTHIA 5.71

oBerezenskii et al. : 2.6 *sqrt(100/Eν)

2.5

2

1.5

1

0.5

0

0

250

500

750

1000

1250

1500

1750

2000

Eν (GeV)Figure 4.10: Deviation of scattering angle between muon neutrinos and generated muons, 1,as a function of neutrino energy.
where p and X0 correspond to muon momentum and radiation length in the standard rock(X0 =10:5 cm), respectively. Fig.4.11 shows the angular deviation of the generated muon direction due
to Coulumb multiple scattering as Eq.(4.10).
Assuming these approximation, 1 and 1mul are simultaneously estimated by comparing thedirection of muons q
entering the detector with the input neutrino direction.
2 , is roughly estimated to be,
2 + 1mul
The deviation, 12  4:1 :12 + 1mul

q(4.11)

These uncertainties smeared the angular distribution of the observed upward through going
muons. These angular uncertainties a ect the total upward through going muon ux by 1.6 %.
The cosmic-ray muons which are contaminating from almost horizontal downward direction are
also included as systematic errors. In this analysis of Kamiokande-III the zenith angle cut is performed
as cos  < 0, but another analysis([32]) used cos  < 0:05. From that analysis the systematic error
from the cosmic-ray background is obtained approximately 010% at most horizontal bin and 0% at
the other bins. In the case of Super-Kamiokande, a zenith angle cut is performed as cos  < 0:1,
then we estimated 3.54 events as cosmic-ray background (see Section 4.4). The systematic error is
04:2% at the most horizontal bin and 0% at the other bins.
There are two more systematic errors which are concerned with the most horizontal bin especiallyin Kamiokande-III. One is the uncertainty of selection eciency which is caused by the di erenceof the estimation method between MC data and real data(non-biased single track cosmic-ray muon
events). At the most horizontal bin the uncertainty of the selection eciency is larger than the otherbins because in the real data the number of events of the most horizontal bin is much less than the:8other bins. The systematic errors from the uncertainty of the selection eciency are +2
08:3 % in themost horizontal bin and 0+01::62 % in the other bins. Another systematic error which is concerned with

CHAPTER 4.

OBSERVED FLUX OF UPWARD THROUGH GOING MUONS

89∆θmul (deg.)

Angular deviation by multiple scattering

2.5

2

1.5

1

0.5

0
1

10

10

2

10

3

Eµ (GeV)Figure 4.11: Angular deviation by Coulumb multiple scattering as a function of muon energy.
the most horizontal bin is the uncertainty of the number of events by the cos  = 00:04 cut. Thisis estimated by varying the cut value cos  = 00:02; 00:03; 00:04; 00:05; 00:06. The systematic:2error of the uncertainty of cos  = 00:04 cut is +18
04:4 %.
Conceivable systematic sources in the determination of the upward through going muon ux issummarized in Table 4.3.
Error source
Error Kam-3 (%) Error S-Kam (%)
alive time
60:4
60:4
e ective area a
60:3
60:1
a
7m
track length cut
60:5
60:1
q
a
2
2
1 + 1mul
61:6
61:6
c
cosmic-ray  contamination
010
04:2
b
selection eciency+0:6; 01:2
1
selection eciency c+2:8; 08:3
1ccos  = 00:04 cut dependence+18:2; 04:4
none
total (i=19 bin)
+1:8; 02:1
+1:7; 01:7
total (i=10 bin)
+18:5; 013:8+1:7; 04:5

Table 4.3: Conceivable systematic errors. a: i=110 bin, b: i=19 bin, c: i=10 bin .

CHAPTER 4.

OBSERVED FLUX OF UPWARD THROUGH GOING MUONS

90

4.6 Observed Flux
4.6.1 Upward Through Going Muon Flux of Kamiokande-II+III
In the case of Kamiokande-II,III, an additional condition of zenith angle cut, cos  < 00:04, is

adapted because of two reasons. One is that the selection eciency of upward through going muonsin the almost horizontal direction is not so good(see Section 4.3). Another is the estimation ofcontamination of cosmic-ray muons is not so clear. After adding the previous conditions, we get
185 events of upward through going muons in Kamiokande-II in 1124 live days, and 188 events in
Kamiokande-III in 1332 live days.
The compilation of Kamiokande-II and Kamiokande-III results in 373 events in 2456 live days.
The total ux of upward through going muons is calculated to be,:07
013 02 01 01
8total = 1.99 6 0.10(stat) +0
00:06 (syst) 210 cm s sr .

The angular distribution of ux is shown in Fig.4.12.

5

UTGM Flux (× 10

-2 -1

-1

cm s sr )6-13

Zenith Angle Dist. of Upward-Through-Going Muon Flux

4

KAM-II + KAM-III, 373 events / 2456 live days

3

2

1

0

-1

-0.9

-0.8

-0.7

-0.6

-0.5

-0.4

-0.3

-0.2

-0.1

0

COS θ

Figure 4.12: Zenith angle distribution of compilation of observed upward through going muonux in Kamiokande-II and Kamiokande-III.

CHAPTER 4.

OBSERVED FLUX OF UPWARD THROUGH GOING MUONS

91

4.6.2 Upward Through Going Muon Flux of Super-Kamiokande
In total 363 live days in Super-Kamiokande, 413 events of upward through going muon events areobserved.
Observed total ux is calculated to be,:01
013 02 01 01
8total = 1.72 6 0.08(stat) +0
00:02 (syst) 210 cm s sr .
Angular distribution of ux is shown in Fig.4.13.

6

-2 -1

-1

cm s sr )Zenith Angle Dist. of Upward-Through-Going Muon Flux

5

UTGM Flux (× 10-13SKAM, 413 events / 363 live days

4

3

2

1

0

-1

-0.9

-0.8

-0.7

-0.6

-0.5

-0.4

-0.3

-0.2

-0.1

0

COS θ

Figure 4.13: Zenith angle distribution of observed upward through going muon ux in SuperKamiokande.

Chapter 5
Expected Flux of Upward ThroughGoing Muons
In order to search for the neutrino oscillations using the upward through going muon, it is necessary
to compare observed upward through going muon ux with expected one which are calculated bynumerically. So it is important to estimate expected upward through going muon ux as carefully
as possible. The method of numerical calculation has been established in the various experiments
which treated neutrino-induced muon. In this chapter, the method to calculate expected upward
through going muon uxes and to apply it for the Kamiokande and Super-Kamiokande detector aredescribed.

5.1 Calculation of Upward Through Going Muon Flux
Neutrino-induced upward through going muons are produced in the following way. Atmospheric
neutrinos are produced by interacting of the primary components of cosmic-ray ux with the earth's
atmosphere, giving rise to pions and kaons which subsequently decay to neutrinos. Produced neutrinos go through the earth and make reaction with nuclei in the rocks near the Kamiokande/SuperKamiokande detector by charged current weak interaction  ( ) + N ! 0 (+ )+ anything .
Produced muons travel in the rocks losing their energy from as far away as several kilometers andpunch through the detector. Therefore, an analytical approach to calculate upward through going
muon ux originating from atmospheric muon netrinos requires basic knowledge as follows,
1. Atmospheric muon neutrino ux
2. Charged current cross section between neutrino and nucleon
3. Muon range in the rock
The calculation procedure was initially developed by M.Honda and M.Mori when they calculated
the upward going muon ux from super nova SN1987A[36]. Some parts were modi ed in order toadapt it to the atmospheric neutrino case.
The ux of induced muons which are generated by parent neutrinos of energy E is given byd2 d2 (E ; cos )
1 dE d
dxdy


92

(5.1)

CHAPTER 5.

EXPECTED FLUX OF UPWARD THROUGH GOING MUONS

93

where d2 =dxdy is the di erential charged current cross section as a function of the Bjorken scaling
parameters, x and y. d2  (E ; cos )=dE d is the di erential spectrum of the parent neutrinos asa function of the neutrino energy, E , and the zenith angle, .
To be detected as through going muons, the produced muon must survive after traveling a
distance X . g(X; E ; Eth ) is de ned as the probability that a muon generated with a energy ofE survives with energy larger than Eth after travering the distance of X . Then the upward going
muon ux at the detector with energy larger than Eth is given by
Z 1
Z 1
d2 d2  (E ; cos )d (Eth ; cos ) Z 1 Z 1dX
dE dy dx 1=
1 dE d1 g(X; E; Eth)
d
dxdy
0
Eth
0
0


(5.2)

The function g(X; E ; Eth ) can be written asg(X; E ; Eth ) = #(R(E ; Eth ) 0 X )(5.3)

where R(E ; Eth ) is the range of the muons in the rock, in other words, a distance that the muontravels while its energy decreases from E to Eth, and #(x) is the Heaviside step function:

1 if x  0(5.4)
0 if x < 0
The integral over X can be simply replaced by R(E ; Eth ), and d (Eth; cos )=d is reduced#(x) =



to be

Z 1
Z 1
d2 d2 (E ; cos )d (Eth ; cos ) Z 1=
dE dy dx 1
1 dE d1 R(E; Eth) (5.5)
d
dxdy
Eth
0
0

Since the neutrino ux and the muon range do not depend on the scaling parameter x, and theintegral over y is independent of the neutrino energy spectrum. equation(5.5) can be transformed

as follows,d (Eth ; cos ) Z 1 Z 1 Z 1 d2
d2 (E ; cos )=1
dx 1 R(E ; Eth )dy 1  
dE
d
dE d
Eth 0
0 dxdy
"

"

#

#

(5.6)

If a function P (E ; Eth ) is de ned as,

1 "Z 1 d2  #P (E ; Eth ) =dx 1 R(E ; Eth )dy
0 0 dxdyd (Eth ; cos )=d can be written as,
d2 (E ; cos )d (Eth ; cos ) Z 1
Zd= E P (E ; Eth) 1
th

dE d

(5.7)
dE

(5.8)

The function P (E ; Eth ) is interpreted as the total cross section convoluting the daughter muon
ranges for neutrinos of energy E , and is also interpreted as the probability that the neutrinos ofenergy E are observed as muons at the detector with energy larger than Eth . One name thisfunction the \observation probability".
It should be noted that this equation is very useful in the actual calculation of the neutrinoinduced muon ux for the following reason. One wants to calculate the upward through going muonux for many assumptions of the neutrino energy spectrum, especially, in the neutrino oscillation

CHAPTER 5.

94

EXPECTED FLUX OF UPWARD THROUGH GOING MUONS

analysis, the muon neutrino ux is a ected by the neutrino oscillation parameters; the mass di erence
squared 1m2 and mixing angle v between the two neutrinos. Therefore the calculation of the
upward through going muon ux should be repeated for a given 1m2 and v . If we calculated allthese processes by the full Monte Carlo simulation, the calculation would take a huge computer
power beyond reach. The neutrino cross sections and the muon range are already included in thefunction P (E ; Eth ), and once this function is calculated, the muon ux with an energy thresholdlarger than Eth can be obtained by the single integral. So this analytic approach is very ecientfrom another viewpoint of avoiding meaningless repetation of the calculation.
In the discussion above,  and  are not distinguished in Eq.(5.8). Indeed, we have to take this
distinction into account, because  has di erent reaction cross section from  . Hence, Eq.(5.8)
should be replace by:1d2  (E ; cos )d2  (E ; cos )d (Eth ; cos  )=P (E ; Eth ) 1+
P (E ; Eth ) 1
dE
d
dE d
dE d
Eth
Z

"

#

(5.9)

The distinction between + and 0 is not necessary, because water Cherenkov detector can notdiscriminate + from 0 .

5.2 Atmospheric Muon Neutrino Flux
Atmospheric neutrino uxes have been calculated from the incident beam of primary cosmic-rays by
Butkevich et al.[38] , Bartol1 [41] ,and Honda et al.[42]In this section, recent status of the atmospheric neutrino uxes calculated by Butkevich, Bartoland Honda are presented. The comparison among their and the uncertainty of ux calculations arealso discussed.

5.2.1 Primary Cosmic-Ray Fluxes
Primary cosmic-ray uxes are relatively well known in the energy region less than 100GeV by which
the low energy atmospheric  - uxes ( < 3 GeV ) mainly produced. Webber and Lezniak havecompiled the energy spectrum of the cosmic-rays for the hydrogen, helium, and CNO nuclei in therange 10 MeV  1,000 GeV for three levels of solar activity. A similar compilation has been madeby others for hydrogen and helium nuclei, which agrees well with what of Webber and Lezniak.
Fig.5.1 shows Observed uxes of cosmic-ray protons, helium nuclei, and CNOs from the compilation of Webber and Leziak.
The geomagnetic eld determines the minimum energy with which a cosmic-ray can arrive at
the earth. For the cosmic-ray nucleus, the minimum energy of cosmic-rays arriving at the earth isdetermined by the minimum rigidity (rigidity cuto ) rather than the minimum momentum. cosmicrays with energy greater than 100 GeV, which are responsible for > 10 GeV atmospheric neutrinouxes, are not a ected by solar activity and by geomagnetic eld. There are few measurementsof the cosmic-ray chemical composition at these energies, especially above 1 TeV. Honda et al.
compiled the available data and parametrized the observed uxes for > 100 GeV with a single powerfunction, and showed the result in Table 5.1.
1

Bartol Research Institute, University of Delaware, Newark, Delaware 19716

CHAPTER 5.

EXPECTED FLUX OF UPWARD THROUGH GOING MUONS

95

Figure 5.1: Observed uxes of cosmic-ray protons, helium nuclei, and CNOs from the compilation of Webber and Leziak. Solid lines are parametrization of Honda, et al. for solar mid,dashed lines for solar min., and dotted lines for solar max. ( gure is taken from [44])

Nucleus
H
He
CNO
Ne-S
Fe

A
(6:65 6 0:13) 2 1002
(3:28 6 0:05) 2 1003
(1:40 6 0:07) 2 1004
(3:91 6 0:03) 2 1005
(1:27 6 0:11) 2 1005

02:75 6 0:020
02:64 6 0:014
02:50 6 0:06
02:49 6 0:04
02:56 6 0:04

Table 5.1: Compiled cosmic-ray spectrum in the form:A(E/100 GeV) .(table is taken from [44])CHAPTER 5.

EXPECTED FLUX OF UPWARD THROUGH GOING MUONS

96

5.2.2 Production and Decay of Hadrons
As cosmic-rays propagate in the atmosphere, they produce 's and K's in interactions with air nuclei.
These mesons create atmospheric  's when they decay as follows:

Acr + Aair !  6; K 6 ; K 0; 1 1 1
+ !
+ + 
+ ! e+ + e + 0 !
0 + 
0 ! e0 + e + 
The calculations of the cosmic-ray protons with air nuclei consists of a number of Monte Carlocodes corresponding to di erent primary energies. Honda et al. employed the NUCRIN Monte Carlocode for the hadronic interaction of cosmic-rays for Elab < 5 GeV, and LUND code { FRITIOF
version 1.6 and JETSET version 6.3 { for 5 GeV < Elab < 500 GeV. Above 500 GeV, the originalcode developed by Kasahara et al. (COSMOS) was used. The K/ ratio is taken 7 % at 10 GeV,
11 % at 100 GeV, and 14 % at 1,000 GeV in laboratory energy.
Except for rare decays, all the decay modes of  and K mesons are considered. Charmed meson
production are ignored, since the contribution of charmed particle to atmospheric neutrinos becomessizable only for E >100 TeV.
In the two body decay of charged  's and K's, the resulting 6 is fully polarized against toward
the direction of  motion in the charged  or K rest frame.

5.2.3 Comparison of Several Calculations
We use the atmospheric neutrino uxes calculated by Butkevich [38], Bartol [41] and Honda [45] inthis analysis.
Fig.5.2 and Fig.5.3 shows energy dependence of atmospheric  + uxes and  / ux ratios
respectively from horizontal direction( cos  = 0:1 ) and vertical direction( cos  = 1:0 ). Fig.5.4
shows zenith angle dependence of atmospheric  + uxes in neutrino energy is 1 GeV and 100
GeV.

5.2.4 Uncertainties of Atmospheric Neutrino Fluxes
The systematic error in the calculation of atmospheric  uxes comes mainly from the incompletenessof the knowledge of the primary cosmic-ray uxes. Even at low energies, where the primary cosmicray uxes are rather well studied, it is dicult to determine the absolute value due to the uncertainties
in the instrumental eciency (  12 % ) and exposure factor ( 2 - 3 % ). In compilation, the error
in the t is  10 % for the nucleon ux at 100 GeV and  20 % at 100 TeV. Assuming  10 %uncertainty below 100 GeV, the systematic error in the atmospheric  - uxes is estimated to be 
10% at < 3 GeV, increasing to  20 % at 100 GeV, and remains almost constant up to 1,000 GeV.
The interaction model is another source of systematic errors. In comparison, the agreement of
the LUND model and the COSMOS code with the experimental data is < 10 %. The authors of the
NUCRIN code claim that the agreement is within 10 - 20 % . The hadronic interaction below 5 GeV
contributes at most 5 % to the production of atmospheric  - uxes at 1 GeV. The systematic error
caused by the hadronic interaction model is estimated to be  10 % above 1 GeV. One-dimensional

CHAPTER 5.

EXPECTED FLUX OF UPWARD THROUGH GOING MUONS

97–

Atmospheric νµ + νµ fluxies
0.1

Butkevich
BartolHonda95

0.08

2

-2

-1

-1

E dN/dE (GeV cm sec sr )

0.12

0.06

3

0.04
cosθ = 0.1

0.02
0
1

-1
-1

10

2

10

3

10

4

10

5

Eν (GeV)

0.12

E dN/dE (GeV cm sec sr )0.1

Butkevich
BartolHonda95

0.08

2

-2

10

0.06

3

0.04
cosθ = 1.0

0.02
0
1

10

10

2

10

3

10

4

10

5

Eν (GeV)Figure 5.2: Energy dependence of atmospheric + uxes. upper gure is from horizontal
direction(cos  = 0:1) and lower one is from vertical direction(cos  = 1:0).
approximation is justi ed only at high energies. It is expected to be accurate above 3 GeV. Since thecalculation of rigidity cuto is very simpli ed in this scheme, this may result in a systematic error in
the absolute value of the atmospheric neutrino uxes of 10 - 20 % at 100 MeV and 5 % at 1 GeV.
The statistics of the Monte Carlo calculation also causes an error in the atmospheric neutrino uxes.
The uncertainty due to the statistics is estimated to be < 5 % up to 100 - 300 GeV for  and , and up to 30 - 100 GeV for e and e , depending on the zenith angle. The errors increase to 
10 % at the highest energy for each kind of  's. Combining all the systematic and nonsystematic
errors, the total error is estimated as 15 % from 1 GeV to 100 GeV, and 20 - 25 % at the highest
energy.

CHAPTER 5.

EXPECTED FLUX OF UPWARD THROUGH GOING MUONS

98–

Atmospheric νµ/ νµ flux ratios–

νµ/ νµ ratio

3.5
3

2.5
2
1.5
1

Butkevich
BartolHonda95cosθ = 0.1

0.5
0
1

10

10

2

10

3

10

4

10

5

Eν (GeV)–νµ/ νµ ratio

3.5
3

2.5
2
1.5
1

Butkevich
BartolHonda95cosθ = 1.0

0.5
0
1

10

10

2

10

3

10

4

10

5

Eν (GeV)Figure 5.3: Energy dependence of atmospheric / ux ratios. upper gure is from horizontal
direction(cos  = 0:1) and lower one is from vertical direction(cos  = 1:0).

CHAPTER 5.

99

EXPECTED FLUX OF UPWARD THROUGH GOING MUONS–

Atmospheric νµ + νµ fluxies
0.1

Butkevich
BartolHonda95

0.08

Eν = 1.0 (GeV)

2

-2

-1

-1

E dN/dE (GeV cm sec sr )

0.12

0.06

3

0.04
0.02
0

-1

-0.9

-0.8

-0.7

-0.6

-0.5

-0.4

-0.3

-0.2

-0.1

0
cosθ

0.1

Butkevich
BartolHonda95

0.08

Eν = 100. (GeV)

2

-2

-1

-1

E dN/dE (GeV cm sec sr )

0.12

0.06

3

0.04
0.02
0

-1

-0.9

-0.8

-0.7

-0.6

-0.5

-0.4

-0.3

-0.2

-0.1

0
cosθ

Figure 5.4: zenith angle dependence of atmospheric + uxes. upper gure shows neutrinoenergy is 1 GeV and lower one is 100 GeV.

CHAPTER 5.

EXPECTED FLUX OF UPWARD THROUGH GOING MUONS

100

5.3 Neutrino Nucleon Cross Section
5.3.1 Cross Section Formula for CC Interaction
The relevant physical process of charged current neutrino interaction with rock is as follows: ( ) + N ! 0(+) + anything(5.10)

where N represents an nucleon in rock. The di erential charged current cross section for thisprocess is expressed in terms of the Bjorken scaling parameter, x = Q2 =2M (E 0 E ) and y =1 0 (E=E ).

2MW4d2  G2F ME
2 xF1 + (1 0 y 0 Mxy )F2 6 (y 0 y )xF3=ydxdy
 (MW2 + Q2 )2
2E
2
"

#

(5.11)

where the last term is positive for  and negative for . 0Q2 is the invariant momentum transfer
between the incident neutrino and outgoing muon. M and MW are the nucleon and intermediateW 6 boson masses. GF is the Fermi constant. And F1(x; Q2 ),F2(x; Q2 ) and F3(x; Q2) are thestructure functions given by quark distribution functions.
For  + n(p) ! 0 + X ,F2 = 2xF1 = 2x [fd (fu ) + fs + fb + fu (fd) + fc + ft]
F3 = 2 [fd (fu ) + fs + fb 0 fu (fd) 0 fc 0 fb ](5.12)(5.13)

And for  + p(n) ! + + X ,F2 = 2xF1 = 2x [fu (fd ) + fc + ft + fd(fu ) + fs + fb ]F3 = 2 [fu (fd ) + fc + ft 0 fd(fu ) 0 fs 0 ft]

(5.14)(5.15)

5.3.2 QCD Parton Distribution Functions
QCD parton distribution function models has been calculated by many physicists and groups. In
previous analysis for upward through going muon, Y.Oyama[50] used quark distribution function
given by Field and Feynman[51] and Eichten et al(EHLQ)[52] to integrate the di erential charged
current cross section. These are out of date and we did not use them. New parametrization ofquark distribution ready to use for high momentum transfer. They are calculated by CTEQ 2 [46],
MRS 3 [47], GRV 4 [48].
Fig.5.5 shows recent experimental results of quark distribution function measured by e-p collider
HERA, comparing with MRS(A),MRS(A0 ),GRV94 prediction. Fig.5.6 shows data by xed target
muon beam experiments of CERN SPS comparing with CTEQ3 prediction.

2
3
4

Coordinated Theoretical/Experimental Project on QCD Phenomenology and Tests of the Standard Model
Martin, Roberts, StirlingGluck, Reya, Vogt

CHAPTER 5.

EXPECTED FLUX OF UPWARD THROUGH GOING MUONS

101

Figure 5.5: ZEUS measurement of F2(x; Q2) in the small x regime compared with the predictionsof the MRS(A) partons, the MRS(A0) and MRS(G) partons and GRV(94) partons ( gure istaken from [49]).

CHAPTER 5.

EXPECTED FLUX OF UPWARD THROUGH GOING MUONS

102

Figure 5.6: Comparison of the CTEQ3 t with F2N data of BCDMS and NMC experiments.
The absolute vertical scale is not labeled since an o set factor has been applied to the various
x bins to avoid overlap. ( gure is taken from [46]).

CHAPTER 5.

EXPECTED FLUX OF UPWARD THROUGH GOING MUONS

103

5.3.3 Comparison with Experimental Data

Using fomula(5.11) and quark distribution functions GRV94(DIS), CTEQ3(M), MRS(A0), we calculate  N and  N total cross section for various neutrino energy. In this calculation, it should betaken care of the valid range of momentum transfer Q2 for each quark distribution function. Thereis relation between x; y and Q2 as follows,

xy =

Q2
2ME

(5.16)

where M is mass of relevant nucleon, and E is energy of incident neutrino. Actually two of x; y; Q2
are free parameter. Therefore, when integrating over x and y, valid range of Q2 make a e ective
contribution to this calculation. Especially for low energy neutrino(E < few GeV ), minimum
momentum transfer Q20 is sensitive to integration. we treat Q2 less than Q20 as constant value Q20 .
CTEQ3(M) and MRS(A0 ) sets Q20 = 4 GeV2 . On the other hand, GRV94(DIS) sets Q20 = 0:4
GeV2 . Each group which make quark distribution function determined the value of Q20 from minimum
Q2 of experimental data they tted.
Fig.5.7 shows calculated total cross section per neutrino energy of  N and  N reactions
together with experimental data as a function of neutrino energy.

5.4 Muon Energy Loss in RockHigh energy muons passing through matter lose energy due to electro-magnetic processes | mainly
ionization (order 2 ), bremsstrahlung (order 3 ), and direct pair production (order 4 ) | and photonuclear interactions. Fig.5.8 shows dE=dx of muons in standard rock 5 . The dash line is dE=dx
contribution of ionization considering density e ect calculated by Sternheimer et al. [53]. The
solid line is dE=dx including all contribution | ionization, bremsstrahlung, direct pair production,
photo-nuclear interactions | calculated by Lohmann et al.[54].
Classically muon energy loss is written as dE=dx = 0 0 E , where ' 2 (MeV/g/cm2 ) and' 3:9 2 1006 (cm2/g). This formula is not accurate enough to obtain a reliable result in theenergy range involved in this analysis. The detailed calculation on muon energy loss in rock is given
by Bezrukov and Bugaev[55] in a polynomial- t form. Also Lohmann et al. give numerical results.
The range of a muon is calculated as

R(E ; Eth ) =

Z

Eth
E

0dE

dE=dx

(5.17)

and results using above energy loss formulas are compared in Fig.5.9. Bezrukov and Lohmann
formulas are almost indistinguishable in the energy range of interest.
In this analysis, Lohmann's dE=dx is taken to calculate muon range.

5Z=11, A=22. Ionization potential and density corrections as for calcium carbonate.

CHAPTER 5.

104

EXPECTED FLUX OF UPWARD THROUGH GOING MUONS

2

0.9
0.8

-38

σT / Eν × 10 cm / GeV–

νµN , νµN CC Total Cross Sections

0.7
0.6
0.5
0.4
0.3
0.2

GRV94 (DIS)
CTEQ3M
,
MRS A

0.1
0

σT / Eν × 10-38 cm2 / GeV

1

10

10

1

2

10

3

10

4

Eν (GeV)

1

0.9

0.9

νN

0.8

0.8

0.7

0.7

0.6

0.6

0.5

νN

0.5
–

–

νN

νN
0.4

0.4

0.3

0.3

0.2

0.1

0

0

0.2

BEBC(1979-1982)

GGM(1979-1981)

BNL(1980-1981)

IHEP-ITEP(1979-1983)

GRV94 (DIS)

CRS(1980-1981)

SKAT(1979)

CTEQ3 M

FNAL(1977-1983)

IHEP-JINR(1995)

MRS A

5

10

15

20

0.1

,

25

30

35

40

0

50

BEBC(1979-1982)

GRV94 (DIS)

FNAL(1977-1983)

CTEQ3 MCCFRR(1984) 20-200 GeV average

MRS A

75

100

125

,

150

175

200

225

250

Eν (GeV)Figure 5.7: N and N total cross section per neutrino energy using GRV94(DIS),
CTEQ3(M), MRS(A0) quark distribution functions. Minimum momentum transfer is set to
Q20 = 4 GeV2 for CTEQ3M and MRSA0, Q20 = 0.4 GeV2 for GRV94DIS. Various experiment
data is also ploted on it. Range of neutrino energy is 0 < E < 40 GeV in left gure, and40 < E < 250 GeV in right one.

EXPECTED FLUX OF UPWARD THROUGH GOING MUONS

Muon dE/dx in standard rock

10

-1

-2

dE/dx (GeV g cm )-1

Lohmann et al.
ionization loss only
(Sternhrimer et al.)

-2

10

-3

10
1

10

10

2

10

3

10

4

Muon Energy (GeV)

Figure 5.8: dE=dx in standard rock

Muon range in standard rock2

muon range (g/cm )CHAPTER 5.

10

6

10

5

10

4

10

3

10

2

α + βE
Bezrukov and Bugaev
Lohmann et al.

10

1
1

10

10

2

10

3

10

4

10

5

10

6

Eµ (GeV)Figure 5.9: Range of muon in the standard rock.

105

CHAPTER 5.

EXPECTED FLUX OF UPWARD THROUGH GOING MUONS

106

5.5 Application to Kamiokande/Super-Kamiokande Detector
In Equation(5.9), the analytical method to calculate the expected upward thorough going muon ux
is expressed as a function of threshold muon energy deposited Eth in the detector and zenith angle.
Prior to the calculation, the threshold muon energy Eth corresponding to the threshold muontrack length 7m in this analysis should be estimated. The muon threshold energy in the Kamiokande(II,III) detector was expressed as a function of muon track length x by Oyama[50] as follows expression,
Eth (x) = 0:0025x2 + 0:2x + 0:1975 :(5.18)
From this formula, the energy threshold corresponding to 7m track length is reduced to 1.7GeV.
On the other hand, using Lohmann dE=dx, the relation between muon threshold energy andmuon track length is no more agree with Oyama fomula. Fig.5.10 shows that. The Lohmann's
relation curve is tted by 2-dimensional polynominal function, and we got following expression withtting condition 2 =n = 0:0007165=195,

Eth (x) = 0:0002464x2 + 0:2258x 0 0:02437 :(5.19)

From this formula, the energy threshold corresponding to 7m track length is reduced to 1.6GeV.
While muon track length is less than 20m, Oyama formula is almost agree with Lohmann andBezrukov. This means Oyama formula is useful for energy threshold determination for Kamiokande(II,III) detectors because maximum muon track length in Kamiokande-(II,III) detector is 19.5 m. Nevertheless in Super-Kamiokande, maximum muon track length is 49.5 m, so one must use Eq.(5.19)
for determination of muon energy threshold.
Furthermore it should be noted that the stopping muons were rejected even if their range in thedetector was longer than 7m in this analysis. Accordingly, the e ective minimum track length in thedetector may be longer than 7m and the threshold energy Eth is dependent on muon track length.
This e ect should be taken into account, when one estimates the expected upward through going
muon ux.
When the e ective expected ux is calculated, the grid point method which was used for deriving
the e ective detection area is also applicable in this case. The e ective expected ux, d8 =d , canbe calculated from d(Eth (xi); cos )=d as follows,d8 (cos )
=
d

P

i



d(Eth (xi );cos )# xidPi # xi x0( 0 x0 )( 0 )

(5.20)

where x0 is the minimum track length 7m, xi is the track length from the i-th grid point and #(x)is step function given by Eq.(5.4).
Corresponding e ective energy threshold is hEthi  2:8GeV for Kamiokande-II,III and hEth i 
6:0GeV for Super-Kamiokande. Fig.5.11 shows zenith angle dependence of e ective energy threshold.

CHAPTER 5.

EXPECTED FLUX OF UPWARD THROUGH GOING MUONS

107

Muon Energy Threshold (GeV)

Muon track length .vs. Energy threshold
30

25

20

15

10

5

0

0

10

20

30

40

50

60

70

80

90

100

Muon track length (m)Figure 5.10: The relation between muon threshold energy and muon track length. dashed line
is Oyama fomula, solid line and dash line is calculated using Lohmann and Bezrukov dE=dx
respectively.

Effective Energy Threshold (GeV)

Zenith Angle Dependence of Effective Energy Threshold
8

7

6

Super-Kamiokande<Eth> = 6.0 GeV

5

4

3

Kamiokande-II,III<Eth> = 2.8 GeV

2

1

0

-1

-0.9

-0.8

-0.7

-0.6

-0.5

-0.4

-0.3

-0.2

-0.1

0

Zenith Angle cosθ

Figure 5.11: Zenith angle dependence of e ective energy threshold. Dashed line is KamiokandeII,III and solid line is Super-Kamiokande.

CHAPTER 5.

EXPECTED FLUX OF UPWARD THROUGH GOING MUONS

108

5.6 Calculated Upward Through Going Muon Fluxes
The expected total uxes of upward through going muon is calculated using atmospheric muon neutrino ux { Butkevich, Bartol, Honda95, and parton distribution function { CTEQ3(M), MRS(A0 ),GRV94(DIS), and muon dE=dx in the rock { Lohmman.
Results of calculation are summarized in Table 5.2 for Kamiokande-(II,III), and Table 5.3 forSuper-Kamiokande. Where unit is 10013 cm02 s01sr01.
:18
+0:13
Average value of all is 2.51 +0
00:21 for Kamiokande-(II,III), and 2.05 00:17 for Super-Kamiokande.
The upper errors are estimated by the di erence of maximum expected ux and averaged one andthe lower errors are estimated by the di erence of minimum expected ux and averaged one.
The ux distribution for zenith angle is shown in Fig.5.12 for Kamiokande-(II,III), and in Fig.5.13
for Super-Kamiokande.
CTEQ3(M) MRS(A0 ) GRV94(DIS) average
Butkevich
2.62
2.69
2.53
2.61
Bartol
2.55
2.62
2.46
2.54
Honda95
2.38
2.44
2.30
2.37
average
2.52
2.58
2.43
2.51

Table 5.2: Expected total uxes of upward through going muon in Kamiokande-III calculated
by using various PDFs and  uxes.
CTEQ3(M) MRS(A0 ) GRV94(DIS) average
Butkevich
2.13
2.18
2.04
2.12
Bartol
2.10
2.16
2.02
2.09
Honda95
1.96
2.01
1.88
1.95
average
2.06
2.12
1.98
2.05

Table 5.3: Expected total uxes of upward through going muon in Super-Kamiokande calculated
by using various PDFs and  uxes.

CHAPTER 5.

EXPECTED FLUX OF UPWARD THROUGH GOING MUONS

109

6

-13

-2 -1

-1

UTG-Muon Flux ( × 10 cm s sr )Expected Upward-Through-Going Muon Fluxies (Kam-III)

5

max. expect.: Butkevich-MRSA
4,min. expect.: Honda-GRV94(DIS)

3

2

1

0

-1

-0.9

-0.8

-0.7

-0.6

-0.5

-0.4

-0.3

-0.2

-0.1

0

cos θ

Figure 5.12: Angular distribution of expected upward through going muon uxes in
Kamiokande-III. The dashed line is maximum expectation and dotted line is minimum one.

6

-13

-2 -1

-1

UTG-Muon Flux ( × 10 cm s sr )Expected Upward-Through-Going Muon Fluxies (S-Kam)

5max. expect.: Butkevich-MRSA,
4

min. expect.: Honda-GRV94(DIS)

3

2

1

0

-1

-0.9

-0.8

-0.7

-0.6

-0.5

-0.4

-0.3

-0.2

-0.1

0

cos θ

Figure 5.13: Angular distribution of expected upward through going muon uxes in SuperKamiokande. The dashed line is maximum expectation and dotted line is minimum one.

Chapter 6
Results and Discussions
The observed upward through going muon ux by Kamiokande-II+III and Super-Kamiokade arepresented in chapter 4 and various expected uxes are presented in chapter 5. In this chapter,a possible neutrino oscillation hypothesis using upward through going muons will be tested. The
di erence between the observed ux and hypothetical ones in terms of neutrino oscillations will bequantitatively evaluated by the 2 analysis.

6.1 Comparison between Expected and Observed Flux
without Oscillations
From calculation of previous chapters, we got both observed and expected upward through going
muon uxes. Results of total ux are summarized in Table 6.1, where the units are 10013 cm02 s01 sr01 .
The values of expected uxes are the averaged ones which are calculated from various models of uxes and PDFs(see Section 5.6). The errors of expected uxes correspond to the di erences of themodels. Each model of expected ux has 20% uncertainty which mainly comes from the uncertaintyof the primary muon neutrino uxes.
Fig.6.1 shows expected and observed total uxes of upward through going muons in KamiokandeII+III and Super-Kamiokande, where the inside errors of the expected uxes are the di erences ofthe models and the outside ones are (di . of models) + (20% uncertainty of each model), where
20% uncertainty comes from the uncertainty of primary muon neutrino uxes. Inside errors of
the observed uxes are statistical ones and the outside ones are (stat) + (syst) errors. For both
Kamiokande-II+III and Super-Kamiokande, the average expected uxes are larger than the observed
uxes, but taking into acount the 20% uncertainty in the expected uxes, the observed uxes are
consistent with expected ones.
Furthermore we can compare the gradient of the zenith angle distribution of upward throughgoing muon uxes. Fig.6.2 shows the zenith angle distribution of upward through going muon uxes,
where lled circles are observed data of Kamiokande-II,III(left gure) and blank circles are observed
data of Super-Kamiokande(right gure). The solid, dashed histograms are maximum,minimum
expectation, respectively.
In order to compare the gradient of the zenith angle distribution, the absolute values of expected
uxes are normalized by a scaling parameter, , and comparison is done by the 2 tting method
where 2 is de ned by the following equation,110

CHAPTER 6.

111

RESULTS AND DISCUSSIONS

8

2 =

d8
>
10 >
<
X
d obs 0
i





i
obs

i=1 >
:
>


2
d8 i >
>
=
d exp
9

>
>
;

+ 1 0


2



;

(6.1)

i

i

where index i represents the i-th cosine zenith angle bin, dd8 obs is the observed ux, dd8 exp
qi
is the expected ux, obs is (stat)2 + (syst)2 the error of the observed ux,  is the uncertaintyof the absolute value of the ux ( = 0:2). Fig.6.3 shows a comparison between the observed
upward through going muon uxes and the 2 tted expected uxes in Kamiokande-II+III(left) andSuper-Kamiokande(right). We can see that di erences in the models are absorbed by a scaling
parameter . The results of the tting conditions are 2  27 for Kamiokande-II+III and 2  13
for Super-Kamiokande.

Kamiokande-II+III

hEth i = 2.8 GeV

:18
+0:54
Expected 2:51 +0
00:21 (model) 00:46 (20%)+0Observed 1:99 6 0:10(stat) 00::07
06 (syst)

Super-Kamiokande

hEth i = 6.0 GeV

:13
+0:44
2:05 +0
00:17 (model) 00:38 (20%)+0
1:72 6 0:08(stat) 00::0102 (syst)

Table 6.1: Expected and observed total ux of upward through going muon.

UTG-Muon Flux (× 10-13cm-2s-1sr-1)

Total Flux of Upward-Through-Going-Muon
4

3.5

Expected3

Expected
2.5

Data
Data

2

1.5

1

Kamiokande-II+III<Eth> = 2.8 GeVSuper-Kamiokande<Eth> = 6.0 GeV

0.5

0Figure 6.1: Total ux of upward-through-going muons. Filled circles are the expected and theobserved ux of Kamiokande-II+III (left). Blank circles are the expected and the observed
ux of Super-Kamiokande(right). Inside errors of the expected ux are the di erences of themodels and the outside ones are (di . of models) + (20% uncertainty of each model), where
20% uncertainty comes from the uncertainty of primary muon neutrino uxes. The inside errorsof the observed uxes are statistical errors and the outside ones are statistical + systematic
errors.

CHAPTER 6.

112

RESULTS AND DISCUSSIONS

Upward-Through-Going Muon Flux (S-Kam)
UTG-Muon Flux ( × 10-13cm-2s-1sr-1 )UTG-Muon Flux ( × 10-13cm-2s-1sr-1 )Upward-Through-Going Muon Flux (Kam-II+III)
6

no oscillation
5

Butkev.-MRSA,
Honda95-GRV94
43

6

no oscillation
5

Butkev.-MRSA,
Honda95-GRV94
4

3

2

2

1

1

0

-1

-0.9

-0.8

-0.7

-0.6

-0.5

-0.4

-0.3

-0.2

-0.1

0

0

cos θ

-1

-0.9

-0.8

-0.7

-0.6

-0.5

-0.4

-0.3

-0.2

-0.1

0

cos θ

Figure 6.2: Zenith angle distributions of expected and observed upward through going muonuxes in Kamiokande-II+III(left) and Super-Kamiokande(right). Solid histogram is maximum
expectation and dashed one is minimum expectation.

Upward-Through-Going Muon Flux (S-Kam)
UTG-Muon Flux ( × 10-13cm-2s-1sr-1 )UTG-Muon Flux ( × 10-13cm-2s-1sr-1 )Upward-Through-Going Muon Flux (Kam-II+III)
6

no oscillation
5

Butkev.-MRSA, (α=0.71, χ2=26.8)
Honda95-GRV94 (α=0.82, χ =28.07)
2

4

3

6

no oscillation
5

Butkev.-MRSA, (α=0.77, χ2=13.83)
Honda95-GRV94 (α=0.89, χ =12.93)
2

4

3

2

2

1

1

0

-1

-0.9

-0.8

-0.7

-0.6

-0.5

-0.4

-0.3

-0.2

-0.1

0

cos θ

0

-1

-0.9

-0.8

-0.7

-0.6

-0.5

-0.4

-0.3

-0.2

-0.1

0

cos θ

Figure 6.3: Comparison between observed upward through going muon uxes and 2 tted
expected uxes in Kamiokande-II+III(left) and Super-Kamiokande(right).

CHAPTER 6.

113

RESULTS AND DISCUSSIONS

6.2 Method to Examine Neutrino Oscillations
For analysis assuming neutrino oscillations, a method to calculate the upward through going muonux distorted by neutrino oscillations should be established. The  !  oscillation case will be
discussed rst, and the e !  case will be presented subsequently.
The probability that a neutrino born as  remains  at distance L is given by

Pv ( !  ) = jh j (t)j ij2 = 1 0 sin2(2v0 ) 1 sin2 (withLv = 2:48km 2

L
);
Lv

(6.2)

p GeV
1m2 eV2 ;(6.3)

where v0 is mixing angle between  and  in vacuum oscillations, p is the momentum of neutrino
in unit of GeV and 1m2 in unit of eV2 . As we observe neutrinos produced on the opposite side ofthe earth, the neutrino propagation length, L(km), can be written as

L = 2Rj cos j(6.4)

where R is radius of the earth (6371 km) and  is zenith angle of the neutrino arrival direction.
Pv ( !  ) is calculated as a function of 1m2, sin2 2v0 , pv and .

If neutrinos actually oscillate, the muon neutrino ux from the opposite side of the earth will begiven by d (Eth ; cos )=dE d 1 Pv ( !  ) instead of d (Eth ; cos )=dE d .
The upward through going muon ux without neutrino oscillations is given by Eq.(5.8). If
neutrino oscillations take place, the ux will be modi ed as:

d2  (E ; cos )d (Eth ; cos )osc Z 1=
P (E ; Eth ) 1  
1 Pv ( ! ) dE
d
dE d
Eth

(6.5)

X
d8 (cos)osc X d (Eth (xi ); cos )osc=#(xi 0 x0) = #(xi 0 x0 )
d
d
i
i

(6.6)

Finally, the upward through going muon ux adapted to the Kamiokande/Super-Kamiokandeenergy threshold, d8 (cos)osc=d , is obtained as:!

where x0 is the minimum track length 7m, xi is the track length from the i-th grid point and #(x)is step function given by Eq.(5.4).
In the case of e !  oscillations, the matter e ect must be taken into account, as is discussed
in Section 1.1.2. The upward through going muon ux is given byd (Eth ; cos  )
d

=
+

Z 1 

Eth

P (E ; Eth )P (E ; Eth )d  (E ; cos  )

 2

dE d

d  (E ; cos )

 2

dE d

1 Pm ( !

d2 e (E ; cos ) ) +
Pm (edE d1!  )
e (E ; cos  ) 1 P (1 Pm ( !  ) + d dEm e !  )
d
2





dE

(6.7)

where Pm ( !  ) is de ned byPm ( !  ) = 1 0 Pm (e !  ) = 1 0 sin2 (2m) 1 sin2 (

L
):
Lm

(6.8)

CHAPTER 6.

114

RESULTS AND DISCUSSIONSIn the case of  !  oscillations, Pv ( !  ) = Pv ( !  ), while in the case of e !  ,Pm( !  ) 6= Pm( !  ) because matter e ect act on neutrinos and anti-neutrinos in adi erent way. See Eq.(1.24),(1.25) and Eq.(1.38),(1.39).
To examine agreement between the expected upward through going muon ux with neutrino
oscillations and observed one, 2 is de ned by the following equation,

2 =

8
>
X<

10

i=1 :


d8 i
d obs

>



0

i
obs

92

d8 i >
d osc =>;

where index i represents i-th cosine zenith angle bin,

+ 10


2





(6.9)

;


d8 i
d obs is observed



q

ux,




d8 i
d osc is thei is (stat)2 + (syst)2 error of observed ux,  is the
expected ux with neutrino oscillations, obs
uncertainty of the absolute value of the ux ( = 0:2). The minimum 2 on the 1m2 -sin2 2v
plane is searched for. Then, the 90% and 95% C.L.(con dence level) allowed parameter regions arede ned as follows,12  2(1m2; sin2 2v ) 0 2min
< 12th ;(6.10)
where 12th is set to 4.61(90% C.L.) or 6.17(95% C.L.) because 12 de ned by Eq.(6.10) depends
on only two free parameters, 1m2 and sin2 2v (see [57]). Table 6.2 shows values of 12th for each

C.L. and degree of freedom.

68% C.L.
90% C.L.
95% C.L.

n=1 n=2 n=3 n=4 n=5
1.00
2.71
4.00

2.30
4.61
6.17

3.53
6.25
8.02

4.72
7.78
9.70

5.89
9.24
11.3

Table 6.2: 12th for various degree of freedom(n) and con dence level(C.L.).
Moreover,the combined 2 of Kamiokande-II+III and Super-Kamiokande is calculated as follows,
8

2

=

d8 i
10 >
<
X
d


obs

i=1 :
>

+

8
>
<
>
:

d8 i
d obs


0



i
obs

0

i
obs



92

d8 i >
d osc =>;

92d8 i >
d osc =
>;

Kam-II+III

S-Kam

+ 10






2

:

(6.11)

CHAPTER 6.

6.3

115

RESULTS AND DISCUSSIONS

 $  Oscillations

In the case of  $  oscillations, it is sucient to merely consider vacuum oscillations describedin Section 1.1.1. We use the Bartol model for the primary  ux and GRV94(DIS) for PDF and
Lohmann's table for dE/dx of muons in the rocks to calculate the expected ux. From the comparison
between the observed ux and the expected ux with  $  oscillations, we got the best tted
parameters calculated by Eq.(6.9), Eq.(6.11). The results are summarized in Table 6.3.
Kam-II+III

S-Kam

Combined

15:9
9:59
26:6
1:04
1:15
1:12
2
0
sin 2v
1:00
0:78
1:00
0
2
0
3
0
2
1m 3:98 2 10 2:00 2 10 6:31 2 1003
Table 6.3: The tted parameters of observed and expected upward through going muon uxes
with  $  oscillations in Kamiokande-II+III , Super-Kamiokande and both combined.
2

min

The zenith angle distributions of observed upward through going muon uxes and expected ones
with  $  oscillations are shown in Fig.6.4. The contour lines of the parameters of  $ 
oscillations for the 90, 95 % C.L. are shown in Fig.6.5.

Upward-Throuh-Going Muon Flux (S-Kam)
UTG-Muon Flux ( × 10-13cm-2s-1sr-1 )UTG-Muon Flux ( × 10-13cm-2s-1sr-1 )Upward-Throuh-Going Muon Flux (Kam-II+III)
6

Expected : Bartol-GRV94DISνµ-ντ osci.
χ2 = 15.888
α = 1.04
sin22θ,v = 1.00
,2∆m = 3.98e-03

5

4

no osci.
χ2 = 28.85
α = 0.76

νµ-ντ osci. (combined)
3

6

Expected : Bartol-GRV94DIS4

νµ-ντ osci. (combined)

2

1

1

-1

-0.9

-0.8

-0.7

-0.6

-0.5

-0.4

-0.3

-0.2

-0.1

0

cos θ

no osci.
χ2 = 16.19
α = 0.82320νµ-ντ osci.
χ2 = 9.492
α = 1.15
sin22θ,v = 0.78
,2∆m = 2.00e-02

5

0

-1

-0.9

-0.8

-0.7

-0.6

-0.5

-0.4

-0.3

-0.2

-0.1

0

cos θ

Figure 6.4: Zenith angle distributions of upward through going muon uxes. Filled circles,
blank circles are observed uxes in Kamiokande-II+III, Super-Kamiokande, respectively. The
dashed line is the expected ux without oscillations, solid and dot-dashed ones are expected
uxes with  $  oscillations tted to the Kam-II+III/S-Kam and the combined data,
respectively.

CHAPTER 6.

116

RESULTS AND DISCUSSIONS∆m,2 (eV2)νµ ↔ ντ oscillation, 90,95% C.L. contour
1

10

Combined 95%-1

Combined 90%

10-2

Kam 90%Kam 95%

10-3

SKam 90%
SKam 95%
10

-4

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

2

1

,

sin 2θv

Figure 6.5: 90, 95% C.L. contour lines for  $  oscillations from results of KamiokandeII+III(dotted), Super-Kamiokande(dashed) and Combined(solid). The points marked by lledstars, blank stars and blank triangles represent (sin2 2v0 , 1m0 2) of minimum 2 for KamiokandeII+III, Super-Kamiokande and Combined, respectively.

CHAPTER 6.

6.4

117

RESULTS AND DISCUSSIONS

e $  Oscillations

In the case of e $  oscillations, it is taken into considerations of the MSW e ect describedin Section 1.1.2. We use the Bartol model for the primary  ux and GRV94(DIS) for PDF and
Lohmann's table for dE/dx of muons in the rock to calculate the expected ux. From the comparison
between the observed ux and the expected ux with e $  oscillations, we got the best tted
parameters calculated by Eq.(6.9), Eq.(6.11). The results are summarized in Table 6.4.
Kam-II+III

S-Kam

Combined

21:2
7:20
29:7
0:95
1:13
1:02
2
sin 2v
0:62
0:72
0:63
2
0
2
0
2
1m 1:58 2 10 7:94 2 10 3:16 2 1002
Table 6.4: The tted parameters of observed and expected upward through going muon uxes
with e $  oscillations in Kamiokande-II+III , Super-Kamiokande and both combined.
2

min

The zenith angle distributions of observed upward through going muon uxes and expected ones
with e $  oscillations are shown in Fig.6.6. The contour lines of the parameters of e $ 
oscillations for the 90, 95 % C.L. are shown in Fig.6.7.

Upward-Throuh-Going Muon Flux (S-Kam)
UTG-Muon Flux ( × 10-13cm-2s-1sr-1 )UTG-Muon Flux ( × 10-13cm-2s-1sr-1 )Upward-Throuh-Going Muon Flux (Kam-II+III)
6

Expected : Bartol-GRV94DIS
νe-νµ osci.
χ2 = 21.198
α = 0.95
sin22θv = 0.622∆m = 1.58e-02

5

4

no osci.
χ2 = 28.85
α = 0.76

νe-νµ osci. (combined)
3

6

Expected : Bartol-GRV94DIS

4

νe-νµ osci. (combined)

2

1

1

-1

-0.9

-0.8

-0.7

-0.6

-0.5

-0.4

-0.3

-0.2

-0.1

0

cos θ

no osci.
χ2 = 16.19
α = 0.82320νe-νµ osci.
χ2 = 7.197
α = 1.13
sin22θv = 0.72
2∆m = 7.94e-02

5

0

-1

-0.9

-0.8

-0.7

-0.6

-0.5

-0.4

-0.3

-0.2

-0.1

0

cos θ

Figure 6.6: Zenith angle distributions of upward through going muon uxes. Filled circles, blankcircles are observed uxes in Kamiokande-II+III, Super-Kamiokande, respectively. The dashedline is the expected ux without oscillations, solid and dot-dashed ones are the expected uxes
with e $  oscillations tted to the Kam-II+III,S-Kam and the combined data, respectively.

CHAPTER 6.

118

RESULTS AND DISCUSSIONS∆m2 (eV2)νe ↔ νµ oscillation, 90,95% C.L. contour
1

Combined 90%
Combined 95%
10-1

SKam 90%
SKam 95%

10

10

-2

-3

Kam 90%Kam 95%

10

-4

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

2

1

sin 2θv

Figure 6.7: 90, 95% C.L. contour lines for e $  oscillations from results of KamiokandeII+III(dotted) and Super-Kamiokande(dashed) and Combined (solid). The points marked
by lled stars, blank stars and blank triangles represent (sin2 2v , 1m2) of minimum 2 for
Kamiokande-II+III, Super-Kamiokande and Combined, respectively.

CHAPTER 6.

119

RESULTS AND DISCUSSIONS

6.5 Comparison with Other Experiments
Experimentally there are two type of detectors which observe upward through going muons. One is
the water Cherenkov type detector: Kamiokande, Super-Kamiokande, IMB [12] and the other is the
tracking type detector: Baksan [29], MACRO [30].
In general, the water Cherenkov type detector is able to observe not only vertically upward throughgoing muons but also horizontally upward through going muons because the detector height is higher
than the calorimeter type. On the other hand, the tracking type detector can detect more verticallyupward through going muons than the water Cherenkov type because of the large bottom area ofthe detector.
The summary of upward going muon experiments is presented in Table 6.5, where live time,observed number of muon events, observed uxes, expected uxes, e ective energy thresholds arepresented.
ExperimentBaksan a
MACRO bIMB-I,II c
Kamiokande-II+III d
Super-Kamiokande dLive Time Number of(years)Events
11.94
558
3.06
255
2.53
430
6.73
373
0.99
413

Observed Expected E ective Energy
Flux(3)Flux(3)Threshold
3.2560.19
2.85
1 GeV
3.6260.55
2.85
1 GeV
2.2260.11
2.51
1.8 GeV
1.9960.12
2.46
2.8 GeV
1.7260.08
2.02
6.0 GeV

Table 6.5: Summary of upward going muons. a:Ref.[29], b:Ref.[30], c:Ref.[12] and d: This
work. The errors include systematic errors except IMB-I,II. Expected upward through going muon ux is calculated from Bartol + GRV94(DIS) + Lohhman in Kamiokande-II+III
and Super-Kamiokande and Bartol + MRS(G) + Lohhman in other experiments. (3) unit is10013cm02s01sr01.
The zenith angle distributions of upward through going muons in Baksan, MACRO, IMB-I,II,
Kamiokande-II+III and Super-Kamiokade are shown in Fig.6.8, where the scale of Kamiokande-II+III
is normalized to the scale of Super-Kamiokande.
The 90% C.L. contour lines of CDHSW, IMB(upward stop/through muon), Kam(contained) and
Kam-II+III+S-Kam(upward through muon) for  $  and Gosgen, Bugey, Chooz, Kam(contained)
and Kam-II+III+S-Kam(upward through muon) for e $  oscillations are shown in Fig.6.9. The
90% C.L. allowed regions of this work for both type set oscillations are consistent with those of
Kamiokande(contained).
If we believe the result of Chooz [26], there are no allowed regions in 1m2 > 1003 eV2 fore $  oscillations. It is natural to think that the 1m2 is allowed in lower regions which solar
neutrino experiments indicate because the mass di erence of the leptons, m 0 me , is much lower
than that of the leptons, m 0 m .

Flux (×10

-13

-2 -1

-1

cm s sr )CHAPTER 6.

8

-2 -1

-1

cm s sr )

-13

8

7

7

(a) Baksan

6

6

5

5

4

4

3

3

2

2

1

1

0

Flux (×10120

RESULTS AND DISCUSSIONS

-1

-0.8

-0.6

-0.4

-0.2

0

0

cos θ

8

(b) MACRO

-1

-0.8

-0.6

-0.4

-0.2

0

-0.2

0

cos θ

8

7

7

(c) IMB-I+II

6

6

5

5

4

4

3

3

2

2

1

1

(d)
Kam-II+III (× 0.819)
S-Kam

0

-1

-0.8

-0.6

-0.4

-0.2

0

cos θ

0

-1

-0.8

-0.6

-0.4

cos θ

Figure 6.8: Zenith angle distribution of upward through going muons observed by (a) Baksan,(b) MACRO, (c) IMB-I+II, (d) Kamiokande-II+III and Super-Kamiokande, where the scaleof Kamiokande-II+III is normalized to the scale of Super-Kamiokande. The histogram is expected upward through going muon ux calculated from Bartol + GRV94(DIS) + Lohmann for
Kamiokande-II+III and Super-Kamiokande and Bartol + MRS(G) + Lohmann for the otherexperiments.

CHAPTER 6.

121

RESULTS AND DISCUSSIONS

νe ↔ νµ oscillation, 90% C.L. contour∆m2 (eV2)∆m,2 (eV2)νµ ↔ ντ oscillation, 90% C.L. contour
1

1

GosgenKam contained

CDHSW
10

-1

10

-1

This work

10

Bugey

-2

10

-2

Chooz
Kam containedThis work
10

-3

10

-3

IMB stop/through

10

-4

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

sin2 2θ,v

10

-4

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

sin2 2θv

Figure 6.9: 90% C.L. contour for  $  oscillations(left) and e $  oscillations(right) for
various neutrino experiments. Solid lines are the results of the combined 2 of KamiokandeII+III and Super-Kamiokande of this work and blank triangles are the minimum 2 points ofthis work.

Chapter 7
Conclusion
Muon neutrino oscillations were searched for in Kamiokande-II,III and Super-Kamiokande detectors
by using upward through going muons which are generated by atmospheric muon neutrinos.
The selected upward through going muons were restricted to those with a track length > 7 m,which corresponds to an e ective muon energy threshold, hEthi = 2:8 GeV for Kamiokande-II,III
and hEth i = 6:0 GeV for Super-Kamiokande. The mean energy of primary muon neutrinos is about
 100 GeV.
A total of 373 events were observed in Kamiokande-II+III during 2456 live days, and 413 eventsin Super-Kamiokande during 363 live days. The total ux of Kamiokande-II+III, 8k23
obs , and Supersk
Kamiokande, 8obs are
+0:07
8k23
obs = 1:99 6 0:10(stat)00:06 (syst) ;:01
8skobs = 1:72 6 0:08(stat)+0
00:02 (syst) :On the other hand, the average of expected total uxes are

+0:54
+0:18
8k23
exp = 2:5100:21 (model)00:46 (20%) ;
+0:44
:13
8skexp = 2:05+0
00:17 (model)00:38 (20%) :(units are 10013 cm01 sec01 sr01 )Taking into acount the 20% uncertainty of the expected total uxes, the observed total uxes
are consistent with expected ones.
Furthermore, zenith angle distributions are compared between observed uxes and expected ones
< 1m2 <
with neutrino oscillations.  $  and e $  oscillations in the range of 1003 12eV are searched for. In the comparison, the Bartol neutrino ux, GRV94(DIS) parton distributionfunction and Lohmann dE/dx in rock are used for the expected uxes and the absolute values arenormalized in the range of 20% uncertainty. In result, 90% C.L. allowed regions of sin2 2-1m2
plane in this work are consistent with that of Kamiokande Sub-GeV+Multi-GeV contained events
for both  $  and e $  oscillations. In case of the e $  oscillations, the recent results ofChooz excluded 1m2 > 1003 eV2 regions. It is natural to think that 1m2 is allowed in lower regions
which solar neutrino experiments indicate because the mass di erence of the leptons, m 0 me , ismuch lower than that of the leptons, m 0 m .

122

Appendix A
The Analysis by Another Energy
Threshold
The energy threshold of the upward through going muons is one of the key parameters to determine
their ux and it depends on the track length cut for muons. In this thesis, the track length cut
is performed by the condition, track length > 7m which corresponds to muon energy depositedin water, Eth >1.6 GeV. Actually, we select only \through going muons", so stopping muons are
not taken even if their track lengths are longer than 7m. Then the e ective value of the energy
threshold is higher than 1.6 GeV. It is dependent on the shape and size of detector. In case of
Kamiokande-II,III, the e ective energy threshold is hEthi = 2:8 GeV. In case of Super-Kamiokande,
it is hEth i = 6:0 GeV (see Chapter 5.5).
The cut value of 7m is decided to get better angular resolution (see Fig.3.21, Fig.3.30), but inone sense, it is a conventional value. So it is important to adopt another track length cut and checkthe consistency of both of results.
Fig.A.1 shows track length distributions of upward through going muons in Kamiokande-II+III(left)
and Super-Kamiokande(right). Ploted lled circles and blank circles are observed data of KamiokandeII+III and Super-Kamiokande, respectively, histograms are expected distributions. The gradient of
the distribution is almost at near the cut length, 7m, so it is expected that the ux value will notchange so much if the cut length is shifted slightly longer to 8m. The energy threshold of an 8m
track length cut corresponds Eth > 1:8 GeV.
The e ective areas becomes slightly smaller when the track length cut is changed from 7m to 8m.
Fig.A.2 shows the e ective areas of 7m track length cut(solid lines) and 8m track length cut(dashedlines). The e ective energy threshold becomes slightly larger when the track length cut is changed
from 7m to 8m. Fig.A.3 shows The e ective energy threshold of 7m track length cut(solid lines)
and 8m track length cut(dashed lines).
Table A.1 shows the result of both of 7m and 8m track length cut. The model of expected
uxes are Bartol for  ux, GRV94(DIS) for PDF and Lohmman for dE=dx. The errors of expected
uxes are 20% uncertainty of  ux. The errors of observed uxes include both of statistical andsystematic errors.

123

124

APPENDIX A. THE ANALYSIS BY ANOTHER ENERGY THRESHOLD

Number of Events

Number of Events

100

90

80

50

45

40

70

35

60

30
7m

50

7m

8m

40

20

30

15

20

10

10

5

0

6

8

8m

25

10

12

14

16

18

0

20

5

10

15

20

25

30

Track Length (m)

35

40

45

50

Track Length (m)Figure A.1: Track length distributions of upward through going muons in KamiokandeII+III(left, lled circles) and Super-Kamiokande(right, blank circles). Histgrams are expecteddistributions, where absolute scales are normalized to data.

Effective Area (m2)

Effective Area (7m and 8m track cut)
1400

1200
Skam (7m)
Skam (8m)

1000

800

600

400

Kam23 (7m)200

Kam23 (8m)

0

-1

-0.9

-0.8

-0.7

-0.6

-0.5

-0.4

-0.3

-0.2

-0.1

0

cosθ

Figure A.2: The e ective areas of the 7m track length cut(solid lines) and the 8m track lengthcut(dashed lines).

APPENDIX A. THE ANALYSIS BY ANOTHER ENERGY THRESHOLD

125

Effective Energy Threshold (GeV)

Effective Energy Threshold (7m and 8m track cut)
8

7
SKam (8m, <Eth> = 6.1 GeV)

6

SKam (7m, <Eth> = 6.0 GeV)

5

4
Kam23 (8m, <Eth> = 2.9 GeV)

3

Kam23 (7m, <Eth> = 2.8 GeV)

2

1

0

-1

-0.9

-0.8

-0.7

-0.6

-0.5

-0.4

-0.3

-0.2

-0.1

0

cosθ

Figure A.3: The e ective energy threshold of the 7m track length cut(solid lines) and the 8m
track length cut(dashed lines).

track length cut
e ective energy
thresholdnumber of events

7m
8m
2.8 GeV (Kam23)
2.9 GeV (Kam23)
6.0 GeV (Skam)
6.1 GeV (Skam)
373 (Kam23)
346 (Kam23)
413 (Skam)
404 (Skam)
expected total ux
2.4660.49 (Kam23) 2.3960.48 (Kam23)(10013 cm02 sec01 sr01 ) 2.0260.40 (Skam) 1.9860.40 (Skam)observed total ux
1.9960.12 (Kam23) 1.8460.12 (Kam23)(10013 cm02 sec01 sr01 ) 1.7260.08 (Skam) 1.7160.08 (Skam)

Table A.1: The result of both the 7m and 8m track length cuts. The model of expected uxes
are Bartol for  ux, GRV94(DIS) for PDF and Lohmman for dE=dx. The errors of expected
uxes are 20% uncertainty of  ux. The errors of observed uxes include both of statisticaland systematic errors.

126

APPENDIX A. THE ANALYSIS BY ANOTHER ENERGY THRESHOLD

6

UTG-Muon Flux ( × 10-13cm-2s-1sr-1 )UTG-Muon Flux ( × 10-13cm-2s-1sr-1 )Fig.A.4 shows zenith angle distributions of the upward through going muon uxes in KamiokandeII+III(left) and Super-Kamiokande(right). Observed data for the 7m cut are ploted by circles and forthe 8m cut are ploted by squares. Solid histograms are expected uxes for the 7m cut and dashedones are for the 8m cut.
Fig.A.5 shows 90% CL contour of  $  oscillations(left) and e $  oscillations(right) fromthe results of Kamiokande-II+III(dotted) and Super-Kamiokande(dashed) and Combined (solid).
Both the 7m and 8m track length cuts are shown.
It is found that results of 8m track length cut are consistent with results of 7m track length cut.

Expected :5

Kam-II+III (track length > 7m)
Kam-II+III (track length > 8m)
Data :

4

Kam-II+III (track length > 7m)
Kam-II+III (track length > 8m)

3

6

S-Kam (track length > 7m)
S-Kam (track length > 8m)

3

1

1

-0.9

-0.8

-0.7

-0.6

-0.5

-0.4

-0.3

-0.2

-0.1

0

cos θ

Data :

4

2-1

S-Kam (track length > 7m)
S-Kam (track length > 8m)

20

Expected :

5

0

-1

-0.9

-0.8

-0.7

-0.6

-0.5

-0.4

-0.3

-0.2

-0.1

0

cos θ

Figure A.4: Zenith angle distributions of the upward through going muon ux in KamiokandeII+III(left) and Super-Kamiokande(right). Observed data for the 7m cut are ploted by circles
and for the 8m cut are ploted by squares. Solid histograms are the expected uxes for the 7m
cut and the dashed ones are for the 8m cut.

127

APPENDIX A. THE ANALYSIS BY ANOTHER ENERGY THRESHOLDνe ↔ νµ oscillation, 90% C.L. contour
2∆m (eV )

1

∆m

2

,2

2

(eV )

νµ ↔ ντ oscillation, 90% C.L. contour
1

10

-1

Combined(7m)10-1

Combined(7m)Combined(8m)10Combined(8m)

-2

10

-2

Kam(7m)Kam(8m)SKam(7m) SKam(8m)
10

-3

10

Kam(7m)-3

Kam(8m)SKam(7m)
SKam(8m)

10

-4

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

2

1

,v

sin 2θ

10

-4

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

2

1

sin 2θv

Figure A.5: 90% CL contour of  $  oscillations(left) and e $  oscillations(right) fromresults of Kamiokande-II+III(dotted) and Super-Kamiokande(dashed) and Combined (solid).
Both the 7m and 8m track length cuts are shown.

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