Measurement of the charge ratio of cosmic-ray muons in Super-Kamiokande
素粒⼦物理学研究室 41502208 北川芙⻄⾳

宇宙線ミューオンの電荷比𝑅(𝜇! /𝜇" )は、1 次宇宙線と大気原子核の衝突反応の理解、中間子の生成比の見積もりおよび大気ニュートリノフラックスの理論計算の評価などにおいて重要な物理量となる。ミューオンは、１次宇宙線の組成の主成分を陽子が占めることから、𝜇! がより多く生成されると考えられており、これまでの過去の実験から、300 GeV までのエネルギー範囲において電荷比の平均値は約 1.27 と得られている。一方で、TeV 以上の高エネルギー領域にかけては、１次宇宙線の組成と中間子成分の変化に応じて、電荷比は異なる振る舞いを取ることが予測されている。このため、地下深部にまで到達する高エネルギーミューオンの検出が可能な地下実験において、電荷比の精密測定が試みられている。
本研究では、スーパーカミオカンデ(SK)検出器を用いて 1.3 TeV のエネルギーをもつ宇宙線ミューオンについて電荷比の測定を行った。まず信号事象となる検出器内に静止したミューオンとその崩壊電子を適切に選別するための識別条件を検出器シミュレーションにより定めた。また、媒質中では𝜇" が原子核による吸収を受けるため、見かけの平均寿命が𝜇! と比較して短くなる。そこで、𝜇 ± の水中での寿命の違いを利用して、𝜇 ± の計測数を見積もり、電荷比の算出を行った(図 1)。本解析結果より、2008
年から 2018 年にかけて取得された SK の平均電荷比として、𝑅(𝜇! /𝜇" ) = 1.42 ± 0.02 (統計誤差)を得た(図 2)。この結果は、300 GeV 以下の領域で測定された平均電荷比と比べて大きい。このことは、π/K 中間子による寄与を仮定した理論予測において、TeV 領域におけるミューオン生成がπ中間子の

107
SK4 muon decay-e
SK4 2017 (280.9 days)
Fit resultµ plus 360869.4 ± 6155.1µ minus 213995.8 ± 7372.6

6

10

105
104

2017年のデータ% &! /&" = ). +, ± .. ./(12321. )103

Charge ratio

Number of events/0.1 µsec

みだけでなく、K 中間子からの崩壊にも依存していることを示している。
1.65
1.60
1.55
1.50
1.45
1.40
1.35

2

10

1.30
10

SK-IV yearly data

1.25

1

SK-IV combined

1.20

10-1
0

5

10

15

20
25
30
Time difference [µsec]

図１：SK で取得された𝜇 − 𝑒崩壊事象の時間情報(2017 年のデータ)

1.15

Kamiokande-II

2010

2012

2014

2016

2018

Year

図 2：2008 年から 2018 年までに測定された電荷⽐。
帯で⽰されている領域は平均値。

[業績⼀覧]
・⽇本物理学会(2 回)：2021 年 3 ⽉ 15 ⽇(⼝頭発表)、 2021 年 9 ⽉ 14 ⽇(⼝頭発表)
・新学術「ニュートリノで拓く素粒⼦と宇宙」研究会：2020 年 12 ⽉ 21 ⽇(ポスター発表)
・新学術「地下宇宙」研究会(2 回)：2021 年 5 ⽉ 20 ⽇(ポスター発表)、2021 年 11 ⽉ 26 ⽇(⼝頭発表)
・International Cosmic Ray Conference 2021：2021 年 7 ⽉ 14 ⽇(ポスター発表)
・Proceedings of the 37th International Cosmic Ray Conference 2021 (出版)Measurement of the charge ratio of cosmic-ray
muons in Super-Kamiokande

Hussain Kitagawa
March 2022

Graduate School of Natural Science and Technology(Master’s Course)
Okayama University

Abstract
Cosmic-rays are the nuclei originating from outside the Earth, incident on the high altitude atmosphere. Since their discovery of cosmic-rays in 1910s, their characteristics have been studied
from various aspects of astrophysics and particle physics: the original source and mechanismof their acceleration in the galaxy, components of the incoming cosmic-rays, and the relation
between hadronic interaction and secondary productions in the atmosphere. Cosmic-rays can
be classified into two categories: primary and secondary cosmic-rays. Secondary cosmic-rays
are produced via the interactions of primary cosmic particles with air nuclei at the frontierof the atmosphere. The interaction products, pions, and kaons composing the showers of secondary cosmic-rays mostly decay into penetrating components, muons, and neutrinos whichreaching under the Earth’s surface. These observable particles carry the history of the cascade
production of secondaries directly and enable probing the mechanism of secondary production.
The muon charge ratio, defined as the ratio of the number of positive to negative charge atmospheric muons, is one of the important parameters to understand the hadronic interactions. It
reflects the production ratio of charged pions and kaons and composition of cosmic primaries.
Moreover, since neutrinos are produced in association with muons, the behavior of the muon
charge ratio strongly relates to the energy distributions of atmospheric neutrinos. Thus, themuon charge ratio is necessary for the studies of secondary production and an improved calculation of the atmospheric neutrino fluxes. The muon charge ratio has been widely measured inseveral experiments on surface and underground in diverse energy ranges. In the previous measurements by CMS [1] and OPERA [2] below and above 1 TeV/c, respectively, the charge ratio
was approximately constant at R(µ+ /µ− ) ≈ 1.27 in the momentum range of 10 − 300 GeV/cand increased above this range [3], [4]. In this thesis, the measurement of the muon charge ratiowith the minimum surface muon energy of 1.3 TeV in Super-Kamiokande detector is presented.
Super-Kamiokande is a large water Cherenkov detector located in the Kamioka mine, Gifu
Prefecture, Japan at a depth of 1,000 m (2,700 m of water equivalent (m.w.e.)). The muon
charge increased from a TeV energy region and the precise measurement helped investigate theeffect of kaon contribution.
At first, the production of cosmic-ray muons and the muon charge ratio are introduced in Chapter 1. Chapter 2 provides the fundamental information of Super-Kamiokande detector with its
detection principle via Cherenkov radiation and data acquisition system. The method of event
reconstruction for muons and decay electrons used in the analysis is explained in Chapter 3.
Chapter 4 describes the simulation of muons and decay electrons including detector structure
for a detailed understanding of the detector response and for the determination of the selection criteria. The evaluation and optimization of selection cuts for the control sample and the
distribution of selected muon decay events are shown in Chapter 5. Chapter 6 reports themeasurement of the charge ratio and the discussion of results. Finally the summary is provided
in Chapter 7.
Keywords: Cosmic-ray muons; Muon decay; Charge ratioiAcknowledgement
I would first like to express my gratitude and appreciation for my research supervisor Koshiosan for his enormous support. He allowed me to join the Super-Kamiokande group and mademy research life a wonderful time. He actively provided me the opportunity to join the meetings
on-site and connected me with other researchers to proceed with my research project smoothly.
Without his precious support, it was impossible to conduct my research. Further, I would liketo acknowledge my co-supervisor, Nakano-san from B4, for his valuable guidance throughout
my research. He provided me with the methods and tips to lead me in the right direction. His
suggestions and feedback brought my thoughts to a higher level. I would also like to appreciate
inviting me to the restaurant after the conference. In addition, I would like to appreciate thesupport from the members of Kobe University. During my stay at Kobe, I was given a valuable
opinion about the simulation from Kurashige-san as a developer. He taught me the details of
the simulation tools and the surrounding physics for my further understanding. I would also liketo express my gratitude to an OB of the particle group at Kobe University, Shiozawa-san. My
research base consists of his tremendous works, and without this support, the research has neverbeen accomplished. I would also thank all the members of Kobe university who helped with myresearch. Moreover, I would like to thank the laboratory members. The discussions with Hinosan of specially appointed researcher, Harada-san of D2, Sakai-san of D1, and conversation with
Nakanishi-san of M1, Tano-kun of B4, Tada-kun of B4, Shiraishi-kun of B4, and supports from
Yamamoto-san of a secretary were very helpful and made my research life fulfilling. Finally, I
would like to thank my family for their kind support.

ii

Table of Contents
Abstracti

Acknowledgmentii

Table of Contentsiii

List of Figuresv

List of Tablesviii

1 Introduction
1.1 Cosmic-ray muons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2 Primary and secondary cosmic-rays . . . . . . . . . . . . . . . . . . . . . . . . .
1.2.1 Primary interaction in the atmosphere . . . . . . . . . . . . . . . . . . .
1.2.2 Production of secondary particles . . . . . . . . . . . . . . . . . . . . . .
1.2.3 Production of atmospheric muons . . . . . . . . . . . . . . . . . . . . . .
1.3 Muon spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.4 Muon charge ratio . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.5 Fraction of atmospheric neutrinos . . . . . . . . . . . . . . . . . . . . . . . . . .
1.6 Previous cosmic-ray muon charge ratio measurements . . . . . . . . . . . . . . .

1
1
1
2
3
4
5
6
8
9

2 Super-Kamiokande detector
2.1 Overview of Super-Kamiokande detector . . . . . . . . . . . . . . . . . . . . . .
2.2 Detector structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2.1 Inner detector and outer detector . . . . . . . . . . . . . . . . . . . . . .
2.2.2 20-inch Photo Multiplier Tube (PMT) . . . . . . . . . . . . . . . . . . .
2.2.3 Helmholtz coils . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3 Detection principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.4 Water and air circulation system . . . . . . . . . . . . . . . . . . . . . . . . . .
2.4.1 Water purification system . . . . . . . . . . . . . . . . . . . . . . . . . .
2.4.2 Air purification system . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.5 Data acquisition system in SK-IV . . . . . . . . . . . . . . . . . . . . . . . . . .
2.5.1 QBEE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.5.2 Software trigger . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

12
12
13
13
13
18
19
20
20
22
23
23
24

3 Event reconstruction
3.1 Reconstruction of muon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.1.1 Cleaning cuts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.1.2 Entry point . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

26
26
26
27

iii

3.1.3 Direction reconstruction . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.1.4 Event classification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Reconstruction of decay electron . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2.1 Vertex reconstruction . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2.2 Direction reconstruction . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2.3 Energy reconstruction . . . . . . . . . . . . . . . . . . . . . . . . . . . .

27
29
31
31
32
33

4 Muon and decay electron simulation
4.1 SK detector simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2 Muon simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2.1 Generating muons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.3 Decay electron simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

37
37
37
38
39

5 Event selection
5.1 Data set of control samples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.2 Selection criteria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.2.1 Selection for stopping muon . . . . . . . . . . . . . . . . . . . . . . . . .
5.2.2 Selection for decay electron . . . . . . . . . . . . . . . . . . . . . . . . .
5.3 Selected samples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.3.1 Stopping muon candidates . . . . . . . . . . . . . . . . . . . . . . . . . .
5.3.2 Decay electron candidates . . . . . . . . . . . . . . . . . . . . . . . . . .

45
45
46
47
48
51
51
55

6 Results and discussion
6.1 Calculation of charge ratio . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

62
62
64
64

7 Summary

69

3.2

iv

List of Figures
1.1
1.2
1.3

1.4
1.5

Overview of cosmic-ray production. . . . . . . . . . . . . . . . . . . . . . . . . . 2
Energy spectrum of cosmic-rays. The equivalent lab energies of particle accelerator experiments are indicated on the energy axis [5]. . . . . . . . . . . . . . . . 3
Comparison between the calculation of high muon energy flux and the measured
data. In the high energy region (Eµ ≫ ϵ ≃ 1 GeV), the effects of decay andenergy loss are negligible (Eµ ≈ Eorg ). However, muon energy loss and decay
become important below 200 GeV [5]. . . . . . . . . . . . . . . . . . . . . . . . 7
Fraction of atmospheric muons and muon neutrinos from pions and kaons decay.
Solid lines represent for vertical and dashed lines for the zenith angle 60◦ [5]. . . 9
The prvious measurements of the muon charge ratio as a function of the verticalmuon momentum with a fit of πK model (see Chapter 6) [1]. . . . . . . . . . . . 10

2.1 Schematic view of the SK detector [11]. . . . . . . . . . . . . . . . . . . . . . . . 12
2.2 Schematic view of the PMT support structure [11]. . . . . . . . . . . . . . . . . 14
2.3 Topology of the muon decay event and neutrino event in SK tank. . . . . . . . . 15
2.4 Construction of a PMT (The PMT shown above is the type of Head-on) [14]. . . 15
2.5 20-inch ID PMT [11]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2.6 Quantum efficiency of the ID PMT as a function of wavelength [11]. . . . . . . . 17
2.7 Distribution of 1 p.e. pulse height of ID PMT with 410 nm wavelength light [11]. 17
2.8 Distribution of relative transit time of ID PMT with 410 nm wavelength light [11]. 18
2.9 Schematic view of acrylic and a FRP cases [15]. . . . . . . . . . . . . . . . . . . 18
2.10 Schematic view of Helmholtz coils surrounding the SK tank [15]. . . . . . . . . . 19
2.11 Formation of Cherenkov radiation. . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.12 Schematic view of the water purification system [15]. . . . . . . . . . . . . . . . 22
2.13 Schematic view of the air circulation system [11]. . . . . . . . . . . . . . . . . . 23
2.14 Block diagram of the QTC and its peripheral devices [19]. . . . . . . . . . . . . 23
2.15 Timing chart for the QTC operation. The gradation of color indicates the integrated charge of PMT signals [19]. . . . . . . . . . . . . . . . . . . . . . . . . . . 24
3.1
3.2
3.3
3.4
3.5
3.6

Four types of cosmic-ray muon events observed at the SK detector. . . . . . . . 27
Distribution of two parameters, GOF and Fcone used in Muboy in assessing quality of the fit. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
The histogram of dE/dX distribution for the track length determination of Stopping muons [21]. The stopping point is determined at 25 m. . . . . . . . . . . . 31
Probability density function of the residual time tres, i obtained from LINAC
calibration. The peaks around 40 nsec and 110 nsec are the PMT late-pulses [23]. 32
Vertex resolution as a function of true electron kinetic energy in SK-IV [23]. . . 33
Energy dependence of the angular likelihood function used in direction reconstruction [15]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
v

3.7
3.8

The θi and ϕi coordinates for the i−th hit PMT [15]. . . . . . . . . . . . . . . . 36
The distribution of photo-cathode coverage function S(θ, ϕ): for PMTs on thebarrel of the tank (left) and for PMTs on top and bottom of the tank (right) [24]. 36

4.1

Mountain Ikenoyama topographical map in 50 m grid. The color represents themagnitude of the altitude and the black point is the location of SK detector [29].
4.2 Schematic of vertical cylindrical detector of SK with coordinate system. θ isdefined to be the zenith angle and ϕ is the azimuthal angle. The slant depth χdenotes the traverse distance by muon in the rock between entering position of
the mountain at the specific altitude (m) to the center position of the detetctor
located at the certain depth underground, 330 m from sea level [30]. . . . . . . .
4.3 Cosmic muon energy spectrum at three underground experiment site [26]. The
average muon energy at SK is Eµ = 271 ± 2 GeV. Muon energy spectrum forEµ ≤ 20GeV is almost flat. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.4 Muon flux as a function of zenith and azimuthal angle at SK depth and location
calculated with MUSIC. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.5 Energy distribution of injected muons (left) and its survival probability definedas the ratio of injected muons to stopping muons per 0.5 GeV bin (right). The
minimum muon energy is set at least 0.1 GeV at the generated point of frameof ID to travel inward the ID tank. . . . . . . . . . . . . . . . . . . . . . . . . .
4.6 Generated muon energy and the Z coordinate of its reconstructed decay electronsin MC. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.7 Zenith angle distribution of generated muons in MC calculated from [31]. The
angular integration covers the entire hemisphere. Blue line and red line represent
injected and reconstructed muons, respectively. . . . . . . . . . . . . . . . . . .
4.8 The true decay curve of stopped muons observed in MC. The fitting range is red
solid line and the expected distribution after fitting the function is red dashedline. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.9 The kinetic energy distribution of decay electrons from positive stopping muonsin MC. The true energy distribution is red solid line and reconstructed energy
distribution is black solid line. Events are required to be positioned more than
200 cm away from the ID wall in the SK detector (Fiducial volume cut). . . . .
4.10 Vertex distribution of reconstructed decay electrons in MC. Here, the fiducial
volume cut is applied. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.1

5.2
5.3
5.4
5.5

5.6

Typical ring pattern of a stopping muon in the SK detector (ID and OD). The
gradation of the color in each point represents the amount of charge observed ina PMT. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Typical ring pattern of a decay electron in the SK detector (ID). The gradationof the color in each point represents the amount of charge observed in a PMT. .
The flow chart of µ-e decay event selection for SK-IV data analysis. . . . . . . .
Total charge of stopping muons in the control sample. . . . . . . . . . . . . . . .
Zenith angle distribution of stopping muons in the control sample. θ is defined as
the zenith angle, where cos θ = +1 correspond to vertically downward direction
and cos θ = −1 to vertically upward direction. Almost all stopping muons arecoming from the top side of the SK tank toward the bottom. . . . . . . . . . . .
Vertex distribution of reconstructed decay electrons in the data. Muon reduction
cuts, fiducial volume cut, and time difference cut for decay electrons are applied.

vi

38

39

40
41

41
42

42

43

43
44

46
46
47
48

49
51

5.7

Distributions of the quality of fitting parameters used for decay electrons in thedata (black line) and MC (red line). Muon reduction cuts, fiducial volume cut,and time difference cut are applied. . . . . . . . . . . . . . . . . . . . . . . . . .
5.8 Distributions of reconstructed energy (left) and N50 (right) of decay electrons in
the data (black line) and MC (red line). Muon reduction cuts, fiducial volume
cut, and time difference cut are applied. . . . . . . . . . . . . . . . . . . . . . .
5.9 Distribution of distance between reconstructed stopped position of stopping
muon and vertex of decay electrons in data (black line) and MC (red line).
Muon and decay electron reduction cuts are applied. . . . . . . . . . . . . . . .
5.10 The fraction of muon events per 10 cm distance bin to the total events (left) and
its detecting efficiency (right) as function of distance in MC. The reduction cuts
described in above for both muons and decay electrons are applied. . . . . . . .
5.11 Schematic view of the prompt signal and delayed signal in the time window. . .
5.12 Distribution of decay electron energy and muon lifetime in the data after applying
fiducial volume cut. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.13 Enter position of stopping muon in the SK tank for X-Y (left) and R-Z (right)
coordinates in the data. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.14 Intensity of stopping muons arriving at SK detector as a function of azimuthaland zenith angle in the data. ϕ = 0◦ and ϕ = 90◦ corresponds to east and north,respectively. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.15 Azimuthal and zenith angle distribution of stopping muons in the data. . . . . .
5.16 Distribution of Muboy goodness for stopping muon in the data (black line) andMC (red line). Almost all muon events are distributed around 0.6. . . . . . . . .
5.17 Distribution of track length for stopping muons in the data. . . . . . . . . . . .
5.18 The relationship between track length and total deposit charge (left) and the
distribution of total charge divided by track length (right) for stopping muonsin the data. The constant of proportionality is approximately 25 p.e per unit cm.
5.19 Vertex distribution of selected reconstructed decay electrons in the data. . . . .
5.20 The decay curve of stopped muons observed in sample data. The fitting range
is red solid line and the expected distribution after fitting the function is reddashed line. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.1

6.2

6.3

6.4

Schematic of the muonic atom. A negative muon takes an atomic orbit (K shell)and creates an excited nucleus. The negative muon follows either process, freemuon decay in orbit of nuclear muon capture. . . . . . . . . . . . . . . . . . . .
The decay curve of stopped muons observed each year from 2008 to 2018. The
solid red line was obtained by fitting a function Eq. (6.5). The effective live timeand fitting results (number of positive and negative muons with the charge ratio,
χ2 /d.o.f ) are listed in the legend. . . . . . . . . . . . . . . . . . . . . . . . . . .
Fitting results of decay curve with χ2 (red) probability, d.o.f (green), (χ2 /d.o.f )×100 (blue) (left) and the event rate of decay electron [event/day] (right) obtained
from positive (red) and negative (blue) stopped muons in SK-IV period. . . . . .
Yearly time variation of the charge ratio of 1.3 TeV cosmic-ray muons from 2008
to 2018 together with the result from the Kamiokande-II. The blue band areashows the average of muon charge ratio in SK-IV period and green band is theresult from Kamiokande-II [10]. . . . . . . . . . . . . . . . . . . . . . . . . . . .

vii

52

53

53

54
54
55
56

57
57
58
58

59
60

61

62

66

67

68

List of Tables
2.1
2.2
2.3
2.4
3.1
3.2
6.1

Experimental phases of SK. The live times are the duration of the observation
after removing bad/calibration runs. . . . . . . . . . . . . . . . . . . . . . . . .
The property of 20-inch PMT. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Energy threshold of Cherenkov radiation for various particles in pure water. . .
Summary of trigger threshold of hit PMTs within a 200 nsec window (N200 ) in
SK-IV and its time width. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

13
16
20
25

Number of hit PMTs (Nhit ) and the corresponding limit on the PMT charge of
all selected ID PMTs in Muboy fit. . . . . . . . . . . . . . . . . . . . . . . . . . 28
Lower limits on the number of nearest neighbors (Nnear ) corresponding to thenumber of PMTs selected after the first cleaning cut. . . . . . . . . . . . . . . . 28
The results of N+ , N− and the charge ratio R(µ+ /µ− ). In this study, only
statistical uncertainty is considered. . . . . . . . . . . . . . . . . . . . . . . . . . 67

viii

Chapter 1
Introduction
1.1

Cosmic-ray muonsCosmic-rays are relativistic high-energy particles entering the Earth’s atmosphere, with somereaching the Earth’s surface. Cosmic-rays can be divided into primary cosmic-ray and secondary
cosmic-ray. Primary cosmic-rays have an energy region expanding from a few MeV to 1020 eV,
composed of various ionized nuclei, such as protons (90%) and alpha particles (9%). Primarycosmic-rays produce several particles through various reactions between atmospheric nuclei
while propagating the atmosphere. The particles generated in this process are called secondary
cosmic-rays. These are mostly produced via hadron interactions generating hadrons, suchas pions and kaons in most cases. Conversely, some secondary particles decay into cosmicray muons and neutrinos. The overview of cosmic-ray production can be seen in Figure 1.1.
Regarding the production of muons, kaons, and pions decay into as described into the following
process:
π ± → µ± + νµ (ν̄µ ) (∼ 100%),
K ± → µ± + νµ (ν̄µ ) (∼ 63.5%).

(1.1)

Here, parent pions and kaons mostly decay as following process via the weak interaction:µ± → e± + νe (ν̄e ) + ν̄µ (νµ ).

(1.2)

However, cosmic-ray muons are less likely to interact with matter, and their relativistic effects
extend their lifetime, allowing them to reach deep underground. In the Earth’s atmosphere,
the muon decay length becomes larger than its typical production height (∼ 15 km) above Eµ∼ 2.5 GeV.

1.2

Primary and secondary cosmic-rays

The primary cosim-rays have been measured in several experiments over the energy range
from a few GeV to above 1020 eV. The differential energy spectrum of the cosmic-ray can be
expressed as:
dN∝ N0 E −α [1/m2 s sr GeV],
dE
1

(1.3)

Primary cosmic ray (p, n)Atmospheric nuclei

Mesons (K, π)

𝝅"

𝑲!

𝜸

𝜸

𝜇! 𝜈"

𝜇!

𝝅!

𝝅𝟎

𝒆!

𝒆"

𝜈#

𝜈#̅

𝜇"

Figure 1.1: Overview of cosmic-ray production.
where N is the number of nucleons, N0 is the constant, E is total energy of nucleon, α is thedifferential spectral index set as α ≡ γ + 1, and γ is the integral spectral index. Figure 1.2
shows the energy spectrum of primary cosmic-ray of various particles obtained by both thedirect measurements above the atmosphere and indirect measurements from the air shower
experiments. Here, the parameter α changes according to the energy between 10 GeV and
1015 eV and set as 2.7. The determination of spectral index γ is significant in examining thecomponents of cosmic primaries, and further understanding the interaction mechanism of highenergy particles in the atmosphere.

1.2.1

Primary interaction in the atmosphere

The diffusion equation of particles in the atmosphere is developed to demonstrate the air
shower production depending on the type of particles. The diffusion equation of the primaryparticles (nucleon) in the atmosphere can be described as follows:Z ∞dNN (E, X)
NN (E, X)
NN (E ′ , X) dE ′=−+FN N (E, E ′ )
,
(1.4)
dX
λN (E)λN (E ′ ) E
E
where NN (E, X)dE is the number of nucleons at a slant depth X in the atmosphere withthe energy range, E ∼ E + dE, λN (E) is the mean free path of nucleons in atmosphere, and
FN N (E, E ′ ) is the cross section for nucleon-nucleon interaction between an incident nucleon ofenergy E ′ and outgoing nucleon of energy E. Here, FN N (E, E ′ ) is defined as:

2

Figure 1.2: Energy spectrum of cosmic-rays. The equivalent lab energies of particle accelerator
experiments are indicated on the energy axis [5].

FN N (E, E ′ ) =dσN N (E, E ′ )
E
,
inel
′
dE
σN
N (E )

(1.5)

inel
′
where σN
N is the total inelastic nucleon-nucleon cross section, dσN N (E, E ) is the cross section
for a nucleon of energy E ′ to produce a nucleon with energy E in nucleon-nucleon interaction.
The solution of Eq. (1.4) is obtained by applying Eq. (1.3) as the boundary condition. For
nucleons, the solution is given as:

N (E, X) = C exp (−X/Λ) E −γ+1 ,(1.6)

where C is a constant, and Λ is the attenuation length (g/cm2 ) of nucleon.

1.2.2

Production of secondary particles

The cascade of various hadrons can be expressed as the set of coupled diffusions in principle.
Analytic solutions for the diffusion equations provide the detail and specific information ofparticle flux and energies of secondary particles.
Considering only pions and kaons as the secondary particles, the diffusion equation of pions
can be expressed as the sum of two components, π + and π − as follows:dNπ (E, X)
=−
dX



1
1
+
λπ dπ



Z 1

Nπ (E/xL , X)Fππ (Eπ , Eπ /xL ) dxLλπ (E/xL )
x2L
0
Z 1
NN (E/xL , X)FN π (Eπ , Eπ /xL ) dxL+
λN (E/xL )x2L0Nπ (E, X) +

3

(1.7)

where xL is E/E ′ , λπ is the interaction length of pions in air, and dπ is the decay mean freepath of pions defined as:1
mπ c2 h0
ϵπ
=
≡
.
dπ
Ecτπ X cos θEX cos θ(1.8)

Here, mπ and τπ is the mass and lifetime of a pion, respectively, h0 is the scale height (≈ 6.5km), and c is the speed of light. The first term of Eq. (1.7) represents the decrease in pions via
interaction and decay in the atmosphere, the second term represents the regeneration of pions
and the third term indicates the production of pions via nucleon-nucleon interaction.
For E ≫ ϵπ , the contribution of Eq. (1.8) can be neglected. Then, the solution of Eq. (1.7) is
obtained by applying the boundary condition Nπ (E, 0) = 0 as follows:Λπ
ZN π(exp (−X/Λπ ) − exp (−X/ΛN ))1 − ZN N Λπ − ΛN
where, the attenuation lengths is given as:
Nπ (E, X) = NN (E, 0)λπ,1 − ZππλN
ΛN ≡.
1 − ZN N

(1.9)

Λπ ≡

(1.10)The pion flux in Eq. (1.9) reaches the maximum at X ∼ 140 g/cm2 (∼ 15 km) and then,decreases according to the attenuation length [5]. Here, ZN N is the spectrum-weighted moment
for a nucleon to produce another nucleon defined as:
Z 1(xL )γ−1 FN N (xL )dxL .
(1.11)ZN N ≡0For γ = 1, Z-moment represents the average ratio of the energy between the incident and outgoing particles in a nucleon-nucleon interaction. Due to the steep primary comic-ray spectrum,for γ > 1, the contribution of xL to the Z-moment becomes small (xL → 0). However, if
xL is large enough, the outgoing particle carries the most energy of incident particle. In thiscase, the fluxes only depend on the particles generated in the forward region of the interaction
frame.

1.2.3

Production of atmospheric muons

Muons are produced in the atmosphere at a high altitude and carry the original informationof early shower production. A relatively long time of 2.2 µsec and a small cross section of themedium render muons detectable at the Earth’s surface. Muons are produced by the decayof unstable secondary particles, including pions and kaons, and charmed particles. The main
decay processes producing muons are shown in Eq. (1.1). In addition to the decay modes ofEq. (1.1), high energy muons can be produced by the decay of charmed mesons (D± , D0 ).
Here, for unstable particles, the two major processes, namely nuclear interactions, and decay
processes, become competing while propagating through the atmosphere. These processes
rely on the particle lifetime and the cross section of the atmosphere depending on its density.
Charmed particles have a small production cross section in the nucleon-nucleon interaction,
which becomes important in over 10 TeV energy region due to the relatively short lifetime ofcharmed mesons and hadrons. Hence, as the secondary particles, pions and kaons can mainly
4

be considered as the primary particles to produce muons. The energy limit of the muon andneutrino from two-body decay in the lab frame is given as:
m2µ≤ Eµ ≤ E,
M 2

m2µ
0 ≤ Eν ≤ E 1 − 2 ,ME(muons)(1.12)(neutrinos)

where M is the mass of a parent meson, and E is the energy of a decaying meson in the lab frame.
For the numerical calculation of the π ± → µ± + νµ (ν̄µ ) decay process, comparing the masses
between the parent pions and muons, the latter has a sufficiently large mass, carrying mostenergy. Therefore, the energy fraction of decay products to the pion are given, as ⟨Eµ ⟩/Eπ =0.79 and ⟨Eν ⟩/Eπ = 0.21, respectively, indicating that the contribution of pions to neutrino isconstrained to that of muons. However, the corresponding kaon decay, numbers are given as⟨Eµ ⟩/EK = 0.52 and ⟨Eν ⟩/EK = 0.48, respectively. Since the mass of kaon is much larger thanthat of muon, the contribution of kaons to neutrinos becomes significant in the high energy
region (TeV and above) and further understanding of kaon production becomes important forthe precise calculation of atmospheric neutrino fluxes.
Here, the muon production spectrum from the parent meson i (pions or kaons) is given as
follows:
X Z Emax dNi (E, E ′ )Di (E ′ , X)dE ′ ,Pµ (E, X) =
dE
Emin
i

(1.13)

where dNi (E, E ′ )/dE is the muon spectrum produced by particles i with energy E ′ , Emin , andEmax are the minimum and maximum energies of parent mesons producing muons, Di (E ′ , X) isthe spectrum of the parent mesons with energy E ′ at the slant depth X. Assuming the parent
mesons are pions, the function Dπ is given as follows:ϵπΠ(E, X).
(1.14)
EX cos θ
The production spectrum of muons from both components, pions and kaons, is obtained bytaking two forms of Eq. (1.13) as follows:
Dπ (E, X) =ϵπPµ (E, X) =X cos θ(1 − rπ )Z Eµ /rπ0.635ϵK+
X cos θ(1 − rK )Z Eµ /rKEµEµNπ (E, X) dE
EENK (E, X) dE.
E
E(1.15)

2
where rπ, K ≡ m2µ /Mπ,
K , and Mπ, K is the mass of either parent meson, pions or kaons.

1.3

Muon spectrum

The muon energy spectrum at the slant depth X0 is obtained by calculating the effects of energy
loss and decay while propagating in the atmosphere. According to Eq. (1.8), the decay rate ofmuons is given as:

5

dNµ
ϵµ
=
Nµ (E, X),(1.16)dX
EX cos θ
where X is the slant depth along the travel direction of the muon with energy E, and θ isthe zenith angle. The muon energy loss in the atmosphere is given as dE/dX = αatm ≈−2 MeV/g/cm2 . Hence, the muon energy Eµ at the vertical atmospheric depth X0 can be
expressed as:
Eµ = Eorg − αatm (X0 / cos θ − X),(1.17)

where Eorg is the muon energy at the slant depth X. The survival probability P of muon withenergy Eµ and zenith angle θ at the observation level X0 from the original location at the slant
depth X is calculated by integrating Eq. (1.16) as follows:ϵµ /(Eµ cos θ+αatm X0 )Eµ
X cos θP (Eµ , θ, X, X0 ) =X0 Eµ + αatm (X0 / cos θ − X) .


(1.18)

Muon intensity at the observation depth in the energy region for Eµ < ϵπ , is given as:
Z X0 / cos θPµ (Eorg , X) × P (Eµ , θ, X, X0 )dX.

Iµ (Eµ ) =(1.19)

0

Numerical approximation values for the muon energy spectrum is provided as:









−2.7
0.14Eµ1dNµ0.054 ,≈+dEµ
cm2 sec sr GeV 
1.11Eµ1.11Eµ 




1+
1 +

ϵπ (θ)ϵK (θ)(1.20)

where ϵπ (θ) = 115 GeV/ cos θ, and ϵK (θ) = 850 GeV/ cos θ. Here, the contribution from charm
particle and heavier particles are not considered though their contributions are significantly
high in the very high energy region. The measured muon energy flux at the sea level together
with the approximation of high energy muon flux from the calculations are shown in Figure 1.3.

1.4

Muon charge ratio

The muon charge ratio is the ratio of the positive to negative charge atmospheric muons.
The measurement of muon charge ratio can provide a detailed information on the high energy
hadronic interaction in the atmosphere and components of cosmic primaries. The muon charge
directly reflects mechanisms of the production of multiple hadrons, such pions and kaons in theatmosphere and contribute to the determination of spectral index γ in the high energy region.
However, since the data regarding the production of multiple hadrons in the accelerator are
limited to the high energy region, the theoretical understanding of the hadronic interaction isstill insufficient. For these reasons, the muon charge ratio at sea level has been studied and themeasurements have been performed in wide energy ranges. For the theoretical understandingof the muon charge ratio, a simplified model assuming the primary spectrum only consisting of
protons is considered [6]. In the model, the pion production wherein all partilces decaying into
muons is considered (except kaon). Further illustration of the production of secondary particles
is further discussed in [6].

6Figure 1.3: Comparison between the calculation of high muon energy flux and the measured
data. In the high energy region (Eµ ≫ ϵ ≃ 1 GeV), the effects of decay and energy loss are
negligible (Eµ ≈ Eorg ). However, muon energy loss and decay become important below 200
GeV [5].
The cross sections of pions in a proton-proton interaction using Eq. (1.5) can be expressed asfollows:
Fpπ± (Eπ , Ep ) =Eπ dσpπ± (Eπ , EP )
,
inel
σpp
dEπ

(1.21)

where Ep and Eπ are the energies of the primary proton and secondary pion in the laboratory
inel
frame, and σpp
is the total inelastic proton-proton cross section. Applying the hypothesis of theenergy spectrum of Eq. (1.3) for primary particles, the pion energy spectrum can be expressedas:Z
dNπ (Eπ )(const) ∞ −(γ+1)±Π (Eπ ) ==E
Fpπ± (Eπ , Ep )dE.
(1.22)dEπ
Eπ
Eπ
Defining xL = Eπ /E and applying the same analogy as the spectrum-weighted moments from
Eq. (1.11), Eq. (1.22) can be simplified expressed as:
Π± (Eπ ) = (const)Eπ−(γ+1) Zpπ± ,(1.23)

where Zpπ± is defined as:Z 1
Zpπ± =(xL )γ−1 Fpπ± (xL )dxL .

07(1.24)

Consequently, the muon charge ratio Rµ can be obtained as follows:
Rµ ≡

Zpπ+
Π+ (Eπ )=.
Π− (Eπ )Zpπ−(1.25)The remarkable features in this model are described as follows:• Only pions contribute to the muon production. Rµ is independent of the muon energy.
• Rµ > 1: As the primary particles consist mainly of protons, π + are more likely to be
produced than π − .
• Rµ depends on the γ of the primary spectrum.
• Rµ depends only on the projectile particle, not the target. Forward moving fragmentswith x > 0 contribute to Zpπ+ , and Fpπ+ (x > 0) relies on the projectile particle.
The numerical estimate for the charge ratio is derived by including the proton and neutroncomponents as cosmic primaries.
Considering only muons produced from the decay of pions, the muon charge ratio can be
expressed as follows:Rµ =1 + δ0 AB
fπ +=,1 − δ0 AB
1 − fπ +(1.26)

where δ0 ≡ (p0 − n0 )/(p0 + n0 ) is the fraction of proton excess to the total particles (protonand nucleon) at the top of the atmosphere, and A and B are, described as:(Zpπ+ − Zpπ− ),(Zpπ+ + Zpπ− )(1 − Zpp − Zpn )B≡,(1 − Zpp + Zpn )A≡(1.27)

respectively, and fπ+ = (1 + δ0 AB)/2 is the fraction parameter of positive muons to the decayof charged pions. The contribution from K ± to the muon charge ratio must also be considered.
More K + is generated than K − in the process, p + air → Λ + K + + anything, which isdifficult for K − production. K − can only be produced with an additional baryon and a strange
particle creating a difference in the Z-moment between the pions (Zpπ+ = Znπ− ) and kaons (ZpK + ≫ ZnK − ≈ ZpK − ). The behavior of the charge ratio depends on the relative production
ratio of kaons to pions. The contribution of kaons gradually increases as compared to that ofpions at the critical energy range for pions, Eµ > ϵπ / cos θ. Thus, the kaon production canaffect the muon charge ratio especially at a higher energy range.

1.5

Fraction of atmospheric neutrinos

The improved contribution of kaon to the atmospheric neutrino production becomes considerably important as the energy increases. As described in the previous section, the fractional kaon
production becomes significant above O(102 ) GeV to a few TeV energy range corresponding toϵπ < E cos θ < ϵK . Neutrinos from kaon decay carry almost half the energy of kaons, and thisfraction is much large than that of pions, as mentioned in Section 1.2.3. Figure 1.4 shows thefraction of muons and neutrinos from the pion and kaon decays, respectively.
8

Figure 1.4: Fraction of atmospheric muons and muon neutrinos from pions and kaons decay.
Solid lines represent for vertical and dashed lines for the zenith angle 60◦ [5].

1.6Previous cosmic-ray muon charge ratio measurements

The measurements of the muon charge ratio have been performed in various experiments, as
shown in Figure 1.5. The previous results of the charge ratio are obtained in the wide energy
range from GeV to TeV. From previous measurements by Compact Muon Solenoid (CMS),
L3+C below 1 TeV/c and Utah, MINOS, OPERA above 1 TeV/c, the charge ratio is approximately constant at R(µ+ /µ− ) ≈ 1.27 in the momentum range of 10 − 300 GeV/c and increases
in the higher momentum region. The increase in the charge ratio in the TeV energy region
is considered a contribution from kaon decay to muon production, as explained in previous
sections. Here, the overview of the past experiments is expressed below:• CMS experiment [1]
CMS is one of the detectors installed at the Large Hadron Collider at CERN. The main
physics target is searching for new signal in proton–proton collisions; however, it is extensively used to measure cosmic-ray muons. At the detector center, the magnetic field
is parallel to the central axis (along the beam direction) of the solenoid. The muon momentum is reconstructed by measuring the muon trajectory curvature. CMS collected
data in two cosmic-ray runs in 2006 and 2008, and performed most precise measurementof the charge ratio in the momentum region below 0.5 TeV/c.
• L3 + C experiment [7]The precise muon magnetic spectrometer of the L3 detector is located at the LEP collider
at CERN. The L3 inner detector is surrounded by the scintillator array, magnet, anddrift chamber. After passing through the rock overburden, muons are measured by the
scintillator and then, the direction is bended while traveling the drift chamber. The track
position is measured in the drift chamber, and its momentum is obtained by determining
its track radius. The absolute muon flux between 20 and 3,000 GeV is measured using
the data collected between 1999 and 2000 for zenith angles ranging from 0◦ to 58◦ .
• Utah experiment [8]The Utah cosmic-ray muon detector is located underground the Masatch Mountains at
9

Figure 1.5: The prvious measurements of the muon charge ratio as a function of the verticalmuon momentum with a fit of πK model (see Chapter 6) [1].
Park City, Utah. The detector consists of cylindrical spark counter arrays filled with argon
and ethylene, and water Cherenkov counter triggering the incoming muons and verticalion magnets with a magnetic field about 16 kG. The muon trajectory is determined fromthe discharge of a cylindrical spark counter. Muons were collected for 765 days withenergies between about 1 and 7.5 TeV at the zenith angle from 40◦ to 80◦ .
• MINOS experiment [9]The MINOS far detector is a steel-scintillator and tracking calorimeter located at a depthof 2070 m.w.e. in the Soudan Underground Laboratory northern Minnesota, originallydesigned for the study of neutrino oscillations with the NuMI beam (Fermilab). Muon
momentum is measured by magnetizing the steel with a toroidal field (about 1.3 T)
along the detector’s beam direction. MINOS recorded data between August 1, 2003 andFebruary 28, 2006 with surface muon energies from 1 to 7 TeV.
• OPERA experiment [2]The OPERA is a hybrid experiment with electronic detectors and nuclear emulsions,located underground at the Gran Sasso Laboratory in Italy at an average depth of 3,800
m.w.e. A dipolar iron magnet is instrumented with resistive plate chambers, and its
magnetic field vertically transverses along the beam axis. The charge and momentum aredetermined via a similar method in other experiments using its magnetic spectrometer.
The muon data corresponding to live time of 113.4 days during 2008 were collected forthe measurement of the charge ratio with the energy range of about 890 GeV to 7 TeV
as a function of vertical surface energy.
• Kamiokande-II experiment [10]The Kamiokande-II (KAMIOKA Nucleon Decay Experiment) is a water Cherenkov detector located 1,000 m underground (2,700 m.w.e) in the Kamioka mine and is the pre10

decessor of the Super-Kamiokande experiment. It was designed to discover the proton
decays and extensively used as it is for a neutrino detector. The operation period staredin 1984 and ended in 1995. The muon charge ration was measured for 978 effective days
from January 1987 to April 1990 with the sea-level momentum of 1.2 TeV/c and foundto be 1.37 ± 0.06(stat) ± 0.01(syst).

11

Chapter 2
Super-Kamiokande detector
2.1

Overview of Super-Kamiokande detector

Super-Kamiokande (SK) is the water Cherenkov detector located 1,000 m underground (2,700
m.w.e) in the Kamioka mine in Japan [11]. The detector consists of a cylindrical stainlesssteel tank. The tank contains 50 kilotons of ultra pure water. Since the detector is locateddeep underground, cosmic ray muons with energy over 1.3 TeV can penetrate the surroundingrocks (2,700 m.w.e) and reach the detector. The fourth phase of the SK experiment (SK-IV)
began in September 2008 with the new electronics and data acquisition systems [12] (details
are given in Section 2.5). The upgrade of electronic systems has enlarged the time window of
event trigger and contributed to the high detection efficiency of the delayed signal of primary
event. Experimental phases of SK are summarized in Table 2.1.

Figure 2.1: Schematic view of the SK detector [11].

12Table 2.1: Experimental phases of SK. The live times are the duration of the observation after
removing bad/calibration runs.
Phase
SK-I
SK-II
SK-III
SK-IV
SK-V
SK-VI1
Start
April 1996 October 2002 July 2006 September 2008 January 2019 July 2020
End
July 2001 October 2005 August 2008
May 2018
July 2020
RunningLive time [days]
1496
791
548
2970
379
ID PMTs11, 146
5, 182
11, 129
11, 129
11, 129
11, 129
OD PMTs1, 885
1, 885
1, 885
1, 885
1, 885
1, 885
PMT coverage [%]
40
19
40
40
40
40

2.2

Detector structure

2.2.1

Inner detector and outer detector

The structure of the SK detector consists of two regions, the inner detector (ID) and the outerdetector (OD) as shown in Figure 2.1. Between the ID and OD, there is a stainless steel support
structure about 60 cm wide to accommodate the PMTs, as shown in Figure 2.2. The ID consistsof a cylindrical stainless steel tank, 33.8 m in diameter and 36.2 m in height. The ID is filledwith 32.5 kton of ultra pure water and installed with 11,129 20-inch ID PMTs facing inward
from the wall surface (inside the super module). In the SK-IV phase, 20-inch ID PMTs cover
40% of the surface area of the ID tank. The space between the ID PMTs is covered with ablack sheet to reduce the unnecessary photon reflection.
The ID tank is surrounded by the OD tank with a thickness of 2.05 m on the upper and lower
sides and 2.2 m on the lateral side. Approximately 17.5 kton of ultra pure water is containedin the OD tank and 1,885 8-inch PMTs are installed outwards on the tank wall (outside thesuper module). The OD wall is covered with white reflective sheet called the Tyvek sheet, used
to prevent light from entering the ID tank. The main purpose of the OD is to identify thecosmic ray muons and to prevent the low energy gamma rays and neutrons, produced in thesurrounding rocks from entering the ID.
The typical neutrino events react with the nuclei in the ID to generate charged particles emitting
Cherenkov light without ringing OD. Conversely, charged particles, such as cosmic ray muons
entering the SK, emit Cherenkov light while traveling through the water, which is detectablein both OD and ID. Therefore, neutrino and cosmic ray muon events can be distinguished byapplying the signal information of OD and ID, as shown in Figure 2.3.

2.2.2

20-inch Photo Multiplier Tube (PMT)

Principle of PMT
To detect the Cherenkov light in water, a highly sensitive optical sensor is required. A PMT is an
optical sensor using the mechanism of photoelectric effect to amplify weak light and converting
it into an electrical current (Figure 2.4). The operation principle of PMT is described as follows:• Photons in a specific wavelength region passing through the PMT’s glass surface emit anumber of electrons according to their energies on photocathode due to the photoelectric1

Gadolinium sulfate dissolved in the ultra pure water [13].

13Figure 2.2: Schematic view of the PMT support structure [11].
effect.
• Photoelectrons are accelerated and converged by the focusing electrode and later amplified
by the dynodes to produce multiple secondary electrons. Here, the inside of PMT isin a vacuum to improve the collection efficiency of photoelectrons and prevent internal
discharge.
• Photoelectrons are accelerated by the electric field in the electron multiplier, and amplified
by the subsequent dynode to create additional secondary electrons.
• The amplified secondary electrons are removed as a signal from the anode.
20-inch PMT
The 20-inch PMT used in the ID tank of SK is a type of Hamamatsu R3600, which is the typeof spherical PMT and about 50 cm in diameter, as shown in Figure 2.5. The property of 20-inch
PMT is listed in Table 2.2. The photocathode of the PMT is made of bialkali (Sb-K-Cs) and14

Figure 2.3: Topology of the muon decay event and neutrino event in SK tank.

Figure 2.4: Construction of a PMT (The PMT shown above is the type of Head-on) [14].
its sensitive wavelength region spans from 280 to 660 nm. The wavelength dependence of thequantum efficiency is the most sensitive around 360 nm, as shown in Figure 2.6. Figure 2.7
shows that the transit time distributed about 2.2 nsec (1σ resolution) for single photoelectron (p.e.) and the one 1 p.e. distribution of the ID PMT with a wavelength of 410 nm lightis shown in Figure 2.8.

15

Table 2.2: The property of 20-inch PMT.
Product Name
Photocathode Area
Photocathode Material
Collection Efficiency
DynodeQuantum Efficiency
Gain
Dark Noise Rate
Timing Resolution
Drift Time
WeightPressure Tolerance

Hamamatsu R3600
50 cm diameter
Bialkali(Sb-K-Cs)
70%
11 Stage Venetian Bind Type21% (360 nm)
107 (1700 - 2000 V)
3 kHz
2.2 nsec(1σ)
100 nsec
13 kg
6 kg/cm2

Figure 2.5: 20-inch ID PMT [11].

16Figure 2.6: Quantum efficiency of the ID PMT as a function of wavelength [11].

Figure 2.7: Distribution of 1 p.e. pulse height of ID PMT with 410 nm wavelength light [11].

17Figure 2.8: Distribution of relative transit time of ID PMT with 410 nm wavelength light [11].
Acrylic and a FRP (Fiber Reinforced Plastic) cases
After the serious accident in 2001, SK has installed a PMT case to protect PMTs from the shock
wave caused by the explosion of PMTs. The surrounding part of the PMT surface is covered
with a highly transparent acrylic case, and the other is covered with the Fiber Reinforced
Plastics (FRP), as shown in Figure 2.9.

Figure 2.9: Schematic view of acrylic and a FRP cases [15].

2.2.3

Helmholtz coils

The photoelectrons generated inside the PMT are affected by the geomagnetic filed, possibly
reducing the collection efficiency. Twenty-six sets of Helmholtz coils are installed around thewater tank to reduce the effects of the geomagnetic field, as shown in Figure 2.10. The original magnetic field was about 450 mG and 32 mG pre and postelectric current application,18

respectively [16].

Figure 2.10: Schematic view of Helmholtz coils surrounding the SK tank [15].

2.3

Detection principle

In the SK detector, the Cherenkov light emitted by the charged particles in water is detected byeach PMT. The Cherenkov light is emitted by the charged particles moving faster than the speedof light in a medium. The schematic view of Cherenkov radiation is shown in Figure 2.11. The
Cherenkov light is emitted in a conical shape at an angle θC satisfying the following equation
relative to the traveling direction of the charged particle:

Figure 2.11: Formation of Cherenkov radiation.

cos θC =
19

1
,
βn

(2.1)

where n is the refractive index of the medium, and β is the fraction of the velocity (= v) of
a particle to the speed of light (= c) in a vacuum (e.g., β = v/c). Since the refractive index
of pure water is n = 1.34, when a charged particle runs at a speed close to the speed of light(β ≃ 1), the Cherenkov angle becomes θC ≃ 42◦ . A charged particle emits light along its
movement direction until it its speed drops below the speed of light, with the light projectedon the wall as a ring is referred to as the Cherenkov ring. The energy threshold of Cherenkov
radiation is given as the following equation:
mc2
mc2
=v
Ethr = p!2 ,u
1 − β2ut1 − 1n(2.2)

Here, at the threshold of Cherenkov radiation (cos θC ≤ 1), β = 1/n. The charged particle withthe energy satisfying Ethr ≤ E can emit the Cherenkov light. The threshold varies according
to the mass of charged particles, as shown in Table 2.3 as for pure water (n = 1.34).
Table 2.3: Energy threshold of Cherenkov radiation for various particles in pure water.
Particle
e±
µ±
π±
K±
p±

Rest Mass [MeV/c2 ]0.5110
105.7
139.6
493.7
938.3

Energy Threshold [MeV]
0.7675
158.7
209.7
741.7
1410

The number of photons dN emitted as Cherenkov light while a particle travels in a medium
with a refractive index n at a velocity β for a small distance dx is given by the following
equation:
d2 N
2παZ 2
1=
1− 2 2
(2.3)
dxdλ
λ2
nβ
where λ is the wavelength of photon, α is fine-structure constant, and Z is charge. The
number of Cherenkov photons emitted from electron per unit cm in the wavelength range,
λ = 300 nm ∼ 600 nm can be obtained by taking the integration of Eq. (2.3) as follow:
Z 1 cm Z 600 nm
1
2πα
(2.4)
N=
dx
dλ 2 1 − 2 2 ≃ 340.
λnβ
0
300 nm2.4

Water and air circulation system

2.4.1

Water purification system

The ultra pure water of the SK detector is generated by purifying the groundwater in the
Kamioka mine. However, groundwater contains several impurities including small dust, metal
ions, bacteria, and radioactive sources. These impurities scatter or absorb Cherenkov light andaffect light transmission in water. In addition, radioactive sources dissolved in water create
20

background events for observations. According to the reasons above, water purification isessential to remove impurities, which reduces the accuracy of the physics measurement. To
maintain high purity, water is constantly circulated inside the detector at a rate of 60 tons/h in
the SK-IV. The water is retrieved from the top of the tank and poured from the bottom after
the purification, as shown in Figure 2.12. The process of water purification system in SK-IV is
explained as below:• 1 µm mesh filter
Removes the dust larger than 1 µm and the radioactive contaminations in water.
• Heat exchanger
The water temperature increases due to the heated pump used in water circulation and
PMTs. The heated water increases the dark rate of PMTs, bacteria in water, and convection in tank. It is required to suppress the water convection to prevent radon mixed
from the wall from spreading to the whole area inside the tank. The water temperature
is adjusted to about 13◦ C.
• Ion exchanger
Removes ions in water, including Ni2+ , Ca2+ , Fe2+ , Co2+ , Cl− and so on.
• UV sterilizer
Radiates ultraviolet (UV) light to water to kill bacteria.
• Radon less air dissolving system
Dissolves Rn free air in water to increase the radon gas removal efficiency.
• Reverse osmosis filter
Removes the contaminants in the water.
• Cartridge polisher
Removes the ions dissolved in water with 99% efficiency.
• Ultra filter
Removes dust particles larger than 10 nm.
• Membrane degasifier
Removes radon and oxygen dissolved in water.
The radon concentration in the ultra pure water after purification is reduced to 1 mBq/m3 orless with the water transparency of about 100 m [17].

21Figure 2.12: Schematic view of the water purification system [15].

2.4.2

Air purification system

Rn produced by Uranium exists around the Kamioka mine and these radioactive sources arecontained in the surrounding rock of the SK detector. In the MeV energy regions, the background events from Rn are dominant; hence, it is necessary to reduce the Rn concentration inboth water and air. In the SK tank, there is some distance between the water surface in thetank and the top side of the detector. The purified air (Rn free air) is passed into the tank toprevent Rn from dissolving into the water. The schematic view of the air purification system
is shown in Figure 2.13, with the process described below:• Compressor
Compresses the air around 7.5 − 8 atm.
• Air filter
Removes dust particles in the compressed air. 0.3, 0.1, and 0.01 µm filters are set, and
0.1 and 0.01 µm filters are used after the process in the carbon column section.
• Air drier
Removes water from the air to improve the Rn removal efficiency.
• Carbon column
Removes Rn in the air by using the carbon column which can absorb Rn.
• Cooled carbon column
Lowers the temperature of carbon column to −40◦ C (changed to −60◦ C during SK-IV
period) to increase the Rn removal efficiency [18].

22Figure 2.13: Schematic view of the air circulation system [11].

2.5

Data acquisition system in SK-IV

A data taking with a new electronics module (QBEE) started from September 2008. The dataacquisition process in SK-IV is discussed below.

2.5.1

QBEE

Since the SK-IV phase, data has been collected by a new front-end electronics module, QBEE (Chargeto-Time-Converter-Based Electronics with Ethernet). The analog signal from the PMT is sent
to the electronics hat at the top of the tank via a 70 m cable and digitized by the frontend electronics module. Hit timing information and the integrated signal charge are sorted inchronological order in the computer and processed as an event by applying software trigger,
and later stored on the disk. In QBEE, the analog signal is converted into a square wave
by the charge-to-time-converter (QTC) and then, its signal is digitized by time-to-digital converter (TDC). Here, each QBEE has height QTCs, and each QTC processes three PMTs (aQBEE processes 24 PMT signals). A block diagram of QTC and its peripheral devices is shownin Figure 2.14.

Figure 2.14: Block diagram of the QTC and its peripheral devices [19].
QTC has three gain range channels, Small, Medium, and Large and the ranges are determined
depending on the detected charge with the relative gain ratio, 1: 1/7: 1/49. The charge range
spans from 0.2 to 2,500 pC. The PMT signal is treated follows steps: The flag (HIT) is issued
when the signal exceeds the certain discriminator threshold. If the HIT is issued, three gates23

are open in the order of charge gate, discharge gate, and VETO gate. The process time fora signal requires approximately 900 nsec. The signal and timing chart of QTC is shown in
Figure 2.15. Here, the rising edge of the square wave is the hit timing, and the pulse widthcorresponds to the amount of charge.

Figure 2.15: Timing chart for the QTC operation. The gradation of color indicates the integrated charge of PMT signals [19].

2.5.2Software trigger

Since the signals are treated with high speed digital processing, all PMT hit information arecollected in a merger PC. Several triggers are set for an event selection depending on the numberof hit PMTs. The number of PMT hits in a time width of 200 nsec defined as N200 , is used,which is triggered when a signal exceeds a certain threshold. Table 2.4 summarizes the typesof software triggers set in the SK-IV phase with the N200 threshold for the trigger conditions,and time width. In the energy region of SLE trigger, the time width is set to 1.5 µsec owing to
the high rate resulting from the radioactive background events. Super High Energy (SHE) and
After-Window (AFT) triggers were newly added for the analysis of supernova relic neutrino
search. The AFT trigger is issued and all hits information within the 500 µsec time window isrecorded if N200 exceeds the SHE trigger threshold.

24

Table 2.4: Summary of trigger threshold of hit PMTs within a 200 nsec window (N200 ) in SK-IV
and its time width.
Trigger TypeSuper Low Energy (SLE)Low Energy (LE)High Energy (HE)Super Low Energy (SHE)
AFT
Outer Detector (OD)

N200 Threshold [hits/200 nsec] Time Width [µsec]
34 → 31 (After May 2015)[−0.5, +1.0]
47[−5, +35]
50[−5, +35]
70 → 58 (After September 2011)[−5, +35]
58[+35, 535]
22 in OD[−5, +35]

25

Chapter 3
Event reconstruction
In the SK experiment, an event reconstruction varies according to its physical target and analysis. Especially for the muon decay analysis, muon events are reconstructed using the specificfitter working differently from that of the decay electrons. In this chapter, the reconstruction
methods for muon and decay electron events are explained, respectively.

3.1

Reconstruction of muon

Cosmic-ray muons observed at SK are classified into six categories: No-fit, Single thoughgoing muon, Stopping muon, Multiple muon-1, Multiple muon-2, or Corner-clipping muon.
Figure 3.1 shows the four types of classified muon events as an example. Each of the muon
events in Figure 3.1 are defined as, (I) Single through-going muons: events penetrating the ID,(II) Stopping muons: events stopping inside the ID, (III) Multiple muons: events with several
tracks, and (IV) Corner-clipping muons: events with a short track length, scraping the edgesof the ID.
The event reconstruction for muon events is performed by the so-called Muboy fitter using
several algorithms to identify as per the muon types. Muboy determines if the event has asingle track or multiples tracks in the ID at the first guess, and then, sees if it stopped in the
ID or not.
Muoby was originally developed by R. Svoboda to fit muon events producing spallation products
and a source of background for the low energy solar neutrino analysis. The Muboy fitter isapplied for both actual and simulated data. Here, the overview and main feature of the Muboy
fitter is discussed and more details can be found in [20], [21].

3.1.1

Cleaning cuts

At first, the cleaning cut is applied for PMT selection. The selection criteria require the PMT’s
charge to be greater than a certain threshold. This threshold is determined based on the numberof hit PMTs (Nhit ) per event. The charge thresholds for the first cleaning cuts are listed inTable 3.1.
After the first cleaning cut is applied, Muboy counts the number of nearest neighbors of each
hit PMT within 10 nsec (Nnear ). Then, Muboy defines its second cleaning cut and selects only
those hit PMTs with Nnear larger than a certain threshold depending on the number of PMTs
after the first cleaning cut, as shown in Table 3.2.

26

Single through-going

Stopping

Multiple

Corner-clipping

Figure 3.1: Four types of cosmic-ray muon events observed at the SK detector.

3.1.2

Entry point

The entry point estimation is performed by selecting the earliest PMT with has at least threenearest neighbors hit within 10 nsec. If no PMT meets the criterion, the required conditions forthe number of nearest neighbors are reduced by one. The entry point is the coordinate of theearliest ID PMT and the entry time is the timing of the earliest PMT hit. The expected exit
point is the center of nine selected PMTs among all remaining PMTs containing the maximum
charge. The direction of the muon track is given by the vector from its entry point to the exit
point. Then, Muboy applies further causality cuts to reduce hits produced by multiple muonsand scattered light.

3.1.3

Direction reconstruction

If a single muon, with known entry point and time, enters the ID, the timing of hit PMTs
cannot get a hit earlier than the travel time of muon in water (Tearly ). However, if there are
multiple muons then there can be early hits than Tearly . The total number of hit PMTs (Nearly )with hit times less than Tearly are counted as Nearly . Here, Tearly is defined as follows:
D
,
(3.1)
v1
where Tentry is the hit time of the entry point PMT, D is the distance between the entry points
Tearly = Tentry +27

Table 3.1: Number of hit PMTs (Nhit ) and the corresponding limit on the PMT charge of all
selected ID PMTs in Muboy fit.
Nhit
> 10000
8000 - 10000< 8000Threshold charge of selected ID PMT
3.0 p.e.
2.5 p.e.
2.0 p.e.

Table 3.2: Lower limits on the number of nearest neighbors (Nnear ) corresponding to the numberof PMTs selected after the first cleaning cut.
Nhit after first cut> 75005000 - 7500
2500 - 5000
500 - 2500
< 500

Nnear cut
5
4
3
2
1

and the hit PMT, and v1 is the muon speed, set as 34.0 cm/nsec. Multiple muons can hit
PMTs less than v1 and can be recognized by counting Nearly . If Nearly > 45, then the eventis classified as a multiple muon-category-1 (Multiple muon-1) and all events of this category
are treated separately to estimate their entry points and directions. In contrast, the PMT hit
produced by the direct light of photons originated from a muon, cannot be hit later than thetravel time of light in water. Here, Tlater is defined as:
D
,
(3.2)
v2
where v2 is the speed of light in water, set as 18.0 cm/nsec. The remaining hit PMTs with hit
times are between Tearly (obtained from Eq. (3.1)) and Tlater are used for direction fitting. The
muon entry point and its direction toward the hit PMTs are in the Cherenkov cone, if each hit
PMT satisfies at least one of the following two conditions:Tlater = Tentry +• d⃗ · ⃗r > 0.74
where d⃗ is a unit vector from the entry point to hit PMT and ⃗r is the muon track direction.
• Distance between the entry point and the hit PMT is less than 2 m.
Once the hit PMTs in the Cherenkov cone are selected, Muboy estimates the direction andrecalculates it until the goodness of fit (GOF) parameter is maximized. The GOF parameteris defined as:
GOF(fcone ) =NXg(δti ),

(3.3)

i=1

where fcone is the number fraction of hit PMTs in the Cherenkov cone, δti (= Texpected − ti ) is
the time residual, ti is the measured time of i−th PMT, and Texpected is the expected time at28

which the PMT should be hit from a given muon track and direction. The function GOF(fcone )takes the following form:
0.9
,
Ccut
0.1fcone + 0.9 − Ccut
fcone > 0.75 : GOF(fcone ) =,1 − Ccutfcone < 0.75 : GOF(fcone ) = fcone(3.4)

where Ccut = 0.74. The function g(δti ) is defined as follows:"#(δti − tmean (qi ))2
δti < 0 or qi > 30 p.e. : g(δti ) = exp −
,
2σ 2


δti
δti > 0 : g(δti ) = exp −
,
λ

(3.5)

where,qi i
σ = 1.690 + 2.514 exp −[nsec],
h 2.453qi i[nsec].
λ = 1.254 + 14.863 exp −
2.316
h

(3.6)

The estimation of direction is iterated around the first guess direction while keeping the entry
point and time fixed until GOF(fcone ) is maximized. Once the best fit direction is determined,
a second iteration is performed for all events. At this point, if the track direction is changed,
the entry time is also allowed to change. The entry time is calculated with hit PMTs occurring
within a 26.7 nsec time window around the track. Then, the GOF(fcone ) is again calculatedwith Ccut in Eq. (3.4), replaced by Ccut = 0.65. After the second maximization, the number of
early hit PMTs under the condition, δti − tmean (qi ) > 250 are counted, when larger than 35.
These muons are categorized as Multiple muon-2. Any other remaining event is classified as athrough-going muon. Additionally, Muboy calculates another goodness parameter for upwardgoing muon reduction produced by the high-energy atmospheric neutrinos by interacting withthe surrounding rock of the detector. The goodness is defined as:
P
qi
(3.7)
Fcone = P ,QiP
whereP qi is the sum of all detected charges remaining after the cleaning cuts explained aboveandQi is sum of all detected charges in the Cherenkov ring. The distribution of both
GOF(fcone ) and Fcone can be found in Figure 3.2.

3.1.4

Event classification

For the categorization of muon events, all Single through-going muons with a track length
shorter than 7 m and the total number of hit PMTs less than 2,000 are examined for the
possibility to be Corner-clipping muons. Corner-clipping muon events scrape the edges of the
ID and have relatively short track lengths. If an entry point is near the top corner and satisfies
the following condition,
29

Events

3

×10

24
22
20
18
16
14
12
10
8
6
4
2
0

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

GOF

Events

3

×10
60
50
40
30
20
10
0

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Fcone

Figure 3.2: Distribution of two parameters, GOF and Fcone used in Muboy in assessing qualityof the fit.

Xentry cos θx + Yentry cos θyq> 0.05,
2
2
Xentry+ Yentry(3.8)

the event is classified as a Corner-clipping muon. Here, Xentry , Yentry are the coordinates ofthe entry position of a muon. Conversely, if the muon enters near the bottom corner and the
distance between the entry and exit points is less than 4 m, then this event is also classified asa Corner-clipping muon.
Then, Muoby then examines the remaining events as to whether they are Stopping muon eventsor not. To identify the Stopping muon events, the number of photo-electrons produced within
2 m (QID ) from the exit point in the ID and within 4 m from the exit point in the OD (QOD ),are counted. An event is classified as a Stopping muon, if it satisfies at least one of the following
requirements:• QID < 200 p.e.
• QID < 400 p.e. and QOD < 30 p.e.
• QOD > 30 p.e. and QID < 150 p.e.
• No OD data and QID < 150 p.e.

Muoby proceeds to calculate the track length of the Stopping muon. The track length isdetermined by considering the observed energy loss via Cherenkov light emission per unit track30

length, dE/dX, corresponding to the amount of detected light per unit track length. Figure. 3.3
shows an example of the dE/dX distribution in 0.5 m bins together with the stopping point.
The stopping point is when dE/dX becomes less than 40% of the average value of the first
1.5 m (Figure 3.3) and the muon track length becomes greater than the peak position of thedE/dX histogram.

Figure 3.3: The histogram of dE/dX distribution for the track length determination of Stoppingmuons [21]. The stopping point is determined at 25 m.

3.2

Reconstruction of decay electron

Low energy events with the total energy below 100 MeV are reconstructed using the hit patternof PMTs (direction), the timing information (vertex), and the number of hit PMTs (energy).
The method of vertex, direction and energy determination are explained as below.

3.2.1

Vertex reconstruction

In the low energy region, the charged particles with the energy of 20 MeV travel only 10 cm
in water. Therefore, the track length is negligible and can be assumed as the occurrence pointof the event. To reconstruct the vertex position, the timing information of each hit PMT isused. The reconstruction of the vertex position is performed by the maximum likelihood fit of
the timing information using an algorithm, called BONSAI (Branch Optimization Navigating
Successive Annealing Iterations) [22]. The residual time of each PMT hit is defined as:
tres, i = ti − ttof − t0 ,

31

(3.9)

where, ti is the hit time of the i−th PMT, the ttof is the time of flight between the vertex
position (x, y, z) to the hit PMT, and t0 is the generated time of the charged particle. Using
the residual time information, the likelihood function is defined as:L(⃗x, t0 ) =NhitXlog {P (tres, i )},(3.10)

i=1

where ⃗x is a candidate vertex position and P (tres, i ) is the probability density function of theresidual time calibrated by the LINAC calibration device, as shown in Figure 3.4. The likelihood
function L(⃗x, t0 ) is calculated at each point while changing ⃗x in the ID, and the position wherethe value becomes the maximum is determined as the vertex position of that event. The numberof Cherenkov photons emitted by a charged particle differs according to its energy, resulting
in the difference in the number of hit PMTs. Therefore, the accuracy of vertex reconstruction
depends on the energy of the charged particle. Figure 3.5 shows the energy dependence of thevertex resolution obtained by the LINAC calibration. Here, the resolution of the reconstructedvertex position is defined as the reconstructed events in the region about 68% (1σ) of the
distribution of the distance between the true and the reconstructed vertex positions.

Figure 3.4: Probability density function of the residual time tres, i obtained from LINAC calibration. The peaks around 40 nsec and 110 nsec are the PMT late-pulses [23].

3.2.2

Direction reconstruction

The Cherenkov light ring pattern is used to reconstruct the direction. The direction is determined by maximizing the likelihood of the following function:
⃗ =
L(d)

N20
X

log {f (cos θi , E)} ×

i=1

32

cos θi,a(θi )(3.11)

Figure 3.5: Vertex resolution as a function of true electron kinetic energy in SK-IV [23].
where d⃗ is the traveling direction of the charged particle, and N20 is the number of PMTs
detecting light within 20 nsec around the time of the event. The distribution of the angularlikelihood function is given in Figure 3.6. f (cos θi , E) is a distribution function of the angle
θi formed between the traveling direction of the charged particle and the observed traveling
direction of photons. cos θi /a(θi ) is a correction term related to the solid angle. Here, θirepresents the incidence angle of photons to the PMT, defined in Figure 3.7. a(θi ) is a functionof the sensitive area of PMT, which depends on the incidence angle θi .

3.2.3

Energy reconstruction

In the low energy region, events with the total energy below 100 MeV emit a few Cherenkov
photons compared with the high energy events; hence, each PMT can only detect a single
photon on average. The charge resolution for 1 p.e. in a PMT is small and can be affectedby gain. The number of PMT hits is used for energy reconstruction instead of using detectedcharge. However, due to the attenuation effect of Cherenkov light in water and backgroundnoises originated from PMTs, the effective number of hit PMTs, Neff is used. Here, Neff iscalculated from the number of hit PMTs within the 50 nsec time window (N50 ) with somecorrections. Neff is defined as follows:

Neff ≡N50 Xi

Xi − ϵitail − ϵidarkS(0, 0)
Nall
×
× exp
×
Nalive S(θi , ϕi )



ri
Lieff




1
×
.
QEi (1 + C × Gi (t))(3.12)The definition of each parameter used in Eq. (3.12) are as follows.
• Xi : Occupancy
When an event occurs near the edge of the fiducial volume and Cherenkov light is emitted
33

Figure 3.6: Energy dependence of the angular likelihood function used in direction reconstruction [15].
toward the ID wall, multiple photo-electrons are detected in a single PMT. Thus, the effectof the multiple photo-electrons for each hit PMT must be considered. The expected
number of photo-electrons detected in a PMT is estimated by considering the occupancy
correction defined as follows:

Xi =


1



 log 1 − x





xi
3.0

i

(xi < 1),

(3.13)(xi = 1),

where xi is the fraction expressed as ni /Ni . Here, Ni is the number of PMTs around thei−th hit PMT and ni is the number of hit PMTs among Ni .
• ϵitail : Late hits
The Cherenkov light reflected by the PMTs or black sheet could enter a PMT outside thetime window 50 nsec. To correct these late signals, the correction factor given below, isapplied:N100 − N50 − Nalive × Rdark × 50 nsec.
(3.14)
ϵitail =N50
Here, N100 is the number of hit PMTs in the 100 nsec time window, Nalive is the numberof operating PMTs, and Rdark is the average dark rate ([hits/nsec]) of all PMTs in eachdata run.
• ϵidark : Dark noise
As irrelevant events to the physics analysis, PMT detects the so-called dark noise, such as
34

thermalized electron events or Cherenkov light emitted by radioactive isotopes containedin PMT’s glass. In the low energy region, the expected observed photoelectrons in each
PMT is assumed as ∼1 p.e./PMT; thus, the photoelectrons caused from dark noise events
are non negligible and these hits must be subtracted from the total hits. The rate of darknoise fluctuates with time and must be corrected according to the operating period. The
correction factor of dark noise is defined as follows:
ϵidark =iNalive × Rdark × 50 nsec × Rdark
.
i
N
50 R
P
dark
N50 ×
N50
i

(3.15)• Nall /Nalive : Dead PMTs
Considering defective PMTs in the ID, the correction scaling factor is applied. Nall is thetotal number of the PMTs (11,129 in SK-IV) and Nalive is the number of alive PMTs.
• S(θi , ϕi ) : Photo-cathode coverage
The detection efficiency varies according to the incident angle to the PMT. Therefore, theeffective photo-cathode area of the i−th PMT must be considered. S(θi , ϕi ) is definedas the correction function, where θi is the incident angle to the i−th PMT and ϕi isthe azimuthal angle, as shown in Figure 3.7. The distribution of S(θi , ϕi ) is shown in
Figure 3.8.
• exp (ri /Lieff ) : Water transparency
Cherenkov light is attenuated by the factor exp (−ri /Lieff ) because of scattering and absorption of photons in water. Here, ri is the distance from the reconstructed vertex to theposition of i−th PMT and Lieff is the effective water transparency in pure water measured
by using decay electrons from cosmic-ray muons.
• 1/(QEi (1 + C × Gi (t))) : Correction of quantum efficiency
Quantum efficiency varies with each PMT; thus, the correction must be considered. Here,QEi is the quantum efficiency for i−th PMT, Gi (t) is the gain of a PMT, and C is thecoefficient set as 0.226 for SK-IV phase.
Considering corrections above, Neff is determined and the energy of the charged particles isreconstructed. The relationship between Neff and the reconstructed energy (MeV) is given by
the following equation obtained by the LINAC calibration, and the result of MC simulation.
 5
P


ai (Neff )i(Neff < 234.1),
i=0Erec =(3.16)

30.0 + 0.131 × (Neff − 234.1) (Neff ≥ 234.1),
where a0 = 7.59×10−1 , a1 = 1.26×10−1 , a2 = −1.99×10−4 , a3 = 2.24×10−6 , a4 = −9.38×10−9
and a5 = 1.39 × 10−11 , respectively.

35

Figure 3.7: The θi and ϕi coordinates for the i−th hit PMT [15].

Figure 3.8: The distribution of photo-cathode coverage function S(θ, ϕ): for PMTs on thebarrel of the tank (left) and for PMTs on top and bottom of the tank (right) [24].

36Chapter 4
Muon and decay electron simulation
To optimize the selection criteria for the data collection of muon decay events, a dedicatedsimulation has been developed. The process and the scheme of the detector simulation for
stopping muon and decay electron is discussed as follows.

4.1

SK detector simulation

To understand the detector’s response to the muon and decay electron process, the SK detector
simulation based on GEANT3 tool-kit [25] was used, called SKDETSIM [16]. This simulation
provides common physics reactions, including decay process, energy loss, scattering and attenuation of Cherenkov photon in water, and the electronic systems response. Thus, to optimize
the selection criteria for cosmic-ray muons observed in the detector, the whole muon decay
process was simulated by using this tool.

4.2

Muon simulation

To generate muons properly, the muon simulation code called MUSIC (Muon SImulation Code)was used. MUSIC is the three-dimensional simulation of the muon propagation through a slant
depth X of a matter, which depends on the topographical location and the thickness of thesurrounding rocks. Figure 4.1 shows the location and altitude of the surrounding mountains of
the SK detector. The slant depth is calculated from the muon entering point at the mountainsurface with the specific altitude and its incoming direction toward the detector, as shown in
Figure 4.2. The simulation provides the average muon energy, flux, and rate by calculating themuon energy loss, angular deviation, and lateral displacement considering muon interactions
with matters, including ionization, pair production, Bremsstrahlung, and inelastic scattering.
The simulation has been performed for several underground experiments, including numerous
testings with the comparison of simulated results against the experimental data. The energy
spectrum and the muon flux at the SK depth are shown in Figure 4.3 and Figure 4.4, respectively. The average muon energy at SK is approximately 271 GeV [26]. The muon flux at SK
is 1.48 × 10−7 cm−2 s−1 and its rate inside the tank is 2.4 Hz [26]. As shown in Figure 4.4, themuon flux varies according to its incoming direction and low flux are reflects the thickness of
the rocks. The precise calculations and complete descriptions can be found in [27], [28], [26].

37Figure 4.1: Mountain Ikenoyama topographical map in 50 m grid. The color represents themagnitude of the altitude and the black point is the location of SK detector [29].

4.2.1

Generating muons

To simulate the muon decay events considering the SK topological structure, the generated
energy range must be adjusted based on muon energy on the ground level. Furthermore, thezenith dependence of injected muons reproduces the actual experimental data. The muon
generation with MUSIC considering the zenith dependence at SK depth has been studied andthe calculations are in good agreement with the data [31]. The procedure of muon injection is
performed as the following steps:• Energy range
The generated muon energies are adjusted to a certain range to simulate stopping muons.
From Figure 4.3, the energy spectrum in Eµ < 20 GeV is almost flat, equivalent to
the energy range of stopping muons. Figure 4.5 (left) shows the injected muon energy
distribution in the simulation. To estimate the number of stopping muons producing
decay electrons in the ID in simulation, its survival probability was examined. The
survival probability is calculated as the number of stopping muon events accompanying
decay electrons divided by the total injected muons. Figure 4.5 (right) shows the survival
probability for stopping muons and most muons with energies greater than 15 GeV are
through-going muons penetrating the ID without decaying. Figure 4.6 explains thatmuons with a higher (lower) energy are more likely to decay in the bottom (top) side of
the ID tank. Thus, the upper limitation of muon energy of 20 GeV is sufficient to only
obtain stopping muons.
• Zenith angle dependence
The zenith angle distribution of generated muons in MC simulation is shown in Figure 4.7
obtained from the calculation of Ref. [31].

38χ (m)

Se
al
ev
el
(0

Altitude (m)m)

SK site height (330 m)

Figure 4.2: Schematic of vertical cylindrical detector of SK with coordinate system. θ isdefined to be the zenith angle and ϕ is the azimuthal angle. The slant depth χ denotes thetraverse distance by muon in the rock between entering position of the mountain at the specificaltitude (m) to the center position of the detetctor located at the certain depth underground,
330 m from sea level [30].

4.3

Decay electron simulation

The stopping muon arriving at the ID in the SK tank typically decays in three bodies accompanied with electrons and two neutrinos, as shown in Eq. (1.2). The decay electron from themuon can be detected as a clear signal and an independent event, also useful in evaluating
the stability of the energy scale in the energy region below 100 MeV. The muon lifetime τµis explained by the V-A (vector-axial vector) theory of weak interaction and know to be approximately 2.197 µsec [32]. Figure 4.8 shows the distribution of the decay time of stoppingmuons in the simulation (without SK detector simulation). The lifetime of muon samples inthe simulation can be measured by fitting the decay curve function to the histogram defined
as:





t
∆t
exp − ,
(4.1)
N (t ∼ (t + ∆t)) = N0 1 − exp −ττwhere N0 is the number of observed muon events, ∆t is the binning width in the histogram set
as 100 nsec, and τ is the lifetime of muon. In the function, N0 and τ are the free parameters to
be fitted. The derivation of Eq. (4.1) is given in Chapter 6. The energy distribution of a decay
electron generated from a muon in a rest frame can be written as follows:
m2µ EG2
dN
4E
=
1−
,
(4.2)
dE
4π 3
3mµ
where G is Fermi coupling constant, and mµ is the mass of muons. According to the conservations of energy and momentum in the rest frame of a muon, the decay electron can
39

Figure 4.3: Cosmic muon energy spectrum at three underground experiment site [26]. The
average muon energy at SK is Eµ = 271 ± 2 GeV. Muon energy spectrum for Eµ ≤ 20GeV isalmost flat.
carry Emax ≃ mµ /2 ∼ 52 MeV at the maximum. Figure 4.9 shows the energy distributionof decay electron from positive muon together with the reconstructed energy in the SK. The
reconstructed energy distribution of decay electrons reaches up to about 70 MeV due to thedetector’s energy resolution. The reconstructed vertex distributions of decay electrons in thesimulation are also obtained, as shown in Figure 4.10. The generated electron events are uniformly distributed in the horizontal (X and Y) and vertical (Z) coordinates respectively.

40

1

×80
10-9

0.9

70

0.8
60

0.7

50

0.6
0.5

40

0.4

30

0.3
20

0.2

10

0.1
0
0

Muon flux [cm-2s-1sr-1]

cosθ-1]θ vs. φ
srcos

50

100

150

200

250

300 350
φ[deg]

0

30

×103

Survival probability

EventsFigure 4.4: Muon flux as a function of zenith and azimuthal angle at SK depth and location
calculated with MUSIC.

25

20

1

0.8

0.6

15
0.4
10
0.2

5

0
0

5

10

0
0

15
20
25
Muon energy[GeV]

5

10

15
20
25
Muon energy[GeV]

Figure 4.5: Energy distribution of injected muons (left) and its survival probability defined as
the ratio of injected muons to stopping muons per 0.5 GeV bin (right). The minimum muonenergy is set at least 0.1 GeV at the generated point of frame of ID to travel inward the ID
tank.

41

Vertex Z[cm]

2000
3500

1500
3000

1000
2500
500
2000
0
1500

-500
-1000

1000

-1500

500

-2000
0

5

10 15 20 25 30 35 40 45 50
Muon energy[GeV]0

Figure 4.6: Generated muon energy and the Z coordinate of its reconstructed decay electronsin MC.

Z noitcerid yobum
×10-1

20.2
0.0
80.18
10.0
60.16
10.0
40.14
10.0
20.12
10.0
10.1
0.0
80.08
00.0
60.06
00.0
40.04
00.0
20.02
00.0
00
00 10.1.0- 20.2.0- 30.3.0- 40.4.0- 50.5.0- 60.6.0- 70.7.0- 80.8.0- 90.9.0- 11Zrid
Muon directioncosθ

Figure 4.7: Zenith angle distribution of generated muons in MC calculated from [31]. The
angular integration covers the entire hemisphere. Blue line and red line represent injected andreconstructed muons, respectively.

42

Events/100[nsec]

×103
14

τµ = 2199.5 ± 4.9 nsec

12

χ2/d.o.f = 90.5 / 81

10
8
6
4
2
0
0

1

2

3

4

×103
5
6
7
8
9 10
Time difference[nsec]

Figure 4.8: The true decay curve of stopped muons observed in MC. The fitting range is red
solid line and the expected distribution after fitting the function is red dashed line.

40 ×10Normalized number of events

-3

35

Rec

30

Init

25
20

15
10
5
0
0

10

20

30

40

50
60
70
80
Electron energy[MeV]

Figure 4.9: The kinetic energy distribution of decay electrons from positive stopping muonsin MC. The true energy distribution is red solid line and reconstructed energy distribution isblack solid line. Events are required to be positioned more than 200 cm away from the ID wallin the SK detector (Fiducial volume cut).

43Events

Events

1800

1800

1600

1600

1400

1400

1200

1200

1000

1000

800

800

600

600

400

400

200

200

0
-2000 -1500 -1000 -50000-2000 -1500 -1000 -500

500 1000 1500 2000
Vertex X[cm]500 1000 1500 2000
Vertex Y[cm](b) Y coordinate
2000

Vertex Y[cm]Events(a) X coordinate

0

1400

40

1500

1200

35

1000

1000

30

500

800

25

0

20

600

-500

15

400

-1000

10

200

-1500

5

0
-2000 -1500 -1000 -5000500 1000 1500 2000
Vertex Z[cm]-2000-2000-1500-1000-500(c) Z coordinate

0500 1000 1500 2000
Vertex X[cm]0(d) X-Y coordinate
×10-3

Vertex Z[cm]

2000

0.35

1500
0.3

1000

0.25

500

0.2

0
-500

0.15

-1000

0.1

-1500

0.05

-2000
-2000-1500-1000-500

0500 1000 1500 2000
Vertex X[cm]0(e) X-Z coordinate

Figure 4.10: Vertex distribution of reconstructed decay electrons in MC. Here, the fiducial
volume cut is applied.
44

Chapter 5
Event selection
Muon charge ratio is measured by using the time information obtained from stopped muons (referred to as the stopping muon) in the ID tank and its decay electron events. The measured
rates of the cosmic-ray muons in the mine are around 2 Hz and approximately 3.56% [24] of
their events consist of stopping muon. To distinguish the muon decay signal events from theobserved data efficiently, the evaluation of the event reconstruction and optimization of selection criteria are needed. In this chapter, the selection procedure of the muon decay candidates
in the data using the optimized variables driven by the MC is described. The results of thedetector simulation with stopping muons and its decay electrons described in Chapter 4 areused to determine the selection criteria. Finally, the selected stopping muon and decay electron
candidates obtained from the data are shown.

5.1

Data set of control samples

Since obtaining the data from April 1st 1996, the observations have been performed in several
phase depending on the detector conditions. In the experimental phase from SK-I to SK-III,
the analog timing module (ATM) was used for the data acquisition. In the ATM system, thecharge leak of the capacity was occurring due to the large pulse height events; time informationof the decay electron event was affected by the prompt muon event whose charge is large. In
the SK-V phase, the calibration data is not enough compared with that of SK-IV, and theuncertainty in the energy determination is twice larger than that of the SK-IV. Therefore, forthe charge ratio measurement, the data analysis was focused on SK-IV data taking period asshown in Table 2.1. The 3 month data (82.4 live time), from November 2017 to January 2018,
was used as a control sample for the evaluation of the reconstruction and the determination of
the selection criteria.
The topological schematic of a stopping muon and its decay electron event observed in the SK
detector are shown in Figure 5.1 and Figure 5.2, respectively. The Cherenkov ring patternsof stopping muons are much clearer than those of decay electrons due to the difference in thenumber of emitted photons while traveling.

45

OD

Top

ID

BottomFigure 5.1: Typical ring pattern of a stopping muon in the SK detector (ID and OD). The
gradation of the color in each point represents the amount of charge observed in a PMT.

Electron energy~24 MeVFigure 5.2: Typical ring pattern of a decay electron in the SK detector (ID). The gradation of
the color in each point represents the amount of charge observed in a PMT.

5.2

Selection criteria

The data selection of stopping muons and decay electrons are processed in the order shownin the flow chart of Figure 5.3. As a first step, the stopping muons are selected among therecorded data based on their timing and topology described in Section 3.1 in Stopping muon
selection. After selecting stopping muon samples, the decay electrons are searched within several
microseconds after the entering time of stopping muons in Decay electron selection. Decay
electron candidates are selected based on its vertex, energy, and fitting quality by evaluating46

with the variables obtained in the MC simulation. The selection criteria for each section are
discussed as below.
82.4 days sample data

Stopping muon selec6on

Decay electron selection

Final sample

Figure 5.3: The flow chart of µ-e decay event selection for SK-IV data analysis.

5.2.1

Selection for stopping muon

Among the cosmic-ray muon events observed in the data, a Single topping muons are used inthis analysis accompanied by the with decay electrons. To select the stopping muon candidates,
several selection criteria are applied to the raw data as follows:• Muon status
Muon events observed in the SK tank are classified in six categories by using a muon
fitter according to its track property, as described in Chapter 3: (0) No-fit, (2) Single
through-going muon, (3) Stopping muon, (4) Multiple muon-1, (5) Multiple muon-2, (6)Corner-clipping muons. In this analysis, a muon event categorized as (3) Stopping muon
are considered.
• Total Q cut
In the low energy region, background contamination originated from charged particles or
γ rays emitted from the radioactive sources existing in the SK detector and surroundingrocks could affect the data analysis [17]. Figure 5.4 shows the distribution of total deposit
charge1 of stopping muon candidates in the data. To reduce irrelevant background eventsfrom the data, events with the total charge ≤ 1500 are rejected.
• Direction cut
Almost all stopping muon events observed at the SK detector are cosmic-ray muons generated directly in the atmosphere. However, there is a possibility that upward-going muon
induced from neutrino interactions can be detected and regarded as background eventsof the cosmic-ray muons [33]. Since an incidence angle of cosimc-ray muons is downward
toward the SK detector, upward-going muons can be distinguished from downward-going1

Total sum of p.e. charge in an event observed by ID PMTs.

47

Events

muons by applying the direction cut. The incidence direction of stopping muons is calculated from its track by using a muon fitter. Figure 5.5 shows the zenith angle distributionof stopping muons, before and after the direction cut. From the zenith angle distributionof cosmic-ray muons as shown in Figure 4.7, events satisfy the upward-going direction
region, cos θ < 0 are rejected in this analysis.

1000

800

600

400

200

0

50

100

×103
150
200
250
Total charge[p.e.]

Figure 5.4: Total charge of stopping muons in the control sample.

5.2.2

Selection for decay electron

From the information of prompt muon events, decay electron candidates are selected by applying
several reduction cuts as follows:• Fiducial volume cut
Low energy events originated from external γ-rays from the radiation source in the structure of the SK detector and surrounding rocks are the major backgrounds for decay electron analysis (∼ 10 MeV). These events can be reduced by requiring the vertex positionof decay electron 200 cm inwards from the ID wall. Figure 5.6 shows the reconstructedvertex√of decay electrons in X-Y, and X-Z coordinates. Events reconstructed in the range,
R = X 2 + Y 2 < 1490 cm and, |Z| < 1610 cm in the cylindrical coordinate system areused in the analysis.
• Ovaq cut
To assess the qualities of the vertex reconstruction, Goodness is defined. The parameter
Goodness is defined as:48

Events

4.5

×103

4
3.5
3
2.5
2
1.5
1
0.5
0
-1 -0.8 -0.6 -0.4 -0.20

0.2 0.4 0.6 0.8 1
cosθ

Figure 5.5: Zenith angle distribution of stopping muons in the control sample. θ is defined as
the zenith angle, where cos θ = +1 correspond to vertically downward direction and cos θ = −1
to vertically upward direction. Almost all stopping muons are coming from the top side of the
SK tank toward the bottom.

PiGoodness =!2 

exp −

ti − ttof, i − t0t − ttof, i − t0
 exp − i √
√
2ω
2σ

!2 
ti − ttof, i − t0√
exp −
2ω

!2 

,

(5.1)

where ω is the resolution of the ti − ttof, i distribution and σ is the timing resolution of
PMT. The parameter Goodness takes a value from 0 to 1, with 1 representing the best
quality case. The parameter DirKS is defined to evaluate the uniformity of the hit PMTs
from its direction. The definition is:
DirKS =

max {̸ uniform (i) − ̸ data (i)} − min {̸ uniform (i) − ̸ data (i)}
,
2π

(5.2)

where ̸ uniform (i) is the azimuthal angle of the i−th hit PMT assuming that the PMT
hits are uniformly distributed along the Cherenkov ring and ̸ data (i) is that of the actual
hit PMT. DirKS takes a value in the range 0 to 1, and 0 represents the best quality
case. Using the two parameters, Goodness and DirKS above, the parameter Ovaq (Onedimensional variable of Vertex and Angular reconstruction Quality) indicating the qualityof event reconstruction, can be defined as follows:
Ovaq ≡ Goodness2 − DirKS 2 .
49

(5.3)

Here, Ovaq takes a value from −1 to 1, and 1 represents the best quality case. According
to the distribution of Ovaq on MC shown together with that of data in Figure 5.7, events
whose Ovaq is lees than 0.3 are regarded as mis-reconstructed and are rejected in thisanalysis.
• Electron energy and the number of hit PMTs within 50 nsec window(N50 ) cuts
The energy distributions of decay electrons from stopping muons in both data and MC areshown in Figure 5.8. In the low energy region, γ-rays from radioactive sources, including
Rn and µ− nucleon captured by 16 O nuclei are contained as background events. The
exited 16 N nuclei is also produced via the following process:µ− +16 O −→ νµ +16 N(5.4)

The maximum energy of de-excitation γ-rays from exited 16 N nuclei is 10.8 MeV. The
small peak around 6 MeV in Figure 5.8 is due to the γ-rays produced by exited 16 N.
To eliminate low energy events with the background contamination, Eelectron < 8 MeV
is applied for the rejection. As the same analogy, events with N50 < 50 (∼ 8 MeV) arerejected in this analysis.
• Distance cut
The distribution of distance between the stopped position of stopping muons estimatedby the track length from their entrance position in the ID tank using the Muboy fitterand vertex of the decay electrons by using BONSAI fitter peaks around 80 cm gradually
decreases as the distance becomes larger as shown in Figure 5.9. Due to the uncertaintyof the stopping point of muons and vertex resolution of decay electrons, the distance iswidely spread apart from the actual value from 0 cm. Events in the long distance region
are expected to be mis-reconstructed or background events. Hence, the appropriate cut forthe distance was determined by examining the detecting efficiency as a function of distancecut. According to the result in Figure 5.10, the distance cut, with distance < 300 cm has
the detecting efficiency of 98% and seems to be a suitable to select sufficient signal events
for the data analysis.
• Time difference cut
The time difference is obtained by measuring the timing information between prompt
muon events and delayed signal events, expected as the decay electrons. Prompt muon
events fire the LE trigger due to its numerous hit PMTs and save the sub events between
in the time region of [−5, +35] µsec as explained in Section 2.5.2. Since the muon mean
lifetime is about 2.197 µsec (in vacuum), the time window of LE trigger is wide enoughto save signal from decay electrons. Decay electron candidates expected to exist in thetime region [−5, +35] µsec, are searched by performing sub-trigger search (offline) using
the method like SLE trigger. The sub event with 1.3 µsec time window is stored when
the number oh hit PMTs within 200 nsec (N200 ) is larger than 20. Figure 5.11 shows aconceptual diagram of PMT hit timings from the prompt stopping muon and the decay
electron. The two peaks occurring from the parent muon and its decay electron are
distributed in less than 1.3 µsec. Since muon and decay electron are successive events,
decay electron can occur before the mean lifetime of muon, 2.2 µsec from the arrival timeof stopping muon. Figure 5.12 shows the signal of the muon decay event with a timedifference between the muon and the decay electron less than 1.3 µsec. In Figure 5.12,
the signal from a decay electron is apart from muon signal aby bout 800 nsec. A decay
50Vertex Z[cm]2000

Vertex Y[cm]

2000
60

1500
1000

50

500

40

60

1500

0

1000

50

500

40

0

30

30
-500

-500
20

20
-1000

-1000
10

-1500
-2000
-2000-1500-1000-500

0500 1000 1500 2000
Vertex X[cm]

10

-1500

0

-2000
-2000-1500-1000 -500(a) Vertex X-Y

0

500 1000 1500 2000
Vertex X[cm]0(b) Vertex X-Z

Figure 5.6: Vertex distribution of reconstructed decay electrons in the data. Muon reduction
cuts, fiducial volume cut, and time difference cut for decay electrons are applied.
electron event with a shorter time difference can mis-identify hit PMTs from the muon
longer tail hits as a decay electron. These contaminated hit PMTs originated from theparent muons are difficult to distinguish from that of an electron signal. A part ofthe signal tail of a parent muon can be integrated in the time window of sub eventand the event reconstruction can be affected. However, as shown in Figure 5.12, thedecay electrons with a longer time difference are distributed according to the Michel
decay spectrum, in the shorter time region, and their events are mostly distributed below
20 MeV. Thus, to avoid including mis-reconstruction events, the sub events occurring
between 1.3 µsec < t < 30 µsec from a prompt event are only selected.

5.3

Selected samples

5.3.1

Stopping muon candidates

The distribution of selected muon samples in the control samples are evaluated by scanning the
distribution after applying the selection cuts in the previous selection to confirm the data quality. Figure 5.13 shows the reconstructed 2D vertex distribution of stopping √muons in X-Y and
R-Z coordinates, where X and Y correspond to the horizontal plane (R = X 2 + Y 2 ) and Z
corresponds to the vertical direction, respectively. According to Figure 5.13, most muons enterfrom the top of the tank from south, with a relatively shorter slant depth2 . The corresponding
angular distribution for stopping muons is given in Figure 5.14 and Figure 5.15. Muon angular
distribution depends on the topography around Mount Ikenoyama and indicates that the intensity of stopping muons depends on muon travel length in the surrounding rocks. The GOF
parameter, representing the reconstruction quality of muon event, as described in Section 3.1
is distributed as shown in Figure 5.16. Comparing the distribution of the goodness of stopping
2

The altitude of the mountain is about 1,000 m, however, the slant depth can be 650 m which depends on
the incoming direction of the muon.

51

0.25

Normalized number of events

Normalized number of events

0.3
MC
Data

0.2

0.15

0.16

MC
Data

0.14
0.12
0.1
0.08
0.06

0.1

0.04
0.05
0.02
0
0

0
0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Bonsai goodness

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
DirKS

Normalized number of events(a) Goodness(b) DirKS
0.18
0.16

MC
Data

0.14
0.12
0.1
0.08
0.06
0.04
0.02
0
0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Ovaq(c) Ovaq

Figure 5.7: Distributions of the quality of fitting parameters used for decay electrons in thedata (black line) and MC (red line). Muon reduction cuts, fiducial volume cut, and timedifference cut are applied.

52

Normalized number of events

Normalized number of events

0.03

MC
Data

0.025

0.03

0.025

0.02

0.015

0.02

0.015

0.01

0.01

0.005

0.005

0
0

MC
Data

10

0
0

20
30
40
50
60
70
80
Reconstructed electron energy[MeV]

50

100

150

200

250

300 350 400
N50[hit/50nsec]

Normalized number of events

Figure 5.8: Distributions of reconstructed energy (left) and N50 (right) of decay electrons in
the data (black line) and MC (red line). Muon reduction cuts, fiducial volume cut, and timedifference cut are applied.

0.08
MC
Data

0.07
0.06
0.05
0.04
0.03
0.02
0.01
0
0

100 200 300 400 500 600 700 800 900 1000
Distance[cm]Figure 5.9: Distribution of distance between reconstructed stopped position of stopping muonand vertex of decay electrons in data (black line) and MC (red line). Muon and decay electron
reduction cuts are applied.

53

Total integrated ratio [%]Events/Total events [%/10cm]

10
9
8
7
6
5

100
90
80
70
60
50

4

40

3

30

2

20

1

10

0
0

100

200

300

400

500

0
0

600 700 800
Distance [cm]

100

200

300

400

500

600 700 800
Distance [cm]Figure 5.10: The fraction of muon events per 10 cm distance bin to the total events (left) and
its detecting efficiency (right) as function of distance in MC. The reduction cuts described inabove for both muons and decay electrons are applied.

Number of hit PMTs

Prompt event : 𝜇

Sub event“Decay electron
candidate”

Time

𝛿𝑡

start

1.3 µsec

30 µsec

Figure 5.11: Schematic view of the prompt signal and delayed signal in the time window.

54

Reconstructed electron energy[MeV]

60

100

50

80

40
60
30
40
20
20

10
0
0

500 1000 1500 2000 2500 3000 3500 4000
Time difference[nsec]

0

Figure 5.12: Distribution of decay electron energy and muon lifetime in the data after applying
fiducial volume cut.
muons between data and MC, events peak at 0.6 and are in good agreement with each other.
Figure 5.17 shows the distribution of muon track lengths in the ID tank for stopping muon
events. Since the incidence direction of the muon events with the energy range from 100 MeV
to 20 GeV has the a zenith angle dependence; muons with the small (large) energy are more
likely to stop (travel) closer (further) to the enter position of the tank. The relationship between the track length and total deposit charge for a stopping muon is shown in Figure 5.18.
The total deposit charge increases in proportion to its track length and found as approximately
25 p.e. per unit cm corresponding to the muon energy deposit per cm, ∼ 2 MeV/cm obtained
from Bethe-Bloch formula.

5.3.2

Decay electron candidates

The data quality of selected decay electron events after applying the selection cuts are examined by checking the distribution of reconstructed decay electron events for whether the bias
exists in the data or not. The reconstructed vertex position of decay electrons in X, Y and Z
coordinates are shown in Figure 5.19. The selected events distribute uniformly in the horizontal
coordinate (X and Y), while, in the vertical coordinate (Z), events are rather distributed upperside (Z > 0) of the tank. This is because most stopping muon events arrive from the top of thetank and lose their energies while propagating in the water and finally decay before reaching
the lower side (Z < 0) of the tank. The observed decay curve of the time difference is shownin Figure 5.20. The lifetime of selected muon decay samples can be measured by fitting the
decay curve function Eq. (4.1) to the histogram. Applying the fitting function Eq. (4.1) to thehistogram in the range of 1300 nsec to 8500 nsec, the τ is finally obtained as 2048.1 ± 6.5 nsec.
The lifetime obtained from the selected samples is slightly shorter than that obtained from
Figure 4.8 due to the contribution from the shorter lifetime of negative muons.
55

2000

Vertex Z[cm]Vertex Y[cm]

2000

1400

1500

3500

1500

1200

1000

3000

500

2500

0

2000

-500

1500

-1000

1000

200

-1500

500

0

-2000
0

1000
1000
500
800
0
600
-500
400

-1000
-1500
-2000
-2000-1500-1000-500

0500 1000 1500 2000
Vertex X[cm](a) X-Y

250 500 750 1000 1250 1500 1750 2000
R [cm]

0(b) R-Z

Figure 5.13: Enter position of stopping muon in the SK tank for X-Y (left) and R-Z (right)
coordinates in the data.

56

cosθ

1

35

0.9
30

0.8
0.7

25

0.6
20
0.5
0.4

15

0.3

10

0.2
5

0.1
0
0

50

0

100 150 200 250 300 350
φ[deg]

EventsFigure 5.14: Intensity of stopping muons arriving at SK detector as a function of azimuthaland zenith angle in the data. ϕ = 0◦ and ϕ = 90◦ corresponds to east and north, respectively.

9000
8000
7000
6000
5000
4000
3000
2000
1000
0-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

Events

cosθ

1400
1200
1000
800
600
400
200
0

50

100

150

200

250

300

350

φ[deg]

Figure 5.15: Azimuthal and zenith angle distribution of stopping muons in the data.

57

Normalized number of events

0.14

MC
Data

0.12
0.1
0.08
0.06
0.04
0.02
0
0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Muboy goodness

EventsFigure 5.16: Distribution of Muboy goodness for stopping muon in the data (black line) andMC (red line). Almost all muon events are distributed around 0.6.

5000

4000

3000

2000

1000

0
0

5

10

15

20

25

30

35 40 45 50
Track length[m]Figure 5.17: Distribution of track length for stopping muons in the data.

58

500 ×10

×103

600

EventsTotal charge[p.e.]

3

450

25

500

400
350

20

400

300
250

15

300

200
150
100

200

10

100

5

0

0
0

50
0

5

10 15 20 25 30 35 40 45 50
Track length[m]10

20

30 40 50 60 70 80 90 100
Total charge/Track length[p.e./cm]

Figure 5.18: The relationship between track length and total deposit charge (left) and the
distribution of total charge divided by track length (right) for stopping muons in the data. The
constant of proportionality is approximately 25 p.e per unit cm.

59

Events

3000

2500

2500

2000

2000

1500

1500

1000

1000

500

500

0
-2000 -1500 -1000 -50000-2000 -1500 -1000 -500

500 1000 1500 2000
Vertex X[cm](a) X coordinate
EventsEvents3000

0500 1000 1500 2000
Vertex Y[cm](b) Y coordinate

2400
2200
2000
1800
1600
1400
1200
1000
800
600
400
200
0
-2000 -1500 -1000 -5000500 1000 1500 2000
Vertex Z[cm](c) Z coordinate

Figure 5.19: Vertex distribution of selected reconstructed decay electrons in the data.

60

Events/100[nsec]

8

×103

7

τµ = 2047.8 ± 6.6 nsec

6

χ2/d.o.f = 70.0 / 70

5
4
3
2
1
0
0

1

2

3

4

×103
5
6
7
8
9 10
Time difference[nsec]

Figure 5.20: The decay curve of stopped muons observed in sample data. The fitting range is
red solid line and the expected distribution after fitting the function is red dashed line.

61

Chapter 6
Results and discussion
6.1

Calculation of charge ratio

Positive muons are free from nuclear capture and their lifetimes are equal to τµ in vacuum,
which corresponds to τµ = 2.1969811 ± 0.0000022 µsec [32]. However, a few negative muons
are captured by the 16 O nuclei in the target, resulting in a shorter lifetime of negative muonsthan that in vacuum. When a negative muon is stopped in a medium, it replaces the innermost
electron of the nucleus and forms a muonic atom as shown Figure 6.1.

decay

𝛾

𝑒

𝜇!
capture𝜇!

𝑵𝒖𝒄𝒍𝒆𝒖𝒔𝑒

K shell𝑒

Figure 6.1: Schematic of the muonic atom. A negative muon takes an atomic orbit (K shell)and creates an excited nucleus. The negative muon follows either process, free muon decay inorbit of nuclear muon capture.
In the ground state, a negative muon follows one of two competing processes; free muon decay (µ− → e− + ν̄e + νµ ) or nuclear muon capture (µ− + (A, Z) → νµ + (A, Z − 1)). The lifetimeof the negative muon can be expressed as two components:τµ−1− = Λd + Λc ,(6.1)

where Λd is the free decay rate, and Λc is the capture rate of the negative muon by a nucleus,
respectively. Here, the nuclear capture rate is given as,62Λc = Λ1 Z 4 ,(6.2)

where Λ1 is the nuclear capture rate in the hydrogen and Z is the charge of the nucleus (corresponds to the proton number). Consequently, the lifetime of the negative muon depends on the
Z number and it becomes shorter in heaver nuclei (Z ≤ 100) [34]. Measurements of the lifetimesof negative muons have been performed in various media in the previous studies [35]. The mean
lifetime of negative muons in pure water is obtained as τµ− = 1.7954 ± 0.0020 µsec [35]. The
lifetime difference between the positive and negative muons allows the estimation of the ratioof the muons by measuring the time of the muon decay. In general, the number of decayed
particles in time difference dt can be expressed as:
dN (t) = −N λdt,

(6.3)

where N (t) is the number of particles at a time t, λ (= 1/τ ) is the probability that a particle
decay per unit time, and τ is the mean lifetime. The decay function of particle is given as
follows by taking the integral of Eq. (6.3):
Z t
dN
dt,
= −λ
0
N0 N
N (t) = N0 exp (−λt).

Z N

(6.4)

The number of decayed particles in an unit time is:dN (t)
N0t−=
exp − ,
dt
τ
τ

(6.5)

which corresponds to the observed decay electron events from parent muons. Furthermore,considering the binning width ∆t of the histogram, the expected events between t to t + ∆tare:Z t+∆t dN (t)N (t ∼ (t + ∆t)) =
−
dt
dt
t





∆t
t
= N0 1 − exp −exp − .
τ
τ

(6.6)

Here, N0 represents the number of particles at t = 0. Muons measured by the detector, consistof two components, τµ+ and τµ− , therefore, the lifetime spectrum is fitted by the sum of two
exponential functions:




t
∆t
N (t ∼ (t + ∆t)) = N+ 1 − exp −
exp −
τµ+
τµ+





∆t
t
+ N− 1 − exp −
exp −
,
τµ−
τµ−


(6.7)

where ∆t is set to be 0.1 µsec in this study, and N+ , N− are the total number of detected
positive muons and negative muons, respectively. The number of observed negative muon
events seemingly decreases by the factor of nuclear capture in the water, and this effect must63

be considered. Since the water can be assumed as 16 O, the nuclear capture fraction for negative
muons in water is Λc = 0.184 ± 0.001 [35]. Considering nuclear capture effects on negative
muons, the charge ratio R(µ+ /µ− ) can be expressed by:R(µ+ /µ− ) =

6.2

N+
.
N− / (1 − Λc )(6.8)

Results

Finally, using the final samples of muons and decay electrons collected in SK-IV, the charge
ratio of cosmic-ray muons was measured. Due to the small residual magnetic field in the watertank [16], the SK can not distinguish the charge of the cosmic-ray muons in principle. However,using the analysis method described in previous section, the number of positive and negative
muons were obtained from the decay curve. The decay curve of stopping muons obtained fromthe time interval between the parent muon and delayed electron events are shown in Figure 6.2.
From the fitting of the decay curve of µ − e decay events observed in the data taking period in
SK-IV from 2008 to 2018, N+ and N− were obtained for each year, respectively. The results of
N+ , N− and the charge ratio R(µ+ /µ− ) calculated from Eq. (6.8) are listed in Table 6.1 with
χ2 /d.o.f . The fitting results of χ2 /d.o.f are also shown in Figure 6.3 along with N+ and N−
per day. The goodness of the fitting results for each data period are distributed between 0.6
to 1.0 and no biases can be found from these results. The event rates of decay electrons per
day are stable at around 1,400 events/day for the positive muon and 800 events/day for thenegative muon. The yearly variation of the charge ratio in the data period of SK-IV is shownin Figure 6.4. The combined result of the charge ratio in SK-IV is obtained by taking the mean
weighted value of eleven data points of the charge ratio and found to be 1.42 ± 0.02 (statisticaluncertainty only). The yearly charge ratios are distributed within 10% of the mean value. The
combined data points are plotted in Figure 6.4 together with the previous measurement at
Kamiokande-II [10]. The total charge ratio in the SK-IV phase is consistent within the error ofthat in Kamiokande-II.

6.3

Discussion

Comparing the muon charge ratio obtained in this analysis to other experimental results, thecharge ratio becomes higher at around 1 TeV than those measured below 500 GeV. As explainedin Chapter 1, the increase in the charge ratio in the TeV region is due to the contribution
from kaon decay to muon flux. In this energy region, the possibility of kaon decay to µ+
becomes larger than that of pion decay. Models of the muon production from secondaries in
the atmosphere provide a qualitative interpretation of the enhancement of the charge ratio
in the TeV region. One of the simplified models called the πK model gives the numericalvalues by considering several assumptions. This model only considers the contribution fromboth paticles; pion and kaon decay and ignores the contribution from the charm particle whose
effect becomes important above 10 TeV. Furthermore, the energy dependence of the charge
ratio is expressed in the form of Eµ cos θ. However, π/K production ratio is assumed to beenergy independent [36]. In this model, the muon charge ratio is parameterized as a function offπ+ and fK + as defined in Section 1.4 as the fraction parameter of all positive muons from π +and K + decay, respectively. The investigation of the fraction of secondaries has been performed
with several underground experiments for the numerical evaluation and reported both fraction
parameters as fπ+ = 0.55 and fK + = 0.67 [1], [9], [37]. The result obtained from the SK-IV data64

Event/bin

Event/bin

107
SK4 muon decay-e
SK4 2008 (55.9 days)
Fit resultµ plus 76893.1 ± 2260.8µ minus 48760.9 ± 2474.2
Charge ratio 1.287 ± 0.075
χ2/d.o.f = 182.236/ 189

106
105
104

107

105
104

103

103

102

102

10

10

1

1

10-1
0

5

10

15

SK4 muon decay-e
SK4 2009 (274.7 days)
Fit resultµ plus 385766.9 ± 6401.3µ minus 219775.7 ± 7690.5
Charge ratio 1.432 ± 0.055
χ2/d.o.f = 176.246/ 225

106

10-1
0

20
25
30
Time difference [µsec]

5

10

15

20
25
30
Time difference [µsec]

107

Event/bin(b) Decay curve of stopped muons in 2009107

Event/bin(a) Decay curve of stopped muons in 2008
SK4 muon decay-e
SK4 2010 (331.6 days)
Fit resultµ plus 456865.1 ± 7252.0µ minus 268920.7 ± 8836.6
Charge ratio 1.386 ± 0.051
χ2/d.o.f = 176.510/ 228

106
105
104

105
104

103

103

102

102

10

10

1

1

10-1
0

5

10

15

10-1
0

20
25
30
Time difference [µsec]

10

SK4 muon decay-e
SK4 2012 (314.1 days)
Fit resultµ plus 461803.5 ± 7149.6µ minus 242548.0 ± 8642.7
Charge ratio 1.554 ± 0.060
χ2/d.o.f = 170.136/ 231

105
104

105
104

102

102

10

10

1

1
10

15

15

20
25
30
Time difference [µsec]

SK4 muon decay-e
SK4 2013 (328.2 days)
Fit resultµ plus 438973.3 ± 7050.2µ minus 256439.3 ± 8555.4
Charge ratio 1.397 ± 0.052
χ2/d.o.f = 173.918/ 225

106

103

5

10

107

103

10-1
0

5

(d) Decay curve of stopped muons in 20117

Event/bin

Event/bin(c) Decay curve of stopped muons in 2010
106

SK4 muon decay-e
SK4 2011 (321.6 days)
Fit resultµ plus 435989.6 ± 6955.3µ minus 245505.7 ± 8414.8
Charge ratio 1.449 ± 0.055
χ2/d.o.f = 167.355/ 224

106

10-1
0

20
25
30
Time difference [µsec](e) Decay curve of stopped muons in 2012

5

10

15

20
25
30
Time difference [µsec](f) Decay curve of stopped muons in 201365

Event/bin

Event/bin

107
SK4 muon decay-e
SK4 2014 (310.2 days)
Fit resultµ plus 388604.3 ± 6586.0µ minus 243742.3 ± 7980.1
Charge ratio 1.301 ± 0.048
χ2/d.o.f = 204.641/ 223

106
105
104

SK4 muon decay-e
SK4 2015 (334.2 days)
Fit resultµ plus 454297.5 ± 7036.8µ minus 238492.3 ± 8514.4
Charge ratio 1.554 ± 0.061
χ2/d.o.f = 151.534/ 228

106
105
104

103

103

102

102

10

10

1

1

-1

-1

10

0

5

10

15

10

20
25
30
Time difference [µsec](g) Decay curve of stopped muons in 2014
10

SK4 muon decay-e
SK4 2016 (300.4 days)
Fit resultµ plus 395983.1 ± 6458.8µ minus 215994.3 ± 7757.3
Charge ratio 1.496 ± 0.059
χ2/d.o.f = 160.845/ 222

106
105
104

0

105
104

102

102

10

10

1

1

-1

-1

5

10

15

10

20
25
30
Time difference [µsec](i) Decay curve of stopped muons in 2016
Event/bin

15

20
25
30
Time difference [µsec]

SK4 muon decay-e
SK4 2017 (280.9 days)
Fit resultµ plus 360869.4 ± 6155.1µ minus 213995.8 ± 7372.6
Charge ratio 1.376 ± 0.053
χ2/d.o.f = 204.937/ 225

106

103

0

10

107

103

10

5

(h) Decay curve of stopped muons in 20157

Event/bin

Event/bin

107

0

5

10

15

20
25
30
Time difference [µsec](j) Decay curve of stopped muons in 2017

7

10

SK4 muon decay-e
SK4 2018 (119.1 days)
Fit resultµ plus 152802.2 ± 3428.5µ minus 84687.7 ± 3873.4
Charge ratio 1.472 ± 0.075
χ2/d.o.f = 140.129/ 203

106
105
104
103
102
10
1
-1

10

0

5

10

15

20
25
30
Time difference [µsec](k) Decay curve of stopped muons in 2018

Figure 6.2: The decay curve of stopped muons observed each year from 2008 to 2018. The
solid red line was obtained by fitting a function Eq. (6.5). The effective live time and fitting
results (number of positive and negative muons with the charge ratio, χ2 /d.o.f ) are listed in
the legend.

66

Data (Live time)N+

N−

R(µ+ /µ− )χ2 /d.o.f

2008 (55.9 days)

76893.1 ± 2260.8

48760.9 ± 2474.2

1.287 ± 0.075

182.236/189

2009 (274.7 days)

385766.9 ± 6401.3

219775.7 ± 7690.5

1.432 ± 0.055

176.246/225

2010 (331.6 days)

456865.1 ± 7252.0

268920.7 ± 8836.6

1.386 ± 0.051

176.510/228

2011 (321.6 days)

435989.6 ± 6955.3

245505.7 ± 8414.8

1.449 ± 0.055

167.355/224

2012 (314.1 days)

461803.5 ± 7149.6

242548.0 ± 8642.7

1.554 ± 0.060

170.136/231

2013 (328.2 days)

438973.3 ± 7050.2

256439.3 ± 8555.4

1.397 ± 0.052

173.918/225

2014 (310.2 days)

388604.3 ± 6586.0

243742.3 ± 7980.1

1.301 ± 0.048

204.641/223

2015 (334.2 days)

454297.5 ± 7036.8

238492.3 ± 8514.4

1.554 ± 0.061

151.534/228

2016 (300.4 days)

395983.1 ± 6458.8

215994.3 ± 7757.3

1.496 ± 0.059

160.845/222

2017 (280.9 days)

360869.4 ± 6155.1

213995.8 ± 7372.6

1.376 ± 0.053

204.937/225

2018 (119.1 days)

152802.2 ± 3428.5

84687.7 ± 3873.4

1.472 ± 0.075

140.129/203

300

2000

Event/day

Event/dayTable 6.1: The results of N+ , N− and the charge ratio R(µ+ /µ− ). In this study, only statisticaluncertainty is considered.

250

1800

Mu plus1600

Mu minus

1400

200

1200
150

1000
800

100
50
0

600
400

c2
d.o.f
(c2/d.o.f)´100

2010

200
2012

2014

2016

0

2018

Year

2010

2012

2014

2016

2018

Year

Figure 6.3: Fitting results of decay curve with χ2 (red) probability, d.o.f (green), (χ2 /d.o.f ) ×100 (blue) (left) and the event rate of decay electron [event/day] (right) obtained from positive (red) and negative (blue) stopped muons in SK-IV period.

67

Charge ratio

1.65
1.60
1.55
1.50
1.45
1.40
1.35
1.30
SK-IV yearly data

1.25
SK-IV combined

1.20
1.15

Kamiokande-II

2010

2012

2014

2016

2018

Year

Figure 6.4: Yearly time variation of the charge ratio of 1.3 TeV cosmic-ray muons from 2008 to2018 together with the result from the Kamiokande-II. The blue band area shows the averageof muon charge ratio in SK-IV period and green band is the result from Kamiokande-II [10].
indicate that the effect of kaon products on the muon charge ratio becomes significant in a few
TeV energy region and in good agreement with the miscellaneous underground experimental
data.

68

Chapter 7
Summary
The muon charge ratio provides the information of hadronic interaction in the atmosphere andallows the precise calculation of atmospheric neutrino flux for further precise measurements of
the neutrino oscillation parameters.
In this thesis, the charge ratio of positive to negative cosmic-ray muons at the energy of 1.3 TeV
was measured through the analysis of the decay curve of stopping muons in the SK detector
at the depth of 2,700 m.w.e. The analysis was performed using the data collected from 2008
to 2018 in the SK-IV period using large statistics, with approximately 2,970 days of live time.
The data sampling of stopping muons and decay electrons was optimized by evaluating theselection criteria and comparing the distribution of the simulated data. The yearly variationof the charge ratio in the SK-IV period was calculated and the combined result of the charge
ratio were obtained by taking the mean weighted value of each data. In conclusion, the charge
ratio was determined as 1.42 ± 0.02 (statistical uncertainty only) with the result consistent
with past underground measurements in a similar energy region. The result suggests a slightincrease in a few TeV regions by comparing with other underground experiments in a diverse
energy range. The increasement in the charge ratio can be sufficiently interpreted by the effectof kaon production and is compatible with a simplified πK model.

69

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