Study of upward-going muons in Super-Kamiokande

Choji Saji
Doctral Program in Fundamental Science and Energy Technology
Graduate School of Science and Technology
Niigata University

1

Acknowledgement
First of all, I would like to express my great appreciation to my adviser, Prof. Kazumasa Miyano. He introduced me to experimental physics and encouragement, Without
his excellent support and encouragement, this thesis would never have been completed.
I would like to thank Prof. N.Tamura and Prof. H.Miyata. They always encouraged
and supported me.
I acknowledge Prof. Y.Totsuka, spokesman of the Super-Kamiokande experiments.
He gave me the opportunity to participate in such an enthusiastic time of the study ofthe neutrino oscillation.
I am deeply indebted to members of upward-going muon analysis group. Especially
I would like to thank Dr. M.Takita. His many instructive suggestion and encouragement were indispensable to progress this study. I would like to thank Dr. A.Okada,Dr. M.Takahata, Dr. M.Yoshida and Mr. K.Nitta. And also wish to thank the members in USA, Prof. R.J.Wilkes, Prof. J.G.Learned, Dr. A.Habig, Dr. S.Matsuno, Dr.
A.L.Stachyra and Mr. D.Shantanu.
I would like to thank all the Super-Kamiokande collaborators for help with the experiments.
At last, I would like to thank all graduate students in High Energy Physics group at
Niigata university.

Abstract
Upward-going muons produced in the surrounding rock by high energy atmospheric neutrinos have been measured for using the large underground water Cherenkov detector,
Super-Kamiokande. During about 1250days data was studied. The experimental upwardgoing muon flux was obtained. The expected upward-going muon flux was estimated byanalytical calculation and Monte Carlo method, respectively.
The νµ → ντ vacuum oscillation is considered as the most probable process. The ν µ → ντoscillation hypothesis with sin 2 2θ ≥ 0.65, 1.9 × 10−3 ≤ ∆m2 ≤ 7.2 × 10−3 is consistent
with the observed zenith angle shape at 90% confidence level.
The observed atmospheric νµ /ν¯µ ratio was measured and the expected ratio was esti+0.39
mated by analytical calculation. The ratio of observed to expected was 0.29 −0.29(stat.) ±0.04(syst.) obtained.

Contents
1 Introduction
1.1 Neutrino oscillation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4
4

2 Atmospheric neutrino
2.1 Atmospheric neutrino production . . . . . . . . . . . . . . . . . . . . . . .
2.2 Atmospheric neutrino in Super-Kamiokande . . . . . . . . . . . . . . . . .
2.3 Upward-going muon in Super-Kamiokande . . . . . . . . . . . . . . . . . .

6
6
6
7

3 Super-Kamiokande detector
3.1 Water Cherenkov image counter . . . . . . . . . . . . . . . . . . . . . . . .
3.2 Detector components . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.3 PMTs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.4 Data acquisition system . . . . . . . . . . . . . . . . . . . . . . . . . . . .

11
11
14
15
18

4 Calibration
4.1 Calibration of PMTs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2 Calibration for low energy events . . . . . . . . . . . . . . . . . . . . . . .
4.2.1 Calibration using neutron capture with Ni . . . . . . . . . . . . . .
4.2.2 Calibration using electron linear accelerator . . . . . . . . . . . . .
4.2.3 µ decay-e event method . . . . . . . . . . . . . . . . . . . . . . . .
4.3 Timing calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.4 Water transparency measurement . . . . . . . . . . . . . . . . . . . . . . .
4.4.1 Water transparency by cosmic-ray µ events . . . . . . . . . . . . . .
4.4.2 Water transparency by laser . . . . . . . . . . . . . . . . . . . . . .

22
22
22
22
24
25
25
29
29
31

5 Data reduction
5.1 Dataset . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.2 Automatic reduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.2.1 Upward stopping muons . . . . . . . . . . . . . . . . . . . . . . . .
5.2.2 Upward through-going muons . . . . . . . . . . . . . . . . . . . . .
5.3 Eye scan with manual fitting . . . . . . . . . . . . . . . . . . . . . . . . . .
5.3.1 Track length cut . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.4 Upward-going muon events . . . . . . . . . . . . . . . . . . . . . . . . . . .

33
33
33
33
38
40
41
44

1

CONTENTS
5.5

2

Backgrounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

6 Observed flux
6.1 Live time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.2 Effective area . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.3 Selection efficiency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.4 Systematic errors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.5 Observed flux of upward going muons . . . . . . . . . . . . . . . . . . . . .

47
47
47
48
50
50

7 Expected flux
52
7.1 Analytical method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
7.2 Atmospheric neutrino flux . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
7.3 Neutrino-nucleon interaction . . . . . . . . . . . . . . . . . . . . . . . . . . 53
7.3.1 Deep inelastic scattering . . . . . . . . . . . . . . . . . . . . . . . . 54
7.3.2 Quasi-elastic scattering and single pion production . . . . . . . . . 56
7.4 Energy loss for muons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
7.4.1 Muon energy loss in rock . . . . . . . . . . . . . . . . . . . . . . . . 57
7.4.2 Muon energy loss in Water . . . . . . . . . . . . . . . . . . . . . . . 57
7.5 Theoretical uncertainties . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
7.5.1 Atmospheric neutrino flux . . . . . . . . . . . . . . . . . . . . . . . 59
7.5.2 The ratio of upward stopping muon flux to upward through-going
muon flux . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
7.6 Expected flux of upward going muons . . . . . . . . . . . . . . . . . . . . . 61
8 Neutrino Oscillation
8.1 Method to examine neutrino oscillations . . . . . . . . . . . . . . . . . . .
8.2 Effect on oscillation analysis from theoretical uncertainties . . . . . . . . .
8.3 νµ → ντ oscillation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

62
62
63
64

9 Approach with Monte Carlo simulation
68
9.1 Monte Carlo method for upward-going muons . . . . . . . . . . . . . . . . 68
9.2 Comparison between MC and analytical calculation . . . . . . . . . . . . . 69
10 Atmospheric ν/ν̄
10.1 Method to estimate atmospheric ν/ν̄ . . . . . . . . . . . . . . . . . . . . .
10.2 Charge separation of muons . . . . . . . . . . . . . . . . . . . . . . . . . .
10.3 Event selection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
10.3.1 Parent upward stopping muons . . . . . . . . . . . . . . . . . . . .
10.3.2 Electron from decayed muon . . . . . . . . . . . . . . . . . . . . . .
10.4 Charge ratio of upward stopping muon . . . . . . . . . . . . . . . . . . . .
10.5 Expected charge ratio . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
10.6 Systematic uncertainties . . . . . . . . . . . . . . . . . . . . . . . . . . . .

72
72
72
74
74
74
81
82
82

CONTENTS

3

10.6.1 The systematic error for absolute energy determination . . . . . . . 82
10.7 Obtained atmospheric ν/ν̄ . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
11 Result and discussions
11.1 Observed upward-going muon flux . . . . . . . . . . . . . . . . . . . . . . .
11.2 Expected upward-going muon flux . . . . . . . . . . . . . . . . . . . . . . .
11.3 Oscillation examination . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
11.4 Monte Carlo approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
11.5 Atmospheric νµ /ν¯µ ratio . . . . . . . . . . . . . . . . . . . . . . . . . . . .

87
87
87
87
88
88

12 Conclusion

89

Bibliography

90

List of Figures
2.1
2.2
2.3
2.4

Schematic view of upward going muon event . . . . . . . . . . . . . . . . . 8
The typical event for upward through-going muon event . . . . . . . . . . . 9
The typical event for upward stopping muon event . . . . . . . . . . . . . . 9
Energy distribution of parent neutrino for upward muons . . . . . . . . . . 10

3.1 Schematic view of Super-Kamiokande . . . . . . . . . . . . . . . . . . . . .
3.2 Cone of Cherenkov light . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.3 The structure of the frame for PMTs . . . . . . . . . . . . . . . . . . . . .
3.4 20inch PMT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.5 The time resolution for 50cm-diameter PMT . . . . . . . . . . . . . . . . .
3.6 Quantum efficiency and spectrum of Cherenkov light . . . . . . . . . . . .
3.7 A typical 1p.e distribution . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.8 The coil to compensate for the magnetic field . . . . . . . . . . . . . . . .
3.9 Block diagram of the ATM module . . . . . . . . . . . . . . . . . . . . . .
3.10 Data acquisition system . . . . . . . . . . . . . . . . . . . . . . . . . . . .

13
14
15
16
17
17
18
19
20
21

4.1
4.2

23

Schematic view of relative energy calibration system . . . . . . . . . . . . .
The gain variation for all PMTs after the high-voltage adjustment (x-axis
shows corrected charge(pC · m2)) . . . . . . . . . . . . . . . . . . . . . . .
4.3 The schematic view of Ni calibration system . . . . . . . . . . . . . . . . .
4.4 Configuration of LINAC calibration system . . . . . . . . . . . . . . . . . .
4.5 Schematic view of timing calibration system . . . . . . . . . . . . . . . . .
4.6 T-Q map of a typical PMT . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.7 Timing resolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.8 Schematic view of cosmic-ray muon method . . . . . . . . . . . . . . . . .
4.9 The light attenuation length(6th July 1996) . . . . . . . . . . . . . . . . .
4.10 Time variation of the water attenuation length . . . . . . . . . . . . . . . .
4.11 Set up of the laser system . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.12 The attenuation length . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.1
5.2
5.3

23
24
25
26
27
28
29
30
30
31
32

The flow chart of event selection. . . . . . . . . . . . . . . . . . . . . . . . 34
Total photo-electron and track length for stopping muons . . . . . . . . . . 36
Schematic view of T OF . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

4

LIST OF FIGURES

5

5.4
5.5
5.6
5.7
5.8
5.9

Total Q cut and maximum Q cut . . . . . . . . . . . . . . . . . . . . . . .
Angular resolution for MC stopping muons . . . . . . . . . . . . . . . . . .
Angular difference depend on the scanner . . . . . . . . . . . . . . . . . . .
Zenith angle and azimuth angle for stopping muons . . . . . . . . . . . . .
Zenith angle distribution of stopping muons . . . . . . . . . . . . . . . . .
The azimuth angle distribution for stopping muons . . . . . . . . . . . . .

40
42
43
44
45
46

6.1
6.2

Schematic view of effective area. . . . . . . . . . . . . . . . . . . . . . . . . 48
Zenith angle dependence of the effective area. . . . . . . . . . . . . . . . . 49

7.1
7.2

The energy spectrum of atmospheric neutrino(ν µ + ν̄µ ) flux . . . . . . . . . 54
The zenith angle distribution of atmospheric neutrino flux in various neutrino energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
νµ N and ν¯µ N total cross section per neutrino energy . . . . . . . . . . . . 56
Total neutrino-nucleon CC cross section . . . . . . . . . . . . . . . . . . . 57
dE/dx in standard rock . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
Range of muon in standard rock . . . . . . . . . . . . . . . . . . . . . . . . 58
Range of muon in water. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
The zenith angle distribution of expected upward-going muon flux . . . . . 60
The zenith angle bin-by-bin error estimated by model difference . . . . . . 60

7.3
7.4
7.5
7.6
7.7
7.8
7.9
8.1

Oscillation result for through-going muon flux. The left figure shows thezenith angle distribution and the right shows the confidence interval. . . . 65
8.2 Oscillation result for the stop/through ratio. The left figure shows thezenith angle distribution of the ratio and the right shows the confidence
interval of the ratio. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
8.3 Zenith angle distribution of upward muon flux. The left figure shows stopping muon flux and the right shows through-going muon one. . . . . . . . . 66
8.4 Confidence interval for combined stopping muon and through-going muon. 67
9.1
9.2
9.3

Upward-stopping muon flux . . . . . . . . . . . . . . . . . . . . . . . . . . 70
Upward through-going muon flux . . . . . . . . . . . . . . . . . . . . . . . 70
The ratio of upward stopping muon flux to upward through-going muon flux 71

10.1 The vertex distribution for low energy events . . . . . . . . . . . . . . . . . 75
10.2 Typical µ decay electron event . . . . . . . . . . . . . . . . . . . . . . . . . 76
10.3 Timing distribution of electron event . . . . . . . . . . . . . . . . . . . . . 77
10.4 The resolution of vertex reconstruction for electron . . . . . . . . . . . . . 79
10.5 The Michel spectrum, solid line shows MC and cross shows data . . . . . . 80
10.6 Schematic view of µ-e decay events for gate pulse . . . . . . . . . . . . . . 81
10.7 Timing distribution for in-gate event . . . . . . . . . . . . . . . . . . . . . 82
10.8 Timing distribution for sub-gate event . . . . . . . . . . . . . . . . . . . . 83
10.9 The Michelle spectrum of µ + decay e+ generated by Monte Carlo simulation 84

LIST OF FIGURES

6

10.10The Ne-tag/Ntotalµ versus Nµ+ /Nµ− . . . . . . . . . . . . . . . . . . . . . 85
10.11The energy distribution of µ decay electron . . . . . . . . . . . . . . . . . . 86
10.12Neutrino-nucleon charged current cross section . . . . . . . . . . . . . . . . 86

List of Tables
3.1
3.2

Specification and performance of the Super-Kamiokande detector . . . . . 12
Critical energy for Cherenkov light emission in water . . . . . . . . . . . . 14

6.1
6.2

The summary for selection efficiency. . . . . . . . . . . . . . . . . . . . . . 49
The summary of systematic error for upward going-muons . . . . . . . . . 51

8.1
8.2

Uncertainties for oscillation analysis. . . . . . . . . . . . . . . . . . . . . . 63
The systematic error of normalization factor for oscillation analysis. . . . . 64

10.1 Vertex reconstructed position difference and resolution by Ni calibration. 78

11.1 The summary of oscillation analysis . . . . . . . . . . . . . . . . . . . . . . 88

7

Chapter 1
Introduction
1.1

Neutrino oscillation

The neutrino is one of leptons that are classified into three types, electron, muon and tau
type. Each type has its own anti-particle. Neutrinos have no charge and are thought tohave zero mass in the standard model. However, there are essentially unsolved problems
in neutrino masses. If neutrinos are indeed massive, then neutrino oscillation may be observed. The oscillation probability, for which one neutrino changes its flavor into anotherdepends on the mass-squared difference of the two neutrino mass eigenstates(∆m 2) andthe mixing angle parameter(θ).
We give a brief review of the neutrino oscillations. The observed neutrinos, |ν µ i and|ντ i, are flavor eigenstates and the massive neutrinos, |ν 1 i and |ν2 i, are the mass eigenstates, respectively. The relation between the flavor eigenstates and the mass eigenstateswith the transformation matrix, U, is written by,











 ν1  νµ 
=
=U


ντ

ν2

cosθ





sinθ   ν1 −sinθ cosθ

ν2

(1.1)



where θ is the mixing angle in vacuum. Time evolution of a state ν(t) is written as,i dtd νj (t) = Ej νj (t) (j = 1, 2),(1.2)

where Ej is the energy of νj , j is the index of mass eigenstates. The neutrino wave
function is written as, ν1 (t) ν2 (t)

 exp(−iE1 t)

=

0

0
exp(−iE2 t)  ν1 (0) 

ν2 (0)





,

(1.3)

where ν1 (0) and ν2 (0) are the mass eigenstates at T = 0 and ν1 (t) and ν2(t) are at T = t.
Eq. 1.3 is rewritten using Eq. 1.1 as, exp(−iE1 t) νµ (t) 
=Uντ (t)00

exp(−iE2 t)8 †  ν1 (0) ,U ν2 (0)

(1.4)

1.1

Neutrino oscillation

9

The probability that a neutrino νµ remains to be νµ is written as,
P (νµ → νµ ) =
=



1 0





 exp(−iE1 t)

U

0

0
exp(−iE2t)exp(−iE1 t)cos2θ + exp(−iE 2t)sin2θ .





2

 † 1 
U 


0

(1.5)
(1.6)

If neutrino mass is sufficiently smaller than their momentum, the following approximation
can be applied,
m2j
,
(1.7)
Ej ≈ p +2pwhere mj is the mass of neutrino νj and p is the momentum of νµ . Finally, the survival
probability can be written as:2

P (νµ → νµ ) = 1 − sin 2θsin

2

L
π
Lν





(1.8)

where ∆m2 = m22 − m21 (eV2 ), and L(km), Lν are the path length and oscillation lengthof neutrino, respectively. L ν is defined as,Lν = 2.48(km) ×
where pν is neutrino momentum in GeV.

pν
∆m2

(1.9)

Chapter 2
Atmospheric neutrino
In this chapter, we report atmospheric neutrino and one detected by Super-Kamiokande
detector.

2.1

Atmospheric neutrino production

The cosmic-ray incident on the atmosphere consists of protons and nuclei. High energy
cosmic-ray interacts with the atmosphere whose major components are oxygen and nitrogen, and some kinds of mesons are generated at the altitude of about 10km hight in
the atmosphere. These mesons produce eventually some neutrinos which are called as theatmospheric neutrinos. The mesons decay as follows,
π±
µ±
K±

→

µ±

νµ (ν̄µ )e±

νe (ν̄e )→µ±νµ (ν̄µ )π0µ±

νµ (ν̄µ )

→

π0

e±

νe (ν̄e )π±µ±νµ (ν̄µ )π±e±νe (ν̄e )

→

→

KL0

→

→

ν̄µ (νµ )Because of rapidly decay of kaons, the kaon decay is important in the higher energy
region.
The uncertainty in the calculation of atmospheric neutrino flux is estimated to beabout 20%, due to the uncertainties in the flux of the primary cosmic-ray and hadronicinteractions.

2.2

Atmospheric neutrino in Super-Kamiokande

The atmospheric neutrino generated in the atmosphere propagates to the detector in
the earth and interacts with water in the detector or rock out of the detector, then10

2.3 Upward-going muon in Super-Kamiokande

11electrons, muons and pions are produced by charged current interaction or neutral current
interaction. The contained event is detected as Cherenkov light by these muons and pions.
Atmospheric neutrinos detected by Super-Kamiokande can be categorized as follows,• Contained event
A contained event is categorized by event topology and also reflects parent neutrinoenergy as follows,
1. Fully contained event
The interaction point of the neutrino is inside the detector, and the generated
particle stops within the fiducial volume of the detector. The neutrino grouped
into this category has about 1 GeV energy.
2. Partially contained event
The interaction point of the neutrino is inside the detector, and generated
particle goes out of the detector. The neutrino grouped into this category has
about 5∼ 10GeV energy.
• Upward-going muon event
Atmospheric neutrinos generated on the opposite side of the earth can propagate
through it and interact with rock surrounding the detector. If the interaction region is within about 5km, the muons generated by charged current interaction may
penetrate the rock to the detector. Muons induced by such neutrinos are observed
having upward-going direction(Fig. 2.1).
The upward-going event is categorized by detecting topology as follows,
1. Upward through-going muon event
The muon events which come from underneath the detector generated in therock. The muon goes through the whole detector. Fig.2.2 is a typical upward
through-going muon event.
2. Upward stopping muon event
The muon events which come from underneath the detector generated in therock. The muon crosses the detector partially and stops in the detector. Fig.2.3
is a typical upward stopping muon event.

2.3

Upward-going muon in Super-Kamiokande

Almost all downward-going muons are cosmic-ray muons which are generated directlyin the atmosphere. Downward-going muons penetrate into the detector but cannot be
distinguished muons induced from neutrinos. Therefore, we select only muons going
upward. Electron neutrinos also interact with rock and upward going electrons can be2.3 Upward-going muon in Super-Kamiokande

12generated. However, they produce showers in the rock and the track length is too short
to reach at the detector.
The average energy of the parent neutrino for these events is about 10GeV for upward
stopping muons and 100GeV for upward through-going muons. The energy distributions
for stopping muons, through-going muons and sum of them are shown in Fig.2.4. According to the figure, the energy region of parent neutrinos for stopping muons is about
1.6GeV to 100GeV. For through-going muons, it is a few GeV to over 10TeV.

Super − KamiokandeEarthupward−going µ

decay−eνµ , νµ_p
He
Figure 2.1: Schematic view of upward going muon event2.3 Upward-going muon in Super-Kamiokande

13

NUM
7
RUN
2930
EVENT
316577
DATE 96-Oct- 8
TIME
9:31:37
TOT PE: 75737.4
MAX PE:
256.5
NMHIT : 8385
ANT-PE:
375.8
ANT-MX:
40.9
NMHITA:
189

RunMODE:NORMALTRG ID :00001011
T diff.:0.219E+05us: 21.9
ms
FSCC:
20000
TDC0: 8899.5
Q thr. :0.0
BAD ch.: masked
SUB EV : 0/ 0

Figure 2.2: The typical event for upward through-going muon eventthe larger display is inner-detector and the smaller display is outer-detector

NUM
30
RUN
3241
EVENT
323918DATE 96-Dec- 7
TIME
14:40:31
TOT PE: 47984.5
MAX PE:
125.1
NMHIT : 7468
ANT-PE:
153.8
ANT-MX:
26.4
NMHITA:
95

RunMODE:NORMALTRG ID :00001011
T diff.:0.583E+05us: 58.3
ms
FSCC:
20000
TDC0: 8905.7
Q thr. :0.0
BAD ch.: masked
SUB EV : 0/ 1

Figure 2.3: The typical event for upward stopping muon event
If neutrinos actually oscillate, the observed upward-going muon flux will deviate fromexpectation and ∆m2 and mixing angle will not be zero. Since the survival probability
depends on neutrino energy Eν and path length L, because the flux of muon neutrinos inthe energy region from a few GeV up to over 10TeV can be searched, upward-going muon

dFlux/dlogE ( ×10-13
cm-2s-1sr-1)

2.3 Upward-going muon in Super-Kamiokande

14

Parent Neutrino Energy Distribution (muon track length > 7m)

0.6

Upward Stopping Muons
Upward Through Muons
All Upward Muons
0.5

0.4

0.3

0.2

0.1

0
1

10

2
10

3
10

4
10
Energy (GeV)

Figure 2.4: Energy distribution of parent neutrino for upward muons. All upward muonsmean sum of stopping muon and through-going muon.

is suitable for oscillation study. As a consequence of oscillations, the flux of upward-going
muons should be affected both in the absolute number of events and in the shape of thezenith angle distribution, with relatively fewer observed events near the vertical than near
the horizontal. This is due to the longer path length of neutrinos from production point
to their observation point.

Chapter 3
Super-Kamiokande detector
The major themes of this detector are experiments on nucleon decay and on neutrino
detection. Super-Kamiokande is located a depth of 2700meters water equivalent in the
Kamioka Observatory in Japan, near the Kamiokande. The schematic view of SuperKamiokande is shown in Fig.3.1. Super-Kamiokande is a cylindrical 50 kton water
Cherenkov image detector. It started running in April 1996.
Since the detector is located deep under ground, the cosmic-ray muon background isreduced. The intensity of cosmic-ray muons at the surface of the earth is about 10 9 m−2 y−1 .
In the Kamioka mine, the intensity is reduced to 10 −5 . The lowest energy of cosmic-ray
muons detected by Super-Kamiokande is about 1.3TeV. The atmospheric muon rate at
Super-Kamiokande is about 2.2Hz. The properties of the Super-Kamiokande are given
briefly shown in Table 3.1. Especially important characteristics are as follows,
1. Due to its large volume, it is suitable for nucleon decay experiment and neutrino
detection experiment.
2. A charged particle’s vertex position, direction, and momentum can be reconstructed
3. A charged particles e, µ, π can be identified.
4. It was good energy resolution with 40% photosensitive coverage of large diameter
PMTs.
5. Since the optical separation between the inner and outer detectors is almost complete, the detector can separate between neutral particles coming from outside of
the tank and charged particles generated by interaction inside the tank.

3.1

Water Cherenkov image counter

The Super-Kamiokande is a ring-imaging water Cherenkov detector. The detector detects
Cherenkov photons emitted in water by charged particles. When the velocity of the
15

3.1

Water Cherenkov image counter

Super-Kamiokande
Total size
Total volume
Number of PMTs

remark42mh × 39mφ
50,000t

11,146( 50cmφ)

inner-detector

1,885(20cmφ)outer-detector

Photosensitive coverage

40%

Timing resolution for PMT

3nsec√
2.6% / E

at 1 photo-electron

2.5%√
16% / Eµ ( < 1GeV)e ( < 20MeV)

50cm

e (10MeV)∼ 10cmp → e+ π 0

Energy resolution

Vertex resolution
Angular resolution
Trigger threshold energy
Analytical threshold energy
e/µ separatione(E(GeV))28◦

e (10MeV)∼ 1◦

through going µ

4 ∼ 5MeV
5MeV
99%

0.03 < pe < 1.33GeV/c
0.2 < pµ < 1.5GeV/c

Table 3.1: Specification and performance of the Super-Kamiokande detector

16

3.1

Water Cherenkov image counter

17

Figure 3.1: Schematic view of Super-Kamiokande
charged particle β exceed the speed of light in the de-electric medium, a charged particle
emits Cherenkov photons. If n is the refraction index of the medium, thennβ ≥ 1, β = v/c(3.1)

Cherenkov photons are emitted forward like a circular cone with a half opening angle θto the direction of motion of the particle(3.2). The opening angle θ is related to n, β asfollows,cosθ =

1
nβ

(3.2)

In this experiment the medium is water(n = 1.33) and the emitted Cherenkov lightis detected by PMTs. The critical energy of each charged particle for Cherenkov lightemission is shown in Table 3.2

3.2 Detector components

18

Particle Critical energy(MeV)
e±

0.766

µ±

157.4

π±

207.9

K±

735.5

Table 3.2: Critical energy for Cherenkov light emission in water

The maximum emission angle of Cherenkov light is about 42 ◦ at β = 1. The numberof photons emitted per unit path length per wave length is,
11
d2 N= 2πZ 2 α(1 − 2 2 ) 2
dxdλ
nβ λ
x

: path length

λ: wave length of Cherenkov lightN: number of photonα: fine structure constant(1/137)

Z: Charge of particle

(3.3)

θ

Cherenkov ring

Figure 3.2: Cone of Cherenkov light

3.2

Detector components

The tank of the Super-Kamiokande detector is 41.4m in height and 39.3m in diameterand the cavity is coated with about 50cm thick concrete and is covered with 8mm thickstainless steel water-tightly. This tank is filled with 50,000m 3 ultra pure water.

3.3 PMTs

19

The PMTs are located on the stainless structure(Fig.3.3), 11146 PMTs in the innerdetector and 1885 PMTs in the outer-detector. The size of PMT is 50cm in diameterin the inner-detector and 20cm in the outer-detector. Inner-detector and outer-detector
are optically separated by black sheet(Fig.3.3). The inner-detector is used for particle
detection while the outer-detector vetoes incoming events from outside of the detector.
The high voltage supply and the data acquisition system are located on the top of thedetector.
Tyvek
8" PMT

Black sheet

20" PMT

Figure 3.3: The structure of the frame for PMTs

3.3

PMTs

The PMT was developed by HAMAMATSU Photonics Company for the Super-Kamiokande
experiment(Fig.3.4). As described earlier, two types of PMTs which differ in diameterare installed in the inner and outer detectors.
• Inner 20-inch φ PMT
The photo-multiplier tubes in Super-Kamiokande are improved version of the PMTs

3.3 PMTs

20~ 70000

cable length
φ 116

φ 254 10φ 82 2glass multi-seal

photosensitive area > φ 460< φ 520

cable

water proof structure( 610 20 )

~ 720

(mm)

Figure 3.4: 20inch PMT
in Kamiokande which were also developed by HAMAMATSU Photonics CO. LTD.
With several improvements in the dinode shape and the bleeder chain were carriedout, we obtained good timing and energy resolution. Timing resolution is improved
from 3nsec in Kamiokande to 2nsec (Fig. 3.5). These improvements affect many
low energy event analysis and we obtain even better energy and vertex resolution.
The quantum efficiency and the spectrum of Cherenkov light are shown in Fig.3.6.
At the peak of the Cherenkov light(λ = 390nm) quantum efficiency is 22%.

3.3 PMTs

Figure 3.5: The time resolution for 50cm-diameter PMT

Figure 3.6: Quantum efficiency and spectrum of Cherenkov light21

3.4 Data acquisition system

22

Fig. 3.7 shows a typical pulse height distribution for 1p.e. This figure shows two
signals, dark noise(around ADC=0) and signal(around ADC=400), which are clearly
divided. Therefore the energy threshold can be decreased to 5MeV. Since electric

Figure 3.7: A typical 1p.e distribution
wires are laid on the surface of the cavity around the detector, the geomagnetic field
is compensated to less than 100G (Fig.3.8)• Outer 20cm φ PMT
20cm-diameter PMTs are installed to the outer-detector. In order to increase lightcollection efficiency, PMTs are attached to a 60cm × 60cm wavelength shifter plate.

3.4

Data acquisition system

Five electronics huts for data acquisition are located on the detector top. Four of thesehuts are for data taking and power supply and one hut controls the event trigger. At first,
one-fourth parts of signal send to workstation in four huts. Next, four parts of data aregathered to control hut and reconstruct to a whole event.
An ATM (Analogue Timing Module) is installed to obtain energy and timing information(Fig.3.9). One ATM deals with signal channels for 12 PMTs and these channels are
independent each other. Each channel consists of discriminator, amplifier, ADC(Analogue
to Digital converter), TDC(Timing to Digital Converter) and memory. Fig.3.10 is theschematic view of the inner detector data taking system. The inner detector consists ofeight parts and each part works independently. One part consists of one workstation,
one VME crate, and six TKO boxes. A signal exceeding the threshold is stored in the3.4 Data acquisition system

F

A

E
E
E

A

E

D

A

F

F

F

G

G

23

G

G

A
B
B
B
B
C
C
C
D

C

coil

current (A)

A

31.35

B

31.35

C

31.35

D

28.20

E

30.10

F

28.60

G

30.10

Figure 3.8: The coil to compensate for the magnetic field
ATM buffer. When a trigger signal from discriminator is available within 1.2µsec, thestored signal in the ATM buffer is summed. The summed signal is called ’HITSUM’ andis treated as global trigger. When this global trigger is available, ADC receives timing,and charge information from the ATM and digitizes them. The digitized signal is received
by FIFO(First In First Out). Since both the short digitizing time with about 5µsec andtwo independent same digitizing systems are performed, the Super-Kamiokande detector achieves the dead time free system. The dead time free system is suitable to collectmultiple reaction like proton decay event and successive event in short time like µ decay
electron events.

3.4 Data acquisition system

PMTSUM
PMTSUMHITSUM

6

HITSUMPMT-INPUT

ANALOG
INPUT

START

TRIGGERFRONT PANEL

TKO BACK-PLANE
TRIGGERHITENCODERTAQ

PROCESSTHRESHOLDSTARTANALOG

QAC

HIT

6
PMT
-INPUTADC

TRIGGER

TRIGGER
INPUT

TRIGGER

EVENT
COUNT

THR.-DACPROTOCOL

MODE

FIFO

Figure 3.9: Block diagram of the ATM module

EVENT-COUNT

24

3.4 Data acquisition system

20-inch PMT
SCH
ATM
x 20
x 240

ATM
GONG

20-inch PMTSCHATMinterface
SUN machine

x 20
x 240

SMP

ATM

SMP

GONG

online CPU(slave)

SMP
SMP
SMP
SMP

x6

Super Memory Partner

VME
20-inch PMT
SCH

SUN machineATM

online CPU(slave)

x 20
x 240

SUN machine

ATM
GONG

online CPU(host)

Analog Timing Module

online CPU(slave)TKO
SUN machine

FDDI

SUN machine

online CPU(slave)
SUN machine

online CPU(slave)
SUN machine

FDDI

online CPU(slave)
SUN machine

online CPU(slave)interface
SUN machine

online CPU(slave)SMP
SMP

SUN machine

online CPU(slave)

SMP
SMP
SMP
SMP

interface

Super Memory Partner

VME
VME

interrupt reg.
TRG

20-inch PMT
SCH
TRIGGERATM

TRIGGER

x 20
x 240

HIT INFORMATION
ATM
GONG

Analog Timing Module

TKO

Figure 3.10: Data acquisition system

PROCESSOR

25

Chapter 4
Calibration
4.1

Calibration of PMTs

Since many PMTs are used to detect Cherenkov light in Super-Kamiokande detector,
energy and timing calibrations are especially important. We describe the relative energy
calibration in following section.
The relative energy calibration is done to obtain the same and adjust the gain foreach PMT so as to output pulse height from each PMTs. Fig.4.1 shows schematic viewof relative energy calibration system.
In relative energy calibration a scintilator ball is used for a light source, because theproperty of scintilator ball is in good agreement with Cherenkov lights. The relative
energy calibration is done by measuring the charge of each PMTs induced by the light of
scintilator ball. The position of scintilator ball is changed vertically in order to calibrate
at various height. In addition, the correction for the acceptance from the light angle
injected to PMTs, the distance between scintilator ball and PMT, and a variation for
intensity of light from the Xe lamp are considered. The variation of the gain for each
PMTs is adjusted so as to have a uniform gain for PMTs by high-voltage adjustment ofeach PMTs. The result for the gain adjustment is shown in Fig.4.2. From Fig.4.2 thegain variation after the high-voltage adjustment was estimated to be 7%.

4.2

Calibration for low energy events

The absolute energy calibration is also important in order to reduce systematic error for
total energy measurement. Therefore it is required to calibrate the energy within a few
percent.

4.2.1

Calibration using neutron capture with Ni

The schematic view is shown in Fig. 4.3. This calibration uses nickel(Ni) for γ source,26

4.2 Calibration for low energy events

27

UV filter ND filter
Xe Flash Lamp

Photo
Diode

SK TANK

Photo
Diode

Optical fiber

ADC

Scintilator Ball
PMT
Scintilator

Monitor20inchPMT

Trigger

Figure 4.1: Schematic view of relative energy calibration system

2000

1750

1500

11146 PMTs

1250

σ= 7.%

1000

750

500

250

0

0

200

400

600

800

1000 1200 1400 1600 1800 2000
2
x 10

corrected Q

Figure 4.2: The gain variation for all PMTs after the high-voltage adjustment (x-axis
shows corrected charge(pC · m2 ))4.2 Calibration for low energy events

28

200mm

130mm

125mm

10mm

30mm

Although the requirement for the absolute energy calibration is mono energy 5 ∼10MeV, such a γ-ray source is not available. Therefore, Ni source which emits several γrays 5 ∼ 10MeV is used to obtain 9MeV γ-ray at thermal neutron capture in Ni(n,γ)Ni∗ .
Since the Super-Kamiokande detector has large volume, absolute energy scale may bestrongly dependent on the detector position. The calibration using neutron capture with
Ni could be carried out at several different position because Ni γ source system is compact
and can be moved.
Also the detector can detect Cherenkov light from secondary electron by Compton
scatter and electron pair creation. It is indirect measurement of electron energy. The
direct measurement is described in next section.

50mm

20mm

Nickel wire + Water

Figure 4.3: The schematic view of Ni calibration system

4.2.2

Calibration using electron linear accelerator

The calibration using electron linear accelerator(LINAC) is direct measurement of energy
for single electron. The LINAC can generate the desired electron beam energy using
collimators and magnets. The electron from LINAC is injected to the Super-Kamiokande
detector. the electron energy is set planned energy between 5-20MeV.
The beam pipe is installed from top of the tank, The measurement can be done atvariable position in inner-detector by changing the beam pipe length. The schematic viewis shown in Fig.4.4.
The advantages using this calibration are direct energy calibration using electron,mono-energetic electron with 5 − 20MeV energy electron. However, it is inconvenient
for study of position and direction dependence. Because it is very large heavy system

4.3 Timing calibration

29

compared with Ni calibration equipment, further LINAC can generate only under going
electron.
beam pipe support structure
magnet 2
magnet 1

LINAC

magnet 3
RockTunnelbeam pipe
76mm

Veto
counter
PMTSuper-Kamiokande tank
Trigger counter(1mm)
Ti window(0.1mm)Figure 4.4: Configuration of LINAC calibration system

4.2.3µ decay-e event method

The µ decay electron event method is useful source of the energy calibration for electrons
around a few tens of MeV. The energy spectrum for µ decay electron is expressed as
Eq.4.1.
dN
G2 2 24Ee=
mµ Ee 3 −
3
dEe
12π
mµ

!

(4.1)

where Ee is the energy of decay electron, G is the Fermi coupling constant, m µ is themass of muon. The maximum energy and mean one of muon decay electron is 53MeV
and 37MeV, respectively.
The energy distribution of muon decay electron is shows in Fig.10.5, a solid line shows
Monte-Carlo result and cross shows data. The difference in the low energy region is found.
The reason of this difference is described in Section 10.3.2.

4.3

Timing calibration

Timing response of each PMT is not uniform. The differences come from fluctuation oftransit time in each PMT and timing response of the discriminator. In order to calibrate

4.3 Timing calibration30

the difference of timing response, the transit time difference and pulse height dependenceof the timing response are measured using the timing calibration system. Fig.4.5 shows
the schematic view of the timing calibration system.
FilterOptical Fiber

Laser

PMT
PMT
Trigger
Diffuser ballElectronics
SK Tank

Figure 4.5: Schematic view of timing calibration system
The diffuser ball is located at the center of the tank. The distance from the positionof each PMT to the diffuser ball is considered in TOF calibration.
Fig.4.6 is the T-Q maps which shows the relation between the mean value of the time
and the charge from the PMT. The T-Q maps are made for each PMT and the measured
time information is corrected using the T-Q maps. The timing resolution after correction
by T-Q maps is shown in Fig.4.7 as 3nsec at one photo-electron.

4.3 Timing calibration

T (nsec)

Typical TQ map
980
970
960
950
940
930
920
910
900
890
880

Q (p.e.)Figure 4.6: T-Q map of a typical PMT

31

4.3 Timing calibration

Time Resolution (nsec)

Time Resolution of Inner PMT
5
4
3
2
1
0

Time Resolution (nsec)

1

10

10

2

Q (p.e.)

5
4
3
2
1
0

1

2

3

4

5

Q (p.e.)Figure 4.7: Timing resolution32

4.4 Water transparency measurement

4.4

33

Water transparency measurement

Since the transparency is about 60m, the water transparency determines the number of
Cherenkov photons at the PMTs, and it affects to detector response. Then it is importantto estimate the water transparency for data analysis.
Two kinds of transparency measurement are done in Super-Kamiokande, one is using
cosmic-ray muon events, another is using the laser system.

4.4.1

Water transparency by cosmic-ray µ events

The cosmic-ray through-going muon from top to bottom into the tank is useful for monitoring the attenuation length. The entrance point and exit point is determined by automatic directional fitting.
The total charge(Qi ) detected by each PMT is described as the following equation(Fig.
4.8),

Qi = const. ×f (θi ) − li
·e L
li

L

: attenuation length

θi: the incident angle for Cherenkov lightf (θi ): acceptance function of PMTli: the flight length of Cherenkov photon

42

µ

θ
L

PMT
Figure 4.8: Schematic view of cosmic-ray muon methodq x travel-length / acceptance(p.e. x m)

4.4 Water transparency measurement

403.5

450

/ 20
5.866
-0.1781E-03

P1
P2

0.8726E-03
0.4873E-06

400

350

Super-Kamiokande Preliminary
300

250

200

150

100

50

0

0

500

1000

1500

2000

2500

3000

3500

travel len. v.s. eff. charge, run002120

4000 4500 5000
travel length(cm)Figure 4.9: The light attenuation length(6th July 1996)

Figure 4.10: Time variation of the water attenuation length34

4.4 Water transparency measurement


35



Fig. 4.9 shows log fQ(θi Li ) plot against l in a typical run. The attenuation length wasestimated to be 105.4 ± 0.5m(error of the fitting) in water from Fig. 4.9.
The attenuation length is estimated for each run. At the just starting this experiment,
the attenuation length was around 60m. After the water purification system is runningand makes very pure water, the attenuation length attained to be about 100m and be
maintained stably.

4.4.2

Water transparency by laser

The directly measurement of water transparency is carried out by using the laser system
which consists of the laser and CCD camera shown as in Fig. 4.11.

Beam Splitter (50:50)
CCD camera
DYE / N2 laser
Lens

Integrating Sphere<< laser box >>Diffuser Ball
2inch PMT<< SK Tank >>Optical Fiber (70m)

Figure 4.11: Set up of the laser system
The light source is a dye laser pumped by a pulsed N 2 laser. The emitted light divides
into two optical fibers, the one is for monitoring PMT which measure the light intensity
directly, the another is guided to the diffuser ball which is located in the water. The
diffuser ball emits the isotropically light. The CCD camera measure the intensity of theattenuated light from diffuser ball. The depth of the diffuser ball is changed from 5m to
30m, Ilaser (the light directly intensity at the laser) and I CCD ( the attenuated intensity at
the CCD ) are measured at each point. The attenuation length(Lλ) is expressed to be4.4 Water transparency measurement36

following equation,
!

ld
ICCD
= const. × exp −,IlaserL(λ)(4.2)

where λ is the wave length of the light. The experimental IICCD
is shown at the wave
laser
length of 400nm in Fig. 4.12. The data is fitted by Eqn. 4.2 and the attenuation lengthis obtained to be 92.2±5.2m.

Figure 4.12: The attenuation lengthChapter 5
Data reduction
The trigger rate for all of Super-Kamiokande data is about 10Hz. But about 80% of
the events are background which is are either low energy electron events or noise events.
These background events are removed by automatically or manually. In this chapter, theevent selection for upward-going muons is described.

5.1

DatasetThe Super-Kamiokande detector started in April 1996. The data earlier than July 1996
was not used for the analysis of stopping muons because the outer-detector was notstable before that time and the reduction for upward-going muons needs outer-detector
information. Thus, the data from July 1996 to April 2000 for stopping muons and from
April 1996 to April 2000 for through-going muons were studied. The live-time for stoppingmuons and through-going muons were 1247days, and 1268days, respectively.

5.2

Automatic reduction

The reduction flow chart is shown in Fig. 5.1. The first reduction, called ’online reduction’,is a high speed and rough process. The second reduction, which is called ’offline reduction’,required high precision and is a more precise process. In eye scan, physicists judge theevent type and reconstruct the track of particle. I will describe these reduction processes.

5.2.1

Upward stopping muons

1st reduction
This reduction applied for rejection low energy γ-ray from radioactivity and through-going
events.
• Total Q cut
A large amount of background, about 80% of all events, are low energy events or37

5.2 Automatic reduction

38

all data

1streduction

2nd reduction

eyescan

finalsam ple

Figure 5.1: The flow chart of event selection.
γ-ray from radioactive sources present in the water or from outside of the detector.
The first reduction applied a total Q cut for rejection these background. Eventswith less than 1000p.e. in total Q were rejected. Total Q means a summation of
the charge from all PMTs for one event.
• Outer-detector cut
Outer-detector cut is applied in order to reject through-going muons. The event rate
due to muons in the Super-Kamiokande is about 2.2Hz. Muon events are categorized
into through-going events and stopping events. The rate for stopping muon event
is about 5% of all muon data.
Through-going muon and stopping muon events are topologically defined by outerdetector information. Through-going muons go through the anti-detector with signals corresponding to entrance and exit points. Stopping muons stopped in thedetector and a single signal is made in the anti-detector. Two kinds of cut are
applied to reject through-going muons.
– Cluster cut
When charged particles go through the PMT layer of the inner-detector andthe outer-detector, Cherenkov light forms some clusters of the active PMT.
Stopping muon and through-going muon have different topological character5.2 Automatic reduction

39of the clusters. The topological difference is in the number of clusters producedin the outer-detector. In the through-going muon case, the muon generated
outside of the outer-detector goes into and through out the outer-detector,
two clusters appear in outer-detector, the entering cluster and the exit cluster.
In the stopping muon case, only one cluster appears, the entering cluster.
Therefore we need to know the number of clusters for each event to discriminate
between through-going muons and stopping ones. At first, we select hit PMTs
with the time between 500nsec and 1300nsec. Then the geometry of the selected
PMTs and the number of these PMTs neighboring each other is estimated anddefined as a cluster candidate. If the cluster candidate has more than 8 PMTs
and the distance between them is less than 8m, then the candidate is judged
as a cluster. The location of clusters is defined as the PMT position withmaximum photo electrons. The events which have two clusters are removed
from the stopping muon data and kept as through going muons.
– Time-window cut
The other method for the separation of stopping and through-going muons
is a time-window cut which uses the difference of time width for PMT hits.
Stopping and through-going muons differ in the time of flight for muon in theouter-detector. The time width for the through-going muon is larger than the
stopping one. The time width is determined as follows. At first, the number of
hit PMTs is counted in the time-window between 800nsec and 900nsec. Next,
the time-window is moved by 1nsec, that is between 801nsec and 901nsec, andthe number of hit PMTs is again counted. This counting of PMTs and movingof the time-window is repeated 500times. The time width is determined as thenumber of 100nsec time-windows where there are more than 15 hit PMTs. An
event whose time width is longer than 240 nsec is judged to be through-going
muons and is removed.
• Detection efficiency
The reduction factor for through-going muons is estimated by using the cosmic-ray
muon data applied with total Q cut. The reduction factor of through-going muons
by outer-detector cut is 96%. However, 76% of through-going muons which could not
be eliminated are events with a shorter track length than 5m (E µ ∼ 1.2GeV) in the
inner-detector. The detection efficiency for stopping muon is estimated to be 98.6%
by time-window cut and 99.9% by cluster cut using cosmic-ray event. Therefore
the combined detection efficiency for stopping muons after all anti-detector cuts is
98.5%. Moreover, an estimation of the detection efficiency for stopping muon iscarried out using Monte-Carlo(MC) data, with neutrino induced muons at higher
energy than 1.6GeV for muon energy. The detection efficiency for stopping muons
by outer-detector cut for MC event is obtained to be 95.9%.

5.2 Automatic reduction

40

Because neutrino induced upward stopping muons could interact with the water of
the anti-detector, the efficiency obtained by using upward stopping muons could be
different with the efficiency obtained by using cosmic-ray muons. Therefore, thedetection efficiency for muons was adopted 95.9% using MC data. The discrepancy
between the MC result with and the cosmic-ray result is included as systematic
error.
2nd reduction• Total Q cut
Total Q(charge) cut for the 2nd reduction is applied to reject short track length
events.
Fig. 5.2 shows the relation of total Q with track length by manual reconstruction
for cosmic-ray stopping muons. Since a short track length event is difficult to reconstruct, events in which its orbit has longer than 7m of track length are collected.
The 7m track length is equivalent to 1.6GeV in energy for a muon.
DATA(p.e.)
10
10000

8000

6000

4000

2000
CUT
1000
2000
3000
4000
5000
(cm)
TOTAL ENERGY VS TRACK LENGTH

Figure 5.2: Total photo-electron and track length for stopping muons. Stopping muonscollected by eye scan were fitted by hand and were determined their track length.
• µ direction cut
Stopping muons are grouped as either cosmic-ray induced or atmospheric neutrino
induced muons. Since cosmic-ray muons are almost downward-going and upward
going muons are almost neutrino induced, downward-going muons are rejected as
background event. The direction of muons is obtained by directional reconstruction(stopping muon fit). The horizontal downward events with cos Θz < 0.12 are5.2 Automatic reduction

41

collected to estimate for contamination of cosmic-ray muons, where Θ z is zenith
angle; cosΘz = −1 is defined as vertically upward. The obtained contamination of
the cosmic-ray muons are in cosΘz < 0.12.
The method of automatic directional reconstruction is as follows,– Determination of entrance point
Since the particle coming from the outer-detector and going into the innerdetector makes entrance cluster at first, the PMTs at the entrance point get
hit earlier than other one. However, some noise events could happen in similar
timing sequence, for example dark noise of PMTs. Therefore, the position of
the earliest hit PMT whose neighbor PMTs get hit within 5nsec is taken to be
the entrance point.
– Directional reconstruction for muons
The direction at the entering point on the inner-detector wall is determinedby using a grid search method. At first, muon direction is approximated fromthe muon entering point to the center of Cherenkov ring which is determinedby a center of mass in charge produced in PMTs. The grid search is carriedout by changing the virtual direction starting from the initial direction and by
minimizing the fitting evaluation function. The fitting function ’goodness’ isas follows,"

1 ti − T i11goodness = 1 × Σ 2 exp −Σ σ2σi
2 1.5σi
i



2 #

(5.1)

where ti is the hitting time for each PMT, σ i is the timing resolution(=3nsec),
and Ti is the hit time of each PMT estimated from T , the entering time ofmuon. Ti is calculated as follows,
lµ n · lph
+
(5.2)
c
c
where c,n are light velocity in vacuum, and the refractive index of water(n =1.344), respectively. l µ is the distance from the entering point to the Cherenkov
emission point along the muon track. l ph is the distance from the Cherenkov
emission point to each PMT(Fig.5.3). Since the fitting result agrees with thedata, when goodness is closest to 1, it means completion of the fitting and gridsearch. Thus, entrance point and direction are obtained.
Ti = T +• Detection efficiency
The 2nd reduction was applied to upward stopping muons generated by MC detector
simulation. The detection efficiency was 100% in all direction. Subsequently, the
2nd reduction was applied to cosmic-ray muons. The result is shown in the table
6.1. The detection efficiency for cosmic-ray stopping muons is almost 100% in all
directions.

5.2 Automatic reduction

42

inner detector

lµ
T

µ
l ph
ti

Figure 5.3: Schematic view of T OF .

5.2.2

Upward through-going muons

1st reduction• Total Q cut
Both low energy and noise events are eliminated with a cut on total Q ≤ 5000p.e.
same as stopping muon reduction.
• Maximum Q cut
The charge of PMTs at exit point must be largest charge because of the no attenuation of photons. Single-ring stopping muons, which has no exit point, are rejectedby this selection. Events having maximum Q larger than 235p.e. are chosen as
through-going events.
• The position dependence of the maximum Q PMT cut
In the through-going muon case, the position of the PMT with maximum Q must be
close to exit point. Therefore, the distance between entrance point and maximum
Q PMTs position(Lent ) is large. On the other hand, the distance between exit point
and maximum Q PMTs position(Lext ) is small. The entrance and exit point isestimated by using same process as stopping muon reduction process. The selection
Lent ≥ 300cm and Lext ≤ 300cm are applied. The event with L ent ≤ 300cm is judged
as stopping muons. The event with Lext ≥ 300cm is classified into the multiple muon
event. Therefore, these two kinds of events were rejected.
• µ direction cut

5.2 Automatic reduction

43

The down-going cosmic-ray muons are eliminated by using same reduction routine
as stopping muons selection. The event with larger goodness than 0.88 is chosen as
the good event. All fitted events are selected by zenith angle with cosθ ≤ 0.1
2nd reduction• Total Q cut
The total Q cut is applied on the events through first reduction in order to make theperformance of the track length reconstruction better. The events with less than
12000p.e. in total Q, corresponding to 5m in track length, were rejected. The total
Q cut and maximum Q cut are shown in Fig 5.4.
• µ direction cut
In the 2nd reduction, two kinds of the directional fit are applied. These are more
precise processes than 1st reductions one. One is based on time-of-flight and otheris based on charge of PMTs.
The following function is defined for the time-of-flight based reduction,
T OF − σ = RM S(T i − T OFi )
i
nlph
lµi
+
T OFi =
c
c

(5.3)
(5.4)

where i is the index of a PMT number, T i is the timing information of i-th PMT.
lµ and lph is described in stopping muons reduction. Fig 5.3 shows the definition oflµ and lph . A smaller T OF − σ means a better muon fitting.
Another function is Rcone , based on charge information. Rcone means the ratio of
hit PMT numbers and is defined as follows,
Rcone = 1 −Nhit42
Nhit70+
NPMT42 NPMT70(5.5)

where Nhiit42 and NPMT42 are the number of hit PMTs and the number of PMTs
in 42◦ Cherenkov cone, respectively. N hit70 and NPMT70 are the number of hit◦◦
PMTs and the number
qof PMTs between 70 cone and 42 cone. The defined threshold of the hit PMT is total Q/100. In equation 5.5, the second term Nhit42 /NP M T 42
shows the filledness of the Cherenkov ring and third term N hit70 /NP M T 70 shows the
sharpness of the edge of the Cherenkov ring. Therefore, if the event is through-going
muon, Nhit42 /NP M T 42 shows a larger value and Nhit70 /NP M T 70 shows a smaller value.
Then, a smaller Rcone means a better muon fitting.

5.3 Eye scan with manual fitting

44

through µ eventsCut

low energy eventsstop µ events

Figure 5.4: Total Q cut and maximum Q cut

5.3

Eye scan with manual fitting

The elimination of short track length events is difficult by means of the automatic directional reconstruction. After the second reduction, edge clipping events still remain whichgo through or along the edge of the inner-detector wall. However, edge clipping eventsare categorized as through-going muon in the reduction process. At the second reduction, because the number of clusters is only one at the edge of the detector, they wereplaced into the same category as stopping muons. Therefore, the eye scan by physicists
was applied to all events remaining after automatic reduction. As mentioned already, thestopping muons are categorized the events with only one cluster and more than 7m track
length in the inner-detector.
The directional resolution for muons becomes better by means of manual reconstruction by physicists. The manual reconstruction is performed by selection of the entrance
and the exit points, the exit point is determined by extrapolation from entrance to stopping direction. The resolution for directional reconstruction manually is estimated to be
1.4◦ using MC events, (Fig. 5.5). Also individual human bias is obtained to be 0.8 ◦ (Fig.
5.6) from angular difference comparing the results of two independent scanners. The contamination of through-going muons with stopping muons is estimated to be lower than
1%. Since the eye scan was applied by two independent groups, the eye scan reduction
could be accomplished with almost 100% efficiency.

5.3 Eye scan with manual fitting

5.3.1

45

Track length cut

Since the directional reconstruction is difficult for short track length events and the background other than muons should be taken into account in the low energy region, eventswith longer than 7m track length(equivalent to 1.6GeV in muon energy) were adopted forfinal data.
The track length of through-going muons is geometrically determined from the entrance and exit points. However, for stopping muons, track length should be estimate
from particle energy at entrance. Algorism of the track length calculation is described
below. Coarse direction and entrance point are obtained by manual reconstruction. Next,
cone with opening angle of 70◦ cone is put on the entrance point in order to obtain roughtotal charge. The charge of all PMTs inside the approximated 70 ◦ cone is summed as thetotal charge. The charge is applied some collections, water transparency and acceptanceby PMT geometry. The momentum is estimated from the total charge and the track
length is calculated from the momentum by using the muon energy loss of Lohmann’s
calculation.

Number of events

5.3 Eye scan with manual fitting

46

MC
50

202
280
1.110
0.8971ID
Entries
Mean
RMS

40

30

20

10

Number of events

0

0

1

2

3

4

5

6

7

8

9
10
∆θ (deg.)

MC
102
280
0.5341E-01
0.8257

ID
Entries
Mean
RMS

35

30

25

20

15

10

5

0

-5

-4

-3

-2

-1

0

1

2

3

4
5
∆Θ z(deg.)

Figure 5.5: Angular resolution for MC stopping muons µ(Eµ > 1.6GeV ). Upper figure
is angle between manual reconstruction and true direction. lower one is zenith angleview(Θgen − Θma ).

5.3 Eye scan with manual fitting

47

MC
200
280
1.102
0.7967

ID
Entries
Mean
RMS

40
35
30
25
20
15
10
5

Number of events

0

0

1

2

3

4

5

6

7

8

9
10
∆θ (deg.)

MC
100
280
0.7214E-01
0.8055ID
Entries
Mean
RMS

35
30
25
20
15
10
5
0

-5

-4

-3

-2

-1

0

1

2

3

4
5
∆Θ z(deg.)

Figure 5.6: Angular difference depend on the scanner, upper figure is angular differenceby two scanner. Lower figure is zenith angle view.

5.4 Upward-going muon events

5.4

48

Upward-going muon events

After all event selection, The final data is selected. The scatter plot of the vertical angle
versus azimuth angle distribution of the final data is shown in Fig. 5.7. Cosmic-ray
contamination is clearly seen in thin rock region between φ = 120 ◦ and φ = 180◦ . Finally,
143 events were observed as upward stopping muons.

cos

DATA
0

-0.2

-0.4

-0.6

-0.8

-1

0

50

100

150

200

250

300
350
[degree]Figure 5.7: Zenith angle and azimuth angle for stopping muons.

5.5

Backgrounds

We considered the reason of background for upward stopping muons.
1. Cosmic-ray muons contamination
The horizontal downward cosmic-ray muons are possibly mistaken as upward-going
muons due to finite resolution of directional manual reconstruction. In Fig. 5.7,
regions with higher density of points, possibly due to this background, are seenin the region vertical angle cosΘ ≈ 0(near horizontal) and azimuth angle between
φ = 120◦ , φ = 180◦ . Since cosmic-ray contamination depends on the amount ofrock surrounding the detector and the shape of Ikenoyama mountain, cosmic-ray
down-going muons penetrate from relatively thin rock direction. In Fig. 5.9, region(2) is thin rock direction, the thin rock region is defined as the region that theazimuth angle between 60◦ and 310◦ , and the zenith angle smaller than −0.1 ofcosine. Region (1) is outside of the region (2). Fig. 5.8 is plotted two region definedas the region (1) and (2) in Fig. 5.9 in zenith angle distribution separately, (1) and5.5 Backgrounds

49(2) region, are normalized in shown azimuth angle respectively. The horizontal line
in center of figure is shown border between downward and upward. Although both
distribution is flat in upward region cosΘ < 0, the number of muons is increasing
exponentially with cosΘ in due to downward-going in the region (2) cosmic-ray
muons.
The number of cosmic-ray contamination into upward region is estimated by extrapolation the distribution of (2) and fitting of exponential function and integrate
the exponential slope in the upward going range as follows,
Number of contamination =Z 0e(a+b) dcosΘ

(5.6)

−1

The number of cosmic-ray muons contamination is estimated to be 22 ± 10 eventsfor upward stopping muons and 10 ± 0.9 events for upward through-going muons.

DATA

=360˚

3
10
Region (1) < 60˚,

Normalized Number of Events /Region (2) 60˚

310˚
310˚

2
10

10

1
-0.1 -0.08 -0.06 -0.04 -0.02

00.02 0.04 0.06 0.08 0.1
cos

Figure 5.8: Zenith angle distribution of stopping muons. The region (1) is thin rock direction, the region (2) is thick rock direction in azimuth angle. Cosmic-ray contaminations
are seen apparently in thin rock region.

Number of Events

5.5 Backgrounds

10

50

2

(1)

(2)

(1)

10

1
0

50

100

150

200

250

300
350
φ [degree]

Figure 5.9: The azimuth angle distribution for stopping muons. The region (1) is thickrock direction, the region (2) is thin rock direction. Three peaks due to contamination
downward-going cosmic-ray are seen in thin rock region.

Chapter 6
Observed flux
The experimental upward-going muons flux(Φobs ) is calculated by using live time(T live),
effective area of the detector(S), and selection efficiency(). The upward-going muons
flux is expressed by the formula,Φobs =

NX
1
1×
2πTlive j=1 (θj )S(θj )(6.1)

where θj is the zenith angle of the j-th upward going muon, 2π is the total solid angle
covered by the detector for upward-going muons, and N is the total number of observed
upward-going muon events. Fluxes of observed upward stopping muons and through-going
muons are estimated from equation 6.1.

6.1

Live time

The detector live time is calculated by following equation,
Tlive = Trun − Tpedestal − Tdead(6.2)

where Tlive is the detector live time, T run is the pedestal event time, T dead is the detector
dead time. The detector dead time means all or a part of event information is lost byelectrical or network trouble. The total detector live time is estimated to be 1247days
for upward stopping muons, and 1268days for upward through-going muons. The difference appeared for the detector live time of upward stopping muons and through-going
because the instability of the detector deteriorates more intensively the events with stopping muons. The live time is about 90% of the true time.

6.2

Effective area

The effective area is the required area in order to get enough information for upward-going
muons which have a given direction, the sensitivity depends on the azimuth angle and the51

6.3 Selection efficiency

52

zenith angle of the path of the muons. In this analysis upward-going muons with longer
than 7m track-length are required. But a non sensitive region is available because of thecylindrical detector. The area which have longer than 7m track length in the detector iscalled the effective area.
The effective area is calculated as follows and the schematic view is shown in Fig
6.1. A large enough plane, having two-dimensional grid points with every 10cm step, isdefined out of the detector. A vertical line on the defined plane is drawn from each grid
point to the detector. If the length of line crossing the detector is longer than 7m, thecorresponding grid point is counted for a part of the effective area. Finally the effective
area is calculated for every zenith angle of the area divided into 100 directions. The resultof the effective area calculation is shown in Fig. 6.2.
projected plane

track length > 7m

muon track

Figure 6.1: Schematic view of effective area.

6.3

Selection efficiency

The selection efficiency is the detection probability for the upward-going muons estimatedfor each zenith angle bin, with the range −1 ≤ cosθ ≤ 0 being divided into 10 bins. The
selection efficiency depends on 1st and 2nd reduction efficiency because the selection
efficiency for eye scan is almost 100%. The result for selection efficiency is shown in table
6.1
The selection efficiency for the most horizontal bin is slightly smaller than any otherbin because of the resolution of automatic fitters. It is possible that true upward-muonsEffective Area (m2)

6.3 Selection efficiency

1500
1400
1300
1200
1100
1000
900
800
700
600
500

-1

-0.9

-0.8

-0.7

-0.6

-0.5

-0.4

-0.3

-0.2

-0.1
0
cosθ

Figure 6.2: Zenith angle dependence of the effective area.

cosθ

-0.95

-0.85

-0.75

-0.65

-0.55

-0.45

-0.35

-0.25

-0 .15

-0.05

stop

1.00

1.00

1.00

1.00

1.00

1.00

1.00

1.00

0.99

0 .98

thru

1.00

1.00

1.00

1.00

1.00

1.00

1.00

1.00

1.00

0 .99

Table 6.1: The summary for selection efficiency.

53

6.4 Systematic errors

54

could be judged as downward-going muons.

6.4

Systematic errors

The systematic errors are categorized into two cases: one which depends on the event
category(whether stopping or through-going), the other is independent of event category.
The independent errors are live time and effective area, while the dependent errors aretrack length and reconstruction efficiency. The uncertainty for cosmic-ray contamination,
which has been described in Section 5.5 and was estimated from down-going muons(0 ≤cosθ ≤ 0.1), is included in the statistical error.

Track length
The reconstructed track length for upward stopping muons is described in Section 5.3.1.
The difference of the track length between reconstructed from total charge of cosmic-ray
stopping muons and calculated from entering point to stopping point with MC is 5%.
When the track length cut with 7m is changed ±5%, the upward stopping muon flux
is changed +4.9% − 4.1%. The systematic error of track length for upward throughgoing muons is determined to be 1% by geometric uncertainty for the Super-Kamiokande
detector.

Reconstruction efficiency
The reconstruction efficiency for upward stopping muons is described as section 5.2.1, thedifference of the efficiency between cosmic-ray muons and MC is 2.6%, and corresponds to
1% for upward stopping muon flux. For the upward through-going muons, the reconstruction efficiency is near 100% as shown in table 6.1. Thus the systematic error is negligible.

Systematic errors for observed flux
Table 6.2 shows the summary of systematic errors for observed upward-going muon flux.
The systematic error for observed upward-going muon flux and through-going muon flux
are estimated to be +5.2%-4.5%, and ±1.2%, respectively.

6.5

Observed flux of upward going muons

In the 1247 live days for upward stopping muons, the observed flux is estimated to be{0.41±0.02(stat.)±0.02(syst.)}×10−13 cm−2 s−1 sr−1 . On the other hand, in 1268 live daysfor upward through-going muons, the observed flux is estimated to be {1.70±0.05(stat.)±6.5 Observed flux of upward going muons

55(upward stopping muon flux)
track length
live time
reduction efficiency
total+4.9% − 4.1%
±1%

±1%

+5.2% -4.5%(upward through-going muon flux)
track length
effective area
live time
total

±0.5%

±0.3%
±1%

±1.2%

Table 6.2: The summary of systematic error for upward going-muons

0.02(syst.)}×10−13 cm−2 s−1 sr−1 . The total upward going muon flux which is a summationof stopping muon flux and through-going muons is {2.11 ± 0.05(stat.) ± 0.03(syst.)} ×
10−13 cm−2 s−1 sr−1 , which is independent of the detector shape. The energy dependence of
upward-going muon is estimated from the ratio of stopping muons to through-going muons
flux, the ratio for stopping muon flux and through-going is 0.24 ± 0.02(stat.) ± 0.01(syst.).

Chapter 7
Expected flux
The expected flux for upward-going muons are estimated for neutrino oscillation analysis.
Neutrino oscillation is deduced by comparison between the observed flux and the expected
one. The expected flux is usually estimated by Monte Carlo approach, but it is difficult in
the upward-going muon case. Since a large amount of neutrino-nucleon interaction near
the detector should be considered, the statistics are not enough for oscillation analysis.
Therefore an analytical method to calculate the expected upward-going muon flux is
adopted.
We give a detailed description of the analytical method to calculate the upward-going
muon flux.

7.1

Analytical method

The atmospheric neutrino induced upward-going muon flux are calculated by using theatmospheric neutrino flux(Φ), the charged current cross section for the neutrino-nucleon
interaction in the rock(σ ν ), and the muon range in the rock. The muon neutrino induced
upward-going muon flux is expressed as follows,d2 σν d2 Φν (Eν , cos θν )
·
(7.1)
dxdy
dEν dΩ
where Eν is the energy of atmospheric neutrino, x, y are Bjorken scaling parameters. The
probability that the muon with energy E µ produced by the neutrino-nucleon interaction
goes through with X(g/cm3 ) in the rock and come into the detector is expressed withg(X, Eµ , cosθν ). The upward-going muon flux with zenith angle θµ is calculated as follows,d2 σν d2 φν (Eν , cos θν )·· g(X, Eµ , Eth )dxdy
dEν dΩ
0
Eµ
0
0
(7.2)
where NA is the Avogadro constant. The θν is nearly equal to θ µ . Therefore θν is taken
to be θµ . The traveling distance is R(E µ , Eth ), over which the muon loses energy from E µ
to Eth in traveling R. The function g(X, E µ , Eth ) is,
dφµ (Eth , cos θµ )
=
dΩ

Z ∞

NA dX

Z ∞

dEν

Z 1

dy

Z 1

dx ·

g(X, Eµ , Eth ) = Θ(R(Eµ , Eth ) − X)
56

(7.3)

7.2 Atmospheric neutrino flux
where Θ is defined as,
Θ(x) =
Z ∞
0



 1

 0

: x≥0

: x<0

g(X, Eµ , Eth )dX = R(Eµ , Eth )

57

(7.4)
(7.5)

The equation (7.2) is rewritten as,
d2 σν d2 φν (Eν , cos θ)·· R(Eµ , Eth )(7.6)
dxdydEν dΩ
0
0""##
Z ∞ Z 1 Z 1 2d σν
d2 φν (Eν , cos θ)=
NA R(Eµ , Eth )dydEν (7.7)dEν dΩ
Eµ
00 dxdyZ ∞d2 φν (Eν , cos θ)P (Eν , Eth )
=
dEν
(7.8)
dEν dΩ
Eµ(7.9)∞
dφµ (Eth , cos θ)
= NA
dEν
dΩ
Eµ

Z

Z 1

dy

Z 1

dx ·

where the function P is defined as,
P (Eν , Eth ) ≡d2 σν
NA R(Eµ , Eth )dy
0 dxdyZ 1 "Z 1
0#(7.10)

The function P is the probability that an neutrino with energy E ν produces muons byneutrino interaction and the muons reach the detector with energy larger than E th . Since
the Super-Kamiokande detector can not discriminate between negative muons from muon
neutrinos and positive muons from anti muon neutrinos, the total upward-going muon
flux is estimated.

7.2

Atmospheric neutrino flux

The energy spectrum of atmospheric neutrino is calculated on the basis of the energy
spectrum of primary cosmic-ray. The calculation of atmospheric neutrino flux needs theinformation about the composition of primary cosmic-ray and hadron interaction. Several
models for the atmospheric neutrino flux are available. Energy spectra for atmospheric
neutrino fluxes by Bartol[13], Volkova[14], Honda[15], and Butkevitch[16] are shown in
Fig.7.1. The zenith angle distribution for various neutrino energy is shown in Fig.7.2
The upward-going muon flux at vertical upward (cosθ = −1) is larger than the one athorizontal(cosθ = 0). The shape of the zenith angle distribution depends on the neutrinoenergy. The shape in low energy is flat while it is steep in high energy.

7.3

Neutrino-nucleon interaction

The atmospheric neutrino interacts with nucleon in the rock and produces muons. The
cross section for charged current(CC) interaction is described in the following. The total

2

-2 -1

-1

E3ν<Φν> (GeV cm s sr )7.3 Neutrino-nucleon interaction

58

0.1
BartolVolkova
Honda
Butkevich

0.09
0.08
0.07
0.06
0.05
0.04
0.03
0.02
0.01
0
1

10

10

2

10

3

10

4

10

5

6

10
10
Eν (GeV)7

Figure 7.1: The energy spectrum of atmospheric neutrino(ν µ + ν̄µ ) flux

cross section for charged current interaction of neutrinos is composed of deep inelastic
scattering(DIS), quasi-elastic scattering(QEL) and single-pion production(1π).
σ CC = σ DIS + σ QEL + σ 1π(7.11)

The cross section for DIS is calculated on the basis of parton distribution function(PDF)
which shows quark distribution in nucleus. The QEL and single pion production are also
taken into consideration.

7.3.1

Deep inelastic scattering

The differential cross section when neutrino with E ν produces muons with Eµ by charged
current interaction is expressed by Bjorken scaling parameter(x = Q 2/2mN Eν y, y =1 − Eµ /Eν ) is shown as follows,y2
G2 mN Eν
mN xy
m4W
d2 σν
2
F
±
y
−
= F
y
xF
+
1
−
y
−
xF3
2
1
dxdy
π
(m2W + Q2 )2
2Eν
2
"





!

#

(7.12)

where the positive and negative signs in the last term correspond to ν and ν̄, respectively,
GF is the Fermi constant, m W the mass of W boson, Q the momentum transfer between
neutrino and muon. F1 , F2, F3 are the structure functions given by quark distributionfunctions fi (i = the flavor of quark) as follows,νµ n(p) → µ− X
F2 = 2xF1 = 2x[fd (fu ) + fs + fb + fū (fd¯) + fc̄ + ft̄]

(7.13)

x 10

59

-1

-3

0.07

0.14

-1

0.08

Eν=1GeV
BartolVolkova
Honda
Butkevich

-2 -1

0.09

Φν (cm s sr GeV )

-1

0.1

-2 -1

-1

Φν (cm s sr GeV )7.3 Neutrino-nucleon interactionEν=10GeV
Bartol

0.12

Volkova
Honda

0.1

Butkevich

0.06
0.08
0.05
0.06

0.04
0.03

0.04
0.02
0.02
0.01

-0.9

-0.8

-0.7

-0.6

-0.5

-0.4

-0.3

-0.2

-6
-1

BartolVolkova
Honda

0.1

-0.9

-0.8

-0.7

-0.6

-0.5

-0.4

-0.3

-0.2

-0.1
0
cosθ

-0.6

-0.5

-0.4

-0.3

-0.2

-0.1
0
cosθ

-9

0.12
Eν=1TeV-2 -1

Eν=100GeV
0.12

x 10

-1

-1

0.14

-1
-2 -1

0

-0.1
0
cosθ

-1

Φν (cm s sr GeV )

x 10

-1

Φν (cm s sr GeV )0

Bartol0.1

Volkova
Honda

0.08

Butkevich

Butkevich

0.08
0.06
0.06
0.04
0.04
0.02

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

-0.9

-0.8

-0.7

Figure 7.2: The zenith angle distribution of atmospheric neutrino flux in various neutrinoenergy60

1W2>1.4GeV2

0.9

GRV94(DIS)-38

2cm /GeV)

7.4 Energy loss for muons

σDIS/Eν (×10

CTEQ3(M)

0.8
0.7

νN

0.6
0.5
0.4
0.3

–

νN

0.2
0.1
0
1

10

10

2

10
Eν (GeV)3

Figure 7.3: νµ N and ν¯µ N total cross section per neutrino energy

F3 = 2[fd (fu ) + fs + fb − fū (fd¯) − fc̄ − ft̄ ](7.14)

F2 = 2xF1 = 2x[fu (fd ) + fc + ft + fd¯(fū ) + fs̄ + fb̄ ](7.15)

F3 = 2[fu (fd ) + fc + ft − fd¯(fū ) − fs̄ − fb̄ ](7.16)ν¯µ p(n) → µ+ X

There are various kinds of the fitting method with the parton distribution function in order
to fit the experimental results. The parton distribution function is limited by minimum
momentum transfer Q 2 for correct calculation . Equation 7.12 shows the deep inelastic
scattering cross section with large momentum transfer Q 2 and large the invariant massof final state hadron system. Fig. 7.3 shows the neutrino-nucleon cross section versus
neutrino energy. The cross section is obtained by using GRV94(DIS)[17].

7.3.2

Quasi-elastic scattering and single pion production

Total charged current cross section are calculated by D.Rein and LM.Seghal taking into
consideration of the quasi-elastic scattering to single pion production. The cross sectionused in this analysis is shown in Fig. 7.4.

7.4

Energy loss for muons

The neutrino-induced muon goes through in the rock with losing their energy. Its travel
length in the rock is calculated to judge whether the muon reaches the detector or not.

61

1

2

σ/E-38ν (10 /cm /GeV)

7.4 Energy loss for muons

0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
-1
10

1

10

10

2

10
Eν (GeV)3

Figure 7.4: Total neutrino-nucleon CC cross section. Dotted line shows the cross sectionof deep inelastic scattering, dashed line shows one of quasi-elastic scattering, dashed-dot
line shows one of single pion production and solid line shows the total cross section

The muon track length in water is calculated for judgment on muons stopping or going
through and on muons traveling over 7m.

7.4.1

Muon energy loss in rock

The muon energy loss in rock is calculated by Lohmann, including contributions frombremsstrahlung, direct pair production and photo-nuclear interactions. Fig. 7.5 shows
dE/dx in standard rock.
The traveling distance for which muons lose energy down to a threshold Eth is calculated from following equation.
R(Eµ , Eth ) =

Z Eth
Eµ

−dE
dE/dX

(7.17)Eth defined in this analysis is 1.6GeV and the range with E th = 1.6GeV is shown in Fig.
7.6

7.4.2

Muon energy loss in Water

The muon energy loss in water is calculated by Lohmann for muon energy larger than
1 GeV. So Lohmann‘s result is applied for energy loss larger than 1GeV, and ionizationenergy loss is applied for smaller than 1GeV. The muon range in water is shown in Fig.
7.7. The 7m track length corresponds to a muon energy of 1.6GeV.

dE/dx (GeV cm2/g)

7.4 Energy loss for muons

10

6

10

5

10

4

10

3

10

2

10

1
1

10

10

2

10

3

10

4

10

5

10
Eµ (GeV)62R(Eµ,Eth) (g/cm )Figure 7.5: dE/dx in standard rock10

6Eth=1.6GeV

10

5

10

4

10

3

10

2

10

10

2

10

3

10

4

10

5

10
Eµ (GeV)6

Figure 7.6: Range of muon in standard rock

62

63

2

Range (g/cm )7.5 Theoretical uncertainties

10

4

10

3

10

2

7m

1.6GeV1

10Eµ (GeV)Figure 7.7: Range of muon in water.

7.5

Theoretical uncertainties

In this section, the theoretical uncertainties for the expected upward-going muon flux are
described. The uncertainties of atmospheric neutrino flux and neutrino-nucleon interaction are main factor.

7.5.1

Atmospheric neutrino flux

The uncertainty of atmospheric neutrino flux comes from the primary-cosmic ray energy
dependence and affects to upward-going muon flux. Although the primary cosmic-ray flux
is well studied at low energy, there is large ambiguity in the determination of absolutevalue. The fitting error of the compilation is obtained to be ∼10% for the nucleon flux at
100GeV and ∼20% at 100 TeV. The uncertainty in the calculation of the upward-going
muon flux is estimated to be approximately ±20% above several GeV.
In addition, the dependencies on neutrino flux model was estimated by comparisonof Bartol model[13] with Honda model [15] and shown in Fig 7.8. The uncertainty ofabsolute flux from model dependence was estimated to be ∼ 10%.
Moreover, the zenith angle bin-by-bin uncertainty was estimated by comparison of
Bartol model and Honda model. Fig. 7.9 shows the zenith angle bin-by-bin error. The
ratio was normalized by the combination of the GRV94 parton distribution function andBartol neutrino flux model.

RatioΦν (GeV-1cm-2s-1sr-1)

7.5 Theoretical uncertainties

64

0.5
0.45

3

0.4
2.5
0.35
0.3

2

0.25
1.5
0.2
0.15

1

0.1
0.5
0.05
0

-1

-0.9

-0.8

-0.7

-0.6

-0.5

-0.4

-0.3

-0.2

0

-0.1
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

0.03

Bin-by-bin errror

Bin-by-bin errror

Figure 7.8: Left figure shows the zenith angle distribution of expected upward throughgoing muon flux and upward stopping muon flux. Right figure shows the zenith angle
distribution of the ratio of upward stopping muon to upward through-going muon flux.
Solid line shows Bartol model and dashed line shows Honda model.

0.02

0.03

0.02

0.01

0.01

0

0

-0.01

-0.01

-0.02

-0.02

-0.03

-0.03

-0.04

-1

-0.9

-0.8

-0.7

-0.6

-0.5

-0.4

-0.3

-0.2

-0.1
0
cos Θ

-0.04

-1

-0.9

-0.8

-0.7

-0.6

-0.5

-0.4

-0.3

-0.2

-0.1
0
cos Θ

Figure 7.9: The zenith angle bin-by-bin error estimated by model difference. Left figure
shows the error of upward stopping muon flux and upward through-going muon flux. Right
figure shows the error of the ratio of upward stopping muon flux to upward through-going
muon flux. These ratio are normalized by the combination of the GRV94 and Bartol
model.

7.6 Expected flux of upward going muons

7.5.2

65

The ratio of upward stopping muon flux to upward throughgoing muon flux

The ratio of upward stopping muon flux to upward through-going muon flux is useful forthe analysis. Because the neutrino flux is approximately defined as (absolute flux)×E −γνand the cross section of the deep inelastic scattering is canceled, the large uncertainty inthe neutrino flux normalization and the neutrino interaction cross section were canceled.
The uncertainty of atmospheric neutrino flux comes from the primary-cosmic ray energy dependence and affects to upward-going muon flux. To estimate the uncertainty of
the upward-going muon flux, the change of energy spectrum for atmospheric neutrino isapplied as follows,
dφν
dφν
→
× Eν±0.05(7.18)dEν
dEν
The uncertainty of the ratio of upward stopping muon flux to upward through-going muon
flux come from the uncertainty of the spectrum index is estimated to be +12.5% -11.4%.
Described as before, the uncertainty of the total charged current cross section only
consider quasi-elastic scattering and single pion production. When the cross section ischanged with ± 15%, the change of upward-going muon flux is estimated to be +4.2%-3.5%.

7.6

Expected flux of upward going muons

The upward-going muon flux are calculated by using Bartol’s atmospheric neutrino flux[13]and GRV94 neutrino-nucleon neutrino interaction[17]. The energy loss for muons are applied with Lohmann’s calculation [18]. The expected upward-going muon flux (E µ >1.6GeV) is
upward stopping muon flux
upward through-going muon flux

: 0.73 ± 0.16(theo.) (×10−13 cm−2 s−1 sr−1 ): 1.97 ± 0.44(theo.) (×10−13 cm−2 s−1 sr−1 )upstop/upthru ratio:+0.0490.37−0.04(theo.)

total upward-going muon flux:

2.70±0.47(theo.) (×10−13 cm−2 s−1 sr−1 )Chapter 8
Neutrino Oscillation
The neutrino oscillation is described in Section 1.1. In this chapter observed and expected
flux are applied to the analysis of neutrino oscillation. ν µ → ντ oscillation is consideredas the most probable process of neutrino oscillation analysis described in Section 1.1.
The three kinds of oscillation result are estimated, for upward through-going muon, theratio of upward stopping muon to upward through-going, upward stopping muon andthrough-going combined.

8.1

Method to examine neutrino oscillations

The basic oscillation examination is done by comparison between the observed result andthe expected. The χ 2 examination is applied to the oscillation analysis in this study. The
χ2 examination is done at the various oscillation parameter, ∆m 2 and sin22θ, and thebest fit parameter at minimum χ 2 is searched.
The basic χ2 definition for oscillation analysis is written in,
2

χ =

X

Nobs − (1 + )Nexp
σNerr

!2

+

X   2

σ

(8.1)

where Nobs is observed value, N exp is expected, N err is the sum of observed and expected
error including statistic and systematic error,  is normalization factor, and σ  is the errorof the normalization factor. We divided the data into 10 zenith angle bin for upward
through-going muon, 5 for the ratio of stopping muon to through-going muon, 15 forcombined upward stopping and through-going muon analysis. The range of the oscillation
parameter is 0 ≤ sin2 2θ ≤ 1 and 10−4 ≤ ∆m2 ≤ 1 examined, this region is divided into
1763 points. The oscillated expected flux are calculated from Equation 1.8 and Equation
1.9.

66

8.2

8.2

Effect on oscillation analysis from theoretical uncertainties

67

Effect on oscillation analysis from theoretical uncertainties

The observed and expected systematic error and normalization factor are described. The
most dominant systematic error comes from the absolute normalization of atmospheric
neutrino flux with error ∼ 20%.
The systematic uncertainties are shown in Table 8.1 and the normalization summary
for oscillation analysis is shown in Table 8.2.

(a)Theoretical uncertainties in flux calculation
typical ν flux normalization
choice of ν flux/PDF±20.0%±10.0%(b)Stop/thru theoretical uncertainties
primary spectrum index+12.5% -11.4%

cross section+4.2% -3.5%

choice of ν flux

±0.9%(c)Experimental sys. errors in through flux
7m track length cut
live time

±0.5%

±1.0%

(d)Experimental sys. errors in stopping flux
7m track length cut+4.9% -4.1%

live time±1.0%(e)Experimental stop/thru sys. errors
7m track length cut
live timestop/thru miss ID

±4.9%

±1.0%

±1.0%

Table 8.1: Uncertainties for oscillation analysis.

8.3 νµ → ντ oscillation

absolute normalization factor σα

22.4%stop/thru relative normalization factor σ β

14.2%

atmospheric K/π ratio normalization factor σ3%

68

Table 8.2: The systematic error of normalization factor for oscillation analysis.

νµ → ντ oscillation

8.3

The three kinds of oscillation analysis results are shown as follows.
The χ2 definition of through-going muon is as follows,χ2 =10(thru)

X
i=1

α
+
σα







dΦ i
dΦ i
) − (1 + α)(1 + 2ε(cosΘi + 0.5))( dΩ)theo (sin2 2θ, ∆m2)
( dΩ

2

q

i
i )2
(σstat)2 + (σsys

ε
+
σε


2

i

2


(8.2)


i

dΦ
is theoretical muon flux,
is observed muon flux in the i-th cosΘ, dΦ
where dΩdΩ theoiiσstat is experimental statistical error, σ sys is bin-by-bin uncorrelated theoretical and experimental systematic errors, α is absolute flux normalization factor with error σ α ,  isvertical/horizontal correlated normalization factor with error σ  . The zenith angle distribution of through-going muon flux and the confidence intervals for νµ → ντ oscillations isshown in Fig.8.1.

The χ2 definition of the ratio of stopping/through-going muon is as follows,5(ratio)

χ

2

=



i

X R


i=1

2
2
2
i
− Rtheo (sin 2θ, ∆m ) × (1 + β) 
 +
r


2
2
i
i
σRstat + σRsys

β
σβ

!2

(8.3)

where Ri is the observed flux ratio in the i-th cosθ, R itheo is the theoretical muon flux
ratio, and β is the relative normalization factor for the ratio of stopping muon flux tothrough-going flux with error σβ . The vertical/horizontal normalization factor is almost
canceled. The zenith angle distribution of the ratio and confidence interval are shown in
Fig.8.2.
The χ2 definition of combined stopping and through-going muon is as follows,

8.3 νµ → ντ oscillationνµ ↔ ντ oscillation(upthru 10bin)

flux (×10

-13

3

2

∆m (eV )Super-Kamiokande (Preliminary) statistical error only
Upward Through Going Muons (>7m) BG subtracted
1405.84 events/1268 livedays (Apr.1996 - Apr.2000)-13-2 -1 -1
Ave. Flux 1.70±0.05±0.02 (×10 cm s sr )Histogram:Expected flux (Bartol-GRV94)
null oscillation
α=-0.156 , ε=0.046
2
χ /ndf=18.7/10, Prob.=4.4%
2
2-3
sin 2θ=0.86 , ∆m =5.2×10
α=0.051, ε=-0.001
χ2/ndf=5.88/8, Prob.=66.1%

2

-2 -1

-1

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

4

69

1

Super-Kamiokande (preliminary)
Bartol-GRV94-Best Fit22
-3
sin 2θ=0.86 , ∆m = 5.2 ×10-1α=0.051,
ε=-0.00110
2
χ /ndf=5.88/8, Prob.=66.1%-Null Oscillation2
χ /ndf=18.7/10, Prob.=4.4%

-2

2

10

1

10

0

10

-3

68%CL
90%CL
99%CL
-4

-1

-0.8

-0.6

-0.4

-0.2

0
cosΘ

0

0.2

0.4

0.6

0.8

1

2

sin 2θ

Figure 8.1: Oscillation result for through-going muon flux. The left figure shows thezenith angle distribution and the right shows the confidence interval.

χ2 =10(thru)

X
i=1




5(stop)

+

X
i=1

α
+
σα


dΦ i
dΦ i
( dΩ
) − (1 + α)(1 + 2ε(cosΘi + 0.5))( dΩ)theo (sin2 2θ, ∆m2)




2

q

i
i )2
(σstat)2 + (σsys

2


dΦ i
( dΦ
)i − (1 + α)(1 + β)(1 + 2ε(cosΘi + 0.5))( dΩ)theo (sin22θ, ∆m2 )
dΩ

q

i
i )2
(σstat)2 + (σsys

β
+
σβ

!2

ε
+
σε


2

2


(8.4)

The zenith angle distribution for stopping and through-going muon flux is shown in Fig.8.3
and the confidence interval is shown in Fig.8.4.

8.3 νµ → ντ oscillation

70

2Super-Kamiokande (Preliminary) statistical error only
Ratio ( Upward Stop/Thru ) 0.241±0.016±0.013
Stop: 321.91 events/1247 livedays
Thru: 1405.84 events/1268 livedays (Apr.1996 - Apr.2000)
Histogram:Expected flux (Bartol-GRV94)
null oscillation
β=-0.335
2
χ /ndf=9.39/5, Prob.=9.4%
2
2-3
sin 2θ=1.0 , ∆m =2.9×10
β=-0.017
χ2/ndf=1.37/3, Prob.=71.2%

0.6

0.4

2

ratio

Zenith Angle Dist. of Ratio (Upward Stop/Thru)∆m (eV )νµ ↔ ντ oscillation(Ratio 5bin)

1

Super-Kamiokande (preliminary)
Bartol-GRV94-Best Fit22
-3
sin 2θ=1.0 , ∆m = 2.9 ×10-1
β= -0.017
102
χ /ndf=1.37/3, Prob.=71.2%-Null Oscillation2
χ /ndf=9.39/5, Prob.=9.4%

-2

10

-3

10

0.2

68%CL
90%CL
99%CL
-4

10

0

-1

-0.8

-0.6

-0.4

-0.2

0

0
cosΘ

0.2

0.4

0.6

cf. Best Fit(unphys)22-3
sin 2θ= 1.04, ∆m = 3.0 ×10

0.8

2

1

sin 2θ

Figure 8.2: Oscillation result for the stop/through ratio. The left figure shows the zenith
angle distribution of the ratio and the right shows the confidence interval of the ratio.

-1-13

flux (×10-13flux (×1014Super-Kamiokande (Preliminary) statistical error only
Upward Through Going Muons (>7m) BG subtracted
1405.84 events/1268 livedays (Apr.1996 - Apr.2000)
Ave. Flux 1.70±0.05±0.02 (×10-13cm-2s-1sr-1)
Histogram:Expected flux (Bartol-GRV94)
null oscillation
α=-0.164 , β=-0.339 , ε=0.052
χ2/ndf=30.2/15, Prob.=1.1%
2
2-3
sin 2θ=1.0 , ∆m =3.2×10
α=0.029 , β=-0.009 , ε=-0.004
2
χ /ndf=7.13/13, Prob.=89.5%-2 -1

Upward Stopping Muons (>7m) BG subtracted
321.91 events/1247 livedays (Apr.1996 - Apr.2000)-13-2 -1 -1
Ave. Flux 0.41±0.02±0.02 (×10 cm s sr )Histogram:Expected flux (Bartol-GRV94)
null oscillation
α=-0.164 , β=-0.339 , ε=0.052
2
χ /ndf=30.2/15, Prob.=1.1%
2
2-3
sin 2θ=1.0 , ∆m =3.2×10
α=0.029 , β=-0.009, ε=-0.004
2
χ /ndf=7.13/13, Prob.=89.5%

-2 -1

-1

cm s sr )Super-Kamiokande (Preliminary) statistical error only

Zenith Angle Dist. of Observed Upward Through Going Muon Flux

cm s sr )Zenith Angle Dist. of Observed Upward Stopping Muon Flux

1.5

3

2

0.5
1

0

-1

-0.8

-0.6

-0.4

-0.2

0
cosΘ

0

-1

-0.8

-0.6

-0.4

-0.2

0
cosΘ

Figure 8.3: Zenith angle distribution of upward muon flux. The left figure shows stopping
muon flux and the right shows through-going muon one.

8.3 νµ → ντ oscillation

71

1

2

2

∆m (eV )νµ ↔ ντ oscillation(upmu 15bin)-1

10Super-Kamiokande (preliminary)
Bartol-GRV94-Best Fit22
-3
sin 2θ=1.0 , ∆m = 3.2 ×10
α= 0.029 ,β= -0.009 , ε= -0.004
2
χ /ndf=7.13/13, Prob.=89.5%-Null Oscillation2
χ /ndf=30.24/15, Prob.=1.1%

-2

10

-3

10

68%CL
90%CL
99%CL
-4

10

0

0.2

0.4

cf. Best Fit(unphys)22-3
sin 2θ= 1.02, ∆m = 3.2 ×10

0.6

0.8

2

1

sin 2θ

Figure 8.4: Confidence interval for combined stopping muon and through-going muon.

Chapter 9
Approach with Monte Carlosimulation
The upward-going muon flux was calculated by analytical calculation method, but there
are no consider for angle of neutrino-nucleon interaction, the multiple scatter of muons,
neutral current effect and the precise detector response. The Monte Carlo approach isgood solution for these problems.

9.1

Monte Carlo method for upward-going muons• Atmospheric neutrino flux
The Bartol model[13] is applied for atmospheric neutrino flux same as analytical
calculation described in Section 7.2. The neutrino energy range is applied from
1.6GeV to 90TeV.
• Neutrino-nucleon cross section
The neutrino-nucleon interaction is applied as same as analytical calculation described in Section 7.3. So the cross section is calculated by using GRV94 parton
distribution function[17]. The only charged current interaction is generated out ofthe detector and both charged current interaction and neutral current are generated
in the detector.
• Area of the neutrino-nucleon interaction point
The area that the neutrino-nucleon interaction is generated is in the spherical shapewith 4km radius, the detector is located at center of the sphere.
• Detector simulator
The GEANT program in CERN program library is used for detector simulationcommonly used in high-energy experiments.

72

9.2 Comparison between MC and analytical calculation73• Muon energy loss
The muon energy loss is applied the Lohmann’s model[18], same as analytical calculation.
• Data selection
The data selection is done by true information, judgment between stop and thru,
and 7m track length cut and zenith angle cut are applied.

9.2

Comparison between MC and analytical calculation

The live time 19.5years Monte Carlo data was generated. The Monte Carlo result andanalytic calculation result are compared in the absolute flux and the zenith angle distribution of the flux.
The comparison result for upward stopping muon flux is shown in Fig. 9.1. The
left-side figure shows the zenith angle distribution for upward stopping muon flux, thecross shows Monte Carlo flux and the solid line shows analytic calculation flux. The absolute flux for upward stopping muon is 0.73 ± 0.02(stat.)(×10−13 cm−2 s−1 sr−1 ) for Monte
Carlo and 0.73(×10−13 cm−2 s−1 sr−1 ) for analytic calculation obtained, both flux are consistent. The right-side figure shows the zenith angle shape difference, upper figure shows
normalized flux distribution and lower figure shows the ratio of Monte Carlo to analytic
calculation, and the shape is found to be consistent.
The comparison result for upward through-going muon flux is shown in Fig. 9.2. The
absolute flux is 1.89±0.02(stat)(×10−13 cm−2 s−1 sr−1 ) for Monte Carlo and 1.97(×10−13 cm−2 s−1 sr−1 )for analytic calculation. The Monet Carlo absolute flux is smaller than analytic calculation with 4%. From The right-side and lower figure, the shape is found to be consistent.
The comparison result for upward stop/through ratio is shown in Fig. 9.3. The
absolute ratio is 0.39±0.01(stat.) for Monte Carlo and 0.37 for analytic calculation, theabsolute difference is 4% which come from the inconsistent of through-going muon result.
The shape is found to be consistent.

9.2 Comparison between MC and analytical calculation

19.5yr upmu MC

stopnormalized flux

-13

-2 -1

-1

cm s sr )1.5flux (×10

74

1

0.75
0.5

1

0.25
0

-1

-0.8

-0.6

-0.4

-0.2

0
cosz

-1

-0.8

-0.6

-0.4

-0.2

0
cosz

ratio

1.2

0.5

1.1
1
0.9

0

-1

-0.8

-0.6
-0.4
stop

-0.2

0
cosΘ

0.8

Figure 9.1: Upward-stopping muon flux

thru

normalized flux

3

3
2
1
0

2

-1

-0.8

-0.6

-0.4

-0.2

0
cosz

-1

-0.8

-0.6

-0.4

-0.2

0
cosz

1.2

ratio

flux (×10

-13

-2 -1

-1

cm s sr )19.5yr upmu MC

4

1.1

1

1
0.9

0

-1

-0.8

-0.6
-0.4
thru

-0.2

0
cosΘ

0.8

Figure 9.2: Upward through-going muon flux

9.2 Comparison between MC and analytical calculation

19.5yr upmu MC

75normalized flux

ratio

0.8

0.4
0.3
0.2

0.6

0.1

ratio

0

0.4

-1

-0.8

-0.6

-0.4

-0.2

0
cosz

-1

-0.8

-0.6

-0.4

-0.2

0
cosz

ratio

1.2
1.1

0.2

1
0.9

0

-1

-0.8

-0.6
-0.4
ratio

-0.2

0
cosΘ

0.8

Figure 9.3: The ratio of upward stopping muon flux to upward through-going muon flux

Chapter 10
Atmospheric ν/ν̄
10.1

Method to estimate atmospheric ν/ν̄The upward-going muons detected in Super-Kamiokande is generated by simple processof neutrino-nucleon charged current interaction as follows,ν + N → µ− + X
ν̄ + N → µ+ + X(10.1)(10.2)

where N is nucleon and X is any particle. Then the relation between atmospheric ν µ /ν̄µand the charge ratio of neutrino induced upward stopping muons is as follows,φνµσν̄N Nµ−=φν̄µσνN Nµ+(10.3)

where φν ,φν¯µ is the atmospheric neutrino flux for ν, ν̄ respectively and σ νN , σν̄N are crosssection of νN, ν̄N in charged current interaction. N µ− and Nµ+ are number of µ− and µ+ ,respectively. We compare experimental data with theoretical value for the atmospheric
neutrino and anti-neutrino ratio as follows,(φν /φν̄ )data(Nµ− /Nµ+ )data=(φν /φν̄ )theo.
(Nµ− /Nµ+ )theo.

(10.4)The theoretical atmospheric neutrino ratio is estimated by using analytical calculation as
follows,RNµ−φν (Eν ) × D × σνN (Eν ) × dEν
R=(10.5)φν̄ (Eν̄ ) × D × σν̄N (Eν̄ ) × dEν̄Nµ+where D is nucleon density of rock and Eν is neutrino energy.

10.2

Charge separation of muons

Although the charged particles is usually separated by magnetic field in accelerator experiments, directly separation of charged particle is impossible for the Super-Kamiokande.
76

10.2

Charge separation of muons77

In this study, charge difference is observed by using lifetime difference of muons inwater. The lifetime for muons in Super-Kamiokande is obtained by observation the timedifference between parent muon and decay electron. Charge ratio for upward going muons
is equivalent to the ratio of the number of µ + to µ− after correction of nuclear absorption
for µ− .
In vacuum, both µ+ and µ− have same lifetime, 2.2µsec [19]. However, µ + and µ− havedifferent lifetime in water because of only µ − captured by medium atom(µ − +p → n+νµ ).
Then µ− has shorter lifetime than µ+ [20]. The lifetime of each muons in water is as follows,τµ− = 1.17954 ± 0.0020µsec
τµ+ = 2.19703 ± 0.00004µsec
Therefore the muon counts of the mixture of two muons which have different lifetimein a water. The number of the particle with lifetime τ is shown as equation(10.6).
t
N (t) = N0 exp −τ

(10.6)

where N0 is number of particle at t=0, and N (t) at t=t.
The two kind of method for estimation Nµ− /Nµ+ are available, the lifetime fitting
method and lifetime ratio method. The former lifetime fitting method is using the decay
curve for time difference between muon and decay electron.
At first decay rate of the particle −dN (t)/dt is expressed as following equation(10.6).
N0t
dN (t)
=
exp −
−
dt
τ
τ




(10.7)In the next step, the number of count(N count) during ∆t is as follows,!

dN (t)
Ncount (t ∼ t + ∆t) =
−
dt
dt
t





t
∆t
= N0 1 − exp −
exp −
τ
τ
Z t+∆t(10.8)The number of count is estimated separately by equation(10.8) for µ − ,µ+ . The total
number of count(Ntotal ) is shown as following equation10.9.
"

!#

!

∆t
t
Ncount (t ∼ t + ∆t) = N µ+ 1 − exp −
exp −
τµ+
τµ+
!#
!
"
t
∆t
exp −
+Nµ− 1 − exp −τµ−τµ−(10.9)The tuning parameters in the equation 10.9 are Nµ− and Nµ+ .
In order to deduce the true charge ratio of muons observed in the SK detector, weshould taken into the µ− nucleon capture in water. There is no electron emission when
nuclear µ− capture occurred. Λ c (the nucleon capture ratio) is defined for oxygen as
follows[20],Λc = 0.184 ± 0.001

10.3 Event selection

78

Finally, the charge ratio of muons(R(µ+ /µ− )) turns out to be,R(µ+ /µ− ) =Nµ+Nµ− /(1 − Λc )(10.10)Since only tagged decay electron events is used in the lifetime fitting method, the large
amount of events are required and it is not suitable for this analysis. Therefore, lattermethod, lifetime ratio method is adopted.
The lifetime ratio method uses all event detected stopping muons. Although the
lifetime ratio method is basically same as lifetime fitting method, the charge ratio is
estimated by integrating the Eqn. 10.9 and Eqn. 10.10.
At first, we defined the ratio of number of µ + and µ− as Nµ+ : Nµ− = 1 : X. The
ratio of the number of total muons to electron tagged events which lifetime is longer thandefinite time (N e−tag /Ntotalµ ) is shown as follows,µ+ (exp(−T /τµ+ ) + Xexp(−T /τµ− )(1 − Λc ))Ne−tag=Ntotalµ1+X(10.11)

where µ+ is the detection efficiency for µ + and T is definite time. The ratio N e−tag /Ntotalµ
is obtained experimentally.

10.3

Event selection

10.3.1

Parent upward stopping muonsContamination of cosmic-ray muons
The contamination of cosmic-ray is described in Section 5.5 The contamination due tothe cosmic-ray muons exists only between −0.1 ≤ cosΘ ≤ 0 (Θ:vertical angle) in zenith
angle direction. Therefore events in the region 60 o ≤ φ(azimuth angle) ≤ 310◦ and−0.1 ≤ cosΘ(Θ : vertical angle) ≤ 0 are removed from the data set.
Fiducial volume cut
Fiducial volume cut is applied on the muon events. This cut requires that stopping
position of muons should be away more than 2m from detector wall as shown in Fig. 10.1.
This cut is useful for rejection γ-ray events which is generated frequency available near
the wall (Fig. 10.1) and affects destructively the detection efficiency for decay electron in
the analysis. Although almost noises near the wall are eliminated by time difference cutby coincidence between parent muon and decay electron, we applied this cut because ofwell known energy resolution for the decay electron in fiducial volume.

10.3.2

Electron from decayed muon

In this section, the direction and vertex reconstruction for µ decay electron is described.
Figure 10.2 is the typical event of µ decay electron.

10.3 Event selection

79

Figure 10.1: The vertex distribution for low energy events. The left-side plot is Z distribution(+18.1m and -18.1m show the top and bottom PMT) and the right-side plot isR(radial direction). The large amount of events are found near inner-detector wall. The
fiducial volume away more than 2m from detector wall is 22.5kton.

Vertex reconstruction
At first, the hit channel is selected for vertex reconstruction as below. The time distribution of hit PMTs is shown in Fig. 10.3. The peak around 900nsec is the signal of electron
event and noise signals by dark current of PMTs are found after 1200nsec. The noise
signal deteriorates the vertex reconstruction. Accordingly it is eliminated as follows.
1. The 200nsec timing window is determined so as to include maximum hit PMTs. In
the Figure10.3, this window is between t 2 and t3 .
2. The number of background hit in the timing window is estimated using following
equation.

Nbackground = (t2 − t3)

Nhit (t1 : t2) + Nhit (t3 : t4)(t2 − t1) + (t4 − t3 )(10.12)

3. 200nsec window is divided to 11 small window and significance for each small window
is calculated as follows,

10.3 Event selection

80

NUM
6
RUN
2592
EVENT
653409DATE 96-Aug-23
TIME
1:49: 5
TOT PE:
523.4
MAX PE:
19.9
NMHIT :
304
ANT-PE:
184.6
ANT-MX:
33.8
NMHITA:
101

RunMODE:NORMALTRG ID :00000011
T diff.: 1.78
us
:0.178E-02ms
FSCC:
20000
TDC0:
681.5
Q thr. :0.0
BAD ch.: masked
SUB EV : 1/ 1

Figure 10.2: Typical µ decay electron event. solid line shows the result of vertex anddirection reconstruction.

signif icance = q

Nsignal
Nbackground(10.13)where Nsignal is equivalence to N hit − Nbackground .
4. Select the small window which has maximum significance.
5. The other small window which the significance is larger than 80% of maximum
significance is selected and use for vertex reconstruction.
Next, reconstruct the vertex point using selected channel by the grid search method
described as Section 5.2.1. The function for vertex reconstruction is Eqn. 10.14 andsearch minimum point for T 2.

2

T =

N
hit
X

t2resolution,i(10.14)i=1

where tresolution,i is the time which difference time between detected time at PMT andtime of flight of photon, tresolution,i is defined as follows,

10.3 Event selection

t1

t2

t3

Figure 10.3: Timing distribution of electron eventt4

8110.3 Event selection

tresolution,i = t0i − (n/c) ×q(x − xi)2 + (y − yi)2 + (z − zi )2 − tc

t0i

: hit time at i-th PMTn: refraction index (n = 1.34 in water)c: velocity of photon in vacuum82(10.15)(x, y, z) : vertex point of decay electron(xi , yi, zi )tc

: location of i-th PMT: average of ti

Goodness is defined by the time resolution of PMT as follows,
t2resolution,i11goodness = 1 × Σ 2 exp −
Σ σ2
σi
2σi2
"

i

#

(10.16)

where σi is time resolution of PMT. The goodness takes the value between 0 and 1. If
tresolution,i is 0, goodness takes 1. The method of grid search is same as muons case anddetermines the point where goodness takes maximum value.
Table10.1 shows vertex reconstruction resolution described in Section 5.2.1. The vertex
position and resolution is estimated from average and σ by Gaussian fitting as figure10.4.
Then σr is estimated from difference distribution(figure10.4) as the region including 68%
of all.
Position (x,y,z)

4x

4y

4z

σx

σy

σz

σr

(35.3,-70.7,-1200)

-0.3

-2.7

-3.0

47.98

49.28

47.66

115.8

(35.3,-70.7,0)

0.7

-0.5

-1.9

47.11

49.23

52.80

114.2

(35.3,-70.7,1200)

-1.0

-0.7

-1.0

46.87

48.17

47.88

113.6

(35.3,-70.7,1600)

-0.6

-2.8

-10.0

46.98

46.40

44.67

111.6

(35.3,-1201,-1200)

-3.4

7.0

-7.0

46.91

47.65

47.40

110.7

(35.3,-1201,0)

-1.1

16.0

-1.1

42.70

43.16

50.08

113.0

(35.3,-1201,1200)

0.6

10.0

2.0

43.74

44.92

48.74

109.4

Table 10.1: Vertex reconstructed position difference and resolution by Ni calibration

Energy spectrum for electron
Characteristics for muon decay is the maximum 50MeV in energy spectrum. The determination of energy is on the different basis between low energy event like electron andhigh energy event like muon. The energy in high energy event is determined by using10.3 Event selection

83

Figure 10.4: The resolution of vertex reconstruction for electron
total charge of hit PMTs. On the other hand, in the low energy case, determined by the
number of hit PMTs. Since the amplifier of PMT is not stable in low energy, definitionof number of hit PMT is better than definition of charge.
In order to reject the dark noise events of PMTs, the corrected number of hit N 50 isdefined which applied corrections of tube weights taking into account of the attenuation,geometry, and charge corrections. The N 50 has a good energy scale in low energy as
characteristic value of the event.
µ decay electron events
Backgrounds for decay electron are radioactivity in water and γ-ray from rock, electric
noise. Following event selection was applied in order to eliminate these background.
µ decay electron Event selection
These electron events are emitted after parent muons released that’s energy and stopped.
Since the lifetime of muon in vacuum is 2.2µsec, events which have time difference between
muon and following event is shorter than 30µsec was collected.
Goodness cut
The definition of goodness was described elsewhere and is used for inspection of quality forvertex reconstruction. This definition is also used for decay electron. For decay electron
selection, goodness is used for elimination of electrical after-pulse due to parent muon.
Event with goodness > 0.5 is taken into data set.

10.3 Event selection

84

Energy cut
Fig. 10.5 shows energy distribution for decay electron of muon. MC and experimental datais consistent with the data around 5MeV except very low energy region. The difference
comes from gamma-ray by radioactivity element in the water and µ − nucleon capture by
nucleus. Stopping µ− is sometimes captured by Oxygen atom in the water and excited to16
N. When excited 16 N goes to ground state, emitted gamma-ray. The maximum energyof emitted gamma-ray is 10.8MeV. Then N 50 > 60 was applied effectively for rejection
these noise events.

Figure 10.5: The Michel spectrum, solid line shows MC and cross shows data

Time difference cut
Since muon decay electron is successive phenomena after stopping muon, there is problemof trigger gate. The trigger time width in Super-Kamiokande detector is 800nsec. The
gate type is classified for three groups because of the relation of muon and electron andgate type as Fig. 10.6. Only sub-gate events of which detection efficiency are highest, areused for this study. The types of trigger gate are explained as follows,
1. in-gate event
The time difference between parent muon and decayed electron is very small. It isdifficult for detection decay electron because of signal tail of parent muon.

10.4 Charge ratio of upward stopping muon85

2. bye-bye event
The event decay electron signal is divided two gate.
3. sub-gate event
After parent muon gate is finished, electron signal appears in the next gate.

mu
e
mu

in-gate eventemu

bye-bye eventesub-gate event

trigger gate
Figure 10.6: Schematic view of µ-e decay events for gate pulse

Detection efficiency for µ + decay e
The detection efficiency for µ + decay e is estimated by Monte Carlo simulation. The
positive electron with Michelle energy spectrum is generated and applied the detector
simulator, the figure is shown in Fig. 10.9. The goodness cut and low energy cut are
applied, the detection efficiency for µ + decay e is estimated to be 98.7 ± 0.04stat.%.

10.4

Charge ratio of upward stopping muon

The 224 upward stopping muons are observed as parent muons and 114 decay electron
which time difference between parent muons and decay electron is larger than 1.2µsec isobserved. The ratio of number of electron tagged event to total muon is,114
Ne−tag== 0.51 ± 0.03(stat.)
Ntotal−µ224(10.17)

10.5 Expected charge ratio

200
175
150
125
100

86

electron

75
50
25
0
600

800

1000

1200

1400

1600

1800

nsec

Figure 10.7: Timing distribution for in-gate event. Both parent muons and decay electron
is available in the gate
Since equation(10.11), Equation(10.17), and the detection efficiency for µ + decay electron described as 10.3.2 are taken into account. The charge ratio for upward stoppingmuons(Eµ ≥ 1.4GeV) is obtained to be
Nµ−
Nµ+

+0.78
= 0.57−0.39(stat.)(10.18)Figure 10.10 shows the relation between Ne-tag/Ntotalµ and Nµ+ /Nµ− .

10.5

Expected charge ratio

The expected charge ratio is estimated by the analytical calculation described as Chapter
7. The expected upward stopping muon flux from atmospheric ν µ and ν¯µ are 0.50 and 0.25
respectively. So the ratio of upward stopping muon flux from ν µ to ν¯µ is 1.98 obtained.

10.6

Systematic uncertainties

10.6.1The systematic error for absolute energy determination

The energy distribution of µ decay electron is shown in Fig. 10.11. The cross shows data
and the slid line shows Monte Carlo data, the difference is found in mean energy with

10.6

Systematic uncertainties

87

700
600
500
400
300
200
100
0
600

800

1000

20
17.5
15
12.5
10
7.5
5
2.5
0

1200

1400

1600

nsec

1800

electron

600

800

1000

1200

1400

1600

nsec

1800

Figure 10.8: Timing distribution for sub-gate event. Upper figure shows parent muonsand lower figure shows decay electron

10.6

Systematic uncertainties

88

20000

10000

0

0

100

200

300

400

nhit50Figure 10.9: The Michelle spectrum of µ + decay e+ generated by Monte Carlo simulation

2.5%. The difference of the mean energy affects to observed atmospheric ν µ /ν¯µ ratio with
13.2%.
Detection efficiency for µ + decay electron
The detection efficiency for µ + decay electron is estimated to be 98.7 ± 0.04stat.% inSection 10.3.2, the statistic error for detection efficiency affects to observed atmosphericνµ /ν¯µ ratio with 0.6%.
The uncertainty of neutrino-nucleon interaction
The total neutrino-nucleon charged current cross section is shown in Fig. 10.12. The
cross points show the various experimental results and the blue line shows the analytical
calculation described in Section 7.3. The ν µ and ν¯µ data are independently fitted by analytical calculation ,respectively. The difference between expected ratio of σ νN to σνN¯ andnormalized theoretical ratio is estimated to be 8%. The effect for observed atmosphericνµ /ν¯µ ratio is 8% obtained.

Ne-tag/N_total

10.7

Obtained atmospheric ν/ν̄

89

0.8

0.7

0.6

0.5

0.4
0

1

2

3

4
Nµ-/Nµ+Figure 10.10: The Ne-tag /Ntotalµ versus Nµ+ /Nµ−

10.7

Obtained atmospheric ν/ν̄

The ratio of observed atmospheric ν/ν̄ to expected one is estimated as follows,(ν/ν̄)obs.
+0.39= 0.29−0.20(stat.) ± 0.04(syst.)(ν/ν̄)theo.
+1.56−0.16 (stat.2σ)(10.19)

10.7

350

Obtained atmospheric ν/ν̄

data

<p>=39.01 MeV/cMC<p>=37.99 MeV/c

Number of events

300
250
200
150
100
50
0

0 10 20 30 40 50 60 70 80 90
Momemtum (MeV/c)

Figure 10.11: The energy distribution of µ decay electron

Figure 10.12: Neutrino-nucleon charged current cross section

90Chapter 11
Result and discussions
The live time 1247 days for upward stopping muons and 1268 days for upward throughgoing muons data were studied, the 345 upward stopping muon events and 1416 upward
through-going muon events were detected.

11.1

Observed upward-going muon flux

The observed upward-going muon flux was estimated from upward-going events. The observed upward stopping muon flux was 0.41±0.02(stat.)±0.02(syst.) (×10−13 cm−2 s−1 sr−1 ),the through-going muon flux was 1.70 ± 0.05(stat.) ± 0.02(syst.) (×10−13 cm−2 s−1 sr−1 ), thestop/through ratio was 0.24 ± 0.02(stat.) ± 0.01(syst.), total upward-going muon flux was
2.11 ± 0.05(stat.) ± 0.03(syst.) (×10−13 cm−2 s−1 sr−1 ) obtained.

11.2

Expected upward-going muon flux

The expected upward-going muon flux was estimated by analytical calculation on thebasis of the Bartol atmospheric flux model, GRV94 parton distribution function for deep
inelastic scatter, energy loss of muons calculated by Lohmann. The expected upward
stopping muon flux was 0.73 ± 0.16(theo.) (×10−13 cm−2 s−1 sr−1 ), through-going muon
flux was 1.97 ± 0.44(theo.) (×10−13 cm−2 s−1 sr−1 ), the ratio of stopping to through-going
muon flux was 0.37 ± 0.05(theo.), the total upward-going muon flux was 2.70 ± 0.47(theo.)(×10−13 cm−2 s−1 sr−1 ) obtained.

11.3

Oscillation examination

The oscillation examination was applied by comparison between the observed upwardgoing muon flux and the expected upward-going muon flux. The result of the oscillation
examination is summarized in Table 11.1, where α, β, γ are described in Chapter8.

9111.4 Monte Carlo approachsin2 θ

∆m2

χ2/d.o.fαβProb.(%)

92

stop/through ratio 5bin
0.00–

9.4/ 5

–

-0.3

–

9.4%

1.04

3.0 × 10−3

1.3/ 3

–

0.0

–

71.8%

2.9 × 10−3

1.4/ 3

–

0.0

–

71.2%

0.00

–

30.2/15

-0.2

-0.3

0.05

1.1%

1.02

3.2 × 10−3

7.1/13

0.03

0.002

-0.01

89.6%

3.2 × 10−3

7.1/13

0.03

-0.01

-0.004

89.5%

0.00

–

18.7/10

-0.16

–

0.05

4.4%

0.86

5.2 × 10−3

5.9/ 80.05–-0.001

66.1%

1.00

stop + through combined 15bin

1.00

through 10bin

Table 11.1: The summary of oscillation analysis

11.4

Monte Carlo approach

The another approach for estimation upward going muon flux applied Monte Carlo method.
The Bartol atmospheric neutrino flux model, GRV94 parton distribution function for deep
inelastic scatter calculation, the energy loss of muon calculated by Lohmann were applied
and the GEANT program in CERN library simulated the detector response. The upwardgoing muon fluxes were estimated by Monte Carlo method, the upward stopping muon
flux was 0.73±0.03(stat.) (×10−13 cm−2 s−1 sr−1 ), the upward through-going muon flux was
1.98 ± 0.02(stat.) (×10−13 cm−2 s−1 sr−1 ), the ratio of stopping muon flux to through-going
was 0.39 ± 0.01(stat.) obtained.

11.5

Atmospheric νµ/ν¯µ ratio

The experimental atmospheric ν µ /ν¯µ ratio was estimated by using the muon lifetime difference in the water.
The experimental charge ratio of negative muon to positive muon(N µ− /Nµ+ ) was ob+0.78
tained to be 0.57−0.39(stat.). and the ratio of observed νµ /ν¯µ to expected from analytical+0.39
calculation was estimated to be 0.29−0.20(stat.) ± 0.04(syst.).

Chapter 12Conclusion
The atmospheric neutrino flux has been measured using about 1250days exposure at theSuper-Kamiokande detector. 345 events for upward stopping muons and 1416 events for
upward through-going muons were detected. The experimental upward-going muon flux
was obtained on the basis of the observed muons. The expected flux without neutrino
oscillation was estimated by analytical calculation.
The oscillation examination was done by comparison between observed upward-going
muon flux and expected one. The ν µ → ντ vacuum oscillation is considered as the mostprobable process. The νµ → ντ oscillation hypothesis with sin 22θ ≥ 0.65, 1.9 × 10−3 ≤∆m2 ≤ 7.2 × 10−3 is consistent with the observed zenith angle shape at 90% confidence
level. We obtained the probability of 89.5% at a best fit parameters at (sin 2 2θ, ∆m2) =(1.0, 3.2 × 10−3 ) for the νµ → ντ oscillation case. On the other hand, we obtained theprobability of 1.1% for the null oscillation case. The Monte Carlo approach was applied
for the realistic expected flux estimation and about 20years upward-going muon eventswere generated. The upward-going muon flux by Monte Carlo was compared with analytical calculation. Although the distribution of the zenith angle was consistent, the absolutevalue of the flux for through-going muon was found 4% difference.
The observed atmospheric νµ /ν¯µ ratio was measured and the expected ratio was esti+0.39
mated by analytical calculation. The ratio of observed to expected was 0.29 −0.29(stat.) ±0.04(syst.) obtained.

93

bibliography[1] K.S.Hirata et al.,Phys. Lett. B205(1988)416[2] M.Takita, Ph.D.thesis, Fac. of Science, Univ. of Tokyo(1989),ICR-Report-186-89-3.
[3] D.Casper et al.,Phys. Rev. Lett. 66(1991)2561[4] R,Beckser-Szendy et al., Phys.Rev. D46(1992)3720[5] T.Kafka Talk at the TAUP 93 Conf., Gran Sasso,Sep.1993[6] F.Ronga et al., Upward-going muons and MACRO, talk of neutrino ’98(1998)[7] Y.Fukuda et al., Phys. Lett. B335(1994)237[8] Ch.Berger et al., Phys.Lett. B227(1989)489.
[9] N.Aglietta et al., Europhys.Lett. 8(1989)611.
[10] Z.Maki, M.Nakagawa and S.Sakata, Prog.Theor.Phys. 28(1962)870.
[11] B.Pontecorvo, Sov.Phys.JETP 26(1968)984.
[12] M.Ambrosio et al. INFN-AE-97-55,Nov. 1997[13] V.Agrawal et al,. Phys.Rev. D53, 1314-1323, 1996
[14] L.V.Volkova, Yad.Fiz. 31(1980)1510 [Sov.J.Nucl.Phys. 31(1980)784].
[15] M.Honda and T.Kajita, K.Kasahara, S,Midorikawa, Phys.Rev. D52(1995)4985.
[16] A.V.Butkevitch, L.G.Dedenko and I.M.Zhelenznykh, Yad.Fiz. 50(1989)90 [Sov.J.Nucl.Phys. 50(1989)142].
[17] M.Glück, E.Reya and A.Vogt, Z.Phys. C67, 433-447, 1995
[18] W.Lohmann, R.Kopp and R.Voss, CERN Report No.85-03(1985) (unpublished).
[19] Review of Particle Properties, Particle Data Group, Phys.Rev., D50(1994)14[20] T.Suzuki, D.F.Measday and J.P.Roalsvig, Phys.Rev., C35(1987)2212[21] A. van der Schaaf et al., Nucl.Phys.A408(1983)573
94